Particulate gravity currents

PGCA01 9/12/08 3:18 PM Page i PARTICULATE GRAVITY CURRENTS Particulate Gravity Currents. Edited by William McCaffrey,...

Author: W. D. McCaffrey | B. C. Kneller | J. Peakall

116 downloads 902 Views 7MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

PGCA01 9/12/08 3:18 PM Page i

PARTICULATE GRAVITY CURRENTS

Particulate Gravity Currents. Edited by William McCaffrey, Ben Kneller and Jeff Peakall © 2001 The International Association of Sedimentologists. ISBN: 978-0-632-05921-8

PGCA01 9/12/08 3:18 PM Page iii

SPECIAL PUBLICATION NUMBER 31 OF THE INTERNATIONAL ASSOCIATION OF SEDIMENTOLOGISTS

Particulate Gravity Currents EDITED BY WILLIAM McCAFFREY, BEN KNELLER AND JEFF PEAKALL

PGCA01 9/12/08 3:18 PM Page iv

© 2001 The International Association of Sedimentologists and published for them by Blackwell Science Ltd Editorial Offices: Osney Mead, Oxford OX2 0EL 25 John Street, London WC1N 2BS 23 Ainslie Place, Edinburgh EH3 6AJ 350 Main Street, Malden MA 02148-5018, USA 54 University Street, Carlton Victoria 3053, Australia 10, rue Casimir Delavigne 75006 Paris, France Other Editorial Offices: Blackwell Wissenschafts-Verlag GmbH Kurfürstendamm 57 10707 Berlin, Germany Blackwell Science KK MG Kodenmacho Building 7–10 Kodenmacho Nihombashi Chuo-ku, Tokyo 104, Japan

distributors Marston Book Services Ltd PO Box 269 Abingdon, Oxon OX14 4YN (Orders: Tel: 01235 465500 Fax: 01235 465555) USA Blackwell Science, Inc. Commerce Place 350 Main Street Malden, MA 02148-5018 (Orders: Tel: 800 759 6102 781 388 8250 Fax: 781 388 8255) Canada Login Brothers Book Company 324 Saulteaux Crescent Winnipeg, Manitoba R3J 3T2 (Orders: Tel: 204 837 2987)

Iowa State University Press A Blackwell Science Company 2121 S. State Avenue Ames, Iowa 50014-8300, USA

Australia Blackwell Science Pty Ltd 54 University Street Carlton, Victoria 3053 (Orders: Tel: 3 9347 0300 Fax: 3 9347 5001)

The right of the Author to be identified as the Author of this Work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

Catalogue records for this title are available from the British Library and the Library of Congress

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the copyright owner. First published 2001 Set by Graphicraft Limited, Hong Kong Printed and bound in Great Britain at the Alden Press, Oxford and Northampton The Blackwell Science logo is a trade mark of Blackwell Science Ltd, registered at the United Kingdom Trade Marks Registry

ISBN 0-632-05921-4 For further information on Blackwell Science, visit our website: www.blackwell-science.com

PGCA01 9/12/08 3:18 PM Page v

Contents

vii Preface 1 Particulate gravity currents: perspectives Jeff Peakall, Maarten Felix, Bill McCaffrey and Ben Kneller Theoretical and numerial approaches 11 Mechanics and simulation of snow avalanches, pyroclastic flows and debris flows Tamotsu Takahashi 45 An analysis of the debris flow disaster in the Harihara River basin Hajime Nakagawa, Tamotsu Takahashi and Yoshifumi Satofuka 65 Theoretical study on breaking of waves on antidunes Yu’suke Kubo and Miwa Yokokawa 71 A two-dimensional numerical model for a turbidity current Maarten Felix 83 Granular flows in the elastic limit Charles S. Campbell 91 Bagnold revisited: implications for the rapid motion of high-concentration sediment flows Stephen Straub Combined theoretical and experimental approaches 113 Downslope flows into rotating and stratified environments Peter G. Baines 121 Two-dimensional and axisymmetric models for compositional and particle-driven gravity currents in uniform ambient flows Andrew J. Hogg and Herbert E. Huppert 135 Ping-pong ball avalanche experiments James McElwaine and K. Nishimura 149 Dam-break induced debris flow Herve Capart, Der-Liang Young and Yves Zech v

PGCA01 9/12/08 3:18 PM Page vi

vi

159

Contents

Experimental approaches Mean flow and turbulence structure of sediment-laden gravity currents: new insights using ultrasonic Doppler velocity profiling Jim L. Best, Alistair D. Kirkbride and Jeff Peakall

173

Turbulence structure in steady, solute-driven gravity currents Clare Buckee, Ben Kneller and Jeff Peakall

189

Experimental evidence for autosuspension Henry Pantin

207

Time- and space-resolved measurements of deposition under turbidity currents Frans deRooij and Stuart Dalziel

219

Field-based approaches Formation of large-scale shear structures during deposition from high-density turbidity currents, Grès d’Annot Formation, south-east France Julian D. Clark and David A. Stanbrook

233

Subaerial liquefied flow of volcaniclastic sediments, central Japan Katsuhiro Nakayama

245

Depositional and eruptive mechanisms of density current deposits from a submarine vent at the Otago Peninsula, New Zealand U. Martin and James D.L. White

261

Deltaic density currents and turbidity deposits related to maar crater rims and their importance for palaeogeographic reconstruction of the Bakony–Balaton Highland Volcanic Field, Hungary Kàroly Németh

279

Synsedimentary deformation in the lower Muschelkalk of the Germanic Basin Katja Föhlisch and Thomas Voigt

299

Index

PGCA01 9/12/08 3:18 PM Page vii

Preface

tinuum needed to describe many natural flows. Accordingly the papers are grouped by dominant research approach. The conference was sponsored by the SEPM, the Geological Society of London and the BSRG, together with Amoco, ARCO, BHP, Chevron, Conoco and Elf to whom we are grateful. We thank the following, together with one anonymous reviewer, for providing technical reviews of the manuscripts for this volume. The high standard of the volume is a tribute to their input.

This volume arises from the conference Sediment transport and deposition by particulate gravity currents hosted by the School of Earth Sciences at Leeds University, UK in September 1998. This three-day meeting was focused on the processes of sediment transport and deposition in geologically or environmentally important gravity currents, and on their resulting products. It attracted 120 delegates drawn from academia and industry, with research interests ranging from turbidity currents in the oceans, lakes and reservoirs to pyroclastic density currents and avalanches; debris flows and lahars; powder snow avalanches; effluent dispersal and ancient gravity current deposits. In addition to covering these topics, the meeting sought to improve the dialogue between numerical and experimental modellers and those with a professional interest in producing better descriptive models of the deposits of particulate gravity currents. Keynote speakers included Peter Baines (downslope flows into stratified environments); Charles Campbell (granular flows); Herbert Huppert (quantitative evaluations of particle-driven flows); Gary Parker (submarine debris flows and turbidity currents) and Tamotsu Takahashi (mechanics and simulation of debris flows, pyroclastic flows and snow avalanches). Four of the keynote speakers were able to contribute research articles to this volume. Another contributor is Dr. Henry Pantin, who, in his 73rd year at the time of writing, is still extremely active as an Honorary Research Fellow at Leeds. A special presentation was made at the conference to Dr. Pantin, in recognition of his role as an elder statesman in the turbidites community, and as a pioneer in the field of autosuspension. The grouping of papers within the volume reflects the integrative aim of the conference. Many natural (and experimental) particulate gravity currents rely on more than one particle support mechanism, and the nature and relative importance of these mechanisms may evolve during flow. Thus, although the subdivision of particulate gravity currents into discrete sub-categories (such as grain flows, turbidity currents or debris flows) provides a convenient descriptive shorthand, it undermines the sense of process con-

Jan Alexander Mustafa Altinaker Jaco Baas Roger Bonnecaze Charlie Campbell Ray Cas Paul Cole Stuart Dalziel Tim Druitt Ian Eames Paul Emms Maarten Felix Armin Freundt Bryce Hand Kolumban Hutter Christopher Kilburn Gregory Lane-Serff

Fabien Lopez Don Lowe Jon Major Alex Maltman Jim McElwaine Dave Mohrig Henry Pantin Jeff Parsons Frans de Rooij Bruno Savoye John Smellie P.K. Stansby Dorrik Stow Stephen Straub Finn Surlyk Tamotsu Takahashi Nigel Woodcock

We thank Tim Druitt for his efforts as conference co-convenor, and for his assistance in the early stages of volume preparation. We would also like to thank Rufus Brunt, Mark Franklin, Sarah Sturgess, Frances Johnson and Susan Lacey for their help in publicizing and organizing the conference. Guy Plint, the IAS Special Publications Editor was unfailingly helpful and efficient. Finally, we thank our families, for their support and forbearance. Bill McCaffrey, Ben Kneller, Jeff Peakall Leeds, July 2000

vii

colour plate 9/12/08 3:20 PM Page 2

Plate 1. Calculated thicknesses of accumulated sediment for Cases 1–5. facing p. 64


(a)

Plate 1. Spatio-temporal surfaces for probes during passage of the density current and subsequent bore propagation. (a) P2, 37 mm above the bed. (b) P4, 97 mm above the bed. Axis labelled x denotes the position of the measuring volume upstream from the first measurement point: this point was set at 9 mm from the probe head in order to avoid interference due to flow stagnation with the transducer. Note the differing velocity ranges for each figure.


facing p. 160

(b)

Plate 1. (continued)



Plate 2. Series of velocity slices at different stages of evolution of the current for (a–g) the forward flow and (h–n) the first reverse flow. Each image is 90-mm high (base at 7 mm) and 86-mm long. The time slices for (a–g) are at t = 15.75, 15.94, 16.13, 16.32, 16.51, 16.88 and 19.91 s, respectively (total time = 4.16 s) whilst for (h–n) these slices are at t = 104.7, 105.5, 105.9, 106.2, 106.6, 107.4 and 108.5 s (total time = 3.78 s). See text for discussion.


Plate 3. Summary spatio-temporal slices for three stages of evolution of the current. The bottom temporal slice in the horizontal plane slice is at y = 7 mm ( y/h h = 0.05) whilst the three temporal slices in the vertical plane are at 16 s (arrival of the head), 106 s (the first type C bore moving upstream after reflection from the downstream end) and 170 s (the first type A bore moving downstream after reflection from the upstream end of the tank).

PGCA02 9/12/08 3:17 PM Page 1

Spec. Publs. int. Ass. Sediment. (2001) 31, 1–8

Particulate gravity currents: perspectives J . P E A K A L L , M . F E L I X , B . M c C A F F R E Y and B . K N E L L E R* School of Earth Sciences, University of Leeds, Leeds, LS2 9JT, UK

ABSTRACT Recent advances in particulate gravity current research are reviewed and future research areas that demand attention are highlighted. In particular, progress in the following areas is discussed: physical modelling; mathematical modelling approaches; fluid entrainment and detrainment; individual flow processes; and new insights from field research. Research on particulate gravity currents has largely been based on individual flow-types, such as turbidity currents, and has often been pursued using a single research approach (e.g. fieldwork, experimental modelling or numerical modelling). There is great scope to integrate further the different approaches to the study of gravity currents, and in so doing rapidly to improve our understanding of these important flows.

INTRODUCTION between two areas of the same tank (e.g. Middleton, 1966a,b; Simpson & Britter, 1979; de Rooij & Dalziel, this volume). Recently, many experimentalists have adopted the use of a fixed-volume input tank which is separate from the main flume (e.g. Baines, this volume; Buckee et al., this volume; Best et al., this volume; de Rooij & Dalziel, this volume; Hogg & Huppert, this volume; Pantin, this volume). Such a set-up can produce currents which, although not strictly steady, run for much longer time periods than lock-exchange currents if the input volume is sufficiently large; these have been referred to as ‘continuous’ currents (e.g. Baines, this volume; de Rooij & Dalziel, this volume). Significant differences between these two modelling approaches are illustrated by de Rooij & Dalziel (this volume) who show that lock-exchange currents have much better defined heads than continuous currents; that deposition is dominated by the head in lockexchange currents and by the body in continuous currents; and that there is an area of reduced deposition immediately downstream of the gate in lock-exchange currents in contrast to continuous currents which show a smooth downstream decrease in sedimentation. These differences can be explained in part by relatively intense turbulence caused by the sudden gravitational collapse of the dense material as the lock-gate is released, which may account for the

The term ‘particulate gravity current’ encompasses many different flow types. In this volume there are descriptions of debris flows, granular flows, snow avalanches, pyroclastic flows and turbidity currents. Such flows are of great importance in nature, because of their potentially destructive capacity, and as agents of sediment transport and deposition. In addition, the study of such flows has important commercial applications in fields as diverse as food processing, effluent dispersal and the removal of dredging waste and mine tailings. This article seeks to synthesize some of the recent advances in our understanding of particulate gravity currents (many of which are detailed in this volume) and to highlight some potentially important future research areas.

ADVANCES IN PHYSICAL MODELLING Lock-exchange versus external input Gravity current experiments have traditionally been conducted using a lock-exchange methodology where a fixed volume of fluid is released by lifting a gate * Present address: Institute for Crustal Studies, Girvetz Hall, University of California, Santa Barbara, CA 93101, USA.


1

PGCA02 9/12/08 3:17 PM Page 2

2

J. Peakall et al.

decreased sedimentation immediately downstream of the lock-gate, and by the countercurrent of ambient fluid filling the lock-box which appears to influence the dynamics of the head. In general, the use of lockexchange experiments appears to have led to an overemphasis on the importance of the head in turbidity currents and a corresponding underemphasis of the body and tail. Indeed, both the collapse of a static mass of fully suspended sediment and the relative magnitude of the countercurrent produced in most lock-box experiments are highly unrealistic of gravity current initiation in most natural situations. It is therefore suggested that use of an external input tank is more appropriate for modelling both natural surgetype and continuous currents, with the relative volume of the input tank determining the flow-type. New experimental methodologies The measurement of detailed spatial and temporal changes in the flow properties of particulate gravity currents has proved to be extremely difficult (see Kneller & Buckee, 2000 for a review). However, a number of new techniques offer great scope for advancing our understanding of the structure of gravity currents and its influence on sedimentation. The most detailed turbulence measurements of a turbidity current to date are presented by Buckee et al. (this volume), using laser Doppler anemometry (LDA) on steady currents whose refractive index (RI) was matched to that of the ambient fluid in order to minimize rates of data dropout from the LDA (technique after McDougall, 1979). Previous studies of turbulence in turbidity currents have been limited by flow unsteadiness, imperfect RI matching and small experimental scale (Kneller et al., 1997, 1999). Buckee et al. (this volume) used a large external tank to produce continuous currents. Achieving flow steadiness entailed (a) minimization of gravity current reflection from the far wall of the tank, and (b) the effect of progressive volume expansion of the current as a result of mixing with the ambient fluid. These problems were solved by pumping dense fluid out of the main tank at a rate higher than the input (balanced to allow for the volume expansion) and by topping up the ambient fluid to maintain a constant total fluid depth. Together with the introduction of particle imaging velocimetry (PIV ) which enables full two- and three-dimensional flow fields to be captured (Prasad & Adrian, 1993; Adrian, 1996), these experimental techniques should allow major advances to be made in our understanding of the turbulence structure of low-concentration turbidity currents.

The use of optical measurement techniques such as LDA and PIV is restricted to low concentration flows (of just a few weight-percent suspended particulates) because of increasing opacity in higher concentration flows. A significant problem is that low concentration flows represent only one small part of the gravity current continuum and experiments must seek to examine both higher concentrations and assess the application of lower density or saline-flow analogues. An important new technique that enables velocity measurement of higher concentration flows is ultrasonic Doppler velocimetry profiling (UDVP), whose first use in sediment-laden gravity currents is shown by Best et al. (this volume). UDVP probes instantaneously measure velocity at 128 points along a single profile and can be multiplexed to provide two-dimensional flow fields (Best et al., this volume) or to produce two- or threecomponent velocities at a point or series of points. A similar technique is acoustic Doppler velocimetry (ADV) in which at-a-point three component velocities can be measured in particulate currents. Hogg and Huppert (this volume) have used ADV in concentrations of approximately 1% by weight but the technique should be capable of working at higher concentrations. There is also potential for measuring particle concentration using UDVP and ADV by analyisng back-scatter intensity, as shown in the use of fieldbased acoustic Doppler current profilers (Thorne et al., 1991; McLelland et al., 1999). A further technique that holds significant promise for the study of concentration sediment-laden flows is ultrasonic imaging (Crapper et al., 2000). High-concentration granular flows pose even greater measurement challenges. McElwaine and Nishimura (this volume) have addressed this problem by creating flows composed of ping-pong balls, which are larger and lighter than most particles and are therefore easier to study using cameras, and less destructive to measurement equipment such as pressure sensors. In addition to obtaining improved measurements of flow structure in gravity currents, it is important to obtain accurate measures of sedimentation. Many laboratory experiments have measured net sedimentation after a run (e.g. Hogg & Huppert, this volume; Pantin, this volume), however, periodic or continuous measurement of sedimentation during a run is required in order to link sedimentation to temporal variations in the dynamics of the flow. A technique based on measurement of the electrical resistance of the sediment layer has been developed and demonstrated for gravity currents, which enables the thickness of an evolving sediment layer to be measured at a rate of up to 0.33 Hz

PGCA02 9/12/08 3:17 PM Page 3

3

Perspectives (de Rooij et al., 1999; de Rooij & Dalziel, this volume). If assumptions are made about the packing density of particles then sediment fluxes may be obtained. However, this technique is limited to monodisperse particles because the packing, shape and size of the grains affects the resistance (de Rooij et al., this volume). Further advances in measuring sediment fluxes from particulate gravity currents is critical for understanding the controls on deposition in gravity currents.

MATHEMATICAL MODELLING APPROACHES Although each type of gravity current flow is different, many flow aspects can be described using similar approaches, as can be seen in the mathematical descriptions in use in the papers in this volume. Methods using dimensionless numbers such as Reynolds and Froude numbers have long been utilized for turbidity currents (e.g. Middleton, 1966a,b), and are still applied to describe the flow structure of gravity currents, including turbidity currents (Kubo & Yokokawa, this volume; Pantin, this volume), and oceanic density currents (Baines, this volume). These methods are based on the use of depth-averaged variables, where it is assumed that one value of a variable can be used to characterize flow behaviour across the entire flow depth. Depthaveraged values are also used in box models (e.g. Hogg & Huppert, this volume) where it is further assumed that one value can sufficiently characterize the flow in the horizontal direction. Experimental results of de Rooij and Dalziel (this volume) show that the latter assumption is not strictly correct and that deposition rates vary significantly in the horizontal direction at any given point in time; box models have, however, been shown to predict well the final deposits of lock-exchange laboratory currents, except near the point of initiation (Huppert, 1998; de Rooij & Dalziel, this volume). Depth-averaged variables are also commonly applied in models where mass and momentum conservation equations are used to describe flow. Models using these equations generally only differ in the stress relationships used to describe flow rheology, with the other terms remaining the same for different flow types. Several of these models of different flow types are described by Takahashi (this volume), some of which are depth-averaged and some of which are not. Models using depth-averaged equations seem to work well for snow avalanches, pyroclastic flows and debris flows (Capart et al., this volume; Nakagawa et al., this volume; Takahashi, this volume) but only as long as

the moving particulate mass is vertically well mixed. For other flow types, such as granular flows and turbidity currents, it is less apparent that depth-averaged approaches work. Results from experimental pingpong ball avalanches (McElwaine & Nishimura, this volume) as well as numerical model results (Campbell, this volume; McElwaine & Nishimura, this volume; Straub, this volume) clearly show a significant change in flow properties in the vertical direction for such flows. The mathematical models used by these authors are based on the rapid granular flow theory, the basics of which are explained by Straub (this volume). This is not a depth-averaged approach and it seems to describe well the observed flow structure of these gravity currents. Measurements of vertical and horizontal flow structure in experimental turbidity currents (Best et al., this volume; Buckee et al., this volume) also indicate significant variation both horizontally and vertically. Similar trends in turbidity current flow properties are shown by the mathematical model results of Felix (this volume) which are also not depthaveraged. Because these observed variations in flow structure cannot be described by depth-averaged models, it is unclear whether the simplifying assumptions generally made in such models can be justified. Furthermore, since models of turbidity currents are generally not compared to observations of actual turbidity currents, as such data are very rare, it is unclear how well such depth-averaged models describe natural scale flows; comparisons have, however, been made between the distribution of natural scale turbidites and some depth-averaged model output (Fukushima et al., 1985; Dade & Huppert, 1994).

FLUID ENTRAINMENT AND DETRAINMENT Gravity currents are multiphase flows, and understanding how fluid phases are entrained and detrained is a critical problem in accurately modelling them. If depth-averaged mathematical models are used, explicit functions need to be incorporated in the model to account for these processes. Although Takahashi (this volume) models both entrainment and detrainment of hot and cold air for pyroclastic flows, models for other flow types generally do not take both processes explicitly into account. For example, although entrainment is a standard component of depthaveraged turbidity current models (e.g. Parker et al., 1986), detrainment is not modelled. Models using higher order turbulence closure (Stacey & Bowen,

PGCA02 9/12/08 3:17 PM Page 4

4

J. Peakall et al.

1988a; Eidsvik & Brors, 1989; Felix, this volume) do not need to explicitly incorporate these effects. Depthaveraged models only work well as long as the different phases in the gravity current are well mixed. When the different phases separate due to entrainment or detrainment, the model assumptions break down and model results are poor (e.g. Capart et al., this volume; Nakagawa et al., this volume). In that case it might be better to model several layers of different characteristics, as done by Takahashi (this volume). In subaerial debris flows, fluid can be entrained through absorption, or detrained through unlocking of the fluid and solid phases. Nakagawa et al. (this volume) model the absorption of surface water into a debris flow and show that their modelling approach is largely insensitive to absorption-related volume concentration changes of up to 3%. This is in contrast to debris flow experiments where variations in weight concentrations of < 1% (i.e. volume concentrations < 3%) can make a large difference to the strength and competence of flows (Hampton, 1975). Further research is required to identify whether these differences between physical and numerical modelling are due to the importance of physical parameters other than competence and strength, or whether modifications are required to the theoretical modelling approach. Similarly, there is a limited understanding of detrainment in debris flows. Experimental and theoretical studies of turbidity currents suggest that entrainment at the upper boundary is a continuous process for most turbidity currents (Lüthi, 1980; Laval et al., 1988; Stacey & Bowen, 1988a) and in conjunction with downslope flow deceleration and flow continuity considerations leads to models predicting ever-increasing flow depths downstream in confined systems (Komar, 1977; Stacey & Bowen, 1988a,b). Little consideration has been given to the process of detrainment, where there is a net loss of fluid from the gravity current to the ambient, although continuous overspill has been suggested as a major detrainment process in leveed aggradational submarine channels (Peakall et al., 2000). Baines (this volume) uses physical modelling to examine the influence of ambient stratification on entrainment and detrainment in gravity currents and demonstrates that there is net detrainment from the current along most of the profile, reaching a peak as the current density approaches the local value of the ambient (see also de Rooij & Dalziel, this volume). The entrainment/ detrainment calculations of Baines (this volume) are based on differencing the initial and final density profiles in the tank, and therefore cannot address the

actual processes of detrainment, which must await further research. However, one identified detrainment mechanism is that of a countercurrent which continuously strips material from the upper surface through interfacial eddies (Hogg & Huppert, this volume). An illustration of the importance of improved understanding of entrainment and detrainment in gravity currents is that of flow along long-distance submarine channels. These channels may run for thousands of kilometres (e.g. Klaucke et al., 1998; Lewis et al., 1998) and turbidity currents are believed to be able to traverse them in a state of autosuspension (see Pantin, this volume). However, existing theoretical models of autosuspension (Parker et al., 1986) predict that flows are unable to traverse the low gradients in the distal parts of these channels unless the fluid entrainment algorithm is modified so as to greatly reduce entrainment (Pantin, pers. comm.). Possible mechanisms that may lead to reduced net entrainment and/or net detrainment in such systems include overspill (Peakall et al., 2000); the possible stripping of material by bottom currents in a manner analogous to that described by Hogg and Huppert (this volume); and longitudinal variations in detrainment/entrainment where there is net detrainment towards the rear of the flow as fluid turbulence is dissipated, in contrast to entrainment at the head.

INDIVIDUAL FLOW PROCESSES Many natural (and experimental) particulate gravity currents rely on more than one particle support mechanism, and the nature and relative importance of these mechanisms may evolve during flow. Nevertheless, understanding individual processes within particulate gravity flows is of critical importance for future advances in the interpretation of gravity flow deposits, and for predictive modelling of such flows. New work is extending, and in cases overturning, some longheld concepts. For example, Nakayama (this volume) argues that autofluidization can take place within some volcaniclastic liquefied flows as a result of large variations in grain density. These results illustrate the potential for discovering new flow processes in volcaniclastic systems and are in conflict with the interpretation of Lowe (1976a) who concluded that autofluidization of uniformly sized sediment is impossible, although the effect of grain density variations was not considered. Straub (this volume) uses rapid granular flow theory to reassess Bagnold’s (1954) experiments on the

PGCA02 9/12/08 3:17 PM Page 5

5

Perspectives grain-inertia regime, and Lowe’s (1976b) sedimentological interpretation of these experiments. Numerical simulations demonstrate that dynamic internal friction angles should be of the order of 12–17° rather than the 33° or greater chosen by Lowe (1976b) which represents the static angle of internal friction (Straub, this volume). Recalculating Lowe’s (1976b) original expressions with the internal friction angles of Straub (this volume) suggests that sand-sized grain flows may be several times thicker than originally suggested. Perhaps more significantly, such granular flows may move across slopes as low as 12–17°, much lower than the angle of repose slopes inferred by Lowe (1976b). Granular flow simulations of long run-out avalanches suggest that in certain circumstances, much thicker granular flows can form, and that these are able to move on very low slopes (see Straub, this volume). The greatly reduced friction implied by these simulations may be caused by a low friction, low-concentration basal layer (Campbell, 1989), or by a breakdown of the grain-inertia regime at the base of such flows characterized by shear stress increasing disproportionally to the normal stress (Campbell et al., 1995; Campbell, this volume). The change in the shear stress to normal stress ratio is inconsistent with both rapid granular flows (characterized by large shear rates and small solid concentrations) and quasistatic flows (characterized by small shear rates and large concentrations), but is characteristic of flows with high shear rates and large concentrations (solid fraction > 0.5) (Campbell, this volume). Straub (this volume) also raises a further mechanism for generating thick granular flows, citing the observation that the basal normal stress of a granular material can become independent of the flow depth. However, this is likely to be highly unusual in natural systems as to date this phenomenon has only been observed where the depth of the flow is significantly greater than the flow width (Jaeger & Nagel, 1992). The ongoing debate on sandy debris flow processes (e.g. Shanmugam, 1996; Hiscott et al., 1997) further illustrates the need for more detailed knowledge of individual flow processes. Although the granular flow studies referred to above demonstrate that granular flows may be more mobile than has previously been thought (Bagnold, 1954; Lowe, 1976b), they are only able to traverse low angle slopes for relatively short distances as a consequence of the large amounts of potential energy released during flow down initially very steep slopes (Campbell, 1989; Campbell et al., 1995). Shanmugam (1996) has suggested, based on the work of Hampton (1975), that the addition of more

than 2% clay (by weight) to an otherwise sandy granular flow may provide matrix strength. Such flows have been proposed as an explanation for massive sand deposits that are observed tens to hundreds of kilometres from their putative sources (Shanmugam, 1996). However, the 2%-by-weight value is based on extrapolation of experimental results which never had less than 10% by weight clay (Hampton, 1975). Furthermore, even if small amounts of clay provide matrix strength, it is unknown whether the clay is able to lower the dynamic internal angle of friction sufficiently to enable sandy debris flows to cross very low-angle slopes. A greater understanding of the role of small amounts of cohesive material on the properties of otherwise granular flows is clearly required, and the physical and numerical modelling of these flows is an important future research area.

FIELD RESEARCH Geological fieldwork on gravity current deposits is important for its ability to provide new insights and new fields of research; of which several examples are presented in this volume. Clark and Stanbrook (this volume) detail soft-sediment deformation structures immediately below thick-bedded graded sandstone beds which they interpret in terms of basal shear generated by high concentration bedload layers at the base of waning high-density turbidity currents. If the association between substrate deformation and normal grading in the suprajacent bed proves to be widespread, and if similar deformation features are not produced by other depositional mechanisms such as sandy debris flows, then the presence of basal shear structures may be a useful field criterion for determining the genesis of some apparently massive deepwater sands (i.e. where grading is present but not visible in the field). The evidence presented by Clark and Stanbrook (this volume) of basal shear generated by high-concentration bedload layers suggests that shear structures might also be developed within the basal parts of turbidite beds (whether deposited by steady depletive flows or by waning flows). Although the presence of shear-related seams at the base of massive sandstones are recognized by Shanmugam (1997) as indicating debris flow emplacement, the reasoning detailed above suggests that the presence of shearrelated fabrics reflects the rheological regime during deposition, and need convey no information about the long-range sediment transport mechanism. A second advance in understanding gravity current processes is

PGCA02 9/12/08 3:17 PM Page 6

6

J. Peakall et al.

the recognition that the eruption flux and magmatic gas content controlling the column dynamics of subaqueous eruptions may be more important in determining the nature of the final deposits than the mode of fragmentation (Martin & White, this volume). Improvements in process understanding often reveal that certain deposit types have been underrecognized in the rock record. Föhlisch and Voigt (this volume) recognize laterally extensive debrites and slide deposits in marine carbonate mudstones which do not display the characteristic morphology of slides and debrites such as head scarps, channels, and fans, perhaps as a consequence of their rheology and low depositional slopes. These observations suggest that mass flow deposits in marine carbonate mudstones are underrecognized and that new criteria for the recognition of such deposits need to be used. Similarly, subaerial liquefied flows in volcaniclastic sediments have gone unrecognized until recently (Nakayama, this volume). A number of additional parameters have been identified in gravity current deposits for constraining palaeotopography and palaeogeomorphology. The orientation of vein structures in debrites and slides can be used to delineate basin morphology and the epicentre positions of triggering earthquakes (Föhlisch & Voigt, this volume). In addition, the sense of shear in recumbent folds and imbricate stacks found below thickgraded sandstone beds can be used to interpret slope directions where the deformation was caused by sandstone emplacement (Clark & Stanbrook, this volume). Finally, Németh (this volume) provides an example of how detailed reconstruction of gravity current deposits can reveal information about the palaeogeography of an area.

CONCLUSIONS This paper has detailed a number of significant advances that have recently been made in the study of particulate gravity currents and has highlighted a number of important research topics for the future. In physical modelling, the use of external input tanks to generate more continuous flows has both improved the realism and extended the range of flows that can be studied. When combined with the benefits of a host of new measurement techniques, the next few years promise a revolution in our knowledge of the flow- and suspended-sediment structure of low-concentration gravity currents, and the first quantitative experimental results on the structure of many higher-concentration flows. However, one topic that requires further study

is the degree to which laboratory-scale experiments can be accurately upscaled, and therefore the degree of confidence that should be placed on experimentallybased predictions. Importantly, advances in fieldbased measurement technology have paralleled those described above for small-scale experimental facilities, and for the first time provide the potential for recording detailed temporal and spatial measurements of the flow structure of natural gravity currents, at least for the low-concentration end of the gravity current continuum. Such field-derived datasets are vital to enable quantification of scaling effects in laboratory experiments and allow detailed testing and validation of numerical models. Furthermore, if coupled with sediment sampling or subsurface characterization programs, they would provide a direct link between natural flow processes and sedimentary deposits. Mathematical models of gravity currents can broadly be divided into two groups, depth-averaged models and those with depth-varying parameters. Although rapid progress has been made with both approaches, a problem common to both is the difficulty in testing these models and their underlying assumptions. Models that have been tested, such as depth-averaged box models, have generally relied on comparison with small-scale experiments, reinforcing the requirement that the degree of scale-induced effects in such experiments be examined. Datasets of high spatial and temporal resolution are now urgently required for detailed testing of depth-varying gravity current models. Another important future research area is the modelling of entrainment and detrainment in gravity currents, with experiments required to enable explicit incorporation of these effects into depthaveraged models and to test the implicit incorporation of these effects in many depth-varying models. As demonstrated in this paper, geological fieldwork also has a central role in providing new insights into individual flow processes. However, to test the depositional results of numerical models there remains a need for field data that adequately document spatial trends in the distribution and grain size of individual flow events, and for which palaeotopography is sufficiently well constrained that its effect upon gravity current deposition may be assessed. In summary, although considerable progress is being made across a range of particulate gravity current processes, our understanding of many aspects of such flows is still rudimentary in comparison with other sedimentary environments such as fluvial, aeolian and coastal systems. In large part, this is a result of the complexity of these multiphase flows and until

PGCA02 9/12/08 3:17 PM Page 7

Perspectives recently the comparatively limited amount of research undertaken on gravity currents. Equally significant, however, is that in nature such flows tend to be both infrequent and destructive, hindering attempts to measure them. As a consequence, gravity currents have been predominantly studied through experimental, numerical or field-based approaches, and often with one technique being used in isolation. Furthermore, different classes of gravity current tend to have been studied separately. A major challenge for the next few years is thus to more closely integrate these different approaches and research fields. If this can be achieved, the next decade holds enormous scope for rapid advancement of our understanding of particulate gravity current processes.

ACKNOWLEDGEMENTS We would like to thank Jim Best for reading an earlier draft of this manuscript. We would also like to thank many of our colleagues in the gravity currents research community for the many interesting discussions that have helped to form our perspectives on gravity current research. In particular, we would like to thank Jaco Baas, Jim Best, Clare Buckee, Tim Druitt, Henry Pantin and Gary Parker. However, the authors take full responsibility for any errors in interpretation in this article.

REFERENCES Adrian, R.J. (1996) Laser velocimetry. In: Fluid Mechanics Measurements (Ed. Goldstein, R.J.), pp. 175 –299. Taylor and Francis, Washington DC. Bagnold, R.A. (1954) Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. S. London, Ser. A 225, 49– 63. Baines, P.G. (2001) Downslope flows into rotating and stratified environments. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Best, J., Kirkbride, A. & Peakall, J. (2001) Mean flow and turbulence structure of sediment-laden gravity currents: new insights using ultrasonic Doppler velocity profiling. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Buckee, C., Kneller, B. & Peakall, J. (2001) Turbulence structure in steady, solute-driven gravity currents. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Campbell, C.S. (1989) Self-lubrication for long runout landslides. J. Geol. 97, 653 – 665.

7

Campbell, C.S. (2001) Granular flows in the elastic limit. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Campbell, C.S., Cleary, P.W. & Hopkins, M. (1995) Largescale landslide simulations: Global deformation, velocities and basal friction. J. geophys. Res. 100, 8267–8283. Capart, H., Young, D.-L. & Zech, Y. (2001) Dam-break induced debris flow. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Clark, J.D. & Stanbrook, D.A. (2001) Formation of large-scale shear structures during deposition from highdensity turbidity currents, Grès d’Annot Formation, south-east France. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Crapper, M., Bruce, T. & Gouble, C. (2000) Flow field visualization of sediment-laden flow using ultrasonic imaging. Dyn. Atmos. Oceans 31, 233–245. Dade, W.B. & Huppert, H.E. (1994) Predicting the geometry of channelized deep-sea turbidites. Geology 22, 645 – 648. Eidsvik, K.J. & Brors, B. (1989) Self-accelerated turbidity current prediction based upon (k–ε) turbulence. Continent Shelf Res. 9, 617– 627. Felix, M. (2001) A two-dimensional numerical model for a turbidity current. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Föhlisch, K. & Voigt, T. (2001) Synsedimentary deformation in the Lower Muschelkalk of the Germanic Basin. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Fukushima, Y., Parker, G. & Pantin, H.M. (1985) Prediction of ignitive turbidity currents in Scripps submarine canyon. Mar. geol., 67, 55 – 81. Hampton, M.A. (1975) Competence of fine-grained debris flows. J. sediment. Petrol. 45, 834 –844. Hiscott, R.N., Pickering, K.T., Bouma, A.H., Hand, B.M., Kneller, B.C., Postma, G. & Soh, W. (1997) Basin-floor fans in the North Sea; sequence stratigraphic models vs. sedimentary facies; discussion. AAPG Bull. 81, 662–665. Hogg, A.J. & Huppert, H.E. (2001) Two-dimensional and axisymmetric models for compositional and particle-driven gravity currents in uniform ambient flows. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Huppert, H.E. (1998) Quantitative modelling of granular suspension flows. Phil. Trans. R. S. London, Ser. A 356, 2471–2496. Jaeger, H.M. & Nagel, S.R. (1992) Physics of the granular state. Science 255, 1523–1531. Klaucke, I., Hesse, R. & Ryan, W.B.F. (1998) Morphology and structure of a distal submarine trunk channel: the Northwest Atlantic Mid-Ocean Channel between lat. 53° N and 44° 30′ N. Geol. Soc. Am. Bull. 110, 22–34. Kneller, B. & Buckee, C. (2000) The structure and fluid mechanics of turbidity currents: a review of some recent studies and their geological implications. Sedimentology, 47 (Suppl. 1), 62–94.

PGCA02 9/12/08 3:17 PM Page 8

8

J. Peakall et al.

Kneller, B.C., Bennett, S.J. & McCaffrey, W.D. (1997) Velocity and turbulence structure of density currents and internal solitary waves; potential sediment transport and the formation of wave ripples in deep water. Sediment Geol. 112, 235–250. Kneller, B.C., Bennett, S.J. & McCaffrey, W.D. (1999) Velocity structure, turbulence and fluid stresses in experimental turbulent density currents. J. geophys. Res. 104, 5381–5391. Komar, P.D. (1977) Computer simulation of turbidity current flow and the study of deep-sea channels and fan sedimentation. In: The Sea; Ideas and Observations on Progress in the Study of the Seas; Volume 6, Marine Modeling (Eds Goldberg, E.D., McCave, I.N., O’Brien, J.J. & Steele, J.H.). New York, John Wiley, pp. 603 – 621. Kubo, Y. & Yokokawa, M. (2001) Theoretical study on breaking of waves on antidunes. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Laval, A., Cremer, M., Beghin, P. & Ravenne, C. (1988) Density surges: two-dimensional experiments. Sedimentology 35, 73–84. Lewis, K.B., Collot, J.-Y. & Lallemand, S.E. (1998) The dammed Hikurangi Trough: a channel-fed trench blocked by subducting seamounts and their wake avalanches (New Zealand–France GeodyNZ project). Basin Res. 10, 441–468. Lowe, D.R. (1976a) Subaqueous liquefied and fluidized sediment flows and their deposits. Sedimentology 21, 285–308. Lowe, D.R. (1976b) Grain flow and grain flow deposits. J. sediment Petrol 46, 188 –199. Lüthi, S. (1980) Some new aspects of two-dimensional turbidity currents. Sedimentology 28, 97–105. McDougall, T.J. (1979) On the elimination of refractiveindex variations in turbulent density-stratified liquid flows. J. Fluid Mech. 93, 83–96. McElwaine, J. & Nishimura, K. (2001) Ping-pong ball avalanche experiments. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. McLelland, S.J., Ashworth, P.J., Best, J.L., Roden, J. & Klaassen, G.J. (1999) Flow structure and transport of sand-grade suspended sediment around an evolving braid bar, Jamuna River, Bangladesh. In: Fluvial Sedimentology VI (Eds Smith N.D. & Rogers J.), IAS Special Publication 28, pp. 43–57. Martin, U. & White, J.D.L. (2001) Depositional and eruptive mechanisms of density current deposits from a submarine vent at the Otago Peninsula, New Zealand. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Middleton, G.V. (1966a) Experiments on density and turbidity currents. Part 1. Motion of the head. Can. J. Earth Sci. 3, 523–546. Middleton, G.V. (1966b) Experiments on density and turbidity currents. Part 2. Uniform flow of density currents. Can. J. Earth Sci. 3, 627– 637.

Nakagawa, H., Takahashi, T. & Satofuka, Y. (this volume) An analysis of the debris flow disaster in the Harihara River basin. Nakayama, K. (2001) Subaerial liquefied flow of volcaniclastic sediments, central Japan. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Németh, K. (2001) Deltaic density currents and turbidity deposits related to maar crater rims and their importance for palaeogeographic reconstruction of the Bakony–Balaton Highland Volcanic Field, Hungary. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Pantin, H.M. (2001) Experimental evidence for autosuspension. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Parker, G., Fukushima, Y. & Pantin, H.M. (1986) Self-accelerating turbidity currents. J. Fluid Mech. 171, 145–181. Peakall, J., McCaffrey, B. & Kneller, B. (2000) A process model for the evolution, morphology, and architecture of sinuous submarine channels. J. sediment Res. 70, 434 – 448. Prasad, A.K. & Adrian, R.J. (1993) Stereoscopic particle image velocimetry applied to liquid flows. Exp. Fluids 15, 49– 60. Rooij, F. de & Dalziel, S.B. (2001) Time- and spaceresolved measurements of deposition under turbidity currents. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Rooij, F. de, Dalziel, S.B. & Linden, P.F. (1999) Electrical measurement of sediment layer thickness under suspension flows. Exp. Fluids 26, 470 – 474. Shanmugam, G. (1996) High-density turbidity currents: are they sandy debris flows? J. sediment Res. 66, 2–10. Shanmugam, G. (1997) The Bouma sequence and the turbidite mind set. Earth Sci. Rev. 42, 201–229. Simpson, J.E. & Britter, R.E. (1979) The dynamics of the head of a gravity current advancing over a horizontal surface. J. Fluid Mech. 94, 477– 495. Stacey, M.W. & Bowen, A.J. (1988a) The vertical structure of density and turbidity currents: theory and observations. J. geophys. Res. 93, 3528–3542. Stacey, M.W. & Bowen, A.J. (1988b) The vertical structure of turbidity currents and a necessary condition for selfmaintenance. J. geophys. Res. 93, 3543–3553. Straub, S. (this volume) Bagnold revisited: implications for the rapid motion of high-concentration sediment flows. Takahashi, T. (2001) Mechanics and simulation of snow avalanches, pyroclastic flows and debris flows. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller, J.J. Peakall) Spec. Publs. int. Ass. Sediment, No. 31, Blackwell Science, Oxford. Thorne, P.D., Vincent, C.E., Hardcastle, P.J., Rehman, S. & Pearson, N. (1991) Measuring suspended sediment concentration using acoustic backscattering devices. Mar. Geol. 98, 7–16.

PGCC01

10/12/08

4:23 PM

Page 9

Theoretical and numerical approaches


PGCC01

10/12/08

4:23 PM

Page 11

Spec. Publs. int. Ass. Sediment. (2001) 31, 11– 43

Mechanics and simulation of snow avalanches, pyroclastic flows and debris flows T. TAKAHASHI Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 6110011, Japan

ABSTRACT The common and differing aspects of the mechanisms of granular flow in air (dry avalanche), snow avalanche, Merapi-type pyroclastic flow and inertial and viscous debris flows are discussed. Constitutive relations for these flows are obtained based on the common effects of particle encounters, particle migration and large-scale turbulence, as well as considering the peculiarities of particular classes of flow. In addition, particle diameter variation models and erosion and deposition process models are introduced. Thus, models for the numerical simulation of these phenomena under natural conditions are constructed. Applications of the simulation models to dry and wet snow avalanches that occurred in Japan, and to a pyroclastic flow (block and ash flow) at Fugendake, Unzen volcano, Japan agreed well with the actual results, not only in reproducing the areas affected but also some characteristics such as velocities, dominant particle diameters in snow avalanches and thickness of the hot ash cloud in the pyroclastic flow. A unified theory for inertial debris flow is given, which can well explain the quantitative characteristics of stony, immature, hybrid and muddy debris flows. A new Newtonian fluid model is introduced for viscous debris flow, in which the effects of particle collision and turbulence are minimal. This new model explains well the aspects of a debris flow that occurred in Jiangjia gully, China, such as the mechanism of highly concentrated particle dispersion through the entire depth, and flow and deposition processes. Classification of water and sediment mixture flows is given in three-dimensional space characterized by the solids concentration, viscoplasticity of the material, momentum exchange by particle collision and turbulence.

INTRODUCTION Hopfinger, 1983; Sampl, 1993). The lower layer has a concentration of dense solids which moves downstream in a somewhat orderly manner, colliding and jostling. The upper layer is composed of fine particles supplied from the lower layer and has dilute concentration. Particles in the upper layer are suspended by the fluid turbulence and flow downstream as a density current. Resemblance in the behaviour of snow avalanche and pyroclastic flow comes from the intrinsic common mechanism in the lower layer that is the main body of the flow. The commonly operating mechanism is the stresses due to collisional impact, frictional contact motion and turbulent mixing of particles. Regarding the main body as a kind of continuous non-Newtonian fluid or a rigid mass (e.g. Salm, 1966) may be to miss the essential mechanism of the flow, and it is important to view the mechanism as granular flow (although there are some particular factors that characterize the respective flow types).

Debris flow, pyroclastic flow and snow avalanche are the typical subaerial mass-flows that arise on steep slopes. The reason why these multiphase flows can reach distant flat areas of low slope is that once they develop, the solids phase in the flows moves as particles and the void space between particles is held large to minimize the resistance to flow during their motion. Therefore, in discussion of the mechanics of flow, interaction between fragments and /or interaction between particles and the interstitial fluid must be properly taken into account. There is some history of investigation of debris flows from this point of view (e.g. Takahashi, 1991; Iverson, 1997) but for pyroclastic flows and snow avalanches few such investigations are available. Snow avalanche and pyroclastic flow consist of quite different materials, cold snow and hot ash, respectively, but their behaviour while in motion shows some common characteristics. When fully developed, both have a two-layered structure (Sheridan, 1979;


11

PGCC01

10/12/08

4:23 PM

Page 12

12

T. Takahashi

Viscous Debris Flow Laminar dispersion Viscosity Clay, Silt, Coarse Particles

Inertial Debris Flow Buoyancy Buffer Water Snow Avalanche Pyroclastic Flow Crushing Fluidization Upward Gas Flow

Cohesion Dry Granular Flow

Slip

Inelastic collision

Air

Turbulence Skeleton stress Granular Material Landslide Slip Slip surface liquefaction Earth Block

Summarizing the aforementioned phenomena, and adding those of plain granular flow in air (a kind of debris avalanche) and landslides, the author considers that subaerial mass-flows can be classified from the mechanical point of view as shown in Fig. 1. Four cards representing granular flow, pyroclastic flow, snow avalanche and inertial debris flow, respectively, overlap a great deal. The overlapping parts are the commonly operating mechanism, i.e. the stresses due to inelastic collision, turbulence and quasi-static skeleton stress, although those are not necessarily equally dominant. The protruding part of each card is the peculiarity of the mechanism in the phenomenon. Namely, to explain the mechanism of the pyroclastic flow, besides the commonly operating stresses, the effect of crushing of particles (especially for the case of the Merapi-type pyroclastic flow which is defined as one that starts from collapse of a lava dome (Boudon et al., 1983) ) and the existence of the fluidized layer caused by emission of gas from the material itself (Sparks, 1978) should be taken into account. For snow avalanche, besides the commonly operating stresses, the effects of cohesion that produces snowballs and the slip on the snow surface may be important (Mears, 1980). For inertial debris flow, besides the commonly operating stresses, buoyancy effect and presumably

Fig. 1. Subaerial mass flows; their mechanical resemblance and differences. The terms written in italic letters mean the principal material or medium comprising respective flows, and those written in roman letters are the dominant mechanisms operating in respective flows. The overlapping parts of the cards indicate the commonly operating mechanism; the stresses due to inelastic collision of the particles, turbulence and enduring contact between the particles (skeleton stress) are commonly operating in the respective phenomena, and the mechanisms written on the protruding parts characterize the particular phenomena simultaneously operating with the common mechanisms. The small overlapping of the card for landslide with that for dry granular flow and that for inertial debris flow with that for viscous debris flow represent less common mechanism between these phenomena, respectively. The order, shape, size and the direction of protrusion of each card have no particular meaning.

the buffer effect to moderate particle collisions should be important. If a debris flow contains much fine sediment, such as clay and silt, viscosity in the interstitial fluid becomes very large and turbulence is necessarily reduced. Then, the effect of particle collision becomes minimal and instead the laminar dispersion mechanism (Phillips et al., 1992) should operate. Therefore, the common mechanism between viscous debris flow and inertial debris flow is only the buoyancy, and the viscous debris flow has little in common with debris avalanche, pyroclastic flow and snow avalanche. Hidden mechanisms behind this figure are the processes of initiation and deposition. The Merapi-type pyroclastic flow initiates from collapse of a lava dome and snow avalanche starts from sliding of a snow layer (Hopfinger, 1983). Debris flows often initiate from landslides (Takahashi, 1991). Therefore, to thoroughly understand these phenomena, detailed modelling is necessary of the processes of initiation of collapse from a previously stable or metastable state, and of crushing of aggregate while in motion into fragments that comprise the flow material; however, many problems are left to be studied. Herein, models are discussed for the developed flow and deposition of dry and wet snow avalanches, Merapi-type pyroclastic flow and inertial as well as

PGCC01

10/12/08

4:23 PM

Page 13

13

Mechanics and simulation viscous debris flows (based on the premise that the mechanism of the phenomena can be grasped by the concept shown in Fig. 1), and some results of simulations of the phenomena are outlined, based on the author’s recent investigations.

CONSTITUTIVE EQUATION FOR GRANULAR FLOW Figure 2 is a schematic diagram of a granular flow that is comprised of intensively colliding particles. In such a flow, momentum transport arises by the inelastic collisions of particles and by the plunging of particles in one layer into other layers. The stress produced by the first phenomenon is called the ‘collision stress’ and the latter is the ‘kinetic stress’. Concerning the collision stress, many investigations have been undertaken during the past decade or two (e.g. Hutter & Rajagopal, 1994). Among others, Gidaspow (1994) gives the following equation: Pc = 2σC 2 go (1 + e ) TI − −

8σC 2 dp go (1

4σC 2 dp go (1 + e )

+ e)

5 π

3 π T ∇s ν

T ∇ ⋅ νI

(1)

where, σ is the density of the particle, C the volume concentration of the particles, e the restitution coefficient, dp the particle diameter, ν the velocity vector, I the unit tensor, ∇s ν the deformation tensor, T the granular temperature and go is the radial distribution function (Savage, 1988). There is a relation between go and Bagnold’s linear concentration (Bagnold, 1954); λ, as follows: λ = go − 1 –13

(2)

−1

where λ = {(C /C ) − 1} and C is the concentration * * of the solids when packed.

The first term on the right-hand side of Eq. (1) is the pressure, the second term is the solids phase viscous stress and the third is the shearing stress. Gidaspow (1994) discussed the kinetic stress at a dilute condition (go 1) by analogy with the mixing length theory in fluid mechanics. Takahashi and Tsujimoto (1997a) extended his theory to a dense condition in a plain shear flow and obtained τk =

σdp 3go

(3)

where du/dz is the mean velocity gradient in the direction of depth. The shearing stress due to particle collision; τc, i.e. the third term of Eq. (1), exceeds τk when solids concentration is larger than 12% if C = 0.65 * and e = 0.85 as usual in flow of sand in air. Granular temperature, T, represents the vibration energy of particles. The energy conservation equation under steady state is given by ∂ ⎛ ∂T ⎞ ∂u −γ=0 ⎜D ⎟ + ( τc + τk ) ∂z ⎝ i ∂z ⎠ ∂z

(4)

where Di is the coefficient of diffusion and γ is the dissipation rate of granular temperature due to inelastic collision. If the first term of Eq. (4) and τk are neglected, one obtains (Takahashi & Tsujimoto, 1997a) T =

d 2p

1 ⎛ du ⎞ ⎜ ⎟ 15 1 − e ⎝ dz ⎠

2

(5)

Neglect of the diffusion and kinetic stress terms results in a linear decrease in granular temperature toward the surface of the flow. This tendency does not strictly accord with the computer simulation of Campbell and Brennen (1985) or the flume experiments of Takahashi and Tsujimoto (1997a) where the granular temperature decreases as a somewhat concave downward curve. However, herein, Eq. (5) is considered to be applicable at least for a first approximation.

Momentum transfer by migration of τk particles

m n cn

mn+1cn+1

m1 c1

Fig. 2. Momentum transfer in granular flow. mn: the mass of the particle; cn and c′n: the velocity of the particles before and after collision, respectively; u: the average velocity of the particles at a certain height in the mean flow direction.

T ⎛ du ⎞ ⎜ ⎟ π ⎝ dz ⎠

u m1c'1

Momentum transfer τc by collision m2 c2

m2 c'2

PGCC01

10/12/08

4:23 PM

14

Page 14

T. Takahashi pd = pc =

⎛ du ⎞ 2 1+ e 2 C go σd p2 ⎜ ⎟ 15 1 − e ⎝ dz ⎠

2

(7)

Previous investigations (Campbell, 1990) agree in the respect that both τd and pd are proportional to (du/dz)2. The coefficients of proportion in respective investigations, however, differ. If we write τd as 2

⎛ du ⎞ τd = fi σd 2p ⎜ ⎟ tan φ ⎝ dz ⎠

(8)

where φ is the internal friction angle, fi ’s in some previous investigations (Bagnold, 1954; Shen & Ackermann, 1982; Jenkins & Savage, 1983) are as shown in Fig. 3. As is clear in this figure, Eq. (6) has similar characteristics and magnitude to previous theoretical and experimental results.

SNOW AVALANCHE Constitutive relations

Fig. 3. The coefficient, fi , in the dynamic shearing stress, τd , as a function of C/C , where τd = fi σd 2p(du/dz)2 tan φ, σ is * the density of the particle, dp the diameter of the particle, u the velocity in the main flow direction, z the height from the bed, C the average solids concentration, C the solids * concentration when packed, and e is the coefficient of restitution of the particle.

Then, from Eqs (1), (3) and (5) the dynamic shearing stress; τd , and the pressure; pd , due to particle motion in a gravity-driven granular flow down an inclined plane are written as follows: τ d = τc + τk ⎛ du ⎞ 1 ⎪⎫ 1 ⎪⎧ 4 σd p2 ⎜ ⎟ = ⎨ C 2 go (1 + e ) + ⎬ 3go ⎪⎭ 15π (1 − e ) ⎝ dz ⎠ ⎩⎪ 5

2

(6)

Herein, only surface snow avalanches are discussed. Even in this category, classification into dry snow avalanche and wet snow avalanche is possible, so that both phenomena are discussed from a standpoint that the snow temperature is the dominant factor to divide those two. In the case of a large-scale dry avalanche, it sometimes develops into a powder avalanche (Hopfinger, 1983). Even in such a case, previous research confirms the existence of a dense flow layer in the lower part, in which particle collision plays a dominant role (Sampl, 1993). Therefore, discussion is developed based on the granular flow model (Takahashi & Tsujimoto, 1999). A snow particle, which is an aggregate of a few ice crystals and air, is assumed to be spherical. The diameter of the snow particle is about 2 mm and the density is about 0.1 g cm−3. The snow avalanche is considered to be comprised of a granular flow of a mixture of snow particles and snowballs as illustrated in Fig. 4. The shearing stress and pressure in such a granular flow would be described by the sum of the dominant stresses as follows: τ = τs + τcs + τcl + τkl

(9)

p = ps + pcs + pcl

(10)

τs = ps tan φ

(11)

where τs is the shearing stress due to continuous contact between granules, τcs the shearing stress due to collision between snow particles, τcl the shearing stress

PGCC01

10/12/08

4:23 PM

Page 15

15

Mechanics and simulation

Fig. 4. Avalanche material as a mixture of snow particles and snowballs.

due to collision between snowballs, τkl the kinematic shearing stress due to momentum transport by the migration of snowballs, ps the pressure due to continuous contact between granules, pcs the collision pressure between snow particles, pcl the pressure due to collision between snowballs. The stresses due to the collision between snowballs and snow particles and by the migration of snow particles are neglected. The internal friction angle would be different for snow particles and for snowballs, as given by the following equation: φ=

(1 − C l )Gs φss + Cl φsl F

(12)

where Cl is the concentration of snowballs, Gs the local snow particle concentration in the domain in which no snowballs are contained, F the average solids concentration of the mixture of snowballs and snow particles, φss and φsl are the internal friction angles between snow particles and between snowballs, respectively. The quasi-static pressure and shearing stress can operate only when the solids concentration is larger than a threshold value, C3, which would depend on composition of particle sizes, but it is around 0.5 (Bagnold, 1966). Therefore, herein, we describe ps = αsFσg(H − z) cos θ

Fσ = ρm = (1 − Cl )Gs ρps + Cl ρpl

τcs =

1 4 15 1 + es 2 ⎛ du ⎞ G s2 gos ρps d ps ⎜ ⎟ gol 75 π 1 − es ⎝ dz ⎠

τcl =

;C ≤F *

2 ⎛ du ⎞ C l2 gol ρpl d pl ⎜ ⎟ ⎝ dz ⎠ 1 − el

4 15 1 + el 75 π

τkl =

2 ρpl d pl

3gol

⎛ du ⎞ ⎜ ⎟ 15π (1 − el ) ⎝ dz ⎠

(14)

(16)

2

(17)

2

(18)

If uniform distribution of the solids concentration is assumed, the balance of force equation for a steady uniform flow is described as τ = ρm g(H − z) sin θ

*

2

where subscripts s and l represent the quantities corresponding to the snow particles and snowballs, respectively. The kinematic stress is from Eq. (6):

; F ≤ C3 ; C3 ≤ F ≤ C

(15)

where ρps is the density of the snow particles and ρpl is the density of the snowballs. From Eq. (6), τcs and τcl are written as

(13)

where H is the depth of flow, θ the slope angle and αs is the ratio of the quasi-static skeleton pressure to the total pressure. This is assumed as ⎧0 ⎪ 0.2 ⎪⎪ ⎛ F − C ⎞ 3 αs = ⎨⎜ ⎟ ⎪ ⎝ C * − C3 ⎠ ⎪1 ⎪⎩

The exponent value 0.2 is determined so as to make the static pressure ps about 80% of the total pressure when the flow is composed only of snow particles and the volume concentration is 0.56. The following equation is satisfied:

(19)

Then the velocity distribution is obtained as follows: u* =

2 3K

⎡K 32 − {K (1 − Z )}32 ⎤ + u* sl ⎢⎣ ⎥⎦

(20)

PGCC01

10/12/08

4:23 PM

Page 16

16

T. Takahashi 2

⎛H⎞ ⎛ ρm α s tan φ ⎞ K = ⎜ ⎟ ⎜1 − ⎟ (21) ρpl{k1 + k2 + k3} ⎝ d pl ⎠ ⎝ tan θ ⎠ k1 =

k2 =

4 15 1 + el 75 π

1 4 15 gol 75 π k3 =

1 − el

C l2 gol

2 ρps d ps 1 + es G s2 gos 2 ρpl d pl 1 − es

1 3gol 15π(1 − el )

(22)

(23)

(24)

where u* = u / gH sin θ , Z = z/H and u*sl is the nondimensional slip velocity at the bed. When tan θ ≤ αs tan φ, K = 0 and the motion becomes a slip of a rigid body if possible. If snowballs are formed by condensation of snow particles and if the density of snowballs is r times that of snow particles, Gs = Fso − Cl (r − 1)

(25)

is satisfied, where Gso is the concentration of snow particles before snowballs are produced and it would have a value a little larger than that at which the internal friction angle of cohesionless granular material goes to zero. Therefore, according to Bagnold (1966), it would have a value between 0.5 and 0.6. Referring to Johnson and Jackson (1987), Takahashi and Tsujimoto (1997b) obtained the following slip velocity: u*sl = Uslc + Uslf Uslc =

Uslf =

4 3 C (1 + es )Gso * φ′ π

(26)

(1 − αs ) tan θ 2(1 + es ) gosGso

2 3 C (tan θ − tan δ) αs * φ′ πgos

(27)

2(1 + es ) gos Gso (1 − αs ) tan θ (28)

where Uslc and Uslf are the slip velocities caused by the collision stress and the friction stress, respectively, φ′ the coefficient of momentum loss when snow particles hit the bed, and δ the friction angle between snow particles and the bed. The values of φ′ and δ become small when snow temperature is low and the bed is smooth. Formation of snowballs One of the characteristic differences between dry and wet snow avalanches is due to the difference in the

content of snowballs. The author considers that snowballs are formed when adhesive snow particles collide. If collision of two particles having the diameters i and j, respectively, produces one particle having the diameter k, the particle production rate is written as ∞ d nk 1 = ∑ N ij − ∑ N ik 2 i + j =k dt i =1

(29)

where nk is number of diameter k particles per unit volume and Nij is the frequency of collision in unit time and per unit volume between the particles whose diameters are i and j. When the concentration of snow particles is large, particle motion is laminar and the frequency of collision is given by N ij =

( dpi + dpj )3 du nn 6 dz i j

(30)

If the local area in which the collision takes place is composed of nearly uniform particles with diameter dpm, from Eqs (29) and (30), the change in total particle number is given by d n∞ du 2 = − d 3pm n2∞ dt dz 3

(31)

By introducing a temperature dependent coefficient ET that represents the stickiness of the snow particles and therefore controls the snowball production rate, Eq. (31) becomes d n∞ du 4 = − Vt ET n dt dz ∞ π

(32)

3 n∞ /6) is the solids volume in unit where Vt (= πd pm volume of the mixture. If ET at snow temperature Ts is given by

⎛ T ⎞ ET = exp ⎜ − s ⎟ , a = const ⎝ a ⎠

(33)

variation of total number of particles is written as ln

n∞ ( 0 ) 4 du = Vt ET t n∞ π dz

(34)

where n∞(0) is the particle number when t = 0. If the mixture is comprised of only the snow particles and the snowballs n∞ =

6Cl 6Gs + 3 3 π d pl πd ps

(35)

PGCC01

10/12/08

4:23 PM

Page 17

17


Fig. 5. Motion of the forefront of an experimental avalanche on a newly deposited snow layer. Entrainment of the new snow layer is evident.

is satisfied. From Eqs (25) and (35) one can obtain temporal change of Cl corresponding to the snow temperature. Erosion and deposition Flume experiments were performed to obtain the erosion and deposition characteristics. An experimental flume 5-m long and 10-cm wide was used. The upstream 1-m reach of the flume has a very smooth bed that was used to accelerate the avalanche material dumped on the bed (about 3 litres of new snow) and the remaining 4-m reach has a rougher bed that was used as the decelerating zone. For the experiment to investigate the erosion process, a 4-cm thick new snow layer was laid on the bed about 1-m long in the most downstream reach. The flume was set to a longitudinal slope of 35°. The behaviour of the avalanche was recorded through the transparent glass wall by a highspeed video. Figure 5 shows the aspect of the forefront of an experimental avalanche that moves by compressing and eroding the new snow layer. From such an experimental result we made a model of erosion velocity as follows: DU ie = n Lh

(36)

where Dn is the thickness of the new snow layer, U the velocity of the snow avalanche and Lh is the length of the front part of the avalanche. As for the deposition velocity, an experiment was carried out without the downstream new snow layer and setting the flume gradient to 30° to satisfy the condition K = 0 in Eq. (20). Figure 6 shows the velocity distribution in the process of deposition. The velocity

U (m s−1) Fig. 6. The velocity distribution in the process of deposition on a fixed or compacted snow layer. The velocity decreases with the characteristic movement of a rigid body. t is the time elapsed from the moment of supply into the flume.

near the bed decreases at first; the slip velocity, however, is still remarkable, and the velocity decreases gradually showing the characteristic movement of a rigid body. Therefore, it would be possible to assume the whole flow stops in a short time ∆t as soon as the velocity becomes smaller than a threshold value Uth. id = −

H ∆t

(37)

The condition K = 0 is approximately satisfied when the slope is lower than 38°. Numerical simulation The fundamental equations for planar two-dimensional flow are as follows: Conservation of mass: ∂H ∂(UH ) ∂(VH ) + + =i ∂t ∂x ∂y

(38)

Momentum conservation for x and y directions: ∂(UH ) ∂(U 2 H ) ∂(UVH ) + + ∂t ∂x ∂y

= gH sin θbxo − gH cos θbxo

∂( H + zb ) τ bx − ∂x ρm

∂(VH ) ∂(UVH ) ∂(V 2 H ) + + ∂t ∂x ∂y

(39)

PGCC01

10/12/08

4:23 PM

Page 18

18

T. Takahashi = gH sin θbyo − gH cos θbyo

∂( H + zb ) τ by − ∂y ρm

(40)

Conservation of particle number in unit volume: ∂( n∞ H ) ∂( n∞UH ) ∂( n∞VH ) + + = inb − Ir H ∂t ∂x ∂y (41) Change in the surface height: ∂zb +i =0 ∂t

(42)

where U and V are the x and y components of the mean velocity, respectively, θbxo and θbyo are the slope angles of x and y axes, respectively, τbx (= ρm fU U 2 + V 2 ) and τby (= ρm fV U 2 + V 2) are the resistance at the bottom of the flow, i is the erosion or deposition velocity and when deposition takes place ( U 2 + V 2 ≤ Uth )i = id and when erosion occurs i = ie, nb is the particle number in unit volume in the new snow layer (when erosion occurs) or in the flow (when deposition occurs), Ir is the changing rate of the total particle number and it is given by ⎛ T ⎞ 2 du Ir = − d 3pm n2∞ exp ⎜ − s ⎟ 3 dz ⎝ a ⎠

(43)

The resistance coefficient f is given by ⎛ 4 ⎞ 4 f =⎜ K + KUsl + U sl2 ⎟ 5 ⎝ 25 ⎠

−1

(44)

Note that the momentum conservation equations (39) and (40) are obtained by depth-averaging the Reynolds equations under the assumption of shallow flow and hydrostatic pressure distribution. If the flow has a much larger horizontal scale than vertical, the shallow flow condition will be met. If local slope change of the terrain is abrupt the hydrostatic pressure assumption may fail at that position, but at other places it should be a reasonable assumption as is usually adopted in hydraulics (Vreugdenhil, 1994). These equations moreover neglect the horizontal stress terms. In a shallow, horizontally large scale flow, the horizontal stress operating on planes normal to the bed should be small in comparison to the stress operating on the horizontal plane. Applicability of these momentum equations is obvious for water flood flow (Takahashi & Nakagawa, 1987) and for debris flow (Takahashi, 1991). Stable and accurate integration of the fundamental system of equations can be fulfilled by the finite differ-

ence staggered leap frog method, in which the depth of flow is calculated at the centre of the grid mesh on n, n + 2, n + 4, . . . time steps and U and V are calculated at the centre of the grid line on n + 1, n + 3, n + 5, . . . time steps, and for the convection terms upwind discretization is used. The detailed description of the method can be found elsewhere (Takahashi & Nakagawa, 1987). These systems of fundamental equations are applied to reproduce two actual snow avalanches. The first one is the Maseguchi dry snow avalanche that occurred in 1986 in Niigata prefecture, Japan (Kobayashi, 1986). The values necessary for the calculation were obtained by referring to the theoretical considerations, flume experiments (Takahashi & Tsujimoto, 1999) and the field investigations at that time. They are Gso = 0.56, dps = 0.002 m, dpl = 0.1 m, es = 0.4, el = 0.6, a = 0.8, φ′ = 0.04, δ = 2°, Ts = −6°C and Dn is equal to 2.5 m at the altitude higher than 600 m and lower than that it is 0.5 m. The upstream boundary condition is not known, but a constant discharge of 24 m2 s−1 is given for 10 seconds with the width 100 m and depth 2.5 m. Figure 7 compares the calculated traces and flowing area at 200 seconds after occurrence with the actual traces. Calculation seems to well reproduce the actual phenomena. The calculated velocity and content of snowballs at the front of the avalanche are depicted in Fig. 8. The maximum velocity on steep slopes reaches almost 60 m s−1 and considering the principal ingredient is snow particles, the avalanche should be a powdery one. This is in agreement with majority opinions from the field survey (Kobayashi, 1986). The second application is for the wet snow avalanche at Hakubadake, Japan, in 1992 (Terada et al., 1993). The values used in the calculation are φ′ = 0.125, δ = 5° and others are the same as the former case. From the results of field surveys, Dn is set at 2.0 m at the altitude higher than 2200 m and 0.2 m lower than that (Terada et al., 1993). The situation at 300 seconds after initiation is shown in Fig. 9. The calculated trace well reproduces the actual one. The total volume of the avalanche at this time is calculated in the simulation as 4.2 × 105 m3 and this means the volume reaches five times that of the supplied material (0.66 × 105 m3). The actual deposit volume measured in the field was about 3.9 × 105 m3. Figure 10 shows the calculated velocity and content of snowballs in the front part of the avalanche. After acceleration up to 40 m s−1, it decelerates and travels a long distance with a velocity of about 10 m s−1. This agrees with the eye-witness record that the velocity

PGCC01

10/12/08

4:23 PM

Page 19


19

Fig. 7. The actual and the calculated avalanche traces of the Maseguchi dry avalanche are shown 200 seconds after the onset. The shaded area is the actual area that the avalanche passed, the circles are the calculated path, and the triangles are those parts still flowing with a velocity faster than 0.5 m s−1.

was about running pace. The snowballs are quickly produced and reach a steady concentration of about 45%. The actual avalanche also contained many snowballs.

MERAPI-TYPE PYROCLASTIC FLOW Mechanism of flow

Fig. 8. The position (elevation), velocity and content of snowballs, cl, at the front by the calculation (Maseguch dry avalanche). The content of snowballs at the distal end of flow is small, so that it should be classified as powdery avalanche.

The processes of occurrence and running down of Merapi-type pyroclastic flow (block and ash flow) can be modelled as shown in Fig. 11 (Takahashi & Tsujimoto, 2000). Immediately after collapse of the lava dome, it is a granular flow comprising huge blocks of lava (Ui et al., 1993). These blocks become finer and finer in a comparatively short time due to explosion and crushing by interparticle or particle-bed collisions. At that time, particles emit plenty of gas. Soon, particles become so fine and the upward gas flow becomes so strong that the particles become

PGCC01

10/12/08

20

4:23 PM

Page 20

T. Takahashi

Fig. 9. The actual and the calculated avalanche traces of the Hakubadake wet avalanche. The aspect 300 seconds after the onset. The shaded area is the actual area that the avalanche passed, the circles are the calculated path, and the triangles are those parts still flowing with a velocity faster than 0.5 m s−1.

Fig. 10. The position (elevation), velocity and content of snowballs, cl, at the front by the calculation (Hakubadake wet avalanche). The content of snowballs at the distal end of flow is about 40%, so that it should be classified as a wet avalanche. The concentration of snowballs in the actual flow was also high.

sustained by the uplifting force of the gas flow. This phenomenon is called fluidization. Fluidization must be the main cause of high mobility of pyroclastic flow on a far gentler slope than the angle of repose of the particles. Note that the simple dry granular flow (e.g. flow of sand, small debris avalanche) can occur only on slopes steeper than about 20°, and dry sand flow forms a cone whose slope is steeper than 35°. Sparks et al. (1978) suggested that without sufficient emission of gas the particles prevailing in pyroclastic flow (1 mm or more in diameter) cannot be suspended even though turbulence in the flow is strong. Generally, pyroclastic flow has three parts: from lower to upper positions, the main body, the hot ash cloud layer and the plume. The main body has a high particle concentration and it travels along topographical depressions. It is the source of material supplied to the hot ash cloud layer. The hot ash cloud layer is more dilute and the mean particle diameter is finer

PGCC01

10/12/08

4:23 PM

Page 21

21

Mechanics and simulation ity is obtained as Ug u

2 Hf 5 dp

=

*

K1

(45)

gH f sin θ , ⎛ αs tan φ ⎞ f5 F (46) 1 − ⎜ ⎟ f2 + f22 ⎝ tan θ ⎠ f2 π where no slip on the bed is assumed and Ug is the mean K1 =

Fig. 11. Process model of a Merapi type pyroclastic flow (block and ash flow). It begins with the collapse of a lava dome, and soon the collapsed blocks are crushed by the impulse of the interparticle or the particle-bed collision and the self-explosion due to the effect of the enclosed volcanic gas. In the early stage, because the diameters of the particles are very large and the strength of the upward gas flow is insufficient to sustain the particles, granular flow occurs. Simultaneously with the progress of the breaking of the large particles into fragments in granular flow, the strength of the upward gas flow increases, and soon the particles become suspended by the upward gas flow. This is the beginning of fluidized flow stage. Within the fluidized layer, turbulence is very intense even though its intensity is insufficient to suspend the particles by turbulence alone. Some finer particles are transported higher above the bed to make the hot ash cloud layer.

than in the main body. It flows as a density current. The plume looks like a cumulonimbus cloud. It develops even after stoppage of both the main body and the hot ash cloud. It is less important from the disaster mitigation point of view. In the early granular flow stage, the effect of upward gas flow is negligible and the constitutive equations for granular flow introduced earlier are applicable. Especially in this stage, the granular temperature is low and particles are moving in order with continuous contact; the skeleton stress that is independent of velocity is dominant. Herein, such quasi-static pressure and shearing stress are considered as similar to the snow avalanche case described by Eqs (11), (13) and (14). However, in the pyroclastic flow the exponent of {(F − C3)/(C − C3)} is changed to 0.5 instead * of 0.2, because the best fit curve for the velocity distribution in flume experiments using polystyrene and glass beads was obtained by using a value of 0.5 (Takahashi & Tsujimoto, 1997a). Uniform particle concentration is assumed in considering the dynamic stress, τc and τk. Then, from Eqs (6), (11), (13), (14) and the balance of force equation (19), the cross-sectional mean veloc-

velocity, Hf the depth of the flow, u = * f1 = 4εF 2go, f2 = 8εF 2go /( 5 π ), f22 = 1/( 3go π ), f5 =

F = Cst− ε)F 2g / ;θ ≤ φ⎫ and ε = (1 + e)/2. 48ε(1 o ⎪ The relationship between concentration⎪⎬ ⎫⎪ ⎧ e average 1 −the −1 ⎪ 12(1 − e ) − φ(Takahashi θ = tan + 2 flow is given ; θ ≥ φ⎪ ⎬ + φby ⎨ and the slope in granular sp 5π F go (1 + e ) π ⎪ ⎪⎩ ⎪⎭ ⎭ & Tsujimoto, 1997a)

(47) where Cst has a constant value between 0.51 and 0.56, and φsp is the angle of repose for spherical particles. To analyse the process of crushing of a lava block into smaller particles, at first the lava dome is considered as an aggregate of big boulders and they are assumed to be crushed into smaller particles by interparticle collisions in laminar flow. Frequency of collision in unit time per unit volume is given by N=

4 3 ∂u 2 d n 3 p ∂z

(48)

where n is the number of particles per unit volume. If rate of increase in particle number; In , is assumed proportional to N, In = β

48C 2 ∂u π 2 d 3p ∂z

(49)

where β is the coefficient representing fragility of particles upon collision. From Eq. (49) and the mass conservation equation, one will know the change in particle diameter. The process of crushing is, however, considered to end when particle diameter becomes dfc, which is the critical diameter to give rise to fluidization. Within the main body of the pyroclastic flow, emission of gas from the particles generates upward gas flow. Therefore, as shown in Fig. 12, the main body is separated into the basal layer in which some of the particle load is sustained by the quasi-static skeleton stress and the upper layer in which the particles are

PGCC01

10/12/08

4:23 PM

Page 22

22

T. Takahashi Z

upper layers can be regarded as constant between 40 and 45%. In the fluidized upper layer, macro turbulence involving particles and gas dominates and the shearing stress is given as

H

p

Hs

Ho clou t ash d la yer

H

f

h zb

Mai

n bo

hc

Upw

ard

gas

⎛ ∂u⎞ τf = σC 2 ⎜ ⎟ ⎝ ∂z ⎠ flow

dy

F dep reshly osit ed l aye r

Fluid ized laye Bas r al la yer

Z0

X

Fig. 12. The structure of the main body of a pyroclastic flow and the notation that appears in the equations. The main body of pyroclastic flow consists of the basal layer (thickness: hf ) and the fluidized layer (thickness: Hf − hf ≡ Hs − h). In the basal layer the upward gas flow is insufficient to completely suspend the particles.

completely sustained by the upward gas flow. The ratio of the collision stress to the turbulent mixing stress is proportional to (dp /)2 where is the mixing length and it may have the scale of the thickness of flow under the effect of upward gas flow, so that if the thicknesses of the basal layer and that of the upper layer are far larger than the particle diameter as in the fluidized flow stage, the collision stress would become negligible. In Fig. 12, z = zb is the surface of the fresh deposit from the pyroclastic flow and z = 0 is the original ground surface. Details of the stresses and resistance law in this kind of flow can be found elsewhere (Takahashi et al., 1995; Takahashi & Tsujimoto, 2000); the essential points are described below. The height of the boundary between the base and the upper layers measured from the original ground surface; h is given by gd 2p cos θ 1650bνC

(50)

where b is the rate of gas emission from unit mass of particles (weight ratio) and ν is the kinematic viscosity of the gas. According to a laboratory flume experiment using silica sand of 0.08 mm in diameter in which gas emission is given from the material itself (Takahashi et al., 1995), C throughout the basal and

(h ≤ z ≤ H )

(51)

where the mixing length is = κ (z − zb), H is the flow depth measured from the original ground level and H = Hf + zb where Hf is the thickness of the main body and κ is the von Kármán’s constant. The quasi-static pressure in the basal layer that is transmitted through the momentary skeleton is ps = Cσg( h − z ) cos θ −

θ bx0

h=

2

1650bνσC 2 h2 − z 2 2 d p2

(52)

Therefore, the shearing stress in the basal layer is written as 2

⎛ ∂u ⎞ τ f = σC 2 ⎜ ⎟ + ps tan φ ⎝ ∂z ⎠

( zb ≤ z ≤ h) (53)

From these equations one obtains the following equation for resistance to flow for the entire main body: Uf u*

=

hf π2 π ⎞ h f π ⎫⎪ 1 ⎧⎪⎛ − ⎟ − ln + ⎬ (54) ⎨⎜1 + Hf κ ⎪⎩⎝ 8 2 ⎠ Hf 2 ⎪⎭

where Uf is the mean velocity of the main body, u = gHf sin θ , Hf and hf are the depths of the main * body and the base layer measured from the height z = zb. The height at which the operating shearing force and the shearing resistance balance is ⎡ tan θ h0 = h ⎢1 − − ⎢ tan φ ⎣

⎛H ⎞ ⎪⎫ ⎤ tan θ ⎪⎧ tan θ + 2⎜ − 1⎟ ⎬ ⎥ ⎨ tan φ ⎪⎩ tan φ ⎝ h ⎠ ⎪⎭ ⎥ ⎦ (55)

Deposition and erosion Although a pyroclastic flow can transport highly concentrated sediment, it will be overloaded when it reaches flatter areas where it will deposit the excess load. Figure 13 illustrates such a depositing process in which the hatched part deposits in the next increment of time. The equation of motion of this hatched volume is written as

PGCC01

10/12/08

4:23 PM

Page 23

23


the dynamic collision force. Therefore, Eq. (58) will give a depositing speed too large. Then, for the granular flow stage, the following formula will be used: ig = αs if

(59)

Numerical simulation Fundamental equations for a planar (2-D) flow are essentially the same as those for the snow avalanche, i.e. Eqs (38)–(42), once the expression of the shearing stress on the bottom is changed appropriately. Namely, for the case of granular flow: dp ≥ dfc, Fig. 13. The depositing process of a pyroclastic flow. The part lower than h0 in which the operating shear stress is smaller than the shearing resistance will deposit. The shaded layer has already deposited and the hatched part will deposit in the next increment of time. The boundary between the basal layer and the fluidized layer is at height h and the boundary between the main body and the hot ash cloud layer is at height Hs.

τbx = αs ρm gHf cos θx tan φ + ρm f Uf U 2f + V f2 (60) τby = αs ρm gHf cos θy tan φ + ρm f Vf U 2f + V f2 (61) for the case of fluidized flow: dp ≤ dfc,

dU dm m 0 =F + u dt dt r

(56)

where m is the mass of the hatched volume, U0 the velocity of this volume, F the external force that operates on the hatched volume and ur is the relative velocity of the depositing part observed from the main flow. The force operating on the hatched volume is 2

⎛ z ⎞ 1 F = − σFgh ⎜1 − b ⎟ cos θ tan φ ∆ x h⎠ 2 ⎝ + σFg( Hs − zb ) sin θ ∆ x

1 2U0

(62)

τby = ρm fVf U 2f + V f2

(63)

where θx and θy are the x and y components of the slope angles of the surface of the deposit, respectively, Uf and Vf are the x and y components of the mean velocities, respectively, and f is the resistance coefficient of the main body that is given by the following equations: in the case of granular flow:

(57)

Substituting this into Eq. (56) and considering that when zb coincides with h0 the volume stops, the rate at which the depositing surface rises, if , is obtained as if =

τbx = ρm fUf U 2f + V f2

2 ⎡1 ⎛ ⎤ ⎞ ⎢ gh ⎜1 − zb ⎟ cos θ tan φ − (H − z )g sin θ⎥ s b ⎢2 ⎝ ⎥ h⎠ ⎣ ⎦

(58) Similar consideration is applicable to the erosion speed. Consequently, when if > 0, deposition arises and when if < 0, erosion occurs. Although Eq. (58) is obtained for fluidized flow, similar consideration must be applicable even in the stage of granular flow. In the granular flow stage, however, some parts of the particle load are supported by

f =

25 K ⎛ dp ⎞ ⎜ ⎟ 4 F ⎝ Hf ⎠

2

(64)

−2 ⎧ ⎧ 2 hf2 hf π15 π ⎞e )F π1⎫⎪ ⎪ κ 2 ⎪⎛1 4 1 + g ( Hf + − ⎟ −o ln ; h f ≤(65) + H + 2⎬ ⎪ K⎨⎪⎜⎝= 8 2 ⎠ Hf ⎪⎭(1 − e ) f ⎪ ⎩ 75 1 − e 3 g 15 π ( ) π f =⎨ o −2 ⎪ ⎛ 2⎞ π 2 1 of + the⎟ fluidized flow: ; h f ≥ Hf in the⎪ κ case 8⎠ ⎪⎩ ⎜⎝

(66) To make possible the simulation of the hot ash

PGCC01

10/12/08

4:23 PM

Page 24

24

T. Takahashi

cloud layer coupling with the main body, an incompressible continuous fluid model (a reasonable assumption as long as flow is subsonic) for the hot ash cloud layer is introduced below. The schematic structure of the hot ash cloud layer and the notation are shown in Fig. 14. The mass conservation equation is given as ∂Hp ∂t

+

∂ (Up Hp ) ∂x

+

∂ (Vp Hp )

= we − wf

∂y

∂t

+

∂(U 2p Hp ) ∂x

+

∂ (UpVp Hp ) ∂y

(68)

=

∂ ⎛ ∂Up Hp ⎞ ∂ ⎛ ∂Up Hp ⎞ ε ε + ∂x ⎜⎝ p ∂x ⎟⎠ ∂y ⎜⎝ p ∂y ⎟⎠ gH p2 ∂ρmp ∂( zb + Hp ) − f U 2 − f (+ θbyo− −UgHp cos θbyo (69) + UpgH − pUsin )U p pi p b f f ∂y 2ρmp ∂y +

∂ (Vp Hp ) ∂t

+

∂ (UpVp Hp ) ∂x

+

∂(V p2Hp ) ∂y

=

fpi Up, Vp, Cp Hp

ws

Hot ash cloud layer we2

fb

where we1 is the air entraining velocity at the upper surface, we2 the gas supplying velocity from the main body and we3 is the detraining velocity of the ash cloud as the plume that becomes lighter than the ambient air. 2 gH p ∂ρmp ∂( zb + Hp ) for x and y − The momentum + gHpconservation sin θbxo − gHequations p cos θbxo ∂x 2ρmp ∂xare; directions ∂ (Up Hp )

we1 we3

(67)

where Hp is the thickness of the hot ash cloud layer, Up and Vp are the x and y components of the mean velocity of the hot ash cloud layer, respectively, we is the rate at which the upper surface of the hot ash cloud layer rises, and wf is the rate of rise of the boundary between the main body and the hot ash cloud layer. Because the thickness of the main body is far thinner than the ash cloud layer, and because virtually no gas emitting particles are supplied from outside in the discussion on the motion of ash cloud layer, wf can be considered as zero. The rising velocity of the upper surface of the ash cloud layer should be written as we = we1 + we2 − we3

Z

Hf

Main body

zb

Freshly deposited layer

Uf, Vf, Cm

Es

Fig. 14. The notation that appears in the equations for the hot ash cloud layer.

+

∂ ⎛ ∂Vp Hp ⎞ ∂ ⎛ ∂Vp Hp ⎞ εp ε + ⎜ ⎟ ∂x ⎝ ∂x ⎠ ∂y ⎜⎝ p ∂y ⎟⎠

+ fpiV p2 − fb (Vp − Vf ) Vp − Vf

(70)

where εp is the coefficient of eddy viscosity. Conservation of solids phase is given as ∂ (C p Hp ) ∂t

+

∂ (U pC p Hp ) ∂x

+

∂ (VpC p Hp ) ∂y

=

max(we2 − ws, 0)Ffcu − max(ws − we2, 0)Cp − max(we3 − ws, 0)Cp +

∂ ∂x

⎧⎪ ∂ (C p Hp ) ⎫⎪ ∂ ⎧⎪ ∂ (C p Hp ) ⎫⎪ ⎨K x ⎬+ ⎨K ⎬ ∂x ⎪⎭ ∂y ⎪⎩ y ∂y ⎪⎭ ⎪⎩

(71)

where ws is the settling velocity of the particle, fcu is the ratio of the particle fraction that is transportable to the ash cloud layer from the main body, Kx and Ky are the coefficients of diffusion and they are assumed to have the same value as εp. The air entraining velocity is given by the following equation (Fukushima & Kaneko, 1997): we1 = αe ( θs /90°) U 2p + V p2

(72)

where θs is the gradient of the surface of the ash cloud layer as shown in Fig. 15, and αe is a coefficient that Fukushima and Kaneko (1997) give as αe = 0.1 ~ 0.4. The gas production rate in unit time from a unit

; PGCC01

10/12/08

4:23 PM

Page 25


ws =

θs

Hot ash cloud

θ

Fig. 15. The schematic shape of the front of a pyroclastic flow and the definition of θs in Eq. (72).

2 gd pf ⎛σ ⎞ − 1⎟ ⎜ 18ν ⎝ ρa ⎠

25 (78)

where dpf is the mean diameter of the particles in the hot ash cloud layer and v is the kinematic viscosity of fluid in the hot ash cloud layer. This system of equations is applied to the reproduction of an actual case of a Merapi-type pyroclastic flow that occurred on June 3rd, 1991, at Fugendake, Unzen volcano, Japan, and which killed 43 people.

area of the main body is FσHf b, and therefore the gas supplying velocity; we2 is given by the following formula: we 2 =

FσHf b ρa

(73)

where ρa is the density of the air which depends on temperature Tp (K) and pressure Hpa (torr) in the ash cloud layer as ρa =

Hpa 1.293 1 + 0.00367(Tp − 273) 760

(74)

Because the vertical acceleration of the plume is −g(Ta − Tp)/Ta, in which Ta is the temperature of the ambient air, we3 is given as we 3 =

2gHp ( 0.5Tp − Ta ) Ta

(75)

where 0.5Tp is used instead of Tp to consider the effect of temperature drop toward the upper part. For the eddy viscosity, εp =

κu p Hp * 6

(76)

is assumed (Lane & Kalinske, 1941). Particle entraining velocity Es from the main body is written as Es = we2 × F × fcu

(77)

The mean particle diameter in the deposit of the pyroclastic flow at Unzen volcano is about 1 mm and that in the pyroclastic surge or volcanic ash is smaller than 0.1 mm. The particle size analysis of the pyroclastic deposit shows the content of 0.1 mm or finer particles is a few percent (Ishikawa et al., 1994). Therefore, fcu would be about the same order. The settling velocity of the particle is given by Stokes’ law as:

Fig. 16. The actual and the calculated traces of the main body and the hot ash cloud of the pyroclastic flow that occurred on June 3rd, 1991 at Fugendake, Unzen volcano, Japan. The solid line encloses the area passed by the actual hot ash cloud and the broken line encloses the area passed by the actual main body, the open circles show the calculated trace of the hot ash cloud and the closed circles are the calculated trace of the main body. The actual trace of the main body almost overlaps with the calculated result.

PGCC01

10/12/08

26

4:23 PM

Page 26

T. Takahashi

Fig. 17. The calculated position (elevation), velocity and the maximum diameter of the particles at the front of the main body of the Unzen pyroclastic flow versus time. The diameter of the particles rapidly decreases to the final value, 1 mm, that is assumed to be the critical diameter to produce fluidization. Fig. 18. The calculated position (elevation), velocity and the thickness of flow at the front of the hot ash cloud of the Unzen pyroclastic flow versus time. The hot ash cloud first appeared about 30 seconds after collapse of the lava dome, its thickness initially increasing rapidly, then more gradually towards a maximum of 80 m, before gradually decreasing. By comparing with Fig. 17, one can see that the hot ash cloud reaches a far larger velocity than that of the main body, but it rapidly decelerates and stops soon after the main body stops (about 400 seconds after the onset).

The volume of the collapsed lava dome is estimated as about 500 000 m3 (Nakada, 1993). Some of the many parameters and coefficients in the equations can be estimated by theoretical considerations and field investigation, but others must be determined by trial and error. For the time being, as a reasonable choice, the following values are adopted. The mean diameter in the hot ash cloud, dpf = 0.1 mm; the critical diameter for fluidization, dfc = 1.0 mm; temperature, Tp = 400 °C; kinematic viscosity of the volcanic gas, ν = 0.0002 m2 s−1; gas emission rate, b = 0.0001 s−1; κ = 0.7, β = 0.02, fpi = 0.1, fb = 0.1, fcu = 0.005, φ = 38.5°, C = 0.65, * αs = 0.3 and C3 = 0.51. The mechanism of collapse of the lava dome is not yet clear, but here we assume the pyroclastic material of net volume 500 000 m3 is supplied by volume concentration of 0.43 (bulk volume 1.16 × 106 m3) from a grid used in the calculation (width 50 m) in 100 seconds. The discharge is assumed to linearly increase to the maximum Qmax = 23 250 m3 s−1 at 50 seconds and then linearly decrease to zero at 100 seconds. The initial particle diameter just after collapse of the dome is

assumed to be 5 m. These values except the supplied net volume are merely assumptions, but the duration of supply is estimated from the video records and others. Figure 16 shows the calculated traces of the main body and the hot ash cloud just after cessation of the pyroclastic flow, and the affected area. The calculated hot ash cloud spreads laterally at the debouchment of the gully more than in the actual case, but as is evident in the figure, the general tendency of the width and length is well reproduced. Moreover, the calculated deposit areas of the main body and in reality almost coincide. Figure 17 shows the temporal changes in the velocity, the maximum particle diameter and the position (altitude) of the front of the main body. One can see that at the front the maximum diameter becomes about 1 mm in 30 seconds and the crushing process ends by that time. The front velocity maintains 20 to 30 m s−1 until it arrives at the position of No. 1 fall and then it gradually decreases. The actual velocity variation in this particular pyroclastic flow was not measured, but on other occasions similar velocities

PGCC01

10/12/08

4:23 PM

Page 27

27

Mechanics and simulation were measured (Ishikawa et al., 1994a). Figure 18 shows the temporal variations in the velocity, the thickness of the flow and the position (altitude) of the front of the hot ash cloud layer. It arises just after the end of the particle crushing process and gradually accelerates until it comes to the debouchment of the gully at an altitude of about 250 to 300 m, where the maximum velocity reaches 70 to 80 m s−1. After that it decelerates drastically. The more or less horizontal characteristic of the ‘elevation’ curve in the figure at the early stage reflects slow movement of the front, and then the curve becomes steeper than the corresponding curve in Fig. 17 reflecting the fact that the front of the hot ash cloud travels with greater velocity than the main body. Note that the abscissa of the figure is not distance but time, so that the elevation curve does not represent the longitudinal topography. For this particular pyroclastic flow, the velocity of the hot ash cloud was estimated by the fact that it toppled concrete electricity poles and trees. According to those data the velocity was around 20 – 60 m s−1 and this value confirms that the results of the calculation are valid. The results of analysis of another pyroclastic flow show that it travels at about 20 m s−1 (Ishikawa et al., 1994a). The calculated thickness of the developed hot ash cloud layer is around 50 m and this value also coincides with other pyroclastic flows (Ishikawa et al., 1994b).

Consider a flow as shown in Fig. 19 that is generated by supplying water from upstream onto a steeply sloping bed composed of non-cohesive particles. In the upper layer over the depth h2, particles are suspended by turbulence in the fluid, and in the lower layer particles move in order, nearly parallel to the bed. The stresses that may exist in the lower layer are the particle collision stress, the kinetic stress, and viscous and turbulent stress in the interstitial fluid. Moreover, if the solids concentration is more than the threshold, C3, particles move in continuous contact so that the quasi-static skeleton stress also operates. Hence, the balance of force equations in the lower layer in the direction of the main flow, x, and perpendicular to it, z, are written as ⎧⎪ 4 1 ⎫⎪ 1 ps tan φ + ⎨ C 2 go (1 + e ) + ⋅ ⎬ 3go ⎪⎭ 15π (1 − e ) ⎩⎪ 5 2

⎛ du ⎞ ⎛ du ⎞ du σd 2p ⎜ ⎟ + µa + ρT 2 ⎜ ⎟ dz ⎝ dz ⎠ ⎝ dz ⎠

C dz + ρg(h − z) sin θ H

= ( σ − ρ) g sin θ

ps +

⎛ du ⎞ 2 2 1+ e C go σd p2 ⎜ ⎟ 15 1− e ⎝ dz ⎠

2

(σ − ρ)Cg cos θ dz ≡ p′ h2

(80)

z

INERTIAL DEBRIS FLOW Constitutive relations

h1

h2

(79)

z

=

H

2

θ

Fig. 19. A schematic illustration of inertial debris flow. In the upper layer (h1) particles are suspended by turbulence and in the lower layer (h2) particles move in order nearly parallel to the bed with colliding and jostling.

where ρ is the density of the interstitial fluid, µa the apparent viscosity of the fluid that takes account the effects of high solids concentration, ρT the apparent fluid density that takes account of the weight of suspended particles, the turbulent mixing length, p′ the excess pressure surplus to the static fluid pressure that is transmitted particle to particle. The scale of turbulence in the lower layer would be of the order of the small distance between the particles (different from the basal layer in pyroclastic flow in which the upward gas flow is effective), and µa is also small because the material does not contain fine cohesive particles. Therefore, the third and fourth terms on the left-hand side of Eq. (79) are negligible in comparison to the second term. In this case, ρT is equal to ρ, i.e. density of plain water. To transmit the quasi-static pressure, ps, particles must always be in contact even though their relative position continuously changes. This condition requires the solids concentration should be larger than C3. The

PGCC01

10/12/08

4:23 PM

Page 28

28

T. Takahashi

value of C3 depends on composition of particle sizes, but it is around 0.5 as mentioned earler. If solids concentration at the bottom of the flow is C and it * decreases upward, u at the bottom is 0 because no flow can exist at the concentration C . Therefore, at * the bottom, the dynamic pressure pd that is given by Eq. (7) is zero and all the excess pressure p′ should be borne by the static pressure ps. At the height where C becomes equal to C3, ps becomes zero. Vertical distribution of ps is not yet known, but to represent the characteristics mentioned the following function is assumed. ⎧ C − C3 p′ ⎪ ps = ⎨ C − C 3 * ⎪0 ⎩

; C > C3

sB 2C −B dC tan θ = dZ K 2 (1 − Z + sF )

(82)

2

where Z = z/H, u′ = u / gH , s = (σ − ρ)/ρ,

tan φ − K1 , C − C3 *

1

f (C ) =

C dZ , h2 / H

(85)

where K4 = − B′ = 6

5 1− e 1 , 3 15π 1 + e 1− e 15π

⎧⎪ 5 1 1 ⎫⎪ ⎨1 + ⎬ − tan θ 2 ⎪⎩ 12 1 + e C go2 ⎪⎭

2

⎛ du ⎞ ⎛ du ⎞ 4 2 1+ e C go σd 2p ⎜ ⎟ + ρT 2 ⎜ ⎟ 5 ⎝ dz ⎠ ⎝ dz ⎠ 15π (1 − e )

ρ g sin θ dz

2

H

=

(86)

T

z

where ρT = σC + (1 − C)ρ. The effect of τk is included in the term of the turbulent mixing. Distribution formula of is unknown, but herein, simply = κz is assumed, where, κ is von Kármán’s constant. Therefore, non-dimensional expression of velocity distribution in the upper layer is

1

(83)

F=

2

The balance of force equation in the upper layer (z > h2) is given by

⎛ du ′ ⎞ 1 1 ⎜ ⎟ = K 3 2 {1 − f (C )} (1 − Z + sF ) C go B ⎝ dZ ⎠

K2 =

(84)

⎛ du ′ ⎞ 1 1 (1 − Z + sF ) ⎜ ⎟ = K3 2 C go B ′ ⎝ dZ ⎠

; C ≤ C3

We can obtain the solids concentration and velocity distributions in the flow from Eqs (79), (80) and (81). Namely, in the region of C ≥ C3 (lower part of the lower layer), neglecting the third and fourth terms on the left-hand side of Eq. (79), and moreover neglecting (–13 go) in the curly bracket of the second term; i.e. τk because the solids concentration in this layer is very large:

1− e , 15π

⎛ sB ′ 2C ⎞ 3 2 ⎜ tan θ − B ′⎟ C go ⎝ ⎠ dC = dZ K 4 ( go + 2) (1 − Z + sF )

(81)

Velocity and solids concentration distributions

K1 = 6

and the velocity distributions are

C − C3 C − C3 *

B = f (C) tan φ + K1{1 − f (C)} − tan θ, 2

⎛ ρ ⎞ ⎛ H ⎞ ⎛ 15 1 − e ⎞ K3 = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ sin θ ⎝ σ ⎠ ⎝ dp ⎠ ⎝ 2 1 + e ⎠ In the region of C < C3 within the lower layer (upper part of the lower layer), f (C ) = 0, and in this layer τk should be taken into account. Therefore, the solids

( sC + 1) sin θ d Z 2 ⎛ du ′ ⎞ Z ⎜ ⎟ = 2 ⎝ dZ ⎠ 4 2 (1 + e ) σ ⎛ dp ⎞ 2 C go ⎜ ⎟ + ( sC + 1) ′ 5 15π (1 − e ) ρ ⎝ H ⎠ (87) where ′ = κZ. In the upper layer, particles are suspended by turbulence, so that like the normal suspended load equation, the following solids concentration distribution is assumed (Rouse, 1937); wso

⎧⎪ ⎫⎪ κ u*1 ξo C =⎨ ⎬ C a ⎪⎩ z + ξ o − ( H − h1) ⎪⎭

(88)

where u 1 = gh1 sin θ , ξ o = −wsCa /{κu 1 (dC /dz )z = h }, 2 * *

PGCC01

10/12/08

4:23 PM

Page 29

Mechanics and simulation wo is the settling velocity of the particle and Ca is C at height ξ = ξo. Concerning the determination of the boundary between the lower and the upper layer, the following criteria are adopted. 1 If ws /u is larger than 1, no suspended particle exists. * In this case h2 = H. Herein, u = gH sin θ . * 2 The critical height above which the second term on

Fig. 20. Solids and velocity distributions in inertial debris flows for different particle diameters and channel slopes under a constant unit width discharge, q = 53.7 cm2 s−1 which corresponds to author’s experiments. (a): dp = 0.017 cm; (b): dp = 0.030 cm; (c): dp = 0.066 cm; (d): dp = 0.201 cm.

29

the left-hand side of Eq. (86) becomes larger than the first term (i.e. τt ≥ τc) is considered as z = h2. Namely, the minimum z that satisfies the following inequality is set as h2.

PGCC01

10/12/08

4:23 PM

Page 30

30

T. Takahashi κ 2{( σ /ρ − 1)C + 1}Z 2 4 2 (1 + e ) ⎛ σ ⎞ ⎛ dp ⎞ C go ⎜ ⎟⎜ ⎟ 5 15π (1 − e ) ⎝ ρ ⎠ ⎝ H ⎠

2

≥1

(89)

3 In the region z ≥ h2, C ≤ C3. The non-dimensional value h′2 is the minimum Z that simultaneously satisfies the criteria 2 and 3. Comparison with the experimental data Figure 20 shows the results of calculation of solids concentrations and velocity distributions under a constant discharge (q = 53.7 cm2 s−1) using the above mentioned system of equations, in which the parameters are, tan φ = 0.7, C = 0.6, C3 = 0.5, e = 0.3, κ = 0.4, * σ = 2.65 g cm−3, ρ = 1.0 g cm−3. When the diameter of particles is large (dp = 0.201 cm), and if channel slope is very steep (θ = 17.5° and θ = 15°), a high concentration of solids is present up to the surface of the flow. These are ‘stony debris flows’. The kink that is obvious on the curves of velocity and solids concentration distributions corresponding to 17.5° and 15° slopes, respectively, indicates the conditions at which the concentration of solids falls below C3. If channel slope is mild (gentler than 12.5°), the solids concentration abruptly becomes small high in the flow. This is the case of an ‘immature debris flow’. The velocity distribution curve whose channel slope is 12.5° has a kink very close to the surface of the flow. This means that a slope of 12.5° is the threshold between stony debris flow in the steeper region to immature debris flow in the gentler region. When particle diameter is small (e.g. dp = 0.017 cm), a suspension layer appears in the upper part, and solids concentration is large up to the surface of the flow. This is the case of a ‘hybrid debris flow’, and if the suspension layer appears as low as the channel bottom, it is a ‘muddy debris flow’. Both the velocity distribution curves for dp = 0.017 cm and dp = 0.03 cm corresponding to the channel slope 17.5° in Fig. 20 have evident kinks, below which the flow is more or less laminar and above which flow is turbulent, so that these are the hybrid type flow. These characteristics agree with the experimental results presented by Takahashi (1991). In the calculation performed to obtain Fig. 20, however, the restitution coefficient of the particles is set as 0.3 which, though very small in comparison to that in air (around 0.8), is necessary to explain the experimental trend. In the analyses of saltation of bed load particles in water, similar restitution values to that in air are often used. However, in debris flow, because the solids concentration is very high and the small void space hampers free

Fig. 21. Comparison of the present theory with the experimental results on inertial debris flow obtained by Hirano et al. (1992). Solid line is obtained by the theory.

exchange of particles with water, a kind of buffering effect would arise. As already shown in Fig. 3, a small restitution coefficient leads to smaller fi than that obtained by Bagnold’s experiment. This necessarily means that pc cannot bear all the load and some part of the load is supported by ps at the lower part of the flow especially when the sediment concentration is large. It must be further noted that Eqs (6) and (7) cannot be applied for a concentration lower than about 0.1. Therefore, for the calculation of the case that corresponds to immature debris flow, the trend in variation of solids concentration around 0.1 is extrapolated until it becomes zero. Figure 21 is a comparison of the calculated velocity distribution with the experimental results obtained by Hirano et al. (1992), in which debris flows were generated by supplying water from upstream onto the water saturated erodible bed flume 20 cm in width and 7 m in length. Bed slope was held constant, 14°, and the relative depth, H/dp was changed by changing the supplying water discharge and particle size. One can see that the theory can explain the experimental trend rather well, particularly for the cases of large relative depth. Figure 22 compares the theoretical transport concentrations Ctr (= Qs /QT , Qs ; sediment discharge, QT ; total discharge of water plus sediment) with the author’s experimental results, in which debris flows

PGCC01

10/12/08

4:23 PM

Page 31

31


seem to change along the dotted lines. In the smaller particle cases the calculated lines split into two parts corresponding to mud flow in the flatter slope region and hybrid flow in the steeper flow region, respectively. Theoretical calculations for the mud flow region, however, extend to an excessively steep channel slope, and those for the hybrid flow region give smaller concentrations than the experiments. These facts suggest that further research is necessary on the concentration formula of suspended sediment. The divergence of the theoretical curves may be brought about by the neglect of some terms in Eq. (79), so that more critical estimations of respective terms are necessary. The broken line is for the cases of no suspension (for large particles), i.e. for stony debris flow and immature debris flow. It coincides well with Takahashi’s (1991) previous equilibrium concentration formulae, i.e. for stony debris flow Fig. 22. The transport concentration of debris flow versus channel slope for different particle diameters. The solid lines are theoretically obtained ones corresponding to different regimes, i.e. hybrid debris flow, muddy debris flow, immature debris flow and stony debris flow, and the dotted lines connecting the respective solid lines show the transition between the two regimes. The continuous broken line shows Takahashi’s previous equations (Takahashi, 1991) for the equilibrium concentration of immature and stony debris flows.

were generated by abruptly supplying water from upstream onto the water-saturated erodible bed in a flume 9.9 cm in width and 10 m in length. Bed slope was changed, and QT was held constant by trial and error. Four kinds of materials 0.201, 0.066, 0.030 and 0.017 cm in the median diameters were used for the erodible bed whose densities were between 2.64 and 2.66 g cm−3 and the internal friction angles between 38.5° and 39°. The debris flow that flowed out from the outlet of the flume was sampled by a bucket, and QT and Ctr were measured. As shown in the figure, the theory can well explain the tendency that the smaller the particle diameter, the larger the transport concentration becomes under a certain channel slope. The solid line calculated for the particle diameter dp = 0.201 splits into three parts that respectively correspond to immature debris flow, hybrid debris flow and stony debris flow from the smaller channel slope to large, and the dotted lines connecting these three curves mean the transition between the different flow types. The transition arises over a small slope change and in this region it is difficult to determine by calculation which of immature, hybrid or stony debris flows actually occurs. In any case, the experimental data

C∞ =

ρ tan θ ( σ − ρ) (tan φ − tan θ)

(90)

and for immature debris flow Ctr = 6.7C 2∞

(91)

VISCOUS DEBRIS FLOW Mechanism to disperse particles and the solids and velocity distributions When a debris flow contains fine cohesive particles, the viscous stress becomes more important than the stress due to particle collisions. This is because the viscosity of the slurry that forms the interstitial fluid is very high and, in addition, the apparent viscosity of the entire material becomes very large due to the effect of highly concentrated coarse particles. In a highly viscous flow, turbulence decreases and particles cannot be suspended by the fluid turbulence. How coarse particles can disperse in highly viscous laminar flow while experiencing minimal collisions is the main interest from the fluid mechanical point of view. Takahashi (1993) claimed that the approach of two particles embedded in adjacent shearing surfaces causes the perpendicular force proportional to d 2p f ′(C) ∇8 due to squeezing of interstitial fluid, where 8 is the deformation rate, and f ′(C) is a function of C. Developing this consideration to the space with a concentration gradient, the following formula is obtained. d 2p f ′(C) ∇(8C) ∝ ( –12 )ρCD ( –π4 )d 2p w 2

(92)

PGCC01

10/12/08

4:23 PM

Page 32

32

T. Takahashi

where w is the velocity of the fluid perpendicular to the main flow and CD is the drag coefficient. For a laminar flow, CD = 24/Re = 24µ/ρwdp, then 1 w∝ d f ′(C ) ∇( 8 C ) 3πµ p

(93)

where µ is the viscosity of the interstitial fluid. Assuming f ′(C ) ~ C and the particle velocity immediately equalizes with that of the ambient fluid, the vertical particle flux due to encounter of particles is given as Nc = −Kcd 2p (C 2 ∇8 + C 8 ∇C )

Nµ = −K µ a

a

d 2p d µa ∇C µa dC

(95)

where µa is the apparent viscosity of the entire mixture and Kµ is a constant. a Particle settling flux due to gravity is given, assuming Stokes’ law, as 2 d 2p ( σ − ρ) g cos θ Ns = − C g (C ) 9 µ

(96)

where g(C) is a hindrance function to account for highly concentrated group settling. Herein, g (C ) =

1−C η

(97)

is used, where η is the relative viscosity and according to Krieger (1972) η=

µa ⎛ C⎞ = ⎜1 − ⎟ µ ⎝ C *⎠

−1.82

(98)

Therefore, the particle conservation equation can be written as ∂C = −∇ ⋅ ( Nc + Nµa + Ns ) ∂t

In the steady state, Eq. (99) is written as

2 d 2p ( σ − ρ)g cos θ + C g(C ) = 0 µ 9

(99)

(100)

The stress balance Eq. (79) in this case would be given by neglecting all the terms except µa(du/dz) in the left-hand side of the equation. Therefore, the relationship between the shearing stress and the viscous stress at height z is given by ⎛ ρu 2 8 =⎜ * ⎝ µa

(94)

where Kc is a constant. Phillips et al. (1992) obtained the same equation via different reasoning. Phillips et al. (1992) also discussed the possibility of the effect of spatially varying viscosity that arises from spatially varying particle concentration. Namely, since the apparent viscosity is larger in the higher concentration region than in the lower concentration region, the resistance to migration into larger concentration region is larger than that into smaller concentration region. This effect gives rise to the particle flux as 8C 2

d2 ⎛ d8 dC ⎞ 2 p d µa dC + C8 Kcd p2 ⎜C 2 ⎟ + K µa 8 C dz ⎠ µa dC dz ⎝ dz

⎧ ⎞ ⎪⎛ z⎞ ε ⎟ ⎨⎜⎝1 − H ⎟⎠ + H ⎠⎪ ⎩

⎫ ⎪ C dz⎬ ⎪ z ⎭

h

(101)

where ε=

σ−ρ ρ

(102)

From Eqs (100) and (101), the particle concentration distribution function is written as dC = dZ −

2 ε (1 − C ) + C (1 + εC ) 9 Kc tan θ

−1 ⎧ ⎧ ⎛ Kµ ⎞ C ⎛ C ⎞ ⎫⎪ ⎪ ⎪ a − 1⎟ ⎨1 + 1.82 ⎜ ⎜1 − ⎟ ⎬ ⎨(1 − Z ) + ε C ⎠ ⎪⎪ ⎝ Kc ⎠ C* ⎝ * ⎭⎩ ⎩⎪

⎫ ⎪ C dZ ⎬ ⎪⎭ z

1

(103) Figure 23 shows the calculated distributions of the particle concentration under the conditions of C = * 0.6, Cb (the concentration at the bottom of flow) = −3 0.57 and ρ = 1.38 g cm (ε 0.9), and C = 0.72, Cb * = 0.70 and ε = 0.77. These conditions correspond to the flume experiment and the actual debris flow at Jiangjia gully, respectively, which will be referred to below. As for the constants, Kc = 0.5, Kµ a = 0.75 are adopted referring to the results of experiment by Phillips et al. (1992). Note that their experiments used neutrally buoyant particles and therefore to determine the viable values more experiments using heavy particles are necessary. As is clear in the figure, when the channel slope is very steep, very high concentration is maintained close to the surface and, with decreasing channel slope, although high concentration is maintained in the lower region, concentration abruptly decreases in the upper region. The particle concentration becomes large and uniform as ε becomes smaller and smaller. If the debris

PGCC01

10/12/08

4:23 PM

Page 33


33

Fig. 23. The theoretical distribution of the coarse particle concentrations in viscous debris flows on various channel slopes. (a): the results obtained for the conditions in the author’s experiments; (b): the results obtained for the conditions in Jiangjia gully, China.

flow material is comprised of a wide range of particle sizes from very fine clay to large stones, it is a difficult problem to decide up to what size particles can be considered to form the interstitial slurry. The typical viscous debris flow that occurs at Jiangjia gully, Yunnan, China has a wide sediment size distribution (the maximum size is, however, about 10 cm) and it passes over a gently sloping reach having a very high solids concentration; it must therefore have small ε value. It is possible that particles larger than the clay size effectively form part of the interstitial slurry. The velocity distribution can be obtained by solving Eq. (101), but if one assumes a uniform concentration throughout the depth, and takes the boundary condition at Z = 0, u′ = 0, into account, one obtains u′ =

ρ gH H sin θ ⎛ F⎞ ⎜1 − ⎟ µ C ⎠ ⎝ *

1.82

⎛ 1 ⎞ (1 + εF ) ⎜ Z − Z 2 ⎟ 2 ⎠ ⎝ (104)

where F is the cross-sectional mean concentration. The resistance to flow is therefore written as U 1 ρu* = (1 + εF )H u* 3 µa

(105)

where U is the cross-sectional mean velocity. Takahashi et al. (1997) carried out some flume experiments, in which a steel movable slope flume 10 m in length, 9 cm in width and 40 cm in depth was used. One side of the flume was a transparent glass wall through which the behaviour of the flow could be observed. The bed was roughened by glueing gravel 3 mm in diameter to the bed. Prior to an experimental run, 25 000 cm3 of the debris flow material was stored in a hopper set 7.5 m upstream of the outlet of the flume. Then, the hopper was opened to supply the material to the flume. A high-speed video mounted alongside the flume 2 m downstream of the hopper recorded the flow structure with 200 frames a second.

Fig. 24. The particle size distributions of the materials in Jiangjia gully and in the experiments.

The discharge from the hopper gradually decreased with the reduction of material left in the hopper, and shortly after all the material in the hopper was released a certain thickness of deposit on the bottom of the flume remained nearly parallel to the bed surface. The well-mixed characteristics of the debris flow material in Jiangjia gully was well simulated (except for the component larger than 10 mm) by mixing silica sand and kaolin. The particle size distributions of the materials in Jiangjia gully (the component smaller than 20 mm) and in the experimental one are compared in Fig. 24. Although debris flows in Jiangjia gully are said to flow with the solids volume concentration as much as 0.53 to 0.85 (Wu et al., 1990), the debris flow in the experiment could not reach the outlet of the flume if the solids concentration was higher than 0.58, and if it was lower than 0.55 the material easily segregated in the hopper before feeding into the flume. Therefore, many experimental runs were conducted under the constant solids concentration of 0.56 to 0.57. The debris flow material in Jiangjia gully has a bimodal distribution and it lacks the fraction whose particle diameter is around 0.1 mm. Therefore, in this material the particles smaller than 0.1 mm may be considered as material that compose the slurry. Similarly, if the particles smaller than 0.1 mm in the experimental material are considered to compose the slurry, the slurry has an apparent density of 1.38 g cm−3

PGCC01

10/12/08

34

4:23 PM

Page 34

T. Takahashi

Fig. 25. Temporal changes in the velocity distribution, the flow depth and the thickness of deposit for viscous debris flow in the 11° channel slope case. The upper horizontal line is the surface of the flow and the lower horizontal line is the surface of the deposit.

Fig. 26. The calculated velocity distributions (parabolic curves) and the experimental results (circles) for viscous debris flow. (1): on the deposit layer; (2): on the rigid bed.

(volume concentration of the solids is 0.226). The slurry in the experimental material itself was well modelled as a Bingham fluid having a viscosity and yield strength of 0.036 Pa·s and 11.8 Pa, respectively. Figure 25 shows the temporal change in the velocity distribution, the flow depth and the deposit thickness in an experimental run. The deposit layer gradually thickness and the vertical velocity gradient becomes smaller with decrease in the flow depth without making a solid plug near the surface of the flow. These characteristics cannot be explained by the so far widely accepted Bingham fluid model, in which stoppage of flow, and therefore the production of sediment accumulation of a certain thickness on the channel

bed, occurs abruptly when the flow depth becomes shallower than the plug thickness. The author’s observations indicate that no plug exists in the actual viscous debris flow in Jiangjia gully. The viscosity of the slurry, µ, is affected by the electrical force between very fine particles and cannot be estimated by Krieger’s formula. It must be found from experimental data at present. As for determining η, the deformable space not occupied by solids is important. Therefore, in taking C and F values in Eq. * (104), the entire solids concentration including coarse particles and clay is used. The velocity distributions given by Eq. (104) are compared with the experimental data in Fig. 26, in which the experimentally obtained

PGCC01

10/12/08

4:23 PM

Page 35

35

Mechanics and simulation Deposition and erosion of viscous debris flow

Fig. 27. The stress–strain relationship of a slurry in viscous debris flow at Jiangjia gully (solid line), and the estimation of viscosity as a Newtonian fluid (broken line). Note that the estimate of 4.0 Pa as the yield strength was obtained by linear extrapolation from the relationship at large strains; the actual relationship in the small strain region has generally the tendency to curve towards smaller yield strengths.

Although viscous debris flow can transport highly concentrated sediment, a flow that is fully developed on a very steep channel upstream must be over loaded when it comes down to a flatter reach. It will deposit the excess load on the flatter bed. Figure 28 illustrates such a depositional process in which the hatched part deposits in the next unit of time. The equation of motion of this hatched volume is given by Eq. (56). If one writes the equilibrium coarse particle concentration in the reach under consideration as Ce and assumes a uniform solids concentration through the entire depth, the excess pressure that is transmitted through coarse particles and operates on the bottom of the hatched volume is (σ − ρT)(C − Ce)gH cos θ. Fluid dynamic resistance, –12 ρT fU 20, also operates on the bottom of the volume, where f is Darcy– Weisbach’s resistance coefficient. The shearing stress operating on the same plane is ρT gH sin θ. Then, the external force F is written as F = −(σ − ρT)(C − Ce)gH cos θ tan φ ∆x

Fig. 28. Depositing process model for a viscous debris flow that is similar to the case of pyroclastic flow.

values of µ = 0.036 Pa·s, F 0.56 and C 0.59 are * used. Let us apply the theory to the Jiangjia gully debris flow. According to the observational data (Wu et al., 1990), volume concentration of the fine particles in the slurry is about 30% and the volume concentration of the coarse particles in the slurry is about 55%. Thus, the densities of slurry and the entire debris flow material are, respectively, ρ = 1.5 g cm−3 and ρT = 2.13 g cm−3; i.e. ε = 0.77. The total solids concentration in the debris flow material is F = 0.685. Estimation of C has * rather high sensitivity; nevertheless, from the previous examinations (Wu et al., 1990), C = 0.72 is adopted. * Eq. (98) gives η = 246. Previous rheological tests for the slurry having solids concentration of about 30% regard the slurry as a Bingham fluid that has a viscosity of µB = 0.02 Pa·s and a yield stress of τB = 4 Pa. The stress and strain relationship is as shown in Fig. 27. The strain in the actual debris flow is around 10 s−1, so that from the slope of the broken line in the figure the viscosity as a Newtonian fluid is µ = 0.42 Pa·s. Consequently, the apparent viscosity of the entire debris flow material is 103 Pa·s. Assuming H = 100 to 200 cm, θ = 3° and ε = 0.77 in Eq. (105), one obtains U/u = 5 to 14. This order value coincides with the * observation in the field (Takahashi & Arai, 1997).

− –12 ρT fU 20 ∆x + ρT gH sin θ ∆x

(106)

Mass variation of the hatched volume is given by dz dm = −ρT b ∆ x dt dt

(107)

If the depositing part that is separated from the hatched part is assumed to stop immediately, the relative velocity u is given as u = −U0

(108)

Substituting these formulae into Eq. (56), one obtains dU0 1 = dt h0 − zb

⎧⎪ ( σ − ρT )(C − C e ) gH cos θ tan φ ⎨− ρT ⎩⎪

+ gH sin θ −

⎫ 3 2 dzb U + U ⎬ Re dt 0 ⎭

(109)

where Re is the Reynolds number (= ρTUH/µa), and because the flow is laminar f = 6/Re is satisfied. Let us assume the velocity of the hatched volume becomes zero when zb coincides with h0 and during that process the velocity decreases linearly. Then, the depositing speed i (= dzb/dt) is, writing the time necessary for this process as Td , i = (h0 − zb)/Td . Because Td = −iU0 /(h0 − zb), the following formula is obtained:

PGCC01

10/12/08

4:23 PM

Page 36

36

T. Takahashi dU 0 U0 =− i dt h0 − zb

(110)

The relationship between the mean velocity of the debris flow, U, and U0 depends on the velocity distribution, but, here, simply the following relation is assumed: U0 = αU

(α < 1)

(111)

Consequently, i=

gH sin θ ⎪⎧ ( σ − ρT ) (C − Ce ) tan φ ⎨ 2αU ⎪⎩ ρT tan θ −1+

3µaU ⎪⎫ ⎬ ρT gH 2 sin θ ⎪⎭

(112)

In laminar flow, the second and third terms in the curly bracket in Eq. (111) are equal. Therefore, i=

3µ a ( σ − ρT ) (C − Ce ) tan φ 2α HρΤ2 tan θ

(113)

According to the author’s aforementioned experiment, when a debris flow surge front passes over the layer deposited by a previous surge, the entire layer begins to move again together with the arriving surge. This phenomenon happens because the deposit still has almost the same properties as the moving material. The freshly deposited layer is barely stable due to a weak binding among the coarse particles, but it is easily destabilized by the arriving surge. The viscous debris flow material is originally so highly concentrated that the volume change from moving stage to deposit is very small. Iverson (1997) claimed that even the coarse particles in a debris flow irreversively settle toward the bottom, though the time necessary for all the particles to settle is normally longer than the duration of the debris flow itself. Therefore, particles are apparently dispersed as long as the debris flow continues. This reasoning seems to be persuasive, but it fails to explain how a newly deposited layer is remobilized and entrained by a newly arriving surge. On the other hand, the laminar dispersion mechanism introduced herein can explain such a phenomenon if the operating shear becomes sufficient to disperse the coarse particles. Removal of the deposit by an arriving surge is confirmed in Jiangjia gully. There, sometimes, even when no surge arrives, minimal disturbance leads to dilatation of the deposit layer giving rise to a bore. The deposit maintains its original very soft condition over several days and it is difficult to walk on. The deposit gradually loses water by evaporation and infiltration, and becomes stiff.

where θb is the original bed slope. The deposit layer is assumed to be immediately eroded to the depth of original bed as soon as a surge front arrives. Then deposition proceeds at the speed i given by Eq. (113). The flume experiments used to check the validity of the routing method were conducted using the same set up as above. Just after the hopper was emptied to generate the first surge, the same quantity (25 000 cm3) and quality (the same material as in the previous experiments) of the debris flow material was prepared again in the hopper. Then, five to six minutes after the termination of the first surge, the second surge was introduced from the hopper to the now mobile bed of the flume. In some cases, a third, fourth and fifth surge were produced with the same procedure. In the experiment, only the channel slope was changed to vary the conditions. The surface level of the flow was measured at two positions; 3 m and 6 m from the upstream end, respectively, using laser deflection sensors. The discharge variation from the hopper was calibrated by capturing the discharge directly with a calibration box. The outflow discharge from the flume was also measured using the calibration box. The experimental data are compared with the calculation using Eqs (114) and (115) in Fig. 29. In the calculation the following values are used: Ce = 0.41 corresponding to the channel slope of 11°, α = 0.9, tan φ = 0.7, ρT = 1.92 g cm−3, σ = 2.65 g cm−3, ρ = 1.38 g cm−3 and µa = 6.2 Pa·s. Figure 29(a) shows the results for the first surge, which travelled on a rigid bed (no deposit layer exists before arrival of the surge). The translation velocities of the front both in the calculation and in the experiment are the same at 73.7 cm s−1 and it takes 4 seconds to travel between the two stations. Apart from the calculated front stage being a little higher than the experimental result, the general tendency of stage variation is well reproduced by the calculation. With time, both calculated and experimental stages approach constant heights at respective stations. This means the flow has already stopped. The calculated time of stopping and the thickness of deposit coincide well with the experiment. Figure 29(b) shows propagation of the second surge that travelled on the deposit made by the first surge. There are some differences in the front arrival time between the calculation and the experiment, but it is probably due to difference in setting of the starting time, because the time lags between the two stations are almost the same, about 2.7 seconds. Stage variation in the experiment has dual peaks, whereas in the calculation it has only one peak. This probably

PGCC01

10/12/08

4:23 PM

Page 37

37


Fig. 29. Comparison between the calculated and experimental flow stages and deposit surface stages for viscous debris flow in the case of 11° channel slope. (a): the first surge on the rigid bed; (b): the second surge that travelled on the deposit made by the first surge.

occurred due to difficulty in controlling supply discharge in the experiment. The hydrographs of the first and second surges in the calculation are identical. Except for this discrepancy, the calculation generally explains well the experimental results. The time required for the first surge to travel the 3 m reach is 4 seconds and that for the second surge is only 2.7 seconds. The translation velocity of the first surge is almost equal to the mean cross-sectional velocity at the peak. Therefore, the existence of the deposit contributes, in this case, to making the front velocity faster than the mean velocity of the following part. Note that the final stages of the surface of the deposit after the first surge and after the second surge are almost the same. This is one of the peculiar behaviours in the Jiangjia gully as well.

CLASSIFICATION OF DEBRIS FLOWS The discussion earler suggests that the total shearing stress, τ, in a sediment laden plain shear flow would be given by the following equation:

τ = τy + τµ + τc + τk + τs + τm

(116)

where τy is the yield strength of the interstitial fluid if any, τµ the viscous stress due to deformation of the interstitial fluid, τc the collisional stress, τk the kinetic stress, τs the static stress and τm the turbulent mixing stress. It is already clear that all the terms in Eq. (116) cannot simultaneously dominate and the dominant terms change depending on solids concentration, particle diameter and other hydraulic conditions. Namely, the dominant term is different in different flow regimes. If we write τy + τµ = τv, τc + τs = τg, τk + τm = τt, τv is considered to represent the shear stress due to viscoplasticity of the material, τg is that due to contact of coarse particles in the material and τt is that due to turbulent mixing and migration of coarse particles. Therefore, under a constant solids concentration, the ratios, τv /τ, τg /τ and τt /τ should be the controlling factors for the behaviours of the debris flow. Based on these considerations and the various kinds of debris flows mentioned in earlier discussion, a classification scheme and existence criteria of the

PGCC01

10/12/08

38

4:23 PM

Page 38

T. Takahashi

Fig. 30. The criteria for the existence of various sediment motions in which the interstitial fluid is water or slurry. In the ternary diagram describing the classification of debris flow (0.02 < C < C3), the ratio of the two end members at the apexes τt /τ and τc /τ; τt /τc, is the function of (H/dp)2, so that as the relative depth of the flow becomes the larger, the larger the relative effect of turbulence becomes compared to the effect of collision. The ratio τc /τµ (τ ν is approximated as τµ ) represents the Bagnold’s number which classifies the flow as inertial or viscous, and the ratio τµ /τt is the inverse of Reynolds number. Therefore, the three axes of the ternary diagram represent relative depth, Bagnold’s number and Reynolds number, respectively. The region where Bagnold’s number is large and the relative depth is small is that for stony debris flow occurrence. In the region where Bagnold’s number and Reynolds number are small, viscous debris flow occurs, and in the region where the relative depth and Reynolds number are large, turbulent muddy debris flow occurs. Thus, the areas close to the three apices are occupied by the existence areas of stony, viscous and muddy debris flows, respectively, and the rest of the area in the triangle should be shared by immature and hybrid debris flows. The boundary lines shown in the figure are arbitrary. The shape of the boundary lines and the area shared by the respective type of flows should change with C.

debris flows are given in Fig. 30. The vertical axis represents the mean coarse particle concentration in the flow. If coarse particles are not present (at the lowest point on the vertical axis), the flow is water or slurry flow. In water flow, almost all the shear stress is normally shared by the turbulent Reynolds stress and the larger the viscosity the larger the viscous stress becomes. Therefore, the flow regime changes along the lowest horizontal axis. The ratio of the end members

of this axis, τµ /τt , is simply the Reynolds number. Therefore, Reynolds number changes along this axis. When solids concentration becomes larger but still less than about 0.02 (Takahashi, 1991), the flow starts to contain bed load or suspended load depending on the turbulence and viscosity. The particle collision stress appears but it is small. As mean coarse particle concentration becomes larger but is smaller than C3, the flow becomes a debris

PGCC01

10/12/08

4:23 PM

Page 39

39

Mechanics and simulation flow. In this case possible dominant stresses are particle collision stress, turbulent mixing stress and the viscous stress, because under such a concentration the quasi-static stress cannot be dominant. Therefore, subclassification of the debris flow is possible on the ternary phase diagram as shown in the figure. If the concentration of the coarse particles is lower than about 0.25, stony debris flow cannot appear (Takahashi, 1991), so that the area occupied by immature debris flow in the ternary phase diagram must be larger in the lower solids concentration. Thus, the boundaries of the subclassification in the triangular domain depend on the particle concentration, and how they change is not yet clear. Highly concentrated turbulent or laminar muddy flows that lack coarse particles are sometimes called ‘hyperconcentrated flow’, in which the particle collision stress is minimal. The ratio τt /τc can be approximated as fn(C, e) (H/dp)2, where fn(C, e) means a function of C and e. The ratio τc /τµ represents the Bagnold’s number which classify the flow is viscous or inertial (Bagnold, 1954). Therefore, the three axes of the ternary diagram represent Reynolds number, relative depth and Bagnold number, respectively. As concentration becomes larger, it exceeds C3. In this region, collision, turbulent and viscous stresses become small and instead the quasi-static Coulomb stress becomes dominant and the flow becomes quasistatic motion. And if concentration becomes larger than C2, dislocation of the particles cannot take place and the material becomes rigid.

CONCLUSION Representative subaerial rapid geophysical flows, i.e. debris flow, pyroclastic flow and snow avalanche are discussed from the fluid dynamic point of view as twophase flows. The common features in inertial debris flow, pyroclastic flow and snow avalanche are the existence of the dynamic stresses caused by granular collisions, migration of particles to other layers and turbulent mixing, and the quasi-static skeleton stresses in the lower very densely concentrated part. The evident difference between granular flow in the air and inertial debris flow is of course the difference in interstitial fluid, and in the latter the buoyancy effect plays some role. Another difference that has become clear is the effect of buffering due to narrowness of void spaces which hinders free escape of water in the latter. The main differences between granular flow and Merapi-type pyroclastic flow are that crushing of

granules into fragments and upward gas flow emitted from the material itself play important roles in the latter. The main differences between pyroclastic flow and snow avalanche are that production of big balls by condensation and slip velocity on the bed play important roles in the latter. The strong erosion of the new snow layer may be another important point, because erosion by the pyroclastic flow is not conspicuous in the flow which occurred at Unzen volcano, or in the authors experiments (Takahashi et al., 1995). Variations of particle diameters in the pyroclastic flow and snow avalanche can be similarly estimated by assessing frequency of encounter between particles. The particle diameter in pyroclastic flow becomes small by fragmentation on collision, and that in snow avalanche becomes large by adhesion on collision. Viscous debris flow is different not only from inertial debris flow but also from granular flow, pyroclastic flow and snow avalanche, because the effect of inelastic collision of particles does not play an important role in viscous debris flow. Dispersion of particles in very viscous laminar flow is intrinsic in the viscous debris flow. The model presented here is a Newtonian fluid model that is different from the previous widely accepted Bingham fluid model. This new model can explain the experimental results as well as the natural debris flows in Jiangjia gully, China. The numerical simulation models for pyroclastic flow and snow avalanche are applied to the actual phenomena and their validity is confirmed. However, they have many coefficients to be determined by experiments and by compilation of actual phenomena. Of course, refinement of theories is also indispensable. The processes of initiation and development of these phenomena are still vague, so that forecasting the place, time and scale of the phenomena remains difficult. Development of viable models is urgently required. Other representative subaerial geophysical flows, i.e. debris avalanche and motion of landslide need investigation from a similar point of view.

ACKNOWLEDGEMENTS This chapter is a review of the recent research that has been done in the research section for sediment disaster of the Disaster Prevention Research Institute, Kyoto University under the leadership of the author. Enthusiastic cooperation of staff and students is highly appreciated. Ben Kneller kindly revised the manuscript and helped to improve the paper by giving numerous

PGCC01

10/12/08

4:23 PM

Page 40

40

T. Takahashi

suggestions. Financial support was obtained from the Ministry of Education, Science, Sports and Culture, Japan.

g go h

NOMENCLATURE

hf h0

a b C Ca CD Ce Cl Cst Ctr C3 C * C∞ F Gs Gso dfc dp dpf dpm Di Dn e Es ET f fb fcu fi fpi F

a constant value rate of gas emission from unit mass of particles volume concentration of the particles solids concentration at a reference height drag coefficient equilibrium concentration volume concentration of snowballs solids concentration of granular flow on the slopes θ ≤ φ transport concentration of solids threshold concentration to generate quasistatic pressure concentration of solids when packed equilibrium concentration in the stony debris flow average concentration of the mixture of snowballs and snow particles concentration of snow particles in the domain no snowball exists concentration of snow particles before snow-balls are produced critical diameter to give rise to fluidization particle diameter mean diameter of the particles in the hot ash cloud mean diameter coefficient of diffusion of the granular temperature thickness of the new snow layer restitution coefficient particle entaining velocity of the hot ash cloud coefficient representing the stickiness of the snow particles frictional coefficient frictional coefficient at the interface between the main body and the hot ash ratio of particle fraction that is transportable to the ash cloud from the main body coefficient to describe the dynamic shear stress in granular flow frictional coefficient at the upper surface of the hot ash cloud external force operating on a volume

H Hf Hp Hpa Hs i id ie if ig I In Ir Kc Kx Ky Kµ a l ′ Lh m n nb nk n∞ N Nc Nij Ns Nµ a p′ pcl

acceleration due to gravity radial distribution function height of the boundary between the base and upper layers in pyroclastic flow thickness of the base layer height at which the operating shearing force and the resistance balance depth of flow thickness of the main body of pyroclastic flow thickness of the hot ash cloud layer pressure in the hot ash cloud layer height of the surface of the flow measured from the original bed erosion or deposition velocity deposition velocity erosion velocity depositing velocity of pyroclastic flow depositing velocity of pyroclastic flow in the granular flow stage unit tensor rate of increase in particle number by collision changing rate of the total particle number a numerical coefficient x-wise coefficient of diffusion y-wise coefficient of diffusion a numerical coefficient subscript to represent the quantities corresponding to the snowballs mixing length non-dimensional mixing length length of the front part of the avalanche mass of the volume that deposits in the next increment of time number of particles in unit volume at granular flow stage of pyroclastic flow particle number in unit volume in the new snow layer or in the flow number of diameter k particles in unit volume total particle number frequency of particle collision in unit time per unit volume in pyroclastic flow vertical particle flux due to encounter of particles frequency of collision in unit time and in unit volume between particles whose diameters are i and j particle settling flux vertical particle flux due to heterogeneous concentration excess pressure surplus to the static fluid pressure pressure due to collision between snowballs

PGCC01

10/12/08

4:23 PM

Page 41

Mechanics and simulation pcs pd ps Qmax Qs QT r s t T Ta Td Tp Ts u u′ ur us u*

collision pressure between snow particles dynamic pressure due to collision of particles pressure due to continuous contact between granules maximum discharge sediment discharge total discharge of water plus sediment density ratio of the snowballs to the snow particles subscript to represent the quantities corresponding to the snow particles time granular temperature temperature of the ambient air time necessary for deposition temperature in the hot ash cloud snow temperature mean velocity in the direction of the main flow nondimensional velocity (= u/ gH ) relative velocity of depositing volume observed from the main flow surface velocity of flow non-dimensional velocity of snow avalanche (= u/ gH sin θ )

u * u p * u*sl U Uf Ug Up Uslc Uslf Uth U0 v V Vf Vp Vt w

shear velocity shear velocity in the hot ash cloud layer non-dimensional slip velocity at the bed cross-sectional mean velocity of flow in the direction of the main flow mean velocity of the main body in the main flow direction cross-sectional mean velocity of pyroclastic flow in granular flow stage x component of the mean velocity of the hot ash cloud layer slip velocity caused by the collision stress slip velocity caused by the friction stress threshold velocity for beginning of deposition velocity of a volume that deposit in the next increment of time velocity vector cross-sectional mean velocity of flow in y direction mean velocity of the main body of pyroclastic flow in y direction y component of the mean velocity of the hot ash cloud layer total solids volume in unit volume of the avalanche material velocity of fluid perpendicular to the main flow

we we1 we2 we3 wf ws x y z zb Z α αe αs β γ 8 δ ε εp η θ θb θbxo θbyo θs θx θy κ λ µ µa ν ρ ρa ρm ρmp ρpl

41

rate at which the upper surface of the hot ash cloud layer rises air entraining velocity at the upper surface of the hot ash cloud gas supplying velocity from the main body to the hot ash cloud separating velocity of the ash cloud as a plume rate of rise of the boundary between the main body and the hot ash cloud settling velocity of the particle coordinate in the direction of the main flow horizontal coordinate in the direction perpendicular to x height from the bottom of flow thickness of the deposit non-dimensional height (= z/H ) a numerical coefficient a coefficient to determine we1 ratio of the quasi-static pressure to the total pressure coefficient representing fragileness of particles upon collision dissipation rate of granular temperature due to inelastic collision deformation rate friction angle between snow particles and the bed relative weight of particles in fluid coefficient of eddy viscosity relative viscosity slope angle on which flow appears original bed slope slope angle of x axis slope angle of y axis gradient of the surface of the ash cloud at the front x component of the slope angle of the surface of deposit y component of the slope angle of the surface of deposit Kármán constant linear concentration viscosity of the interstitial fluid without particle load apparent viscosity of the interstitial fluid kinetic viscosity of the pyroclastic gas density of the interstitial fluid density of air apparent density of the flowing material density of the hot ash cloud density of snowball

PGCC01

10/12/08

4:23 PM

Page 42

42 ρps ρT σ τ τbx τby τc τcl τcs τd τf τg τk τkl τm τs τt τv τy τµ φ φ′ φsl φsp φss

T. Takahashi density of snow particle apparent density of fluid that suspends particles density of the particle total shearing stress in flow frictional resistance stress in the x direction at the bottom of flow frictional resistance stress in the y direction at the bottom of flow shearing stress due to particle collision shearing stress due to collision between snowballs shearing stress due to collision between snow particles dynamic shearing stress shearing stress in the fluidized layer shear stress due to contact of coarse particles kinetic stress due to plunging of particles kinetic shearing stress due to migration of snowballs turbulent mixing stress shearing stress due to continuous contact between granules shear stress due to turbulent mixing and migration of coarse particles shear stress due to viscoplasticity of the material yield strength of the interstitial fluid viscous stress due to deformation of the interstitial fluid internal friction angle of the granular material coefficient of momentum loss when snow particles hit the bed internal friction angle between snowballs angle of repose for spherical particles internal friction angle between snow particles REFERENCES

Bagnold, R.A. (1954) Experiments on gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. S. London, Ser. A 225, 49– 63. Bagnold, R.A. (1966) The shearing and dilatation of dry sand and the ‘singing’ mechanism. Proc. R. S. London, Ser. A 295, 219–232. Boudon, G., Camus, G., Gourgaud, A. & Lajoie, J. (1993) The nuée-ardente deposits of Merapi volcano, central Java, Indonesia: stratigraphy, textural characteristics and transport mechanism. Bull. Volcanol. 55, 327–342. Campbell, C.S. (1990) Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57–92. Campbell, C.S. & Brennen, C.E. (1985) Chute flows of granular material: some computer simulations. Trans. ASME J. Appl. Mech. 52, 172–178.

Fukushima, Y. & Kaneko, Y. (1997) Decelerating inclined thermals with suspended solid particles. Ann. J. Hydraul. Engineering, JSCE 41, 537–542 (in Japanese). Gidaspow, D. (1994) Multiphase Flow and Fluidization. Academic Press, London. 467 pp. Hirano, M., Hashimoto, H., Fukutomi, A., Taguma, K. & Pallu, M.S. (1992) Nondimensional parameters governing hyperconcentrated flow in an open channel. Proc. Hydraul. Engineering, JSCE 36, 221–226 (in Japanese). Hopfinger, E.J. (1983) Snow avalanche motion and related phenomena. Ann. Rev. Fluid Mech. 15, 47–76. Hutter, K. & Rajagopal, K.R. (1994) On flows of granular materials: review article. Continuum Mech. Thermodynam. 6, 81–139. Ishikawa, Y., Yamada, T., Kawashita, S., Matsumoto, T., Takahashi, T. & Otsuzuki, H. (1994a) Sedimentary structure and physical properties of the pyroclastic flow deposits produced at Mt. Unzen volcano on June 8, 1991. Special Report on Volcanic Sabo Research of Mt. Unzen, JSECE 1, 71–78 (in Japanese). Ishikawa, Y., Yamada, T. & Yajima, S. (1994b) Kinematic characteristics of the main body and the hot ash cloud in pyroclastic flow at Unzen. Special Report on Volcanic Sabo Research of Mt. Unzen, JSECE 1, 24 –30 (in Japanese). Iverson, R.M. (1997) The physics of debris flows. Rev. Geophys. 35-3, 245–296. Jenkins, J.T. & Savage, S.B. (1983) A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 187–202. Johnson, P.C. & Jackson, R. (1987) Frictional–collisional constitutive relations for granular materials with application to plane shearing. J. Fluid Mech. 140, 223–256. Kobayashi, S. (Principal Investigator) (1986) Studies on the Maseguchi snow avalanche disaster occurred in Noumachi, Niigata Prefecture. Special Report on Natural Disaster, B-60-8, 90 pp (in Japanese). Krieger, J.M. (1972) Rheology of monodisperse lattices. Adv. Coll. Int. Sci. 3, 111–136. Lane, E.W. & Kalinske, A.A. (1941) Engineering calculations of suspended sediment. Trans. AGU 22, 603– 607. Mears, A.I. (1980) A fragment-flow model of dry-snow avalanches. J. Glaciology 26, 153–163. Nakada, S. (1993) Nature of pyroclastic flows; an example of eruptions at Mt. Fugen, Unzen volcano. Tsuchi-ToKiso, JSSMFE 41-3, 37– 42 (in Japanese). Phillips, R.J., Armstrong, R.C., Brown, R.A., Graham, A.L. & Abbot, J.R. (1992) A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Physics Fluids, A 4-1, 30 – 40. Rouse, H. (1937) Modern conceptions of the mechanics of fluid turbulence. Trans. Am. Soc. Civ. Eng. 102, 463– 543. Salm, B. (1966) Contribution to avalanche dynamics. IASHAIHS Pub. 69, 199–214. Sampl, P. (1993) Current status of the AVL avalanche simulation model. Numerical simulation of dry snow avalanches. Proc. ‘Pierre Beghin’ International Workshop on Rapid Gravitational Mass Movements, Cemagref, Geneva, pp. 269–276. Savage, S.B. (1988) Streaming motions in a bed of vibrationally fluidized dry granular material. J. Fluid Mech. 194, 457– 478.

PGCC01

10/12/08

4:23 PM

Page 43

Mechanics and simulation Shen, H.H. & Ackermann, N.L. (1982) Constitutive relations for fluid–solid mixtures. J. Eng. Mech. Div. Am. Soc. Civ. Eng. 108, 748 –763. Sheridan, M.F. (1979) Emplacement of pyroclastic flows: A review. Geol. Soc. Am. Spec. Pap. 180, 125 –136. Sparks, R.S.J. (1978) Gas release rates from pyroclastic flows: an assessment of the role of fluidization in their emplacement. Bull. Volcanol. 41, 1–9. Sparks, R.S.J., Wilson, L. & Hulme, G. (1978) Theoretical modeling of the generation, movement, and emplacement of pyroclastic flows by column collapse. J. geophys. Res. 83-B4, 1727–1739. Takahashi, T. (1991) Debris Flow. Monograph of IAHR. Balkema, Rotterdam, 165 pp. Takahashi, T. (1993) Fluid mechanical modelling of the viscous debris flow. Proc. ‘Pierre Beghin’ International Workshop on Rapid Gravitational Mass Movements, Cemagref, Geneva, pp. 313–321. Takahashi, T. & Arai, M. (1997) Mechanism of viscous debris flow. Proc. Intern. Symp. on Natural Disaster Prediction and Mitigation, Disaster Prevention Research Institute, Japan, pp. 231–238. Takahashi, T. & Nakagawa, H. (1987) Hazard zone mapping in respect to the damage to wooden houses due to breaking of levee. Bull. Disas. Prev. Res. Inst., Kyoto Univ. 37-2, 59–90. Takahashi, T. & Tsujimoto, H. (1997a) Mechanics of granular flow in inclined chute. J. Hydraulic, Coastal and Environmental Eng., JSCE 565/II-39, 57–71 (in Japanese). Takahashi, T. & Tsujimoto, H. (1997b) Dynamics of the

43

snow avalanche. Annu. Disas. Prev. Res. Inst., Kyoto Univ. 40 B-2, 409– 424 (in Japanese). Takahashi, T. & Tsujimoto, H. (1999) Granular flow model of avalanche and its application. J. Hydrosci. Hydraulic Eng., JSCE 17, 47–58. Takahashi, T. & Tsujimoto, H. (2000) A mechanical model for Merapi type pyroclastic flow. J. Volcanol. Geothermal Res., 98, 91–115. Takahashi, T., Satofuka, Y., Kondo, M. & Takeuchi, R. (1995) Fluid dynamics of the pyroclastic flow. Proc. Intern. Sabo Symposium, Tokyo, pp. 1–8. Takahashi, T., Nakagawa, H., Satofuka, Y. & Tomita, T. (1997) Mechanics of the viscous type debris flow (2) – Flume experiments using well-graded sediment mixture. Annu. Disas. Prev. Res. Inst., Kyoto Univ. 40 IDNDR S. I., 173–181 (in Japanese). Terada, H., Fujisawa, K., Ooura, J., Ogawa, K. & Usuki, N. (1993) A large scale avalanche at the Hakuba snow patch occurred on May 4th 1992. J. Japanese Soc. Snow Ice 55, 183–189 (in Japanese). Ui, T., Sumita, M. & Geological Party of Joint University Research Group (1993) Generation of Merapi-type pyroclastic flows. Observations of the 6th dome of Unzen volcano, Japan. Kazan 38-2, 45–52 (in Japanese). Vreugdenhil, C.B. (1994) Numerical Methods for ShallowWater Flow. Kluwer Academic Publishers, Dordrecht. 261 pp. Wu, J., Kan, C., Tian, L. & Zhan, S. (1990) Debris Flow Observational Study at Jiangjia gully, Yunnan, China. Science Publisher, Beijing 251 pp (in Chinese).

PGCC02 9/12/08 3:16 PM Page 45


An analysis of the debris flow disaster in the Harihara River basin H . N A K A G A W A , T . T A K A H A S H I and Y . S A T O F U K A Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji City, Kyoto 611-0011, Japan

ABSTRACT A severe debris flow disaster occurred on the Harihara River, Japan about midnight of July 10, 1997. Twenty-one people were killed, and 18 houses destroyed by this flow which was caused by a massive slope failure with a sediment volume of approximately 160 000 m3 following heavy rainfall. Residents were aware that an extraordinary phenomenon was occurring in the river but did not take refuge, resulting in the large number of human casualties. A numerical simulation model is developed to explain the behaviour and the depositional processes of the debris flow. In this model, a debris flow is considered to move as a continuous fluid until just before it stops, and a system of depth-averaged two-dimensional momentum and mass conservation equations for fluid flow are used which take into account variation in the concentration of the sediment fraction owing to the deposition of sediment and variation in the discharge. To estimate the amount of sediment deposited, deposition equations were introduced for a fully developed debris flow, an immature debris flow and a turbulent flow. The models are in general agreement with actual phenomena such as the sediment deposition area and thickness of the deposit.

INTRODUCTION Although research has yet to fully determine these factors, it is important that residents of sediment-hazardprone areas be aware of simulations made under various landslide conditions. A numerical model that simulates the behaviour and depositional processes of a debris flow is presented and is applied to the debris flow that occurred in the Harihara River basin.

A severe debris flow occurred in the Harihara River basin, Izumi City, Kagoshima Prefecture, Japan, about midnight of July 10, 1997. Twenty-one people were killed, and 18 houses destroyed by this flow which resulted from a massive slope failure of about 160 000 m3 (Minami et al., 1997) following heavy rainfall of 275 mm on the day before, recorded at the Izumi Meteorological Observatory (Shimokawa & Jitousono, 1998). The slope on which the landslide occurred had an average angle of 26°, and the slope catchment area was apparently not large enough to hold a large volume of water. Before the disaster, on May 13, 1997, an earthquake (M6.2) had occurred in the northwestern part of Kagoshima Prefecture. Many surface landslides occurred on mountain slopes near the Harihara district, and unfailed slopes in the Harihara district probably were affected by the strong seismic forces. To determine why such a large landslide occurred only on this slope in response to the heavy rainfall requires investigations in many areas of research. If the number of such disasters are to be decreased, the past timing, occurrence, and scale of landslides need to be known and compared with rainfall records.

SUMMARY OF THE DEBRIS FLOW DISASTER The Harihara River basin covers an area of 1.55 km2 and contains two primary channels. The 2.3-km long main channel originates at an altitude of 445 m and debouches into the Yatsushiro Sea (Fig. 1). The channel is incised to an altitude of approximately 50 m where a relatively large debris fan has developed. The community of Harihara is located on this debris fan. A part of the fan and surrounding low-angle slopes were intensely cultivated with mandarin orange trees. The Sabo dam was under construction in the main channel (Fig. 1), and the bulk of the dam had been completed prior to the disaster (Fig. 2). A deep-seated


45

PGCC02 9/12/08 3:16 PM Page 46

46

H. Nakagawa et al.

Fig. 1. Topography and key physical features of the Harihara River basin.

Fig. 2. Sabo dam under construction on the Harihara River before the disaster. The dimensions of the dam are 85-m wide and 14-m high. An irrigation pond is present just upstream of the dam. A deep-seated landslide occurred at the site enclosed by the broken line (photo courtesy of Kokusai Aerial Photo Co. Ltd.).

landslide occurred on the mountain slope (shown in Figs 1 and 2) at about 00:45 on July 10, 1997. Only one-half of the sediment volume of the deep-seated landslide was estimated to be caught by the Sabo dam and half discharged past the dam (Kagoshima Prefectural Government, 1997). Figures 3 and 4 show the Harihara district just before and after the disaster, respectively. The partly constructed Sabo dam and several houses situated just downstream of the dam can be seen. The deep-seated landslide occurred at the place shown by broken lines in Fig. 2. The landslide’s dimensions were about 200m long, 80-m wide, and the maximum depth was 27 m (Moriwaki et al., 1998). Geologically, the slope was predominantly composed of weathered andesite underlain by tuff breccia. A large part of the landslide mass consisted of this weathered andesite. The sliding mass first surged down along the Harihara River then flowed into an irrigation pond with a maximum storage capacity of about 9000 m3 (the normal capacity has been estimated as 3500 m3 (Hashimoto et al., 1997) or 6000 m3 (Shimokawa et al., 1998)) (Fig. 2). The landslide mass not only pushed this water out of the pond but may also have absorbed some of it. The landslide underwent transformation to a debris flow and overtopped the Sabo dam, flooding and depositing sediment on the Harihara fan, destroying 18 houses and killing 21 people. A large part of the landslide was deposited on the top of the irrigation pond, and a significant amount of sediment was deposited in a high mound on the left bank (looking downstream) just upstream of the Sabo dam (Fig. 5) as the debris

PGCC02 9/12/08 3:16 PM Page 47

Debris flow disaster at Harihara

Fig. 3. Harihara district just before the disaster (photo courtesy of Kokusai Aerial Photo Co. Ltd.).

Fig. 4. Harihara district just after the disaster (photo courtesy of Kokusai Aerial Photo Co. Ltd.).

Fig. 5. High mound formed by deposits from the landslide mass on the left bank (looking downstream) just upstream of the Sabo dam.

Fig. 6. Water flooded area on the right bank (looking downstream) of the main channel just downstream of the Sabo dam.

47

PGCC02 9/12/08 3:16 PM Page 48

48

H. Nakagawa et al.

N Flood Marks 80

60 50m

70

Harihara River Sabo Dam

60

70 0

50m

Fig. 8. Damaged Harihara Sabo dam. Photo taken from downstream.

flow partially ran up onto the opposite side of the valley. The elevation at the top of this mound is about 82 m above mean sea level, whereas the elevation of the dam’s crown is about 64 m. After the disaster, several dead fish, which had been displaced from the pond were found on the top of the mound. Reportedly, 80 000 m3 of sediment (bulk volume) was trapped by the Sabo dam and an additional 80 000 m3 of sediment was deposited downstream of the dam (Minami et al., 1997). Figure 6 shows the area flooded by water on the right bank just downstream of the Sabo dam. There was no sediment deposition on the bank but bamboo bushes and herbaceous plants were uniformly flattened in a downstream direction. Figure 7 shows the

Fig. 7. Measured high-water marks on the right bank. The highest flood mark is about 76 m above mean sea level (contour interval is 2 m).

Fig. 9. Destroyed house located on the left side (looking downstream) of the Harihara fan. There are some large scattered boulders around the house, but they have not formed a significant thickness of deposit.

measured high-water marks (as determined from the vegetation), the highest ranging from 74 to 76 m. A part of the right wing of the dam was broken and swept away by the debris flow, and this reduced the 64-m elevation of the dam’s crown by 1 m (Fig. 8). Taken together, these findings suggest that clear water was suddenly discharged through the dam, in front of the debris flow, but it is not certain whether the water came from the irrigation pond or was related to rainfall runoff. A house on the left side of the Harihara fan was destroyed (Fig. 9). As there are few large stones or sediment in the remnants of the building (some rather large boulders can be seen in Fig. 9, however these

PGCC02 9/12/08 3:16 PM Page 49


Fig. 10. A house near the river whose first floor is deeply buried by a large amount of sediment transported by the debris flow. The thickness of deposits is about 3 m.

Fig. 11. Calculated water discharge at the dam site and the observed hourly rainfall intensity at the Izumi Meteorological Observatory (Shimokawa & Jitousono, 1998).

boulders were scattered around the house and did not accumulate deep deposits suggesting that these boulders were transported by the flood flow and not by the debris flow), it is suggested that the house was destroyed by flood water discharged from the Sabo dam. In contrast, a house near the Harihara River was buried by a large amount of sediment transported by the debris flow (Fig. 10).

FLOOD RUNOFF ANALYSIS If the water discharge in the Harihara River was great enough to affect the behaviour of the landslide mass, its effects would have to be considered in the simulation model. A flood runoff analysis was therefore made to estimate the water discharge at the dam site. The following kinematic wave runoff model (Tachikawa et al., 1997) was used; ∂hw 1 ∂ + {qb( x )} = r ( x, t ) cos θ( x ) ∂t b ( x ) ∂x q=

sin θ( x ) m hw nm

49

(1)

(2)

where x is the distance measured from the top end of the sub-basin along a slope surface for the sub-basin; b(x) the width of the slope at distance x; hw the water depth; q the water discharge per unit width; r the rainfall intensity; t the time; θ(x) the inclination of the slope at distance x; nm the Manning’s roughness coefficient taking the value of 0.2; and m the constant, value –53 . Kanamaru and Takasao (1976) showed that

the Manning’s roughness coefficient of a channel temporarily distributed on the mountain slope is 1 −–3 0.01–0.03 m s and that of a slope with plants is 1 0.3–0.4 m−–3 s. As the objective basin is well covered with plants and there are some channels temporarily formed by a heavy rainfall, a little smaller value (0.2) than 0.3–0.4 is used as the Manning’s roughness coefficient. Because the surface layer of the slope was saturated by antecedent rainfall, all precipitation in the 27 hours prior to the landslide was considered to contribute to the surface flow (Fig. 11). As there is no water-level gauge and no measurements on this river, base flow discharge is not known. The base flow is taken as zero because water discharge of the Harihara River is usually very small. The catchment area at the Sabo dam site was 0.475 km2. Rainfall data recorded at the Izumi Meteorological Observatory of the Japan Meteorological Agency was used for the calculations (Shimokawa & Jitousono, 1998). The calculated water discharge at the dam site and hourly recorded rainfall intensity are given in Fig. 11. Very intense rainfall occurred from 10:00 to 11:00 (62 mm h−1) and from 16:00 to 17:00 (55 mm h−1) on July 9th. The calculated water discharge corresponds closely to the variation in rainfall due to the small size of the catchment area. Water discharge was negligible when the debris flow occurred (July 10, 0:45), which agrees with eye-witness evidence given by residents (July 9, 22:45) (Kagoshima Prefectural Government, 1997). The calculated results show that the landslide mass was influenced little by the

PGCC02 9/12/08 3:16 PM Page 50

50

H. Nakagawa et al.

river water; therefore, the effect of river water on the behaviour of the debris flow is not considered in the calculations described later.

SIMULATION OF THE DEBRIS FLOW Flow regime Debris flows have been classified and modelled in a number of different ways (e.g. Takahashi, 1991; Iverson, 1997; Major, 1997). The approach of Takahashi (1991), where debris flows are subdivided into stony, immature, and turbulent phases, is used herein. A stony debris flow is a flow in which coarse grained particles are dispersed in water or clay slurry. The dense concentration of coarse particles in the flow necessarily induces frequent interparticle collisions, which cause dispersion of particles throughout the flow depth. The equilibrium concentration can be defined by considering the stress balance between the bed and the moving layer (Takahashi, 1991): ρ tan θ C∞ = ( σ − ρ) (tan φ − tan θ)

Basic equations As the transformation of the landslide into a debris flow has not been clarified yet, it is supposed that a system of basic equations for a stony debris flow, an immature debris flow, and a turbulent flow was applicable to the behaviour of the landslide from the onset of its movement. The debris flow is considered to move as a continuous fluid until just before it stops. A system consisting of momentum and mass conservation equations for fluid flow is used, that takes into account variation in discharge and the volumetric concentration of the sediment in the fluid, owing to erosion and deposition. The basic equations used to calculate the development and deposition of the debris flow and its flooding are depth-averaged two-dimensional momentum and continuity equations; ∂M ∂(uM ) ∂(vM ) +β +β ∂t ∂x ∂y

(3)

where C∞ is the equilibrium solid concentration of the flow, that is, the concentration when a debris flow undergoes neither deposition nor erosion, ρ the density of clear water, tan θ the channel slope or the energy gradient of the flow, σ the density of sediment particles, and φ the internal friction angle of the bed materials. To disperse the particles throughout the entire flow layer by the actions of interparticle encounters and repulsions, comparatively high concentrations are required. However low concentrations are predicted from Eq. (3) when the channel inclination is very small. At these low concentrations the coarse particles are no longer dispersed throughout the entire depth, but are concentrated in the lower part of the flow. Above this particle mixture layer a water layer appears which may contain suspended sediment; producing ‘an immature debris flow’ in the terminology of Takahashi (1991). If the effect of friction by interparticle contact is smaller than the effect produced by collision for flows with solid concentrations lower than those of ‘immature debris flows’ then the stress allotted by the large-scale turbulent mixing of the fluid mass incorporated with the solid and water in the total shear stress becomes significant. In this flow regime, the flow can be treated as ‘a turbulent flow’ (Takahashi, 1991).

∂( zb + h) τ bx − ρT ∂x

(4)

∂( zb + h) τ by − ρT ∂y

(5)

∂h ∂M ∂N + + = i{C + (1 − C )Sb } + r * * ∂t ∂x ∂y

(6)

= gh sin θbx 0 − gh cos θbx 0 ∂N ∂(uN ) ∂(vN ) +β +β ∂t ∂x ∂y = gh sin θby0 − gh cos θby0

The continuity of the coarse particle fraction sustained in the flow by the action of particle encounters is: ∂(CL h) ∂(CL M ) ∂(CL N ) ⎪⎧ iC*L =⎨ + + ∂t ∂x ∂y ⎪⎩ iC DL *

(i ≥ 0 ) (i < 0 ) (7)

∂{(1 − CL )CF h} ∂{(1 − CL )CF M } + ∂t ∂x +

∂{(1 − CL )CF N } ⎪⎧i (1 − C*L ) C*F =⎨ ∂y ⎪⎩i (1 − C DL ) CF *

(i ≥ 0 ) (i < 0 ) (8)

where M = uh and N = vh are the x and y components of the flow flux; u and v the x and y components of the mean velocity; g is the gravitational acceleration; h the flow depth; zb the erosion or deposition thickness measured from the original bed elevation; θbx0 and θby0 are

PGCC02 9/12/08 3:16 PM Page 51

51

Debris flow disaster at Harihara the x and y components of the inclination of the original bed surface; ρT is the specific density of the debris flow; β the momentum correction coefficient; τbx and τby are the x and y components of the resistance to flow; i is the erosion (≥ 0) or deposition (< 0) velocity; C the total solid fraction in the bed; Sb the degree of * saturation in the bed (applicable only for erosion, when deposition takes place substitute Sb = 1); r the water supply intensity from the side per unit length of the channel (unidirectional flow case); CL the volume concentration of the coarse fraction in the flow; CF the volume concentration of the fine fraction in the interstitial fluid; C L and C F are the volume concen* * trations of the coarse and fine fractions in the original bed; and C DL is the volume concentration of the * coarse fraction in the static bed produced by deposition of the debris flow. The bottom resistance (not only the channel bed but also the surface of the flooding area) for a twodimensional flow is described as follows. For a fully developed stony debris flow (Takahashi, 1991): 2

τbx =

u u 2 + v2 ρT ⎛ dL ⎞ 1 ⎜ ⎟ 8 ⎝ h ⎠ {C + (1 − C ) ρ /σ}{(C /C ) 3 − 1}2 L L m *DL L (9) 2

τby =

v u 2 + v2 ρT ⎛ dL ⎞ 1 ⎜ ⎟ 8 ⎝ h ⎠ {C + (1 − C )ρ /σ}{(C /C ) 3 − 1}2 L L m *DL L (10)

An immature debris flow occurs when CL is less than 0.4C L (Takahashi, 1991). Equations (13) and * (14), which are a form of the Manning formula, apply when CL is less than 0.02 or h/dL is larger than 30. The momentum correction coefficient, β, is equal to 1.25 for a stony type debris flow and 1.0 for both an immature debris flow and a turbulent flow (Takahashi, 1977). The equation of erosion for an unsaturated bed (Takahashi et al., 1992) is 1

⎧⎪ ⎛ tan φ ⎞ ⎫⎪ 2 σ − ρm CL ⎜ = K sin θ ⎨1 − − 1⎟ ⎬ ⋅ ρm ⎝ tan θ ⎠ ⎪⎭ ⎪⎩ gh i

3 2

⎛ tan φ ⎞ h − 1⎟ (CT∞ − CL ) ⎜ dL ⎝ tan θ ⎠

(15)

and for a saturated one (Takahashi et al., 1987), i = δe

CT∞ ⋅ C − CT∞ *

⎧⎪ CL ρm tan φ − (CT /CT∞ )(CL∞ /CL )( ρ/ρm ) tan θ ⎫⎪ qT ⎬ ⎨1 − CL∞ ρ tan φ − tan θ ⎪⎭ dL ⎪⎩

(16) where K and δe are constants; CT the volume concentration of the total solid fraction; and qT the flow discharge per unit width;

For an immature debris flow (Takahashi, 1991):

CT∞ =

ρ tan θ ( σ − ρ) (tan φ − tan θ)

(17)

C L∞ =

ρm tan θ ( σ − ρm ) (tan φ − tan θ)

(18)

2

τbx =

ρT ⎛ dL ⎞ 2 2 ⎜ ⎟ u u +v 0.49 ⎝ h ⎠

τby =

ρT ⎛ dL ⎞ 2 2 ⎜ ⎟ v u +v 0.49 ⎝ h ⎠

(11)

2

(12)

For a turbulent flow; τbx =

τby =

2 u u 2 + v2 ρgnm 1

where CT∞ is the equilibrium of the total solids, and CL∞ the equilibrium solid concentration of a stony debris flow. For immature debris flows and bed load, the respective equilibrium solid concentrations are

(13)

h3 2 v u 2 + v2 ρgnm

h

1 3

(14)

where dL is the mean diameter of the coarse fraction; ρm the density of the interstitial muddy fluid (ρm = σCF + (1 − CF)ρ, ρT = σCL + (1 − CL )ρm ); nm the Manning’s roughness coefficient for a debris flow on the deposition or erosion surface; and tan θ the energy gradient given by tan θ =

2 + τ 2 /ρ gh. τ bx by T

2 CS∞ = 6.7C T∞

(19)

CB∞ = qB∞ /(hu)

(20)

where qB∞ is the equilibrium bed load transport rate. In this study the equation used (Takahashi, 1987) is qB∞ 1

{( σ /ρ − 1)gd L3} 2 3

τ2 *

1 + 5 tan θ cos θ

= τ ⎞⎛ τ 8 ⎛ 1 − γ 2 *c ⎟ ⎜ 1 − γ * c ⎜ ⎜ τ* τ* ⎠ ⎝ f ⎝

⎞ ⎟⎟ (21) ⎠

PGCC02 9/12/08 3:16 PM Page 52

52

H. Nakagawa et al.

where f is the resistance coefficient; τ and τ c are the * * non-dimensional tractive and non-dimensional critical tractive forces; and γ2 =

0.85 − 2σ tan θ/( σ − ρ) 1 − σ tan θ/( σ − ρ)

(22)

Equation (19) is applicable only when CS∞ has a value less than CT∞, as calculated by Eq. (17) (Takahashi et al., 1992). The following deposition equations (Takahashi et al., 1992) are introduced. For stony debris flows; ⎛ i = δd ⎜1 − ⎜ ⎝

u 2 + v2 pve

⎞ C −C L ⎟ L∞ ⎟ C DL ⎠ *

u 2 + v2

(23)

where δd and p are constants, and ve is the equilibrium velocity that characterizes a flow that is neither undergoing erosion or deposition: 2 ve = 5dL

1 1 ⎫ 2 ⎧ 3 ⎡ g sin θ ⎧ ρm ⎫⎤ ⎪⎛ C DL ⎞ ⎪ 3 e ⎢ ⎨ CL + (1 − CL ) ⎬⎥ ⎨⎜ * ⎟ − 1⎬ h 2 a sin C α σ ⎢⎣ ⎩ ⎭⎥⎦ ⎪⎝ L ⎠ ⎪ ⎩ ⎭

(24) where α is the dynamic internal angle of friction (tan α = 0.6); a the constant to describe the constitutive relations (a = 0.04, see Bagnold (1954) and Takahashi (1991)); and θe the critical slope angle that is neither undergoing erosion or deposition derived from Eq. (18) as ⎛ C ( σ − ρm ) tan φ ⎞ θe = tan −1 ⎜ L ⎟ ⎝ CL ( σ − ρm ) + ρm ⎠

(25)

The equation for the bed surface elevation is ∂zb /∂t + i = 0

(26)

The debris flow begins to decelerate from the point where the channel slope changes. Deposition starts only after the front travels some distance downstream from the point where the velocity becomes less than pve, in which p (< 1) is a numerical constant introduced to take into account the difference between the location where actual deposition takes place and the point where the velocity becomes less than ve . This adjustment in the deposition distance may account for the effects of inertial motion of the debris flow. For an immature debris flow and a turbulent flow: i = δd′

CS∞ − CL C DL *

u 2 + v2

(27)

i = δd′′

CB∞ − CL C DL *

u 2 + v2

(28)

where δ′d and δ″d are constants. As deposition is the dominant phenomenon in the Harihara disaster, only the depositional process is considered. Conditions for calculations The assumptions for the landslide conditions are that the landslide mass is a continuous fluid with a volumetric coarse sediment fraction of 52% and that of fine sediment fraction of 0% (total sediment fraction is 52%), and that initiation of movement occurs at the start of calculation. A series of permeability tests were carried out by Kitamura et al. (1998). The porosity of the sediment samples (weathered andesite) taken at the landslide site was estimated about 0.5 by these tests. So that the initial sediment concentration of the landslide mass would be 0.5 (= 1.0 − 0.5). As it is assumed that the sediment samples contain large boulders and fresh rocks, porosity must take a smaller value, i.e. initial sediment concentration must be a little larger than 0.5. As the precise value of the initial sediment concentration is not known, a slightly larger value of 0.52 than 0.5 was used in the calculations. Using a sediment fraction of 52%, substantial sediment volume in the landslide mass, Vslp, is estimated as Vslp = 160 000 × 0.52 = 83 200, while observed substantial volume of sediment in deposits, Vdep, is estimated as Vdep = [80 000 (upstream from the Sabo dam) + 80 000 (downstream from the Sabo dam) ] × C DL * (= 0.65) = 104 000 m3. In order to make Vslp coincide with Vdep, the initial landslide volume is modified to 200 000 m3 (= 160 000 × 104 000/83 200). This value is a little larger than that of estimated by Minami et al. (1997) but the landslide shape and scale is very similar to the actual shape. So that the landslide mass has 104 000 m3 of substantial sediment and 96 000 m3 of water initially. The landslide’s behaviour, flooding and depositional processes were computed using a twodimensional numerical simulation and a staggered first order up-wind finite difference scheme. The grid sizes are ∆x and ∆y = 5 m, and the time interval ∆t = 0.01 s. The numerical constants and the other parameters adopted in the calculation were p = –13 , δd = δ′d = δ″d = 0.1 (Takahashi et al., 1992). Other sediment properties were estimated from field survey findings; C DL = 0.65, * CF = 0.0, dL = 10 cm, tan φ = 0.75, σ = 2650 kg m−3, ρ = 1000 kg m−3, and nm = 0.04. One important parameter is the volume of water behind the Sabo dam immediately prior to the landslide. Some workers constructing the Sabo dam noted that the water level in the Sabo dam reached about

PGCC02 9/12/08 3:16 PM Page 53

Debris flow disaster at Harihara Table 1. Conditions for calculations Case 1

3500 m3 of water stored in the irrigation pond.

Case 2

3500 m3 of water stored in the irrigation pond and 10 000 m3 of water behind the Sabo dam.

Case 3

3500 m3 of water stored in the irrigation pond and 10 000 m3 of dry sediment behind the Sabo dam.

Case 4

9000 m3 of water stored in the irrigation pond and 200 m3 of water behind the Sabo dam.

Case 5

3500 m3 of water stored in the irrigation pond and no Sabo dam present.

60 m in altitude in the midst of the heavy rain (about 16:00 on July 9th) (Kagoshima Prefectural Government representative, pers. comm.). This water level corresponds to a water volume of 10 000 m3 behind the Sabo dam, but it is unclear whether the water volume had been maintained or not until the landslide occurred. As there were several open outlet holes in the dam body and the landslide occurred about 8 hours later, a large volume of the water stored behind the dam must have outflowed prior to the landslide. Hirano and Ogawa (1999) estimated the water volume behind the Sabo dam as only 200 m3 when the landslide occurred. Five different scenarios were considered (Table 1). Case 1 in which 3500 m3 of water was stored in the irrigation pond; Case 2 in which 10 000 m3 of water was stored in the Harihara Sabo dam in addition to 3500 m3 of water in the irrigation pond. Case 3 in which 10 000 m3 of sediment only was stored behind the Sabo dam and 3500 m3 of water in the irrigation pond; Case 4 in which 9000 m3 of water was stored in the pond and 200 m3 of water behind the dam; and Case 5 in which no Sabo dam existed (imaginary case). Calculated results and discussion The calculated amounts of sediment deposited and areas flooded by water for Cases 1–5 are shown at 1, 2, and 3 minute(s) after the landslide (Figs 12–16). Several houses are shown to have been destroyed by a debris flow only one minute after the landslide occurred. The flow therefore moved approximately 600 m in 60 seconds giving a mean velocity over the 8–9° slope of 10 m s−1 (note: maximum slope angle is given by the distance (800 m) and difference of elevation (155 m) between the top of landslide (185 m) and front of flow (30 m) and is about 11°). Such speeds are not atypical of debris flows as observed in the Name

53

River in Japan, where front velocities of debris flows were usually observed to be larger than 10 m s−1 on slopes in excess of 10.1° (Ishikawa, 1985). Water flooding is much greater in those cases where large volumes of water were assumed to be present in either the irrigation pond (Case 4) or behind the Sabo dam (Case 2). This water may have been incorporated into the debris flow and then expelled as the debris flow came to a halt (as assumed in the numerical simulations), or have been pushed ahead of the debris flow (some influence of the latter is suggested by the presence of dead fish on top of the mound downstream of the irrigation pond). The fluidity of the debris flow would be expected to be higher if the bulk of the water was absorbed rather than pushed ahead of the flow, and this might be expected to produce faster inundation times and larger areas of flow inundation. However, there is a remarkable similarity in timing and areal distribution between all the simulations, irrespective of whether additional volumes of ponded water were present or not. One explanation for this is that absorption of the additional volumes of water (13 500 m3 (Case 2); 9200 m3 (Case 4)) would not significantly change the volumetric sediment fraction. The initial volumetric sediment fraction is assumed to be 52% (104 000 m3 of substantial sediment and 96 000 m3 of water) and where an additional 13 500 m3 of water is added (Case 2), the volumetric sediment fraction only changes to 49%. Similarly, the additional 9200 m3 of water in Case 4 would only change the volumetric sediment fraction to 50%. The additional 10 000 m3 of sediment in Case 3 would not change the volumetric sediment fraction because the sediment only contributes to the increase of bed elevation behind the Sabo dam (the sediment is not entrained into the debris flow). Although, absorption, mixing and subsequent expulsion of additional water cannot be confirmed by field observations, the simulations suggest that changes in the volumetric sediment fraction of up to 3%, through water absorption (or sediment entrainment) do not significantly alter the timing or areal distribution of debris flow in this example. Calculated maximum areas of sediment deposition and water flooding 20 minutes after the landslide (i.e. the final stage of the debris flow) are shown in Fig. 17. The actual areas of sediment deposition and water inundation are shown in the Case 1 example. The calculated sediment deposition area in each case corresponds comparatively well with the area in which houses were destroyed and is mainly along the Harihara River. However, all of the model runs show a radial distribution on the fan, rather than the strongly

PGCC02 9/12/08 3:16 PM Page 54

54

H. Nakagawa et al.

Case 1

100

t = 1 min 90 80

20

60 30

40

N

70

50

Sabo Dam JR Kagoshima Line 10m Sediment deposited Water flooded House destroyed

Route 3

Case 1

t = 2 min

Case 1

t = 3 min

0

200

(m)

Fig. 12. Calculated sediment deposition (dark grey) and water flooded area (light grey) (Case 1). Black spots are destroyed buildings.

400

PGCC02 9/12/08 3:16 PM Page 55

55


Case 2

100

t = 1 min 90 80

20

60 30

40

N

70

50


Route 3

Case 2

t = 2 min

Case 2

t = 3 min

0

200

(m)


400

PGCC02 9/12/08 3:16 PM Page 56

56

H. Nakagawa et al.

Case 3

100

t = 1 min 90 80

20

60 30

40

N

70

50


Route 3

Case 3

t = 2 min

Case 3

t = 3 min

0

200

(m)


400

PGCC02 9/12/08 3:16 PM Page 57

57


Case 4

100

t = 1 min 90 80

20

60 30

40

N

70

50


Route 3

Case 4

t = 2 min

Case 4

t = 3 min

0

200

(m)


400

PGCC02 9/12/08 3:16 PM Page 58

58

H. Nakagawa et al.

Case 5

100

t = 1 min 90 80

20

60 30

40

N

70

50


Route 3

Case 5

t = 2 min

Case 5

t = 3 min

0

200

(m)


400

PGCC02 9/12/08 3:16 PM Page 59

59

Debris flow disaster at Harihara Case 1 Maximum flooded and deposition area 100 90 80 70 60 50 40 30

20

N

Sabo Dam JR Kagoshima Line Route 3

10m Sediment deposition

Water flooded

Actual deposition area by debris flow 0

200

(m)

400

Actual flooded area by water flow

Case 2 Maximum flooded and deposition area


Fig. 17. Calculated maximum areas of sediment deposition (dark grey) and water flooding (light grey).

PGCC02 9/12/08 3:16 PM Page 60

60

H. Nakagawa et al.



Fig. 17. (cont’d)

asymmetric distributions of the actual flows. The calculated height of the sediment deposited on the left bank just upstream of the Sabo dam is about 82 m, which satisfactorily matches the actual situation, especially in Case 4. Prior to the simulations it was expected that the sediment deposition area in Case 5 would be wider than in the other cases, but the results were nearly the same. This is because the bed slope between the foot of the landslide and the site of the Sabo dam is low enough to promote deposition of part of the landslide mass, and therefore a large amount of sediment is deposited in this reach in every case. The flood marks shown in Fig. 7 were expected to be reproduced by the simulations,

but calculated depths only reached altitudes of about 70 m, and the value is nearly the same for each case. Accuracy of the elevation map used in the calculation and imperfections in the calculation itself account for this discrepancy. Calculated sediment accumulations for each case give deposit thicknesses of more than 5 m just upstream of the Sabo dam, and about 2–5 m along the Harihara River downstream of the dam, whereas there is less than 1 m of deposition on the left side of the fan (Plate 1 facing p. 64). These values satisfactorily express the sediment deposition shown in Figs 9 and 10. Figure 18 shows the calculated thickness of the sediment deposit along the Harihara River near the Sabo dam. Just

PGCC02 9/12/08 3:16 PM Page 61


Fig. 18. Calculated thicknesses of sediment deposits along the Harihara River near the Sabo dam for Cases 1–5.

61

PGCC02 9/12/08 3:16 PM Page 62

62

H. Nakagawa et al.

Table 2. Comparison of the calculated and observed sediment volumes of the deposits Deposit

Calculated (m3)

Observed (m3) Case 1 104 000

Case 2 104 000

Case 3 104 000

Case 4 104 000

Case 5 104 000

59 020

58 695

59 345 (+6500)**

55 120

57 720

Captured by the Sabo dam

6500 5 4 13 000 6* 4 32 500 7

Deposited on the Harihara debris fan

52 000

44 980

45 305

44 655

48 880

46 280

104 000 Remaining at the landslide site Deposited on the river bed

* Upstream of the Sabo dam. ** Sediment volume of 6500 m3 in substance had been stored behind the Sabo dam as an initial condition. This volume contributes only to heightening the bed elevation behind the dam.

upstream of the dam it is more than 10-m deep in Case 3, which is very similar to Cases 1, 2 and 4 in which sediment was deposited to the height of the water course of the Sabo dam. The observed data, also shown in the figure, are in good agreement with the calculated results. Table 2 shows a comparison between the calculated and observed (Minami et al., 1997) sediment volumes of the deposits. The observed sediment volumes deposited downstream and upstream of the Sabo dam are both about 52 000 (deposit volumes = 80 000 m3). The calculated results show similar volume distribution of sediment, except for Case 3 in which 10 000 m3 of sediment had been stored in the Sabo dam before the landslide. Even in the case in which no Sabo dam existed (ideal case), similar sediment volumes were deposited upstream and downstream of the site of the actual Sabo dam.

not. One explanation for this is that absorption of the additional volumes of water would not significantly change the volumetric sediment fraction (Cases 2 and 4). Although, absorption, mixing and subsequent expulsion of additional water cannot be confirmed by field observations, the simulations suggest that changes in the volumetric sediment fraction of up to 3%, through water absorption (or sediment entrainment) do not significantly alter the timing or areal distribution of debris flow in this example. The simulation model, however, is not precise enough because it is not clear how the bottom shear stresses of the landslide mass changes when the mass becomes a debris flow. Moreover, there are limitations when expressing the mixing process of the debris flow with water using the depth-averaged two-dimensional model and questions as to whether all the ponded water is actually mixed in with the debris flow. These are urgent problems that must be solved.

CONCLUSIONS ACKNOWLEDGEMENTS How the debris flow was generated and developed from the landslide mass is unclear, but the depositional process of the flow is considered to be satisfactorily expressed by the numerical simulation model presented here. Judging from the calculated results, the case that gives the most probable explanation of the debris flow phenomena is Case 4 which agrees with the estimation of Hirano and Ogawa (1999) that there was very little water (200 m3) behind the Sabo dam immediately prior to the landslide. However, there is a remarkable similarity in timing and areal distribution between all the simulations, irrespective of whether additional volumes of ponded water were present or

This work was financially supported by Grant-in-Aid for Scientific Research No. 09600003 (Prof. Etsuro Simokawa, Kagoshima University) from the Japanese Ministry of Education, Science, Culture and Sports. We thank Dr. Yasuto Tachikawa, Associate Professor, DPRI, Kyoto University and Mr. Yasuhiro, Sato, Undergraduate Student, Kyoto University for their help in the flood runoff analysis, Dr. Motoyuki Ushiyama for the rainfall data, and the many other persons who kindly made useful suggestions. We also are grateful to Kokusai Aerial Photo Co. Ltd. for providing the aerial photos.

PGCC02 9/12/08 3:16 PM Page 63


NOTATION a

b(x) CB∞ CF CL CL∞ CS∞ CT CT∞ C

*

C

*F

C

*L

C

*DL

C∞ dL f g h hw i K m M, N nm p q qT

numerical coefficient (= 0.04) to describe the constitutive relations (Bagnold, 1954; Takahashi, 1991) width of the slope at distance x equilibrium solid concentration of the bed load volumetric concentration of the fine particles in the interstitial fluid volumetric concentration of the coarse particles in the flow equilibrium solid concentration of the stony debris flow equilibrium solid concentration of the immature debris flow volumetric concentration of the total solids equilibrium concentration of the total solids volumetric concentration of the solids in the bed volumetric concentration of the fine particles in the original bed volumetric concentration of the coarse particles in the original bed volumetric concentration of the coarse particles in the static bed produced by debris flow deposition equilibrium solid concentration of the flow mean diameter of the coarse particles resistance coefficient gravitational acceleration flow depth water depth in flood runoff analysis erosion (≥ 0) or deposition (< 0) velocity numerical constant which appears in the erosion formula constant (–53 for Manning’s formula) respective flow discharge per unit width in the x and y directions Manning’s roughness coefficient 1 (0.04 m−–3 s) on the bed for the turbulent flow numerical constant (–13 ) which appears in the deposition formula water discharge per unit width which appears in the flood runoff analysis flow discharge per unit width

qB∞ r r Sb t u, v ve Vslp Vdep x, y zb α β γ δd δ′d δ″d δe θbx0, θby0 θ

63

equilibrium bed load transport rate rainfall intensity water supply intensity from the side per unit length of channel degree of saturation in the bed time respective mean velocity components in the x and y directions equilibrium velocity where a flow is neither depositing nor eroding sediment volume in the landslide mass observed substantial volume of sediment in deposits coordinates of the flow erosion or deposition thickness measured from the original bed elevation dynamic internal angle of friction (tan α = 0.6) momentum correction coefficient parameter which appears in the bed load formula numerical constant of the equation of deposition for a stony debris flow numerical constant of the equation of deposition for an immature debris flow numerical constant of the equation of deposition for a turbulent flow numerical constant of the equation of erosion respective inclination of the original bed surface in the x and y directions energy gradient given by 2 /ρ gh tan θ = τ 2bx + τ by T

θe θ(x) ρ ρm ρT σ τbx, τby τ * τ c * φ

equilibrium angle of the bed surface with neither deposition nor erosion inclination angle of the slope at distance x from the top of a slope which appears in the flood runoff analysis density of the clear water density of the interstitial fluid with fine sediment particles (= ρ(1 − CF) + σCF) density of the flow (= ρm(1 − CL) + σCL) density of the coarse and fine particles respective shear stresses on the bed surface in the x and y directions non-dimensional tractive force non-dimensional critical tractive force angle of internal friction of sediment particles on the bed

PGCC02 9/12/08 3:16 PM Page 64

64

H. Nakagawa et al.

REFERENCES Bagnold, R.A. (1954) Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R.S. London, Ser. A 225, 49 – 63. Hashimoto, H., Hirano, M., Hikita, M., Paku, K. & Tamamatsu, J. (1997) Debris flow behaviour and depositional properties in the Harihara River, Izumi City, July 10, 1997. Proceedings of the 16th Conference of the Japan Society for Natural Disaster Science, pp. 103 –104 (in Japanese). Hirano, M. & Ogawa, S. (1999) Premonitory phenomena of the debris flow disaster at the Harahara River on June 10, 1997. Hidden lesson behind the disaster. J. Japan. Soc. Erosion Control Eng. 51, 27–34 (in Japanese). Ishikawa, Y. (1985) Debris flows in the Name River. Shin Sabo 121, 1–7 (in Japanese). Iverson, R.M. (1997) The physics of debris flows. Rev. Geophys. 35, 245–296. Kagoshima Prefectural Government (1997) Materials for discussion: the 1st Committee on the Debris Flow in the Harihara River, Kagoshima Prefectural Government Issued, 1.1–3.3 (in Japanese). Kanamaru, A. & Takasao, T. (1976) Hydrology. Asakura Shoten, p. 223 (in Japanese). Kitamura, R., Ochiai, H., Kobayashi, T., Yakabe, H., Takada, M. & Abe, H. (1998) Mechanism of a landslide occurred at Harihara, Izumi City, Kagoshima Prefecture, 1997. Research Report on Natural Disasters, Supported by the Japanese Ministry of Education, Science, Culture and Sports (Grant No. 09600003), 31–38 (in Japanese). Major, J.J. (1997) Depositional process in large-scale debrisflow experiments. J. Geol. 105, 345 –366. Minami, N., Yamada, T. & Mizuno, H. (1997) Sediment volume and the deposition area of the debris flow in the Harihara River, July 10, 1997, Izumi City, Kagoshima Prefecture, Japan. J. Japan. Soc. Erosion Control Eng. 50, 81–82 (in Japanese).

Moriwaki, H., Sato, T. & Chiba, M. (1998) Report on the Harihara River debris flow disaster of July 10, 1997 in Kagoshima Prefecture, Japan. Natural Disaster Research Report No. 35. National Research Institute for Earth Science and Disaster Prevention, Science and Technology Agency, Japan, pp. 1– 69 (in Japanese). Shimokawa, E. & Jitousono, T. (1998) A study of the change from a landslide to debris flow at Harihara, Izumi City, southern Kyushu. J. Natural Disaster Sci. 20, 75 – 81. Shimokawa, E., Jitousono, T. & Ogawa, S. (1998) Debris flow disaster in the Harihara River, Izumi City. Research Report on Natural Disasters, Supported by the Japanese Ministry of Education, Science, Culture and Sports (Grant No. 09600003), 19–30 (in Japanese). Tachikawa, Y., Haraguchi, A., Shiiba, M. & Takasao, T. (1997) Development of a distributed rainfall-runoff model based on Tin-based topographic modeling, J. Hydraulic, Coastal Environment Eng. JSCE No. 565/II-39, 1–10 (in Japanese). Takahashi, T. (1977) Mechanism of the occurrence of muddebris flows and their characteristics in motion. Ann. Disaster Prevention Res. Inst. Kyoto Univ. 20, 405 – 435 (in Japanese). Takahashi, T. (1987) High velocity flow in steep erodable channels, Proceedings of 22nd Congress of IAHR, Lausanne, pp. 42–53. Takahashi, T. (1991) Debris Flow. Balkema, Rotterdam, p. 165. Takahashi, T., Nakagawa, H. & Kuang, S.F. (1987) Estimation of debris flow hydrograph on a varied slope bed. Proceedings of the Corvallis Symposium on Erosion and Sedimentation in the Pacific Rim, IAHS Publ. No. 165, pp. 167–177. Takahashi, T., Nakagawa, H., Harada, T. & Yamashiki, Y. (1992) Routing debris flows with particle segregation. J. Hydraulic Eng. Am. Soc. Civ. Eng. 118, 1490 –1507.

PGCC03 9/12/08 3:16 PM Page 65


Theoretical study on breaking of waves on antidunes Y . K U B O * 1 and M . Y O K O K A W A † *Department of Geology and Mineralogy, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan † Department of Earth and Space Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan

ABSTRACT Breaking of waves on antidunes is modelled by considering the hydraulics of a supercritical flow over an obstacle. A supercritical flow tends to convert an initially flat bed into bedwaves, which in turn prevent smooth passage of the flow. The flow is blocked, and a hydraulic jump is formed over the crest of the bedwaves. A turbulent hydraulic jump moves upstream with a cloud of suspension, resulting in deposition on the upstream side of the antidunes. With enough sediment transport, upstream-dipping laminae or lenticular structures are formed. This cycle of events observed in antidune sedimentation is matched with a hydraulic description of flow over an obstacle.

INTRODUCTION ton, 1965; Allen, 1966, 1982). The stationary waves on the free water surface develop in-phase with antidunes and tend to break rather frequently. The interaction with the bed may be so strong that the breaking waves obliterate the antidunes and level the bed (Kennedy, 1963). Several authors have pointed out that the breaking of waves, as well as the differential deposition, can give rise to the upstream migration of antidunes (Middleton, 1965; Allen, 1966, 1982).The mechanism of the breaking of waves, however, has not been emphasized in antidune sedimentation, despite its importance as a reworking process of bed configuration and as a secondary process of the upstream migration of antidunes. This study gives a theoretical description of the process of wave-breaking on antidunes formed on sand beds in open-channel laboratory flumes. Hydraulics of the flow over an obstacle are explored to quantify the flow conditions for breaking antidunes. General consideration is given to deposition caused by the breaking of waves, and the applicability of the model to antidunes formed in turbidity currents is then discussed.

Antidunes are bedforms produced under high flow intensities of the upper flow regime, and are characterized by possible upstream migration against the flow. The possible upstream migration of antidunes is in contrast with downstream migration of other wavy bedforms such as dunes and ripples, and is due to the formation of upstream-dipping laminae on the stoss side of the bed undulations. Unusual upstreamdipping laminae and hummocky cross-stratification in turbidites, therefore, have been attributed to antidune sedimentation, and used as an indicator of palaeoflow conditions (Walker, 1967; Skipper, 1971; Prave & Duke, 1990; Yagishita, 1994). The upstream migration takes place as a result of differential deposition on upstream and downstream flanks of antidunes. Kennedy (1963) theoretically obtained a stability field for antidunes by considering differential deposition by a steady flow on a sinusoidal bed. His equations were successfully applied to antidunes both in laboratory flumes and rivers. In open-channel laboratory flumes, antidunes have been described to be accompanied by breaking waves at the water surface (Gilbert, 1914; Kennedy, 1963; Middle-

THE BREAKING OF WAVES 1

Present address: Deep-Sea Research Department, Japan Marine Science and Technology Centre (JAMSTEC) 2-15 Natsushima-cho, Yokosuka 237-0061 Japan.

There have been many descriptions of antidunes since Gilbert (1914), based mainly on experiments on


65

PGCC03 9/12/08 3:16 PM Page 66

66

Y. Kubo and M. Yokokawa

Fig. 1. Converted pictures of the breaking of waves on an antidune, captured at 0.1 s intervals. Flow is from right to left. Darker tone indicates higher concentration of sediment. Note the upstream migration of sediment along with the breaking wave. Reproduced from Yokokawa et al. (1999). See the original article for details.

sand beds in open-channel laboratory flumes. It has been commonly observed that the breaking of surface waves occurs rather frequently and cyclically, and is associated with destruction and reestablishment of antidunes (Middleton, 1965; Allen, 1966, 1982; Yokokawa et al., 1999). Middleton (1965) described the process of wave breaking on antidunes in a laboratory flume. He indicated that the wave breaking, as well as the differential deposition, gave rise to upstream migration of antidunes and to lens-like sedimentary structures in sand-sized sediments. Allen

(1966) identified four modes of deposition from antidunes, one of which is characterized by the breaking water waves and resultant upstream-dipping laminae. Yokokawa et al. (1999) described the effect of wave breaking over antidunes in a laboratory flume. They interpreted that lenticular structures in their antidune cross-stratification were formed during wave breaking, based on observation with a video camera. According to these authors, a series of events is observed in antidune sedimentation, which begins with a smooth flow and ends with the breaking of surface waves.

PGCC03 9/12/08 3:16 PM Page 67

67

Breaking of waves on antidunes Figure 1 shows the typical sequence of events observed in the experiments of Yokokawa et al. (1999). Antidunes were formed in well-sorted fine sand with wavelengths of 40 –100 cm and wave heights of 1– 2 cm. The mean flow velocity was 84 cm s−1, the mean flow depth was 4 cm and the slope of the flume floor was 1.5°. Breaking of surface waves was observed, though the critical conditions for wave breaking could not be estimated due to the unstable and non-uniform nature of the flow. In Fig. 1, antidunes and surface waves, which are initially in-phase and almost stationary, grow in size gradually under a smooth flow. The growing antidunes then disturb the smooth passage of the flow. The flow becomes unstable, and breaking of the wave takes place on the upstream flank of the associated antidune. Just before breaking, the wave moves upstream generating intense turbulence. A cloud of sediment is taken into suspension, and a substantial part surges upstream with the breaking wave. The suspended sediment is deposited on the upstream side when the wave decays and the turbulence diminishes. The flow is smooth again with the formation of the new antidune and returns to the start of the cycle. In order to describe this sequence of events repeated in antidune sedimentation, the hydraulics of the flow in the vicinity of a mound are explored (Fig. 2). Smooth flow Antidunes grow in size under a smooth flow (Fig. 2a). The flow velocity uc and depth dc at the crest of the mound are determined from those on the upstream side, ua and da, by u 2a u2 + gd a = c + g ( d a + z ) 2 2

(1)

uada = ucdc

(2)

where z is the height of the mound and g is the acceleration due to gravity. Equation (1) describes conservation of energy and is derived from the Bernoulli equation. Equation (2) represents mass1 conservation. Using the Froude number Fr = u/(gd) –2 instead of the flow velocity, Eqs (1) and (2) are unified as D3

⎛ Fr2 z ⎞ 2 Fra2 − ⎜ a +1− D + =0 2 da ⎟⎠ ⎝ 2

(3)

where D = dc /da is the dimensionless flow depth. The Froude number represents the ratio of the flow velocity divided by the maximum wave celerity. Flow is

Fig. 2. Definition sketches for notation for typical sequence of events in antidune sedimentation. (a) Smooth flow with a mound; (b) hydraulic jump moving upstream; (c) deposition on the upstream side of the mound.

termed ‘supercritical’ or ‘subcritical’ when its Froude number is larger or smaller than unity, respectively. Equation (3) gives the change of the flow depth on the mound. When the bed is flat (z = 0), Eq. (3) has solutions for positive D as (Fig. 3a), D=1 ⎛ 8 ⎞ D = ⎜1 + 1 + 2 ⎟ Fra ⎠ ⎝ The first solution (D = 1) obviously yields no variation in the flow conditions. The value of the second solution becomes larger than unity (dc > da) in a supercritical flow (Fra > 1) and smaller than unity (dc < da) in a subcritical flow (Fra < 1). A transition of the flow depth from supercritical to subcritical conditions is called a ‘hydraulic jump’.

PGCC03 9/12/08 3:16 PM Page 68

68


Fig. 3. The cubic expression for the left-hand side of Eq. (2), shown for four different values of the height of the topographic elevation, z. (a) Eq. (2) has positive solutions for z = 0; (b) the solutions are changed as a slight z is induced; (d) Eq. (2) has no real solutions for positive D for sufficiently large z through the critical condition (c) determined by Eq. (3).

With the mound on the bed, the solutions for D are changed with the height of the mound, z (Fig. 3b). In a supercritical flow, the two solutions for D both become larger than unity (dc > da). The water surface is, therefore, in-phase with the bed topography, and the flow velocity is decreased on the mound. The deceleration, and the resultant difference of the rate of erosion (or deposition), is likely to further develop the mound on an erodible bed. In a subcritical flow, the two solutions for D both become smaller than unity (dc < da). The flow becomes faster and thinner on the mound. The mound is unlikely to grow due to the larger rate of erosion on the mound. When the mound grows sufficiently large in a supercritical flow, Eq. (3) has no real solutions for positive D (Fig. 3d). The critical condition (Fig. 3c) is expressed as ⎛1 3 2 ⎞ z0 = d a ⎜ Fra2 − Fr a3 + 1⎟ 2 ⎝2 ⎠

(4)

Frc = 1 where z0 is the critical height of the mound. Hydraulic jump When the height of the mound exceeds z0, Eq. (3) yields no appropriate solutions. The flow can no longer attain smooth and steady conditions so that a hydraulic jump with energy loss must form on the

Fig. 4. Relative height of the mound (z/da) as a function of upstream Froude number (Fra) calculated from Eq. (4) for 1 different values of dimensionless jump speed, C = c/( gda)–2. Hysteresis is observed within region BAC determined by z = z0 in Eq. (3) and C = 0.

upstream side of the mound (Fig. 2b) (Long, 1954). While the flow dynamics are complicated due to the turbulent and unstable nature of the hydraulic jump, a simple description for such flow is obtained by assuming the flow to be hydrostatic except at the jump with conservation of momentum (e.g. Baines, 1995), as (ua + c)da = (ub + c)db

(5)

ubdb = ucdc

(6)

(ua + c )2 =

gd b ⎛ d ⎞ 1+ b⎟ ⎜ da ⎠ 2 ⎝

(7)

ub2 u2 + gd b = c + g( d c + z ) 2 2

(8)

u 2c = gdc

(9)

where c is the adverse speed of the jump. Equations (5) and (6) represent mass conservation. Equation (7) represents momentum conservation through the hydraulic jump. Equation (8) is the Bernoulli equation behind the jump. Equation (9) is the critical condition at the crest of the mound. The solution of these equations is shown in Fig. 4 for the upstream jump speed rendered dimensionless by C=

c gd a

Within region BAC in Fig. 4, the flow can be either supercritical everywhere or accompanied by an

PGCC03 9/12/08 3:16 PM Page 69

Breaking of waves on antidunes upstream jump (Baines, 1995). The two possible states imply that there is hysteresis in the system, which seems to be consistent with a cycle of events observed in antidune sedimentation. Assuming that the height of the mound is increased in a supercritical flow and decreased in a subcritical flow, the flow conditions should be expected to show the following temporal changes. When the height of the mound exceeds its critical value (curve AB), a hydraulic jump is formed at the crest of the mound. The jump moves upstream (c > 0), and a subcritical flow is imposed between the jump and the mound. In the subcritical flow, however, the mound tends to be depressed because of differential erosion or deposition. The height of the mound is decreased until the jump becomes stationary relative to the mound (curve AC). Destruction of the mound is possibly promoted by turbulence generated within the stationary or slowly migrating hydraulic jump. The hydraulic jump disappears when the height of the mound is further depressed and the jump speed becomes negative. After the disappearance of the hydraulic jump, which is expressed in the breaking of waves, the flow is again supercritical everywhere and returns to the start of the cycle. Deposition caused by the hydraulic jump Through the above cycle of events, upstream transport of sediment can take place in the vicinity of the mound. Some of the sediment eroded from the mound is held in suspension by turbulence within the hydraulic jump, and is transported along with the upstream migrating jump. As the hydraulic jump disappears (Fig. 2c), the suspended sediment is deposited on the upstream side of the mound, resulting in the possible formation of upstream-dipping laminae on the upstream flank or lenticular structures in the troughs. It should be noted, however, that a weak hydraulic jump is unlikely to contribute to upstream transport of sediments. The hydraulic jump in a flow with Froude number near unity is easily destroyed by slight erosion at the crest of the mound. The flow is expected to be fairly stable, so differential deposition would be the dominant process in the development of antidunes. In fact, a hydraulic jump takes the form of a smooth undular bore when the change of Froude number through the jump is slight (Binnie & Orkney, 1955; Baines, 1995). The observed structures of bores are described as smooth when the upstream Froude number Fra > 1.26, partially turbulent when 1.26 < Fra

69

< 1.55 and fully turbulent when Fra > 1.55 (Binnie & Orkney, 1955). Sedimentary structures formed by the hydraulic jump can be identified in antidune stratification, if a sufficient volume of sediment is transported and deposited. Deposition takes place both on the upstream flank and in the troughs of antidunes, depending where the hydraulic jump disappears. Deposits caused by the breaking of waves, therefore, might not show a direct relationship to the wavelength of antidunes. Middleton (1965) observed in his flume experiments that the breaking of waves gave rise to lens-like structures whose lengths were somewhat shorter than the wavelength of the antidunes. Hand et al. (1969) found partial filling of the troughs in antidune structures in the Mount Toby conglomerate, and attributed it to deposition by breaking waves. He showed that groups of steep antidunes were separated by other less-steep antidunes, and suggested the less-steep antidunes marked sites of wave breaking. Application to density currents It is possible that a hydraulic jump is formed at a fluid interface in two- or multi-layer systems (Baines, 1988). Experimental work of Hand (1974) showed that antidunes and associated breaking waves were formed by density currents in two-layer systems. While the upper and lower fluids interact in a complicated way in the turbulent hydraulic jump, the simplest approach to describe the two-layer systems in which the overlying fluid is sufficiently deep and at rest is to replace the Fra in Fig. 41 with the densiometric Froude number Frd = u/(g′d ) –2, where g′ = g∆ρ/(ρ + ∆ρ) represents reduced gravity, ρ is the density of the ambient fluid and ∆ρ is the density difference between the underflow and overlying fluid (Komar, 1971). The relationships illustrated in Fig. 4 predict that a similar process of wave breaking (hydraulic jump) should be observed at the upper interface of the underflow. There are several reports of antidune structures produced by turbidity currents (Walker, 1967; Skipper, 1971; Prave & Duke, 1990; Yagishita, 1994). The interpretations are, however, commonly questioned, based on hydrodynamic reasons (e.g. Pickering & Hiscott, 1985). From the theoretical work of Hand et al. (1972), it follows that antidunes in two-layer systems have wavelength (L) to flow depth (d) ratios of L/d > 12.6, implying a centimetre-scale flow thickness for decimetre-scale antidunes. In order to explain the small flow thickness, Hand et al. (1972) suggested that the antidunes in turbidites might be attributed

PGCC03 9/12/08 3:16 PM Page 70

70


to a thin internal layer of a differing density within the turbidity current. The solutions in Fig. 4 and consideration of the genesis of turbulent hydraulic jumps also suggest that the flow depth should be the same order as the antidune heights.

DISCUSSION The sequence of events observed with waves breaking on antidunes is successfully described by the formation and disappearance of a hydraulic jump in a supercritical flow. The solutions obtained are, however, only approximate due to the assumptions involved in the derivation of the equations and the simplifications in the descriptions of the systems in the vicinity of the mound. Only a single mound has been considered in this study, whereas antidunes are a train of such mounds on a relatively steep slope. The model must be extended to a curvilinear and sloped bed to include quantitatively the features of antidune configurations. It should be emphasized, though, that the formation of a hydraulic jump is mainly controlled by the height of the mound and independent of the wavelength and shape of antidunes, if the incoming flow is steady. The conditions of the incoming flow are continuously changing with the growth and the decay of the first mound in the train. When the flow accelerates on the lee-side downslope of the first mound, the incoming flow will temporarily attain a higher Froude number than the average value. Such variation of flow conditions is likely to result in further development of antidunes and a possible hydraulic jump. The flow conditions for the formation of the hydraulic jump should not be determined only by averaged flow conditions. The dynamics of a hydraulic jump are well understood as an upstream effect generated by an obstacle in a stratified flow (for a review see Baines, 1995). However, a turbulent and unstable hydraulic jump on an erodible bed cannot be described quantitatively. Further investigations are required to quantify the interaction of the hydraulic jump with the bed configuration and to obtain the relationship with sedimentary structures.

ACKNOWLEDGEMENTS We particularly thank F. Masuda, B. Kneller and M. Felix for their constructive reviews of earlier ver-

sions of this paper. We also thank T. Sakai and N. Endo for help with the flume experiments in Osaka University.

REFERENCES Allen, J.R.L. (1966) On bed forms and paleocurrents. Sedimentology 6, 153–190. Allen, J.R.L. (1982) Sedimentary Structures: Their Character and Physical Basis, Vol. 1. Elsevier, Amsterdam. 593 pp. Baines, P.G. (1988) A general method for determining upstream effects in stratified flow of finite depth over long two-dimensional obstacles. J. Fluid Mech. 188, 1–22. Baines, P.G. (1995) Topographic Effects in Stratified Flows. Cambridge University Press, Cambridge. 482 pp. Binnie, D.J. & Orkney, J.C. (1955) Experiments on the flow of water from a reservoir through an open horizontal channel. II. The formation of hydraulic jumps. Proc. R.S. London, Ser. A 230, 237–246. Gilbert, G.K. (1914) The transportation of debris by running water. U.S. Goel. Surv. Prof. Pap. 86, 1–263. Hand, B.M. (1974) Supercritical flow in density currents. J. sediment. Petrol. 44, 637– 648. Hand, B.M., Wessell, J.M. & Hayes, M.O. (1969) Antidunes in the Mount Toby conglomerate (Triassic), Massachusetts. J. sediment. Petrol. 39, 1310 –1316. Hand, B.M., Middleton, G.V. & Skipper, K. (1972) Antidune cross-stratification in a turbidite sequence, Cloridorme Formation, Gaspe, Quebec. Sedimentology 18, 135–138. Kennedy, J.F. (1963) The mechanics of dunes and antidunes in erodible-bed channels. J. Fluid Mech. 16, 521–544. Komar, P.D. (1971) Hydraulic jumps in turbidity currents. Geol. Soc. Am. Bull. 82, 1477–1488. Long, R.R. (1954) Some aspects of the flow of stratified fluids. II. Experiments with a two-fluid system. Tellus 6, 97–115. Middleton, G.V. (1965) Antidune cross-bedding in a large flume. J. sediment. Petrol. 35, 922–927. Pickering, K.T. & Hiscott, R.N. (1985) Contained (reflected) turbidity currents from the Middle Ordovician Cloridorme Formation, Quebec, Canada: an alternative to the antidune hypothesis. Sedimentology 32, 373–394. Prave, A.R. & Duke, W.L. (1990) Small-scale hummocky cross-stratification in turbidites: a form of antidune stratification? Sedimentology 37, 531–539. Skipper, K. (1971) Antidune cross-stratification in a turbidite sequence, Cloridorme Formation, Gaspe, Quebec. Sedimentology 17, 51– 68. Walker, R.G. (1967) Upper flow regime bed forms in turbidites of the Hatch Formation, Devonian of New York State. J. sediment. Petrol. 37, 1052–1058. Yagishita, K. (1994) Antidunes and traction-carpet deposits in deep-water channel sandstones, Cretaceous, British Columbia, Canada. J. sediment. Res. A64, 34 – 41. Yokokawa, M., Masuda, F., Sakai, T., Endo, N. & Kubo, Y. (1999) Sedimentary structures generated in upper-flowregime with sediment supply: Antidune cross-stratification (HCS mimics) in a flume. In: Proceedings of an International Workshop on Sediment Transport and Storage in Coastal Sea–Ocean System (Tsukuba, Japan), STA(JISTEC) & Geol. Surv. Japan, Tsukuba, pp. 409 – 414.

PGCC04

10/12/08

4:27 PM

Page 71


A two-dimensional numerical model for a turbidity current M. FELIX School of Earth Sciences, University of Leeds, Leeds, UK

ABSTRACT A 2-D numerical model for flow of and deposition from a turbidity current was developed to simulate processes on a natural scale and topography for flows with sand-size sediment. The velocity of the bulk fluid (sediment + water) is calculated from a momentum equation that is derived from the Navier–Stokes equation using the hydrostatic and boundary layer approximations. Sediment concentrations are calculated with advection– diffusion equations for each grain size and turbulence is taken into account by using the Mellor–Yamada level 2–12 second-order closure model. The effect of the presence of particles on the turbulence other than through buoyancy is incorporated through a drag term that leads to an extra dissipation term in the turbulent kinetic energy equation and in the turbulent lengthscale equation. The equations are solved numerically using a finite volume method on a non-staggered grid with a BDF method. Model results are shown for two flows with different initial concentrations, showing the behaviour of the model.

INTRODUCTION to model turbidity currents. All of these models are for a steady current and describe the vertical flow structure but not the horizontal structure and hence cannot model spatial distribution of deposits. In the model presented here, both horizontal and vertical flow structure are described. In this model, a turbidity current forms from a failed mass of sediment that is initially at rest. For low volumetric particle concentrations (less than O(10−3) according to Hinze, 1972), the influence of the particles on the water can be ignored. If the concentration is higher, which is not unlikely in turbidity currents, influences such as buoyancy effects and dissipation need to be modelled. According to Bagnold (1966), particle–particle interactions can be ignored for concentrations up to 9% for non-cohesive particles, but are important for concentrations higher than that. To deal with high sediment concentrations, the multiphase flow approach is used here to derive the influence of the presence of particles on the turbulence. Particle–particle interactions are modelled using a concentration-dependent viscosity. The level 2–12 turbulence model of Mellor and Yamada (1982, from

Turbidity currents are the principal means by which coarse clastic sediments are transported and deposited in the deep sea, but unfortunately natural turbidity currents are hard to observe and study. One way of gaining a better understanding of the flow processes is by using mathematical models. Most mathematical models of turbidity currents that have been developed are depth-averaged or algebraic models (Parker et al., 1986; Mulder et al., 1998 and references in these papers), which can help in understanding some aspects of the flow behaviour (e.g. ignition). Flow structure, including vertical distributions of velocity, sediment concentration and turbulence, however, cannot be modelled using these approaches. To model this, a non-depth-averaged model together with a turbulence model needs to be used. A few such models have been described in the literature. Hinze (1960) used a constant turbulent mixing coefficient proportional to the maximum velocity to describe the turbulence. Stacey and Bowen (1988) used an algebraic model dependent on velocity and gradient Richardson number. Eidsvik and Brørs (1989) used a k − ε turbulence model and Brørs and Eidsvik (1992) used a Reynolds stress model


71

PGCC04

10/12/08

4:27 PM

Page 72

72

M. Felix

now on referred to as the Mellor–Yamada model) is adapted to account for the influence on turbulence by the presence of particles. The boundary layer approximation (long and thin flow) and the hydrostatic pressure approximation (pressure equals weight of fluid) are used to simplify the model. Flow results are shown for a low concentration flow and for a high concentration flow. Both runs are shown to illustrate the model behaviour. No historic flow described in the literature has been modelled and the model still needs to be tested using measured values of flow variables, such as velocity, sediment concentration and turbulent kinetic energy.

∂ρk c k u ki u kj ∂ρ kc k u ki + ∂t ∂x j = − ck

∂pk ∂ 2 u ki + ρk ck g i + c k µ k + Mk ∂xi ∂x 2j

where ρk = (constant) density of phase k, uki = ith velocity component of phase k, pk = pressure of phase k, gi = ith component of constant vector of gravitational acceleration, µk = viscosity of phase k and Mk = interfacial interaction term of phase k. The interfacial terms Mk (k = 1, . . . , n) represent all the forces that the water exerts on the particles and the term M0 represents all the forces that the particles exert on the water, such that n

MULTIPHASE FLOW APPROACH

∑ Mk = 0.

In the multiphase flow approach all the different phases in the flow (in this case water and n different particle classes) are treated as interpenetrating continua that interact on their boundaries. Momentum and mass conservation equations are given for all the different phases in the flow and the interaction is represented by an interfacial interaction term in the momentum equations that results from spatial averaging. This approach has been used to model sediment–water interaction for a number of different applications. Ishii and Zuber (1979) and Greenspan and Ungarish (1982) used this method to model settling of particles from high concentrations, but in their models no turbulence is involved. The method has also been used to model turbulent flows for low particle concentrations (Pourahmadi & Humphrey, 1983; Rizk & Elghobashi, 1985, 1989; Kulick et al., 1994; Sato et al., 1996) and for low to high concentrations (Elghobashi & Abou-Arab, 1983). The different phases have volume fractions ck such that n

∑ ck = 1.

k =0

In determining the terms Mk (k = 1, . . . , n), it is assumed here that the drag force is the most important force of the water on the particles and that all other forces can be ignored. Experiments (Lee, 1987; Littman et al., 1996) show that the drag coefficient of a particle in fully turbulent flow is smaller than in quiescent water; due to the turbulence, the wake behind the particle is completely eliminated and the influence of the viscous layer (which is what causes the drag) around the particle is diminished by high fluid velocity fluctuations. The Stokes relation (e.g. Graf, 1971) for the drag force is still valid for a particle in a turbulent flow (independent of particle Reynolds number) if the viscosity µ 0 is replaced by the effective viscosity µe felt by the particle in the turbulent flow (an expression for µ e will be given later). Using this pseudo-Stokes relation, the interfacial drag term for the particles (k = 1, . . . , n) is now given by Mk =

ck F 4 d k3 D π 3

=

9 µe c (u − uk ), 4 d k2 k 0

where dk = particle diameter and FD = drag force.

k =0

In the present model k = 0 is the water phase and k = 1, . . . , n are the particle phases. To model the interaction of the phases, the mass and momentum conservation equations for all the phases are spatially averaged (Ishii & Zuber, 1979; Greenspan & Ungarish, 1982; Drew, 1983; Buyevich, 1995). For each phase (k = 0, . . . , n) the mass and momentum conservation equations are given by ∂ρk ck ∂ρk ck uki + = 0, ∂t ∂xi

(2)

(1)

TURBULENCE MODEL To incorporate the Mellor–Yamada turbulence model, the mass and momentum equations are Reynolds averaged where the instantaneous value of a variable f is split up in f = F + f ′, where F is the time averaged part of f and f ′ is the fluctuating part of f. After Reynolds averaging and

PGCC04

10/12/08

4:27 PM

Page 73

73

2-D numerical model for a turbidity current applying the order of magnitude analysis used by Mellor and Yamada (1982) and Galperin et al. (1988), the momentum equation for phase k is ∂ρkCkUkiUkj ∂ρkCkUki + ∂t ∂xj ∂ρkCk u i′u ′j ∂P ∂2Uki = −Ck k + Ck ρk gi + µkCk − + Mk , ∂xi ∂x 2j ∂xj (3) with the interfacial term (k = 1, . . . , n) given by Mk =

9 µe c (U − Uk ). 4 d k2 k 0

Here it has been assumed that the term µ ck′ uk′ is negligible. It is assumed that turbulence is a result only of the water motion (phase 0), so the equation for the turbulent kinetic energy q2 and the equation for the turbulent lengthscale l will be derived from the water momentum equation only. The equation for turbulent kinetic energy now becomes, after applying the closure assumptions and the order of magnitude analysis of the Mellor– Yamada model (see also Galperin et al., 1988): ∂C 0q 2 ∂C 0U0kq 2 + ∂t ∂xk ∂ ∂q 2 = C 0lqSq ∂xk ∂x k ⎛ ∂U0i q3 ⎞ − 2 ⎜C 0 u i′ u ′j + gi u i′ c 0′ + ⎟ + Mq , ∂x j B1l ⎠ ⎝ where Sq and B1 are empirical constants given by the Mellor–Yamada model (Sq = 0.2 and B1 = 16.6). The interfacial term Mq is given by ⎛ 9 C ⎞ M q = constant ⋅ ⎜ − q2 µe ∑ 2k ⎟ . ⎜ 4 ⎟ k =1 d k ⎠ ⎝ n

This term represents dissipation of turbulent kinetic energy due to the particle drag (buoyancy effects caused by the presence of particles are represented by the term gi u i′ c0′ ). For low concentrations several authors (Elghobashi & Abou-Arab, 1983; Pourahmadi & Humphrey, 1983; Kulick et al., 1994) have used the multiphase flow approach to adapt a k − ε turbulence model for the presence of particles. From their experiments, Kulick et al. (1994) found that dissipation due to the presence of particles is of the same importance as viscous dissipation and in their model they use the same constant

of proportionality for the viscous dissipation and for the dissipation due to the presence of particles. A constant of proportionality of O(1) was also found by Sato et al. (1996) in their experiments. Using the same constant for the two types of dissipation gives ∂C 0q 2 ∂C 0U0kq 2 + ∂t ∂xk

⎡⎛ ∂U ⎞ 2 ⎛ ∂U ⎞ 2 ⎤ ∂ ∂q 2 01 02 ⎥ = C K + 2C 0 ⎢⎜ +⎜ ⎟ ⎢⎝ ∂x 3 ⎟⎠ ∂x 3 0 q ∂x 3 ⎝ ∂x 3 ⎠ ⎥ ⎣ ⎦ n ∂ρf Ck 9 2 q3 q µe ∑ 2 , + 2g 3 KH − 2C 0 − 4B1 B1l ∂x 3 k =1 d k where the turbulent mixing coefficient is Kq = qlSq and KH is a turbulent mixing coefficient described below. Following a similar procedure to adapt the turbulent length scale equation (Mellor & Yamada, 1982) results in ∂C 0 q 2 l ∂C 0U 0kq 2 l + ∂t ∂xk

⎡⎛ ∂U ⎞ 2 ⎛ ∂U ⎞ 2 ⎤ ∂ ∂q 2 l 01 02 ⎥ = + lE1 ⎢⎜ +⎜ C K ⎟ ⎢⎝ ∂x 3 ⎟⎠ ∂x 3 0 q ∂x 3 ⎝ ∂x 3 ⎠ ⎥ ⎦ ⎣ 2⎤ ⎡ 3 ∂ρf ⎛ l ⎞ q + lE1 g 3 KH − C 0 ⎢1 + E2 ⎜ ⎟ ⎥ ∂x 3 B1 ⎢ ⎝ κL ⎠ ⎥ ⎦ ⎣ n Ck 9 2 − q l µe ∑ 2 , 4B1 k =1 d k where E1 and E2 are empirical constants (E1 = 1.8 and E2 = 1.33), κ is the von Kármán constant (κ = 0.4) and L is vertical distance from the bottom. Bulk fluid flow To further simplify the equations, only the motion of bulk fluid flow will be modelled, rather than all the different velocities of the different phases. The bulk fluid velocity Uf is defined by n

Uf =

∑CkUk ,

k =0

and bulk fluid density by n

ρf =

∑C k ρk .

k =0

Assume now that all the particle velocities are the same as the bulk fluid velocity except in the vertical direction, where they differ by the settling velocity wsk (k = 1, . . . , n)

PGCC04

10/12/08

4:27 PM

Page 74

74

M. Felix Uki = Uf i − wskδi 3,

where δij is the Kronecker delta (δij = 1 if i = j and δij = 0 if i ≠ j). Note that this results in an upward motion of water due to the settling of particles. DeVantier and Larock (1983) take a similar approach for low concentrations of one particle class. Near the bottom, particles will not have the same velocity as the water because the water velocity will go to zero (no slip at the boundary) while the particle velocity does not go to zero. It is assumed that the difference will be small enough not to be significant.

Limits on GH are given by Galperin et al. (1988) as −0.28 ≤ GH ≤ 0.0233. Volume conservation equation Mass and momentum conservation equations can now be written in terms of bulk fluid properties by adding equations for all different phases (k = 0, . . . , n). Dividing by individual densities before adding leads to ∂Uf

= 0,

i

∂xi Closure terms For the Mellor–Yamada model with the boundary layer approximation, the closure terms are (Mellor & Yamada, 1982; Galperin et al., 1988) − u ′w ′ = KM

∂U , ∂z

− v ′w ′ = KM

∂V , ∂z

− c ′w ′ = KH

∂C 0 . ∂z

The turbulent mixing coefficients KM and KH are given by KM = − 13

ql

B1

− A1A2GH [(B2 − 3A2 )(1 − 6A1/B1 ) − 3C1(B2 + 6A1)] , [1 − 3A2GH (6A1 + B2 )] (1 − 9A1A2GH )

A 2 (1 − 6 A1 / B1 ) KH = ql . 1 − 3A 2GH (6 A1 + B2 ) In these equations

Momentum equation Adding the momentum equations for all the phases (k = 0, . . . , n) and using the closure described above gives a momentum equation in terms of the bulk fluid velocity ∂ρf Uf ∂t =−

+

∂ρf Ufi Ufj ∂x j

ρf − ρ0 ∂Ufi ∂P ∂ ( µr + K M ) , + gi + ∂xi ρ0 ∂x3 ∂x3

where µr = viscosity of the water–particle mixture (see below) and P is the hydrostatic pressure. The influence of the particles on the fluid motion is present via the density and viscosity terms and in the turbulent mixing coefficient that is determined using the turbulent kinetic energy and the turbulent lengthscale. The change in viscosity due to the presence of particles is given by an empirical equation. Pressure

2

⎛ Nl ⎞ GH = − ⎜ ⎟ , ⎝ q⎠ where N2 = −

g ∂ρf , ρ0 ∂x3

is the Brunt–Väisälä frequency and ρf is total fluid density (water and particles). Empirical constants given by Mellor and Yamada (1982) are (A1, A2, B1, B2, C1) = (0.92, 0.74, 16.6, 10.1, 0.08). In stably stratified flows the turbulent lengthscale l is limited according to (Blumberg et al., 1992) l≤

0.53q . N

The hydrostatic pressure is given by

zt

P = Pt + g3

ρf dz, zb

where Pt is the pressure at the top of the domain (taken to be zero since only pressure differences are needed) and zt and zb are respectively the top and bottom coordinates of the domain. The computational domain needs to be such that no mass flux across the upper boundary takes place since this would change the value of the hydrostatic pressure there. Mass conservation equations for sediment The concentrations for all the different particle classes

PGCC04

10/12/08

4:27 PM

Page 75

75

2-D numerical model for a turbidity current are now given by mass conservation equations in terms of the bulk fluid velocity (ρk is assumed to be constant and can be divided out)

ive viscosity µe felt by a particle in a turbulent flow is taken to be

∂Ck (Ufi − wsk δ i 3 ) ∂Ck ∂C ∂ + = KH k . ∂t ∂xi ∂x3 ∂x3

where µr is the viscosity of the water–particle mixture and KH is the turbulent mixing coefficient for particles given by the Mellor–Yamada model. This viscosity will be used in the terms due to particle drag in the turbulent kinetic energy equation and the turbulence lengthscale equation.

The particle settling velocities wsk are calculated using the empirical equation of Dietrich (1982). Hindered settling effects are taken into account using the equation of Richardson and Zaki (1954). As in the model of Adams and Weatherly (1981), the mixing coefficient KH is used in the sediment equation and the same KH is used for all particle classes. Viscosities

µe = µr + KH,

Boundary conditions The boundary conditions for the fluid at the bed are as given by Mellor and Yamada (1982) and Blumberg and Mellor (1987): Ui = 0.

Krieger and Dougherty (1959) gave the following relationship to determine the viscosity of a particle–water mixture dependent on particle concentration c and maximum packing concentration cmax µr ⎛ c ⎞ = ⎜1 − µ0 ⎝ c max ⎟⎠

− w ′ui′ = τbi , 2

q2 = B1–3 u 2τ , b

−2.5cmax

,

where c = (1 − c0) is total sediment concentration. In a sediment mixture with different grain sizes, cmax = 0.62. The viscosity µr takes all particle–particle interactions into account. Prasad and Kytömaa (1995) found in their experiments that the relative viscosity due to particle presence is independent of shear rate in the range where shear-thickening does not take place. Turbulence therefore does not change the relative viscosity. By just changing the viscosity, the mixture is assumed to behave in a Newtonian manner. For monodisperse flows of volumetric concentration up to about 50% and for mixtures that contain several grain sizes with volumetric concentration up to about 60%, nonNewtonian shear thickening behaviour does not take place (see review by Barnes, 1989). For monodisperse particles, 50% concentration is approximately cubic packing and for a mixture of several grain sizes, 62% concentration is maximum packing concentration, so non-Newtonian behaviour is probably not relevant to the problem addressed here. Ishii and Zuber (1979) and Greenspan and Ungarish (1982) take a similar approach to using the viscosity of the water–particle mixture in a multiphase flow approach. However, in their models, all the different phases are modelled, not just the bulk fluid as here. By analogy with the diffusion term (µr + KM) in the momentum equation for bulk fluid motion, the effect-

q2l = 0, 1

where uτb = (τ 2b)–2 /ρ0 is the bottom shear velocity and τb is the bottom frictional stress given by τb = ρ0CD |Ub|Ub, where Ub is the velocity vector at the lowest grid cell near the bottom. The bottom drag coefficient CD is given by −2

⎡1 ⎤ CD = ⎢ ln ( zb /z 0 )⎥ , κ ⎣ ⎦ where z0 is the roughness coefficient and zb is height of lowest grid cell. As in Blumberg and Mellor (1987), the maximum value of CD = 0.0025 and CD as given by the above equation is used. Depending on the bottom shear stress, the boundary condition for the sediment will be either erosion or deposition. To determine whether net erosion or deposition will take place the Shields diagram can be used to determine a critical shear stress for a certain grain size. The equation given by van Rijn (1982) based on the diagram is used here to calculate the critical stress. If the critical shear stress τcr is larger than the bottom shear stress τb, deposition will take place and the boundary condition is KH

τ ∂C = −Cws b . τcr ∂x3

Erosion is modelled using the empirical entrainment function Esk of Garcia and Parker (1991). Using this function, the Reynolds stress wfk ′ ck′ at the bed is

PGCC04

10/12/08

4:27 PM

Page 76

76

M. Felix wfk ′ ck′ = wsk Esk .

This function is valid for several grain sizes present in the bed and takes armouring into account.

COORDINATE TRANSFORMATION To simplify the discretization, a coordinate transformation is used to map the equations from a topography-following curvilinear coordinate system to a computational equidistant coordinate system, in which discretization will then take place. Deposition and erosion will change the topography during flow, but to take this into account, the coordinate system would have to be remapped in every time step. To avoid this, the influence of deposition and erosion on the topography will be ignored. Using the chain-rule of differentiation, a differential equation ∂Fj ∂f + = S, ∂t ∂x j where Fj is the flux and S is a source term, can be written in terms of the computational coordinate system (X1, X2, X3) as ∂Xi ∂Fi ∂f + = S. ∂t ∂x j ∂Xj Derivatives in the flux term (e.g. diffusion terms) and in the source term (e.g. turbulence production terms in the turbulent kinetic energy equation) are written in terms of the computational coordinate system using the chain-rule of differentiation. For mass conservation of the discretized equations, it is preferable for the equations to be discretized in conservative form. Viviand (1974) showed that the transformed differential equations can be written in conservative form as ∂⎛ f⎞ ∂ ⎛ ∂Xj Fi ⎞ S = , ⎜ ⎟+ ∂t ⎝ D ⎠ ∂Xj ⎜⎝ ∂xi D ⎟⎠ D

fluxes are defined on the cell faces. The values of the variables are defined at the cell centre and need to be determined at the cell faces by using values of the neighbouring cells. For cell boundaries that have normals in the slope-parallel X1 direction, the SHARP algorithm of Leonard (1988) is used. This method is used to prevent oscillatory behaviour of the solution that can occur if large gradients are present. For cell boundaries that have a normal in the X3 direction, the values on the cell faces are calculated using the average of the values of the cells adjacent to the boundary. Using the SHARP algorithm in this direction actually led to more oscillatory solutions than using the average. The pressure is calculated using the midpoint rule of integration. Time discretization The equations, of the form ∂φ = H( φ ), ∂t were discretized using the second-order Backward Differentiation Formula for a variable time step (Hairer et al., 1993). In the method used here, only part of H is calculated implicitly; the horizontal advection terms are calculated explicitly since the non-linear SHARP method is used in that direction. This limits the time step according to the CFL condition for the horizontal direction, but the much more limiting CFL condition for the vertical direction (as a result of high settling velocities near the bed where the cells are small) and the viscous limit, are avoided. For the time step ratio ωn +1 =

where ∆t n = t n − t n−1, the discretization is φ n +1 −

(4)

where D is the Jacobian of the transformation.

DISCRETIZATION The spatial derivatives are discretized on a non-staggered grid using a finite volume method described by Zang et al. (1994). In this method, all the (Cartesian) variables are defined in the cell centre and all the volume

∆t n +1 , ∆tn

(1 + ωn +1 )2 n ω 2n +1 φ + φ n −1 1 + 2ωn +1 1 + 2ωn +1

= ∆ t n +1

1 + ωn +1 ( H n +1 + (1 + ω n +1 )H En − ωn +1H En−1), 1 + 2ωn +1 I

where HI is the part of the equation discretized implicitly and HE is the part of the equation discretized explicitly. The explicit part is a second-order accurate approximation of that part of the equation at time t n+1. In the momentum equation and in the particle equations, horizontal advection is calculated explicitly. In the TKE and in the lengthscale equations, horizontal advection and turbulent production terms are cal-

10/12/08

4:27 PM

Page 77

77

2-D numerical model for a turbidity current culated explicitly. All the other terms are calculated implicitly.

coarse sand respectively. The initial sediment distribution in the bed (along the whole flow trajectory) is the same as in the column. The bed roughness coefficient z0 was set to 10−3 m. One test run was with low initial concentration, one with high initial concentration. In the low concentration flow, all three grain sizes had an initial volumetric concentration of 1% (so total concentration was 3%) while in the high concentration flow, the initial volumetric concentration was 10% for all grain classes (so total concentration was 30%). Calculations were performed on a 400 × 150 grid, which was equidistant in the horizontal direction and logarithmic in the vertical direction. The figures are shown at a time during the flow process, before flow has ceased and deposition is completed. No attempt has been made to model a historic flow described in the literature, only general results are shown. The flow behaviour (as shown in Figs 1 to 4) was rather different for the two different flows. In these figures it can be seen that the low concentration flow is much slower than the high concentration flow. The high concentration flow is shown 600 s after initiation of flow and the low concentration flow is shown 2000 s after initiation. The velocity maximum for both flows is at the front of the flow, a little behind the head. For the low concentration flow, the maximum is confined to a smaller area than in the high concentration flow. The values of the velocity are the same order of magnitude as those reported by Kuenen (1952) for the Grand Banks flow and by Piper and Savoye (1993) for the Var Canyon. In both these cases, velocity was calculated from times of telephone cable breaks. Neither flow has problems flowing over the obstacle, which is of the same order as the flow thickness. The high concentration flow is thicker than the low concentration flow, but neither are of the order of hundreds of metres reported in the literature. The reason for this is probably that in the flows modelled here, no mud is present and the sand will only form the bottom

Numerical boundary conditions For the downstream and top boundaries, a zero second derivative condition perpendicular to the boundary is used. For the upstream boundary, a reflection boundary condition is used: the flow on both sides of the boundary are mirror images. This gives a zero-horizontal velocity on the boundary. Values for concentration (Ck) and turbulence (q2 and l) are extrapolated from interior cells using a second-order accurate method. Flow domain Of all the boundaries of the flow domain, only the bottom one is a physical boundary. All other boundaries are in the water and the boundary conditions are purely numerical. To capture the velocity profile near the bottom, where there is a large gradient due to the no-slip boundary condition, the grid is logarithmic in the vertical direction.

TEST RUNS To get an idea of the behaviour of the model, two tests were run for different initial concentration over the same topography. The flow starts from rest on a slope, flows downslope onto a flat surface, over an obstacle (10-m high, 1500-m long) and then across a flat surface (Fig. 1). At time t = 0, three different grain classes are distributed evenly (so there is no particle concentration gradient) throughout a column 50-m high and 1000-m long. The grains all have a density ρsk of 2650 kg m−3 and diameters are 10−5, 10− 4 and 2 × 10−3 m which correspond to silt, very fine sand and very

Velocity u (m/s) 80

x3 (m)

PGCC04

60 40 0.5

Fig. 1. Velocity contour plot for low concentration case (total initial concentration 3%), 2000 s after initiation of flow. Contours are in m s−1.

1

1.5

20 2

0 0

1

2

3

4 x (km) 1

5

2.5

6

7

10/12/08

4:27 PM

Page 78

78

M. Felix Velocity u (m/s) 80

x3 (m)

60 40 2

4

8

Fig. 2. Velocity contours for high concentration case (total initial concentration 30%), 600 s after initiation of flow. Contours are in m s−1.

7

Fig. 3. Total concentration contour plot for low concentration case, 2000 s after initiation of flow. Initial total concentration was 3%. Contours are volume fraction.

6

20 1

2

3

4

5

10

0 0

9

8

6

7

x1 (km) Volumetric sediment concentration c 80

3

x (m)

60 40 0.001 0.002

20

0.003

04 0.0 5 0.00

0 0

1

2

3

4

5

6

x1 (km) Volumetric sediment concentration c 80 60 x3 (m)

PGCC04

40 0.0

20 0 0

1 0.03

5 0.0 0.07 0.09

1

2

3

4

5

x1 (km)

part of the flow. The flow thicknesses here are of the same order as the thicknesses reported by Menard (1964) and as found by Shepard et al. (1977) and Piper and Savoye (1993) for the sandy part of turbidity currents in canyons (for which the 2-D approach is most valid). Shepard et al. (1977) found in the Abra Canyon (Philippines) flow thicknesses on the order of 30 m or less. Piper and Savoye (1993) found erosion up to 30 m above the thalweg in the Var Canyon (Mediterranean) and deposition up to 50 m above the floor. The Var Canyon flows are thicker than these modelled here,

6

7

8

Fig. 4. Total concentration contour plot for high concentration case, 600 s after initiation of flow. Initial concentration case was 30%. Contours are volume fraction.

but the floor of the Var Canyon has a high roughness coefficient. If the roughness coefficient z0 is increased in this model, flow thickness increases due to the increased turbulent kinetic energy q 2 at the bed. The sediment distribution in the low concentration flow (Fig. 3) shows a distinct maximum near the head. In the high concentration flow (Fig. 4), the concentration remains high along the entire base of the flow (a maximum is present near the head but it is not as distinct as in the low concentration case). The high concentration flow is more stratified over its entire

10/12/08

4:27 PM

Page 79

79

2-D numerical model for a turbidity current

Deposit of grains 0.5 mm diam

2

10

0.001 0.01 0.1

0

1

10

2

10

10

2

10

3

0.001 0.01 0.1

10

4

10

4

10

Deposit of grains 0.05 mm diam

0

10

thickness (m)

thickness (m)

PGCC04

0

2

4 6 x1 (km)

8

10

Fig. 5. Final deposits from a uniform grain size of 0.5 mm for different initial volumetric concentrations. The gap in the line for initial concentration 0.01 is due to net erosion. Deposit thickness is without porosity or compaction.

flow length than the low concentration flow, except at the nose. To look at final deposits, experiments were run with a uniform grain size for two different grain sizes (0.5 mm and 0.05 mm) for different initial concentrations (0.001, 0.01 and 0.1). The other conditions (topography and initial size of sediment column) were kept the same as described above. The results from these runs are shown in Figs 5 and 6. These are the deposits when the flow has stopped flowing and deposited all its sediment or when the flow has left the flow domain and no more deposition takes place in the domain. For the lowest concentration (0.001) of the coarser grains (Fig. 5), sediment deposits almost immediately. Some net erosion is observed but due to the low concentration and high settling velocity, not much horizontal momentum is generated. For an initial concentration of 0.01, a lot of sediment is deposited at the end of the slope and before the obstacle. The higher concentration flow is not so clearly influenced by this and flows much further than the other two. Due to the hindered settling, the flow keeps sediment in suspension longer thus generating more momentum and it is able to flow further but deposits get very small towards the end of the domain shown. Flows with the smaller grain size (Fig. 6) all travel at least the entire length of the domain shown, and, for the higher concentration flows, deposit a large amount further than 10 km from the initiation point. Some minor influence of the topography can be seen for the lowest concentration flow (0.001), but the

0

2

4 6 x1 (km)

8

10

Fig. 6. Final deposits from a uniform grain size of 0.05 mm for different initial volumetric concentrations. Deposit thickness is without porosity or compaction.

other two flows do not seem to be influenced much by topography. For the topography used in these tests, which is of the same order, but smaller than flow thickness, topography seems to influence the deposits mainly for the low concentration flows, where flow velocities are fairly low. If flow velocities are still fairly high at the time of deposition (as is the case for the high concentration flows), the turbidity currents flow over the topography without being influenced by it very much.

CONCLUSIONS A two-dimensional mathematical model to simulate turbidity currents is described. The model calculates both vertical and horizontal flow structure of a turbidity current through time. No attempt has been made to reproduce a specific flow as described in the literature, but results from runs for a low concentration and a high concentration flow show that model results compare well qualitatively with flows observed in nature. Flow thicknesses and velocities are of the same order of magnitude as flows observed in nature, but further tests are still needed to assure that the results are correct quantitatively as well. The adaptation of the Mellor–Yamada model needs to be compared with data from measured high concentration flows. Deposits also need to be tested. For this, deposits described in the literature need to be used.

PGCC04

10/12/08

4:27 PM

Page 80

80

M. Felix

ACKNOWLEDGEMENTS I would like to thank Don Lowe, Bob Street and Steve Monismith for their help with this work. Clare Buckee, Jeff Peakall, Ben Kneller and Rob Felix are thanked for useful discussions on this paper. Financial support for this work was given by Chevron.

NOTATION Variables indicated with capital letters are timeaveraged parts, primed variables are fluctuating parts and variables indicated with small letters are not Reynolds averaged. Mellor–Yamada model empirical constants: (A1, A2, B1, B2, C1, E1, E2, Sq) = (0.92, 0.74, 16.6, 10.1, 0.08, 1.8, 1.33, 0.2) CD Ck d, dk Es FD g KM KH Kq l L Mk N Pk q2 t uτb Ub Uki Ufi wsk xi z0 zb δij κ µe µk

bed drag coefficient volume fraction of phase k sediment grain diameter sediment entrainment function drag force on a particle constant of gravitational acceleration (= 9.8) turbulent mixing coefficient for momentum equation turbulent mixing coefficient for mass equation turbulent mixing coefficient for q2 and q2l equations turbulent macroscale vertical distance from the wall interfacial drag force on phase k Brunt–Väisälä frequency pressure of phase k turbulent kinetic energy time bottom shear velocity velocity at the grid point closest to the bottom ith component of velocity of phase k ith component of bulk fluid velocity settling velocity of particle phase k coordinate roughness coefficient grid point closest to the bottom Kronecker delta von Kármán constant (= 0.4) effective viscosity felt by particle in turbulent flow viscosity of phase k

µr ρf ρk τb τcr

viscosity of water–sediment mixture density of water and sediment mixture density of phase k bottom shear stress critical shear stress for erosion

REFERENCES Adams, C.E. & Weatherly, G.L. (1981) Suspended-sediment transport and benthic boundary-layer assumptions. Mar. Geol. 42, 1–18. Bagnold, R.A. (1966) An approach to the sediment transport problem from general physics. U.S. Geol. Surv. Prof. Pap. 9, 970 –1073. Barnes, H.A. (1989) Shear-thickening (‘dilatancy’) in suspensions of nonaggregating solid particles dispersed in Newtonian liquids. J. Rheol. 33, 329–366. Blumberg, A.F. & Mellor, G.L. (1987) A description of a three-dimensional coastal ocean circulation model. In: Three-dimensional Coastal Ocean Models (Ed. Heaps, N.J.). AGU Coastal Estuarine Sci. 4, 1–16. Blumberg, A.F., Galperin, B. & O’Connor, D.J. (1992) Modeling vertical structure of open-channel flows. J. Hydraul. Eng. 118, 1119 –1134. Brørs, B. & Eidsvik, K.J. (1992) Dynamic Reynolds stress modeling of turbidity currents. J. Geophys. Res. 97, 9645– 9652. Buyevich, Y.A. (1995) Interphase interaction in fine suspension flow. Chem. Eng. Sci. 50, 641– 650. DeVantier, B.A. & Larock, B.E. (1983) Sediment transport in stratified turbulent flow. J. Hydraul. Eng. 109, 1622–1635. Dietrich, W.E. (1982) Settling velocity of natural particles. Water Res. Res. 18, 1615–1626. Drew, D.A. (1983) Mathematical modeling of two-phase flow. Ann. Rev. Fluid Mech. 15, 261–291. Eidsvik, K.J. & Brørs, B. (1989) Self-accelerated turbidity current prediction based upon (k − ε) turbulence. Continent. Shelf Res. 9, 617– 627. Elghobashi, S.E. & Abou-Arab, T.W. (1983) A twoequation turbulence model for two-phase flows. Physics Fluids, 26, 931–938. Galperin, B., Kantha, L.H., Hassid, S. & Rosati, A. (1988) A quasi-equilibrium turbulent energy model for geophysical flows. J. Atmos. Sci. 45, 55 – 62. Garcia, M. & Parker, G. (1991) Entrainment of bed sediment into suspension. J. Hydraul. Eng. 117, 414 – 435. Graf, W.H. (1971) Hydraulics of Sediment Transport. New York, McGraw-Hill. 513 pp. Greenspan, H.P. & Ungarish, M. (1982) On hindered settling of particles of different sizes. Int. J. Multiphase Flow, 8, 587– 604. Hairer, E., Nørsett, S.P. & Wanner, G. (1993) Solving Ordinary Differential Equations. I. Nonstiff Problems, 2nd edn. Springer-Verlag, Berlin. 528 pp. Hinze, J.O. (1960) On the hydrodynamics of turbidity currents. Geologie en mijnbouw, 39e jaargang, 18–25. Hinze, J.O. (1972) Turbulent fluid and particle interaction. Progr. Heat Mass Transfer 6, 433– 452. Ishii, M. & Zuber, N. (1979) Drag coefficient and relative velocity in bubbly, droplet or particulate flows. Am. Inst. Chem. Eng. J. 25, 843– 855.

PGCC04

10/12/08

4:27 PM

Page 81

2-D numerical model for a turbidity current Krieger, I.M. & Dougherty, T.J. (1959) A mechanism for non-Newtonian flow in suspension of rigid spheres. Trans. Soc. Rheol. 3, 137–152. Kuenen, Ph.H. (1952) Estimated size of the Grand Banks turbidity current. Am. J. Sci. 250, 874 – 884. Kulick, J.D., Fessler, J.R. & Eaton, J.K. (1994) Particle response and turbulence modification in fully developed channel flow. J. Fluid Mech. 277, 109 –134. Lee, S.L. (1987) Particle drag in a dilute two-phase suspension flow. Int. J. Multiphase Flow 13, 247–256. Leonard, B.P. (1988) Simple High-Accuracy Resolution Program for convective modelling of discontinuities. Int. J. Numerical Methods Fluids 8, 1291–1318. Littman, H., Morgan, M.H. & Paccione, J.D. (1996) A pseudo-Stokes representation of the effective drag coefficient for large particles entrained in a turbulent airstream. Powder Technol. 87, 169–173. Mellor, G.L. & Yamada, T. (1982) Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Physics 20, 851– 875. Menard, H.W. (1964) Marine Geology of the Pacific. McGraw-Hill, New York. 271 pp. Mulder, T., Syvitski, J.P.M. & Skene, K.I. (1998) Modeling of erosion and deposition by turbidity currents generated at river mouths. J. Sediment. Res. Sect. A: Sediment. Petrol. Process. 68, 124 –137. Parker, G., Fukushima, Y. & Pantin, H.M. (1986) Selfaccelerating turbidity currents. J. Fluid Mech. 171, 145–181. Piper, D.J.W. & Savoye, B. (1993) Processes of late Quaternary turbidity current flow and deposition on the Var deep-sea fan, north-west Mediterranean Sea. Sedimentology 40, 557–582.

81

Pourahmadi, F. & Humphrey, J.A.C. (1983) Modeling solid–fluid turbulent flows with application to predicting erosive wear. Physico Chem. Hydrodynam. 4, 191–219. Prasad, D. & Kytömaa, H.K. (1995) Particle stress and viscous compaction during shear of dense suspensions. Int. J. Multiphase Flow 21, 775–785. Richardson, J.F. & Zaki, W.N. (1954) Sedimentation and fluidisation: part I. Trans. Inst. Chem. Eng. 32, 35–53. Rizk, M.A. & Elghobashi, S.E. (1985) The motion of a spherical particle suspended in a turbulent flow near a plane wall. Physics Fluids 28, 806 – 817. Rizk, M.A. & Elghobashi, S.E. (1989) A two-equation turbulence model for dispersed dilute confined two-phase flows. Int. J. Multiphase Flow 15, 119 –133. Sato, Y., Hishida, K. & Maeda, M. (1996) Effect of dispersed phase on modification of turbulent flow in a wall jet. J. Fluids Eng. 118, 307–315. Shepard, F.P., McLoughlin, P.A., Marshall, N.F. & Sullivan, G.G. (1977) Current-meter recordings of lowspeed turbidity currents. Geology 5, 297–301. Stacey, M.W. & Bowen, A.J. (1988) The vertical structure of density and turbidity currents: theory and observations. J. Geophys. Res. 93, 3528 –3542. Van Rijn, L.C. (1982) Sediment transport, part I: bed load transport. J. Hydraul. Eng. 110, 1431–1456. Viviand, H. (1974) Formes conservatives des équations de la dynamique des gaz. La recherche Aérospatiale 1, 65– 66. Zang, Y., Street, R.L. & Koseff, J.R. (1994) A non-staggered grid, fractional step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. J. Computat. Physics 114, 18–33.

PGCC05 9/12/08 3:15 PM Page 83


Granular flows in the elastic limit C. S. CAMPBELL Department of Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, USA

ABSTRACT This chapter describes recent computer simulation studies into the rheological behaviour of granular materials in the regime that lies between the quasi-static and rapid-flow regime. This study was a result of studies of landslides, hopper flows and the ‘phase change’ (i.e. the change between solid-like and fluid-like behaviour) all of which indicated that the shear-to-normal stress ratio (the effective friction coefficient for the material) increased with shear rate. The results presented herein account for those observations by demonstrating that the stress ratio varies with a dimensionless parameter created by scaling the shear rate with the stiffness of the interparticle contacts. The results indicate that in dense regimes, the stresses themselves scale with the stiffness indicating that they are generated by the elastic response of particle networks. Such speculation is supported by studies that show that the normal stresses are strongly dependent on the interparticle friction coefficient which affects the ability of internal elastic particle structures to support load. Finally, estimates are made of the regions of particle concentration for which the elasticity of the material is important.

INTRODUCTION However, it eventually became apparent that very few granular flows, especially those under Earth’s gravity, were actually rapid granular flows. The first indication of this came from studies of the phase change between fluid-like and solid-like behaviour of granular materials (Zhang & Campbell, 1992; Campbell & Zhang, 1992; Campbell, 1993) which indicated that this transition could not be described in rapid granular flow ideas but was instead demonstrated a quasi-static yield behaviour. However, later studies on hoppers (Potapov & Campbell, 1996), which are clearly not rapid granular flows, indicated that the general accepted assumption of quasi-static yield could not explain the stress state within the hopper. Finally, large-scale computer simulations of landslides (Campbell et al., 1995) indicated that the ratio of shear-to-normal stress on the base of the slide appeared to vary with the shear rate (i.e. the velocity gradient at the slide’s base) even though both rapid flow and quasi-static flow theories suggest that this ratio should be independent of shear rate. Examinations of both the hopper and phase change data also support the notion that the stress ratio

This chapter represents a portion of my lecture entitled Granular Flows – A Somewhat Personal Perspective. That lecture more or less reviewed the body of my work on granular flows from a historical point of view, ending with a presentation of the previously unpublished work presented in this chapter. It began with a brief description of quasistatic flow theory (see the review by Jackson, 1983), that preceded any of my contributions and then went on to a broad description of very early work that supported the theory of rapid granular flows (see the review by Campbell, 1990 and the references therein) a point of view that a class of granular flows could be analysed using the techniques derived from the kinetic theory of gases. In such systems, transport is controlled by a ‘granular temperature’ which is a measure of the kinetic energy contained in the random motion of the individual particles. Most of the information presented in the lecture as derived from computer simulations (see the reviews in Campbell 1986, 1997b). Early work showed these ideas can account for measurements of heat, mass and momentum transfer (Campbell & Gong, 1986; Campbell, 1989, 1997a; Wang & Campbell, 1992).


83

PGCC05 9/12/08 3:15 PM Page 84

84

C.S. Campbell

increases with the local shear rate. However, in none of these cases was the spatial resolution of the velocity profile adequate to make a quantitative determination of this shear rate required to evaluate this speculation. This led to the current investigations which were designed to quantitatively determine the effect of shear rate on the stresses in dense granular flow and, in particular, on the shear-to-normal stress ratio. Like most of the above, this will be done with computer simulations of systems of 1000 spheres. The simulation will use the soft-particle technique (see Campbell, 1997b) in which the interaction between particles is modelled as a parallel linear spring and dashpot in the direction along the particle centres for as long as they remain in contact. The spring serves to push the particle surfaces apart and the dashpot dissipates the collisional energy. The spring has an associated stiffness k (which will be varied in the following data) while the dashpot coefficient is varied to keep a constant coefficient of restitution (the ratio of recoil to impact velocity for a binary collision in the centre of mass). The Young’s modulus of a bulk material consisting of many such particles are proportional to this stiffness (Bathurst & Rothenburg, 1988). In the direction tangential to the contact point the particles are connected with a frictional slider in parallel with another spring with an associated friction coefficient µ. (i.e. as the particle surfaces move relative to one another in the direction tangential to the contact point, the tangential spring will load until the tangential force reaches µ times the normal force, at which point the surfaces slip relative to one another against a force equal to µ times the normal force). The particles are confined in a control volume bounded in all directions by periodic boundaries. This means that as a particle passes through one periodic boundary, it re-enters from the opposite side with exactly the same position and velocity with which it left; this simulates a situation where the control volume and every particle within it are periodically repeated infinitely many times up and downstream, so that as a particle passes out the downstream boundary, it passes into the downstream periodic image and is replaced by a particle entering from the upstream periodic image. These will be rheological studies, and it is necessary to induce a uniform shear within the control volume. To do this, the periodic images above and below the control volume in the y-direction are set in motion with fixed velocity in the manner originally used by Lees and Edwards (1976).

RESULTS Now the hopper, landslide and phase change simulations referred to above, all indicate that the apparent friction coefficient τxy /τyy, increases with the shear rate γ. At the outset, this presents a dimensional problem as γ has units of (time)−1, while the stress ratio τxy /τyy is dimensionless and can thus only be a function of dimensionless parameters. Now, the inverse shear rate represents the only time-scale in rapid granular flow theory while no time-scale appears in quasi-static theories. Thus some other parameter must be introduced into the analysis which contains units of time by which the shear rate can be scaled. The only possible candidates comes from the particle contact model and as the particle surface friction and the coefficient of restitution are dimensionless, the only remaining possibility is the contact stiffness, k. This suggests a dimensionless parameter of the form: k ρd 3 γ 2 where ρ is the particle density and d is the particle diameter. Note that this is similar to the parameter B studied by Babic et al. (1990). Other parameters were explored, e.g. the shear scaled by the binary collision time, but there was no apparent advantage over the parameter proposed above. The first test of whether this is an appropriate parameter is to examine its effect on the stress ratio τxy /τyy. Figure 1 shows a plot of τxy /τyy vs. k/ρd 3 γ 2 at a constant solid concentration, ν = 0.5 (i.e. 60% of a unit volume is solid material while the rest is void) particle surface friction coefficient µ = 0.6, for three different coefficients of restitution, ε = 0.1, 0.7 and 1.0. A relatively large concentration ν = 0.6 is used because all of the cases discussed above, phase-change, hoppers and landslides operate near the shearable limit, (about ν = 0.62). Checks on this parameter by working with different diameters, shear rates and stiffnesses that yielded the same k/ρd 3 γ 2, showed that the resultant stress ratios overlapped exactly which indicates that no other parameter makes an appearance in this problem. Now, from the form of k/ρd 3 γ 2 it can be seen that increasing the stiffness k, moves one from left to right on the figure while increasing the shear rate, γ, moves one from right to left. One can see that for the lower values of k/ρd 3 γ 2, the values of the stress ratio τxy /τyy, drop with increasing k/ρd 3 γ 2 (i.e. with increasing k or with decreasing, γ). For large values of k/ρd 3 γ 2

PGCC05 9/12/08 3:15 PM Page 85

Elastic granular flows

85

Fig. 1. The ratio of shear to normal stress, τxy /τyy , as a function of the parameter, k/ρd 3 γ 2. All of this data was taken from 1000 particle computer simulations at a constant solid concentration, ν = 0.6 and particle surface friction, µ = 0.5 for three different values of the coefficient of restitution, ε. Note that τxy /τyy decreases with k/ρd 3 γ 2 (i.e. increasing k, decreasing γ) eventually approaching a constant value indicative of quasi-static behaviour.

(i.e. large k, small γ) the stress ratio becomes constant which is indicative of quasi-static behaviour. Note that the coefficient of restitution ε, is only important at small k/ρd 3 γ 2 which might be expected as this corresponds to conditions of large shear rate, γ, making the flows more rapid. Note that while the effect of γ fits the basic understanding of quasistatic (small γ) and rapid-flow (large γ) behaviour, the effect of the contact stiffness, k is somewhat contradictory. Large k implies a shorter

Fig. 2. The shear stress scaled by the particle stiffness, τxyd /k for the data plotted in Fig. 1. Note that the data scales with the stiffness indicating that the stresses are generated by the elasticity of the material.

binary contact time which approaches the instantaneous contact time assumption implicit in rapid-flow theories, yet increasing k (leading to large k/ρd 3 γ 2) corresponds to more quasi-static behaviour. Figures 2 and 3 show the stresses scaled by the stiffness, τxyd/k and τxyd/k. These data were generated for two different values of the stiffness k although this is not apparent because the values overlay one another almost identically. Notice that both quantities decrease with k/ρd 3 γ 2, becoming constant at large

PGCC05 9/12/08 3:15 PM Page 86

86

C.S. Campbell

Fig. 3. The normal stress scaled by the particle stiffness, τxyd/k for the data plotted in Fig. 1. Note that, again, the data scales with the stiffness indicating that the stresses are generated by the elasticity of the material.

Fig. 4. The effect of particle surface friction on the stress ratio τxy /τyy, all from 1000 sphere simulations at a concentration, ν = 0.6. Note that while τ xy /τyy varies strongly with µ, there is no direct relationship between the two.

k/ρd 3 γ 2, just as for the stress ratio τxy /τyy. Note also that both quantities vary with the coefficient of restitution ε, in a manner qualitatively similar to the stress ratio although the shear stress τxy, demonstrates a stronger effect of ε than does τyy. The effect of the particle surface friction is shown in Figs 4 and 5. For a rapid granular flow, the surface friction largely affects only the energy dissipation; so that the larger the surface friction, the larger the energy dissipation and, consequently, the smaller the

granular temperature and all of the associated transport rates. But overall, this effect is weak. However, Fig. 4 shows a relatively strong effect of the surface friction on the stress ratio τxy /τyy. As expected, τxy /τyy increases with µ. But notice that while there is a significant change in τxy /τyy by increasing µ from 0.1 to 0.5, there is a relatively minor change by going from µ = 0.5 to µ = 1.0, that data for which nearly overlap in the ‘rapid’ region (small k/ρd 3 γ 2). (Note: curve fitting shows that all the lines will collapse

PGCC05 9/12/08 3:15 PM Page 87

Elastic granular flows

87

Fig. 5. The effect of particle surface friction on the scaled normal stress τxyd/k for the data presented in Fig. 4. Surprisingly, there is a very strong effect of µ on the normal stress. In fact the normal stress almost disappears in the quasi-static limit (large k/ρd 3 γ 2). This indicates that the stress is supported by elastic particle networks whose strength depends on their structural integrity which is strongly affected by µ.

together by plotting (τxy /τyy)µ−0.085 as a function of k/ρd 3 γ 2.) Much more interesting is the effect on the normal stress τxyd/k plotted in Fig. 5. In particular, in the quasistatic limit, (large k/ρd 3 γ 2,) the low friction, µ = 0.1 shows almost no generated normal stress (in actuality these stresses are about 40 times smaller than those for µ = 0.5 or 1.0). This indicates the ultimate source of the stresses in this elastic limit may well be the structures that form within a static granular material (see for example, Drescher & De Josselin de Jong, 1972 and Cundall & Strack, 1979). The strength of the structures will be strongly affected by the interparticle friction. For the small friction case, these structures are weak and can only support very weak forces before the structure fails. Now the ν = 0.6 concentration used for all of these examples, is smaller than the concentration of a random close pack of uniformly sized spherical particles. As a result, it is possible that particles need not be in intimate contact and thus may not be able to support an applied normal stress. This appears to be what is happening for the µ = 0.1 cases shown in Fig. 5, i.e. at large values of k/ρd 3 γ 2, elastic structures do not form and little normal force can be supported. Normal forces can be supported for small k/ρd 3 γ 2 (the ‘rapid’ limit) as the large shear rates drive particles together apparently forming internal elastic networks. The effect shown here cannot be simply related to the friction coefficient, µ. Unlike Fig. 4, it is not possible

to collapse these curves together by plotting anything of the form τxyd/k f (µ) as a function of k/ρd 3 γ 2; this can be easily seen as, while the stresses for µ = 0.1 are about 40 times smaller than the µ = 1.0 stresses a k/ρd 3 γ 2 = 106, they are only about 1.8 times smaller at k/ρd 3 γ 2 = 100. The fact that the shear rate affects the stresses at all within these ‘elastic’ flows, indicates an inertial effect. This may seem somewhat confusing because the elasticity of a material depends on deformation and not on deformation rate and from that point of view should be shear rate independent. However, from a more basic point of view, this effect should be anticipated. Simply put, the larger the shear rate, the larger the particle impact velocities, the larger the particle deformation and the larger the generated elastic forces. Finally, one might suspect that one could use the elastic scalings τd/k, to determine the regions where the elastic region begins and the rapid-flow region ends. In other words, if the material scales as τd/k, the material should be operating in the elastic limit, and if it scales with the rapid-flow scaling, τ /ρd 2γ 2, it is obviously in the rapid-flow limit. However, one cannot make that judgement because in order to compare the behaviour under the same circumstances, the simulation should be run at constant k/ρd 3 γ 2, and at constant k/ρd 2γ 2, τ/ρd 2 γ 2 = τd/k · k/ρd 3 γ 2, i.e. the two scalings are simply proportional to each other. However, a rapid flow should be relatively

PGCC05 9/12/08 3:15 PM Page 88

88

C.S. Campbell

Fig. 6. Normal stress data presented in the rapid granular flow scaling τ/ρd 2 γ 2 as a function of the solid concentration, ν for values of the elastic parameter k/ρd 3 γ 2 covering the range 100 < k/ρd 3 γ 2 < 107. True rapid flows should be relatively independent of k/ρd 3 γ 2 and a significant vertical spread in the data is indicative that the material is operating in the elastic regime. The above data indicate that elastic effects begin to be significant for concentrations of ν = 0.5 and above.

independent of k/ρd 3 γ 2. (Hwang and Hutter, 1995, showed that there will still be a small effect of elasticity due to the effect of the binary collision time on the transport rates.) Figure 6 shows a plot of τxy /ρd 2 γ 2 as a function of the solid concentration ν, for a wide range of k/ρd 3 γ 2. A vertical spread in the data indicates a strong effect of k/ρd 3 γ 2 and indicates that the material is operating in the elastic regime. At small ν (the region smaller than ν = 0.5) there is very little effect of k/ρd 3 γ 2 on the stresses. Beyond that region, there is a strong effect of k/ρd 3 γ 2 indicating that such flows operate in an elastically dominated region.

increase with the shear rate. In that region, the stresses scale with the interparticle stiffness indicating that the forces are supported by networks of particles in intimate contact. The fact that the stresses vary with the shear rate at all appears to be a first-order inertial effect, i.e. that the larger the shear rate, the larger the impact velocities, the larger the particle deformation and the larger the generated elastic forces. Finally, the results can be used to estimate the limits of ‘rapid’ granular flow theory in that, the stresses appear to be relatively independent of the elasticity for concentrations smaller than ν = 0.5, and strongly dependent on k/ρd 3 γ 2 for larger concentrations.

CONCLUSIONS NOMENCLATURE This chapter starts to investigate the intermediate range of granular flow that lies between the rapid flow (which dominates at large shear rates and small solid concentrations) and the quasi-static limit which dominates at large concentrations and small shear rates. Previous studies of landslides, hopper flows, and of the ‘phase-change’ between solid and fluid behaviour indicate that the stress ratio τxy /τyy increased with the shear rate γ. Dimensional analysis provided an appropriate dimensionless parameter, k/ρd 3 γ 2 that scales the shear rate with the interparticle stiffness, k. The computer simulations shown in this chapter were designed to generate uniform shear rates and were thus designed to study this behaviour in a rheological context. The simulations demonstrated that the stress ratio τxy /τyy decreases with k/ρd 3 γ 2 and thus, does indeed

d k γ ε µ ν ρ τxy, τyy

particle diameter interparticle contact stiffness shear rate coefficient of restitution particle surface friction coefficient solid fraction particle density shear and normal stresses

REFERENCES Babic, M., Shen, H.H. & Shen, H.T. (1990) The stress tensor in granular shear flows of uniform. deformable disks at high solids concentrations. J. Fluid Mech. 219, 81–118.

PGCC05 9/12/08 3:15 PM Page 89

Elastic granular flows Bathurst, R.J. & Rothenburg, L. (1988) Micromechanical aspects of isotropic granular assemblies with linear contact interactions. J. Appl. Mech. 55, 17. Campbell, C.S. (1986) Computer simulation of rapid granular flows. Proceedings of the 10th National Congress of Applied Mechanics, Austin, Texas, June 1986, pp. 327–338. ASME, New York. Campbell, C.S. (1989) The stress tensor for simple shear flows of a granular material. J. Fluid Mech. 203, 449– 473. Campbell, C.S. (1990) Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57–92. Campbell, C.S. (1993) The transition from fluid-like to solidlike behavior in granular flows, Powders and Grains 1993: 2nd International Conference on Micromechanics of Granular Media (ed. C. Thornton) pp. 289 –294. Campbell, C.S. (1997a) Self-diffusion in granular shear flows. J. Fluid Mech. 348, 85–101. Campbell, C.S. (1997b) Computer simulation of powder flows. Powder Technology Handbook, 2nd edn. (Eds Gotoh, K., Masuda H. & Higashitani K.) Marcell Dekker, New York, pp. 777–794. Campbell, C.S. & Gong, A. (1986) The stress tensor in a two-dimensional granular shear flow. J. Fluid Mech. 164, 107–125. Campbell, C.S., Cleary, P. & Hopkins, M.A. (1995) Large landslide simulations: global deformation, velocities and basal friction. J. geophysical Res. 100, 8267–8283.

89

Campbell, C.S. & Zhang, Y. (1992) Interfaces between fluidlike and solid-like behavior in granular flows. In: Advances in Micromechanics of Granular Materials. Proceedings of the Second U.S./Japan Seminar on the Micromechanics of Granular Materials, Potsdam, New York, August 5–9, 1991 (Eds Shen, H.H., Satake, M., Mehrabadi, M., Chang, C.S. & Campbell, C.S.), Elsevier, Amsterdam, pp. 261–270. Cundall, P.A. & Strack, O.D.L. (1979) A discrete numerical model for granular assemblies, Geotechnique 29, 47–65. Drescher, A. & De Josselin de Jong, G. (1992) Photoelastic verification of a mechanical model for the flow of a granular material. J. Mech. Phys. Solids 20, 337–351. Hwang, H. & Hutter, K. (1995) A new kinetic model for rapid granular flow. Cont. Mech. Therm. 7, 357–384. Jackson, R. (1983) Some mathematical and physical aspects of continuum models for the motion of granular materials. In: Theory of Dispersed Multiphase Flow (Ed. Meyer, R.E.), Academic Press, New York. pp. 291–337. Lees, A.W. and Edwards, S.F. (1976) The computer study of transport processes under extreme conditions, J. Phys. C: Solid State Phys. 5, 1921–1929. Potapov, A.V. & Campbell, C.S. (1996) Computer simulation of hopper flows. Physics Fluids A, 8, 2884 –2894. Wang, D.G. & Campbell, C.S. (1992) Reynolds’ analogy for a shearing granular material. J. Fluid Mech. 244, 527–546. Zhang, Y. & Campbell, C.S. (1992) The interface between fluid-like and solid-like behavior in two-dimensional granular flows. J. Fluid Mech. 237, 541–568.

PGCC06 9/12/08 3:15 PM Page 91


Bagnold revisited: implications for the rapid motion of high-concentration sediment flows S. STRAUB* Center for Nonlinear Studies and Geoanalysis Group, MS B258, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

ABSTRACT This chapter reviews the work of Bagnold (1954) on the grain-inertia regime, and its application to sediment mass flows in the geological literature. The review is based on new computer experiments using a discrete particle model which can simulate Bagnold’s Couette flow experiments as well as open flows of granular material. The experiments show that there are no significant differences in stress development between the confined flow situation of Bagnold’s laboratory experiments and unconfined granular flows, i.e. the basic mechanical behaviour is identical. The assumptions and conclusions of Lowe (1976), whose quantitative considerations on the relevance of the grain-inertia regime had enormous impact on the appraisal of sediment mass flows, are discussed with regard to the new experimental results and today’s better knowledge of granular physics. This leads to new conclusions which deviate seriously from the old concepts of high-concentration mass flows: there is evidence that grain-inertia flows have a distinct internal flow structure, they have low angles of dynamic friction (approx. 12°–17°), that gives them great mobility, and they can be much thicker than was estimated by Lowe (1976).

INTRODUCTION (Sanders, 1965; Stauffer, 1967; Middleton, 1969; Middleton & Hampton, 1973; Hsü, 1975, 1978, 1989; Lowe, 1976; McTigue, 1979; Takahashi, 1980; Vallejo, 1980). Altogether, Bagnold’s work had much influence on the way high-concentration sediment flows and their deposits were discussed in the geological literature. Unfortunately the relations between his experimental results and the motion of natural granular flows have never been explicitly investigated. Instead, assumptions have been made on how Bagnold’s results can be utilized to understand natural sediment flow behaviour. The aim of this chapter is to close the gap between our knowledge on the grain-inertia regime in laboratory test devices and natural rapid flows of granular material. The appropriate tool for this study is a discrete particle model which can be used to set up different granular flow simulations. One set of experiments performed for this study re-investigates Bagnold’s work to provide a better understanding of

Rapid shearing of cohesionless granular solids produces intergranular stresses which are proportional (i) to the square of the shear rate, (ii) to the square of the particle diameter, (iii) to the density of the particles and (iv) to the particle concentration. It was Bagnold (1954) who first discovered the significance of this mechanical behaviour, which is distinct from granular flow mechanics at lower shear rates. He called this the grain-inertia regime. From the results of his laboratory experiments and simple physical arguments he inferred semi-empirical constitutive equations which can predict the stresses in a sheared fluid–solid mixture for a given set of parameters. He also was the first to use his results to explain sedimentary flow phenomena (Bagnold, 1954, 1956, 1966, 1973). Other authors recognized the potential of Bagnold’s results and incorporated them into their line of argument * Present address: H.A.N.D. ATC Systems Development, Siemensstrasse 11, 65205 Wiesbaden, Germany.


91

PGCC06 9/12/08 3:15 PM Page 92

92

S. Straub

his semi-empirical equations. In the second part of the study, unconfined granular flows are investigated to compare Bagnold’s experiments to a setup which resembles more closely a natural sediment flow.

A REVIEW OF GRAIN-INERTIA FLOW Bagnold’s experiments Bagnold (1954) studied the mechanical behaviour of identical, neutrally buoyant, spherical particles (made of a mixture of paraffin wax and lead stearate) in a ‘Newtonian’ fluid (water or water–alcohol mixture)

(a)

under shear. In his coaxial cylinder rheometer (Fig. 1a) the dispersion was contained between an outer rotating cylinder, exerting tangential stress on the dispersion, and a fixed inner drum with a torque spring to measure the tangential stress component conducted through the dispersion. The inner drum, filled with water and covered with a rubber membrane, was connected to a manometer to measure pressure changes developing in the sheared mixture. This so-called ‘Couette flow’ experiment is a standard setup to study the mechanical behaviour of fluids under constant shear conditions, which was applied for the first time to investigate granular dispersions. Bagnold recorded the mean normal pressure PZ and

Manometer Torque spring

Rotating outer drum Fixed inner drum

Wax particles Rubber membrane

Rotation axis

(b)

bD

D

s

Layer B

δU

Layer A U(y) z

x

(c) ui

α

δU

(d)

∆M α ∆M = ∆M cos α z ∆Mx = ∆M sin α

Fig. 1. (a) Sketch of Bagnold’s coaxial cylinder rheometer. (b) Bagnold’s theoretical approach: well-ordered layers of equidistant particles are sheared with a constant shear rate; (c) particles of adjacent layers collide with the relative collision velocity δU; (d) the resultant momentum change is ∆M, which corresponds to the impact velocity ui, with its normal component ∆MZ and its tangential component ∆MX.

PGCC06 9/12/08 3:15 PM Page 93

93

Bagnold revisited the mean tangential stress TXZ acting on the inner drum for different particle concentrations and different rotation speeds of the outer cylinder. Depending on the concentration and the shear rate he found that different regimes develop in the dispersion: the macroviscous, the transitional, and the grain-inertia regime. At the low shear rates of the macroviscous regime PZ and TXZ appeared to be a linear function of the shear rate dU/dz (see Jenkins & McTigue, 1990). The flow is dominated by the fluid’s viscosity and the suspended solids simply increase the viscosity of the Newtonian fluid. With increasing shear rates and for solid concentrations > 9% the dispersion passes through a transitional regime into the grain-inertia regime, where PZ and TXZ are proportional to the square of the shear rate. Here the stress propagation depends predominantly on the momentum transfer by particle–particle collisions and the viscosity of the interstitial fluid has lost its influence. This grain-inertia regime is studied here in detail, because this is the regime one expects in high-concentration rapid sediment flows where the particle concentrations are high and extreme shear rates are present. Bagnold’s analytical approach Bagnold’s (1954) model for collisional stress in the grain-inertia regime is based on the assumption of a simple rectilinear shear flow geometry. In his approach he replaced the axisymmetric coordinates of the coaxial cylinder rheometer by Cartesian coordinates, which is an adequate approximation since the curvatures considered are sufficiently small. The grains in this flow are arranged in distinct, well-ordered layers, each layer moving with its local mean velocity, due to a uniform shear strain dU/dz between the moving outer cylinder and the inner drum (Fig. 1b). From simple physical arguments, which follow below, the mean normal pressure PZ arising from the collision of particles in adjacent layers should be proportional to the average momentum change 〈∆M 〉 during a collision, the average collision frequency 〈k〉 for a particle, and the number of particles N per area in a layer: PZ = 〈∆M 〉 〈k〉N.

(1)

The particles in this model are supposed to be rigid elastic spheres of uniform diameter D, density ρ, and mass m. They are assumed to be evenly spaced with a distance bD from particle centre to particle centre. If the resulting particle separation is s, we have bD = s + D

(2)

or b=

s 1 + 1 = + 1, λ D

(3)

where λ = D/s represents the ‘linear concentration’, which is Bagnold’s method of describing concentrations in his model. The solid fraction ν ν=

VP , VT

(4)

which is the relation of the volume of all particles VP to the total volume VT, can also be expressed by the linear concentration ν=

ν0 ν0 . = 3 b3 ⎛ 1 ⎞ ⎜ + 1⎟ ⎝λ ⎠

(5)

ν0 is the maximum solid fraction when s = 0 (λ = ∞), which in a three-dimensional packing of uniform spheres is c. 0.65. In a rapidly sheared dispersion each layer A moving with speed u is overtaken by the adjacent layer B moving with speed u + δU (Fig. 1b). Every particle of layer B colliding with a particle of layer A has therefore a relative collision velocity δU in the direction of the flow (Fig. 1c). Thus, the collision results in a momentum change of the order of ∆MZ normal to the stress plane (Fig. 1d) ∆MZ = mδU cos α.

(6)

The collision frequency k of a particle in layer B encountering particles in layer A can be estimated from the relative mean velocity δU and the average particle separation s as k = f( λ )

δU , s

(7)

where f(λ) is an unknown function of the linear concentration which takes into account that for large λ, i.e. for densely packed flows, geometrical constraints should have an influence on the collision frequency. For low linear concentrations (λ 0.3–0.4 of some of the sediment mass flows. These flows are predominantly small volume flows, i.e. they have volumes typically smaller than 10−2–10−3 km3. Heim (1932) observed that volumes smaller than 10−3 km3 do not behave as coherent flows, instead they move as individual rocks saltating down steep hillsides. Hence, there is a critical volume for the coherent flow of a rock avalanche. Below this critical volume the avalanche moves as a rock fall rather than a flow. The actual limiting factor behind this critical volume is the load acting on the basal shear zone, which is counteracted by the dispersive pressure. In Couette flows the dispersive pressure increases with increasing shear rate (see Eq. 22), in open-channel flows the shear zone expands until the dispersive pressure equates the confining load. For small loads and rapid shear, however, the dispersive pressure exceeds the confining load and the coherent flow disintegrates into individual rocks: it transforms into a rock fall (Straub, 1994). It is likely that most of the small volume flows in Fig. 13 are governed by a flow regime at the limits of rapid granular flow, which is characterized by significantly higher coefficients of internal friction. Besides, for

PGCC06 9/12/08 3:15 PM Page 107

107

Bagnold revisited some prehistoric deposits in Fig. 13 the flow velocity is unclear (see Shaller, 1991) and it is possible that this figure contains data from the deposits of some slow sediment mass flows. These flows move with an internal friction near the static angle of friction and their mechanical behaviour cannot be compared to rapid granular flows. It is not the intention of this paper to compare experimental data and rock avalanche data quantitatively. This task is unsatisfying because the commonly-applied, oversimplified approach introduced by Heim (1932) produces impressive data, nonetheless unsuitable for a physical analysis. The misinterpretation of Bagnold’s (1954) experimental data, on the other hand, has led to the misconception that pure grain-inertia flows are not very mobile (Bagnold, 1966; Lowe, 1976). Consequently, there was only one conclusion to draw: it is impossible that large rapid sediment mass flows move as grain-inertia flows. It is the intention of this paper to prove that this conclusion is wrong. It has been demonstrated here that (i) Couette flows and open-channel flows in the rapid granular flow regime show similar mechanical behaviour and (ii) this applies very well to rapid natural sediment mass flows. The exact values of the dynamic coefficient of friction µdyn determined in the computer experiments are of minor importance, for example the coefficient of restitution ε could have been adjusted to better fit Bagnold’s results. However, µdyn is always significantly lower than the angle of repose and this is a dynamic property of the flow, not of the granular material (Straub, 1996). There are some natural flow phenomena, e.g. lahars, pyroclastic flows, meteor impact structures, brittle deformation along faults, to which the above discussion can be extended. Consequently, all multiphase flows with high solid concentrations have to be reconsidered. A brief discussion of the rapid granular flow regime in lahars can be found in Straub (2000). Another example is the motion of high-concentration pyroclastic flows, which has been controversially discussed by various authors (see references in Druitt, 1998). The significance of the rapid granular flow regime for high-concentration pyroclastic flows is demonstrated by Straub (1994, 2000) and Straub and Valentine (1998).

concentration mass flows. The modern computational methods presented here permit the reappraisal of old concepts of sediment mass flows. It is clear that some of these concepts have to be revised: the grain-inertia regime is probably one of the most important mechanisms in the motion of sediment mass flows. It is the central mechanism in grain flows, which are much more mobile than it is usually assumed, and it must contribute in every highly sheared concentrated sediment flow, such as fluidized and matrix-supported flows.

ACKNOWLEDGEMENTS I thank G.A. Valentine and R. Mikulla for helpful comments and discussions. Constructive reviews by H.M. Pantin and C.S. Campbell improved this manuscript. The computations for this study were made possible by using the Avalon parallel-computing workstation cluster of the Center for Nonlinear Studies, Los Alamos National Laboratories.

NOMENCLATURE αint αapp αdyn β ε fij fXZ fZZ γ λ µint µapp µdyn ν, ν0

CONCLUSIONS This study reviewed the work of Bagnold (1954) and its significance for the sedimentology of high-

ρ τc, τc XZ, τc ZZ

angle of internal friction (in general) apparent angle of friction defined by Heim (1932) angle of dynamic friction = angle of internal friction of a rapid granular flow slope angle coefficient of restitution stress function shear component of the stress function normal component stress function shear rate linear concentration coefficient of internal friction (in general) apparent coefficient of friction defined by Heim (1932) coefficient of dynamic friction = coefficient of internal friction of a rapid granular flow solid fraction, solid fraction of the maximum packing particle density collisional stress, shear collisional stress, normal collisional stress

PGCC06 9/12/08 3:15 PM Page 108

108 τij τs, τs XZ, τsZZ τ XZ τ ZZ ai D E e g H hw k L m mz N PZ PZZ p R T TXZ U u, v, w u′, v′, w′ ui V VP VT 〈〉 〈H 〉 〈L〉 〈∆M〉 〈k〉 〈u〉

S. Straub stress tensor streaming stress, shear streaming stress, normal streaming stress shear stress normal stress

〈z〉

empirical parameter in Bagnold’s (1954) model particle diameter kinetic energy of the relative velocities of two colliding particles unit vector gravitational acceleration total drop height in Heim’s (1932) model distance between upper and lower wall in the Couette experiment collision frequency total horizontal travel distance in Heim’s (1932) model particle mass mass above a plane in depth z in a sediment flow number of particles mean normal pressure in Bagnold’s (1954) model normal load momentum during a particle–particle collision particle radius granular temperature defined by Ogawa (1978) mean tangential stress in Bagnold’s (1954) model velocity of the upper wall in the Couette flow experiment velocity in x-, y-, z-direction deviatoric velocity in x-, y-, z-direction impact velocity volume particle volume total volume

Bagnold, R.A. (1954) Experiments on a gravity-free dispersion of large solid spheres in a Newton fluid under shear. Proc. R. S. London, Ser. A 225, 49–63. Bagnold, R.A. (1956) The flow of cohesionless grains in fluids. Proc. R. S. London, Ser. A 249, 235–297. Bagnold, R.A. (1966) The shearing and dilatation of dry sand and the ‘singing’ mechanism. Proc. R. S. London, Ser. A 295, 219–232. Bagnold, R.A. (1973) The nature of saltation and of ‘bedload’ transport in water. Proc. R. S. London, Ser. A 332, 473–504. Campbell, C.S. (1982) Shear Flows of Granular Materials. Ph.D. thesis. CalTech, Pasadena, California. Campbell, C.S. (1986) The effect of microstructure development on the collisional stress tensor in a granular flow. Acta Mech. 63, 61–72. Campbell, C.S. (1989) The stress tensor for simple shear flows of a granular material. J. Fluid Mech. 203, 449– 473. Campbell, C.S. (1990) Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57–92. Campbell, C.S. (1993a) Boundary interactions for twodimensional granular flows, part I: flat boundaries asymmetric stresses and couple stresses. J. Fluid Mech. 247, 111–136. Campbell, C.S. (1993b) Boundary interactions for twodimensional granular flows, part II: roughened boundaries. J. Fluid Mech. 247, 137–156. Campbell, C.S. & Brennen, C.E. (1985a) Computer simulation of granular shear flows. J. Fluid Mech. 151, 167–188. Campbell, C.S. & Brennen, C.E. (1985b) Chute flows of granular material: some computer simulations. J. Appl. Mech. 52, 172–178. Campbell, C.S. & Gong, A. (1986) The stress tensor in a twodimensional granular shear flow. J. Fluid Mech. 164, 107–125. Campbell, C.S. & Gong, A. (1987) Boundary conditions for two-dimensional granular flows. Proc. Sino–U.S. Int. Symp. Multiphase Flows Hangzhou China 1, 278–283. Druitt, T.H. (1998) Pyroclastic density currents. In: The Physics of Explosive Volcanic Eruptions (Eds Gilbert, J. & Sparks, R.S.J.). Geol. Soc. Lond. Spec. Publ. 145, 145–182. Heim, A. (1932) Bergsturz und Menschenleben. Fretz und Wasmuth, Zürich. Howard, K. (1973) Avalanche mode of motion: implications from lunar examples. Science 180, 1052–1055. Hsü, K.J. (1975) Catastrophic debris streams (sturzstroms) generated by rockfalls. Geol. Soc. Am. Bull. 86, 129–140. Hsü, K.J. (1978) Albert Heim: Observations on landslides and relevance to modern interpretations. In: Rockslides and Avalanches. 1. Natural Phenomena (Eds Voight, B. & Pariseau, W.G.). Elsevier, Amsterdam, pp. 71–93. Hsü, K.J. (1989) Physical Principles of Sedimentology. Springer, Berlin. Jaeger, H.M. & Naegel, S.R. (1992) Physics of the granular state. Science 255, 1523–1531. Jenkins, J.T. & McTigue, D.F. (1990) Transport processes in concentrated suspensions: The role of particle fluctua-

average over all particles drop height of the centre of mass travel distance of the centre of mass average momentum change average collision frequency average flow velocity = velocity of the centre of mass

height of the centre of mass in a granular flow

REFERENCES

PGCC06 9/12/08 3:15 PM Page 109

Bagnold revisited tions. In: Two Phase Flows and Waves (Eds Joseph, D.D. & Schaeffer, D.G.). Springer, New York, pp. 70–79. Jenkins, J.T. & Savage, S.B. (1983) A theory for the rapid flow of identical smooth nearly elastic particles. J. Fluid Mech. 130, 187–202. Lowe, D.R. (1976) Grain flow and grain flow deposits. J. sediment. Petrol. 46, 188–199. Lucchitta, B.K. (1979) Landslides in Valles Marineris Mars. J. geophys. Res. 84, 8097–8113. Lun, C.K.K., Savage, S.B., Jeffrey, D.J. & Chepurniy, N. (1984) Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256. Lun, C.K.K. & Savage, S.B. (1985) The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials. Acta Mechan. 63, 15–44. McEwen, A.S. (1989) Mobility of large rock avalanches: Evidence from Valles Marineris, Mars. Geology 17, 1111–1114. McTigue, D.F. (1979) A Nonlinear Continuum Model for Flowing Granular Materials. Ph.D. thesis, Stanford University, California. Middleton, G.V. (1969) Grain flows and other mass movements down slopes. In: The New Concepts of Continental Margin Sedimentation (Ed. Stanley, D.J.). AGI Short Course Lecture Notes Philadelphia, USGS Professional Paper. gmb1–gmb14. Middleton, G.V. & Hampton, M.A. (1973) Sediment gravity flows: mechanics of flow and deposition. In: Turbidites and Deep-Water Sedimentation (Eds Middleton, G.V. & Bouma, A.H.). SEPM Pac. Sec. Short Course Notes 1–38. Nemec, W. (1990) Aspects of sediment movement on steep delta slopes. Spec. Publs int. Ass. Sediment. 10, 29–73. Ogawa, S. (1978) Multitemperature theory of granular materials. In: Proceedings of US–Japan Symposium on Continuum Mechanics and Statistical Approaches to the Mechanics of Granular Media (Eds Cowin, S. & Satake, M.), Gakujutsu Bunken Fukyu-kai, Tokyo, pp. 208–217. Sanders, J.E. (1965) Primary sedimentary structures formed by turbidity currents and related resedimentation mechanisms. In: Primary Sedimentary Structures and their Hydrodynamic Interpretation (Ed Middleton, G.V.). SEPM Spec. Pub. 12, 192–219.

109

Scheidegger, A.E. (1973) On the prediction of the release and velocity of catastrophic rockfalls. Rock Mechan. 5, 231–236. Scheller, E. (1971) Beitrag zum Bewegungsverhalten grosser Bergstürze. Eclogae Geol. Helvetiae 64, 195–202. Shaller, P.J. (1991) Analysis and Implications of Large Martian and Terrestrial Landslides. Ph.D. thesis, CalTech, Pasadena, California. Shanmugam, G. (1996) High-density turbidity currents: are they sandy debris flows? J. sediment. Res. 66(1), 2–10. Shen, H.H. & Ackermann, N.L. (1984) Constitutive equations for a simple shear flow of a disk shaped granular material. Int. J. Eng. Sci. 22, 829–843. Siebert, L. (1984) Large volcanic debris avalanches: Characteristics of source area deposits and associated eruptions. J. Volcanol. geothermal Res. 22, 163–197. Staufer, P.H. (1967) Grain-flow deposits and their implications Santa Ynez Mountains California. J. sediment. Petrol. 37, 487–508. Straub, S. (1994) Rapid Granular Flow in Subaerial Pyroclastic Flows. Doctoral thesis, University of Würzburg, Germany. (In German.) Straub, S. (1996) Self-organization in the rapid flow of granular material: evidence for a major flow mechanism. Geolog. Rundschau 85, 85–91. Straub, S. (1997) Predictability of long runout landslide motion: implications from granular flow mechanics. Geolog. Rundschau 86, 415– 425. Straub, S. (2000) Discrete particle modeling of volcanogenic high-concentration mass flows. In: Capriclous Earth: Models and Modeling of Geologic Processes and Objects (Eds Glebovitsky, V.A. & Dech, V.N.). Theophrastus publications. St. Petersburg, pp. 123–136. Straub, S. & Valentine, G.A. (1998) 3-d modelling of rapid granular flows and its application to high-concentration pyroclastic flows. In: Particulate Gravity Currents Conference (Ed. Kneller, B.). University of Leeds, Leeds. Takahashi, T. (1980) Debris flow in prismatic open channel. J. Hydraul. Div. Am. Soc. Civ. Eng. No. HY3 Proc. Paper 15245, 381–396. Vallejo, L.E. (1980) Mechanics of mudflow mobilization in low-angled clay slopes. Eng. Geol. 16, 63–67.

PGCC07 9/12/08 3:15 PM Page 111

Combined theoretical and experimental approaches


PGCC07 9/12/08 3:15 PM Page 113


Downslope flows into rotating and stratified environments P. G. BAINES CSIRO Atmospheric Research PMB 1, Aspendale 3195, Australia

ABSTRACT Downslope flows in the deep ocean that contain or are driven by sediment are affected by two external factors: the Earth’s rotation and the oceanic ambient density stratification. Here, various experiments that model the effects of these factors on downslope gravity currents are described. If the environmental fluid is stratified, mixing at the upper part of the current causes detrainment of fluid from the current as well as entrainment into it, with the latter process being the weaker of the two. Consequently, the current loses fluid and the nett volume flux of the downflow decreases with increasing distance downslope. With rotation, the Coriolis force deflects the main downflow alongslope to the right (left) in the Northern (Southern) hemisphere. More direct downslope flow occurs within a thin Ekman layer close to the boundary, where friction reduces the effect of the Coriolis force. Eddies may form in the alongslope current due to vortex stretching, and three possible mechanisms for eddy formation have been identified. These processes are relevant to deep downslope flows in the Arctic ocean, in the Greenland Sea, and around Antarctica, some of which drive the deep ‘thermohaline’ ocean circulation and ventilate the deep water. There is some evidence of sedimentary fans at the foot of many of these deep downflow regions, so that sediment transport in these currents is significant for geology, but whether it is also important for ocean dynamics is uncertain.

INTRODUCTION dense AABW moves northward along the bottom of the deep ocean basins and trenches, where it slowly mixes with the overlying water, to northern latitudes. By continuity, there is a corresponding equal southward flow at shallower depths that returns to the surface at high latitudes in the Southern Ocean. The extent to which sediment transport by these deep ocean downflows is significant or important, either as a mover of sediment or as a dynamic effect on the oceanic stratification, is generally, as yet, unknown. However, there is evidence that it is important in some specific regions such as in the Sulu Sea (Quadfasel et al., 1990) where it affects the deep stratification, in the Greenland Sea and Arctic Ocean (Fohrmann et al., 1998), and particularly around Antarctica. Here some observations of Antarctic downslope flows are first described, followed by some intrinsic properties of the effects of the Earth’s rotation and of ambient stratification on downslope flows, based on observations from laboratory experiments. Dense downslope flows in the ocean have small density

Gravity currents have the property of being able to lift sediment and transport it (e.g. Middleton, 1993), thereby affecting both the distribution of sediment on the sea floor and the dynamics of the downflowing fluid. This chapter surveys some properties of the flow of dense fluids down slopes in two-layer systems, for small to moderate slope angles (i.e. less than about 15°). It concentrates on the consequences of ambient density stratification and the Earth’s rotation, with some speculations on the effects of suspended sediment when combined with these factors. The dynamics of these systems are relevant to downslope flows in the ocean, and we focus here on the specific example of flows around Antarctica. The dynamics of such flows affect the production and movement of Antarctic Bottom Water (AABW), which is the densest water mass at the bottom of the main ocean basins. These downflows are important for climate dynamics, because the continual production of this dense water mass drives a deep circulation or ‘conveyor belt’ that helps to overturn and ventilate the deep oceans. The


113

PGCC07 9/12/08 3:15 PM Page 114

114

P.G. Baines

differences relative to the ambient fluid, and the resulting velocities are not large. Consequently, the Earth’s rotation in the form of the Coriolis force is important for the overall structure of deep downflows in the ocean.

OBSERVATIONS OF DOWNSLOPE FLOWS AROUND ANTARCTICA The Antarctic continental shelf has a depth of 400 m or more, and the bottom slopes upward away from the continent toward the shelf break. Dense water is formed on this shelf and collects in the depressions. The basic process is driven by the freezing of sea water in winter, causing formation of sea ice and the consequent rejection of salt into the water below. This results in cold salty water that sinks and collects in the depressions on the shelf. Around much of the continent this occurs beneath leads and coastal polynyas where the ice is continually driven offshore by katabatic winds pouring off the continent (Baines & Condie, 1998). In the largest source region of AABW, the Weddell Sea, the process is more complex; brine rejection occurs beneath leads on the continental shelf in the southern Weddell Sea, seaward of the Ronne Ice Shelf (Gill, 1973; Killworth, 1983). This water mixes with cold water from the melting underside of the Ice Shelf, with cold, fresher water in a coastal current over the continental slope from the east, and with Circumpolar Deep Water offshore, forming a soup of dense bottom water. The whole then moves northward and downslope on the western shelf and slope of the Weddell Sea (Muench & Gordon, 1995). When a sufficient amount of dense water has accumulated on the shelf, it spills over the shelf break onto the continental slope and tends to flow downslope under the influence of gravity, but is deflected alongslope by the Coriolis force. The detailed processes by which dense fluid moves downslope are complex, and depend on bottom friction and Ekman drainage and eddy formation. A schematic diagram of the chain of processes involved is shown in Fig. 1. Around much of Antarctica where downslope flows occur, the continental slope is punctuated with many canyons, which can negate the effects of rotation and channel the flow of dense fluid more directly downslope. Large canyons are apparently absent from the western Weddell Sea, but elsewhere they can provide conduits for dense fluid to leave the continental shelf. Since the Antarctic canyons are not well resolved, and the flows within

them are unobserved, their downslope transports are largely inconspicuous. The role of suspended sediment in present-day Antarctic downslope flows is not clear. There are extensive depositional fans in the eastern Weddell Sea, notably the Crary Fan (Kuvaas & Kristoffersen, 1991) and the Weddell Fan (Anderson et al., 1986), but these do not seem to be actively evolving under present interglacial conditions. Instead, there is evidence for erosion of sediment by downslope flows in canyons in and near these fans. In the Ross Sea, Jacobs et al. (1970) reported nephelometer observations showing the presence of suspended sediment in the dense bottom layer at a number of locations over the continental slope. This cold ‘nepheloid layer’ was also high in oxygen concentration, and its depth ranged from 400 to 1000 m. Whether or not the concentration of suspended sediment in this water is large enough to cause a dynamically significant increase in the effective fluid density is unknown.

PROPERTIES OF ROTATIONALLY DOMINATED GRAVITY CURRENTS Experiments on downslope flow of dense fluid released into a homogeneous fluid in a rotating environment have been described by Lane-Serff and Baines (1998). These experiments have been extended to include stratified environments, with application to specific oceanic situations, by Lane-Serff and Baines (1999). The discussion in this section is focussed on the physics of the homogeneous situation, which includes many of the important phenomena that occur in both situations. Experiments were carried out with apparatus as shown in Fig. 2. Dense salty water (with density in the range 1.01–1.05 g cm−3) was released from an elevated source tank into a tank full of fresh water on a rotating turntable, to flow over a weir or sill onto the plane slope (slope angle θ to the horizontal, having values 5.7° and 15.7°). Rotation periods ranged from 3 to 20 s per revolution, with clockwise (Southern hemisphere) direction. These flows are approximately laminar, and are not significantly affected by small scale turbulence where it occurs. In a typical experiment, the dense fluid debouches over the sill and flows a short distance down the slope, before being deflected to the left by the Coriolis force to flow along the depth contours. There is little further flow down the slope by the main body of this dense fluid. However, more direct downslope flow occurs in a thin boundary layer (thickness 1–2 mm) that develops beneath this main

PGCC07 9/12/08 3:15 PM Page 115

115

Rotating and stratified environments Antarctica Ice sheet Pack ice Polynya

Turbulent convection Downslope flow

Eddies Ekman Drainage

Continental shelf

Canyon plume Continental slope

Fig. 1. A schematic view of dense water production over the continental shelf through convection beneath a polynya, formation of a dense body of water on the shelf, and its descent down the continental slope with eddy formation and Ekman drainage. An alternative route is via a descending plume in a submarine canyon (from Baines & Condie, 1998).

Source fluid

Shelf Weir

θ Slope

Shelf

Source fluid

Walls

Mesh Weir

Fig. 2. Schematic diagram of the rotating tank and slope used for the rotating experiments.

(a) Side view

(b) Plan view

PGCC07 9/12/08 3:15 PM Page 116

116

P.G. Baines

Fig. 3. Plan view of a rotating experiment with small Γ(= 0.034). The source of dense (dyed) fluid is at the bottom right-hand corner, and the depth increases toward the top of the page. The main flow is seen to be along the constant depth contours at the top of the slope, and the Ekman drainage flow covers most of the slope. Waves may be seen on this thin flow (thickness 1–2 mm) that have ‘roll-wave’ character (roll waves are described in Baines, 1995).

body. This boundary layer is termed an ‘Ekman layer’ (e.g. Gill, 1982) because it is dominated by rotation, and in these experiments it carries most of the downslope flow. It connects the state of zero fluid velocity at the sloping bottom to the state of inviscid balance between the buoyancy and Coriolis forces that applies to the dense fluid above it. The important parameters for these experiments are the volume inflow rate Q0, the fractional density difference between that of the dense fluid and that of the environment, ∆ρ/ρ, the bottom slope tan θ, the total depth at the top of the slope, D, and the fluid viscosity, ν. The depth d of the dense layer on the slope, and the horizontal scale of this layer, L (taken to be the Rossby deformation radius) are then of the order 1

d ~ (2Qf /g′ ) –2,

1

1

3

L = (g′d ) –2 /f ~ (2Qg′ ) –4 f − –4 , (1)

where g′ = g∆ρ/ρ, and f is the Coriolis parameter, or frequency, defined by f = 2Ω, where Ω is the angular velocity of the tank. In the real ocean, f = 2Ω sin(latitude), where Ω is the angular velocity of the Earth’s rotation. f is negative in the Southern hemisphere, but its magnitude is implicit in Eq. (1). L is the lengthscale in which the flow adjusts toward geostrophic balance, and an important parameter in these experiments is the ‘stretching parameter’, Γ = L tan θ/D. Figure 3 shows the plan view of an experiment looking downslope, as viewed from the left in Fig. 2b. Here

Fig. 4. Same as for Fig. 3, except that Γ(= 0.13) is now larger. The main body of dense fluid now has the form of a succession of cyclonic eddies that occupy the whole water column, with the dense fluid as (relatively) anticyclonic domes at the bottom.

Γ = 0.034, and the main ‘inviscid’ current (denoted by the dark, dyed fluid) can be seen flowing along the top of the slope, with the thin Ekman layer downflow draining fluid from it. The latter is characterized by the conspicuous dark lines denoting ‘roll waves’ (Baines, 1995; Jiang & Garwood, 1996) on this thin viscous layer, which has a clear-cut right-hand edge visible in Fig. 3. In many experiments these flows are complicated by eddies which form in the main body, and a typical example is shown in Fig. 4, where Γ = 0.13. These eddies consist of a ‘dome’ of dense lowerlayer fluid, with cyclonic rotation in the environmental fluid above. The ultimate fate of this trapped dense fluid depends on what happens to the eddies. The time interval between eddies, and their strength, depends on the parameter Γ. The vorticity in the eddies is approximately proportional to Γf, and the time interval between formation of eddies decreases with increasing Γ, as shown in Fig. 5. Formation of an eddy disrupts the Ekman layer drainage process, as illustrated by the comparative absence of the latter in Fig. 4. There are three possible mechanisms that can cause these eddies (Lane-Serff & Baines, 1998). They all depend on the stretching of the total water column above the dense fluid, which stretches the vortex lines and hence ‘spins up’ the column. This column stretching can be realised in three different ways, the relative importance of which are still unclear. Firstly, it is possible that the initial downslope motion of the debouched dense fluid can carry the whole column downslope with it by the ‘Taylor column’ effect

PGCC07 9/12/08 3:15 PM Page 117

117

Rotating and stratified environments

form) to slowly move downslope. The first two mechanisms operate immediately the dense fluid flows onto the slope, whereas the Ekman drainage process acts 3 1 over a longer lengthscale (= (Q f 2 )/g ′ tan θ( 2ν) 2 ), and takes a correspondingly longer period of time to be effective (Lane-Serff & Baines, 1998). For this reason, this last may be the most significant in practice. To date, no eddies of this type have been observed in the Antarctic environment. However, eddies with a periodicity of approximately two days and a similar character to those described here have been observed regularly in the ocean in the deep overflow passing through the Denmark Strait in the North Atlantic (Bruce 1995; Krauss, 1996).

+

++

14

+

12

++

8

6

+ + ++

Interval ( Tint /T )

10

4

EFFECTS OF A STRATIFIED ENVIRONMENT ON GRAVITY CURRENTS

2

0 0

0.1

0.2

0.3

0.4

0.5

T

Fig. 5. Time interval between one eddy and the next, as a function of the ‘stretching parameter’ Γ. Where no interval could be calculated it is plotted as 0. The solid line has been fitted to the data and has the form 0.19/Γ + 2.4. Different symbols denote different experimental conditions, as described in Lane-Serff and Baines (1998).

(Pedlosky, 1979). Secondly, the inflowing dense fluid quickly adjusts to a state of geostrophic balance (mostly between Coriolis force and buoyancy), and this generally involves a decrease in vertical thickness and flattening of the lower layer, which thereby stretches the thickness of the upper layer. Thirdly, the drainage of dense fluid downslope in the bottom Ekman layer removes it from the region of the alongslope flow, and this stretches the total fluid column there. From the well-known properties of Ekman layers, if the dense fluid has anticylonic relative vorticity (which it acquires in the geostrophic adjustment process), the Ekman layer drains dense fluid from the region where this vorticity is present. But as the whole column becomes more cyclonic by this stretching, this anticyclonic vorticity in the body of the dense fluid disappears, and the Ekman drainage is then fed from the upslope side of the dense current. This removal of fluid from the upslope side then causes the whole body of dense fluid (whether in eddy or alongslope current

Extensive experiments on downslope flows into a nonrotating homogeneous environment were carried out by Ellison and Turner (1959), who showed that the downflow caused the dense fluid to mix with the environment, thereby increasing the volume of dense fluid, which in turn causes the volume of downslope flowing fluid to increase. They identified this process as turbulent entrainment, by analogy with turbulent jets and plumes, and showed that the increase in the total volume flux Q (per unit width) with downslope distance s could be described by the equation dQ = E (Ri )U , ds

(2)

where U is the mean downslope velocity in the current, and E is the entrainment constant. E was found to be a function of the local Richardson number (Ri), Ri =

g ′d 3 cos θ , Q2

(3)

where d is the mean depth of the downflow and θ is the slope angle. The additional downflowing fluid was located above the main initial current, and extended further above it with increasing downslope distance. For deep downslope flows in the ocean, such as those found around Antarctica, the environment is not homogeneous and the density varies with depth. Experiments that investigate the effects of this density variation have been described by Baines (1999), and are summarized here. In a typical experiment, dense fluid was released from a continuous two-dimensional

PGCC07 9/12/08 3:15 PM Page 118

118

P.G. Baines

source at the top of a long slope, into a large, effectively two-dimensional (x – z) tank filled with uniformly density-stratified fluid. The density of the initial stratification equalled that of the initial inflow at a depth near the bottom of the tank, at some distance down the slope. The flow reached an approximately steady state for a period of (typically) several minutes. The inflow was then stopped, and the final density profile was measured after all motion in the tank had ceased. By taking the difference between the initial and final density profiles, one may determine where the inflowing fluid went (in overall terms), and what redistribution of the initial stratification took place. In these flows it is possible to make a reasonably clear distinction between the main downslope flowing gravity current, and the ambient fluid above it. An example is shown in Fig. 6. The mean depth of the downflow is observed to be remarkably uniform with distance downslope, which greatly assists the analysis. Mixing takes place at the upper boundary of this current, causing an exchange of fluid between the current and the fluid above it. This may be described as entrainment of environmental fluid into the current, and detrainment of fluid from it. The technique of differencing the initial and final density profiles, and assuming a constant exchange process during the experiment, enables the measurement of the nett downflow as a function of distance s, and the nett inflow and outflow from the current. The entrainment

and detrainment may be specified by respective coefficients Ee and Ed. The important parameters in these two-dimensional experiments were the slope angle θ, the inflow rate Q0, the density difference of the inflowing fluid relative to the ambient fluid at the inflow level, ∆ρ/ρ, the depth below the inflow where the ambient density equals the inflow density, De, and the density gradient of the ambient fluid as measured by the buoyancy frequency g′ N, given by N 2 = , where g′ = g∆ρ/ρ. The behaviour De of the flow is governed by three dimensionless parameters – the slope angle θ, the Reynolds number Re and the parameter M0, defined by M0 =

Q0 ( g0′ De3 )

1 2

=

Q 0N 3 , g0′ 2

Re = Q 0 / ν.

M0 is based on the inflow conditions, but M(s) may also be defined as a function of downslope distance s by using the local values of Q(s), N and g′ in Eq. (4). Examples of profiles of inflow and outflow from the current as a function of depth Z = z/D (where z = −s sin θ) obtained from the procedure described above, for a slope angle of 6° and a range of M0 values, are shown in Fig. 7. The details of these profiles will be discussed elsewhere. They indicate that some nett detrainment from the current occurs along most of the profile, and the main outflow is found near Z = −1.0, where the remaining fluid reaches its ambient density. If there were no entrainment this outflow level would lie at Z = −1.0, and the elevation of this outflow above Z = −1.0 is a measure of the nett entrainment into the current. This level rises and entrainment increases with increasing M0. These observations, plus others taken at other slope angles, indicate that a suitable model that may be used to describe the bulk behaviour of these downflows in steady state may be written as dQ = ( Ee − Ed )Q /d , ds d Q2 = Gd sin θ − ( k + CD ) Q 2 /d 2 , ds d

Fig. 6. An example of a typical downslope gravity current in a stratified environment, in the central part of the slope some time after the passage of the head of the current. The inflowing fluid has been dyed with fluorescein, and the flow has been illuminated by a vertical sheet of laser light to show a vertical section. Note the sharp upper boundary of the main current, the wisps of detrained fluid, and the presence of detrained fluid in the environment.

(4)

dG = −N 2 sin θ − EeG /d . ds

(5) (6) (7)

These equations describe the variation of the downslope fluxes of fluid volume, momentum and buoyancy respectively, in the current close to the wall. Here d is the local thickness of the current, G = g′, CD is a drag coefficient for the sloping bottom, and k is a

PGCC07 9/12/08 3:15 PM Page 119

Rotating and stratified environments

119

Fig. 7. A range of observed velocity profiles representing flow into (positive V ) and out of (negative V ) the downflow, as a function of depth beneath the source. Each profile represents the mean horizontal flow towards and away from the slope for a given run. The slope angle is 6°, and the values of M0 are shown. Here V = u /U0, where u is the horizontal inflow/outflow velocity, and U0 is the mean velocity in the current at the top of the slope. The depth Z = −1 denotes the level at which the ambient stratification equals the inflow density, and the large ‘spike’ just above it (for small M0) denotes the main outflow. The fact that V is negative for most Z values, for most runs, indicates that detrainment from the downflow dominates over entrainment into it.

corresponding drag term due to momentum exchange with the overlying fluid that is not associated with fluid exchange. The coefficients Ee and Ed are found to be functions of the local values of M(s) and Ri(s), and forms for these will be described in a forthcoming article.

CONCLUSIONS The presence of the Earth’s rotation and environmental stratification can each cause major changes to the bulk properties of downslope-flowing gravity currents. The Earth’s rotation may have little effect on flows that are sufficiently rapid, or sufficiently short. Sedimentary gravity currents are often very rapid

events, so that rotation should not be significant for events that last for much less than a pendulum day, but the environmental stratification effect should still be significant, particularly if the flow takes place over a substantial depth range. For the extreme case of deep flows in the ocean, such as those around Antarctica, both are important. If submarine canyons are absent, which is the case (apparently) on the continental slope of the western Weddell Sea, the descending path is mainly alongslope (i.e. nearly horizontal) with more direct downslope flow occurring in the Ekman drainage layer below. There is some evidence that the suspension of sediment in such a downflowing current may affect the dynamics of such flows by increasing the density, but the magnitude and frequency of this effect is unknown at present. There is,

PGCC07 9/12/08 3:15 PM Page 120

120

P.G. Baines

however, clear evidence that rotation can affect the distribution of deposited sediment.

NOMENCLATURE CD drag coefficient d thickness of the dense layer D total depth of the fluid E, Ee entrainment coefficient Ed detrainment coefficient f Coriolis parameter g acceleration due to gravity G = g′ = g∆ρ/ρ reduced gravity k ‘drag coefficient’ for overlying fluid L horizontal lengthscale N buoyancy frequency of the stratification U fluid speed Ri Richardson number Re Reynolds number s downslope distance Γ = L tan θ/D stretching parameter θ slope angle ν kinematic viscosity ρ fluid density ∆ρ difference between the density of the downflow and that of the environment

REFERENCES Anderson, J.B., Wright, E. & Andrews, B. (1986) Weddell Fan and associate abyssal plain, Antarctica: morphology, sediment processes, and factors influencing sediment supply. Geo-Mar. Lett. 6, 121–129. Baines, P.G. (1995) Topographic Effects in Stratified Flows. Cambridge University Press, Cambridge, 482 pp. Baines, P.G. (1999) Downslope flows into a stratified environment—structure and detrainment. In: Mixing and Dispersion in Stably Stratified Flows. Proceedings of the 5th

IMA Conference on Stratified Flows, Dundee (Ed. Davies, P.A.), pp. 1–21. Baines, P.G. & Condie, S. (1998) Observations and modelling of Antarctic downslope flows: a review. In: Ocean, Ice and Atmosphere: Interactions at the Antarctic Continental Margin, AGU Antarctic Research Series Vol. 75 (Eds Jacobs, S.S. & Weiss, R.), pp. 29– 49. Bruce, J.G. (1995) Eddies southwest of Denmark Strait. Deep-Sea Res. 1 42, 13–29. Ellison, T. & Turner, J.S. (1959) Turbulent entrainment in stratified flows. J. Fluid Mech. 6, 423 –448. Fohrmann, H., Backhaus, J.O., Blaume, F. & Rumohr, J. (1998) Sediments in bottom-arrested gravity plumes: numerical case studies. J. Phys. Oceanog. 28, 2250 –2274. Gill, A.E. (1973) Circulation and bottom water production in the Weddell Sea. Deep-Sea Res. 20, 111–140. Gill, A.E. (1982) Atmosphere–Ocean Dynamics. Academic Press, London, 662 pp. Jacobs, S.S., Amos, A.F. & Bruchhausen, P.M. (1970) Ross Sea oceanography and Antarctic bottom water formation. Deep-Sea Res. 17, 935 –962. Jiang, L. & Garwood, R.W. (1996) Three-dimensional simulations of overflows on continental shelves. J. Phys. Oceanog. 26, 1214–1233. Killworth, P.D. (1983) Deep convection in the world ocean. Rev. Geophys. Space Phys. 21, 1–26. Krauss, W. (1996) A note on overflow eddies. Deep-Sea Res. 1 43, 1661–1667. Kuvass, B. & Kristoffersen, Y. (1991) The Crary Fan: A trough-mouth fan on the Weddell Sea continental margin, Antarctica. Mar. Geol. 97, 345–362. Lane-Serff, G.F. & Baines, P.G. (1998) Eddy formation by dense flows on slopes in a rotating fluid. J. Fluid Mech. 363, 229 –252. Lane-Serff, G.F. & Baines, P.G. (1999) Eddy formation by overflows in stratified water. J. Phys. Oceanog., in press. Middleton, G.V. (1993) Sediment deposition from turbidity currents. Ann. Rev. Earth Planet. Sci. 21, 87–114. Muench, R.D. & Gordon, A.L. (1995) Circulation and transport of water along the western Weddell Sea margin. J. geophys. Res. 100, 18503 –18515. Pedlosky, J. (1979) Geophysical Fluid Dynamics, SpringerVerlag, Berlin, 624 pp. Quadfasel, D., Kudrass, H. & Frische, A. (1990) Deepwater renewal by turbidity currents in the Sulu Sea. Nature 348, 320 –322.

PGCC08

10/12/08

4:28 PM

Page 121


Two-dimensional and axisymmetric models for compositional and particle-driven gravity currents in uniform ambient flows A. J. H O G G * and H. E. H U P P E R T † * Centre for Environmental and Geophysical Flows, School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW; and † Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

ABSTRACT The propagation and dispersion of compositional and particle-driven gravity currents intruding into a uniform ambient flow are analysed. Both two-dimensional (line source) and axisymmetric (point source) releases are investigated. Appropriate theoretical models to describe both the fluid motion and resulting deposit density distribution are developed. We show that the behaviour of the current depends on a nondimensional parameter that represents the ratio of the advected flux of the ambient flow to the downward particle flux to the boundary. Relationships for the maximum upstream penetration distance are derived in terms of that parameter. The theoretical results are compared successfully with data from laboratory experiments for the upstream and downstream lengths of the current as functions of time, the final density of deposit on the floor and the maximum upstream penetration of the current. A few numerical examples are discussed.

INTRODUCTION atmosphere. The results will thus be relevant to pollution discharge in channels (Huppert, 1997), turbidite formation in the oceans (Dade & Huppert, 1994) and the dispersion of either ash from a volcanic cloud (Ernst et al., 2001) or metalliferous particles from hydrothermal plumes (Baker et al., 1995). The chapter considers two different geometrical situations: two-dimensional and axisymmetric. Put another way, we consider the influence of a uniform ambient flow on a line release and a point source release of particulate matter. The resulting flow could be followed (in all probability) by solving the relevant shallow-water equations. Instead we develop simpler, but powerful ‘box-models’ in which horizontal variations within the current at any particular time are ignored (Huppert & Simpson, 1980; Dade & Huppert, 1994). There is generally good agreement between the results of such box models and the more rigorous shallow-water approach (Hallworth et al., 1998). In addition, the link between such integral models and the shallow-water equations has been recently uncovered by Hogg et al.

Particle-laden flows play an important role in many different natural and industrial situations. Due to the density difference between the particles and surrounding fluid, the particulate matter is not only advected with the flow but also sediments through it. This relative motion both influences the evolution of the flow and also determines the deposition pattern of the particulate matter. Most studies of particulate flows have concentrated on situations in which the ambient fluid is quiescent, and much has recently been uncovered about the flow of particle-laden gravity currents intruding into a less dense, otherwise quiescent ambient (see, for example, Simpson, 1997 or Huppert, 1998 for reviews of these situations). The aim of the present contribution is to consider the advection by and sedimentation from a particle-laden current intruding into an ambient which is moving with a uniform horizontal speed far from the current. Such a flow could result in a channel from an applied horizontal pressure gradient, or could represent tidal motions in the ocean, or winds in the


121

PGCC08

10/12/08

122

4:28 PM

Page 122

A.J. Hogg and H.E. Huppert

(2000). In the box-model formulation, the presence of an ambient flow can be eliminated by considering a frame of reference that translates uniformly downstream with velocity U. Within that frame the current spreads as if in a quiescent medium, in either a twodimensional or axisymmetrical fashion as appropriate. However, especially in the axisymmetric situation, the radial spreading in a uniformly translating frame can lead to rather complicated patterns when viewed in the fixed frame (of the stationary floor). Here we first consider the evolution of compositional currents where the density difference is due to a dissolved component, such as salt. This sets up part of the theoretical framework needed to analyse particledriven currents, which are then described in detail. Experimental verification of the quantitative theoretical results is then presented, followed by a similar treatment of theoretical aspects of the axisymmetric situation. The main conclusion of our study, along with a brief summary and some practical applications, appear in the discussion and conclusions.

of the current is represented by an evolving series of rectangular boxes. In the following subsections, following Hallworth et al. (1998), we derive the box models appropriate to compositional and particledriven flows in the presence of a uniform mean flow, which inhibits propagation upstream and promotes propagation downstream. Compositional currents We consider the instantaneous intrusion of a finite volume V of fluid with a density ρc into an ambient fluid of lower density ρa. The intruding fluid spreads in a two-dimensional manner along the horizontal boundary underlying the ambient fluid, its motion being driven by gravity acting on the density difference. The ambient fluid flows uniformly in a horizontal direction. The gravity currents spread in both the downstream (x) and the upstream (y) directions, as indicated in Fig. 1(a). On the assumption that there is no entrainment of ambient fluid into the gravity current, the conservation of volume may be written as

TWO-DIMENSIONAL BOX MODEL

l ≡ x + y = A/h,

The dynamics of large-scale intrusions of dense fluid, spreading over an impermeable, rigid horizontal boundary, are modelled by balancing inertial and buoyancy forces. Such a theoretical approach has been employed by a number of investigations of buoyancy-driven phenomena in both laboratory and natural environments (Simpson, 1997). It is necessary to include the influence of viscous forces only when the current has spread over a considerable distance and becomes sufficiently thin that an appropriate Reynolds number is less than 2.25 (Bonnecaze et al., 1993). The motion of buoyancy-driven intrusions is predominantly horizontal since the vertical accelerations are negligible. This observation has been exploited in the development of mathematical models of the flow which successfully reproduce the experimental measurements (e.g. Rottman & Simpson, 1983; Bonnecaze et al., 1993). However it is possible to derive a simpler representation of the flow which avoids the need for numerical solution of partial-differential equations and yet produces results in good agreement with the experiments. The intrusion is analysed on the assumption that at all times the salt concentration, or particle distribution, and the height of the current are uniform in the horizontal direction (e.g. Dade & Huppert, 1994; Hallworth et al., 1998). The resulting description is termed a box-model representation because the shape

where l is the total length of the current, A is the volume per unit width and h is the thickness of the current, which is assumed to be uniform along the current in this box-model formulation. Conditions on each of x and y (or more accurately, their rates of change) link the frontal velocity with both the ambient velocity and the local wave velocity based on the excess density of the current. (These are the Froude number condition which was studied theoretically by Benjamin (1968) and experimentally by Huppert and Simpson (1980).) Hence we may write that 1 dx = U + Fr( g ′ h) 2 dt

and

(1)

1 dy = −U + Fr( g ′ h) 2 , dt (2a, b)

where g′ ≡ (ρc − ρa)g/ρa is the reduced gravity of the current, U is the mean velocity experienced by the current in the x direction and Fr is the Froude number which is assumed constant and equal to 1.19 (Huppert & Simpson, 1980). Previous studies have suggested that the velocity U is a factor of 0.6 times that of the vertically-averaged ambient flow (Simpson & Britter, 1980 and see the experimental section below). Integrating Eqs (1) and (2), subject to the initial conditions that x = y = 0, we obtain 2

x = Ut + γt 3

and

2

y = −Ut + γt 3 , (3a, b)

PGCC08

10/12/08

4:28 PM

Page 123

123

Compositional and particle-driven gravity currents

Fig. 1. (a) Schematic diagram of the experimental apparatus (not to scale). (b) Profiles of the horizontal downstream ambient fluid velocity as a function of depth, measured by acoustic Doppler velocimetry at various distances either side of the release position along the medial plane of the flow channel (in the absence of gravity currents). The structure in the profile at y = 3 m is due to the proximity of this section to the input diffuser box.

where 2 3

1 3

γ = 21 (3Fr ) ( g ′A) .

(4)

By differentiating Eq. (3b) we predict that the maximum distance upstream that the current propagates is given by –16 Fr2(g′A/U 2), which except for the premultiplicative constant can be quite simply obtained by dimensional analysis. Inspection of Eq. (3a, b) indicates that there is a timescale τc = (γ/U )3,

(5)

which corresponds to when the buoyancy-related and ambient velocities are comparable. For times less than τc the current propagates mainly due to buoyancy, and the effects of the external flow make a smaller contribution; for times in excess of τc external flow effects are more important than the buoyancy. In the experiments described below, this timescale is approximately 200 s, whereas a typical duration of an

experimental run was around 60 s, beyond which the gravity current became too thin and weak to be distinguished against the background flow. (After this time the Reynolds number of the current was also too small for the above theory to be appropriate: using the results in appendix B of Bonnecaze et al. (1993), obtained by balancing forces, we can evaluate the 7 appropriate Reynolds number as (A2/γ 2 ν)t − –3, where ν is the coefficient of kinematic viscosity. Bonnecaze et al. (1993) show that viscous forces dominate the flow when this Reynolds number falls below a value of approximately 2). It was thus impossible for us to test robustly both of the terms of Eq. (3) (in the form written). It is therefore worthwhile concentrating on the alternative representation 2

l ≡ x + y = 2γt –3

and

z ≡ x − y = 2Ut, (6a, b)

which evaluate the evolution of the length and twice

PGCC08

10/12/08

4:28 PM

Page 124

124


Fig. 2. (a) Non-dimensional length Lc, and (b) non-dimensional position of the centroid Zc, plotted against nondimensional time Tc, for compositional currents of 50 g, (), 200 g () and 400 g () of salt dissolved in 2 litres of water, released into an ambient flow. The theoretical predictions are shown as solid lines in each graph.

the displacement of the centroid from its initial position. Introducing the lengthscale lc = γ 3/U 2

(7)

and the non-dimensional variables Lc = l/lc

Zc = (x − y)/lc

and

Tc = t/τc , (8a, b, c)

we plot all our experimental data in Fig. 2 and compare these to our theoretical predictions 2

Lc = 2T c–3

and

Zc = 2Tc .

fully requires the velocity U to be 0.6 U, the value of the mean ambient flow. The statement that a gravity current only experiences 0.6 of the mean flow has been suggested previously by Simpson and Britter (1980) and in this study we find further strong experimental evidence of this relationship (see below). We note that Eq. (9) indicates that the total length of the current increases as the two-thirds power of the time while the centre of the current propagates downstream at 0.6 times that of the ambient velocity.

(9a, b)

We note that the non-dimensionalization of the experimental data which allows it to be collapsed so success-

Particle-driven currents The development of a box model for the propagation

PGCC08

10/12/08

4:28 PM

Page 125

125

Compositional and particle-driven gravity currents of a monodisperse particle current is similar to that for a compositional current. However its density is progressively reduced due to particle sedimentation, which in turn leads to a reduction in flow speed. The Froude number conditions are now written as 1 dx = U + Fr( g p′ φh) 2 dt

1 dy = −U + Fr( g p′ φh) 2 , dt (10a, b)

and

where φ is the volume fraction of particles and g′p ≡ (ρs − ρa)g/ρa is the reduced gravity of the particulate phase which has density ρs. The sedimentation of particles from the vertically well-mixed current though a basal viscous boundary layer is modelled by a constant settling velocity, Vs, on the assumption that the deposited particles are not re-entrained d( φA) = −Vs φl . dt

(11)

The system of Eqs (10) and (11), together with an expression for the conservation of mass (Eq. 1), are integrated subject to the initial conditions x = y = 0 and φ = φ0. Identifying the dimensionless length and timescales 2

⎡10Fr( g ′ φ A3 ) 12 ⎤ 5 p 0 ⎥ l∞ = ⎢ ⎥ ⎢ Vs ⎦ ⎣

and

τ∞ =

5A l∞Vs (12a, b)

which are then used to define L, Z and T as the nondimensional variables for the length, twice the displacement of the centroid and time, we obtain the relationships (in terms of the variable of integration s) 5 φ = (1 − L 2 ) 2, φ0

T =

0

5 2

(13a)

(13b)

and Z = ΛT,

(13c)

which are graphed in Fig. 3 along with our experimental data. In these expressions there remains a single nondimensional parameter 10UA Λ= 2 . l ∞Vs

φ dt,

η( x ) = ρpVs

1 2

1s− dss ≡ F (L) L

In contrast to the compositional situation (see above), the current ceases (φ = 0) when l = l∞, a value that is independent of the ambient flow speed, though we see from Eq. (13b) that this length takes (theoretically) an infinite time to achieve. The magnitude of Λ represents the influence of the mean flow on the runout of the gravity current and is proportional to the ratio of the mean flow to the settling velocity of the particles. More precisely, if we re-write A as the product of the runout length l∞ and the height of the fluid layer when this length is attained, A = h∞l∞, then the parameter Λ may be seen to represent the ratio of the horizontal flux of fluid, Uh∞, to the vertical settling flux of particles, Vs l∞. When the settling flux is large compared to the flux of the mean flow (and thus Λ is much less than unity), the evolution of the gravity current is only weakly affected by the motion of the ambient. Conversely, when Λ is much greater than unity, the gravity current is strongly influenced by the mean flow. The distribution of the deposit arising from these particle-driven gravity currents may be calculated from the box model analysis on the assumption that the particles sediment out of the current uniformly along its length. When there is no ambient flow, the gravity current propagates symmetrically in the upstream and downstream directions until it attains the maximum length (l∞). In contrast, when there is an ambient flow, the centroid of the current is advected downstream (cf. Eq. (13c)). Thus the resulting deposit is asymmetric about the initiation line of the twodimensional current and extends over a considerable distance downstream. The deposit, expressed as the integrated mass flux per unit area delivered to the bottom while the current is overhead, is given by

(14)

tf

(15)

ts

where the limits of this integral correspond to the times at which deposition starts and finishes, denoted by ts and tf, respectively. We reiterate that a box model of the gravity current is being used in which there is uniform sedimentation along its entire length. Furthermore the length of the box is increasing whilst its centroid is advected downstream. Therefore at a particular location deposition starts when a front of the current first passes and ceases when the rear of the current is swept by. Substituting Eq. (12a, b) into Eq. (15), we obtain the implicit relationship η( x ) =

3 5φ0 ρp A ⎡ 2L 2 L4 ⎤ L f ⎢ ⎥ , − l∞ 4⎥ ⎢⎣ 3 ⎦ Ls

(16)

PGCC08

10/12/08

126

4:28 PM

Page 126


Fig. 3. (a) Non-dimensional length L plotted against nondimensional time T, and (b) non-dimensional position of centroid z¯, plotted against ΛT/U, for particle currents with various masses and sizes of particles (given by legend) initially suspended in 2 litres of water, released into an ambient flow. The solid curve in (a) is the theoretical relationship given by Eq. (13b). The solid curve in (b) is the best-fit straight line through all the data, the gradient of which determines U.

where Ls and Lf are the values of the dimensionless length L at non-dimensional times of Ts and Tf , which correspond to the dimensional times ts and tf . We plot some illustrative profiles of the deposit as a function of position for a range of values of Λ in Figs 4 and 5. As notes above, when there is no ambient flow, U = Λ = 0, the deposit distribution is symmetric about the point at which the suspension of particles is released. However, as the magnitude of the ambient flow increases relative to the settling velocity of the particles, the profiles become increasingly asymmetric. Using this simple model of the deposit, we can straightforwardly calculate numerically the maximum upstream distance, d+, over which the current propagates, as a function of Λ. We present the results in

Fig. 6 and find very good agreement between the theoretical curve and the experimental observations (except at Λ = 0, corresponding to zero flow, the situation in which the maximum upstream point is not as sharply defined, as is seen from Fig. 4). From Eqs (12c) and (16) it is possible to calculate asymptotic representations of this maximum upstream distance in the regimes Λ > 1. Using these expansions we may formulate an approximate composite expansion which is given by d+ = l∞

1 2

+ ( 15 log Λ − π8 ) Λ + 1+

3 4 Λ 25

1 2 Λ 50

,

(17)

which is indistinguishable from the full numerical solution.

PGCC08

10/12/08

4:28 PM

Page 127


Fig. 4. Non-dimensional deposit thickness as a function of position for a range of values of Λ = 10UA/(Vs l 2p ).

Fig. 5. Plots of the experimentally measured deposit profiles and the theoretical predictions for four different values of Λ = 10UA/(Vsl 2p).

127

PGCC08

10/12/08

4:28 PM

Page 128

128


Fig. 6. The maximum nondimensional upstream distance d + /lp as a function of Λ = 10UA/(Vs l 2p ). The calculation of the theoretical curve is described in the text.

EXPERIMENTS Experimental set-up The experimental apparatus used is shown schematically in Fig. 1(a). A 9.4-m long Perspex channel having a rectangular cross-section 26-cm wide and 50-cm high was filled with water to a depth H of 28.7 cm. A uniform ambient flow was established by pumping the water at a fixed rate in a continuous loop via a hose connecting inlet and outlet diffuser boxes situated at either end of the channel, thereby giving a working flow section of 8.4 m. Each diffuser box comprised a flared section packed with 1-cm diameter plastic balls and a horizontally aligned honeycomb section, designed to introduce and withdraw the flow evenly across the whole cross-sectional area of the channel. Profiles of the flow velocity as a function of depth were measured at several distances along the medial plane of the working section using a Sontek acoustic Doppler velocimeter (Lane et al., 1998). This nonintrusive device focuses an acoustic beam on a 0.5-cm3 fluid sample volume and digitally translates the reflected signal into three mutually perpendicular velocity components, which we orientated to coincide with the major axes of the channel. Both the vertical and horizontal cross-stream velocity components were negligible. The horizontal downstream velocity components at various positions are presented as velocity profiles in Fig. 1(b). Each profile displays a fairly uniform velocity, U, averaging 2.9 cm s−1 in the interior of the flow which reduces in value to zero at the channel floor through a lower boundary layer, approximately 2-cm thick. A reduction in flow velocity is also apparent as the free surface is approached. Integration of these flow profiles yielded an average volumetric flux

of 1850 cm3 s−1, which corresponds to a Reynolds number of approximately 7000. The conventional lock-release method of instantaneously initiating a gravity current of fixed volume into a stationary ambient fluid was impossible to achieve in the present situation without severely disrupting the ambient flow. An alternative release mechanism was therefore designed whereby a fixed volume of dense fluid, initially held in a reservoir above the midpoint of the channel, was allowed to drain rapidly (in less than 1 s) into the flow stream through a 3-cm diameter tube positioned just beneath the free surface. The emergent jet of dense fluid inevitably entrained a significant volume of ambient fluid during its descent and subsequent lateral deflection upon impinging on the solid channel floor. On testing our release mechanism in quiescent ambient conditions we found that the jet split equally and extended roughly 30 cm either side of the central release position before buoyancy forces began to dominate the motion. Entrainment of ambient fluid during the early momentum-dominated phase was measured to cause a dilution of the released fluid by a factor of approximately 20. This estimate was achieved by trapping a released current between vertical barriers, positioned either 50 or 100 cm on either side of the entry point. When confined in this manner, the dense flow eventually settled to form a layer of constant composition. By measuring the height of this layer, its volume could be calculated and compared with the initial volume. In each case our measurements indicated that the released fluid was diluted by a factor of 20 ± 2 through entrainment of ambient fluid during an initial phase in which the dynamics of the jet is dominated by the momentum of the intruding fluid, rather than its buoyancy. In the calculation of the length and time scales for

PGCC08

10/12/08

4:28 PM

Page 129

Compositional and particle-driven gravity currents the compositional currents (Eqs (5) and (7)), the initial area, A, does not occur separately from the initial total buoyancy, g′A. The buoyancy is conserved under mixing and so the dilution described above does not affect the non-dimensionalization. However for the particledriven currents, the initial area does occur separately from the total buoyancy (in Eq. (12) for example) and so a knowledge of the initial dilution is vital for the scaling of these experimental results. Measurements were made of the horizontal distance to the front of the current from the release point as functions of time in both the downstream (x) and upstream (y) directions by marking the position of the nose of the current at 3-second intervals. In the case of particle-driven gravity currents, the final distribution of sedimented particles was measured by recovering the mass of particles within a 5-cm-wide strip across the width of the tank at various distances from the release point. Compositional currents Compositional currents of different initial densities were generated by releasing 2 litres of water containing 50 g, 200 g and 400 g of dissolved salt into the ambient flow, resulting in initial values of g′ of 17.1, 64.4 and 121 cm s−2 respectively. Solutions of each concentration were also released into a quiescent ambient for comparison. The currents released into a uniform ambient flow advanced both upstream and downstream, but were markedly asymmetrical. In the downstream (x) direction, the current was noticeably thicker than its counterpart in a static environment, and propagated with an increased velocity. As distance from the release point increased, the velocity of the current gradually decreased to a value approaching 0.6 times the mean ambient velocity (see above). In the upstream (y) direction, the current was significantly retarded by the opposing ambient flow, and eventually came to rest. Prior to final arrest, the current profile was observed to undergo a transition from the typical head and tail of a gravity current intruding quiescent surroundings into a much thinner wedgeshape within the lower boundary layer. Once in this form, dense fluid was continually stripped away from the upper surface of the arrested wedge by the action of interfacial eddies. In Fig. 2, we plot the relationships (9) and note that the agreement between the theoretical predictions and the experimental data is good. Hence the relationships (3) for the upstream and downstream position of the current can be used with confidence.

129

Particle-driven currents Particle-driven gravity currents were generated by releasing well-mixed suspensions of silicon carbide particles in water. These particles are fairly monodisperse, non-cohesive and have a density ρs = 3.217 g cm−3. As a precaution, a small amount of Calgon was added to the suspension to prevent particle agglomeration. Three different particle sizes were used, with mean diameters of 23, 37 and 53 µm. Details of the size distribution with each grade are reported in Huppert et al. (1991). For each particle size, experiments were run with four different initial particle masses of 50 g, 100 g, 200 g and 400 g suspended in 2 litres of water. Upon release, the particle-driven gravity currents propagated with decreasing velocity in both the x and y directions while simultaneously depositing a sediment layer over the channel floor until all the particles had settled out, whereupon the current ceased to exist. Velocities of the current at any point achieved by each flow were observed to increase monotonically with increasing initial mass of suspended sediment, and the current attained progressively longer maximum distances from the release point with decreasing particle diameter. The currents released into an ambient flow were markedly elongated in the downstream direction. The development of an arrested wedge of dense fluid in the upstream direction was not as noticeable as that seen in the compositional currents, since particles quickly sedimented from thinned flows which are formed in the slow moving lower boundary layer of the opposing stream. We plot in Fig. 3 the theoretical curves (Eqs 13b, c) and the experimental data, non-dimensionalized according to the scaling suggested by the box model. We note that the non-dimensionalization collapses the experimental data and that there is very good agreement with the theoretical predictions, again confirming the power of the box model approach. The relationship (13c) suggests that the position of the centroid should depend linearly on ΛT. However, as noted above, we are uncertain as to the exact value of U to use in the definition of Λ, although previous smaller studies (Simpson & Britter 1980) have indicated that U = 0.6U, where U is the mean velocity in the channel. Figure 3(b) presents experimental data on the position of the centroid against ΛT/U. From the gradient of the fitted curve we find that U = 1.8 cm s−1 which corresponds to U = 0.62U. In our experiments, the velocity in the boundary layer attains 62% of the mean ambient velocity at a distance of approximately

PGCC08

10/12/08

130

4:28 PM

Page 130


7 mm from the floor. Most of the currents were considerably thicker than this. It is our opinion that the ratio of around 0.6 (between U and U ) is independent of the thickness of the current as long as this is considerably larger than the thickness of the boundary layer. Once all the particles had settled out, the final length of the deposited layer was recorded, and its mass distribution measured by ‘vacuuming up’ the sediment using a siphon tube within a 5 cm × 25 cm rectangular ‘pastry cutter’ placed over the layer at specific intervals. The mixture was collected in a beaker, the water decanted and the particles dried and weighed to determine the mass of deposit per unit area. As a check on the sampling method, the total mass of sediment was recovered by integrating the measured deposition profile and was generally found to be within 1% of the initial value. We compare some of the experimentally measured deposit profiles with the theoretical predictions for four values of Λ in Fig. 5. The agreement is seen to be very good and in particular the asymmetry predicted by the theory is accurately reflected by the data.

AXISYMMETRIC CURRENTS We now develop a theoretical model of the radial spreading of an intrusion of relatively dense fluid within a uniform ambient flow. In the absence of an ambient flow, the dense fluid propagates radially away from its source. Its rate of propagation may be modelled using a box-model approach which has been demonstrated to yield good agreement with experimental measurements for both compositional currents (Huppert & Simpson, 1980) and particle-driven currents (Dade & Huppert, 1995; Bonnecaze et al., 1995). In this section we extend the box-model analysis to incorporate the effect of a unidirectional flow. The spreading of dense fluid is no longer radial and we calculate the locus of points which correspond to the position of the front of gravity current. The gravity current never reaches positions outside this locus. Axisymmetric models of gravity currents may be equally well applied to flows within an angular sector, provided that the boundaries of the sector have only a negligible influence. (This is equivalent to requiring that the lengthscale associated with the gravity current be much larger than that of the boundary layer.) Hence this study may be applied to discharges of dense fluid at the boundary of a relatively wide channel flow, such as a river or an estuary.

Guided by the success of the approach outlined above, in the following we develop box models of gravity currents driven by a compositional difference or the suspension of particles in the presence of a mean flow. Our emphasis is to establish the extent of the region within which the intrusion will propagate and, for particle-driven gravity currents, to calculate the distribution of the deposited sediment. Compositional currents We consider the instantaneous intrusion of a finite volume of fluid with a density, ρc into an ambient fluid of a lower density, ρa. The intruding fluid spreads along the horizontal boundary underlying the ambient fluid, its motion being driven by gravity. The ambient fluid flows uniformly in a horizontal direction. We define horizontal coordinate axes such that the x-axis is aligned with this uniform flow and the y-axis is perpendicular to the flow. As in the two-dimensional case, it is convenient to formulate the equations governing the evolution of the flow in a frame of reference which moves with the velocity of the centroid of the intrusion. In this frame the gravity current spreads radially in the form of a uniform disk whose radius and height change with time. Denoting the radial distance from the centroid by r, the conservation of fluid volume of the current may be expressed by r 2h = V,

(18)

where h is the height of the current and V is twice the volume per unit (radian) angle, which remains constant during the evolution of the current. (The total volume is πV.) The rate of radial expansion of the gravity current in this moving frame is given by 1 dr = Fr( g ′h) 2 , dt

(19)

where g′ ≡ (ρc − ρa)g/ρa is the reduced gravity of the current and Fr is the frontal Froude number, which is assumed constant. Substitution of Eq. (18) into Eq. (19) and integration, subject to the initial condition that r = 0 at t = 0, yields 1

1

r = ( 2Fr( g ′V ) 2 t ) 2 .

(20)

Denoting the position of the centroid by I we write I = Ut,

(21)

where U is the velocity experienced by the gravity current in response to the ambient flow. (For the

PGCC08

10/12/08

4:28 PM

Page 131

131

Compositional and particle-driven gravity currents two-dimensional gravity currents it was established that U was 0.6 of the mean flow.) At this stage it is convenient to identify the following time and length scales which may be used to non-dimensionalize the relationships (20) and (21), 1

ta = 2Fr(g′V ) –2 /U 2

1

d (Vφ) = −Vs φ r 2 . dt

1

The timescale ta corresponds to the time at which the velocity of the gravity-driven motion and the uniform flow are comparable, while the lengthscale ra is the downstream distance moved after this time. Writing the dimensionless variables as Ra = r/ra, Xa = I/ra and Ta = t/ta, we find that Xa = Ta.

and

1

⎛ φ ⎞2 Vs r 4 1 , ⎜φ ⎟ = 1− ⎝ 0⎠ 8Fr( g ′p φ0V 3 ) 2

We may therefore determine the boundary of the region within which the current propagates. In parametric form it is given by evaluating the full derivative of Eq. (24) with respect to Ta and setting dXa /dTa and dYa /dTa to zero, which yields X=τ−

–12

and

Y =τ− 2

–14 .

1

⎡ 8Fr( g ′ φ V 3 ) 12 ⎤ 4 p 0 ⎥ rp = ⎢ ⎥ ⎢ Vs ⎦ ⎣

Y 2 = X + –14 .

(26)

Particle-driven currents

r 2h = V and

(27)

tp =

4V Vs rp2

to non-dimensionalize the governing equations. The lengthscale rp corresponds to the radius of the particledriven intrusion at which the volume fraction vanishes. Defining the dimensionless variables as Rp = r/rp, Xp = X/rp, Tp = t/tp and Φ = φ/φ0 and integrating the equations of motion, we obtain 1

Rp = tanh–2Tp,

(32a)

Φ = sech4Tp

(32b)

Xp = λTp.

(32c)

and

In these expressions there remains a single dimensionless parameter, λ=

A conceptually similar model may be developed to describe the gravity-driven flow of a particle-laden intrusion within a uniformly flowing ambient. This motion, however, is complicated by the sedimentation of particles from the current to the lower boundary which progressively reduces the density, and hence the buoyancy-induced propagation velocity is also reduced. Box-model equations for the conservation of fluid volume and the rate of radial propagation may be formulated in a frame of reference moving with the centroid. They yield

and

(31)

(25)

where (X, Y ) are non-dimensional co-ordinates. In these non-dimensional variables the region is given by the parabola

(30)

where φ0 is the initial volume fraction of particles. Hence we use the length and timescales

(23)

(24)

(29)

Thus from Eqs (28) and (29) we find that

Hence the boundary of the spreading gravity current, expressed in terms of a fixed frame of reference in which Xa and Ya are dimensionless horizontal coordinates, is given at any particular time by (Xa − Xa)2 + Y 2a = R a2.

(28)

Particle settling is modelled in an analogous manner to the two-dimensional case

ra = 2Fr( g′V ) –2 /U. (22)

and

Ra = T a–2

1 dr = Fr( g ′p φ h) 2 . dt

4UV , r 3pVs

(33)

which measures the magnitude of the horizontal flux of fluid against the vertical settling flux. (It is analogous to the parameter Λ introduced above.) These equations govern the temporal evolution of the radius and volume fraction of particles of the particle-laden cloud in a frame moving with the centroid. In a fixed frame of reference, the perimeter of the cloud is given by (X − λTp) 2 + Y 2 = R 2p.

(34)

The region within which the gravity current propagates can now may be calculated. It corresponds to the region within which particles are deposited from the

PGCC08

10/12/08

4:28 PM

Page 132

132

A.J. Hogg and H.E. Huppert η( x, y ) = tp ρpVs φ0

Tf

Φ dτ,

(37)

Ts

where Ts and Tf are the dimensionless times at which sedimentation starts and finishes at this particular location. If the dimensionless position, (X, Y ), falls outside of the locus given by Eq. (35a, b) there is no deposit. Conversely if the position lies within the locus the deposit is given implicitly by R

η( X , Y ) =

Fig. 7. The boundary of the region within which the particledriven gravity current propagates for λ = 0.5, 1, 2.

current. The boundary of this region, in parametric form, is given by X =−

sech 2 τ + λτ 2λ

(35a)

and 1

⎛ sech 4 τ ⎞ 2 Y = ⎜ tanh τ − . 4λ2 ⎟⎠ ⎝

(35b)

We plot these loci in Fig. 7. The maximum upstream distance, Xm, propagated by the current may be calculated by finding the value of the parameter τ for which Y = 0 (Eq. 35b) which is then substituted into (Eq. 35a). The distance, Xm, may be evaluated in the regimes of weak (λ > 1) ambient flows relative to the initial buoyancy-induced speed of propagation. We find that Xm = −1 + –12 λ(1 + ln 2 − ln λ) − –14 λ 2 + O (λ4) =−

⎛1⎞ 1 1 + + O⎜ 9 ⎟ 4 λ 192λ5 ⎝λ ⎠

(λ > 1). (36b)

The distribution of the deposit arising from the flow of the particle-driven intrusion may be calculated in a manner analogous to the two-dimensional case. It is assumed that the particles sediment out of the flow uniformly throughout the area covered by the particleladen cloud. Therefore at a dimensional location (x, y) the deposit measured in mass per unit area is given by

4Vρp ⎡ 2 R 6p ⎤ p ⎥ , ⎢R − 3 ⎦⎥ rp2 ⎢⎣ p Rs

(38)

where the terms within the square bracket are evaluated at the upper and lower limits of Rf and Rs which are the dimensionless radii of the intrusion corresponding to the times Tf and Ts, respectively. In Fig. 8 we plot contours of the deposit, noting that the degree of asymmetry is a function of the parameter λ.

DISCUSSION AND CONCLUSIONS We have investigated the flow and dispersion of small, heavy particulate matter in a uniform horizontal flow. We analysed the current using the box model approximation which yields a simple analytic formulation. We considered both two dimensional (line source) and axisymmetric (point source) releases of the particles and experimentally verified our quantitative results for the former geometry. In the two-dimensional situation the flow varies with the parameter Λ = 10UA/(l 2∞Vs), as given by Eq. (14), which represents a ratio of the advective flux of the flow to the downward particle flux within the flow. The maximum upstream penetration is given (to a high degree of approximation) by Eq. (17). The particle concentration within the current as a function of time and position is given by Eq. (13) and the final deposition density by Eq. (16). In the situation where the particle dispersion is unconfined (axisymmetric), the governing parameter is λ = 4UV/(r3pVs) and the horizontal extent of the current is given by Eq. (35) with the deposit density represented by Eq. (38). The upstream penetration against the main flow can have important (and to some people surprising) consequences. Ms Mary-Louise Timmermans (HEH’s graduate student) tells us that when she and her medical father visited Dawson City, Yukon in Northern Canada, he was approached by the town authorities about an outbreak of disease which seemed to occur after the town’s sewerage outlet was discharged into

PGCC08

10/12/08

4:28 PM

Page 133


133

diameter and have an excess density of 1 g cm−3, which implies a settling velocity in water of 0.022 cm s−1. A suspension of such particles with concentration 5% by volume and cross-section (in the two-dimensional situation) of 100 m2 leads to a length-scale l∞ of 1.1 km. In a uniform flow (such as the tides, for example) of 10 cm s−1, Λ = 0.25 and the particle-laden current propagates upstream over a distance d+ of 200 m. If, instead, a total of 103 m3 of suspension of volume concentration 5% is instantaneously released from a point source in the same uniform flow the radial scale rp is 180 m, λ = 0.2 and the upstream penetration length is 120 m.

ACKNOWLEDGEMENTS We are very grateful to Dr. Mark Hallworth for helping us with the experiments and their analysis and for preparing the figures for this publication. A.J.H. acknowledges the support of a grant from the Nuffield Foundation (NUF-NAL).

REFERENCES

Fig. 8. (a) Contours of the distribution of the deposit per unit area and the (thicker) bounding curves for (a) λ = 0.5, (b) λ = 1 and (c) λ = 2. The deposit has been non-dimensionalized with respect to 4Vρp /r 2p. The contours are shown at intervals of 0.1 in the range 0–0.6 for (a) and (b), and 0–0.4 for (c).

the town river but downstream of the water-supply intake. Ms Timmermans had no difficulty in explaining the problem! In order to give some quantitative feeling for our results, consider now the following examples. For each of them consider the particulate matter to be 20 µm in

Baker, E.T., German, C.R. & Elderfield, H. (1995) Hydrothermal plumes over spreading-center axes: global distributions and geological inferences. Physical, chemical, biological and geological interactions within seafloor hydrothermal systems. Geophys. Monograph 91, 47–71. Benjamin, T.B. (1968) Gravity currents and related phenomena, J. Fluid Mech. 88, 223–240. Bonnecaze, R.T., Hallworth, M.A., Huppert, H.E. & Lister, J.R. (1995) Axisymmetric particle-driven gravity currents. J. Fluid Mech. 294, 93–121. Bonnecaze, R.T., Huppert, H.E. & Lister, J.R. (1993) Particle-driven gravity currents. J. Fluid Mech. 250, 339–369. Dade, W.B. & Huppert, H.E. (1994) Predicting the geometry of channelised deep-sea turbidites. Geology 22, 645–648. Dade, W.B. & Huppert, H.E. (1995) Runout and finesediment deposits of axisymmetric turbidity currents. J. geophys. Res. 100, 18597–18609. Ernst, G., Hogg, A.J. & Huppert, H.E. (2001) Deposit densities resulting from explosive umbrella clouds influenced by the wind. (In preparation.) Hallworth, M.A., Hogg, A.J. & Huppert, H.E. (1998) Effects of external flow on compositional and particle gravity currents. J. Fluid Mech. 359, 109–142. Hogg, A.J., Ungarish, M. & Huppert, H.E. (2000) Particledriven gravity currents: asymptotic solutions and box models. Eur. J. Mech. 19, 139 –165. Huppert, H.E. (1997) Buoyancy effects in fluids. Part III Mathematical Tripos Examination. Huppert, H.E. (1998) Quantitative modelling of granular suspension flows. Phil. Trans. R. Soc. A 356, 2471–2496.

PGCC08

10/12/08

134

4:28 PM

Page 134


Huppert, H.E. & Simpson, J.E. (1980) The slumping of gravity currents. J. Fluid Mech. 99, 785–799. Huppert, H.E., Kerr, R.C., Lister, J.R. & Turner, J.S. (1991) Convection and particle entrainment driven by differential sedimentation. J. Fluid Mech. 226, 349–369. Lane, S.N., Biron, P.M., Bradbrook, K.F., Butler, J.B., Chandler, J.H., Crowell, M.D., McLelland, S.J., Richards, K.S. & Roy, A.G. (1998) Measurements of river channel flow processes using acoustic Doppler

velocimetry. Earth Surface Processes & Landforms, 23, 1247–1267. Rottman, J.W. & Simpson, J.E. (1983) Gravity currents produced by instantaneous releases of heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–100. Simpson, J.E. (1997) Gravity Currents in the Environment and the Laboratory. Cambridge University Press, Cambridge. Simpson, J.E. & Britter, R.E. (1980) A laboratory model of an atmospheric mesofront. Q. J. R. Met. Soc. 106, 485–500.

PGCC09 9/12/08 3:14 PM Page 135


Ping-pong ball avalanche experiments J . M c E L W A I N E and K . N I S H I M U R A Institute of Low Temperature Science, University of Hokkaido, North 19 West 8, Kita-Ku, Sapporo 0060-0819, Japan

ABSTRACT Ping-pong ball avalanche experiments have been carried out for the last three years at the Miyanomori ski jump in Sapporo, Japan, to study three-dimensional granular flows. Up to 550 000 balls were released near the top of the landing slope. The balls then flowed past video cameras positioned close to the flow, which measured individual ball velocities in three dimensions and air pressure tubes at different heights. The flows developed a complicated three-dimensional structure with a distinct head and tail, lobes and ‘eyes’. ‘Eyes’ have been observed in laboratory granular flow experiments and the other features are similar not only to snow avalanches, but also to other large-scale geophysical flows. The velocities attained showed a remarkable increase with the number of released balls. A power law for this relation is derived by similarity arguments. The air pressure data is used to deduce the structure of the air flow around the avalanche and, in conjunction with the kinetic theory of granular matter, to estimate the balance of forces in the avalanche head.

INTRODUCTION used makes direct observation of individual particles difficult, and only a few similarity parameters are typically satisfied (Keller, 1995). For example, no laboratory experiments have yet been carried out in which a dense granular flow becomes a turbulent suspension by entraining the ambient fluid, though in some experiments (Rzadkiewicz et al., 1997) a small number of the grains may enter suspension. Instead experimental models of powder snow models in water tanks use a denser fluid or a premixed turbulent suspension. Laboratory granular flows also rarely exhibit the complex three-dimensional structure which is characteristic of avalanches and other large geophysical flows. For these reasons, for the last five years, large scale granular flow experiments have been carried out using golf balls and ping-pong balls. The first experiments were carried out on long (20–30 m) chutes and more recently on the Miyanomori ski jump. Ping-pong balls are particularly suitable, since they reach terminal velocity in only a few metres, so fully developed flows occur even on short slopes. These experiments have been described in several papers (Nishimura et al., 1996, 1998; Keller et al., 1998). The aim of these experiments is to elucidate the

Snow avalanches have been measured and observed in the Shiai valley, Kurobe since 1989. Though there are partial data on the internal velocity distribution for both dense and powder parts (Gubler, 1987; Kawada et al., 1989; Nishimura et al., 1989, 1993a; Dent et al., 1994; Nishimura & Ito, 1997) the data are insufficient to constrain and discriminate between current avalanche models (for a recent survey of current models, see Harbitz (1999)) and thus insufficient to allow a quantitative understanding of the dynamics and internal structure of snow avalanches. The poor quality of the data is because of the unpredictability, scarcity and intense destructive power of avalanches. Avalanches can be modelled in the laboratory using granular materials on inclined planes, usually in water for powder avalanches (Tochon-Danguy & Hopfinger, 1975; Hopfinger & Tochon-Danguy, 1977; Beghin & Brugnot, 1983; Hermann et al., 1987; Beghin & Olagne, 1991; Keller, 1995) or air for dense avalanches (Hutter, 1991; Hutter et al., 1995; Nishimura et al., 1993b; Greve & Hutter, 1993; Greve et al., 1994). Laboratory experiments are much easier to perform than field experiments and are, usually, easily repeatable. However, the small size of the granular particles


135

PGCC09 9/12/08 3:14 PM Page 136

136

J. McElwaine and K. Nishimura

dynamics of two-phase granular flows rather than to directly extrapolate the results to snow avalanches. The experiments provide detailed data and provide insights on the physically significant dynamical processes controlling avalanches. The hope is that this will lead to a theory of snow avalanches based on physical processes with no free parameters. The kinetic theory of granular matter provides only poor agreement with experiments (Jenkins & Savage, 1983; Haff, 1983; Lun et al., 1984; Jenkins & Richman, 1988; Johnson et al., 1990; Anderson & Jackson, 1992; Jenkins, 1994), but does provide a theoretical framework for discussing stresses in granular flows. Another approach is the direct simulation of granular flows using the discrete element method (Campbell & Brennen, 1985; Campbell & Gong, 1986; Cleary & Campbell, 1993; Campbell et al., 1995; Hanes et al., 1997). These simulations have increased the understanding of granular flows, including two-phase flows, but these simulations have not yet accurately dealt with particles strongly coupled to fluids or threedimensional anisotropic flows. The ping-pong ball avalanches can be described by well known equations. The air flow obeys the Navier–Stokes equations and individual ping-pong balls follow Newton’s laws, whereby the force on a particle is a function of gravity, particle–particle contacts, particle–ground contacts and air drag. The no-slip boundary condition between particles and the air flow determine the drag force. For small numbers of particles at low Reynolds numbers in closed domains these equations can be directly solved (Glowinski et al., 1996; Hu, 1996; Blackmore et al., 1999), but for this experiment it is currently impossible, because of the large number of particles and the large range of length and time scales. Particle–particle collisions occur over time intervals of order of 10−3 s whereas the duration of the flow in these experiments is around 30 s. The length scales in these experiments are given by the length of the ski jump (160 m), the volume of the flow ( 3 1 m3 = 1 m), the diameter of the balls (0.038 m) and the compression of the balls during collisions (10−3 m). This chapter discusses two complementary approaches for describing the experiments. The first is to consider the flow as a single object moving down the ski jump and to use similarity arguments to deduce gross features of the flow. The second approach is to use two-phase flow equations that couple the Navier–Stokes equation for the air flow to the kinetic theory equations for the ball flow using an (empirical) drag force.

EXPERIMENT The experiments were undertaken at the Miyanomori ski jump (the normal hill for the 1972 Olympics) in Sapporo, Japan. The landing slope was 160-m long and 60-m high (Fig. 1) and covered with an artificial surface. Standard ping-pong balls with a diameter of 38 mm and mass of 2.48 g per ball, were placed in a large box, 15 m or 30 m from the top of the landing slope. Flow was initiated by opening the hinged door on the front of the box. The balls then flowed down the slope and past the measurement sensors, which were all placed near the middle of the slope 100 m down from the top (83 m from the front of the box). The experimental procedure is described in more detail in Nishimura et al. (1996). Ball measurements A video camera was set pointing perpendicularly down at the slope (Fig. 2) at a height of 0.82 m. Balls closer to the lens of the camera appear larger than those which are further away. Thus the z-coordinate of a ball can be calculated by measuring the size of a ball in the video picture, since all the balls have the same diameter. The x- and y-coordinates of a ball are given directly by its position in the video frame. Comparing positions for the same ball from adjacent video fields gives the three-dimensional velocity of 1 a ball (time-averaged over the 60 s between video fields). The necessary camera corrections and detailed method are described in Keller et al. (1998). Air measurements Measuring air velocity in particulate flows is very difficult. In snow avalanches ordinary meteorological anemometers are inaccurate because of the snow particles, and are usually destroyed (Kawada et al., 1989; Nishimura et al., 1989). Nishimura et al. (1996) and Nishimura & Ito (1997) have developed the use of pressure measurements (sampling frequency 1 kHz) for inferring air speed. A tube connected to a pressuredifference sensor is set so that the open end points downwards, perpendicular to the main flow direction. Note that this is not a Pitot tube since each tube has only one opening and the pressure difference is measured with respect to the air pressure some distance from the flow. Bernoulli’s law then gives ∆p = − –12 ρaν 2,

(1)

PGCC09 9/12/08 3:14 PM Page 137

137

Ping-pong ball avalanche experiments

Fig. 1. Cross-section of the landing slope of the Miyanomori ski jump. Marked distances are measured from the top of the landing slope.

slope and at the bottom of the slope. As can be seen in Fig. 3 the leading edge of the avalanche is clearly visible. The position of the front was measured and used to calculate the front velocity.

z

y x

RESULTS

v v

Fig. 2. Schematic of the video camera for measuring ball positions.

where ∆p is the pressure difference, ρa is the air density and ν is the air speed parallel to the slope. However, this equation is only valid when the air flow is perpendicular to the end of the pipe and the local static pressure is known. Also the sensor itself disturbs the flow. The Reynolds number for the flow around the tube is approximately 10 000 (wind speed 10 m s−1, tube diameter 0.01 m), so the flow will be partially turbulent around the sensor. The interaction of the pressure sensor with the flow coupled with the rapid pressure fluctuations as a result of the turbulent flow field would lead to inaccurate measurements, if solely based on Eq. (1). Therefore the pressure tubes were calibrated by measuring the static pressure depression in a wind tunnel over a range of velocities. Four of these air pressure sensors were placed 100 m down the slope at heights of 0.01 m, 0.15 m, 0.3 m and 0.45 m. Front position Several video cameras were placed to the side of the

When the door of the box was opened, the balls at the front of the box rapidly accelerated down the slope (Figs 3 and 4). The front velocity was much larger than the tail velocity (the last balls took several seconds to leave the box). For a 550 000 ball flow, the front of the flow accelerated approximately linearly with distance until it reached a speed of 18 m s−1 after 65 m, whereas the balls in the tail had a speed of only a few metres per second—similar to the speed of a single ball. The front velocity was roughly constant for the next 30 m until the slope angle started to decrease. This large disparity in speed between head and tail caused the flow to elongate so much that at times it covered more than half the slope. The flows can be separated into three distinct regions: a short, high, fast moving head; a longer, lower body moving at the same speed; and a very long tail moving much more slowly, consisting of separated balls. Other macroscopic features of the flow are interesting but hard to quantify. At the beginning of the flow there are often several waves within the flow which move faster than the body and coalesce in the head (Nishimura et al., 1998). Another obvious feature are two roughly circular regions of reduced flow height, symmetrically located about the flow centreline, a little behind the head, called ‘eyes’ after Nohguchi et al. (1997). They can be seen on the third line up from the bottom of Fig. 3 as the darker regions. Similar patterns have been reported in laboratory granular flow experiments with styrene foam particles (Nohguchi,

PGCC09 9/12/08 3:14 PM Page 138

138

J. McElwaine and K. Nishimura different scales suggests that the mean velocity fields and flow structure are similar in all these experiments. The ‘eyes’ may represent a pair of vortices shed by the head, but only a detailed quantitative analysis of ball velocities can confirm this. In the tail the balls are not distributed evenly but tend to cluster, because of inelastic collapse. (As density in a granular flow increases, the collision rate increases thus increasing dissipation and reducing granular pressure. The density thus continues to increase and the collision rate diverges, so that a group of particles can come to rest in continuous contact in finite time.) Front velocity In Nohguchi (1996), granular flow experiments with styrene particles were performed and the front velocity was observed to increase with the number of balls. Similar increases were observed in these experiments. Nohguchi (1996) deduced that the maximum front velocities, u, for flows which vary only in the number of balls, N, should scale according to 1

u ∝ N –6.

Fig. 3. Front view of a 550 000 ball avalanche at the Miyanomori ski jump. The horizontal lines are 5-m apart and the lowest one is 90 m from the top of the landing slope.

Fig. 4. Side view of the head of a 5 550 000 ball avalanche at the Miyanomori ski jump.

1996) and with ice particles (Nohguchi, pers. comm.). In these experiments the particles are around 1 mm in diameter and the flows contain 1000–100 000 particles. For such a feature to exist in experiments of such

(2)

In order to derive Eq. (2) the drag force was assumed to be a linear function of the flow velocity. However, the result can be obtained without this assumption as follows. The critical assumption is that there is only one significant lengthscale given by 1

1

L(x) ∝ V –3 ∝ dN –3,

(3)

where V is the volume of all the balls and d the ball diameter. The implied constant of proportionality in Eq. (3) is constant between experiments with different numbers of balls, but is not constant along the slope, i.e. the height and width of the flow at any given position (x) scales with the number of balls. Equation (3) will not be true initially when the input box size is important nor will it be true where the flow is only a few balls thick. However, all the flows with more than ≈ 10 000 balls are observed to rapidly reach a selfsimilar shape in around 10 m and the flows are many particle diameters thick, except in the tail. The effective gravitational acceleration on the flow is g* = g(1 − ρa /ρb)(sin θ − µ cos θ ), where θ is the angle of the slope, µ the friction with the slope, g the acceleration due to gravity and ρa and ρb are the air and ball densities, respectively. After the initial surge from the box the flow is close to its equilibrium velocity, i.e. it is accelerating/decelerating slowly, so inertia can be ignored. The Reynolds number for the air flow

PGCC09 9/12/08 3:14 PM Page 139


139

Fig. 5. Front velocities at the k point for different sized avalanches.

is of the order 106 so air viscosity can be ignored. Under the length scale assumption (Eq. 3) the nondimensional density ratio ρaV/(Nm), where m is the mass of a single ball, is constant for different sized flows since V ∝ N. Therefore air density ρa need not be further considered as a dimensional variable since it can be substituted by m/L3. The dependence on the box size and ball diameter has already been discussed which leaves only three variables g*, L and u. Thus the only dimensionless combination that can be formed containing the front velocity is the densiometric Froude number Fr( x ) =

u 2 (x ) . L( x )g*( x )

(4)

This must be constant for different flows thus 1

u ( x ) = L( x )g*( x )Fr( x ) ∝ N 6 , –13

(5)

since L ∝ N . In Nishimura et al. (1998), the front velocity was measured between the k point and the p point (where the slope angle, 36°, is roughly constant and steepest, see Fig. 1). The remarkably good fit between this equation and experiment is seen in Fig. 5 and provides additional justification for Eq. (3). As expected, the error is worse for small flows, since they rapidly spread into single thickness layers with two significant length

Fig. 6. Ball heights in a 200 000 ball avalanche calculated as the balls are advected beneath a fixed video camera. The lines show the ball trajectories from one field to the next.

1

scales dn–2 (width and length) and d (height). The height is likely to be the significant length scale in this range so for small flows we expect the velocities to be independent of flow size. Flow structure and ball velocities By analysing the video film (Keller et al., 1998), individual particle positions can be calculated, and by identifying balls between adjacent video frames, particle velocities are obtained. Figure 6 shows the perpendicular positions and velocities for a 200 000 ball flow as the particles are advected beneath the camera. 1 The time interval of one profile is 17 ms ( 60 s). This technique, however, cannot see through ping-pong balls, and in the dense head (volume fraction ≈ 0.2) only the balls from the top 0.2 m can be identified thus there is a blank region, marked passage of the head, in Fig. 6 where there is no data. If the structure of any feature in the _ow of size l is changing slowly with respect to l/E, where E is the mean flow velocity, then we can regard this data as providing a cross-section through the flow in the direction of mean velocity, in this case down the slope. For Fig. 6, the mean flow velocity is 15 m s−1 thus 0.1 s corresponds to 1.5 m. The head of 1-m long, 0.4-m high followed by a body 0.2-m high is visible. The full flow (not shown in Fig. 6) has a body of approximately

PGCC09 9/12/08 3:14 PM Page 140

140


Fig. 7. Vertical profile of ball downslope (x) velocity. The height and velocity of each data point are an average over 50 balls calculated from video camera measurements of the front and body.

The central part of the ski jump is composed of downslope pointing bristles and experiments with individual balls show that it is totally inelastic—the balls bounce to no observable degree—so that horizontal momentum cannot be converted to vertical momentum by collisions with the slope, but only in collisions with other balls. Since vertical motion will rapidly decay through ground collisions, a priori, one might have expected a high density flow where the balls are in continuous contact, with very small velocity fluctuations. This is indeed what happens initially when the balls slump out of the box. However, this dense flow state is unstable and as the flow accelerates the velocity fluctuations increase and the density decreases. The kinetic theory of granular matter follows that of gases and describes a system by a particle distribution function f, where f(c, x, t) dc dx is the number of particles with velocity c and range dc that are centred at x and range dx at time t. The number density n(x, t) is the integral of f over all velocities and the volume fraction φ (x, t) = n (x, t)πd 3/6. The mean value of any particle property ψ(c, x, t) is defined as 〈ψ 〉 =

constant height 0.2 m and length 10 m followed by the tail of the flow which stretches back to the box and consists of separated balls. The shape of the velocity profile in a steady shear flow is governed by the relative magnitude of the drag forces on the upper and lower surfaces, the body forces and the vertical transport rate of momentum. Figure 7 shows that the mean down-slope (x) velocity of the balls decreases monotonically with height. There is no visible velocity reduction at the base indicating that surface friction is unimportant. The mean velocity slowly decreases in the dense part of the flow by around 1 m s−1 and then very rapidly in the less dense top layer by a further 1–2 m s−1. This diffuse, top layer of saltating balls moving along approximately parabolic trajectories is visible in Figs 4 and 6 and has been discussed in the literature (Johnson et al., 1990). This behaviour is characteristic of dilute energetic flows. In high-density flows, on the other hand, the top surface is well defined to within a particle diameter. The lower mean velocities of these saltating balls is easily explained by the extra air drag they experience since they move in regions of higher relative air velocity. The relative air velocity in the bulk of the flow must be much lower since the flows are in approximate equilibrium and move up to four times faster than the terminal velocity of an individual ball.

1 n ( x, t)

f (c, x, t) ψd c.

(6)

The mean velocity field u(x, t) is thus 〈c〉, the fluctuation velocity C = c − u and the second moment of the fluctuation velocity K(x, t) = 〈CC〉. The granular temperature T(x, t) ≡ –31 (Kxx + K yy + kzz ) is the isotropic component of K. The stress tensor (sometimes referred to as the ‘pressure tensor’ (Jenkins & Savage, 1983)) for a granular flow is Θ [C], σ = φρbK + mΘ

(7)

where ρb is the ball density, m is particle mass and Θ [C] denotes the collisional transport of velocity fluctuation. The notation follows Jenkins and Richman (1988), where it is shown that in dilute flows the collisional transport term can be ignored. The dilute approximation consists of retaining only terms that are constant or linear with respect to volume fraction φ and is valid when the strength of mean shear relative to velocity fluctuations is small. This approximation is assumed valid for the rest of the chapter. The square root of the diagonal elements of K are the velocity standard deviations along the coordinate axis and are shown in Fig. 8. The standard deviation is taken over each video field. This shows that the perpendicular (Kzz), and cross-slope (Kyy) velocity deviations are roughly similar in the head and the body, 1.5 m s−1 and 0.5 m s−1 respectively. However, the

PGCC09 9/12/08 3:14 PM Page 141


141

Fig. 8. Ball velocity standard deviation for a 300 000 ball experiment. (The data have been smoothed.)

down-slope velocity deviation (Kxx) is low initially, then increases rapidly in the head to reach a maximum of 2.5 m s−1 before decaying to a roughly constant 0.5 m s−1 in the body. The results are similar for other flows. Kinetic theories (Lun et al., 1984; Anderson & Jackson, 1992) of granular matter often postulate that K is isotropic, i.e. the diagonal stresses Kxx, Kyy and Kzz are identical and the off-diagonal stresses are zero. This is clearly not the case for these flows. Figure 8 shows that the diagonal elements of K are never equal. These data are consistent with video footage in which horizontal velocity structure is visible and with Fig. 7 that shows that there is no appreciable vertical shearing. In the case of steady and uniform flow the mean velocity must be constant and the momentum equation for the flow is ∇ · φρbK + φ∇p = φρbg + φf

(8)

where p is the air pressure, g is gravity and f is the drag force from the air on the balls (Jenkins, 1987). For a free surface to be steady and clearly delineated there is a kinematic constraint that n·K·n vanishes on the surface, where n is the surface normal. That is to say as well as the mean velocity vanishing normal to the surface so must the velocity fluctuation. This term, if nonzero, would result in a diffusion outward from the surface of the volume fraction φ. The top surface in contrast is diffuse, φ slowly decreases with z, and such a condition is not satisfied. Figure 4 shows that the front is indeed very clearly defined which requires that

Kxx = 0. Since K is calculated from averages over two video fields the value at the front is not known, but extrapolating the curve make it plausible that Kxx is indeed zero (Fig. 8). There is also a dynamical requirement given by Eq. 8 that the forces on the front should balance. ∂σxy ∂σxx ∂ σxz ∂p + + +φ = φ fx − φρb g sin θ. ∂x ∂y ∂z ∂x (9) Unfortunately only a dozen balls or so in each video frame can be identified, which does not provide enough data to calculate y and z derivatives, unless averaged over the whole length of the flow. Assuming that the only z dependence of stress is that required to counteract gravity the ∂/∂z terms can be dropped. The ∂/∂y terms should be zero on the centre line (y = 0) of the flow and can also be dropped. The equation is then ∂ σxx ∂p +φ = φ fx − φρb g sin θ. ∂x ∂x

(10)

and can be integrated if fx is known to provide a variant of Bernoulli’s law. This equation will be returned to after a discussion of the air pressure data. Pressure measurements Although in the general case of flow past bodies of arbitrary form the actual flow pattern bears almost no relation to the pattern of potential flow, for streamlined

PGCC09 9/12/08 3:14 PM Page 142

142


Fig. 9. Streamlines for irrotational flow around a stationary sphere.

shapes the flow may differ very little from potential flow; more precisely, it will be potential flow except in a thin layer of fluid at the surface of the body and in a relatively narrow wake behind the body (Landau & Lifschitz, 1987). In particular, in front of the avalanche head the flow will be irrotational since the Reynolds number (Re) is very high (for length of 1 m, velocity 10 m s−1, Re ≈ 106 ). A simple approximation is to assume that the flow field is that of irrotational flow around a sphere (Fig. 9) where the sphere represents the head of the gravity current (cf. Fig. 6) in a stationary frame. The flow field has the required symmetries since it is symmetric about the crossstream ( y = 0) plane and, if the influence of the ground on the air flow is assumed to be small, the flow field can be reflected in the perpendicular (z = 0) plane. A similar approach to the ambient flow around gravity currents was pioneered by von Kármán (1940). He considered the local flow around the point where the head meets the ground and used this to deduce the head angle (60°). This is accurate over distances small compared to the head height. Similar ideas were also discussed in Hampton (1972), but he considered the ambient flow around semi-infinite debris flows, thus his approach is correct over scales large compared to the head height but small compared to the flow length. In contrast, the approach in this chapter is equivalent to retaining the first three terms (up to the dipole) in a multipole expansion and is therefore asymptotically correct. To apply Bernoulli’s theorem, it is most convenient to work in a frame in which the flow field can be approximated as steady. This is true in a frame moving with the same velocity as the avalanche head since the slope angle changes slowly. The velocity distribution around a stationary sphere of radius R in a flow field moving with constant velocity v at infinity is

v( x ) = −v +

⎞ R 3 ⎛ 3x( x ⋅ v ) − v⎟ ⎜ 3 2 2x ⎝ x ⎠

(11)

Using Bernoulli’s theorem the corresponding pressure distribution is ∆p( x ) =

1 R3 ρa ν2 3 2 x

⎡ ⎛ R3 3 R3 ⎞ ⎤ ⎢2 − 3 − sin2 θ ⎜ 4 − 3 ⎟ ⎥ , 4 x x ⎠ ⎥⎦ ⎝ ⎢⎣ (12)

where x cos θ = x·v. Regarding the pressure sensors as fixed on the centre line, the data is not of sufficiently high quality to warrant a more complicated approach, then x(t) = v[(R/ ν) − t], where t = 0 is taken as the time when the front reaches the sensor. The output from a pressure sensor is then ∆p(t ) =

⎛ 1 νt ⎞ ρa ν2 ⎜1 − ⎟ 2 R⎠ ⎝

−3 ⎛ νt ⎞ ⎤ ⎢2 − ⎜1 − ⎟ ⎥ . (13) R⎠ ⎥ ⎢ ⎝ ⎦ ⎣

−3 ⎡

Figure 10 shows the result of fitting this curve to the data from one of the sensors. The equation has three free parameters: the impact time, which is taken as the point of highest pressure; the effective radius R; and the effective velocity ν. The pressure data was sampled at 100 Hz and passed through a 4 ms width Gaussian filter, ρa was taken as 1.2 kg m−3 and an additional correction was applied after calibrating the sensors in the wind tunnel. The velocities implied by the pressure data are shown in Table 1. The lower three sensors are all in rough agreement with the velocity increasing slightly with height. The difference between these velocities and the video derived head velocity (of order 5 m s−1) is the penetration velocity of the air into the head. Unsurprisingly, this decreases with height as the air flows over the avalanche rather than into it. The flow

PGCC09 9/12/08 3:14 PM Page 143

143


Table 2. Comparison of implied radii for 150 000 (R1) and 300 000 (R2) ball avalanches 1

Height (m) 0.01 0.15 0.30 0.45

Fig. 10. Static air pressure change as the front of a 300 000 ball avalanche is advected past the sensor at height 0.3 m. The balls reached the sensor at t = 0 and for t < 0 the line of best fit (least mean squares) is drawn assuming the pressure distribution in front of a sphere (two free parameters effective radius R and velocity ν). Table 1. Comparison of implied velocities for 150 000 (ν1) and 300 000 (ν2) ball avalanches Height (m) 0.01 0.15 0.30 0.45

R1 (m)

R2 (m)

2 3 R1/R 2

2 γ ′R1/R 2

0.76 0.68 0.87 0.98

0.88 0.80 1.02 1.08

1.09 1.06 1.08 1.14

1.02 1.00 1.01 1.07

expanding the surface to second order in the coordinates). The equation fit is influenced by the region of high pressure difference close to the flow front and the length scale measured here is actually the local radius of curvature. Thus 1/R = (1/R1) + (1/R2). Video footage and pictures of the slope shows that the flow front is reasonably approximated by the parabola y = x2/d where d = 5 m and y is the distance from the centre line. Thus in Fig. 3 it can be seen that 5 m back from the front the flow is 10-m wide. The measure radius of curvature in the x–y plane is thus R2 = d /2 = 2.5 m independent of the flow scale. This does not necessarily contradict the scaling hypothesis, because this is a local length scale and1 the width of the flow is still expected to scale as N –3. Thus if R1 scales and R2 is constant the ratio between front radii is R ′ /R =

1

ν1 (m s−1)

ν2 (m s−1)

2 6 ν1 / ν2

7.55 8.21 8.88 6.81

8.16 9.66 10.13 8.96

1.04 0.95 0.98 0.85

(1/R 1 ) + (1/R 2 ) 1/( λR1 ) + (1/R 2 )

⎡ λ −1 ⎤ = λ ⎢1 + ⎥ ⎢⎣ 1 + (R 2 /R 1 ) ⎥⎦

−1

1

where λ = (N ′/N)–3 the length scaling ratio. When λ is close to 1 this can be simplified to R ′ /R = λR2 /(R1+R 2 ) + O [(1 − λ ) 2 ]

velocities from the top sensor (height 0.45 m) are low because it is largely out of the flow in a region of reduced air velocity. The third column of Table 1 compares the air velocities with scaling Eq. (2). The agreement for the lowest three sensors in the flow is very good and provides further evidence in favour of the length scaling hypothesis. Though the calculated velocities match the scaling law reasonably well the radii do not (Table 2). A possible explanation is as follows. The flow field far from the body is that of a dipole imposed on constant flow. The magnitude of the dipole is the surface area of the implied sphere times the velocity πR2u. Close to the front however the flow field, to second order, will be more like that around an ellipsoid (this is the result of

(14)

thus the scaling exponent γ = is altered to γ′ = γ/[1 + (R1/R2 )]. To the same order of approximation R1 can be taken from the flows for either N or N′. The fourth column of Table 2 shows the much better –5 . fit obtained with this analysis R2 /R1 = 2.5 and γ′ = 21 This is a very tentative solution and an explanation is still required as to why the front should have a constant parabolic shape. The definition of the calculated radius is somewhat arbitrary. The pressure data could be equivalently fitted to –1 3

∆p(t) = –12 ρaν 2[1 − (t/t0)]−3 {2 − [1 − (t/t0)]−3} (15) where t0 is a time constant. The radius is then deduced from R = t0ν. The above analysis took ν as the air

PGCC09 9/12/08 3:14 PM Page 144

144


velocity, but a more natural choice would be to use the velocity of the coordinate frame, that is the front velocity u if this is known. Since this velocity has the same scaling this would only result in the implied radii being multiplied by some constant factor, thus the previous discussion is unaffected. Air pressure through the front Within the ping-pong ball flow the steady-state mass and momentum equations for the air flow are ∇ · (1 − φ)v = 0,

(16)

ρa(1 − φ)(v · ∇)v + (1 − φ)∇p = −φf.

(17)

To calculate the pressure inside the flow these equations must be integrated. A simple approximation for the force valid to lowest order in φ is f = ρaν /α 2

(18)

where α = ρa u T2 /( ρb − ρa )g = 0.08 m an air drag length scale and uT is the terminal velocity of an individual ball (7.5 m s−1). An analytic solution is not easy to find. However, if the streamlines are not diverging (or converging) too rapidly the continuity equation can be approximated by (1 − φ)v = v0 over short distances if the streamlines diverge slowly, where v0 is the penetration velocity of the air into the front. This is most likely to be true close to the ground where symmetry suggests there will be a streamline passing straight through the centre of the flow. Substituting this into Eq. (17) and integrating along this streamline x

φ(s) ds + O(φ ), (19)

p = p0 − ν20 ρa φ − ν20 ρa /α

2

0

where s is distance along a streamline. Even if the surface of the flow is sharply defined to within a ball diameter, φ changes more slowly because it is defined as an average over a volume containing many balls. Suppose the ball concentration is 0 outside the flow and φc constant inside the flow. Then φ increases lin1 early from 0 to φc over a width w of order d φc− –3. Then integrating Eq. (19) gives. p − p0 =

⎧2 s (1 + s ) w 2α s ⎪(1 + 2 − w ) 2α ⎩

⎪ − ν 20 φc ρa ⎨

s ≤ w, s ≥ w.

(20)

accurate for small s. Over longer distances the streamlines will diverge, as the air flow is deflected up and out of the avalanche, the velocity will decrease and the pressure will increase (see Fig. 11).

Fig. 11. Air pressure through the front at 0.01 m for 300 000 balls. The front reaches the sensor at t = 0 s.

The rapid decrease in pressure as the flow front goes past the sensor that is predicted by this equation is clearly seen in Fig. 11 (and also Fig. 10). The total pressure drops by 84 Pa from 0 s to 0.035 s. The front velocity is around 15 m s−1 so this corresponds to a distance of 0.5 m certainly much larger than w thus Eq. (20) is ν 20φc is c. 9. From Table 1 the air velocity for this flow at 0.01 m is 8 m s−1 thus ν0 = 7 m s−1 so that φc is c. 0.2. This value for the volume fraction seems reasonable and is much lower than the maximum packing fraction thus justifying the dilute approximation. The x momentum equation for the ping-pong balls (Eq. 8) can now be integrated with the same approximations to give K xx ( x ) = [ p( 0 ) − p( x )]/ρb + x

ρa ν 20 − xg sin θ ρb α

(21)

At t = 0.035 s (using linearly interpolation) Kxx = 1.9 m2 s−2, ( p(0) − p(x))/ρb = 0.96 m2 s−2 and x(ρaν 20 / ρbα) = 4.5 m2 s−2 and xg sin θ = 2.5 m2 s−2. These quantities are of the same order showing that the structure of the head is determined by the balance between air drag, granular stress and gravity. Further back in the body of the avalanche Kxx is approximately constant and the pressure varies only slowly. Since the effect of surface drag appears to be small this implies that the air drag on the top surface balances gravity. A more detailed analysis of Eq. (21) is not appropriate for several reasons. The large fluctuations of the air pressure in the avalanche imply that the flow is turbulent and make interpretation of the air pressure data very difficult since the sensor measures a complicated function of the local velocity and local pressure which

PGCC09 9/12/08 3:14 PM Page 145

Ping-pong ball avalanche experiments can only be simply understood if the direction of the velocity is known. The ball velocity data also contains a lot of noise since in a typical frame, only a dozen balls can be identified. Though mean values of velocity are reasonably accurate derivatives of K are much less so. There is an additional problem that the balls that can be identified may be very special (perhaps only those with low vertical velocity have been sampled for example) possibly leading to systematic errors which have not been estimated. In addition the ball position measurements were taken one metre to the left of the flow centre and the location of the pressure measurements. Despite all these difficulties the data does suggest a number of significant processes within the avalanches.

DISCUSSION Classical work on gravity currents is based on perfectfluid theory and assumes that the effects of viscosity and mixing of the fluids at the interface can be ignored (Benjamin, 1968). A major result of Benjamin (1968) is that, except when a gravity current exactly fills half a cavity, energy dissipation must occur through the formation of a head and turbulent flow behind it. Extensions to the basic theory include lower boundary effects (Simpson, 1972) and a mixing region behind the head (Simpson, 1986), but there is still assumed to be a clear boundary at the front of the current.

Fig. 12. Schematic of the air flow round and through a ping-pong ball avalanche.

145

A complete description of the flow field for a mixture of Newtonian fluids requires only one velocity field. This is because there can be no relative motion (at a point) between two fluids since a no-slip condition holds everywhere, thus the velocity fields for each fluid (where they are defined) must be identical. Thus mixing between fluids is a slow diffusion process and there are often well-defined boundaries. The stability of boundaries is also enhanced by surface tension. However, when one of the fluids is a non-cohesive granular fluid, there is no surface tension and the granular fluid will generally have a distinct velocity field. This is because, although on the surface of each grain, a no slip condition holds, very large velocity gradients can exist across a narrow boundary layer. Thus the difference between the ambient fluid velocity field and the granular velocity field averaged over volumes containing a few grains can be very large. For grains falling in a gravitational field for example, the relative velocity will be of the order of the terminal velocity. The standard gravity current theory (Benjamin, 1968) (correctly) assumes a stagnation point at the front of a gravity current because the velocity of the ambient and the current must be equal. This need not be true for granular gravity currents and the air pressure data shows that there is a significant relative velocity over the width of the head (Fig. 12). The drag force is related to the relative velocity so a large difference between these avalanches and standard gravity

PGCC09 9/12/08 3:14 PM Page 146

146


currents is that the drag is a body force over the head of the avalanche rather than a surface force over the head’s front surface. Analysis of the forces in the head of the avalanches shows that there is an approximate balance of forces on the balls between gravity, granular stress and air drag, and that surface friction is negligible. The air drag is balanced by a large, anisotropic increase in the granular stress and gravity. This increase is a result of an increase in the downslope fluctuation velocity which then leads to an increase in vertical and cross-slope fluctuations through collisions. Though a quantatitive balance of the vertical forces in the head has not been accomplished, the granular stress and the vertical component of the drag are probably both significant and lead to the height of the flow. Air drag may also directly enhance vertical velocity fluctuations. Further back in the body of the avalanche, the granular stresses are constant (downslope) and the height is lower. Since surface drag is negligible, the gravity must be balanced by air drag forces through the top surface. A likely mechanism for this is momentum transfer by the saltating particles. During their high trajectories they have time to exchange considerable horizontal momentum with slowly moving air and when they collide with the main body this momentum will be almost perfectly transfered. In effect there is a drag interaction between the main body and the air flow over the whole height of the saltating balls. Though this has not been quantified, this mechanism of momentum transfer is most likely more efficient than the drag on the upper surface of a smooth gravity current and helps explains why steady flows occur on such steeps slopes even with such a large relative density (ρb /ρa is c. 90). CONCLUSIONS The air pressure distribution in front of the ping-pong ball avalanches is well approximated by irrotational flow around a sphere. This approach could be extended to the flow behind the head by comparing the data with turbulent wake theory, but this is difficult because of the complicated interaction of the pressure sensors with the air flow when the velocity direction is unknown. The implied air velocities scale as the sixth power of the number of balls in agreement with dimensional analysis and the scaling for the ping-pong ball velocities. The length scales implied by the air flow are of the same order of magnitude as the front height, but only obey the scaling law if the shape of the head is assumed to have a constant curvature (in the plane of

the slope). Kinetic theory calculations show a quantitative balance of forces in the head between gravity, granular stress and air drag.

ACKNOWLEDGEMENTS The authors gratefully thank the many people who came to Miyanomori for the experiment, which could not have been done without their help. Furthermore we wish to thank the workshop staff of the Institute of Low Temperature Science who made the equipment. This work was partly supported by grant-in-aid for co-operative research and science from the Japanese Ministry of Education, Science and Culture. One of the authors was supported by an EU/JSPS Fellowship.

NOMENCLATURE α c C d f f Fr g, g g* γ K l L λ m µ n n N p p0 φ φ0 ψ r R R1, R2 ρa

air drag lengthscale ball velocity ball fluctuation velocity ball diameter position and velocity distribution function drag force between balls and air Froude number gravity gravity adjusted for buoyancy, friction and slope angle scaling exponent second moment of ball fluctuation velocity length of a feature in the flow lengthscale of a flow lengthscale ratio mass of an individual ball ball-slope friction surface normal ball number density number of balls in an experiment air pressure air pressure at the flow front ball volume density constant ball volume density in the flow general function of balls distance from centre of flow sphere radius of sphere radii of curvature air density

PGCC09 9/12/08 3:14 PM Page 147

Ping-pong ball avalanche experiments ρb σ s t t0 T θ θx, θy, θz u u E v, ν v0, ν0 w x = (x, y, z)

ball density stress tensor distance along a streamline time time constant granular temperature slope angle rotations of the principal axes of K from the coordinate axes mean ball velocity speed of the flow front mean speed of the flow air velocity and speed relative velocity and speed between air and flow front width of the ball volume density variation local Cartesian coordinates aligned with the slope

REFERENCES Anderson, K.G. & Jackson, R. (1992) A comparison of the solutions of some proposed equations of motion of granular materials for fully developed flow down inclined planes. J. Fluid Mech. 241, 145–168. Beghin, P. & Brugnot, G. (1983) Contributions of theoretical and experimental results to powder-snow avalanche dynamics. Cold Reg. Sci. Tech. 8, 67–73. Beghin, P. & Olagne, X. (1991) Experimental and theoretical study of the dynamics of powder snow avalanches. Cold Reg. Sci. Tech. 19, 317–326. Benjamin, T.B. (1968) Gravity currents and related phenomena. J. Fluid Mech. 31(2), 209–248. Blackmore, D., Samulyak, R. & Rosato, A.D. (1999) New mathematical models for particle flow dynamics. Nonlin. Math. Phys. 6(2), 198–221. Campbell, C.S. & Brennen, C.E. (1985) Computer simulations of granular shear flows. J. Fluid Mech. 151, 167–188. Campbell, C.S., Cleary, P.W. & Hopkins, M. (1995) Largescale landslide simulations: Global deformation, velocities and basal friction. J. geophys. Res. 100, 8267– 8283. Campbell, C.S. & Gong, A. (1986) The stress tensor in a two-dimensional granular shear flow. J. Fluid Mech. 164, 107–125. Cleary, P.W. & Campbell, C.S. (1993) Self-lubrication for long runout landslides: Examination by computer simulation. J. geophys. Res., 98(B12), 21911–21924. Dent, J.D., Adams, E., Bailey, I.J., Jazbutis, T.G. & Schmidt, D.S. (1994) Velocity and mass transport measurements in a snow avalanche. In: International Snow Science Workshop, 30 October–3 November 1994, Snowbird, Utah. American Association of Avalanche Professionals. Snowbird, UT, P.O. Box 49, pp. 347–369. Glowinski, R., Pan, T.W. & Periaux, J. (1996) Fictitious domain methods for incompressible viscous flow around moving rigid bodies. In: The Mathematics of Finite

147

Elements and Applications, Highlight 1996 (Ed. Whiteman, J.R.). John Wiley, Chichester, pp. 155–174. Greve, R. & Hutter, K. (1993) Motion of a granular avalanche in a convex and concave curved chute: Experiments and theoretical predictions. Phil. Trans. R. S. London A 342, 573–600. Greve, R., Koch, T. & Hutter, K. (1994) Unconfined flow of granular avalanches along a partly curved surface. I. Theory. Proc. R. S. London A 445, 399– 413. Gubler, H. (1987) Measurements and modelling of snow avalanche speeds. In: Avalanche Formation, Movement and Effects. Proceedings of the Davos Symposium, September 1986. IAHS, Wallingford, pp. 405– 420. Haff, P.K. (1983) Grain flow as a fluid-mechanic phenomenon. J. Fluid Mech. 134, 401– 430. Hampton, M.A. (1972) The role of subaqueous debris flow in generating turbidity currents. J. sediment. Petrol. 42(4), 775–793. Hanes, D.M., Walton, O., Zakirov, V., Locurto, G. & Bucklin, R. (1997) Observations and simulations of the flow of nearly-ellipsoidal, inelastic particles down a bumpy incline. In: Proceedings of the Third International Conference on Powders & Grains. Balkema, Rotterdam, pp. 459– 461. Harbitz, C.B. (1999) Snow avalanche modelling, mapping and warning in Europe. deliverable no. 4: A survey of computational models for snow avalanche motion. Technical Report 581220-1. Norwegian Geotechnical Institute. Hermann, F., Hermann, J. & Hutter, K. (1987) Laboratory experiments on the dynamics of powder snow avalanches. In: Avalanche Formation, Movement and Effects. Proceedings of the Davos Symposium September 1986. IAHS, Wallingford, pp. 431– 440. Hopfinger, E.J. & Tochon-Danguy, J.-C. (1977) A model study of powder snow avalanches. J. Glaciol. 19(81), 343–356. Hu, H.H. (1996) Direct simulation of flows of solid–liquid mixtures. Int. J. Multiphase Flow 22(2), 335–352. Hutter, K. (1991) Two- and three-dimensional evolution of granular avalanche flow – theory and experiments revisited. Acta Mech. Suppl 1, 167–181. Hutter, K., Koch, T., Plüss, C. & Savage, S.B. (1995) The dynamics of avalanches of granular materials from initiation to runout. Part ii: Experiments. Acta Mech. 109, 127–165. Jenkins, J.T. (1987) Kinetic theory for nearly elastic spheres. In: Physics of Dry Granular Media (Eds Herrmann, H.J., Hovi, J.-P. & Luding, S.). Kluwer, Dordrecht, pp. 353–370. Jenkins, J.T. (1994) Rapid granular flow down inclines. Appl. Mech. Rev. 47(6), 240–244. Jenkins, J.T. & Richman, M.W. (1988) Plane simple shear of smooth inelastic circular disks: the anisotropy of the second moments in the dilute and dense limits. J. Fluid Mech. 192, 313–328. Jenkins, J.T. & Savage, S.B. (1983) A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202. Johnson, P.C., Nott, P. & Jackson, R. (1990) Frictional– collisional equations of motion for particulate flows and their application to chutes. J. Fluid Mech. 210, 501–535. Kawada, K., Nishimura, K. & Maeno, N. (1989) Experimental studies on a powder-snow avalanche. Ann. Glaciol. 13, 129–234.

PGCC09 9/12/08 3:14 PM Page 148

148


Keller, S. (1995) Measurements of powder snow avalanches – laboratory. Surv. Geophys. 16(5/6), 661–670. Keller, S., Ito, Y. & Nishimura, K. (1998) Measurements of the velocity distribution in ping-pong ball avalanches. Ann. Glaciol. 26, 259–264. Landau, L.D. & Lifschitz, E.M. (1987) Fluid Mechanics, volume 6 of Course of Theoretical Physics, 2nd edn. Butterworth-Heinemann, Oxford, p. 16. Lun, C.K.K., Savage, S.B., Jeffrey, D.J. & Chepurniy, N. (1984) Kinetic theories for granular flow: inelastic particles in couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223–256. Nishimura, K. & Ito, Y. (1997) Velocity distribution in snow avalanches. J. geophys. Res., 102(B12), 27297–27303. Nishimura, K., Narita, H. & Maeno, N. (1989) The internal structure of powder-snow avalanches. Ann. Glaciol. 13, 207–210. Nishimura, K., Maeno, N., Kawada, K. & Izumi, K. (1993a) Structures of snow cloud in dry-snow avalanches. Ann. Glaciol. 18, 173–178. Nishimura, K., Kosugi, K. & Nakagawa, M. (1993b) Experiments on ice-sphere flows along an inclined chute. Mech. Mat. 16, 205–209. Nishimura, K., Nohguchi, Y., Ito, Y., Kosugi, K. & Izumi, K. (1996) Snow avalanche experiments at ski jump. In: International Snow Science Workshop, 6–10 October 1996, Banff, Canada. Canadian Avalanche Association, P.O. Box 2759, Revelstoke, B.C., pp. 244 –251.

Nishimura, K., Keller, S., McElwaine, J. & Nohguchi, Y. (1998) Ping-pong ball avalanche at a ski jump. Gran. Matter 1(2), 51–56. Nohguchi, Y. (1996) Avalanche experiments with styrene foam particles. In: Proceedings of the Third International Conference on Snow Engineering (ICSE-3), Sendai, Japan, 63– 68. Nohguchi, Y., Ozawa, H. & Nishimura, K. (1997) 3-d experiments of light granular avalanches. In: Proceedings, 1997 Meeting of Japan Society of Fluid Mechanics, pp. 421– 422. (In Japanese with English abstract.) Rzadkiewicz, S.A., Mariotti, C. & Heinrich, P. (1997) Numerical simulation of submarine landslides and their hydraulic effects. J. Waterway, Port, Coastal, Ocean Eng. 123(4), 149–157. Simpson, J.E. (1972) Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53(4), 759–768. Simpson, J.E. (1986) Mixing at the front of a gravity current. Acta Mech. 63, 245–253. Tochon-Danguy, J.-C. & Hopfinger, E.J. (1975) Simulations of the dynamics of powder snow avalanches. In: Proceedings of the Grindelwald Symposium April 1974 – Snow Mechanics. IAHS, Wallingford, pp. 369–380. Von Kármán, T. (1940) The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615 – 683.

PGCC10 9/12/08 3:14 PM Page 149


Dam-break induced debris flow H . C A P A R T * , D . - L . Y O U N G † and Y . Z E C H ‡ * Fonds National de la Recherche Scientifique and Department of Civil Engineering, Université catholique de Louvain, Belgium; † Department of Civil Engineering and Hydraulic Research Laboratory, National Taiwan University, Taipei, Taiwan, Republic of China; and ‡ Department of Civil Engineering, Université catholique de Louvain, Belgium

ABSTRACT The present report focuses on the hydrodynamic and geomorphic behaviour of dam-break induced debris flows. Based on a review of the literature, we first examine various instances in which abrupt floods released by the breaking of natural or constructed dams have been observed to develop into debris waves, picking up solid material both from the dam body and from the valley floor. We then assess the extent to which modelling and experimental approaches can account for the field observations gathered from these events. The modelling approach considered consists of a shallow-water, single-fluid physical description, while the experiments involve idealized laboratory analogues. Results of these two approaches are compared and discussed for the case of a dam-break induced debris wave propagating past a sudden channel enlargement.

INTRODUCTION Initiation mechanisms for debris flows include the en masse failure of water-saturated slopes (Iverson et al., 1997), the mixing of hot pyroclastic material with water of hydrothermal, glacial or snowmelt origin (Scott, 1988), and the bulking of torrential water flows with eroded sediment. Some of the most destructive flows of the latter category are those resulting from catastrophic dam failures (Costa & Schuster, 1988). A review of the literature indicates that a number of documented dam failures have been associated with debris flows, transporting significant volumes of sediment and causing severe geomorphic changes. Rough evaluations of the volumes of water and transported sediment involved in eight of these events, concerning both natural and constructed dams, are listed in Table 1. These volumetric data show that in many instances, the volume of transported sediment can be of the same order of magnitude as the volume of water drained from the reservoir. In most cases, the transported sediment volume is considerably larger than the volume of material contained in the body of the ruptured dam. For the Klattasine Creek event, for instance, Clague et al. (1985) estimate that only 4000 m3 of the total transported material (3.0 × 106 m3) originated from

the moraine dam. Dam-break events can thus lead to a severe bulking of the flood waters with sediment eroded from the valley bed and banks.

FUNDAMENTAL PROCESSES Field observations gathered from the above-mentioned events point to a series of key processes, strongly coupled with each other. They can be synthesized as follows. A dam-break wave starting out as clear water is likely to rapidly pick up sediment both from the dam body and from the valley floor. This can be particularly intense when channel gradients are steep, and when large volumes of loose material are available, as in the case of moraine deposits. In turn, erosion of the valley bed by the flood current can destabilize the slopes, making even more material available. In many instances, it is possible for the wave to entrain enough sediment to turn into a fully developed debris flow (Costa & Schuster, 1988). This was observed for all the events of Table 1 except for the Aniakchak volcano and Lawn Lake dam-break floods. Conversely, the


149

PGCC10 9/12/08 3:14 PM Page 150

150

H. Capart et al.

Table 1. Dam-break events with documented sediment deposits Location Aniakchak volcano, Alaska

Year

Nature of the dam

Water volume Vw (m3)

Sediment Ratio volume Vs (m3) Vs /Vw

Source

ca. 3400 BP Caldera rim

3.7 × 109

0.5 × 109

0.14

Waythomas et al., 1996

3.0 × 10

6

Quebrada Los Cedros, Peru

1950

Moraine dam

2.0 × 10

1.50

Lliboutry et al., 1977

Klattasine Creek, British Columbia

1971–1973

Moraine dam

1.7 × 106

3.0 × 106

1.76

Clague et al., 1985

Spiral Tunnels, British Columbia

1978

Ice dam

750 000

175 000

0.23

Jackson, 1979

Lawn Lake, Colorado

1982

Earthen dam

830 000

0.33

Jarrett & Costa, 1986

0.17

Blown & Church, 1985

6

280 000

Nostetuko Lake, British Columbia

1983

Moraine dam

6.0 × 10

Centralia, Washington

1991

Concrete dam

13 250

1000

0.08

Costa, 1994

Biescas, Spain

1996

Check dams

800 000

50 000

0.06

Benito et al., 1998

wave can at some point deposit most of its material and continue as a muddy streamflow. This bulking and debulking process has three consequences. Firstly, the incorporation of sediment can completely transform the flow rheology. If a debris flow develops, it can take either the form of a cohesive, slurry-like mudflow or a non-cohesive granular flow. In the first case, the mud matrix plays a key role in the transport of large grains and the whole mixture flows as a viscoplastic fluid. In the second case, frictional and collisional contacts between grains dominate the flow behaviour. In both cases, the mixture rheology differs markedly from that of pure water. Secondly, the erosion and deposition process itself can significantly affect the wave hydrodynamics. The inertia of the entrained material can slow down the wavefront, while deposition of material can temporarily dam a flow reach, as in the 1980 Polallie Creek event (Gallino & Pierson, 1985). Both mechanisms can be sufficient to transform flows from a supercritical to a subcritical regime. They can thus lead to the development of strong backwater effects, raising the upstream flow stage. Thirdly, the bed topography evolves as a result of the erosion and deposition process. This geomorphic action of the flood wave can induce strong feedback effects. In some locations, for instance, severe scour can lead to further acceleration of the flow by increasing the stream gradient. The development of the wave into a debris flow is also associated with a number of other processes. Lateral levees composed of bouldery material constitute a characteristic feature of debris flow deposits (see e.g. Whipple & Dunne, 1992; Costa,

6

1.0 × 10

6

1994). In addition to levee formation, a debris flow can incise the underlying bed closer to the flow centreline. Through geomorphic action, a dam-break wave can therefore shape its own bed and undergo channelization. In summary, dam-break waves can in many instances entrain and deposit large volumes of sediment. This bulking and debulking process can significantly affect (i) the flow rheology, (ii) the wave hydrodynamics, and (iii) the valley morphology. Despite the wide variety of conditions and the vastly different scales encountered in actual events, field studies thus document a number of shared features. It is then natural to wonder whether and to what extent such features could be captured by physically based numerical models or laboratory scale analogues.

PHYSICAL AND MATHEMATICAL DESCRIPTION In what follows, an idealized valley configuration is examined, composed of a narrow upstream reach and a wide downstream plain, separated by a sudden channel enlargement. The narrow reach is barred by a dam, holding a constant level of water upstream. The lateral walls of the channel are rigid, but the bottom is composed of loose granular material. Initially, the bed is horizontal and saturated with water throughout. At some point, the dam is removed, and a dam-break wave develops into a fast debris wave which expands at the enlargement. The geometry approximates the transition between narrow-valley and wide plain often

PGCC10 9/12/08 3:14 PM Page 151

Dam-break induced debris flow encountered in field cases, such as the Klattasine Creek (Clague et al., 1985) and Biescas events (Benito et al., 1998). For the heavy sediments encountered in nature, the formation of debris flows is unlikely over flat or mildly sloped channel beds. In the experiments, a light sediment analogue was chosen in order to obtain sufficient mobility of the granular material over low bed gradients. This trade-off between slope and density was motivated by practical considerations, and means that the experimental results can only be compared with field observations in a qualitative sense. In order to describe the highly transient two-phase flow, two main assumptions are made. Firstly, the flow is considered to be shallow, which reduces the problem to a vertically averaged one. Secondly, the fluid–granular mixture is assumed to behave as a single-phase fluid with varying density and rheological properties. These are the assumptions most commonly made when modelling debris flow (see e.g. Takahashi & Nakagawa, 1994; Hungr, 1995; Han & Wang, 1996). Equations for continuity of the mixture, momentum conservation of the mixture, and continuity of the transported sediment and bed material can then be written. They require a description of the mass and inertia exchange between bed and flow, which is achieved using a non-equilibrium transport formulation. Constitutive relations are required for closure, and these are based on empirical descriptions of rapid (collisional) flows of fluid–grain mixtures in sheet- and debris-flow modes (Capart & Young, 1998). In mathematical terms, this description results in a set of Eulerian partial differential equations with independent variables provided by time and the two horizontal dimensions. The governing equations in one-dimensional form are given in the appendix. The system is hyperbolic and can thus lead to the formation of discontinuities, or shocks, in the flow domain. Solutions are to be constructed numerically, achieved here using a finite-volume scheme which generalizes to two horizontal dimensions the algorithm of Capart and Young (1998). In Fig. 1, a temporal sequence is shown for the flow free-surface and flow–bed interface. A number of qualitative features of these solutions can be pointed out. A steep front is first observed to propagate downstream and radiate from the enlargement. The freesurface presents a non-monotonous longitudinal profile near the initial position of the dam. This is due to the formation of a hydraulic jump, shown by Capart and Young (1998) to result from flow–bed interaction. Secondly, the bed dynamics show that bed

151

erosion is most intense at two locations: at the initial position of the dam, where the water-wave loads itself with sediment in the first instants following the dambreak, and at the corner of the enlargement. Conversely, deposition is most severe in the lateral dead zone downstream of the enlargement, where the aggraded material forms a levee. Interestingly, such results coincide with field observations which appeared in the notebooks of Leonardo Da Vinci (Da Vinci, 1975). Due to feedback from the bed morphodynamics and to added inertia associated with bed material incorporation, the flow upstream of the steep front is subcritical and a backwater effect appears. A characteristic consequence of this backwater effect is to equalize the flow free surface throughout the zone located between the enlargement and the debris front. This stands in sharp contrast with the propagation of a pure fluid dam-break wave over a rigid dry bed (see e.g. Fraccrollo & Toro, 1995). Finally, the global effect of the erosion and deposition is to channelize the flow. While a quantitative assessment remains to be made, these results correspond to a number of qualitative features observed in the field, where erosion and bulking of the flood wave with sediment in the upstream narrow valley is often followed by spreading and deposition at the valley enlargement (Clague et al., 1985; Costa, 1994; Benito et al., 1998). The spreading pattern observed when the flow ceases to be confined appears to be shared with debris and mudflows in general, as exhibited both by field observations (e.g. Whipple & Dunne, 1992) and laboratory experiments (Takahashi, 1991; Coussot & Proust, 1996).

EXPERIMENTS AND COMPARISON OF RESULTS Corresponding experiments were performed in a small-scale laboratory set-up, shown in Fig. 2. The flume is 10-cm wide in the narrow reach, and 20-cm wide beyond the enlargement. The light sediment analogue used for the experiments is composed of white artificial pearls of relative density equal to 1.05 and a diameter of 6 mm. Imaging techniques were used to characterize the flow, using a high-speed (100 Hz) digital camera. The highly repeatable debris flows were filmed a number of times with the camera located at different positions. In Fig. 3, images from a downstream viewpoint are compared with the numerical solutions, for the instants directly following the propagation of the

PGCC10 9/12/08 3:14 PM Page 152

152

H. Capart et al.

(a)

(d)

0.1

0.1

0

0

0.1 1

0.1 1

0.5

0.5

0

0

0.2

0.5

0.2

0.5 1 0

1 0

(b)

(e)

0.1

0.1

0

0

0.1 1

0.1 1

0.5

0.5

0

0

0.2

0.5

0.2

0.5

1 0

1 0

(c)

(f)

0.1

0.1

0

0

0.1 1

0.1 1

0.5

0.5

0

0

0.2

0.5 1 0

0.2

0.5 1 0

Fig. 1. Computed results for the propagation of a dam-break induced debris wave past a sudden channel enlargement: (a) T = 0; (b) T = 0.2 s; (c) T = 0.4 s; (d) T = 0.6 s; (e) T = 0.8 s; (f ) T = 1.0 s. Thin mesh: bed interface; thick line: free surface contour. All dimensions in metres.

debris wave past the enlargement. The constitutive parameters used for the simulations are those derived from the prismatic channel dam-break tests (i.e. no channel enlargement) of Capart and Young (1998), with no additional calibration or adjustment. The obtained agreement is therefore rather remarkable. In Fig. 4, the computed horizontal velocity field is compared with measurements extracted from the

digital images (with the camera placed directly above the enlargement). The velocity measurements were obtained using an original particle tracking algorithm specifically designed for rapid, high-density granular flows. Again, a good agreement is obtained. Does this quantitative agreement hold for the ensuing time? The answer is unfortunately negative. In the experiments, a gradual deceleration of the debris wave

PGCC10 9/12/08 3:14 PM Page 153

Fig. 2. Laboratory set-up: view from upstream. Dimensions are the following: upstream width = 10 cm; downstream width = 20 cm; initial upstream depth above granular bed = 10 cm.

Fig. 3. Comparison of experiments (left) and computations (right) for the debris wave expansion: (a) T = 0.30 s; (b) T = 0.35 s; (c) T = 0.40 s; (d) T = 0.45 s. Axis dimensions in metres.

PGCC10 9/12/08 3:14 PM Page 154

154

H. Capart et al.

1.4

0.2

(a)

1.2

Time [s]

1 0.1

0.8 0.6 0.4

0 0.1

0.2

0.3

0.4

0.2 0

0

0.2

0.4

0.6

0.8

1

Distance [m]

0.2

(b)

Fig. 5. Trajectory of the foremost tip of the debris wave. Dashed line: observed; continuous line: computed.

0.1

0 0.1

0.2

0.3

0.4

Fig. 4. Horizontal velocity field at T = 0.4 s: (a) digital imaging measurements; (b) finite volume computations. Axis dimensions in metres.

is observed, which temporarily forms a quasi-stationary ridge across the width of the ‘plain’, before being set in motion again. This effect is not well described by the computations, as shown in Fig. 5 which plots the propagation of the foremost tip of the debris wave. A closer look at the experimentally obtained images shows that at this stage the flow ceases to behave as a single-phase fluid. As the temporary ridge forms, the fluid and the granular phases unlock, the fluid wave seeping ahead of the granular wave. Similar observations are made in nature, as in the 1996 Biescas dambreak induced debris flow (Benito et al., 1998). For this event, the deposits indicate a self-damming of the wave at the valley enlargement (the apex of an alluvial fan). Boulder lobes were formed, releasing water runout and diverting the ulterior flow out on the allu-

vial fan surface (Benito et al., 1998). The single-fluid assumption therefore appears to break down beyond the first moments of the debris wave propagation across the wider plain, both in the experiments and in the field.

CONCLUSIONS Despite the idealized character of the adopted physical assumptions and conditions, the computational and laboratory experiments described above have been shown to qualitatively capture a number of the features observed in the field for dam-break induced debris flows. These features include the bulking and debulking processes, backwater effects, and channelization. Quantitatively, good agreement between numerical and experimental results is obtained, as long as the mixture behaves like a single-phase fluid. In the experiments, however, it was observed that the single fluid assumption breaks down at a certain stage, beyond which relative motion between fluid and grains is significant. This unlocking phenomenon, associated with the formation of depositional lobes, or ridges, appears to play an important role in field conditions as well. Despite the many idealizations, a qualitative correspondence is obtained between field observations and the experimental analogue.

PGCC10 9/12/08 3:14 PM Page 155

155

Dam-break induced debris flow

ACKNOWLEDGEMENTS The experiments described above were performed by the first author at the Hydraulic Research Laboratory of the National Taiwan University, in the summer of 1997. The assistance of H.H. Liu and the financial support of the Belgian National Fund for Scientific Research (FNRS) are gratefully acknowledged.

APPENDIX: ONE-DIMENSIONAL GOVERNING EQUATIONS In one horizontal dimension, the vertically averaged evolution equations are written (Capart & Young, 1998): ∂Z ∂H ∂( HU ) + = − b, ∂T ∂X ∂T

(1a)

∂(ρHU ) ∂(ρHU 2 + 21 ρgH 2 ) + ∂T ∂X = −I − ρgH

∂Zb − τb , ∂X

(1b)

∂H B ∂ ( H BU ) 1 + = (H eq − HB ) = N bB , (1c) ∂T ∂X TbB B ∂Zb 1 = − NbB . ∂T cb

(1d)

In the above equations, T = time; X = longitudinal coordinate; H = flow depth; U = vertically averaged mixture velocity indices b and B respectively refer to the motionless bed and the moving bed material; Zb = bed elevation; τb = bed shear stress; ρ = ρw [1 + (s − 1)HB /H ] = density of the water–sediment mixture, with ρw = density of water, ρs = sediment density, and s = ρs /ρw = specific gravity of sediment grains; I = amount of X momentum lost to the flow due to deposition, i.e. I = −ρbUNbB /cb if NbB < 0, I = 0 otherwise; HB = sediment load, i.e. volume of moving bed material per unit bed surface; HBeq = sediment load under steady uniform equilibrium condition; NbB = rate of sediment exchange between bed and flow, i.e. net volume of bed material transferred to the flow per unit time and unit bed surface; g = acceleration of gravity; TbB = relaxation time associated with sediment exchange between bed and flow; cb = volume concentration of sediment in the undisturbed bed.

Closure relations corresponding to the conditions of the experiments are detailed in Capart & Young (1998). Equivalent equations in two horizontal dimensions are available from the first author.

REFERENCES Benito, G., Grodek, T. & Enzel, Y. (1998) The geomorphic and hydrologic impacts of the catastrophic flood failure of flood-control dams during the 1996 Biescas flood (Central Pyrenees, Spain). Z. Geomorph. 42, 417– 437. Blown, I. & Church, M. (1985) Catastrophic lake drainage within the Homathko River Basin, British Columbia, Can. Geotech. J. 22, 551–563. Capart, H. & Young, D.L. (1998) Formation of a jump by the dam-break wave over a granular bed. J. Fluid Mech. 372, 165–187. Clague, J.J., Evans, S.G. & Blown, I.G. (1985) A debris flow triggered by the breaching of a moraine-dammed lake, Klattasine Creek, British Columbia. Can. J. Earth Sci. 22, 1492–1502. Costa, J.E. (1994) Multiple flow processes accompanying a dam-break flood in a small upland watershed, Centralia, Washington. U.S. Geol. Sur. Water-Resources Investigations Rep. 94 -4026, p. 24. Costa, J.E. & Schuster, R.L. (1988) The formation and failure of natural dams. Bull. geol. Soc. Am. 100, 1054 –1068. Coussot, P. & Proust, S. (1996) Slow, unconfined spreading of a mudflow. J. Geophys. Res.-Solid Earth, 101, 25217– 25229. Da Vinci, L. (1975) The Notebooks of Leonardo Da Vinci. Dover, p. 433. Fraccarollo, L. & Toro, E.F. (1995) Experimental and numerical assessment of the shallow water model for twodimensional dam-break type problems. J. Hydr. Res. 33, 843 – 864. Gallino, G.L. & Pierson, T.C. (1985) Polallie Creek debris flow and subsequent dam-break flood of 1980, East Fork Hood River basin, Oregon. U.S. Geol. Sur. Water Supply Paper 2273, p. 22. Han, G. & Wang, D. (1996) Numerical modeling of Anhui debris flow. J. Hydr. Eng. 122, 262–265. Hungr, O. (1995) A model for the runout analysis of rapid flow slides, debris flows, and avalanches. Can. Geotech. J. 32, 610 – 623. Iverson, R.M., Reid, M.E. & LaHusen, R.G. (1997) Debrisflow mobilization from landslides. Annu. Rev. Earth Planet. Sci. 25, 85–138. Jackson, L.E., Jr. (1979) A catastrophic glacial outburst flood ( jökulhaup) mechanism for debris flow generation at the Spiral Tunnels, Kicking Horse river basin, British Columbia. Can. Geotech. J. 16, 806 – 813. Jarrett, R.D. & Costa, J.E. (1986) Hydrology, geomorphology and dam-break modeling of the July 15, 1982 Lawn Lake Dam and Cascade Lake Dam failures, Larimer County, Colorado. U.S. Government Printing Office, Washington, p. 78. Lliboutry, L., Morales, A.B., Pautre, A. & Schneider, B. (1977) Glaciological problems set by the control of dangerous lakes in Cordilera Blanca, Peru. I: Historical failure of

PGCC10 9/12/08 3:14 PM Page 156

156

H. Capart et al.

morainic dams, their causes and prevention. J. Glaciol. 18, 239–254. Scott, K.M. (1988) Origins, behavior, and sedimentology of Lahars and Lahar-Runout Flows in the Toutle-Cowlitz River System. U.S. Geol. Surv. Prof. Pap. 1447-A, p. 74. Takahashi, T. (1991) Debris flow. IAHR Monograph. Balkema, Rotterdam, p. 156. Takahashi, T. & Nakagawa, H. (1994) Flood /debris flow hydrograph due to the collapse of a natural dam by overtopping. J. Hydrosc. Hydr. Eng. 12, 41– 49.

Waythomas, C.F., Walder, J.S., McGimsey, R.G. & Neal, C.A. (1996) A catastrophic flood caused by drainage of a caldera lake at Aniakchak Volcano, Alaska, and implications for volcanic hazards assessment. Bull. geol. Soc. Am. 108, 861– 871. Whipple, K.X. & Dunne, T. (1992) The influence of debrisflow rheology on fan morphology, Owens Valley, California. Bull, geol. Soc. Am. 104, 887–900.

PGCC11 9/12/08 3:29 PM Page 157

Experimental approaches


PGCC11 9/12/08 3:29 PM Page 159

Spec. Publs. int. Ass. Sediment. (2001) 31, 159 –172

Mean flow and turbulence structure of sediment-laden gravity currents: new insights using ultrasonic Doppler velocity profiling J . L . B E S T * , A . D . K I R K B R I D E † and J . P E A K A L L * * School of Earth Sciences, University of Leeds, Leeds LS2 9JT, West Yorkshire, UK and † Institute of Environmental and Natural Sciences, Lancaster University, Lancaster LA1 4YB, UK

ABSTRACT The mean and turbulent structure of a sediment-laden gravity current is quantified using ultrasonic Doppler velocity profiling (UDVP). UDVP allows both measurements of velocities in opaque fluids and holistic reconstruction of the one-dimensional flow field. This chapter presents results from the first use of UDVP in a sediment-laden gravity current that illustrate: (i) the velocity maximum in the forward flow lies at approximately 0.3 of the head height above the bed; (ii) the region of the velocity maximum of the forward flow is associated with the lowest turbulence intensities within the current; (iii) the highest turbulence intensities are associated with both near-bed shear and mixing on the back of the head; (iv) return flows created by reflection from topography, in the form of different types of bore, may be fully quantified by UDVP; (v) the scale of turbulence associated with the head appears to be of the same order as the head height, and (vi) past studies of gravity currents that have had to use saline solutions and sediment-free dispersions as a proxy for sediment-laden currents, are in many aspects a good substitute for low-density, sediment-bearing gravity currents. Future use of UDVP holds great promise for study of the interaction between turbulence, sediment transport and deposition in a range of sediment-laden flows, including flows where conventional measurement techniques are difficult or impossible to apply.

INTRODUCTION either relied on use of saline solutions to generate the density excess (e.g. Simpson, 1972; Simpson & Britter, 1979; Huppert & Simpson, 1980; García & Parsons, 1996; see Simpson (1997) and Kneller & Buckee (2000) for comprehensive summaries) or sediment-bearing suspensions to model sediment transport and grainsize segregation (Parker et al., 1987; García & Parker, 1991, 1993; Bonnecaze et al., 1993; García, 1994; Altinakar et al., 1996; Hürzeler et al., 1996). However, logistical difficulties have often arisen in matching these two approaches to study the fluid dynamics of sediment-laden density currents. Whilst considerable progress has been accomplished recently in quantifying turbulence within density currents using refractive-index matched fluids and laser Doppler anemometry (Kneller et al., 1997, 1999; Kneller & Buckee, 2000; Buckee et al., 2000), this methodology cannot be used in opaque, sediment-laden currents which may be eroding, transporting or depositing

The erosion, transport and deposition of sediment by gravity currents is one of the principal methods by which sediments are dispersed into the ocean basins and have been subjects of considerable research over the past 30 years. Understanding and predicting the nature of sediment transport and deposition from density currents relies on a detailed understanding of the fluid dynamics of these currents, which may have multifarious characteristics. For instance, debate concerning the nature of high-concentration turbidity currents, and the role of debris flows, has provoked considerable controversy in recent years (Shanmugam, 1997) and it is now clear that turbulent density currents may be radically influenced by many factors, including flow duration, acceleration/ deceleration profile (Kneller, 1995), bed topography (Edwards, 1993; Edwards et al., 1994) and ambient stratification (Simpson, 1997). Much research investigating the dynamics of turbulent density currents has


159

PGCC11 9/12/08 3:29 PM Page 160

160

J.L. Best et al.

sediment. The application of the results from these experiments to real turbidity currents is therefore somewhat problematic. Although study of flow within sediment-laden currents has been achieved using techniques such as current meters (García & Parker, 1991, 1993; García 1994; Altinakar et al., 1996), these have been incapable of resolving higher frequency turbulent fluctuations (of the order of several to 30 Hz) and do not provide knowledge of the changing flow field. However, development of ultrasonic Doppler methods for quantifying fluid turbulence has seen rapid development in recent years and provides a methodology that can enable quantification of flow within opaque, sediment-laden suspensions. This paper presents details of the methodology and first use of an ultrasonic Doppler velocity profiler (UDVP) in quantifying the mean flow and turbulence structure of a sedimentladen, laboratory density current. Details of the technique, its application and inherent errors are presented with reference to the flow of a finite-volume, surge-type density current.

x = ct/2

(2)

where t is the time lapse between the emission and reception of ultrasound pulses. The maximum distance to which the UDVP may detect, Dmax, is a function of the frequency of repetition of the ultrasound pulses, fprf, and is given by: Dmax = c/2fprf

(3)

Additionally, the maximum detectable velocity, Umax, is determined by the Nyquist sampling theorem fD(max) < fprf /2

METHODOLOGY

(4)

where fD(max) is the maximum detectable Doppler shift frequency, and hence

Principles of Ultrasonic Doppler Velocity Profiling (UDVP)

Umax < cfprf /4fo

Several studies in recent years have outlined the principles and techniques of acoustic Doppler anemometry (Lhermitte & Lemmin, 1990, 1994; Takeda, 1991, 1993, 1995; Takeda et al., 1990, 1992; Tuefel et al., 1992; Tokuhiro & Takeda, 1993; Rolland & Lemmin, 1997; Graf & Yulistiyanto, 1998; Nikora & Goring, 1998). The UDVP used in the current study is described elsewhere (Takeda, 1991, 1993, 1995; Takeda et al., 1990, 1992, 1994; Metflow, 1997) but a brief summary is given below. Acoustic Doppler anemometry relies on use of pulsed echosound echography (Takeda, 1991, 1993) wherein an ultrasound pulse is emitted along a measuring line from a transducer, and the same transducer receives the echo reflected from the surface of small particles suspended within the flow. Flow velocity is given from detection of the Doppler shift in the ultrasound frequency, fD, as particles pass through the measurement volume. The velocity of a particle, U, is given by: U = cfD /2fo

that may be used with the current MetFlow system are 2, 4 or 8 MHz). In this manner, the velocity of a particle can be calculated at one point. However, UDVP uses a beam of ultrasound that is emitted from a transducer to obtain simultaneous velocity measurements along a profile. In order to achieve this profiling, the return signal detecting the shift in frequency is gated at certain return times, and thereby allows measurement of velocity at up to 128 points along the axis of the ultrasound beam. The distance to the location of the sample location, x, is given by:

(1)

where c is the velocity of ultrasound in the fluid being investigated, fD is the Doppler frequency shift and fo is the ultrasound frequency (the ultrasound frequencies

(5)

From Eqs (3) and (5), the following constraints therefore exist for UDVP measurements: Dmax × Umax = c 2/8fo

(6)

Therefore, for a fixed ultrasound frequency, larger distances of penetration and velocity measurement range are given by selecting lower pulse repetition frequencies, but this also results in a lower detectable maximum velocity. The maximum detectable velocity is given by: Umax = c 2/8fo Dmax

(7)

and therefore for an 8-bit accuracy, the velocity resolution, Ures, is given by Umax /127 for a configuration that is directionally sensitive or Umax /255 if the sign of the signal is ignored. Table 1 lists the physical parameters used for the UDVP in the current experiments. The length of each measuring volume along the axis of detection is chosen from preselected multiples of 0.745 mm between 0.745 and 5.215 mm, although this only refers to the longitudinal length of each measurement bin. A window within the profile may be selected and the size of this window may be reduced through selection of a smaller bin size. The cross-beam breadth

PGCC11 9/12/08 3:29 PM Page 161

Sediment-laden gravity currents Table 1. Parameters of the ultrasonic Doppler velocity profiler used in the present experiments Ultrasound frequency, fo

4 MHz

Transducer and probe diameter

5 and 8 mm

Measurement window

9–103.7 mm

Maximum velocity, Umax

±308 mm s−1

Measurement bin length

0.745 mm

Velocity resolution, Ures

2.4 mm s−1

Height of measurement ‘bin’ nearest transducer

5.8 mm

Height of measurement ‘bin’ furthest from transducer

14.1 mm

Ultrasound velocity

1490 m s−1

Sampling time for each profile (and delay time between profiles)

81 ms (15 ms)

Sampling frequency/probe

2.6 Hz

Pulse repetition frequency, fprf

3307 Hz

of each bin at different distances away from the probe is determined by the angle of spread of the beam away from the probe, which is a function of both transducer size and ultrasound frequency. For the 4-MHz, 5-mmdiameter transducers used in this study, the beam spread angle was 2.5°, and this can be used to calculate the cross-beam width of the measurement bins nearest and furthest away from the probes (Table 1). Two assumptions are inherent when applying UDVP in sediment-laden flows. First, it must be assumed that the sediment and fluid travel at the same velocity and there is no slip velocity between the two phases. This assumption is likely to be valid when considering fine, predominantly suspended sediment (Elghobashi, 1994) but may not be valid when larger grains may lag behind the fluid (Hetsroni, 1989, 1993; Gore & Crowe, 1991; Best et al., 1997). Secondly, it must be assumed that the spatial change in density within the current, and therefore change in ultrasound velocity, is less that the error limits within the technique. Initial calibration experiments were conducted using an ultrasonic bed profiler (Best & Ashworth, 1994) at a range of suspended sediment concentrations. The ultrasonic profiler was mounted at a fixed position in a 5-litre glass beaker filled with clear water and the distance to the bottom was measured. Progressively greater amounts of silica flour were added to the water, to yield densities of 1000 to 1065 kg m−3, and the velocity of sound adjusted to give the same depth value as in clear water. In this

161

manner, the velocity of ultrasound in different concentrations of suspended sediment could be assessed. Results from these simple tests showed that the difference in sound velocity between clear water and the sediment concentration used in the present experiments was a maximum of 13 m s−1, this producing an error of approximately 3 mm s−1 with the 4 MHz probes used here (see Eq. 1). This error is within the range of velocity corrections that can be made with the UDVP software (10 m s−1 intervals). Although not conducted in this study, simultaneous measurement of suspended sediment concentration, by either direct or indirect methods, may permit later recalibration of acoustic Doppler data where the density of the flow may change appreciably within the measuring region or during the course of an experiment. Acoustic Doppler methods also have the potential to use the magnitude of the backscatter Doppler signal to quantify the sediment concentration (e.g. see Thorne et al., 1996), if the sediment size and mineralogical properties of the sediment are known. Future development of the technique to provide flow field data on both flow velocity and semi-quantitative sediment concentration holds great promise. In the present experiment, an array of four, 4 MHz ultrasonic probes was used which were multiplexed with the UDVP processor. The time taken for each profile measurement is a function of both the maximum measurement distance and the finite time that is required for data processing and switching between channels. In the current configuration, the sampling time for each profile was 81 ms, the switching time between probes was 15 ms, and hence each probe sampled at approximately 2.6 Hz. Since a source of experimental error may arise from slight misalignment of the UDVP probes, great care was taken in positioning the probes before the experiment, with the error in horizontal and vertical alignment of the four probes being measured at less than 0.5 mm. The probes were mounted on a machined bar that ensured they were both parallel to the flume base and each other. Experimental Setup and Procedure The experiments were conducted in a horizontal tank that was 10-m long, 0.30-m wide and 0.30-m deep (Fig. 1a). This flume tank was filled to a depth of 0.20 m with clear tap water and allowed to reach a constant value at the room temperature (18°C). A suspension of silica flour (see below) was then prepared in an overhead reservoir, located 3.5 m from the tank entrance, which had a capacity of 50 litres.

PGCC11 9/12/08 3:29 PM Page 162

162

J.L. Best et al.

Fig. 1. (a) Schematic experimental setup. (b) Grain-size distribution of silica flour.

The water in the reservoir was also allowed to reach room temperature before the correct amount of silica flour was added to the water to generate a density of 1061.5 kg m−3. The silica flour had a mean grain size, D50, of 9.25 µm (see grain-size distribution, Fig. 1b). Once this sediment had been fully mixed with the water and allowed to stand for no more than 10 s, the gate valve in the supply pipe (Fig. 1a) was opened, allowing the silica suspension to flow into the main tank and generate a sediment-laden density current. The supply pipe was 0.05 m in diameter and introduced the silica suspension onto the bed at the centre of the flume. The density current expanded and generated a two-dimensional, fully-formed gravity current, with distinct head and body, by the time it reached the UDVP measurement section 3.7-m downstream from the supply pipe outlet. Although the experiment concerned a surge-type current, the 50-litre volume of silica flour suspension ensured the initial surge occurred over a period of approximately 80 s, enabling reliable temporal measurements within the head and body of the current. The gravity current then continued down-

channel before encountering a vertical gate at the end of the tank and forming a reversed current, consisting of a series of bores (Pantin & Leeder, 1987; Edwards, 1993; Edwards et al., 1994) that travelled back up the tank and subsequently generated a second reflection. UDVP measurements were allied with digital video camera recording within the test section, enabling the position of the current to be subsequently matched with the UDVP records. Table 2 gives the initial experimental conditions for the present experiment. An array of four UDVP probes, 3.7-m downstream from the supply pipe (Fig. 1a), was used in this study with the probes being positioned 7, 37, 67 and 97 mm above the floor of the tank (termed probes P1–P4 respectively, see Fig. 1a). The set-up parameters of the UDVP (Table 1) were selected so that each probe measured a distance of 95 mm from a starting position 9-mm upstream of the probes. Each probe sampled for a time period (approximately 300 s) that incorporated the movement of the density current down tank, reverse flow from the downstream end of the flume and then a second series of reflected waves that were

PGCC11 9/12/08 3:29 PM Page 163

Sediment-laden gravity currents Table 2. Experimental conditions and summary statistics for the experimental sediment-laden gravity current Initial conditions Ambient depth

0.20 m

163

the current and then above the tail of the main forward flow once the head had passed. For the first return bore and solitary waves, the bottom three probes were largely within the return flow but the top probe was above this height. The structure of the sediment-laden current with respect to these video images will be examined by considering the time series of each probe at one location, the time series of two probes at all spatial locations and then the spatial series of all four probes at several time slices.

Flow temperature

17.65°C

Initial density of sediment-laden current

1061.5 kg m−3

Mean grain size, D50, of silica flour

9.25 µm

Volume of suspension release

50 litres

Summary of measurements Maximum head height, hh (forward flow)

143 mm

Mean velocity of forward flow

115 mm s−1

Time Series

Maximum velocity of forward flow

170 mm s−1

The time series for a measurement point 20.1-mm upstream from each probe (Figs 3–5) show the passage of the sediment-laden gravity current past the measurement section and the subsequent reflections of the current from the downstream (c. 105 s and c. 290 s) and upstream (c. 170 s) ends of the tank. As the forward current passes this measurement position (Figs 2b–h, 3 and 4), the downstream velocity rapidly increases at the bottom probe, reaching a maximum value of 121 mm s−1 some 0.35 s after the current has reached this point. The maximum velocity within the current (169.7 mm s−1) is recorded at P2, 37 mm above the bed. This velocity maximum is at approximately 0.3 of the head height, in agreement with past quantitative measurements (e.g. Altinaker et al., 1996;Kneller et al., 1997, 1999). The temporal lag in the point of maximum velocity between the probes (lag = 1.38 s between probes P1 and P3) reflects the morphology of the head, with the nose of the flow being c. 0.20 m in front of the maximum head height (Fig. 2b). Additionally, the velocity maximum at P4 is detected 4.28 s after P1, with the maximum at P4 being associated with longer-period fluctuations than the lower probes, which are most likely produced by the impact of large Kelvin–Helmholtz instabilities shed off the top of the current (Fig. 2d–g). Upstream flow at P4 before the arrival of the head (t < 13.8 s) depicts the fluid displacement and return flow produced by the density current in this shallow-flow experiment. Once the head of the current has passed the measurement points (> c. 53 s, Figs 3 and 4), the flow at P4 is upstream, being formed by return flow in the ambient fluid, whilst flow at the lower three probes is markedly less turbulent than within the head. Reflections of the current from the bottom end, top end and a second reflection from the bottom end of the tank are clearly evident at approximately 102, 170 and 288 s respectively. Detail of the first reflection (Fig. 5; Fig. 2i) shows that the base of the first wave reaches

Mean tail height, ht (forward flow)

39 mm

Maximum height of first return bore

72 mm

Mean velocity of first return bore

45 mm s−1

Flow Reynolds number, Re Re = (ρsUmeanhh)/µs

13 780

Densiometric Froude number, Fr Fr = Umean/(g′hh)0.5

0.36

Flow Richardson number, Ri Ri = g′d/U 2mean

6.5

generated by interaction with the upstream end of the tank. It should be noted that some influence on the flow structure was created by interference of the first series of reflected waves with the probe mounting supports, but that measurement of the forward gravity current and second series of reflected waves did not involve any disturbance by the probe mounts to the flow.

RESULTS Mean Flow Structure The location of the probes with respect to the forward current and the reflected solitary waves are depicted in a series of video images from the experiment (Fig. 2). These images show that the four probes were located at a height, h, relative to the maximum head thickness, hh (c. 143 mm) of 0.05, 0.26, 0.47 and 0.68. In relation to the thickness of the tail of the current, ht (39 mm), once the main head had passed downtank, the probes were located at relative heights, h/ht of 0.18, 0.95, 1.73 and 2.5. The top probe (P4, 97 mm above bed) was therefore located both within the mixing zone and region of Kelvin–Helmholtz instability at the head of

PGCC11 9/12/08 3:29 PM Page 164

164

J.L. Best et al. Fig. 2. Series of ten video images at different stages of the experiment. The UDVP probes are marked on the left of the images whilst the position of the horizontal location of the measuring volume for Figs 3–5 (20.1 mm from the probe) is shown by an arrow on each image. Flow is from right to left. Times refer to those displayed on subsequent time series plots in Figs 3–5 and Plates 1–3. (a) As density current approaches UDVP probes. (b) As the density current head impacts the UDVP probes near the bed. P4 (97-mm above bed) is still outside the density current and is experiencing upstream flow (see Figs 3 and 4). (c) The top of the density current head. (d) The top of the density current head: note the developing Kelvin–Helmholtz instabilities at the top of the current and that P4 lies above the main forward flow. (e) The top and back of the density current head pass the probes: note further development of the Kelvin–Helmholtz instabilities at the top of the current and that P4 still lies above the main forward flow (see Figs 3 and 4). (f ) Continued growth of large-scale Kevin– Helmholtz instabilities at the back of the current head. P4 is now within the mixing zone at the top of the head of the main forward flow. (g) Further development of Kevin–Helmholtz instabilities. Note both the vortices generated at the upper current interface approximately 40-mm upflow from the arrow and also how the large Kelvin–Helmholtz instability shown in (f ) has advected upstream. This upstream movement is due to the strong return flow at the top of the current, which is generated in this shallow-flow experiment. (h) During passage of the tail of the current. (i) As the first bore moves upstream past the probes: note that P4 lies above the top of the bore, as also shown by its velocity signature (see Figs 3 and 5). The top of the bore shows slight irregular curvature due to mixing at the back of this type C bore (Fig. 6). ( j) The first bore of the second series of solitary waves, reflected from the upstream end of the tank and moving back downstream.

PGCC11 9/12/08 3:29 PM Page 165

165

Sediment-laden gravity currents 180 140 100 4:97 mm

60 20 –20 –60 180 140 100 60

3:67 mm

Velocity (mm s–1)

20 –20 –60 180 140 100 2:37 mm

60 20 –20 –60 180 140 100

i

60

1:7 mm

20 –20

a-f

–60 0

20

g

j

h 40

60

80

100

120

140

160

180

200

220

240

260

280

300

Time (seconds)

Fig. 3. Complete time series at four points in a vertical profile 7, 37 67 and 97 mm above the bed (probes P1–P4, Fig. 1a) at 20.1-mm upstream from the UDVP probes. Arrowed letters refer to video images shown in Fig. 2.

the lowest probe 3.07 s before the bore is detected at P3 (67 mm). This wave can be classified as a strong, type C bore according to the classification scheme of Edwards (1993) and Edwards et al. (1994), since the depth ratio between the mean bore height, db, and the thickness of the lower fluid, dl (the residual flow of the density current tail) had a value of c. 3.8 (see Fig. 6). The first bore had an asymmetrical form with some Kelvin–Helmholtz instability at its rear (Fig. 2i), showing more rapid acceleration of flow at its front than deceleration on its rear side (Fig. 5). Assuming a mean velocity of 45 mm s−1 for this bore (Fig. 5), the wavelength of the bore can be estimated at c. 0.22 m, in good agreement with the wavelength of c. 0.27 m measured from the video images. It is clear that P4 (97 mm

above the bed) was located above the bore (see Fig. 2i; maximum bore height measured from video = 73 mm) and that, as the bore passed this point, fluid was displaced in a downstream direction (Figs 3 and 5): this displacement matches the upstream flow at P3 (67 mm) with no temporal lag. The first flow reversal is shown by a bore with three solitary waves passing P3 (Figs 3, 5 and 7). Additionally, four smaller waves on the tail of this upstream flow are present lower in the flow at t > 150 s at P1 (7 mm from the bed; Figs 3 and 5). Flow at P3 is upstream in these first bores, until after 130 s flow is then directed downstream (Fig. 3), even though flow near the bed is still upstream due to the continuing series of reflections. This differential response of P3 is due to the fact that the amplitude of

PGCC11 9/12/08 3:29 PM Page 166

166

J.L. Best et al. 180 140

4:97 mm

100 60 20 –20 –60 180 140

3:67 mm

100 Velocity (mm s–1)

60 20 –20 –60 180 140

2:37 mm

100 60 20 –20 –60 180 140 1:7 mm

100 60 20 –20

a bcd e f

–60 10

20

g

h 30

40

50

60

70

80

90

100

Time (seconds)

Fig. 4. Detail of time series (10 –100 s) shown on Fig. 3 for the passage of the density current head and tail. Arrowed letters refer to video images shown in Fig. 2.

these waves changes though the wave group of the first reversal, and after 130 s P3 lies above these later, smaller solitary waves. Closer examination of the velocity time series for P3 during the first two waves (Fig. 5) shows both upstream and downstream flow during the passage of these waves, reflecting the location of this measuring volume at the top of the wave train, and hence both within and outside the bores as they translate past this point (Fig. 7). Near the bed (P1 and P2), the upstream flow in the wave train of the first reversal is continuous over a period of c. 65 s (c. 103–170 s), before arrival of the first waves associated with subsequent reflection from the upstream end of the tank (Fig. 2j). This second

reflection is first recorded as two, small amplitude waves: this is evidenced by the out-of-phase relationship between the velocity traces for P1 and P2 with those of P3 and P4 for the first wave (Fig. 2j) and the traces of P1, P2 and P3 with P4 for the second wave. This illustrates that the second reflected wave was of higher amplitude than the first, as evidenced by the differential response of P3. Probe 3 (67 mm) thus lay above the first wave in this second, downstreamdirected wave group, but lay within the second wave. It is also noticeable that the waves in this second wave train had a more symmetrical form than the first upstream reflections: these second reflections were smoother form solitary waves of an intermediate type

PGCC11 9/12/08 3:29 PM Page 167

167

Sediment-laden gravity currents 4:97 mm 60 20 –20 –60

3:67 mm

60 20

Velocity (mm s–1)

–20 –60

2:37 mm

60 20 –20 –60

1:7 mm

60 i 20 –20 –60 100

j 110

120

130

140

150

160

170

180

190

200

210

220

230

240

Time (seconds)

Fig. 5. Detail of time series (100 –240 s) shown on Fig. 3 for the passage of the first reflected bores as they pass upstream past the UDVP probes. Arrowed letters refer to video images shown in Fig. 2.

B or weak type A bore according to the scheme of Edwards et al. (1994; see Fig. 6), with db/dl ratios of 1.8, 2.0 and 2.1 for the three solitary waves. At the end of the velocity record, a third series of reflected waves moving back upstream can be discerned (Fig. 3). These waves are of lower velocity than previous reflections and have a height that encompassed the two probes near the bed, the upper probe again showing an out-of-phase relationship with the lower probes. These are weak type A bores, according to the classification scheme of Edwards et al. (1994), with db/dl values of c. 1.7. It is evident from these velocity traces that flow reversals create a series of bores of different type (Edwards et al., 1994; Fig. 6)

and that the reverse flow may form a more continuous current which, when superimposed with solitary waves, may give the potential to generate oscillatory near-bed flows (Kneller et al., 1997). Spatio-temporal series The above discussion has focused on the time series at four points in the vertical dimension. However, one significant advantage of UDVP is that it can provide such time series for up to 128 points for each probe, thus providing an opportunity to assess the spatial and temporal evolution of the current. Three-dimensional surface plots of the spatio-temporal series from P2 and

PGCC11 9/12/08 3:29 PM Page 168

168

J.L. Best et al.

Fig. 6. Classification scheme of bore morphology (after Edwards et al., 1994).

Residual forward flow Type B

Type C

P3

Type B P2 Reverse flow

P4 (Plate 1a,b) show the potential of UDVP to quantify the evolution of the flow field. Flow 37 mm above the bed (P2) illustrates that most of the coherent flow structures detected by the probe can be traced throughout the 95-mm sampling volume, the slope of the structures in the downstream temporal plane indicating their advection velocity. Downstream propagation of the main density current head proceeds at a mean rate of c. 115 mm s−1, in good agreement with the video (106 mm s−1) but some 32% less than the maximum velocities recorded at this height (cf. Kneller et al., 1999). The waves associated with both the upstream-moving first wave train and the downstream-moving second series of reflections are clearly evident. The upstream and downstream movement of the first waves in both wave groups is clearly discerned if the temporal grid lines are viewed with respect to the velocity peaks of these waves: the first wave of the downstream migrating bores can be clearly seen to become later in the time series relative to the temporal grid line (arrowed on Plate 1a). The spatio-temporal

P4

P1

Fig. 7. Schematic diagram of bore propagation past the measuring volumes for the first flow reversal, illustrating the height of each probe with respect to the first three bores (Figs 3 and 5).

slice at P4 (97 mm above the bed, Plate 1b) shows a radically different response and displays the largescale turbulent eddies associated with the shear layer at the top of the current. Some of these eddies can be seen to change in magnitude as they convect downstream, this representing either their advection out of the measuring volume or their mixing, growth or decay. Passage of the bores both upstream and downstream is shown at P4 by displacement of fluid in the opposite direction to the bore propagation, since P4 is above the top of the bores. These reflections can clearly be seen but are marked by fluid displacement opposite to the bores (compare Plate 1a and b). Turbulence Structure Indications of the turbulence structure within the gravity current may be obtained from examination of the variance and periodicity of fluctuations in each time series (Table 3, Fig. 8; Figs 3–5). Two periods of time, in which flow is quasi-steady, are examined

PGCC11 9/12/08 3:29 PM Page 169

169

Sediment-laden gravity currents Table 3. Summary statistics for the forward flow at two time intervals (see Figs 3 –5) for a measurement volume 20.1-mm upstream from the probe head (see Figs 3–5). Mean velocity, U (mm s−1) and turbulence intensity, σTI, are given for each time window* Probe distance above bed (mm) Time interval T1

7

37

67

97

U = 87.3 σ TI = 17.9% (15.8 –35.8 s)

T1a: U = 155.8 σ TI = 4.1% (15.8–22.9 s)

U = 87.1 σ TI = 13.2% (16.7–53.7 s)

U = 25.0 σ TI = 65.3% (18.8–54.6 s)

U = 12.9 σ TI = 27.2% (75.8–99.8 s)

U = −10.2 σ TI = 29.4% (75.9–99.9 s)

T1b: U = 99.0 σ TI = 6.6% (28.4 – 44.6 s) T2

U = 8.2 σ TI = 48.7% (75.6 –99.6 s)

U = 36.3 σ TI = 5.6% (75.7–99.7 s)

* Figures in parentheses denote time interval of window. 100 T1: U T1: Turb. Intensity T2: U T2: Turb. Intensity

90

0.6

0.5

70 60

0.4

h/h

50

h

height above bed, h, mm

80

0.3

40 30

0.2

20 0.1 10

Fig. 8. Vertical profiles of mean velocity (U ) and turbulence intensity (σTI) for two periods of flow, T1 and T2 (see text and Table 3 for explanation). Mean velocity, U, and height above the bed, h, are also expressed in dimensionless form through division by the velocity maximum, Umax, and maximum head height, hh, respectively.

0 -20

0 0

20

40

60

80

100 120 140 160

velocity, U (mm s -1 ) or turbulence intensity, σ -0.12

(Table 3). Period T1 corresponds to the period of maximum flow velocity, whereas period T2 represents flow in the body/tail of the current before arrival of the first reflection. It is evident that during the passage of the forward flow, the regions of highest turbulence intensity (turbulence intensity, σTI = standard deviation/ mean velocity at-a-point, for the period of time considered) lie both in the shear layer at the top of the current (P4, 97 mm), which attains turbulence intensities up to 65%, and near the wall (P1, 7 mm, σTI up to 49%). Flow at the velocity maximum (P2, 37 mm) has a markedly lower turbulence intensity (4 –7%) than that on either

0.12

0.36

0.6

TI

(%) 0.84

U/U max

side, this supporting past measurements in these regions (Kneller et al., 1999; Kneller & Buckee, 2000; Buckee et al., this volume). Indications of the scale of turbulence within the current can be obtained from the time series at each point (Figs 3–5). Turbulence at the top of the forward current (P3 and P4) has fluctuations up to c. 68 mm s−1 and a periodicity of between 1.95 and 4 s. If these periods are multiplied by the mean advection velocity, and assuming a frozen eddy hypothesis, then the mean length of eddies at the top of the current is c.140 mm. These estimates illustrate that the scale of turbulence at the top of the current, and associated

PGCC11 9/12/08 3:29 PM Page 170

170

J.L. Best et al.

with shear and Kelvin–Helmholtz instability at the head, is roughly equivalent to the thickness of the current, a result confirming the observations of Kneller et al. (1999). Flow field visualization Sequential slicing of the spatio-temporal series for an interpolated volume obtained from all four probes allows the temporal evolution of the flow field to be quantified. Interpolation was achieved by using the time series from each probe at 128 measurement locations and then interpolating these using a kriging routine (kriging used a spherical distribution for each data point) and a grid of 339 768 points (4 × 117 × 726) in Spyglass T3D©. A series of velocity slices at different stages of evolution of the current are shown for the forward flow (Plate 2a–g) and first bore moving upstream (Plate 2h–n), whilst animations of the forward flow and the first reverse flow bore are available at http://earth.leeds.ac.uk/research/seddies/best/ gravity.htm. These interpolated volumes allow the potential to yield unique visualization of a sedimentladen density current. Intrusion of the density current is clearly shown by the advancing high velocity front (Plate 2a,b) and the velocity maximum at c. 0.3 of the current height (Plate 2a–g). The sharp velocity gradient at the top of the current, and reverse flow in the ambient fluid due to fluid displacement in this small flow–depth experiment, are clearly evident and generate a layer of intense shear at the top of the current. As the head passes (Plate 2e–g), large-scale fluctuations in the velocity field again indicate the occurrence of largescale vorticity within the current which is evident on the video (Fig. 2b–h). The velocity maximum at c. 0.3 of the current height becomes more evident within the rear of the head and tail of the current, developing a more uniform structure with a clear velocity gradient both at the bed and upper surface of the tail. The presence of higher velocity fluid near the bed in the initial stages of the intrusion of the density current (Plate 2a–d) bears witness to the greater mixing within the front of the flow than at later stages of the current advance (Plate 2e–g). Movement of the first bore back upstream (Plate 2h– n) is well depicted by the smooth front of the bore (Plate 2i, j), with slight instability at the top of this type C bore being evidenced by the bulges on the top of the flow (Plate 2k). The asymmetry of the bore is also shown by the longer time interval from the front to the top of the bore (Plate 2h–m; c. 2.60 s) than from the

top to the rear of the bore (Plate 2m–n; c. 1.2 s). Fluid displacement downstream by passage of the bore is clearly shown in the ambient flow (Plate 2j–m) and there appears to be little velocity gradient within the bore, a feature also evident within the time series plots (Fig. 5). Combination of these spatio-temporal slices allows construction of a unique summary image of various stages of evolution of the current (Plate 3). This plot shows three spatio-temporal slices in the vertical plane at 15, 106 and 170 s depicting the initial intrusion of the density current downstream (15 s) and propagation of bores moving first upstream (106 s) and then downstream (170 s). The spatio-temporal slice at the base of the plot in the horizontal plane depicts the downstream evolution of the current, with the gradient of the isovels quantifying the celerity of the density current and bores.

CONCLUSIONS This chapter has shown that quantification of the mean, and low-order turbulence, statistics of sedimentladen gravity currents may be uniquely approached through use of ultrasonic Doppler velocity profiling. Ultrasonic Doppler velocity profiling can be used to provide temporal, quantitative analysis of the changing flow field as a gravity current evolves and has great potential for investigating the links between entrainment, transport and deposition in such flows. This chapter has illustrated the first use of UDVP in a simple laboratory configuration with a surge-type current and shown that: (i) the velocity maximum of the forward flow lies at approximately 0.3 of the head height above the bed; (ii) the region of the velocity maximum of the forward flow is associated with the lowest turbulence intensities; (iii) the highest turbulence intensities of the forward flow are associated with both near-bed shear and mixing on the back of the head; (iv) return flows, in the form of different types of bore, may be fully quantified by UDVP, and (v) the scale of turbulence associated with the head appears to be of the same order as the head height. Additionally, these results demonstrate that past studies of gravity currents, that have had to use saline solutions or sediment-free dispersions as a proxy for sedimentladen currents, are a good substitute for low-density, sediment-bearing gravity currents. However, this chapter has only dealt with results detailing solely the downstream component of flow: use of multiple UDVP probes in different orientations

PGCC11 9/12/08 3:29 PM Page 171

171

Sediment-laden gravity currents may yield quantitative information on both vertical and transverse velocities within the current, and if the probes are precisely aligned, two- and threedimensional velocity measurements at several different points. Use of these arrangements has begun in other applications (Rolland & Lemmin, 1997; Graf & Yulistiyanto, 1998) and holds the promise of key advances in modelling the behaviour of sediment-laden density currents, particularly if the fluid dynamics of such flows can be quantified together with ongoing erosion/deposition from within the density current.

ACKNOWLEDGEMENTS Development of the UDVP within sedimentological applications has been made possible through funding from the UK Natural Environment Research Council (NERC grant GR3/10015) to J.B., A.K. and Ian Reid (Loughborough, UK) for which we are extremely grateful. J.P. acknowledges funding from a consortium of oil companies which support the Turbidites Research Group at Leeds. The input and advice of Yasushi Takeda of the Paul Scherrer Institute and Robert Jaryczewski and Graham Hassall at DANTEC UK have been invaluable in developing this multiplexed UDVP system. Discussions with Clare Buckee and Henry Pantin have been most helpful in shaping our ideas during these experiments, whilst constructive reviews by Frans de Rooj, Mustafa Altinakar and Ben Kneller have helped clarify and improve the manuscript.

NOTATION c db dl Dmax D16 D50 D84 fD fD (max) fo fprf Fr

velocity of ultrasound mean bore height thickness of the residual flow of the density current tail maximum depth to which the UDVP may detect 16th percentile of grain-size distribution mean grain size 84th percentile of grain-size distribution Doppler frequency shift maximum detectable Doppler shift frequency ultrasound frequency frequency of repetition of the ultrasound pulses densiometric Froude number

g g′ h hh ht Re Ri t U Umean Umax Ures x ρa ρs σTI µs

acceleration due to gravity reduced gravity (= g(ρs − ρa)/ρa) height above the bed visually measured maximum head thickness (forward flow) visually measured thickness of tail of the forward flow flow Reynolds number flow Richardson number time flow velocity mean velocity of forward flow maximum detectable velocity velocity resolution location of sampling volume density of ambient fluid initial density of sediment suspension in current turbulence intensity (= standard deviation/ Umean) viscosity of sediment suspension in current

REFERENCES Altinakar, M.S., Graf, W.H. & Hopfinger, E.J. (1996) Flow structure in turbidity currents. J. Hydraul. Res. 34, 713 –718. Best, J.L. & Ashworth, P.J. (1994) A high-resolution ultrasonic bed profiler for use in laboratory flumes. J. sediment. Res. A64, 674 –675. Best, J.L., Bennett, S.J., Bridge, J.S. & Leeder, M.R. (1997) Turbulence modulation and particle velocities over flat sediment beds. J. Hydraul. Eng. 123, 1118 –1129. Bonnecaze, R.T., Huppert, H.E. & Lister, J.R. (1993) Particle-driven gravity currents. J. Fluid Mech. 250, 339–369. Buckee, C., Kneller, B.C. & Peakall, J. (2000) Turbulence structure in steady, solute driven gravity currents. In: Sediment Transport and Deposition by Particulate Gravity Currents (Eds McCaffrey, W.D., Peakall, J. & Kneller, B.C.). International Association of Sedimentologists Special Publication. No. 31, pp. 173 –187. Blackwell Science, Oxford. Edwards, D.E. (1993) Turbidity Currents: Dynamics, Deposits and Reversals. Lecture Notes in Earth Sciences 44. Springer-Verlag, Berlin, 173 pp. Edwards, D.E., Leeder, M.R., Best, J.L. & Pantin, H.M. (1994) An experimental study of reflected density currents and the interpretation of certain turbidites. Sedimentology 41, 437– 461. Elghobashi, S. (1994) On predicting particle-laden turbulent flows. Appl. Sci. Res. 52, 309–329. García, M.H. (1994) Depositional turbidity currents laden with poorly sorted sediment. J. Hydraul. Eng. 120, 1240 –1263. García, M. & Parker, G. (1991) Entrainment of bed sediment into suspension. J. Hydraul. Eng. 117, 414 – 435.

PGCC11 9/12/08 3:29 PM Page 172

172

J.L. Best et al.

García, M. & Parker, G. (1993) Experiments on entrainment of bed sediment into suspension by a dense bottom current. J. geophys. Res. 98, 4793– 4807. García, M.H. & Parsons, J.D. (1996) Mixing at the front of gravity currents. Dyn. Atmos. Oceans 24, 197–205. Gore, R.A. & Crowe, C.T. (1991) Modulation of turbulence by a dispersed phase. Trans. Am. Soc. Mech. Engnrs. J. Fluids Engrg. 113, 304 –307. Graf, W.H. & Yulistiyanto, B. (1998) Experiments on flow around a cylinder; the velocity and vorticity fields. J. Hydraul. Res. 36, 637– 653. Hetsroni, G. (1989) Particles–turbulence interaction, Int. J. Multiphase Flow 5, 735–746. Hetsroni, G. (1993) The effect of particles on the turbulence in a boundary layer, In: Particulate Two-Phase Flow (Ed. Roco, M.C.). Butterworth-Heinemann, pp. 244–264. Huppert, H.E. & Simpson, J.E. (1980) The slumping of gravity currents. J. Fluid Mech. 99, 785–799. Hürzeler, B.E., Imberger, J. & Ivey, G.N. (1996) Dynamics of turbidity currents with reversing buoyancy, J. Hydraul. Eng. 122, 230 –236. Kneller, B.C. (1995) Beyond the turbidite paradigm: physical models for the deposition of turbidites and their implications for reservoir description. In: Characterisation of Deep Marine Clastic Systems (Eds Hartley, A.J. & Prosser, D.J.), Geol. Soc. London Spec. Publ. 94, 29– 46. Kneller, B. & Buckee, C. (2000) The structure and fluid mechanics of turbidity currents: a review of some recent studies and their geological implications, Sedimentology, 48, Supplement 1, 62–94. Kneller, B.C., Bennett, S.J. & McCaffrey, W.D. (1997) Velocity and turbulence structure of density currents and internal solitary waves: potential sediment transport and the formation of wave ripples in deep water. Sediment. Geol. 112, 235–250. Kneller, B.C., Bennett, S.J. & McCaffrey, W.D. (1999) Velocity structure, turbulence and fluid stresses in experimental turbulent density currents. J. geophys. Res. 104, 5381–5391. Lhermitte, R. & Lemmin, U. (1990) Probing turbulence by high frequency Doppler sonar. Geophys. Res. Lett. 17, 1549–1552. Lhermitte, R. & Lemmin, U. (1994) Open channel flow and turbulence measurement by high resolution Doppler sonar. J. Atmos. Ocean Technol. 11, 1295 –1308. Metflow (1997) Ultrasonic Velocity Profile Monitor Operation Manual, Model X-3-PSi, Metflow SA, Lausanne, Switzerland, 47 pp and supplements/appendices. Nikora, V.I. & Goring, D.G. (1998) ADV measurements of turbulence: can we improve their interpretation? J. Hydraul. Eng. 124, 630 – 634. Pantin, H.M. & Leeder, M.R. (1987) Reverse flow in turbid-

ity currents: the role of internal solitons. Sedimentology 34, 1143–1155. Parker, G., García, M., Fukushima, Y. & Yu, W. (1987) Experiments on turbidity currents over an erodible bed. J. Hydraul. Res. 25, 123 –147. Rolland, T. & Lemmin, U. (1997) A two-component acoustic velocity profiler for use in turbulent open-channel flow. J. Hydraul. Res. 35, 545–561. Shanmugam, G. (1997) The Bouma Sequence and the turbidite mind set. Earth Sci. Rev. 42, 201–229. Simpson, J.E. (1972) Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53, 759–768. Simpson, J.E. (1997) Gravity Currents: In the Environment and the Laboratory. 2nd edn. Cambridge University Press, Cambridge, 224 pp. Simpson, J.E. & Britter, R.E. (1979) The dynamics of the head of a gravity current advancing over a horizontal surface. J. Fluid Mech. 94, 477– 495. Takeda, Y. (1991) Development of an ultrasound velocity profile monitor. Nucl. Eng. Design 126, 277–284. Takeda, Y. (1993) Velocity profile measurement by ultrasonic Doppler method. In: Experimental Heat Transfer, Fluid Mechanics and Thermodynamics (Ed. Kelleher, M.D. et al.). Elsevier, Amsterdam, pp. 126 –131. Takeda, Y. (1995) Instantaneous velocity profile measurement by ultrasonic Doppler method. Int. J. Japan. Soc. Mech. Eng., Series B 35, 8–16. Takeda, Y., Kobashi, K. & Fischer, W.E. (1990) Observation of the transient behaviour of Taylor vortex flow between rotating concentric cylinders after sudden start. Exp. Fluids 9, 317–319. Takeda, Y., Fischer, W.E., Kobashi, K. & Takada, T. (1992) Spatial characteristics of dynamic properties of modulated wavy vortex flow in a rotating Couette system. Exp. Fluids 13, 199–207. Takeda, Y., Fischer, W.E. & Sakakibara, J. (1994) Decomposition of the modulated waves in a rotating Couette system. Science 263, 502–505. Thorne, P.D., Hardcastle, P.J. & Hogg, A. (1996) Observations of near-bed suspended sediment turbulence structures using multifrequency acoustic backscattering, In: Coherent Flow Structures in Open Channels (Eds Ashworth, P.J., Bennett, S.J., Best, J.L. & McLelland, S.J.). John Wiley, Chichester, pp. 343–358. Tokuhiro, A. & Takeda, Y. (1993) Measurement of flow phenomena using the ultrasonic velocity profile method in a simulated Czochralski puller. J. Crystal Growth 130, 421– 432. Tuefel, M., Trimis, D., Lohmüller, A., Takeda, Y. & Durst, F. (1992) Determination of velocity profiles in oscillating pipe-flows by using laser Doppler velocimetry and ultrasonic measuring devices. Flow Meas. Instrum. 3, 95–101.

PGCC12 9/12/08 3:28 PM Page 173


Turbulence structure in steady, solute-driven gravity currents C . B U C K E E , B . K N E L L E R and J . P E A K A L L School of Earth Sciences, University of Leeds, Leeds LS2 9JT, UK

ABSTRACT High-resolution turbulence data from refractive index-matched gravity currents have been used to quantify the mean flow and turbulence structure in steady, experimental gravity currents. Comparison of this data-set with experimental and theoretical gravity current data from the literature and with turbulent wall jets, reveals several new insights into both subcritical and supercritical flows. Existing data collapse approaches can be improved with the use of a new characteristic lengthscale taken from the wall jet literature. Turbulence production from shear has been quantified and is seen to be most significant in subcritical currents. Density stratification is also shown to be an important control on the distribution of turbulent kinetic energy. A slow diffusion zone (SDZ) characterized by low turbulence intensities and reduced vertical mass transport in the lower part of the current, around the level of the velocity maximum, has been identified, and related to both density stratification and reduced turbulence production around the velocity maximum. The results presented in this chapter should lead to improved understanding of gravity current dynamics and provide a good test for the output of numerical models.

INTRODUCTION cannot be used to quantify the turbulence structure. Recently, the use of laser Doppler anemometry (LDA) in refractive index (RI) matched flows has allowed high resolution velocity data to be collected non-intrusively from simple solute-driven gravity currents enabling the turbulence structure to be examined (Kneller et al., 1997, 1999). However, these were highly unsteady, small-volume, surge-type currents allowing only very short data intervals (less than 5 s) for analysis. Most recently, ultrasonic Doppler velocity profiling (UDVP) has allowed one-dimensional resolution of the flow field in an unsteady gravity current (Best et al., this volume). This chapter aims to describe both qualitatively and quantitatively the vertical distribution of turbulence within experimental gravity currents driven by a solute-derived density difference. The ultimate goal of this research is to improve the understanding of particulate gravity current dynamics, specifically turbidity currents. Fluid turbulence is a primary mechanism for sediment entrainment, suspension and transport, and also indirectly controls deposition from turbidity currents. In order to improve predictions about

Turbulent gravity currents occur in many different environments including subaerial pyroclastic flows in volcanic environments, turbidity currents in lakes and oceans, and thunderstorm outflows in the atmosphere (Simpson, 1997). Small-scale laboratory experiments present one of the best means of studying gravity current processes. In contrast to open-channel flows, there have been relatively few experimental studies on gravity currents and these have been largely confined to unidirectional, surge type flows. Previous work has focused on the morphology of the current (e.g. Middleton, 1996a,b; Allen, 1971; Simpson, 1972; Britter & Simpson, 1978; Simpson & Britter, 1979), mean flow characteristics (e.g. Middleton, 1966b; Parker et al., 1987), and vertical profiles of mean flow properties such as velocity and density in steady currents (e.g. García & Parker, 1993; Altinakar et al., 1996). However, fluid turbulence, which is a primary control on gravity current behaviour, has not been studied in detail, primarily due to inherent technical problems. Flow visualization techniques have been used to study the internal structure in experimental gravity currents (e.g. Pantin & Leeder, 1987; Edwards et al., 1994) but


173

PGCC12 9/12/08 3:28 PM Page 174

174

C. Buckee et al.

turbidite deposits (extent, geometries, grain-size distributions, etc.) using numerical or physical models it is important to be able to answer the following questions; 1 What is the turbulence structure in a gravity current? 2 How does the turbulence structure control sediment entrainment, suspension and deposition? 3 How does the presence of suspended sediment affect the turbulence structure? In order to address these questions a preliminary set of experiments was carried out on non-particulate gravity currents. The rationale for this was threefold: (i) non-particulate currents are the simplest case and thus the best starting point for studying the turbulence structure in the body region of gravity currents; (ii) the presence of particulate matter in the gravity current limits the techniques available for instrumenting the flows and hence reduces the resolution of the experiments; (iii) the data from the non-particulate experiments provide a ‘control’ data-set that can be compared with data from particulate currents. This chapter presents the results of a series of RI-matched experiments that were carried out on steady, turbulent gravity currents. Profiles of at-a-point instantaneous

velocity and density data were collected allowing the quantification of both the mean flow properties and turbulence structure through the depth of the flow.

EXPERIMENTAL EQUIPMENT AND PROCEDURE The experiments were carried out in the Sedimentological Fluid Dynamics Laboratory of the University of Leeds. The flume tank (shown schematically in Fig. 1) is 6-m long, 0.5-m wide and 1.5-m deep. A smooth false floor adjustable slope was erected 0.5 m above the bottom of the tank in order to create a sump in which the dense fluid could collect. Two large mixing tanks (1.8 m3 each) were used to mix solutions for the ambient fluid and the gravity current. The contents of the tanks were pumped into the flume tank, the input rate being controlled by an inverter and monitored by an electromagnetic flow meter. In this way, the current could be input at a quasi-steady rate of 4 × 10−3 m3 s−1 (±0.02 × 10−3 m3 s−1). The dense fluid was introduced through an inlet box with an adjustable gate, allowing the inlet height to be varied. The false floor was 5.5 m

Header tank

Side view

Planform view

Level switch

Perforated pipes

Key Pump

Drainage from base of tanks

Fig. 1. Schematic diagram of the flume tank in which the experiments were carried out.

Flowmeter

PGCC12 9/12/08 3:28 PM Page 175

Turbulence structure in length, and was set at an angle of 2° below horizontal. At the far end of the tank the dense fluid was pumped out through a drain, with the output rate monitored by an electromagnetic flow meter and controlled in the same way as the input pump. Fluid was pumped out of the tank at a higher rate (6 × 10−3 m3 s−1) than that at which it was introduced in order to minimize the effect of reflection of the gravity current head as it impinged upon the far wall of the tank and to account for the volume expansion of the current due to mixing and entrainment of the ambient fluid. A top-up system for the ambient fluid was used to ensure that the total fluid depth remained constant. Two header tanks (0.2 m3 each) fed two perforated pipes running the length of the tank, at the top (Fig. 1), which thus refilled the tank with fresh ambient fluid without setting up secondary circulation cells. Laser Doppler anemometry Laser Doppler anemometry (see Buchhave et al. (1979) for a general description) was used to make non-intrusive, high-resolution measurements of instantaneous, at-a-point, downstream and vertical velocities with a rapid response time. A DANTEC 100-mW argon-ion laser was used in backscatter mode with a 400-mm focal length lens. Downstream velocity (u) and bed-normal velocities (v) were measured using green light (λ 1 = 488 nm) and blue light (λ 2 = 514.5 nm), respectively. The beams were aligned so that they were parallel with u, and perpendicular to the false floor. The laser was operated with a 40-MHz Doppler shift in order to allow measurement of both positive and negative flow along either axis. The LDA was run with a DANTEC particle dynamics processor, which operates on a correlation type process. The gravity current was seeded with traces of poster paint, particles of which have a low settling velocity and are carried passively, reflecting the fluid motions. The laser head was mounted on a high precision traverse system, which allowed accurate positioning of the laser intersection volume to ±1 mm. Measurements were taken 2.9 m from the inlet box, 0.15 m into the flow (transverse to the downstream direction) in order to take measurements in a region in which sidewall effects were minimized. In each experiment, the instantaneous at-a-point two-component velocity was measured for 12 s at each of 11 points (3, 6, 10, 15, 20, 30, 40, 60, 80, 100 and 120 mm) above the false floor. Data acquisition rates of between 15 and 200 Hz were achieved, depending upon the height in the flow and the degree and intensity of mixing at that point.

175

Refractive index matching In a turbulent flow with refractive index variations, the beams of a laser Doppler anemometer are attenuated and refracted. Consequently, the beams only cross in the measuring volume intermittently, and if they do intersect further refraction of the Dopplershifted reflections may occur, which results in a severe loss of signal. In a density driven flow, such as a gravity current, the refractive index may change with the density hence restricting the use of LDA. RI matching, however, allows the use of LDA in such a flow by using two solutes that contribute to the RI and the density of a solution in different proportions (e.g. McDougall, 1979; Alahyari & Longmire, 1994). In these experiments, aqueous solutions of glycerol and monopotassium phosphate were used for the ambient fluid and the dense gravity current respectively. A maximum density difference of 4.78% can be achieved using this combination of fluids without resulting in significant RI variations during mixing (Alahyari & Longmire, 1994). The RI of the solutions was measured to within 0.0001 using a hand-held Leica refractometer. Conductivity probe The conductivity of an aqueous solution of an electrolyte, such as monopotassium phosphate, is directly related to the density of that solution. A four-electrode conductivity microprobe from Precision Measurement Engineering Inc. was used for high-resolution measurement of turbulent density fluctuations in the body of the gravity current (for details see Head, 1984). The probe yields an output voltage that is a linear function of the conductivity of the solution. The conductivity probe was positioned vertically, with the measurement volume at the same level above the floor as the laser measurement volume, but positioned 0.05 m further downstream. This distance was chosen in order to minimize the interference of the probe on the LDA measurements, while being close enough for the density and velocity measurements to be comparable. It was assumed that the lengthscale of macroturbulence in the flow is significantly greater than the horizontal separation of the measurement volumes. The LDA measurement volume and the conductivity probe were raised in parallel during the course of the experiment. Instantaneous conductivity measurements at 800 Hz were recorded directly to a PC. The RI-matched fluids were ideal for use with the conductivity probe as the ambient glycerol solution is non-conductive, providing a simple relationship

PGCC12 9/12/08 3:28 PM Page 176

176

C. Buckee et al. 0.28

Table 1. Experimental conditions for Run LD and Run HD Run LD

Run HD

Initial reduced gravity, g′ (m s−2 ) Depth averaged velocity, U (m s−1) Maximum mean velocity (m s−1) Current thickness, h (m) Reynolds number Bulk Richardson number

0.196 0.15 0.18 0.09 c. 13 000 0.78

0.294 0.14 0.22 0.12 c. 16 000 2.4

0.24 µi (ms–1)

Parameter

0.20

0.16

between the measured conductivity and the gravity current density. The conductivity data (Cout) were converted to absolute densities using an empirical relationship for the density of monopotassium phosphate solution of the form: Density = C1(Cout)2 + C 2(Cout) + C 3

(1)

where the constants C1 (9 × 10 −6 ), C 2 (0.001) and C3 (0.998) were determined empirically using solutions of known density.

8.5

9

9.5

10

10.5

11

Time (s) Fig. 2. Diagram showing rationale behind filtering technique. The shaded regions represent the cut-off points if filtering were based on a temporal average of the entire data set shown here. Small-scale turbulent fluctuations superimposed on large fluctuations would be removed as ‘spikes’ when filtered this way. The fine lines show the cut-off points based on the moving average.

Experiments A series of experiments was carried out in order to investigate the effects of density difference, slope, flow regime (subcritical or supercritical) and bed roughness on gravity current dynamics. This paper primarily concentrates on two of these experiments which had different initial densities, producing subcritical and supercritical currents. In both experiments the inlet gate height was 0.1 m, the slope, 2°, and the total ambient fluid depth above the hydraulically smooth false floor at the measurement point was 0.8 m. In Run LD (low density) the density difference was 1.9% while in Run HD ( high density) the density difference was 2.9% ( Table 1). Simultaneous measurements from the body of the current using the LDA and the conductivity probe began when the head of the current had reached the end of the tank. After 12 s of data acquisition the instruments were moved vertically upwards to the next measurement position, and this was continued until the profile was complete. The duration of each experiment was approximately five minutes.

RESULTS Time series The instantaneous velocity and conductivity time series were filtered using a technique which replaced

points that were more than two standard deviations from the mean (determined using a 10-point moving average) with the average value of the four surrounding data points. This approach was taken in order to ensure that only spikes in the data were removed rather than turbulent fluctuations (Fig. 2). Filtering was necessary as the spikes in the data, although rare (< 5%) were sufficiently large to affect the calculated mean flow properties. The currents were quasisteady, allowing temporal averaging of the entire time series and hence calculation of both the mean flow properties and the instantaneous turbulent deviations from the mean. Temporal averages at each point (hereafter referred to as mean and denoted by an overbar) were taken over the full 12-second data set except in a small number of cases in which a shorter period was used because of anomalously unsteady records. Mean flow properties Downstream velocity (u) There was a close correspondence between the shape of the mean downstream velocity profile in the two currents (Fig. 3a). The maximum mean downstream velocity (u) in Run HD was 0.22 m s−1 (Table 1) with maximum instantaneous velocities (u i) of 0.25 m s−1

PGCC12 9/12/08 3:28 PM Page 177

177

Turbulence structure

Fig. 3. Profiles of (a) mean downstream velocity (u), (b) mean vertical velocity (v) and (c) mean density (7) for Run LD and Run HD. Heights in the flow (y) are normalized by the characteristic lengthscale ( y–12 ) defined as the distance between the bed and the height above the velocity maximum at which the downstream velocity is half the maximum downstream velocity. Velocities are normalized by max (u) and density is normalized by the initial current density, ρo. The horizontal solid line shows the location of Umax for Run LD and the horizontal dashed line shows Umax for Run HD.

at this point, 14% higher than the mean. However, at the base of the current maximum instantaneous velocities up to 0.26 m s−1 were recorded, 18% higher than the mean value at the position of the velocity maximum (Umax). Similarly, in Run LD the maximum mean velocity was 0.18 m s −1 with instantaneous velocities of 0.26 m s−1 at the velocity maximum, 42% higher than the mean, and with instantaneous velocities of the same magnitude at the base of the current. Depth-averaged velocities (U ) and current thicknesses (h) were calculated for each experiment using the following expressions (Ellison & Turner, 1959): Uh =

U2h

∞

∞

=

Re =

ρo Uh µ

(4)

g ′h U2

(5)

Rio =

Where ρo is the initial current density, µ is the dynamic viscosity and g′ is the reduced gravitational acceleration (g′ = g∆ρ/ρa, ρa is the ambient density). Both flows were fully turbulent (Re > 2000), Run LD was found to be supercritical (Rio < 1) and Run HD was subcritical (Rio > 1), based on values of U and h derived as described above. Vertical velocity (v)

u dy

(2)

0

u 2 dy

(3)

0

where y is height above the floor of the tank. The derived values (see Table 1) were used to calculate flow Reynolds number (Re) and bulk Richardson number (Rio):

Mean vertical velocities (v) in both currents were of the same magnitude (−0.007 to 0.001 m s−1) and reached only about 4% of maximum u. Maximum instantaneous vertical velocities (vi), however were an order of magnitude larger than the mean, with a range of approximately ±0.04 m s−1 in both experiments. Figure 3(b) shows the distribution of mean vertical velocity. In both cases average velocities are negative (downwards) which may be due to minor flow non-uniformity as the current undergoes gravitational collapse.

PGCC12 9/12/08 3:28 PM Page 178

178

C. Buckee et al. 1

Density distribution The vertical distributions of mean density (calculated from the measured conductivity) for both experiments are shown in Fig. 3(c). The overall shape of the density profiles is in good agreement with published results from experiments on saline currents and weakly depositional particulate flows (e.g. Ellison & Turner, 1959; Fietz & Wood, 1967; García, 1994). In both cases the flows are density stratified. A dense basal layer with a high density gradient can be observed in the lower part of the flow with less dense, more homogeneous mixed fluid above. The density profile is smooth but the density gradient declines rapidly around the level of the velocity maximum, as observed by García and Parker (1993) and discussed by Peakall et al. (2000). The density gradient at Umax is higher in the denser current (Run HD). In Run LD the maximum at-apoint mean density ( 7) of 1015 kg m−3 was recorded at the base of the current. The maximum mean density in Run HD was 1016 kg m−3. Instantaneous density measurements, however, show that during turbulent fluctuations the instantaneous reduced gravitational acceleration can increase by up to 30% when dense parcels of fluid (approaching the initial density) are mixed upwards in the flow. Turbulence structure The passage of eddies at all scales manifests itself as instantaneous fluctuations in velocity and density, the magnitude of the fluctuation being a measure of the intensity of the turbulence. Reynolds averaging was used to decompose the instantaneous data into mean and fluctuating components in order to examine the instantaneous turbulent deviations from the mean: u′ = ui − u

(6)

v′ = vi − v

(7)

ρ′ = ρi − 7

(8)

Where u′, v′, and ρ′ are the fluctuating components of downstream velocity, vertical velocity and density respectively. Turbulent velocities The second moments of the velocity distribution, a measure of the magnitude of the turbulence, were calculated as follows:

U rms

⎤2 ⎡1 n = ⎢ ∑ ( u i − u) 2 ⎥ ⎥⎦ ⎢⎣ n i =1

Vrms

⎡1 =⎢ ⎢⎣ n

(9)

1

⎤2 ∑ ( v i − v ) 2 ⎥⎥ i =1 ⎦ n

(10)

Where Urms and Vrms are the root-mean-square (rms) values of the downstream and vertical components of velocity respectively, n is the number of observations, u i and vi are the instantaneous velocities and u and v are the time averaged at-a-point velocities. High Urms occurred close to the bed in both experiments (Fig. 4a), 0.024 m s−1 in Run LD and 0.014 m s−1 in Run HD. However, in Run LD, the supercritical current, the maximum Urms (0.0253 m s−1) was observed in the upper part of the flow, whereas in Run HD maximum Urms occurred at the base of the current. In both cases local low values of Urms were observed in the lower part of the flow, just above the level of the velocity maximum (Fig. 4a). Downstream turbulent velocities (Urms ) were significantly higher in Run LD despite the fact that maximum mean velocities are lower. Vrms is similarly distributed in both experiments, with an identical maximum value of 0.099 m s−1 at the base of the current (Fig. 4b). Vertical turbulent velocities are an order of magnitude lower than downstream turbulent velocities. Turbulent density fluctuations The density root-mean-square (ρrms) was calculated as follows (Fig. 4c):

ρrms

⎡1 =⎢ ⎢⎣ n

1

⎤2 ∑ ( ρ i − 7) 2 ⎥⎥ i =1 ⎦ n

(11)

Where ρi is the instantaneous density and 7 the time averaged density at-a-point. The vertical distribution of the fluctuations in density for Run LD is similar to the distributions of Urms in that the maximum ρrms (0.38 kg m−3) occurs in the upper part of the current. Low turbulent density fluctuations are observed close to, or below, the level of the velocity maximum. In Run HD there is a second region of high density fluctuations above Umax. Turbulent kinetic energy The vertical distribution of mean turbulent kinetic

PGCC12 9/12/08 3:28 PM Page 179


179

Fig. 4. Profiles of (a) root-mean-square (rms) downstream velocity (Urms ), (b) rms vertical velocity (Vrms ) and (c) rms density (ρrms ) for Run LD and Run HD. Heights in the flow are normalized by the characteristic lengthscale ( y–12 ). The horizontal solid line shows the location of Umax for Run LD and the horizontal dashed line shows Umax for Run HD.

energy per unit mass (k) for each experiment was calculated using the following expression: k=

1 (u ′ 2 + v ′ 2 ) 2

(12)

The cross-stream component was not measured in these experiments and hence was left out of the calculation. For the purposes of comparison, k has been normalized using the maximum down stream velocity for each current, such that: k* = k /maxu2.

(13)

The distribution of k* (Fig. 5) closely follows the distribution of Urms in both cases as the highest contribution to k* comes from the u component of velocity (i.e. from the mean flow). In Run LD the maximum value of mean k* occurs above the velocity maximum. This result agrees well with published turbulent kinetic energy distributions from lock-exchange, saline gravity currents (Kneller et al., 1999), with theoretical distributions (Eidsvik & Brørs, 1989) and qualitatively with data from plane turbulent wall jets (Launder & Rodi, 1983, and references therein). In Run HD, however, the highest k* occurs at the base of the current; k* is low in the upper part of the current. Low k* is observed in the region just above Umax for both experiments; this phenomenon has been observed in

Fig. 5. Profile of k* (turbulent kinetic energy normalized by the square of the maximum downstream velocity) per unit mass, for Run LD and Run HD. The horizontal solid line shows the location of Umax for Run LD and the horizontal dashed line shows Umax for Run HD.

experimental gravity currents by Kneller et al. (1999), and produced in numerical models (e.g. Eidsvik & Brørs, 1989) but is not seen in wall jets (Launder & Rodi, 1983).

PGCC12 9/12/08 3:28 PM Page 180

180

C. Buckee et al. and maximum stresses occur above the velocity maximum (Irwin, 1973; Launder & Rodi, 1983; Schneider & Goldstein, 1994).

DISCUSSION Mean velocity profiles

Fig. 6. Profile of τR* (Reynolds stress normalized by the square of the maximum downstream velocity per unit mass) for Run LD and Run HD. The horizontal solid line shows the location of Umax for Run LD and the horizontal dashed line shows Umax for Run HD.

Reynolds stresses The mean Reynolds stress per unit mass (τR) at each point in the profile (Fig. 6) was calculated as follows:

τ R = − ( u ′v ′ )

(14)

and normalized such that: τR* = τR/maxu2

(15)

In both experiments, maximum Reynolds stresses occurred at the base of the current, correlating with high turbulence intensities (see turbulent kinetic energy section, above). In Run LD, Reynolds stresses fall to zero near the top of the current. In Run HD Reynolds stresses fall to zero just above the downstream velocity maximum, at the level of minimum k*, and τR* is high and negative before falling back to zero near the top of the current. Previously published results on the distribution of Reynolds stresses in gravity currents (Kneller et al., 1999) show significant temporal variations related to the passage of large coherent structures. In plane wall jets, experimental results show distributions qualitatively similar to the results from Run HD in that they have a central region of negative Reynolds stresses bounded by positive stresses. However, unlike in Run HD, in plane wall jets the zero Reynolds stresses occur below the velocity maximum,

The mean velocity profile observed in experimental and natural gravity currents (e.g. Chikita, 1990; Altinakar et al., 1996; Simpson, 1997) is similar to the velocity profile in turbulent plane wall jets (Launder & Rodi, 1983). Both types of flow are considered to have an inner and outer region divided by the velocity maximum. The inner, wall-bounded, region has a positive velocity gradient and is generally less than half the thickness of the outer region, which has a negative velocity gradient. The level of the velocity maximum is controlled by the ratio of the drag forces at the upper and lower boundaries (Middleton, 1993; Kneller et al., 1997). High levels of drag at the upper boundary (as in wall jets) tend to lower the level of the velocity maximum, while high lower boundary drag (for example, turbidity currents travelling over bedforms) will raise the level of the velocity maximum. In the gravity current literature, the level of the velocity maximum is usually expressed in terms of the ratio of the height of the velocity maximum (ymax) to the total current thickness (h), Kneller et al. (1999) found ymax /h c. 0.2, Altinakar et al. (1996) found ymax /h c. 0.3, and ymax /h c. 0.1 for the experiments described here. However, it is impossible to define the position of the diffuse top of the gravity current visually, and the current thickness must be calculated via a set of moments (Eqs 2 and 3). The resultant value of h is also commonly used to normalize the data. García (1993) found that velocity data from supercritical currents collapse better than for subcritical currents. It can be seen that the velocity distribution above the velocity maximum in the present experiments (Fig. 7a) is approximately linear, as has been observed and discussed by Geyer and Farmer (1989), García (1990, 1993) and Parsons (1998). The accuracy of the calculated current thickness, however, is highly dependent on the number of data points available and the maximum height in the flow from which measurements were obtained, which may explain the large amount of scatter observed in the experiments reported here (Fig. 7a). Furthermore, the physical significance of the current thickness calculated this way is questionable

PGCC12 9/12/08 3:28 PM Page 181


181

Fig. 7. Comparison of data collapse of downstream velocity using characteristic heights, h calculated using depth averaging (a), and y–12 (b). Run LD is represented by open circles, and Run HD by open squares. The remaining data come from the same series of experiments. (c) incorporates, in addition to data shown in (a) and (b), data from Ellison and Turner (1959), triangles; Parker et al., (1987), solid squares; García and Parker (1993), solid circles and García (1994), open diamonds, and includes both subcritical and supercritical currents.

(Stacey & Bowen, 1988a). It may be more appropriate, therefore, to adopt the convention used in the wall jet literature where the height of the velocity maximum is considered in terms of ymax /y–12 where y–12 is defined as the distance between the bed and the height above the velocity maximum at which the downstream velocity is half the maximum downstream velocity. The use of y–12 as the characteristic lengthscale is suggested because it is simple to ascertain accurately, and collapses the data well (Fig. 7b). Figure 7(c) shows the same data set and also includes data taken from the literature, it can be seen that the collapse is still reasonable, despite the wide variety of experimental conditions. The value of y–12 is a function of the downstream velocity gradient above the velocity maximum and so reflects the fundamental properties of each current and can be used to collapse both supercritical and subcritical data sets together. In these experiments, Umax was found to occur at ymax /y–12 = 0.14 in Run HD, and ymax /y–12 = 0.1 in Run LD. As the lower boundary conditions were identical in each experiment, this result suggests that there was greater drag at the upper boundary in Run LD (suppercritical). Recalculated values of some gravity current data and from plane turbulent wall jets from the literature are shown in Table 2. It is apparent from

Table 2 that, within the available data set, there is much greater variety in the position of Umax is experimental gravity currents than wall jets. All the wall jets were supercritical, with no slope and without any density difference, hence the consistent values of Umax and y–12 . However, the experimental gravity currents are more variable, and this may reflect sensitivity to factors such as the effects of density stratification, bed roughness, slope, and the presence of particles which may influence the drag at both boundaries. It has also been shown that the level of the velocity maximum in experimental gravity currents is dependent on the flow Reynolds number (Parsons, 1998). Generally, it is apparent that within a certain experimental configuration, the velocity maxima in supercritical currents occurs nearer to the base of the current than in subcritical currents; this is a consequence of the fact that there is greater drag at the upper boundary in a supercritical current (Ellison & Turner, 1959). Turbulence production There are two primary turbulence generation mechanisms in gravity currents, turbulence production by shear from the mean flow and production by buoyancy. The rate of turbulence generation by shear (G)

PGCC12 9/12/08 3:28 PM Page 182

182

C. Buckee et al.

Table 2. Table showing normalized height (ymax /y–12 ) for various data sets from the literature including gravity currents and plane turbulent wall jets. Some of the data for turbulent wall jets is taken from a summary paper by Launder and Rodi (1981) and references therein Study

Umax at ymax /y–12

Comments

Gravity currents Run LD—this study

0.1

Supercritical Steady. Solute-driven

Run HD—this study

0.14

Subcritical Steady. Solute-driven

Kneller et al., 1997

0.19– 0.5

Subcritical Surge type. Solute-driven

Kneller et al., 1999

c. 0.35

Subcritical Surge type. Solute-driven

Ellison & Turner, 1959

0.25

Supercritical ~ Steady. Solute-driven

García, 1994

0.4 0.3

Subcritical Supercritical Both steady. Sediment laden

García & Parker, 1993

0.34 – 0.55

Supercritical Steady. Sediment laden

Parker et al., 1987

0.07– 0.15

Supercritical Steady. Sediment laden

Stacey & Bowen, 1988b

0.3– 0.46

Supercritical Theoretical. Steady. Sediment laden

0.17

Supercritical

Wall jets Förthmann, 1934 Schwartz & Cosart, 1961

0.16

Supercritical

Schneider & Goldstein, 1994

0.17

Supercritical

is controlled by the vertical gradient in downstream velocity and can be calculated as follows (Nezu & Nakagawa, 1993): G = − u ′v ′

du dy

(16)

While there are a limited number of vertical data points for calculating the velocity gradient in this study, it is assumed that there is sufficient control to capture the broad trends in shear production. A further source of turbulent kinetic energy in density currents is buoyancy. Buoyancy derived turbulence generation can occur even in the presence of a stable stratification if there is an appreciable source of disturbance (Long, 1970), although the turbulence may be patchy and intermittent. As mixed patches of fluid develop, or parcels of fluid are moved up/down the density gradient then there is an increase of potential energy in those patches. This potential energy is con-

verted to turbulent kinetic energy as the density gradient re-equilibrates. Figure 8 shows the calculated rates of turbulence production by shear and buoyancy flux for both currents. In both experiments the highest rate of turbulence production is through shear and occurs at the base of the current, where the velocity gradient and the Reynolds stresses are greatest. A second region of high turbulence production occurs above the velocity maximum in the subcritical current, Run HD (Fig. 8b). Negative production by shear is observed above the velocity maximum in both currents. This occurs because the location of zero Reynolds stress (Fig. 6) does not coincide with Umax. This phenomenon has been observed in turbulent wall jets and in other asymmetrical velocity profiles (Irwin, 1973; Oster & Wygnanski, 1982; Katz et al., 1992). In turbulent wall jets, Reynolds stresses become negative at about

PGCC12 9/12/08 3:28 PM Page 183

183


Fig. 8. The rate of turbulent kinetic energy production by extraction of energy from the mean flow by shear, per unit mass (grey line), and from buoyancy flux per unit mass (black line) plotted with k per unit mass (dotted line) for (a) Run LD and (b) Run HD. The solid horizontal lines show the location of Umax for Run LD and Run HD. Note that the mean turbulent kinetic energy profile is a function of both the turbulence generation mechanisms and the damping of turbulence by density stratification.

60% of the height of the velocity maximum (Launder & Rodi, 1983). In the experiments reported herein, Reynolds stresses reach zero above Umax. This phenomenon has also been observed in weak wall jets in which the jet excess velocity was low compared to the external flow (Erian & Eskinazi, 1964), and boundarylayer turbulence was more vigorous than in the outer layer, resulting in the diffusion of turbulence properties away from the wall (Irwin, 1973). Previously, it has been assumed that Reynolds stresses would be zero at the level of the velocity maximum (e.g. see Stacey & Bowen (1988a,b), and the predictions of the Eidsvik & Brørs (1989) model). Consequently, the present result has implications for assumptions made about the turbulence structure in numerical modelling of gravity currents. Regions of negative turbulence production are regions in which energy is extracted from the turbulence to the mean flow. Physical explanations that have been suggested for this phenomenon are: 1 Paired vortices with opposite senses of vorticity (Zaman & Hussain, 1980). 2 The presence of large, coherent eddies (Katz et al., 1992). 3 Inclination/shearing of eddies in a velocity gradient (Hussain, 1983). 4 Suppression of vortex interaction and inhibition of lateral growth of the shear layer (Oster & Wygnanski, 1982).

Further experiments using flow visualization may clarify the physical behaviour associated with negative turbulence production in gravity currents. Stratification Gravity currents are density stratified (e.g. Fig. 3c) and the stability of the stratification affects the turbulence structure. In a stable stratification turbulent motions are damped and turbulence rapidly dissipates unless there is a supply of energy such as shear from the mean flow (Turner, 1973). Gradient Richardson numbers (Rig) were calculated for each experiment using: −g Ri g =

dρ dy

⎛ du ⎞ ρo ⎜ ⎟ ⎝ dy ⎠

2

(17)

The results (Fig. 9a,b) agree well with the predicted gradient Richardson number profiles in Stacey and Bowen (1988a,b). The gradient Richardson number is necessarily large at the level of Umax where the velocity gradient is zero (Fig. 9a,b). In the upper region, the stratification in Run LD is less stable (on average Rig = 0.17 above Umax) than Run HD (Rig = 0.37 above Umax) although the variance in Run HD is greater. When Rig is greater than the critical value of 0.25

PGCC12 9/12/08 3:28 PM Page 184

184

C. Buckee et al.

Fig. 9. Calculated gradient Richardson numbers for Run LD and Run HD are shown in (a). The horizontal solid line shows the location of Umax for Run LD and the horizontal dashed line shows Umax for Run HD. In (b) a detail of (a) is shown co-plotted with a theoretical curve for Rig (dotted line) for a supercritical current adapted from Stacey and Bowen (1988b). The theoretical curve compares well with the supercritical current (Run LD). It should be noted that the curves calculated from experimental data are less smooth than the theoretical curve due to their poorer vertical resolution. The line Rig = 0.25 highlights the value of the gradient Richardson number above which the flow is said to be locally stably stratified.

(Fig. 9b), energy produced by shear is insufficient to do work against the density gradient and hence is dissipated (Turner, 1973). This means that turbulence is more significantly damped in Run HD, explaining the lower observed turbulent kinetic energy in the upper part of the flow despite the higher production by shear (Figs 5 and 8). This result is in good agreement with García (1993), who observed that gradient Richardson numbers in the upper part of supercritical flows were in the range 0.13–0.23, below the critical value of 0.25 and that subcritical flows were more stably stratified with a range 0.54 – 0.84. Run LD is, in this respect, more like an unstratified turbulent wall jet in which turbulence is not damped by gravitational and buoyancy effects, which may explain the similarity in distribution of turbulent kinetic energy already noted between Run LD and turbulent wall jets. Velocity maximum It has been observed previously that a zone of lowturbulence intensity exists around the level of the velocity maximum in experimental gravity currents (Kneller et al., 1999; Best et al., this volume). The results from the present experiments support these observations, as both subcritical and supercritical flows are characterized by a region of low turbulence in the lower part of the current, around the level of Umax (Fig. 4).

There are two possible explanations for this phenomenon. It has been noted previously that at step, or inflection, in the concentration profile of experimental gravity currents is located at approximately the level of the velocity maximum (e.g. Ellison & Turner, 1959; García & Parker, 1993; Peakall et al., 2000). The strength of the stratification across such a point may be sufficient to dissipate the turbulence significantly and reduce convection of turbulence upwards through the current. It has also been suggested that little turbulence is generated at the level of Umax because, locally, shear production of turbulence tends to zero (i.e. du/dy = 0) (Stacey & Bowen, 1988a; García & Parker, 1993; Kneller et al., 1999). It is difficult to assess from the current data-set, and from examination of results in the literature, whether the low turbulence around Umax is related to the lack of shear production at that point or to the strength of stratification. Furthermore, the step, or inflection, in the density profile and location of the low turbulence do not always coincide with Umax as observed both in these experiments and in examples from the literature. In unstratified plane turbulent wall jets, however, no zone of low turbulence intensity is observed at Umax (e.g. Launder & Rodi, 1983), suggesting that the lack of turbulence production by shear alone is not sufficient to account for this effect in gravity currents. It seems likely that transport of mass in the lower part of the current is retarded by a slow diffusion zone

PGCC12 9/12/08 3:28 PM Page 185

185

Turbulence structure (SDZ), related to the strength of the stratification and the lack of turbulence production by shear at Umax. Therefore particles entrained from the bed become temporarily concentrated below Umax . Similarly, reduced mass transport across the SDZ will also tend to reduce downward mixing of lower-density fluid entrained at the upper boundary. In currents which entrain fluid rapidly at the upper boundary, or in which rapid entrainment of bed sediment occurs, there will be reinforcement of the stratification and the stepped concentration profile may become more pronounced (Peakall et al., 2000). This can be observed in the results of García and Parker (1993); the data show density profiles from an eroding gravity current in which the step becomes more pronounced with distance along the erodible floor because the flow is not in equilibrium. Conversely, depositing turbidity currents or steady, uniform gravity currents may reach a smooth equilibrium density profile, with time, as particles diffuse across the SDZ (Peakall et al., 2000). In the present experiments, although the density profile is not significantly ‘stepped’, the stratification in the lower part of the flow is still significant and reduced turbulent velocities are still observed in that region (Fig. 4). Further experiments are required on both equilibrium and non-equilibrium gravity currents to study the relationship between the strength of the stratification and the rate of mass transport across the SDZ.

CONCLUSIONS The data presented in this chapter are, perhaps, the most detailed description of the turbulence structure of experimental gravity currents yet produced. Comparison of these data with those in the experimental gravity current and turbulent wall jet literature suggests the following: 1 The characteristic height, y–12 , is an appropriate lengthscale against which to normalize gravity current data because it is simple to ascertain, reflects the fundamental dynamic properties of the flow and produces good data collapse for both subcritical and supercritical currents. 2 The vertical gradient in downstream velocity is shown to be one of the principal controls on the turbulence structure of gravity currents. Calculated turbulence production by shear was found to be significant, particularly in subcritical currents. 3 Calculated gradient Richardson numbers for supercritical and subcritical currents were found to agree

well with published values from García (1993) and predicted distributions for supercritical currents in Stacey and Bowen (1988a,b). Stable stratification was shown to damp the turbulence generated in the upper part of subcritical currents significantly. 4 The concept of a ‘slow diffusion zone’ (SDZ) in the lower part of the current, around the velocity maximum, has been re-examined in the light of new data presented here, supporting previous hypotheses that turbulence intensities are reduced in this region. These results add to our understanding of gravity current dynamics but leave many questions unanswered. Further experimental work is needed to support the data presented here and to extend it by including experiments on particulate currents. The results given here provide a ‘control’ data-set that can be compared with data from particulate currents, leading to a better understanding of the dynamics of turbidity currents and the affect of suspended sediment on turbulence structure; an important issue in fluid dynamics. These data also provide a means of testing complex mathematical models (e.g. Stacey & Bowen, 1988a,b; Eidsvik & Brørs, 1989) and the results produced so far are encouraging (e.g. Fig. 10).

ACKNOWLEDGEMENTS Thanks are due to all those who assisted with the laboratory experiments, especially Rufus Brunt and Mark Franklin. We are indebted to Jim Best, Maarten Felix and Henry Pantin for constructive conversations and to Jaco Baas and Jeff Parsons for their helpful reviews of the manuscript. The work was carried out during tenure of a NERC studentship to the first author, and was partially supported by an industrial consortium including Arco, Amerada-Hess, an Amoco, BG, BHP Petroleum, BP, Chevron, Conoco, Elf, Enterprise, Fina, Mobil, Shell and Texaco, to whom we express out sincere gratitude.

LIST OF SYMBOLS C1,2,3 Cout g g′ G h k k*

constants conductivity probe output acceleration due to gravity reduced gravitational acceleration rate of turbulence generation by shear calculated current thickness turbulent kinetic energy per unit mass normalized turbulent kinetic energy

PGCC12 9/12/08 3:28 PM Page 186

186 max u n Re Rio Rig u u′ ui U Umax Urms u v v′ vi Vrms v y y–12 y ymax ρ ρ′ ρa ρc ρi ρo ρrms 7 τR τR* µ

C. Buckee et al. maximum mean downstream velocity number of observations Reynolds number bulk Richardson number gradient Richardson number downstream velocity fluctuating component of downstream velocity instantaneous downstream velocity depth averaged downstream velocity position of maximum downstream velocity root-mean-square downstream velocity temporally averaged downstream velocity vertical velocity fluctuating component of vertical velocity instantaneous vertical velocity root-mean-square vertical velocity temporally averaged vertical velocity height of the velocity maximum height at which the downstream velocity is half the maximum velocity, above Umax. vertical height above tank floor height of the velocity maximum above the floor density fluctuating component of density ambient fluid density current density instantaneous density initial current density root-mean-square density temporally averaged density Reynolds stress per unit mass normalized Reynolds stress dynamic viscosity

REFERENCES Alahyari, A. & Longmire, E.K. (1994) Particle image velocimetry in a variable density flow: application to a dynamically evolving microburst. Exp. Fluids 17, 434 – 470. Allen, J.R.L. (1971) Mixing at turbidity current heads, and its geological implications. J. sediment. Petrol. 41, 97–113. Altinakar, M.S., Graf, W.H. & Hopfinger, E.J. (1996) Flow structure in turbidity currents. J. Hydraul. Res. 34, 713–718. Best, J.L., Kirkbride, A. & Peakall, J. (2001) Mean flow and turbulence structure of sediment-laden gravity currents: new insights using ultrasonic Doppler velocity profiling. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller & J. Peakall) Spec.

Publs. int. Ass. Sediment. No. 31, pp. 159–172, Blackwell Science, Oxford. Britter, R.E. & Simpson, J.E. (1978) Experiments on the dynamics of a gravity current head. J. Fluid Mech. 88, 223–240. Buchhave, P., George, W.K.J. & Lumley, J.L. (1979) The measurement of turbulence with the laser–Doppler anemometer. Ann. Rev. Fluid Mech. 11, 443–503. Chikita, K. (1990) Sedimentation by river-induced turbidity currents; field measurements and interpretation. Sedimentology 37, 891–905. Edwards, D.A., Leeder, M.R., Best, J.L. & Pantin, H.M. (1994) On experimental reflected density currents and the interpretation of certain turbidites. Sedimentology 41, 437– 461. Eidsvik, K.J. & Brørs, B. (1989) Self-accelerated turbidity current prediction based upon (k-ε) turbulence. Continent. Shelf Res. 9, 617–627. Ellison, T.H. & Turner, J.S. (1959) Turbulent entrainment in stratified flows. J. Fluid Mech. 6, 423 – 448. Erian, F. & Eskinazi, S. (1964) The wall-jet in a longitudinal pressure gradient. Syracuse University Research Institute, Mech. and Aerospace Engineering Dept. report ME 937– 6410F, 31 pp. Fietz, T.R. & Wood, I.R. (1967) Three dimensional density current. J. Hydraul. Div., Proc. ASCE 93, 1–23. Förthmann, E. (1934) Über turbulente Strahlausbreitung. Ingenieur Arch. Ber. 5, 42–54. García, M.H. (1990) Depositing and Eroding Sedimentdriven Flows: Turbidity Currents. Unpublished Ph.D. thesis, University of Minnesota, St. Anthony Falls Hydraulic Laboratory, 179 pp. García, M.H. (1993) Hydraulic jumps in sediment-driven bottom currents. J. Hydraul. Eng. 119, 1094 –1117. García, M.H. (1994) Depositional turbidity currents laden with poorly sorted sediment. J. Hydraul. Eng. 120, 1240 – 1263. García, M.H. & Parker, G. (1993) Experiments on the entrainment of sediment into suspension by a dense bottom current. J. geophys. Res. 98, 4793 – 4807. Geyer, W.R. & Farmer, D.M. (1989) Tide-induced variation of the dynamics of a salt wedge estuary. J. Phys. Oceanogr. 19, 1060 –1072. Head, M.J. (1984) The Use of Miniature Four-Electrode Conductivity Probes for High Resolution Measurement of Turbulent Density or Temperature Variations in Salt-Stratified Water Flows. Unpublished Ph.D. thesis, University of California, San Diego, 211 pp. Hussain, A.K.M.F. (1983) Coherent structures – reality and myth. Phys. Fluids 26, 2816 –2850. Irwin, H.P.A. (1973) Measurements in a self preserving plane wall jet in a positive pressure gradient. J. Fluid Mech. 61, 33 – 63. Katz, Y., Horev, E. & Wygnanski, I. (1992) The forced turbulent wall jet. J. Fluid Mech. 242, 577– 609. Kneller, B.C., Bennett, S.J. & McCaffrey, W.D. (1997) Velocity and turbulence structure of density currents and internal solitary waves; potential sediment transport and the formation of wave ripples in deep water. Sediment. Geol. 112, 235–250. Kneller, B.C., Bennett, S.J. & McCaffrey, W.D. (1999) Velocity structure, turbulence and fluid stresses in experi-

PGCC12 9/12/08 3:28 PM Page 187

Turbulence structure mental turbulent density currents. J. geohpys. Res. 104, 5381–5391. Launder, B.E. & Rodi, W. (1981) The turbulent wall jet. Prog. Aerospace Sci. 19, 81–128. Launder, B.E. & Rodi, W. (1983) The turbulent wall jet – measurements and modelling. Ann. Rev. Fluid Mech. 15, 429–459. Long, R. (1970) A theory of turbulence in stratified flows. J. Fluid Mech. 42, 349–365. McDougall, T.J. (1979) On the elimination of refractiveindex variations in turbulent density-stratified liquid flows. J. Fluid Mech. 93, 83–96. Middleton, G.V. (1966a) Experiments on density and turbidity currents. Part 1. Motion of the head. Can. J. Earth Sci. 3, 523–546. Middleton, G.V. (1966b) Experiments on density and turbidity currents. Part 2. Uniform flow of density currents. Can. J. Earth Sci. 3, 627–637. Middleton, G.V. (1993) Sediment deposition from turbidity currents. Ann. Rev. Earth Planet. Sci. 21, 89–114. Nezu, I. & Nakagawa, H. (1993) Turbulence in OpenChannel Flows. International Association of Hydraulic Research, Monograph Series. Balkema, Rotterdam, 281 pp. Oster, D. & Wygnanski, I. (1982) The forced mixing layer between parallel streams. J. Fluid Mech. 123, 91–130. Pantin, H.M. & Leeder, M.R. (1987) Reverse flow in turbidity currents; the role of internal solitons. Sedimentology 34, 1143–1155. Parker, G., García, M.H., Fukushima, Y. & Yu, W. (1987) Experiments on turbidity currents over an erodible bed. J. Hydraul. Res. 25, 123–147.

187

Parsons, J.D. (1998) Mixing Mechanisms in Density Intrusions. Unpublished Ph.D. thesis, University of Illinois at Urbana-Champaign, 239 pp. Peakall, J., McCaffrey, W.D. & Kneller, B.C. (2000) A process model for the evolution, morphology and architecture of sinuous submarine channels. J. sediment. Res. 70, 434 – 448. Schneider, M.E. & Goldstein, R.J. (1994) Laser Doppler measurement of turbulence parameters in a twodimensional plane wall jet. Phys. Fluids 6, 3116–3129. Schwartz, W.H. & Cosart, W.P. (1961) The twodimensional wall jet. J. Fluid Mech. 10, 481– 495. Simpson, J.E. (1972) Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53, 759–768. Simpson, J.E. (1997) Gravity Currents: In the Environment and the Laboratory, 2nd edn. Cambridge University Press, Cambridge, 244 pp. Simpson, J.E. & Britter, R.E. (1979) The dynamics of the head of a gravity current advancing over a horizontal surface. J. Fluid Mech. 94, 477– 495. Stacey, M.W. & Bowen, A.J. (1988a) The vertical structure of density and turbidity currents: theory and observations. J. geophys. Res. 93, 3528–3542. Stacey, M.W. & Bowen, A.J. (1988b) The vertical structure of turbidity currents and a necessary condition for selfmaintenance. J. geophys. Res. 93, 3543 –3553. Turner, J.S. (1973) Buoyancy Effects in Fluids. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge, 368 pp. Zaman, K.B.M. & Hussain, A.K.M.F. (1980) Vortex pairing in a circular jet under controlled excitation. Part 1. General jet response. J. Fluid Mech. 101, 449.

PGCC13 9/12/08 3:28 PM Page 189


Experimental evidence for autosuspension H. M. PANTIN School of Earth Sciences, University of Leeds, Leeds LS2 9JT, UK

ABSTRACT An autosuspension current may be defined as a particle-driven gravity flow, which can persist indefinitely without an external supply of energy. The ultimate criterion for autosuspension must, therefore, be lack of net deposition, a criterion which is reinforced if the current is also capable of erosion. A series of experiments has been carried out to investigate the physical conditions required for the onset of autosuspension, and to demonstrate autosuspension in the laboratory. Gravity currents were generated in a tubular channel (5.1-cm i.d. Perspex) with a gradient of c. 20°, the dense fluid employed being either salt solution, or a suspension of industrial silica flour with a modal grain size dmode of about 20 µm. Prior to a run, a series of roughness elements were placed in the tube, and a test bed (almost always silica with a dmode of about 20 µm) was introduced as a saline suspension. The dense fluid forming the current was placed in a Perspex header tank, located at the top of the tube, the whole apparatus being immersed in a deep tank of water. The fluid in the header tank was released and allowed to run over the test bed, measurements being made of the amount of sediment taken into suspension, together with the flow velocity. The most significant results were obtained in a 1.75-m long tube, with a silica flour suspension in the header tank and using a series of deflector plates, constructed to impart a right-handed swirl to the flow in the tube. Four lines of evidence indicate that autosuspension was achieved, at least on a limited scale: 1 after a run, the test bed revealed a sharp erosional upper surface; 2 about 30% of the suspension load of the flow consisted of test-bed sediment, entrained and incorporated into the flow; 3 at its highest level, the proportion of sediment in subsamples from the flow was found to exceed that in the original suspension in the header tank; and 4 current velocity measurements showed evidence of acceleration.

BACKGROUND AND OBJECTIVES OF RESEARCH A number of mathematical models of turbidity flow have been developed in recent years which predict autosuspension under physically reasonable conditions (e.g. Pantin, 1979, 1986, 1991; Parker, 1982; Akiyama & Stefan, 1985; Fukushima et al., 1985; Parker et al., 1986; Stacey & Bowen, 1988; Eidsvik & Brørs, 1989; Fukushima & Parker, 1990). Furthermore, good evidence of autosuspension has been obtained in natural systems (e.g. Inman et al. (1976)), interpreted by Fukushima et al. (1985) and Pantin (1986); Prior et al. (1987), interpreted by Pantin (1991); Hughes Clarke et al. (1990). Prior et al. (1986) have described channeling due to erosion by high-density hyperpycnal

An autosuspension current may be defined as a particledriven gravity flow, which can persist indefinitely without any external supply of energy. The ultimate criterion for autosuspension must, therefore, be lack of net deposition. This criterion is reinforced if the current is also capable of erosion. Mathematical analysis shows that when a certain combination of velocity and sediment content is exceeded, and if a sufficient supply of erodible sediment is available, the flow will ‘ignite’, i.e. a positive feedback process will be set up, by which increasing velocity causes net erosion and thus an increase in particle concentration; this in turn causes increasing velocity, and so on.


189

PGCC13 9/12/08 3:28 PM Page 190

190

H.M. Pantin

flows, a phenomenon entirely consistent with autosuspension. However, it has been clear for some time that our understanding of this process would be greatly improved if it could be reproduced in the laboratory, and studied under controlled conditions. The broad aim of the project described herein has been to design a simple experimental apparatus to allow investigation of the physical conditions required for the onset of autosuspension, by means of a series of experimental flows in which the controlling parameters were likely to favour that phenomenon. Such experiments would serve as a basis for testing and improving existing theoretical treatments of autosuspension and form a background for future quantitative measurements of currents undergoing autosuspension. It is well established from mathematical and numerical modelling ( Pantin, 1979; Parker et al., 1986) that a high gradient, fine (non-cohesive) sediment, and a large flow thickness are all conducive to autosuspension. Flow thickness is severely limited in laboratory experiments, and it is thus obviously desirable to use a high gradient for the experimental flows, together with material with as fine a grain size as possible without encountering problems associated with sediment cohesion. These are the limiting factors on which the design of the apparatus and experiments were based. Medium-silt sized material (usually 20-µm industrial silica flour) was employed, together with a bottom gradient of 20°, since any greater angle would have run the risk of approaching the angle of rest of the sediment. This material was introduced, in suspension, in to a flow tube and allowed to settle, thus forming the test bed. A charge of relatively dense fluid, contained in a header tank, was then run over the test bed, and the amount of entrained sediment measured, together with the velocity of the flow. It was decided at an early stage to use a tube as the conduit, rather than a channel with an open top. Such an arrangement would eliminate the need for measuring or controlling the entrainment of ambient water, besides rendering the average velocity uniform throughout the length of the tube, thereby simplifying greatly the task of relating sediment entrainment to velocity. The most important long-term aim of these experiments was not only to demonstrate erosion by turbidity flow (or saline analogue), but to entrain sufficient suspended load to raise the specific gravity of the fluid to such an extent that measurable acceleration would be observed. Acceleration due to the incorporation of bed material is an essential feature of ignitive autosuspension (Parker et al., 1986).

EXPERIMENTAL APPARATUS AND TECHNIQUES Outline A major part of this research involved the design, construction, testing and modification of an experimental rig to generate and monitor autosuspensive currents. Experiments were carried out in two phases, the apparatus used for Phase 1 being substantially modified for Phase 2, although the technique remained essentially the same. Apparatus for Phase 1 experiments The channel used for the experiments was a Perspex tube, 1 m in length and 5.1-cm internal diameter, which was tilted at an angle of 20° to the horizontal and mounted on a frame (Fig. 1). The upper end of this flow tube was joined to an L-tube, via a verticallymounted valve, to a Perspex header tank of 6.86 litres capacity. An auxiliary tank with a capacity of 1 litre was attached to the upper part of the header tank, but separated from the latter by a lock gate. The lower end of the flow tube was joined to a second valve, connected to the first by a metal bar, so that both valves could be opened and closed simultaneously by a lever. Inside the flow tube, approximately 0.1 m from its end, a small T-tube (internal diameter 1 cm) was mounted within the main tube (Fig. 1). This T-tube was connected to a system of plastic tubing, which could be used either for filling the flow tube with ambient water (or other fluid) needed for an experiment, or most importantly for siphon-sampling the fluid within the flow tube during a run. The T-tube could be raised or lowered within a range of about 2 cm. Experiments were carried out with the rig immersed in water, contained in a ‘deep’ tank measuring 3 m in length, 1.6 m in depth, and 0.3 m in width, and constructed of 1-cm thick Perspex sheet within a steel framework. An overhead electric hoist above the tank enabled the rig to be removed from the tank for preparation at the beginning of a run, or for subsequent cleaning. A reservoir tank (Fig. 1) was used to supply water or saline (NaCl) solution, as necessary, to fill the flow tube prior to a run. The outlet tap on the reservoir was connected to a Y-tube by plastic tubing, the other two branches of the Y-tube being connected, respectively, to the T-tube, and to the sampling siphon, the latter consisting of a long plastic tube with the lower end closed

PGCC13 9/12/08 3:28 PM Page 191

191

Autosuspension Funnel Taps

Reservoir

Ω-Tube(s)

Lock gate

WATER Valve lever

LEVEL

Header tank (6.86 litres)

Auxiliary tank (1 litre) Upper valve (51mm I.D.)

Connecting rod operating Plastic tubing (9mm I.D.)

Flow tube (51mm I.D.)

T-tube

Framework

Lower valve (51mm I.D.) 0

cm

20

Tap 20°

Fig. 1. Schematic diagram of experimental apparatus, Phase 1.

by a tap. The flow through the different branches of the Y-tube was controlled by a series of taps, to allow free passage as desired between any two of the three items. Experimental procedure The preparation of a run was as follows. Whilst the deep tank was filled with mains water, the flow tube was prepared in several stages. Firstly, the rig was raised, and the flow tube supported in a horizontal position, at the top of the tank. The connecting rod was then detached from the lower valve. The flow tube was then removed from the rig, a special roughness insert placed on the bottom, and the tube replaced. The lower valve, filled with water and closed at the far end with a bung, was emplaced, and the upper valve opened. A test-bed was then prepared, by running a suspension of the chosen material (initially magnetite powder in the range 3 –15 µm, but usually 20-µm silica flour)

into the flow tube, through a funnel inserted into the header tank. The quantity of material was chosen to produce a test-bed thickness ranging from c. 2 to 5 mm. The greater thicknesses were employed for the higher velocities, to ensure that some test bed always remained after the run. In the earliest experiments, the suspending fluid was water, later replaced by a saline solution. Any remaining space in the flow tube was then filled from the reservoir with the same fluid, up to and just over the upper valve; when saline was used, this was chosen to have the same specific gravity as the dense fluid to be used later in the header tank (Fig. 1). The upper valve was then closed, the connecting rod re-attached, and the entire rig lowered into the deep tank. The brackets at the sides of the frame ensured that the header tank was squarely mounted, and that the flow tube rested at the desired angle of 20°. The level of ambient water in the main tank was allowed to rise to approximately 1 cm below the top of the header tank.

PGCC13 9/12/08 3:28 PM Page 192

192

H.M. Pantin

The header tank and auxiliary tank, with the lockgate out, were then filled with water; two hydrostatic links were then inserted over the end of the auxiliary tank, and the lock-gate replaced. These hydrostatic links consisted of two water-filled Ω-tubes, which acted as siphons during the runs, and allowed ambient water from the main tank to run into the header tank itself, replacing fluid lost from the header tank as the dense fluid passed into the flow tube. For consistency, the rig was then left undisturbed for about 1.5 hours to ensure that the test bed had settled as uniformly as possible. The plastic tubes connecting the T-tube, the reservoir, and the sampling siphon were checked for any airlocks, which were removed by running reservoirtank fluid (water or saline) through the system. The charge for the header tank, consisting of a chosen relatively dense fluid, was then prepared in a separate vessel. In some early runs, this consisted of a suspension of 20-µm silica flour in water, but in most experiments it consisted of a saline solution, often coloured with KMnO4 to improve definition. The header tank, separated from the auxiliary tank by the lock-gate, was then emptied, and filled with the charge of heavy fluid. The sampling siphon was then re-checked for airlocks, and a video camera set up, both to visualize the movement of the fluid through the test section and to enable calculation of mean velocities. To run the experiment, two operators were necessary. One operator raised the lock-gate separating the header tank and the auxiliary tank, and immediately opened both valves. At the same time, the second operator took a series of samples from the flow tube with the sampling siphon. After all apparent motion in the flow tube had ceased (c. 30 – 40 s), the video was stopped and repositioned to provide a close-up of the test-bed following the run.

EXPERIMENTAL RESULTS: PHASE 1 Initial experiments A series of six initial runs were conducted, using a suspension of silica flour in water as the charge, the modal grain size of the silica being about 20 µm and the concentration about 350 g l−1. The test bed in these initial runs consisted of magnetite powder (in the range 3–15 µm). This material was chosen for its supposed ease of extraction from the siphoned samples after the run, in spite of its much higher specific gravity (5.18) than silica (2.65).

Powerful permanent hand-magnets were used for removing the magnetite from the much larger concentrations of silica contained in the sample. However, the recoveries of magnetite were low (in the order of 0.4% of the sample) and there were unexpected difficulties in obtaining a clean separation of the magnetite and silica, due to adhesion of the finer magnetite grains. Therefore, it was decided to use saline solutions as the charge (an analogue of turbidity flow) with silica itself as the test bed, allowing the take-up of silica to be measured by simple filtration. Current velocities, being uniform along the tube, were obtained by video measurements of the travel time of the headertank fluid along the tube. This was straightforward when the fluid was coloured with KMnO4, although in most other cases the fluid front could be distinguished by a sudden increase in turbulence and sediment content. Roughness elements Mechanisms exist by which sediment can be entrained into suspension due to the presence of self-generated, rather than imposed, roughness elements on the bottom. These include: (a) the effect of local, thin, highvorticity streaks in the viscous sublayer, associated with velocities high enough to raise individual grains from the bed (Kaftori et al., 1995); or (b) lofting of bedload or a sediment-rich viscous sublayer by ejections from the fluid boundary layer (Hogg et al., 1994; García et al., 1996). However, it was found almost immediately that the presence of imposed roughness elements, of whatever kind, always produced an improvement in sediment entrainment. A series of roughness elements were therefore tested. These were mounted on strips of plastic or other material with curved cross-section in order to fit inside the tube, and emplaced prior to the introduction of the test bed. Except where indicated, the roughness elements were placed on the bottom of the tube. Eleven types of roughness element were constructed and tested, in succession: (i) pairs of small angular pebbles, median diameter 3–7 mm (symmetrically placed, with further small pebbles added progressively); (ii) narrow spanwise ridges of wood quadrate; (iii) total small-pebble cover; (iv) pairs of short, upward-pointing nails; (v) low-amplitude plaster of Paris ripples (three in number); (vi) a line of long nails, projecting from the top of the tube; (vii) highamplitude plaster of Paris ripples (six in number); (viii) a series of spheres approximately 1.74 cm in diameter, mounted on downward-projecting nails: (ix) a

PGCC13 9/12/08 3:28 PM Page 193

Autosuspension

193

(a)

(b) Fig. 2. These photos show deflector plates of the types used at the end of the Phase 1 experiments, and on a regular basis during Phase 2. (a) Shows the blue-coloured silica test bed in place before a silica-suspension run (Phase 2), together with part of the 31-deflector configuration used for most of the Phase 2 experiments. (b) Shows the scour pits remaining after a silicasuspension run; note the sharp interface between the eroded remains of the blue-coloured test bed (dark) and the overlying unstained silica from the header tank, deposited by the tail of the current (light).

sloping plate, tilted towards the far end of the tube; (x) plastic grills (four in number); and (xi) static deflectors (four or eight in number). These deflectors were propeller-shaped, being made from discs of shim brass cut into eight sectors, which were then twisted anticlockwise to give a right-handed swirl to the current (Fig. 2). The main sediment entrainment mechanism which it was hoped to enhance was the lofting of bed material by strong vortices associated with scour pits around roughness elements, assisted by increased boundarylayer turbulence in the wake of such obstructions; these considerations applied to types (i)–(iv) above. Turbulence concentrated at, or directed towards the boundary layer, without roughness elements in the test bed itself, were also tried (types (vi), (viii) and (ix)). The artificial ripples were used to test the possible entrainment effect of spanwise bedforms, for example (a) the lofting of bed material by fluctuating turbulent pressures at the reattachment point of the flow, downstream of a zone of separation (Kostachuk & Church, 1993); or (b) lofting of bed material in a high-speed boundary layer travelling over a convexity (Nezu & Nakagawa, 1993). The plastic grills (x) were used to produce both scouring round the base and increased general turbulence, besides imposing a more regular pattern of turbulence on the flow. The deflector plates (xi) were designed to increase scouring, as well as elevating the sediment-bearing boundary layer by forcing it up the side of the tube. This would greatly reduce the turbulent energy needed to overcome gravity during the process of sediment entrainment.

Saline solutions Evaluation During the early runs, saline solutions with specific gravities of about 1.035 (immersed S.G. 0.035) were used in the header tank, producing current velocities in the region of 16–18 cm s−1. It was soon found that higher immersed S.G. gave not only higher velocities, but considerably higher entrainment of bed material, and it was therefore decided to conduct a series of comparative runs (PFEX23–60), alternating 60% saturated saline (specific gravity 1.12) with saturated saline (specific gravity 1.20), and using different roughness elements. In all cases, the fluid used in the emplacement of the test bed was saline, of the same specific gravity as that in the header tank. In some runs, the siphoned sample taken during the run was collected in a single 2-litre plastic container, but in other cases a series of 250-ml beakers were used, to provide a time/ accumulated weight profile of the sampled suspension. Initially, the T-tube was located with its axis about 1 cm above the base of the flow tube, but was later raised to the centre of the flow tube. The aim of these experiments was (a) to find the most effective way of entraining sediment from the test bed, by determining the sediment concentration (C ) in the samples; (b) to convert this to a rate of entrainment, and thence to calculate the equilibrium sediment concentration (Ce ) which would have been attained if entrainment had continued, up to the point

PGCC13 9/12/08 3:28 PM Page 194

194

H.M. Pantin

where this process was counteracted by the settling of sediment. Early results Current velocities for the series PFEX23–60 varied from 30 to 45 cm s−1 for the 60% saturated saline, to 43–63 cm s−1 for the saturated saline. Sediment from the test bed was entrained into suspension in all cases, although the quantities varied considerably. Bedforms observed after the runs were limited to (a) a faint current lineation, and (b) scouring around the roughness elements (see Fig. 2b). No spanwise bedforms, e.g. ripple marks, were observed in any of the runs. Video observations indicated that the main source of suspendedsediment entrainment was the scour pits around the roughness elements. Significant movement of bedload was observed in all runs, but was not investigated further. The volume of siphoned fluid containing the abstracted silica was measured or estimated, and the silica removed by filtration, being then washed, dried, and weighed. The concentration of sediment C in the siphoned samples was obtained by calculating equivalent dry weights for a volume of 1 litre; the value of C was then used to estimate an equilibrium suspendedsediment concentration Ce (i.e. with entrainment balancing sediment settling) in the following way. For an arbitrary concentration of silica Ca in the flow tube, the weight W of silica in a vertical segment of unit thickness (Fig. 3a) will be given by W = Xf sec 20° Ca , where Xf is the free cross-section; that is, the cross-section of the tube (πD2/4, where D is the tube diameter), minus the cross-section (Xb ) taken up with the test bed and roughness elements. Mass balance thus gives dW/dt = Xf sec 20° dCa /dt. If there were no entrainment of sediment, the rate of settling out from

the segment may be regarded as the settling flux of sediment grains through the level of maximum diameter (equal to the tube diameter D; Fig. 3b); this would give dW/dt = −DCaVg, where Vg is the settling velocity of the sediment grains. Combining the above two equations, we obtain dCa /dt = −DCaVg cos 20°/Xf . The rate of sediment entrainment Es can be obtained from the measured concentration in the samples, provided that certain corrections are applied. If there were no settling, and if no considerations of available fluid space were involved, we would have Es = C/Tr, where Tr is the run time for the sample in question (Tr = L/Vr, where L is the run length (i.e. 100 cm) and Vr is the run velocity in cm s−1). However, various corrections have to be made. In the first place, allowance must be made for the ‘choke effect’, which assumes that other things being equal, the rate of sediment entrainment is proportional to the fluid space available; the rate of entrainment would then be C(1 − N)/Tr , where N is the proportion of solid already in suspension. Secondly, allowance must be made for the amount of sediment lost by settling out during the actual run, causing the measured value of C to be less than would otherwise be the case; this effect is represented by the factor (1 + δ), with δ = DVg cos 20°(Tr /2Xf ). This would give Es = C(1 − N)(1 + δ)/Tr . Finally, allowance must be made for the small choke effect of the sediment already entrained during the course of the run; this effect is represented by the factor 1/(1 − Nc /2), where Nc is the value of N in the siphon sample, giving Es = C(1 − N)(1 + δ)/[(1 − Nc /2)Tr ]. In the present case, concentrations are measured in g l−1. Taking the silica density as 2650 g l−1 gives N = Ca /2650 and Nc /2 = C(1 + δ)/5300. Balancing settling against entrainment, and reducing the above formulae, we finally have Ca = Ce = A/[B + (A/2650)], where Ce is

Fig. 3. Vertical segment of the flow tube, 5.1-cm diameter, sloping at 20°. (a) in profile, and (b) in cross-section.

PGCC13 9/12/08 3:28 PM Page 195

195

Autosuspension

using both the earlier (lower) position of the T-tube, and the later (higher) position are included, since the change in setting showed no significant difference. These early results showed a great deal of scatter. It appeared that the high scatter was due to significant variations in the turbulence structure from one flow to another, this being inherited both from the initial opening of the upper valve and the flow through the L-tube upstream of the flow tube. It was therefore decided to attempt to homogenize turbulence within the flow, by placing a series of four plastic grills at strategic positions within the flow tube. Passage of flow through these grills would both damp out larger turbulent eddies inherited from the valve or L-tube, and would impose their own turbulent structure on the flow as it passed above the test bed. Later results Fig. 4. Equilibrium suspended sediment concentration (Ce ) in grams dry weight per litre; results for 100% saturated saline versus 60% saturated saline, for pairs of samples from series PFEX23–60 (Phase 1).

the equilibrium concentration, A = CXfVr sec 20°(1 + δ), and B = 100DVg [(1 − C /5300)]. Xf is measured in cm2, Vr and Vg in cm s−1, and both D and L in cm. The above calculations were normalized, as far as possible, by using throughout a settling velocity (Vg ) appropriate to plain water (c. 1 cm min−1). The available data do not permit corrections for possible variations in settling velocity with increased concentration, nor for possible variations in sediment entrainment, for a given velocity, with changes in fluid density. A series of values of Ce for the two salinities, using pairs of measurements with the same roughness element, were plotted against one another (Fig. 4); this plot includes results from all roughness elements, up to and including the sloping plate (roughness insert (ix) above), and allows for the variable Xb. Results

A series of 24 runs (PFEX63-88), with six runs at each of four different salinities (15%, 30%, 60% and 100% saturated), were conducted with the grills in place, and in the absence of other roughness elements. Fluid volumes and weights of silica from siphon sampling were converted to actual sediment concentrations (C ) and to equilibrium suspended-sediment concentrations (Ce ) the same way as in the earlier series (PFEX23-60). In addition, flow velocities and sample concentrations were averaged for each of the four sets of results (Vav and Cav , Table 1), and standard deviations (SD) of the concentrations were determined. It was found that the ratio of SD to the average concentration (Table 1) varied little for the three highest velocities (0.067, 0.069 and 0.071), although that for the lowest velocity was somewhat higher (0.094). The results, although showing some degree of scatter, were far more consistent than for the series PFEX23–60, and clearly demonstrated the increase in Ce with increasing flow velocity (Fig. 5). Reynolds and densimetric Froude numbers for these flows are given in Table 2.

Table 1. Salinities, average flow velocities (Vav ), average sediment concentrations (Cav ) in g (dry weight) l−1, standard deviations (SD) of C, and ratios of SD to Cav, for the four groups of Phase 1 siphon samples PFEX63-88 Samples from series PFEX63-88 (six samples/group) Group 1 Group 2 Group 3 Group 4

Salinity (% saturated) of header-tank fluid

Average velocity (Vav ) (cm s–1)

Average concentration of sediment (Cav ) in sample group

Standard deviation (SD) of sample concentrations (C)

SD/Cav

15 30 60 100

17.13 21.65 29.45 42.12

0.966 1.725 4.397 8.012

0.091 0.123 0.305 0.536

0.094 0.071 0.069 0.067

PGCC13 9/12/08 3:28 PM Page 196

196

H.M. Pantin

Fig. 5. Equilibrium suspended sediment concentration (Ce ) in grams dry weight per litre, versus flow velocity; 24 samples from series PFEX63-88 (Phase 1).

The Ce results were likewise averaged (C eav ) for each of the four groups, and plotted against the average flow velocities Vav , an empirical curve being drawn through the four C eav points obtained (Fig. 6). The resulting curve shows a concave-up initial portion, approximating to a 2.5 power relationship between the first two points, grading into an approximate 3.0 power between the second and third points. This agrees at least qualitatively with published suspendedsediment entrainment formulae (Parker et al., 1986; García & Parker, 1991). In contrast, the curve becomes slightly convex-up between the third and fourth points, corresponding to an apparent power-law decrease to about 1.7. Published curves do not predict convexity until the state of catastrophic equilibrium is approached, this being a state in which the current has reached the limit of possible sediment entrainment (Parker, 1982). Data quoted by Parker et al. (1986) indicate that catastrophic equilibrium occurs when N reaches a value of c. 0.3; in the present case, measured volume concentrations of solid are much lower than this, and it seems very unlikely that the limit of sediment entrainment has been approached. Additional data for higher velocities will be necessary to resolve the situation. To obtain a relationship between the velocity in the tube and the concentration of a suspension with a specific gravity (σ) equal to that of the saline solution employed, the salinities were converted to equivalent

Fig. 6. Concentration–velocity phase plane, with curves for equilibrium (C eav ) and equivalent (Cs ) suspended-sediment concentration, derived from the data in Fig. 5, and plotted against flow velocity (Vav ). The crossing point of the two curves (X) is the ‘ignition point’, while ‘AGL’ marks the autosuspension generation line (Pantin, 1979; Parker et al., 1986). The AGL is shown diagrammatically, sloping downward from left to right, but the available data are not sufficient to determine its actual gradient. Arrows denote direction of trajectories, showing the mutual variation of sediment concentration and flow velocity with time in different parts of the phase plane. Flows in area A decay; flows in area B burgeon, being in a state of autosuspension.

dry-sediment concentrations (Cs ) using the formula Cs = (σ − 1) × 1000 × (2.65/1.65) g l−1, where 2.65 and 1.65 are respectively the dry and immersed specific gravities of silica. An empirical curve of Cs against the velocity (Vav ) was then plotted (Fig. 6). Surprisingly, although the velocity showed an obvious increase with concentration, the data did not agree well with a quadratic stress curve, i.e. the ‘drag coefficient’ varied with the velocity. Possibly, the entrainment of sediment itself may affect the overall drag. Further experiments will be needed to resolve this problem. Figure 6 shows the Cs, Vav curve superimposed on the C eav, Vav curve. The Cs, Vav curve cuts the C eav, Vav curve at point X, corresponding to a velocity of about 24 cm s−1, and with Cs = C eav, i.e. about 108 g l−1. The plot constitutes a phase plane, which can be used to predict the behaviour of any flow with a given combination of sediment concentration and velocity (below). Subsequent to the experiments with the grills, a short series (PFEX89-93) was carried out using the

PGCC13 9/12/08 3:28 PM Page 197

Autosuspension deflectors (roughness insert (xi) above). Sediment concentrations obtained in the siphoned samples were of the order of five times greater than those obtained with the grills, and it was decided to conduct further experiments with these deflectors. This was an essential part of the second phase of experiments. Interpretation of results: Phase 1 Sediment entrainment mechanisms The main process of sediment entrainment into suspension appears to be, both in Series PFEX23-60 and PFEX63-88, the lofting of bed material by strong vortices associated with scour pits around the roughness elements, this being assisted by increased turbulence in the wake of the obstructions. Entrainment by highvorticity streaks in the viscous sublayer may also have played a significant part in all cases; the observed current lineations must represent small-scale bottom irregularities which, combined with longitudinal vortices, would be expected to promote entrainment of bedload into the sublayer. Again, the repeated sediment-bearing ejections seen during a run would obviously carry the contained sediment into suspension, although measurements of the actual concentration of sediment in these ejections would be needed to estimate their effectiveness in transporting suspended sediment from the viscous sublayer to the body of the flow. To the list must be added the specific mechanism of the deflector plates, i.e. the forcing of the boundary layer up the left-hand side of the tube by the imposed swirl. Entrainment by processes associated with spanwise bedforms was probably not significant, since no ripples or other similar bedforms were observed during the experiments. The presence of some kind of inhibition to suspended-sediment entrainment at higher velocities is indicated by the relatively modest increase in Ce from the 60% saturated to the 100% saturated saline in the samples shown in Fig. 4. The same tendency is emphasized by the 3.0 power law in the centre of the velocity range for the samples shown in Fig. 6, as opposed to the 4–5 or 9–10 power laws implied by recent models (Parker et al., 1986; García & Parker, 1991), and is even more true of the falling-off of the power law towards the top of the velocity range (Fig. 6). A high density contrast at the top of a sediment-laden viscous sublayer might lead to a failure of Kelvin–Helmholtz instability (Turner, 1973); this would inhibit the process of ejection from the boundary layer. The ‘choke’ effect cannot be responsible for the observed entrain-

197

ment inhibition, since the relatively low concentrations of sediment would not make this effect significant. Prediction of autosuspension At all points in the phase plane above the C eav, Vav curve (Fig. 6), sediment will tend to deposit, and below this curve sediment will be entrained. Correspondingly, at all points above the Cs, Vav curve, the flow will burgeon, and below this curve will decelerate. These effects will govern the direction of trajectories indicating the variation in sediment content and velocity of any given flow. The intersection point of the two curves (X, Fig. 6) defines two significant fields of behaviour: at lower velocities, the C eav, Vav curve lies below the Cs , Vav curve, and at higher velocities lies above it. X is in fact the ignition point, and AGL marks the autosuspension generation line, dividing flows which move towards area A and decay, from those which move towards area B and accelerate, being then in a state of autosuspension (Parker et al., 1986). The available data are not sufficient to predict whether or not the state of catastrophic equilibrium would be reached. Conclusions from Phase 1 Three principal conclusions can be drawn from the Phase 1 experiments: 1 An experimental methodology has been developed, which demonstrates erosion of bed material at a greater rate than sediment would settle out from a turbid flow; the system is therefore capable of generating autosuspension under suitable, physically-realisable conditions. 2 The results suggest the existence of a phase plane which predicts acceleration and autosuspension at relatively high velocities and specific gravities, but deceleration and decay at lower values. 3 Autosuspension may be inhibited at high sediment concentrations, possibly due to damping of Kelvin– Helmholtz instability.

EXPERIMENTAL APPARATUS AND TECHNIQUES: PHASE 2 Redesign of apparatus It was obvious, after the conclusion of the experiments with the original rig, that the existing design would need considerable modification. The rates of entrainment

PGCC13 9/12/08 3:28 PM Page 198

198

H.M. Pantin

Fig. 7. Modified experimental apparatus, Phase 2.

with the roughness elements employed, up to and including the grills, produced relative increases in specific gravity of no more than c. 0.5%, which would produce relative accelerations of only some c. 0.25%. Such low accelerations would be far too small for reliable measurement, being only of the same order as turbulent fluctuations. Since the amount of sediment entrained, at least initially, would be roughly proportional to the length of the flow tube, it was clearly desirable to make the flow tube as long as possible, although the extension was limited by the size of the tank. It was also imperative to increase the flow time considerably, to improve the period of observation and thereby to obtain more reliable velocity measurements; this would be done by enlarging the header tank. Finally, it was necessary to employ new and far more efficient roughness elements, to increase the rate of entrainment by something like an order of magnitude. The redesigned apparatus is shown in Fig. 7. The length of the flow tube was increased by inserting another 0.75-m section of Perspex tube (this being limited by the depth of the tank), and the capacity of the header tank was increased to 24.3 litres. Due to the weight of the longer rig, the framework was modified, increasing the angle of slope from 20° to 24°. The roughness elements chosen were static deflector plates, similar to those employed at the end of the Phase 1 experiments, and which had given such promising results. Experimental procedure: Phase 2 The experimental procedure used for the redesigned rig was essentially similar to that in Phase 1, although a number of modifications were made. Since the greatest increase in sediment entrainment in the Phase 1 experi-

ments was found between saline solutions of specific gravity 1.06 and 1.12, respectively, it was decided, initially, to use only fluids of specific gravity 1.06, so that any increase in velocity would have the maximum effect. Again, since the timing of the arrival of the header-tank fluid would only give a single approximate reading of the current velocity, it was decided to use a current meter on the intake of the Ω-tube (only one such tube was found necessary at specific gravity 1.06). The equipment used was a Nixon flow meter, which depends on electronic impulses from a miniature propeller, and can only be used in clear water (i.e. not within a suspension). This arrangement was found to work reasonably well, although the Nixon has a stepped readout (one reading per second at best), and the degree of resolution at the observed velocities (up to 17–18 cm s−1) was no more than adequate. It is important to remember that due to the presence of deflectors and the test bed, the free cross section of the flow tube will be of the order of 80% of the cross section of the intake tube. The actual velocities within the flow tube will, therefore, be correspondingly higher. However, the valves have the same diameter as the intake (c. 5 cm), and output velocities will be about the same as those at the intake. The test bed consisted in all cases of silica flour with a dmode of 20 µm, a larger quantity being used than in the Phase 1 experiments, giving a thickness of about 1 cm on the bottom of the flow tube. The initial runs were carried out using saline solutions of specific gravity 1.06 as the heavy fluid, since observations in the tube were relatively easy, and extraction by filtration of the entrained sediment in the samples was straightforward. The washed and filtered samples were dried and then weighed. In the early stages of experimentation, using saline solutions, a total of 16 deflectors were used, placed about 10 cm apart. In most of these runs, all the deflectors were designed to give a right-handed swirl to the flow, although two runs were made using alternate right-handed and left-handed deflectors. The number of deflectors (right-handed) was later increased to 31, placed about 5 cm apart; this change increased the drag in the flow tube, but improved sediment entrainment. The 31-deflector configuration was used for all the later experiments, in which the heavy fluid was silica suspension. After the initial runs with saline solution had been carried out (BB1-19), the saline was replaced by a suspension of silica with a specific gravity of 1.06 (BB20-37 and BB43). Whilst increasing the degree of realism in terms of turbidity flow, the use of a silica

PGCC13 9/12/08 3:28 PM Page 199

Autosuspension suspension posed the problem of distinguishing the entrained bedload from the header-tank material in the siphon samples. Magnetite had already been found unsatisfactory, and would be even more so in the larger quantities required for the Phase 2 experiments. Powdered lime of suitable grain size would have been a very good test-bed material, as the proportion of CaCO3 could have been easily estimated by weighing before, and after, treatment with acid. However, no source of suitable powdered lime could then be located. It was therefore decided to try an optical method. The test bed (Fig. 2) was prepared by staining the silica with an aqueous solution of Methylene Blue dye (a very strong adsorbate with a deep blue colour), and the ‘blueness’ (colour saturation) of the samples determined by illumination with a sodium lamp. Light-meter readings of the blue-stained silica were consistently about one-half that of the plain silica, and the proportions of ‘blue’ and ‘white’ could be estimated with readings from the extracted samples (although the curve of blue : white ratio against meter reading was by no means linear). When preparing the blue test bed, care was necessary to introduce enough Methylene Blue to stain all of the silica, but to make sure, in turn, that all of the dye had dissolved in the water. Undissolved dye in the test bed would obviously result in staining of some of the ‘white’ silica from the header tank. Sample volumes removed by siphon were usually of the order of 1000–1500 ml, although larger numbers of smaller samples (150–240 ml) were sometimes taken to give a more accurate profile of the quantities extracted. Weights and volumes of the extracted samples were measured, to provide values for the sediment concentration (C ). It was decided to use wet weights for these runs with silica suspension in the header tank, as drying all of the many samples would have been excessively time consuming. Tests indicated that a specific gravity of about 1.92 was a suitable value to use for the water-saturated silica flour. Such a procedure is obviously less accurate, but very much faster, than dry weighing. In all cases, both with saline and silica suspension, the turbidity flow emerged from the lower valve as a negatively-buoyant forced plume of suspended material (Fig. 8). A series of video recordings were made of this, as well as of the test bed itself after the current had passed. It was found that during the later stages of the flow, as the velocity was decreasing, a thin, lowvelocity trail of bedload began to emerge from the base of the valve exit; for this reason, a rectangular box was attached below the outlet in some runs, to retain the

199

Fig. 8. The plume emerging from the lower valve, during a silica-suspension run (Phase 2).

bedload output. The lowest part of the suspension plume tended to be trapped as well as the bedload, but the procedure nevertheless gave an indication of the maximum amount of bedload to be expected during a run. For the purpose of comparison, and to estimate the accuracy of the sampling method, a few runs were carried out with the usual silica suspension in the header tank, but no test bed (BB38-40). Experimental results: Phase 2 Earlier results (saline flows) Due to the greatly enlarged header tank, combined with greater drag from the roughness elements, the time span of an individual run was typically of the order of 3 min, roughly 40–50 times as long as runs in Phase 1 of the experimental programme. Concentrations of sediment in the siphon samples were much higher than in the earlier series, improving to c. 30 g l−1 dry weight with 16 deflectors and to c. 45 g l−1 dry weight with 31. The two runs made with 16 alternate right- and lefthanded deflectors showed no improvement, but rather a very slight decrease in the amount of entrainment. Experimental velocities were very significant; with 16 deflectors, there was an almost immediate acceleration (over 1–2 s) up to some 10–11 cm s−1, followed by a slower acceleration (over about 20 s) up to some 13–14 cm s−1. The velocity was maintained at near maximum for perhaps 50–60 s, after which the current slowly decayed to zero over some 3 min. With 31 deflectors, the first acceleration (over 1–2 s) reached about 8 cm s−1, and the second acceleration (over 25–30 s) reached about 11 cm s−1; the velocity was maintained

PGCC13 9/12/08 3:28 PM Page 200

200

H.M. Pantin

Table 2. Approximate Reynolds and densimetric Froude numbers for Phase 1 and Phase 2 Runs. Values for Phase 1 are based on Vav. Values for Phase 2 show the range for flows with test bed during the ‘second acceleration’, except for BB40 (no test bed), where the range for the pulsating flow is given. The kinematic viscosity of the fluid is taken throughout as 1.2 × 10 −6 m2 s−1 (output velocity used) Re Phase 1 (PFEX63-88) Group 1 Group 2 Group 3 Group 4 Phase 2 Saline solutions, with test bed Silica suspensions, with test bed BB40 (no test bed)

Fr (densimetric)

7.28 × 103 9.20 × 103 1.25 × 104 1.79 × 104

0.99 1.25 1.20 1.33

4.25−5.95 × 103

0.58−0.81

2.98−4.68 × 103

0.40− 0.64

3.15 −4.55 × 103

0.43−0.62

near maximum for about 60 –70 s, decreasing to zero after some 4 min. Reynolds and densimetric Froude numbers for these flows are given in Table 2. In all of these runs, the flow emerged from the lower end of the tube as a negatively buoyant forced plume. Initially, all of the transported sediment appeared to be in suspension, but after some 90–100 s, a thin trail of bedload appeared at the base of the outlet, raining more or less vertically downward. Examination of the test bed after a run showed deep scour pits around the deflectors. Typically, these took the form of a short lunate pit on the downstream side of a deflector, and a more extensive bilobate pit on the upstream side, the right-hand lobe (looking downstream) being lunate and the left-hand elongated. Later results (silica-suspension flows) The results obtained from the runs with silica suspension may be divided into five groups. 1 After a run, deep scour pits were again found around the deflectors, and showed essentially the same characteristics as those obtained in the earlier series. However, a new and important feature was revealed. The walls of the scours (Fig. 2) were in many cases sufficiently steep to prevent the slow tail of the current, which was still depositing (white) silica, from blanketing them. An unconformity could be clearly seen in each case, with the eroded surface of blue test bed overlain by white material from the header tank. 2 Comparison of weights of test bed originally prepared with weights remaining after the run (the remainder in

the flow tube, the contents of the bedload box, and small quantities lost while introducing the test bed) showed that some 33–37% had been removed from the flow tube. The content of the bedload box proved to be only some 10–11% of the missing weight, and could not possibly account for the missing sediment. Moreover, the remainder in the flow tube and in the bedload box contained significant proportions of header-tank material, showing that the true percentage of bed material removed must be higher than the figures quoted. 3 The proportion of sediment in samples from the flow was found, at its highest level, to exceed that in the original suspension in the header tank by some 16– 33%. Optical examination showed that the sediment in these samples contained at least 22–24% of blue testbed material, corresponding to a relative increase of about 30% over the original sediment concentration. Samples from runs with the normal silica suspension, but with no test bed, showed a decrease in concentration of 1–6%, relative to the header-tank concentration. This indicates a small degree of undersampling. 4 Current velocities showed similar time changes to those observed in the earlier saline flows; a rapid first acceleration to some 7 cm s−1 lasting 1–2 s, followed by a more gradual second acceleration (over 20–40 s) up to some 10 cm s−1 (Fig. 9a–c). There was typically, also, a late surge with a peak of 10–11 cm s−1, lasting for some 10 s (Fig. 9a–c). The velocity was maintained at near maximum for about 40–50 s, after which the current slowly decayed to zero after some 30–40 s. Reynolds and densimetric Froude numbers for these flows are given in Table 2. As in the earlier series, the flow emerged as a negatively-buoyant forced plume (Fig. 8). The trail of bedload appeared some 75 s into the run. For comparison, a velocity–time graph is presented for BB40, a silica-suspension run with deflectors but no test bed, and with no dummy inserts to represent the volume normally occupied by the test bed (Fig. 9d). The graph shows a rapid acceleration to a value of c. 9 cm s−1; this evidently corresponds to the ‘first acceleration’ of the runs with a test bed, although the velocity is significantly higher (runs with a test bed achieved only c. 7 cm s−1). After the initial acceleration, the BB40 record shows only a pulsation, with a period of c. 15 s and an amplitude of c. 1.25 cm s−1, with no evidence of an overall second acceleration. Reynolds and densimetric Froude numbers for this flow are given in Table 2. 5 Calculation of the equilibrium suspended-sediment concentration Ce for run BB34 gave a value of 448 g l−1 (dry weight), compared with the initial (header tank) concentration Ch of 96 g l−1.

PGCC13 9/12/08 3:28 PM Page 201

201

Autosuspension 12 11

12

E

D

(a)

(b)

BB26

BB27

11

C

C

10

10

velocity (cm s-1)

velocity (cm s-1)

D 9

AB

8 7 6

8 7 6

5

5

4

4

3 -10

0

10

20

30

40

50

60

70

80

90

100

110

120

130

3 -10

140

0

10

20

30

40

50

60

time (s)

70

80

90

100

110

120

130

140

time (s)

12

12

(c)

C

11 10

D

(d)

BB29

E

AB

8 7 6

9

X

7 6 5

4

4

0

10

20

30

40

50

60

70

80

90

100

110

120

130 140

time (s)

A

8

5

3 -10

B

10

9

BB40

11

velocity (cm s-1)

velocity (cm s-1)

E

AB

9

3 -10

0

10

20

30

40

50

60

70

80

90

100

110

120

130 140

time (s)

Fig. 9. Velocity–time curves, obtained with a Nixon current meter for three silica-suspension runs with test bed (BB26, BB27 and BB29, Phase 2), and one without test bed (BB40, Phase 2). In (a–c), A—B is the first acceleration, B—C the second acceleration, and D—E the late surge. (d), in contrast, shows clearly an initial acceleration, corresponding to the first acceleration in (a–c), but no steady increase in velocity corresponding to the second acceleration of (a–c); the record shows only a pulsation, with a period of c. 15 s and an amplitude of c. 1.25 cm s−1.

Interpretation of results: Phase 2 The runs using silica suspension in the header tank provide clear evidence that autosuspension was achieved. Several lines of evidence are available, which can be grouped into two distinct subsets: erosion, sediment entrainment and increased suspended load; and observed flow acceleration. Erosion, sediment entrainment and increased suspended load The lunate pits on both sides of the deflectors are obviously the result of erosion, probably by eddies caused by the obstruction, analogous to scour around bridge piers; the elongated upstream pits, on the other hand,

are probably due to eddies near the obstruction, combined with a high-velocity core extending upstream from the gap between the lowermost plates of the deflector. The unconformity seen in the scour pits, between the remains of the blue test bed and the overlying white header-tank sediment, clearly demonstrate that erosion occurred during the passage of the current through the flow tube. The white sediment above the unconformity must have been laid down by the progressively-slowing tail of the current. The missing test-bed material revealed by the ‘test-bed’ budget described earlier can only have been removed in suspension, other possibilities having been excluded. This demonstrates that bed material was entrained into suspension.

PGCC13 9/12/08 3:28 PM Page 202

202

H.M. Pantin

The major factor in sediment entrainment into suspension was no doubt the eddies associated with the scour pits, combined with the swirl of the current. Even if the material initially formed part of the bedload, the sediment-laden boundary layer would be transported up the left-hand side of the tube, and greatly reduce the gravitational damping which would have occurred with a horizontal bed. The increased proportion of suspended sediment in the most concentrated samples, relative to the proportion in the original suspension, further indicates that sediment entrainment was achieved on a significant scale. The much higher value of equilibrium suspendedsediment concentration (Ce ) in BB34, as compared with the initial (header tank) concentration Ch, shows that if Ce were attained, other things being equal, the flow would accelerate considerably. This acceleration cannot be estimated accurately, since the specific gravity/velocity data obtained for the Phase 1 series PFEX63-88 (Figs 5 and 6) do not extend to velocities as low as those encountered in the Phase 2 experiments. However, assuming a quadratic stress law, the velocity would increase by a factor of about 2.16. Assuming a monotonic relationship between the rate of suspended-sediment entrainment and velocity, the rate of entrainment would also increase, and the system would be in the ignitive state.

pits that form in the runs with a test bed are highly irregular, and it would be quite possible for irregularities of this kind to increase, rather than reduce the drag in the tube. In fact, the increase in specific gravity appears to be insufficient to account for the whole of the observed acceleration (see below). Assuming a quadratic stress law, an increase in the intake velocity from 7.4 to 9.325 cm s−1 indicates, approximately, a 1.59-fold increase in immersed specific gravity, considerably higher than the 30% increase indicated by the optical measurements. The siphon-sample evidence from runs with no test bed indicates that a small degree of undersampling may be taking place, but this would scarcely affect the relative proportions of header-tank and testbed material. The reason for the typically-observed late-velocity surge (Fig. 9a–c) is not immediately apparent. It may be due to the flow encountering more-easily erodible layers in the test bed; improved quality control of the test-bed material may show whether or not this is the case. The lack of a second acceleration in the run with no test bed (Fig. 9d) is the result to be expected if that acceleration were due to sediment entrainment, cross-section enlargement, or a combination of the two. Possible explanations for the 15-s pulsation are beyond the scope of this chapter.

Observed flow acceleration

Independent check on specific gravity and velocity from plume characteristics

The first acceleration is presumably due to the rapid attainment of equilibrium between gravity and drag within the flow tube. The immersed specific gravity of the saline solutions and silica suspensions would be 1.06 − 1.00 = 0.06, giving a reduced gravitational acceleration of 0.06 × g = 0.59 m s−2. Allowing for the gradient of the tube, the initial acceleration would be c. 0.59 × sin 24° = 0.24 m s−2, which would lead to virtual equilibrium within 1–2 s. The second acceleration could, in principle, be due partly to the increased specific gravity of the flow as a result of the entrainment of test-bed sediment, and partly to a reduction in the drag coefficient, due to the reaming-out of the tube by erosion. The relatively high velocity following the initial acceleration in the run with no test bed (BB40; Fig. 9d) is probably due simply to the greater free cross-section, as opposed to the runs with a test bed. However, caution is needed here, since it does not automatically follow that an enlarged cross-section will reduce the drag coefficient; the scour

An independent check on the relative magnitudes of the specific gravity and the velocity of the flow can, in principle, be obtained by consideration of the flow emerging from the lower valve into the ambient water. This flow is a forced negatively-buoyant plume, whose trajectory and other properties are a function of the initial conditions, i.e. the immersed specific gravity of the flow, velocity, radius, and angle to the horizontal, at the point of emergence. A theoretical treatment of forced angled plumes has been developed by LaneSerff et al. (1993); their model relates the plume angle to the velocity, radius, and reduced gravity of the plume, and explores the change in plume angle with distance along trajectory from the source. Their model is discussed in terms of positively-buoyant plumes, but can equally be applied to negatively-buoyant ones, as in the present case. The application of their model to the emergent plume in the case of run BB34 (Phase 2) is described in the Appendix.

PGCC13 9/12/08 3:28 PM Page 203

Autosuspension

INTERPRETATION OF PHASE 1 AND PHASE 2 RESULTS: SUMMING UP The results obtained in the Phase 1 experiments show that provided certain reasonable assumptions are fulfilled, in particular the rate of settling-out of sediment from suspensions of equivalent specific gravity to the saline solutions employed, autosuspension is possible. The results from the Phase 2 experiments demonstrate the autosuspension phenomenon itself. Above all, it is important to emphasize that the energy needed to entrain sediment, in both the Phase 1 and Phase 2 experiments, was provided by gravity alone, within the system (apart from the turbulent energy needed to prevent suspensions in the header tank from settling out prior to the run). There was no external input of energy, such as a stirring mechanism, employed during the progress of the run. The use of a tube as the conduit, rather than an open channel, obviously simplified analysis of the results, there being no entrainment of ambient fluid or longitudinal changes of flow thickness or velocity to consider. However, repetition of some of the foregoing experiments, using for example a channel with rectangular cross-section, open at the top to the ambient fluid, would be highly desirable for the purpose of comparison.

APPENDIX Lane-Serff et al. (1993) describe the behaviour of a plume using equations which may be written as dt/ds = 1 1 q/(t 2 + 1) –2 and dq/ds = (t 2 + 1)–4, where t = tan θ, θ being the angle of the plume centre line to the horizontal; q is the non-dimensional mass flux; and s is the non-dimensional distance along the plume centreline. The foregoing equations can easily be modified to give the changes in q and s as functions of q and θ, the result 7 being dq/dθ = 1/(q cos–2 θ) and ds/dθ = 1/(q cos3 θ). The initial conditions (denoted by the1 subscript 1 5 s) are given by qs = (λ/(2α) –2 )( g s′ Rs /Vs 2) –2 (cos θs )− –4, where λ and α are constants (1.1 and 0.1 respectively), and θs, g′s , Rs, and Vs are the plume angle, reduced gravity, radius, and velocity at the source of the plume. Using the above expressions, (q,θ) trajectories or (s,θ) trajectories can be computed for any initial combination of q or θ (initial s is normally taken as zero). Lane-Serff et al. (1993, Fig. 8) shows a (q,θ) phase plane derived in this way. To obtain the real centreline distance S, we need only multiply the non-dimensional length

203

s by the1 lengthscale of 1the system LA, given by 3 LA = (λα –2 )−1 Rs(g′s Rs /Vs 2)− –2 (cos θs ) –4. The above theory has been applied to the plume emerging at different stages during run BB34 of the Phase 2 experiments. Centreline trajectories were estimated from two groups of still frames, A and B, taken 5–6 s and 22–23 s respectively after release of the flow. For both A and B, θs was taken as 24°. For group A, the value of θ increased from 24° to 58° in a centreline distance of 5.8 cm, while for group B, θ increased from 24° to 61° in 5.8 cm. A series of calculations were made for the BB34 plume, using the Lane-Serff et al. (1993) model. It was assumed that the initial value of the immersed specific gravity (∆ s ) was approximately the same as that within the lower part of the flow tube, in the neighbourhood of the T-tube. On this basis, ∆ s, for group A was taken as 0.06 (i.e. that of the saline emplaced with the test bed), while that for B was taken as 0.078, i.e. 30% higher (corresponding to the percentage increase in sediment content indicated by the colorimetric tests). This gave g′s = g∆ s values of 58.86 and 76.52 cm s−2 respectively. The measured flow velocity at the intake was respectively 7.4 and 9.325, but a correction was necessary for both plume-source velocity and radius, since it was observed in both cases that the outflowing plume did not completely fill the tube. The gap for stage A was about 1.346 cm, and for stage B about 1.215 cm. The outflow velocity was corrected for the reduced cross-section, while the effective radius of the source was taken as that of a circular cross-section with the same (reduced) area. The values in question were Rs = 2.215 cm and 2.279 cm for stages A and B, with Vs = 9.807 cm s−1 and 11.217 cm s−1 respectively. As it turned out, it was impossible to match the observed trajectories of either group A or group B, without assuming a further correction, i.e. the entrainment of ambient water near the outlet. This was most clearly shown in relation to group A, where little sediment had been entrained and the specific gravity would be expected to be similar to the original saline. In this case, it was necessary to assume a dilution factor of 0.68 to match the observations. In the case of the group B trajectory, it was necessary to assume a dilution factor of 0.78 to match the plume, if ∆ s was indeed 0.078. An obvious source of ambient water is that separating the outflowing plume from the top of the outlet; this layer of water does not extend as far as the Perspex flow tube, and must therefore take the form of a wedge, analogous to the salt wedge found in estuaries.

PGCC13 9/12/08 3:28 PM Page 204

204

H.M. Pantin

Although entrainment of water from this ambient wedge is entirely plausible, there appears to be no independent way of estimating the rate of entrainment at the interface, under these particular conditions; models developed for free plumes, or for wall plumes in a deep ambient, are not appropriate in the present case. Lacking such a model, the plume data alone cannot, in the case of group B, provide a definitive value for the initial specific gravity (∆ s ); the value obtained depends on the assumed dilution factor, and vice versa, e.g. an initial value of 0.09 for ∆ s and a dilution factor of 0.75 also match the plume data. One can say only that the values of 0.078 for ∆ s and 0.78 for the dilution factor are not out of the question.

µm s

micrometres seconds

Labels AGL X

autosuspension generation line ignition point

Parameters D dmode g L α λ

diameter of flow tube (5.1 cm) modal diameter of sediment acceleration due to gravity (9.81 m s−2 ) run length in flow tube plume parameter (value 0.1) plume parameter (value 1.1)

ACKNOWLEDGEMENTS The writer would like to express his very best thanks to the following members of Leeds University: to Phil Fields and Tony Windross (School of Earth Sciences Workshop), who constructed the apparatus; to Neil Woodhouse (School of Earth Sciences), who assisted in the experiments throughout Phase 1 and the beginning of Phase 2; to Alison Manson and David Appleyard (School of Geography), who helped greatly in the preparation of Fig. 1 and the various graphs; to Paul Pitts (Media Services), for major assistance with data analysis; and to the other members of the Sedimentology Group (School of Earth Sciences), especially Jim Best, Rufus Brunt, Clare Buckee, Mark Franklin, Bill McCaffrey, and Jeff Peakall, who contributed many useful ideas, and often provided much-needed technical assistance. He would also like to thank two reviewers, G.F. Lane-Serff and I. Eames, whose pertinent comments pointed the way to many improvements in the content of the chapter. The construction of the experimental apparatus and the deep tank was supported by the UK Natural Environment Research Council (NERC Grant GR9/1479).

NOMENCLATURE Units cm g l m mm

centimetres grams litre metres millimetres

Variables C Ca Cav Ce C av e Ch Cs Es Fr g′ g′s LA N Nc q qs Rs Re s s(subscript) SD t Tr Vav Vr Vg Vs W

sediment concentration by dry weight in siphon samples arbitrary value of C average value of C for group of samples equilibrium value of C average value of Ce for group of samples header-tank concentration of sediment suspended-sediment concentration equivalent to saline sediment entrainment rate densimetric Froude number reduced gravity reduced gravity at plume source plume lengthscale volume fraction of solid in suspension volume fraction of solid in siphon sample non-dimensional mass flux of plume value of q at plume source radius of plume at source Reynolds number non-dimensional axial distance along plume value of variable at plume source standard deviation time (main paper), tan θ (appendix) run time for a given sample average velocity for group of samples run velocity settling velocity of sediment grains axial velocity of plume at source weight of sediment in vertical segment of unit thickness

PGCC13 9/12/08 3:28 PM Page 205

Autosuspension Xb Xf δ ∆s θ θs σ

cross-section of tube taken up by bed material and roughness elements free cross-section in flow tube factor representing sediment lost by settling during run immersed specific gravity of plume at source plume angle relative to horizontal value of θ at source of plume specific gravity of saline solution, or equivalent sediment suspension

REFERENCES Akiyama, J. & Stefan, H. (1985) Turbidity current with erosion and deposition. J. Hydraul. Eng. 111, 1473–1496. Eidsvik K.J. & Brørs, B. (1989) Self-accelerated turbidity current prediction based upon (k-ε) turbulence. Continent. Shelf Res. 9, 617– 627. Fukushima, Y., Parker, G. & Pantin, H.M. (1985) Prediction of ignitive turbidity currents in Scripps Submarine Canyon. Mar. Geol. 67, 55–81. Fukushima, Y. & Parker, G. (1990) Numerical simulation of powder-snow avalanches. J. Glaciol. 36, 229–237. García, M. & Parker, G. (1991) Entrainment of bed sediment into suspension. J. Hydraul. Eng. 117, 414 – 435. García, M., Niño, Y. & López, F. (1996) Laboratory observations of particle entrainment into suspension by turbulent bursting. In: Coherent Flow Structures in Open Channels (Eds Ashworth, P.J., Bennett, S.J., Best, J.L. & McLelland, S.J.), John Wiley, Chichester, pp. 63–86. Hogg, A.J., Huppert, H.E. & Soulsby, R.L. (1994) The dynamics of particle-laden fluid elements. In: Sediment Transport Mechanisms in Coastal Environments and Rivers (Eds Bélorgey, M., Rajaona, R.D. & Sleath, J.F.A.), Proceedings of the EUROMECH 310 Colloquium (Le Havre, France, 13–17 September, 1993), World Scientific Publishing Co. Pte. Ltd. PO Box 128, Farrer Road, Singapore 9128, pp. 64 –78. Hughes Clarke, J.E., Shor, A.N., Piper, D.J.W. & Mayer, L.A. (1990) Large-scale current-induced erosion and deposition in the path of the 1929 Grand Banks turbidity current. Sedimentology 37, 613–629.

205

Inman, D.L., Nordstrom, C.E. & Flick, R.E. (1976) Currents in submarine canyons: an air–sea–land interaction. Ann. Rev. Fluid Mech. 8, 275–310. Kaftori, G., Hetsroni, G. & Banerjee, S. (1995) Particle behaviour in the turbulent boundary layer. 1. Motion, deposition, and entrainment. Phys. Fluids 7, 1095–1121. Kostaschuk, R.A. & Church, M.A. (1993) Macroturbulence generated by dunes: Fraser River, Canada. Sediment. Geol. 85, 25–37. Lane-Serff, G.F., Linden, P.F. & Hillel, M. (1993) Forced, angled plumes. J. Hazard. Mater. 33, 75–99. Nezu, I. & Nakagawa, H. (1993) Turbulence in OpenChannel Flows. Balkema, Rotterdam, 281 pp. Pantin, H.M. (1979) Interaction between velocity and effective density in turbidity flow: phase-plane analysis, with criteria for autosuspension. Mar. Geol. 31, 59–99. Pantin, H.M. (1986) Triggering of autosuspension by a periodic forcing function. In: River Sedimentation, Vol. III (Eds Wang, S.Y., Shen, H.W. & Ding, L.Z.), Proceedings of the Third International Symposium on River Sedimentation (Jackson, Mississippi, USA, 31 March to 4 April, 1986), School of Engineering, The University of Mississippi, Mississippi, USA, pp. 1765–1770. Pantin, H.M. (1991) A model for ignitive autosuspension in brackish underflows. In: Sand Transport in Rivers, Estuaries, and the Sea (Eds Soulsby, R. & Bettess, R.), Proceedings of the EUROMECH 262 Colloquium (Wallingford, UK, 26–29 June 1990), A.A. Balkema, PO Box 1675, 3000 BR Rotterdam, the Netherlands, pp. 283–290. Parker, G. (1982) Conditions for the ignition of catastrophically erosive turbidity currents. Mar. Geol. 46, 307–327. Parker, G., Fukushima, Y. & Pantin, H.M. (1986) Selfaccelerating turbidity currents. J. Fluid Mech. 171, 145–181. Prior, D.B., Bornhold, B.D., Wiseman, W.J. Jr. & Lowe, D.R. (1987) Turbidity current activity in a British Columbia fjord. Science 237, 1330–1333. Prior, D.B., Yang, Z.-S., Bornhold, B.D., Keller, G.H., Lin, Z.H., Wiseman, W.J. Jr., Wright, L.D. & Lin, T.C. (1986) The subaqueous delta of the modern Huanghe (Yellow River). Geo-Mar. Lett. 6, 67–75. Stacey, M.W. & Bowen, A.J. (1988) The vertical structure of turbidity currents and a necessary condition for selfmaintenance. J. geophys. Res. 93, 3543–3553. Turner, J.S. (1973) Buoyancy Effects in Fluids. Cambridge University Press, Cambridge, 368 pp.

PGCC14 9/12/08 3:27 PM Page 207


Time- and space-resolved measurements of deposition under turbidity currents F . D E R O O I J and S . B . D A L Z I E L Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

ABSTRACT This chapter describes experiments measuring the instantaneous sediment deposition rates for a turbidity current. These measurements were obtained using a recently developed technique capable of very accurate measurements of sediment layer thickness, from which the concentration in the current could be derived. Three types of experiments were performed: lock-releases, lock-releases in a confined environment and continuous releases. It was found that turbidity currents from a lock-release form a head, where the particle concentration is highest and under which the largest deposition occurs. The initial decrease in particle concentration within the head is induced by the entrainment of ambient fluid. A maximum in the final layer thickness occurs when the current slows down and the turbulence has decayed to favour detrainment rather than entrainment. Deposition dominates the flow at later stages and the concentration decreases. When the current encounters a vertical wall it is reflected and deposits on top of the deposit formed earlier. A turbidity current from a continuous source attains a steady state with a fixed run-out length. The data suggest that the concentration in the steady state decays exponentially away from the source leading to a bed thickness with a similar exponential profile.

INTRODUCTION • spatially resolved • time-resolved To meet these requirements we developed a technique measuring the vertical electrical resistance of the layer (de Rooij et al., 1999). The present chapter describes the application of this technique to laboratory experiments on two-dimensional turbidity currents, both from a finite volume release and from a continuous release.

The formation of sediment layers by deposition from turbidity currents is an important topic in sedimentology. Many laboratory experiments have been carried out to investigate turbidity currents in various geometries and ambient conditions (see e.g. Bennett & Bridge, 1995, Alexander & Morris, 1994, Bonnecaze et al., 1993 and many earlier papers referred to by Middleton, 1993). In these experiments it is often desirable to obtain accurate measurements of the sediment layer thickness. Various techniques have been described in the literature (see e.g. Bonnecaze et al., 1993, Garcia, 1994, Best & Ashworth, 1994 and Ernst et al., 1996), each with its specific limitations. In our experiments we wanted to obtain instantaneous measurements of sediment deposition under a propagating turbidity current, so that we could analyse the deposition rate from different parts of the current. This required a technique to measure the sediment layer thickness which is • non-intrusive • independent of the presence of suspended sediment

EXPERIMENTAL ARRANGEMENT The experiments were performed in a rectangular tank with a flat floor 25.4 cm × 1000 cm, filled with tap water to a depth of 26.5 cm. The experiments were recorded with a video camera, mounted on a linear traverse parallel to the tank at a distance of 4 m. Three sets of experiments were carried out, and their details are listed in Tables 1 and 2. In the first set of experiments we studied the deposition from a lock-release turbidity current. The lock


207

PGCC14 9/12/08 3:27 PM Page 208

208

F. de Rooij and S.B. Dalziel

Table 1. Overview of the parameters of the experiments with a lock-release Experiment

dpart (µm)

φ0 (‰)

L0 (cm)

Tank length (cm)

d7 d8 d12 d13 d14 d15

13 13 37 37 37 37

1.21 2.38 2.33 2.71 2.75 2.33

20 20 10 10 10 10

1000 1000 1000 1000 1000 1000

d2 d3 d4 d5 d6

13 13 13 13 13

2.98 1.65 0.90 0.53 1.16

20 20 20 20 20

250 250 250 250 250

Lock-releases

Confined lock-releases

Table 2. Overview of the parameters of the constant-flux experiments Experiment c4 c5 c6 d9 d16

dpart (µm)

φ0 (‰)

Q (cm3 s −1)

Tank length (cm)

23 23 37 13 37

4.51 4.51 2.17 1.15 2.26

58 15 17 20 29

1000 1000 1000 1000 1000

was formed by placing a removable Perspex gate with foam seals around its edges vertically in the tank at either L0 = 10.0 cm or 20.0 cm from the end wall. A measured amount of silicon carbide particles was added to the lock, to yield an initial volume concentration φ0. The particles, ordinarily used for industrial polishing, had a density ρp of 3.217 g cm−3 and a narrow size distribution around a mean diameter of dpart = 37 µm. At the start of an experiment the water and particles in the lock were stirred vigorously to bring all the particles into suspension and the gate was rapidly but carefully lifted. In the second set of experiments we studied the deposition from a reflecting lock-release turbidity current by placing a rigid end wall in the tank at 230 cm from the front of the lock. In these experiments the lock length L0 was 20 cm, and particles with a mean diameter dpart = 13 µm were used. In the third set of experiments we studied turbidity currents generated by a continuous flux of particleladen fluid into the tank. The continuous flux was provided by a reservoir, where the particles were kept in suspension by a recirculating pump and a mechanical stirrer. A second pump was used to maintain a constant fluid level in a separate section of the reservoir, which ensured a constant volume flux. The particle-laden fluid entered the experimental tank via a

1-cm diameter nozzle, aimed downwards from 20 cm above the tank bottom and 3 cm from the end wall. Although this specific inlet configuration does not reflect natural conditions, the ensuing flow becomes two-dimensional at approximately 15 cm downstream from the end wall and is representative of natural planar currents. The technique we apply to measure the sediment deposition is based on the electrical resistance of the layer in the vertical direction being a monotonic function of the thickness of the layer. Since the resistance of the sediment that we use in our experiments is much larger than the resistance of tap water, the depth of the overlying fluid, which may contain a relatively dilute suspension, is unimportant (de Rooij et al., 1999). We can therefore simply measure the resistance between electrodes under the sediment layer on the bottom of the tank and a long reference electrode in the fluid at the top of the tank, as indicated in Fig. 1. Since the electrical field lines are highly concentrated in the vicinity of the bottom electrode (Fig. 2), a small increase of the sediment layer thickness will yield a relatively large increase of the resistance. The size, shape and packing of the particles will also affect the resistance, limiting the use of this technique to monodisperse particles. For the present experiments, layer thickness

PGCC14 9/12/08 3:27 PM Page 209

209

Deposition under turbidity currents Reference electrode

imately 1 s to obtain a set of measurements, which is much faster than the timescale for the flow so the data may be viewed as being instantaneous. A careful calibration was carried out for each specific batch of particles used, with resistance measurements taken at each electrode for five known sediment layer thicknesses. To obtain these layers the tank was divided into 50-cm sections and measured quantities of particles were added. Each section was thoroughly stirred, the particles were allowed to settle and the section dividers were removed before the measurements were taken. The calibration curve for each electrode was obtained from these measurements with the fitting process described in de Rooij et al. (1999). The small offset due to temperature variations was corrected for using the resistance measurements taken just before the start of each experiment. The accuracy of the system is approximately 0.5 mg cm−2. (Note that in this chapter all layer thicknesses are expressed as a mass per unit area.) To obtain a linear layer thickness the particle packing would need to be quantified, but this falls beyond the scope of the present study.

3 × 22 bottom electrodes

Fig. 1. Sketch of the experimental tank and the location of the electrodes.

E Fluid

Sediment

D

RESULTS

Perspex 6 mm Stainless steel

Lock-release currents

Fig. 2. Electrical field lines through the sediment layer in the vicinity of a bottom electrode.

A typical example of a current formed from a lockrelease is shown in Fig. 3. Shortly after release a pronounced head is formed, followed by a thinning tail. The temporal build-up of the sediment layer under such a lock-release turbidity current is shown in Fig. 4. The sequence of dashed and dotted curves represent the layer thickness along the length of the tank x, each curve at intervals of 6 s following the release of the current. The curves show how the horizontal extent of the sediment layer increases during each 6-s interval, reflecting the propagation of the turbidity current. The propagation of the head is also indicated by the ticks at the top of the graph. The horizontal distance between subsequent curves is gradually reduced, indicative of the slowing propagation of the current. Visual

measurements were taken using 66 bottom electrodes with a diameter of 6 mm, arranged on a rectangular grid with 22 rows of three electrodes (see Fig. 1). The downstream distance between the rows is 10 cm. The spanwise distance between the electrodes is 7.5 cm, with the middle electrode located on the centre line of the tank. The experiments showed no significant differences between the measurements at the electrodes in the same row, and the data presented in following sections are therefore spanwise averages. Using a computerized data acquisition system, measurements were taken every 3 s at all 66 locations. It took approx-

z (cm)

25.0 20.0 15.0 10.0 5.0 0.0

Fig. 3. An image of the flow from a lock-release (d13), 24 s after release.

0.0

10.0

20.0

30.0

40.0

50.0

60.0

x (cm)

70.0

80.0

90.0

100.0 110.0 120.0

PGCC14 9/12/08 3:27 PM Page 210

210


Fig. 4. Evolution of the sediment layer thickness under a turbidity current from a lock-release (experiment d14). The curves represent subsequent sediment layer thickness profiles at 6-s intervals. The corresponding position of the current nose is indicated by the marks at the top of the graph. The smooth solid curve represents the approximated numerical solution described by Bonnecaze et al. (1996).

observations indicate that the current comes to a halt at a run-out length of approximately xf = 190 cm. The growth of the thickness of the sediment layer at each point is initially relatively rapid. However, approximately 18 s after the current reaches some point x, the curves at that point x coincide and the layer thickness reaches an asymptotic value. The final layer thickness profile can be compared with the smooth solid curve, indicating the algebraic approximation to the numerical solution of a single-layer turbidity current model suggested by Bonnecaze et al. (1996). The run-out length is predicted reasonably well by this model, but close to the source significant discrepancies in the layer thickness are seen. The model does not include the effects of a return flow in a shallow ambient, which leads to the development of a pronounced head mentioned above. This effect is especially strong for the relatively short and tall locks used in our experiments. All our experiments exhibited a rather pronounced maximum in the layer thickness at some distance away from the source, a feature that has also been observed in experiments by others (see e.g. Bonnecaze et al., 1993; Gladstone et al., 1998). Further information about the deposition is shown in Fig. 5, where the rate of deposition (expressed as the speed at which the layer thickness increases) is plotted against the position x along the length of the tank. The deposition rate is calculated from five layer thickness measurements at a spacing of 3 s, effectively yielding a centred average over 12 s. Curves are shown at 6-s intervals, with the sudden increase from zero corresponding to the front of the turbidity current. However, the curves also show that the bulk of the deposition

takes place over a limited length behind the front of the current, of approximately 60 cm. This refutes the assumption sometimes made in turbidity current models that the current deposits equally over its full length, stretching back to the source. Instead, it shows that the deposition from the head of the current is dominant and largely determines the final sediment distribution. Visual observation of turbidity currents shows this head as a clearly distinguishable feature (see Fig. 3), with a depth more than twice that of the tail and a length comparable with the 60 cm where most deposition occurs. The particles deposit from the current in a layer adjacent to the lower boundary, where vertical velocities are reduced due to viscosity and the blocking effect of the wall. As we are in a régime where resuspension does not occur, the growth of the sediment layer thickness D is proportional to the particle concentration φ and the settling velocity of the particles Vs , yielding the equation dD = −φVs dt

(1)

This means that the deposition rate dD/dt at each location, shown in Fig. 5, provides information about the overlying particle concentration, independently of the depth of the current. At early times, the deposition rate is relatively low. It takes a finite time and distance before the suspension released from the lock has formed a pronounced head. During this collapse phase some entrainment of ambient fluid may take place, which gives rise to a reduced particle concentration and thus a reduced deposition rate. At approximately 60 cm from the front of the lock a maximum in the final layer thickness is seen, which

PGCC14 9/12/08 3:27 PM Page 211

Deposition under turbidity currents

211

Fig. 5. Sediment deposition rates at 6-s intervals for the lock-release turbidity current in experiment d14. The corresponding positions of the current nose are indicated by the marks at the top of the graph.

corresponds with the location where the deposition rate from the head of the current shows a maximum. A possible cause for the increased deposition rate may be a diminishing intensity of the turbulence in the current. The turbulence, initially generated by the release and later by the shear at the top of the current, will keep the particles in suspension and keep the current well mixed. Diminished turbulence may result in detrainment of interstitial fluid at the top of the current, leading to an increased particle concentration φ and thus indirectly to an enhanced deposition rate (Eq. 1). The turbulence within the current has recently become a topic of detailed investigations (Kneller et al., 1997; Best et al., this volume). It is hoped that those new measurements, combined with a further series of experiments currently in progress, will lead to an enhanced understanding of the effects of turbulence of the deposition. Another phenomenon that may contribute to a maximum in the final layer thickness is the deceleration of the current. The longer residence time of the current head over the surface further away from the source yields a larger total deposit. Yet the deposition reduces the particle concentration in the head of the current exponentially, reducing the deposition rate as was seen in Fig. 5. This counterbalances the increased residence time at larger distances from the source, leading to the decreasing total sediment deposition at long distances shown in Fig. 4. Lock-release currents in a confined environment Figure 6 shows the temporal build-up of the sediment layer under a turbidity current that was reflected off a

rigid vertical boundary. The current was formed from a lock-release as described above, but to generate a current with a longer run-out length a longer lock and smaller particles were used. The initial evolution is similar to a current in an unconfined environment: a large, depositing head propagates along the tank. The head is approximately 150-cm long, much longer than in the experiment described above because of the larger size of the lock. However, we still observe the slow layer growth close to the source once the head has propagated away. Even before the current reaches the end wall this leads to a broad maximum in the deposit some distance away from the lock, similar to the ‘infinite’ currents described above. When the head of the current reaches the vertical boundary, it is reflected and propagates back to the lock. Since not all particles in the tail behind the head have been deposited, some interaction is observed between the head and the shallow sedimenting tail. However, the interaction is not as strong as for a reflecting saline current, which forms a backwards propagating bore on the saline tail, rather than a clear head structure (see also Edwards et al., 1994). The reflected current deposits on top of the sediment layer already formed. For the region near the location of the release the layer thickness asymptotes to a constant value after the first passage of the head, and when the head passes for the second time the layer thickness starts increasing again. This becomes particularly clear from Fig. 7, where the deposition rate at three different locations is plotted as a function of time. The first passage of the head gives rise to a very pronounced maximum in the deposition rate, after which the deposition rate decreases until a smaller

PGCC14 9/12/08 3:27 PM Page 212

212


Fig. 6. Evolution of sediment layer thickness under a lock-release current in a confined environment (experiment d2). The curves show sediment layer thickness profiles at 10-s intervals.

Fig. 7. Time-evolution of the sediment deposition rate at three positions along the length of the tank, for a confined lock-release turbidity current (experiment d2).

increase in the deposition rate indicates the passage of the reflected head. The trace at 125 cm even exhibits a weak third peak, resulting from the second reflection. The net effect of the confined environment on the final sediment distribution is that the part of the usual sediment distribution that would be beyond the end wall is deposited onto the earlier part of the sediment distribution, but it is not simply a linear addition of the reflected deposition profile. The enhanced turbulent entrainment at the end wall and interaction with the shallow tail change the characteristics of the flow and the deposition rate, as described by Pantin and Leeder (1987). Unfortunately, our present experiments

do not provide measurements of the deposition close to the wall. To investigate the deposition near walls and obstacles, and to allow comparison with earlier studies (Edwards et al., 1994; Alexander & Morris, 1994), future experiments would need to be carried out with a closer spacing of electrodes. Continuous release currents The sediment deposition resulting from a continuous release of a suspension is shown in Fig. 8. Curves indicating the deposit are shown at 35-s intervals. Initially, the horizontal extent of the sediment layer increases

PGCC14 9/12/08 3:27 PM Page 213

213

Deposition under turbidity currents

Fig. 8. Evolution of the sediment layer thickness under a current generated by a continuous suspension flow at x = 0 cm (experiment d16). The curves represent subsequent sediment layer thickness profiles at 35-s intervals.

with time, although the current does not have a welldefined head as in the lock-release experiments. The propagation slows down and the current reaches its run-out length xf at approximately 140 cm. From this time on, the sediment layer continues building up along the full length of the current, but the sediment does not spread further. The sediment layer profile is approximately self-similar, and the deposition rate at each point x along the length of the tank is approximately independent of time. An image of this steady-state flow from a continuous release is shown in Fig. 9 for a qualitatively similar experiment. Figure 10 shows the time-averaged deposition rate during the steady state. Rapid deposition occurs close to the source, and less deposition further away. As we saw from Eq. (1), this implies that the concentration in the current decays similarly away from the source. An estimate of the expected decay rate can be made by using the visual observation that the current depth h is approximately constant along its

-10.0

0.0

10.0

20.0

30.0

40.0

50.0

length, as shown in Fig. 9. The velocity within the current is then u = dx/dt = Q/bh, where Q represents the volume flux and b the width of the tank. The deposition dD from Eq. (1) equals the loss of particles from the current h dφ, which yields bV dφ = −φ s . dx Q

(2)

Integrating this equation yields an exponential decay of the concentration away from the source, with a decay rate bVs /Q. The two straight lines in Fig. 10 indicate this exponential decay, with two different decay rates. The first one is based on a volume flux Q calculated from purely jet-like entrainment of ambient fluid during the descent of the suspension from the orifice to the tank floor, and the second one is calculated from purely plume-like entrainment. The experiments appear to indicate an exponential decay close to these suggested decay rates. Note that due to viscous effects, we do not expect an exponential decay at large

60.0

70.0

80.0

90.0 100.0 110.0 120.0

x (cm) Fig. 9. An image of the flow from a continuous release of particle-laden fluid through four inlets at the level of the horizontal tank floor. The apparent inclination is a result of an improperly aligned camera. The inlet configuration is slightly different from the experiments described in the text, and this image serves to provide only a qualitative impression of the flow.

PGCC14 9/12/08 3:27 PM Page 214

214


Fig. 10. Logarithmic plot of the sediment deposition rate in a constant-flux experiment, averaged from 260 s to 385 s after the start of the experiment (d16). The solid line shows the deposition rate based on the volume flux with jet entrainment between orifice and tank bottom, the dashed line shows the deposition rate with plume entrainment.

distances from the source, as is demonstrated by the finite extent xf of the current. However, it is clear that more experiments need to be carried out, including measurements of h and Q, to establish the exponential decay firmly and to allow a useful comparison between theory and experiment. These experiments are currently under way.

CONCLUSIONS We have applied a recently developed experimental technique to measure sediment deposition to laboratory experiments on turbidity currents in a long rectangular tank. The technique, based on measuring the vertical electrical resistance of the sediment layer, yields spatially and temporally resolved measurements of the sediment layer thickness under turbid flows. We have obtained detailed information about the temporal build-up of the sediment layer under a lockrelease turbidity current. The deposition from the head of the current was most important in determining the final sediment distribution, and the effects of the head will have to be included in further mathematical models of turbidity currents. Shortly after release, the rate of deposition is relatively low because of entrainment, which reduces the particle concentration. During the subsequent stage the deposition rate increases. A possible source of this increase is the change over from entrainment of ambient fluid to detrainment of interstitial fluid when the turbulence is reduced. A maximum in the final layer thickness is observed, to which the decreasing propagation velo-

city of the current also contributes. In the final stage the deposition reduces the particle concentration exponentially and the layer thickness quickly trails off at greater lengths. The experiments have also confirmed that on encountering a vertical end wall, the turbidity current is reflected and propagates in the opposite direction, depositing sediment on top of the layer formed earlier. Turbidity currents from a continuous release of a suspension were shown to reach a steady state soon after extending to their run-out length, with deposition rates decreasing away from the source. An exponential decay in the suspended particle concentration is suggested by the measurements, but further experiments are needed to establish the precise decay rate.

ACKNOWLEDGEMENTS The financial support from Yorkshire Water for this research project is gratefully acknowledged. We thank David Page-Croft for his help in designing and building the experimental equipment. We are grateful to J. Alexander, F. López and W. McCaffrey for useful comments on the draft of this paper.

NOMENCLATURE b dpart D E

tank width particle diameter sediment layer thickness electric field

PGCC14 9/12/08 3:27 PM Page 215

Deposition under turbidity currents h L0 Vs x xf z φ φ0 ρp

current depth lock length particle settling velocity length along the tank current run-out length height above the tank floor particle concentration initial particle concentration particle density REFERENCES

Alexander, J. & Morris, S. (1994) Observations on experimental, non-channelized high-concentration turbidity currents and variations in deposits around obstacles. J. sediment Res. 64, 899–909. Bennett, S.J. & Bridge, J.S. (1995) An experimental study of flow, bedload transport and topography under conditions of erosion and deposition and comparison with theoretical models. Sedimentology 42, 117–146. Best, J. & Ashworth, P. (1994) A high-resolution ultrasonic bed profiler for use in laboratory flumes. J. sediment Res. A64, 674–675. Best, J., Kirkbride, A. & Peakall, J. (2001) Mean flow and turbulence structure of sediment-laden gravity currents: new insights using ultrasonic Doppler velocity profiling. In: Particulate Gravity Currents (Eds W.D. McCaffrey, B.C. Kneller & J. Peakall) Spec. Publs. int. Ass. Sediment., No. 31, pp. 159–172. Blackwell Science, Oxford.

215

Bonnecaze, R.T., Huppert, H.E. & Lister, J.R. (1993) Particle-driven gravity currents. J. Fluid Mech. 250, 339–369. Bonnecaze, R.T., Huppert, H.E. & Lister, J.R. (1996) Patterns of sedimentation from polydisperse turbidity currents. Proc. R. S. London, Sel. A 452, 2247–2261. Edwards, D.A., Leeder, M.R., Best, J.L. & Pantin, H.M. (1994) On experimental reflected density currents and the interpretation of certain turbidites. Sedimentology 41, 437– 461. Ernst, G.G.J., Sparks, R.S.J., Carey, S.N. & Bursik, M.I. (1996) Sedimentation from turbulent jets and plumes. J. geophys. Res. 101 (B3), 5575–5589. Garcia, M. (1994) Depositional turbidity currents laden with poorly sorted sediment. J. Hydr. Eng. 120, 1240–1263. Gladstone, C., Phillips, J.C. & Sparks, R.S.J. (1998) Experiments on constant volume gravity currents: propagation and sediment deposition. Sedimentology 45, 833–843. Kneller, B.C., Bennett, S.J. & McCaffrey, W.D. (1997) Velocity and turbulence structure of density currents and internal solitary waves: potential sediment transport and the formation of wave ripples in deep water. Sediment Geol. 112, 235–250. Middleton, G.V. (1993) Sediment deposition from turbidity currents. Ann. Rev. Earth Planet. Sci. 21, 89–114. Pantin, H.M. & Leeder, M.R. (1987) Reverse flow in turbidity currents – the role of internal solitons. Sedimentology 34, 1143 –1155. Rooij, F. de, Dalziel, S.B. & Linden, P.F. (1999) Electrical measurement of sediment layer thickness under suspension flows. Exp. Fluids 26, 470 – 474.

PGCC15 9/12/08 3:26 PM Page 217

Field-based approaches


PGCC15 9/12/08 3:26 PM Page 219


Formation of large-scale shear structures during deposition from high-density turbidity currents, Grès d’Annot Formation, south-east France J . D . C L A R K and D . A . S T A N B R O O K Department of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK

ABSTRACT Sedimentary shear structures at the base of thick-bedded graded sandstone beds in the Grès d’Annot turbidite system of south-east France suggest that the deformation was related to the emplacement (transport and deposition) of the overlying thick-bedded sandstones. Examples presented in this chapter from the Trois Evêchés outcrop area, where such structures are well developed, range from rip-up clasts, clastic injection structures, folded beds, recumbent folded strata, and imbricate stacks composed of sandstone beds from the underlying deposits. Grain fabric analysis presented in this chapter suggests that the transport and deposition process of the overlying sandstones was largely that of a high-density turbidity current. We postulate that high shear stress at the base of flow caused the observed deformation and, locally at the seafloor–sediment interface, rapid flow freezing.

INTRODUCTION through time as various parts of the basin floor were successively infilled (e.g. Stanley, 1967; Bouma & Coleman, 1985; Elliott et al., 1985; Ravenne et al., 1987; Bouma, 1990; Pickering & Hilton, 1998). The southern Alps basin was 160-km long, 80-km wide and orientated NW–SE. The basin was primarily fed with sediment derived from the Corsica–Sardinian Massif (part of the Estérel-Thyrenide mountain chain) in the south and distributed northwards into the fully marine ‘nummulitic sea’, which extended into Germany. Other uplifted massifs (e.g. the Pelvoux massif and Argentéra-Mercantour massifs) provided sediment to other turbidite systems active at this time (Keunen et al., 1957; Stanley, 1961; Bouma, 1962; Stanley & Bouma, 1964). The sandstone bodies in the Grès d’Annot are typically sheet-like, interbedded with laterally extensive shale-rich units, and form dramatic onlaps onto topography created in the Marnes Bleues Formation. In addition, major channel-fill and scour-fill sandstone bodies and channel-levee systems have been identified (examples are described in Stanley, 1967; Bouma, 1988, 1990; Sinclair, 1994; Clark & Pickering, 1996; Clark & Good, 1998; Pickering & Hilton, 1998).

Geological background The Grès d’Annot (Annot Sandstone) is part of an Eocene – Oligocene succession that can be recognized throughout the Alpine foreland basins. This succession commences with deposition of a nummulitic limestone, followed by calcareous marls, and finally by deep-water sandstones. The Grès d’Annot outcrops largely in the southern part of the French Alps, between Nice and Barcelonnette. The deep-water clastic deposits of the Grès d’Annot turbidite system have been the focus of many studies, and were first interpreted to be the deposits of turbidity currents (Kuenen et al., 1957). The study of the Grès d’Annot in the Peïra-Cava area provided the description of the sequence of sedimentary structures known as the Bouma Sequence (Bouma, 1962). Subsequent work has integrated structural and sedimentological interpretations of the basin (Apps, 1987), and focused on detailed sedimentological and stratigraphic interpretations (Ravenne et al., 1987; Pickering & Hilton, 1998). The Grès d’Annot is up to 1200-m thick (Inglis et al., 1981), and is interpreted as having formed parts of a largely deep-water clastic system that developed


219

Barrême

Peyresq

Thorame Haute

La Valette

Annot

Colmars

Allos

Entrevaux

0

Guillaumes

St Antonin

20 km

Peïra-Cava

Contes

(Crystalline & Permian-Triassic Qiuartzite)

Argentera Massif

Basement Grès d'Annot Palaeocurrents ANNOT

Ventimiglia

FRANCE

Menton

ITALY

N

NICE

BARCELONNETTE

MAP AREA

GAP

Fig. 1. Geological map of the southern Alps basin (after Elliott et al., 1985; Apps, 1987). Palaeocurrents have been summarized from Sinclair (1994) and Bouma and Coleman (1985). Note the Trois Evêchés area in the NW of the map.

Digne

Barcelonnette

Trois Evêchés

EmbrunUbaye Thrust Sheets

Base Tertiary

Grès d'Annot

220

Montagne de Chalufy

Dormillouse

Sub-Brianconnais Thrust Sheets

PGCC15 9/12/08 3:26 PM Page 220

J.D. Clark and D.A. Stanbrook

PGCC15 9/12/08 3:26 PM Page 221

Large-scale shear structures A detailed programme of mapping and characterization of sandstone architecture and facies was undertaken in the Trois Evêchés area, Alpes de Haute Provence. During this study, a number of genetically related sedimentary shear structures were identified, and their description, and an interpretation of the depositional processes involved, form the subject of this chapter. The shear structures can be recognized in all parts of the Grès d’Annot, but several particularly well-exposed examples from the Trois Evêchés outcrop area are presented in this chapter. The Trois Evêchés area forms the central part of a 30-km section of continuous exposure between Montagne de Chalufy in the south and Dormillouse in the north (Fig. 1). The Grès d’Annot in this area is extremely well exposed around the flanks of the summit of Tête de l’Estrop (2961 m), from the onlap onto the Marnes Bleues Formation at its base, to the overlying Schistes à Blocs Formation at the foot of the Trois Evêchés summit. Facies descriptions The depositional sequence in the Trois Evêchés area comprises distinct packets of beds with different lithological and sedimentary characteristics. These depositional units have been mapped out in the area (Fig. 2). The clastic sequence is characterized by thick-bedded sandstone and conglomerate units, shale-rich horizons, and units comprising interbeds of largely medium-bedded sandstones and thin shales. Note that the majority of the units are continuous across the 1–2 km of continuous exposure in the area, with little change in thickness. The facies of these units are shown in the sedimentary log (Fig. 3). The thick-bedded units form ‘Barres’ (topographic ridges formed by the coarse-grained intervals) and are here given informal names for correlation purposes (see Figs 2 and 3). These units comprise amalgamated graded and non-graded sandstone and conglomeratic beds. The bases of the beds are nearly always erosive, and to some extent exhibit minor channel or scour surfaces, and the sands contain abundant shale rip-up clasts. Structures indicative of rapid dewatering, such as dishes, pillars and fluidization pipes, are almost ubiquitous in the sandstones of these thick-bedded units. Cross-lamination and low-angle lamination are relatively common in the coarse-grained sandstones. The shale-rich horizons are distinct packets of thinbedded sandstone and shale, but may contain one or two thicker sandstone beds within the sequence. These units are, in places, intensely bioturbated and the thin sandstones are commonly rippled. At least two of

221

these horizons in the area can be traced laterally to significant channel-fill sandstones, implying that these units may in part represent laterally equivalent levees of thick-bedded channel sandstones. The interbedded units comprise medium- and thickbedded graded and non-graded sandstones interbedded with shale, silty shale and packets of very-thin bedded sandstone and shale. Many of the sandstone beds show S1, S2 and S3 divisions of the high-density turbidite model proposed by Lowe (1982). The deposits in the Trois Evêchés area have previously been interpreted as accretionary lobe and interlobe deposits in an overall outer-fan setting (Elliott et al., 1985). The high degree of erosional events and the association of large-scale channel and levee facies in this succession suggest that these deposits accumulated in a variable sedimentary environment, analogous to a mid-fan setting in more fan-like systems. In this setting, sediment supply into this area fluctuated in a relatively abrupt manner to produce distinct, stacked depositional units which included amalgamated sandy and gravelly sheet sandstones with minor channels, rarer large-scale channels with shale-rich levees, and proximal parts of sheet-like lobes.

SEDIMENTARY SHEAR STRUCTURES Description Soft-sediment deformation structures are evident in many parts of the Grès d’Annot turbidite system. Some of these features, located in proximity to onlap surfaces, are clearly related to local slope instabilities resulting from topographic relief on the basin floor. Such deformation structures include slumps and slides at a variety of scales. There is, however, a subset of soft-sediment deformation structures observed throughout the Grès d’Annot exposures, which appear to have been produced by bed-parallel shear at the seafloor–sediment interface, caused by the deposition of overlying thick-bedded coarse-grained high-density turbidites. The deformation ranges from small-scale shale rip-ups (essentially type A3 rip-up clasts of Johansson & Stow, 1995), to complex ‘bulldozing’ of seafloor sediment. Commonly, material from the coarse-grained thick-bedded sandstone beds can be observed to have been forcibly injected downwards, forming clastic sills between partially attached interbedded sequences. This type of clastic injection process, associated with coarse-grained deep-water deposits, may be similar to that described by Beaudoin

tes eyt V s Le 0

500

Low San er Th dsto ick ne Beds

e

ton

nds

Sa

e

Ba rre Sa nd sto ne

"M Sa iddle" nds B ton arre e

Upper Thick Beds Sandstone

s re

Approximate line of measured section (Fig. 3)

Topographic ridges

Faults

Shale-rich unit (or interval of poor exposure)

Mixed thick-, medium- and thinbedded sandstone interval

Amalgamated sandstone and pebbly sandstone unit

ck hi e eT ton l d id s M and S

Fig. 2. Map of the Trois Evêchés/Tête de l’Estrop outcrop area, showing the correlatability of distinct lithostratigraphic units.

e

ton

nds

Sa

6m

291

ton

ds

an

pS

tro

Es

Gr os se

ds

Cr te de Cha br i

tes

Ve yt

Chabri res Sandstone

2916 m

Tête de l'Estrop (2961m)

P Sa etite nd B sto ar ne re

2798 m

Be

Marnes Bleues

T te de Chabri res

S

2802 m

222

2927 m

Est

r op

n

ti o ec

N

PGCC15 9/12/08 3:26 PM Page 222


PGCC15 9/12/08 3:26 PM Page 223

Grosse Barre Sst

300 m

Petite Barre Sst

Upper Thick-Beds

2916 m Sst

600 m

100 m

Middle Thick-Beds

Estrop Sst

Middle Barre Sst

400 m

200 m

Lower Thick-Beds

Grosse Barre Sst

500 m

0m

sh. vf f m c vc gran. Lithostratigraphic Units Amalgamated sandstone and pebbly sandstone unit Mixed thick-, medium- and thinbedded sandstone interval Shale-rich unit (or interval of poor exposure)

sh. vf f m c vc gran.

Lithology

Base of Bed

Sandstone

Planar

Shale

Erosive

Thin-bedded sands and shale

Loaded

Poor Exposure

Amalgamated

sh. vf f m c vc gran. Sedimentary Structures Planar Stratification Cross Stratification Dish Structures Pipe Structures

sh. vf f m c vc gran. Palaeocurrents

Granules/ dispersed granules Large clasts/ deformed beds Bioturbation Current ripples Shale Clasts

Fig. 3. Sedimentary log through part of the Trois Evêchés section. See Fig. 2 for location of section.

Palaeocurrent orientation indicators Palaeocurrent direction indicators

PGCC15 9/12/08 3:26 PM Page 224

224


TB

R

I

U

U

Fig. 4. Photograph of a thick-bedded turbidite sandstone with large-scale deformation of underlying sediments. The coarse sand and granule-grade sediment from the thick-bedded turbidite (TB) can be seen injected (I) between interbedded undeformed sandstone and shale layers (U). The thick-bedded turbidite has caused local erosion and rotation (R) of the underlying sediments.

et al. (1985), and occurs during sedimentation, rather than from post-depositional burial and compaction. A progression of this type of deformation can be seen in Fig. 4, where interbedded sandstones and shales have been partially lifted up by the emplacement of the thick-bedded sandstone. A range of deformation structures is evident at one particular stratigraphic horizon, which has been studied in detail in order to assess the depositional processes involved in producing these deformation structures. Shear structures and deformation of previouslydeposited beds can be seen at the base of a thickbedded sandstone unit (named here the ‘Lower Thick Beds’) which can be traced for over 3 km in the Trois Evêchés area. Over all of this distance, the unit consists of four thick, graded sandstone beds separated by thin, discontinuous intervals of silty shale and very thin-bedded turbidites and shows very little change of thickness. Principal grain-size components in the thick beds range from coarse sand to granule grade, with varying amounts of finer-grained matrix. All four beds are characterized by an overall upwards-fining grain size, but contain subtle, discontinuous grain-size breaks within each bed. The base of the unit is characterized by unusually large intraclasts of a distinctive 45-cm-thick turbidite which occurs in situ immediately beneath the thick-

bedded unit. At various locations, this underlying sandstone bed has been removed, deformed and incorporated into the basal part of the thick-bedded unit. Soft-sediment deformation structures observed in this interval include recumbent folds and imbricate stacks (Fig. 5). The sense of shearing in the fold and imbricate stack structures are consistent with palaeocurrent direction. In terms of understanding the depositional processes which caused the deformation, we can say that the deformation occurred in the very last stages of sediment transport, as many of the deformation structures show minor amounts of lateral translation, and relatively complete beds are preserved as clasts at the base of overlying thick-bedded sandstones. The preservation of complete turbidite sandstone beds,

Fig. 5. (opposite) Examples of large-scale shear structures. (a) Photograph of imbricate clasts of a sandstone bed which lies in situ immediately below the thick-bedded granular sandstone at the base of the ‘Lower Thick Beds’ interval. The upper clast is inverted (as shown by the way-up arrows). The sense of shear is consistent with palaeocurrent direction (towards the left). (b) Photograph of four imbricate slices of the same sandstone bed observed at the base of the ‘Lower Thick Beds’ interval. The way-up of the clasts is indicated by the arrows. The sense of shear is consistent with palaeocurrent direction (towards the left).

PGCC15 9/12/08 3:26 PM Page 225

Granular sandstone

In situ

(a)

Granular sandstone

(b)

Shale No Exposure

0

0.5 1 m

IN SITU BED

Fig. 5. (contd.) (c) Photograph of part of a complex imbricate stack of inverted and right-way-up clasts, all of which are from the same turbidite sandstone bed, which can be found in situ just to the right of this photograph. (d) Sketch of the deformation structure shown in (c), showing the way-up of the clasts (where identified) and the in situ sandstone bed to the right. The sense of shearing in this structure is towards the left, consistent with palaeocurrent direction.

(d)

Granule grade sandstone (thick-bed)

Right Way-up Sst. Inverted Sst.

226

(c)

PGCC15 9/12/08 3:26 PM Page 226


PGCC15 9/12/08 3:26 PM Page 227

Large-scale shear structures shale layers and interbedded sequences within the thick sandstones suggest that the near-surface seafloor sediments were relatively consolidated. Localized downward injection of coarse-grained sand between underlying beds, along planes that have been deformed immediately upcurrent, suggest that the deformation is a consequence of the transport and deposition of the overlying thick-bedded sandstones, rather than a slump that has been partially eroded and overlain by subsequent deposits. We can only attempt to piece together the geometry and kinematics of the complex deformation structures observed (e.g. Fig. 5) by inferring that the structures described in the chapter represent a possible continuum of genetically related deformation processes. The continuum of deformation structures related to the emplacement of high-density flows includes simple substrate erosion, substratum injection of coarsegrained material from large-volume flows and the development of imbricate slices of the substratum. Development of the imbricate stacks may have involved formation and subsequent shearing of recumbent folds in the sea-floor sediments. This would result in an idealized sequence of alternating right-way up, and inverted, slices. An alternative process involves progressive overturning and shearing at the downcurrent margin of an erosion surface in the underlying beds, forming individual slices which were transported over short distances at the base of the flow, until coming to rest to form imbricate stacks. This would ideally result in a sequence of overturned slices. The imbricate slices at the base of the thick-bedded sandstones shown in Fig. 5 comprise predominantly inverted slices of the underlying turbidite sandstone, although some of the slices are the right-way up. We therefore suggest that both processes are equally probable. In addition, since there may have been erosion of these structures at the base of the rapidly depositing flows, coincident with their formation, the present-day outcrops may not be fully capable of resolving the kinematics of the deformation in more satisfactory detail. The rotation and peel back of underlying beds is unlikely to happen at the base of a high density turbidity current without the substrate being at least semi-consolidated. There has been little work on establishing the mechanics of deformation in recently deposited material. We suspect that the interbedded nature of the substratum is an important feature in the formation of some of the deformation structures. In addition, the thin shale layers may have been important décollement surfaces in the formation of the stacked imbricate slices. Further experimental

227

work and field observations in this area may substantiate our explanations of the soft deformation phenomena described above. The deformation structures described above are related to the transportation and depositional processes of the thick-bedded sandstones. The following part of this chapter, therefore, focuses of the description and process interpretation of the thick-bedded sandstones that overly the deformation structures that we describe above.

THICK-BEDDED SANDSTONE FACIES The evidence presented above suggests that the deformation structures are genetically related to the emplacement of the thick-bedded sandstones. The question remains, however, as to what processes were involved in the transportation and deposition of the thick-bedded sandstones, and which of them caused this deformation in the final stages of deposition. Detailed petrographic and natural gamma-ray studies of the thick-bedded sandstones were undertaken to aid field observations in establishing the processes involved in the deposition of the thick-bedded sandstones and their related deformation of the sea floor. Grain fabric Grain fabric studies in deep-water clastic sediments have proved useful in evaluating the depositional processes, at least during the final stages of deposition (e.g. Parkash & Middleton, 1970; Walker, 1975; Hiscott & Middleton, 1980). Hiscott and Middleton (1980) show from fabric analysis of sandstone beds from the Ordovician Tourelle Formation, that grains from turbidite sandstones have a strong a-axis fabric, parallel to flow direction, and imbrication angles of approximately 10° or less. Arnott and Hand (1989) show from flume-tank experiments that the steepness of imbrication angle increases with rate of sediment fallout. Clasts in debris flow deposits show no preferred orientation, wide dispersion of orientations, including high orientations, with a mean around a horizontal aaxis fabric (Lindsay, 1968). Hiscott et al. (1997) state that from outcrop observations, debris flows have poorly organized or polymodal a-axis fabrics. In order to characterize the grain fabric of the thickbedded sandstone units, ten orientated samples were collected from the sandstone bed at the base of the Lower Thick Beds Unit, from a location that did not incorporate deformed beds. From each sample, two

x. ( qu art z)

grain dimensions (mm) (of largest quartz grains)

.25 .5 .75 1 1.25 1.5 1.75 2 2.25 2.5

270

225

180

a-axis orientation (palaeocurrent alignment)

315

(c)

0

50

100

150

200

250

N 264˚

0

-50

084˚

imbrication angle

50

(d)

0

50

100

150

200

250

300

height above base (cm)



(e)

0

50

100

150

200

250

300

0

vector magnitude

10 % 20% 30% 40% 50% 60% 70%

a-axis imbrication

a-axis orientation

Fig. 6. Grain fabric analysis from bed 1 in the ‘Lower Thick Beds Unit’. (a) Sedimentary log with the location of the samples used in the analysis. (b) Long-axis length of the largest grains of each sample. (c) Variation of apparent a-axis fabric measured in thin sections cut parallel to the bedding. This shows the variability in local current direction during the deposition of the bed. (d) Variation in apparent a-axis imbrication angle derived from thin sections cut perpendicular to bedding, in an orientation parallel to mean palaeocurrent direction (264°). Mean palaeocurrent was determined from sole marks on the base of the bed. The graph shows imbrication angle increases upwards through the bed, and then decreases. At the top of the bed, the imbrication angle is reversed (see text for discussion). (e) Vector magnitude of the a-axis and imbrication angle data. This plot shows that grain alignment in both the a-axis fabric and the imbrication angle is generally well developed at the base of the bed, poorer in the centre of the bed, but with a general upwards increase in the development of the fabric.

(b)

file

pro

(a)

0

ma ize

0

ns

i gra

50

100

150

200

250

300

228

location of samples


300

PGCC15 9/12/08 3:26 PM Page 228


PGCC15 9/12/08 3:26 PM Page 229

Large-scale shear structures thin sections were cut: one in the plane parallel to bedding (horizontal section), and a section perpendicular to bedding plane (vertical section), cut parallel to the mean palaeocurrent direction ascertained from sole structures. The length and orientation of the apparent long and short axes of the largest quartz grains in each thin section were measured. In the orientation and imbrication analysis, only those grains with an aspect ratio higher than 1.25 were considered. Grain-size distribution In each sample, the apparent long axis of at least 80 of the largest quartz grains was measured. The box plot in Fig. 6b shows the skewed distribution of grain sizes from each sample. The maximum grain size and the upper 5% of the sampled population show a general decrease in quartz grain long axis length from the base to the top of the bed. The median and minimum values shown by the box plot are not statistically representative, since only the longest quartz grains were measured. These results provide quantitative evidence of a near-perfect fining-up profile of maximum grain size, implying deposition from a waning current. Grain orientation relative to palaeocurrent direction Apparent a-axis (i.e. long axis) orientations were measured from thin sections cut in the plane parallel to bedding ( horizontal sections). Only the largest quartz grains, with an aspect ratio greater than 1.25, were used in this analysis. The mean a-axis orientation was plotted against height within the bed and as rose diagrams depicting the spread of data (Fig. 6c). The calculated vector magnitudes for these orientation data range from 19% to 48% (Fig. 6e). Mean a-axis orientation in each sample varies from 012°/192° to 122°/302°, with a mean for the whole bed of 086°/266°. Most studies of grain orientation in turbidites demonstrate a preferred a-axis orientation aligned parallel to palaeocurrent direction (e.g. McBride, 1962; Parkash & Middleton, 1970 and references therein; Hiscott & Middleton, 1980). Other studies have shown that some turbidites can show a preferred grain orientation perpendicular to sole structures (Basset & Walton, 1960; Bouma, 1962; Ballance, 1964) and also oblique to sole structures (Spotts, 1964; Spotts & Weser, 1964; Scott, 1967). In the samples from the bed in this study, the mean a-axis orientation (086°/266°) is essentially parallel or sub-parallel to the mean palaeocurrent directions measured from sole marks (264°). The mean orientation data from the ten samples throughout the

229

bed suggest that, throughout deposition, the flow was being deflected, and switched direction at least four times. The likely cause of this variability is probably related to flows interacting with sea-floor topography. The a-axis orientation data show that the magnitude of palaeocurrent swing in the bed decreases with each successive switch of direction. Imbrication Imbrication of grain a-axes is also a common feature of turbidite deposits, although the degree of imbrication and the angle of imbrication may vary from base to top in individual beds. Hiscott & Middleton (1980) found imbrication from 30 samples taken from turbidites in the Tourelle Formation ranging from 16° to 65°, with a mean of 30°. Turbidites from the Miocene Topanga Formation show a range of imbrication angle from 6° to 22°, with a mean of 14° (Spotts & Weser, 1964). Onions & Middleton (1968) found that 80% of thin sections from turbidites from the Normanskill formation showed an imbrication between 15° and 25°. All the samples from the Lower Thick Beds showed preferential apparent a-axis orientation in the vertical thin sections cut in a plane parallel to mean palaeocurrent. The calculated vector magnitudes for imbrication data ranges from 30% to 68% (see Fig. 6e). The angle and direction of imbrication, however, varied upwards throughout the bed. Most of the bed shows an upcurrent-dipping imbrication, but the upper 20% of the bed shows a horizontal to downcurrent-dipping preferred a-axis orientation (Fig. 6d). The upcurrentdipping section has an imbrication angle of 11°–29° with a mean of 16°, while in the upper part of the bed, the angle of preferred a-axis orientation varies from 0° to 27° (dipping downcurrent). In all samples, there is a general pattern showing an improving definition (i.e. higher vector magnitudes) as imbrication angle increases. There is also a trend showing increased confidence in palaeocurrent orientation (i.e. increase in vector magnitude of a-axes in the horizontal sections) with increased angle of imbrication. Commonly, grain fabric analysis of turbidite beds shows natural upcurrent-dipping imbrication, but downcurrent-dipping imbrication, as seen in the uppermost sample taken from the sampled bed has also been observed. Negative imbrication in parts of turbidite sandstones has been documented from the Polish Carpathian Flysch (Dzulynski & Slaczka, 1958), the Cretaceous of the Coast Ranges, western California (Colburn, 1968), Grès d’Annot at PeïraCava (Bouma, 1962) and the Cloridorme Formation

PGCC15 9/12/08 3:26 PM Page 230

230


(Parkash & Middleton, 1970). Colburn (1968) proposed that the reversal of imbrication can be explained by a backwash in current from a reflecting slope, similar to the turbidity current flow reflectance process described by Pickering & Hiscott (1985). There are no sedimentary structures at the top of the Lower Thick Beds, however, to support the idea of flow reversal from a reflected current. Hiscott and Middleton (1980) found ‘negative’ downcurrent-dipping imbrication in 24% of samples taken from structureless sandstones from the Tourelle Formation. This differs from the samples taken from stratified sandstones, which showed a consistent upcurrent-dipping imbrication. Furthermore, downcurrent and shallow imbrication samples were largely found in the middle portions of the structureless sandstones from the Tourelle Formation, and negative imbrications were found to be characterized by poor preferred a-axis orientations in plan. In contrast, however, the sandstone beds of the Lower Thick Beds Unit are not structureless and contain horizontal lamination, and dish structures. The cause of the downcurrent dipping imbrication seen in the uppermost sample at the top of the thickbedded sandstone remains unclear, but one possibility is that it resulted from grain fabric disruption caused by escaping fluid during fluidization of the bed immediately following deposition. Gamma profile Figure 7 shows a photograph of part of the Lower Thick Beds with natural gamma-ray logs superimposed. The gamma-ray logs were collected using a

portable gamma-ray spectrometer. Total counts per minute were recorded from two vertical profiles (sample spacing of 20 cm) and a horizontal profile (sample spacing of 50 cm). The vertical profiles (survey lines 1 and 2) clearly show the shale and siltyshale interbeds between the main sandstone beds of the thick-bedded unit. The contact between bed 1 and bed 2 contains an interval of shale at section 1 and no shale at section 2. Both profiles, however, show an increase in the gamma-log at the boundary. The vertical profiles of individual thick beds (beds 1–4) show an overall upward decrease in the total counts recorded. This may be due to the increased upward sorting within each of the thick beds, resulting in a decrease of fine-grained clay material in the sandstone matrix. These data support visual outcrop observations of the degree of sorting throughout the bed. The considerable variability in the horizontal gamma-log, taken at a constant height within bed 3, is possibly a result of numerous ‘floating’ shale clasts observed within this part the bed. The shale clasts become more prominent towards the right of the photograph. Process interpretation The sandstone beds of the Lower Thick Beds Unit (beds 1–4 in Fig. 7) are coarse-sand to granule grade, with varying amounts of finer-grained matrix. All four beds show a general upwards fining in grain size with subtle, discontinuous grain-size breaks (see Fig. 6a), a preferred a-axis orientation in plan and, in all but one sample, a relatively steep angle of a-axis imbrication. The vertical gamma-log profiles suggest an upwards

14 12 10

Bed 2

m 8 6 4

Bed 1

Survey line 3

3000 Total counts/ min.

Survey line 1

Bed 3

4000

Survey line 2

Bed 4

D

2 0 3000

4000

5000

Total counts/min.

6000

U

3000

4000

Total counts/min.

Fig. 7. Photograph of part of the ‘Lower Thick Beds’ interval with vertical and horizontal gamma logs overlain. Note the deformed imbricate stacked slices (D) of the distinctive turbidite bed that can be found undeformed (U) in situ immediately below the ‘Lower Thick Beds’ interval. The gamma profiles clearly show an increase in the number of total counts recorded where the thin shale or interbedded very thin-bedded turbidites and shale intervals occur. Note also the upward decreasing profiles for individual beds in the majority of the gamma profiles (see text for discussion).

PGCC15 9/12/08 3:26 PM Page 231

231

Large-scale shear structures increase in sorting within each of the thick beds. These observations are consistent with the deposition of suspended material from a waning, high-density turbidity current. The lowermost sandstone bed of the Lower Thick Beds Unit commonly contains large coherent clasts of the underlying deposits. These deformed beds may be overturned (as indicated by reverse grading, sole structures on the upper surface, and ripple structures), folded, or arranged stacked in imbricate stacks. The observed field relationship between these deformation structures and the overlying deposits strongly suggest that the deformation resulted from, and occurred during, the deposition of the overlying high-density turbidite deposits (see Figs 4 and 5). A possible mechanism for this deformation is basal shear resulting from high concentration bedload layers (traction carpets) at the base of high-density turbidity currents (Lowe, 1982; Hiscott & Middleton, 1980; Sohn, 1997). During deposition, mean flow velocity decreases, dampening turbulence and increasing current density at the base of the bed, which results in a high concentration inertia layer at the base of flow. Immediately prior to deposition transition from turbulent to laminar flow occurs near the base of the active flow resulting in increased shear along the seabed. The increased shear in turn causes deformation of the underlying beds, incorporation of underlying sediment as clasts within the flow, and further rapid deceleration of the flow (flow freezing). The range in type and style of shear structures observed beneath thick-bedded turbidite sandstones elsewhere in the Grès d’Annot, suggest that this process may be a relatively common deformation process associated with the deposits of highdensity turbidity currents.

CONCLUSIONS The Trois Evêchés outcrop area consists of a sequence of distinct sedimentary units comprising amalgamated sheet sandstones with minor channels, and rarer largescale channels with laterally equivalent shale-rich levee units. In some of the thick-bedded sandstones of the amalgamated sheet sandstone units, there is evidence to suggest that the deposition of these beds caused largescale deformation of underlying beds, producing a suite of deformation structures involving relatively undeformed clasts of interbedded sandstone and shale. The deformation is likely to have occurred in the final stages of flow transportation, since the large intraclasts

are relatively well preserved. Outcrop observations, confirmed by grain fabric studies of thin sections from an individual thick-bedded sandstone unit, suggest that the principal sediment transport was by highdensity turbidity current processes, rather than debris flow processes. The grain fabric analysis showed a general upward-fining profile, preferential long-axis alignment subparallel to palaeocurrent direction, and inclined grain imbrication, all of which are typical of deposition from turbidity currents. In addition, the gradual upwards reduction in natural gamma-ray emission, as shown by the gamma profiles from these thick-bedded sandstones, suggests that there is an upward reduction in the proportion of clay in the sandstone matrix, due to improved sorting. Upwardimproved sorting has been noted in turbidites where the basal parts of the beds have been rapidly deposited with little grain size sorting, whereas the upper parts of the beds have been deposited under more stable flow conditions, allowing better grain sorting. It is proposed that the complex deformation of the underlying beds was caused by the high shear stress between sediment transported in the high-density flow and the seabed. Although we cannot determine the absolute shear stress or flow densities from the rock record, the variability in scale and type of deformation reflects variable flow velocity and density of turbidity currents in this part of the basin.

ACKNOWLEDGEMENTS This work presented in this chapter was undertaken during field studies in the Trois Evêchés area for the Genetic Units Project at Heriot-Watt University, and we are grateful to the following companies for their support: Amerada Hess, Amoco, Arco, BG, Chevron, Conoco, Exxon, and Mobil. Some of the ideas presented in this chapter developed from discussions in the field with representatives from the companies mentioned above. The authors are also very grateful for field assistance and contributions in the field from Caroline Hern, Andy Gardiner, Tim Good, and Jamie Pringle. Andy Gardiner, Dorrik Stow, and Alex Maltman are thanked for their contributions in reviewing and improving earlier drafts of the manuscript.

REFERENCES Apps, G. (1987) Evolution of the Grès d’Annot Basin, SW Alps. Ph.D. thesis. University of Liverpool.

PGCC15 9/12/08 3:26 PM Page 232

232


Arnott, R.W.C. & Hand, B.M. (1989) Bedforms, primary structures and grain fabric in the presence of suspended sediment rain. J. sediment. Petrol. 59, 1062–1069. Ballance, P.F. (1964) Streaked-out mud ripples below Miocene turbidites, Puriri Formation, New Zealand. J. sediment. Petrol. 34, 91–101. Basset, D.A. & Walton, E.K. (1960) The Hell’s Mouth Grits, Cambrian greywacke in St. Tudwal’s Peninsula, North Wales. Q. J. geol. Soc. London 116, 85 –110. Beaudoin, B., Friès, G., Parize, O. & Pinault, M. (1985) L’origine des injections sableuses: les sills et dykes albiens du Ravin de la Baume, Bevons (Alpes de Haute-Provence). Comptes Rendus 301, Série II, 407– 410. Bouma, A.H. (1962) Sedimentology of Some Flysch Deposits: A Graphic Approach to Facies Interpretation. Elsevier, Amsterdam. Bouma, A.H. (1988) Migratory and braided channels in Eocene submarine fan deposits in the French Maritime Alps. Am. Ass. Petrol. Geologists Bull. 72, 990. Bouma, A.H. (1990) Clastic depositional styles and reservoir potential of Mediterranean basins. Am. Ass. Petrol. Geologists Bull. 74, 532–546. Bouma, A.H. & Coleman, J.M. (1985) Peïra-Cava Turbidite system, France. In: Submarine Fans and Related Turbidite Systems (Eds Bouma, A.H., Normark, W.R. & Barnes, N.E.), Springer-Verlag, New York, pp. 217–222. Clark, J.D. & Pickering, K.T. (1996) Submarine Channels: Processes and Architecture. Vallis Press, London, 231 pp. Clark, J.D. & Good, T.R. (1998) Geometric variability within deep-water clastic sheet-systems: an example from the Annot Sandstone, SE France. Annual AAPG Convention, Salt Lake City, Utah, AAPG Extended Abstracts 1, No. A126. Colburn, I.P. (1968) Grain fabrics in turbidite sandstone beds and their relationship to sole mark trends on the same beds. J. sediment. Petrol. 38, 146 –158. Dzulynski, S. & Slaczka, A. (1958) Directional structures and sedimentation of the Krosno Beds (Carpathian flysch). Ann. Soc. Géol. Pologne 28, 205 –259. Elliott, T., Apps, G., Davies, H., Evans, M., Ghibaudo, G. & Graham, R.H. (1985) A structural and sedimentological traverse through the Tertiary foreland basin of the external Alps of southeast France. In: Field Excursions Guidebook for the International Association of Sedimentologists Meeting on Foreland Basins, Fribourg (Eds Allen, P.A. & Homewood, P.), Blackwell Scientific Publications, Oxford, pp. 39–73. Hiscott, R.N. & Middleton, G.V. (1980) Fabric of coarse deep-water sandstones, Tourelle Formation, Quebec, Canada. J. sediment. Petrol. 50, 703–722. Hiscott, R.N., Pickering, K.T., Bouma, A.H., Hand, B.M., Kneller, B.C., Postma, G. & Soh, W. (1997) Basinfloor fans in the North Sea: sequence stratigraphic model vs. sedimentary facies: discussion. Am. Ass. Petrol. Geologists Bull. 81, 662–665. Inglis, I., Lepvraud, A., Mousset, E., Salim, A. & Vially, R. (1981) Étude sédimentologique des Grès d’Annot (région de Colmars les Alpes et du Col de la Cayolle. ENSPM Rèf. 29765. Johansson, M. & Stow, D.A.V. (1995) A classification scheme for shale clasts in deep water sandstones. In:

Characterization of Deep Marine Clastic Systems (Eds Hartley, A.J. & Prosser, D.J.), Spec. Publ. Geol. Soc. London 94, 221–241. Kuenen, Ph.H., Faure-Muret, A., Lanteaume, M. & Fallot, P. (1957) Observations sur les flyschs des Alpes Maritimes Française et Italiennes. Bull. Soc. Géologique France 6, 11–26. Lindsay, J.F. (1968) The development of clast fabric in mudflows. J. sediment. Petrol. 38, 1242–1253. Lowe, D.R. (1982) Sediment gravity flows: II. Depositional models with special reference to the deposits of highdensity turbidity currents. J. sediment. Petrol. 52, 279– 297. McBride, E.F. (1962) Flysch and associated beds of the Martinsburg Formation (Ordovician), Central Appalachians. J. sediment. Petrol. 32, 39–91. Onions, D. & Middleton, G.V. (1968) Dimensional grain orientation of Ordovician turbidite greywackes. J. sediment. Petrol. 38, 164 –174. Parkash, B. & Middleton, G.V. (1970) Downcurrent textural changes in Ordovician turbidite greywackes. Sedimentology 14, 259–293. Pickering, K.T. & Hiscott, R.N. (1985) Contained (reflected) turbidity currents from the Middle Ordovician Cloridorme Formation, Quebec, Canada: an alternative to the antidune hypothesis. Sedimentology 32, 373–394. Pickering, K.T. & Hilton, V.C. (1998) Turbidite Systems of Southeast France. Vallis Press, London, 229 pp. Ravenne, C., Vially, R., Riche, P. & Tremolieres, P. (1987) Sedimentation et tectonique dans le bassin marin Eocene superieur-Oligocene des Alpes du sud. Rev. l’Institute Francais Petrole 42, 529–553. Scott, K.M. (1967) Intra-bed paleocurrent variations in a Silurian flysch sequence, Kirkcudbrightshire, Southern Uplands of Scotland. Scot. J. Geol. 3, 268–281. Sinclair, H.D. (1994) The influence of lateral basinal slopes on turbidite sedimentation in the Annot sandstones of SE France. J. sediment. Res. 64, 42–54. Sohn, Y.K. (1997) On traction-carpet sedimentation. J. sediment. Res. 67, 502–509. Spotts, J.H. (1964) Grain orientation and imbrication on Miocene turbidity current sandstones, California. J. sediment. Petrol. 34, 229–253. Spotts, J.H. & Weser, O.E. (1964) Directional properties of a Miocene turbidite, California. In: Turbidites (Developments in Sedimentology, 3) (Eds Bouma, A.H. & Brouwer, A.). Elsevier, Amsterdam, pp. 199–221. Stanley, D.J. (1961) Etudes sedimentologiques des Grès d’Annot et de leurs equivalents lateraux. Rev. l’Institute Francais Petrole 16, 1231–1254. Stanley, D.J. (1967 ) Comparing patterns of sedimentation in some modern and ancient submarine canyons. Earth planet. Sci. Lett. 3, 371–380. Stanley, D.J. & Bouma, A.H. (1964) Methodology and palaeogeographic interpretation of flysch formations: a summary of studies in the Maritime Alps. In: Turbidites (Developments in Sedimentology, 3) (Eds Bouma, A.H. & Brouwer, A.). Elsevier, Amsterdam, pp. 34 –64. Walker, R.G. (1975) Generalized facies model for resedimented conglomerates of turbidite association. Bull. geol. Soc. Am. 86, 737–748.

PGCC16 9/12/08 3:24 PM Page 233


Subaerial liquefied flow of volcaniclastic sediments, central Japan K. NAKAYAMA Department of Geoscience, Shimane University, Matsue, 690-8504 Japan

ABSTRACT Subaerial liquefied flow of reworked volcaniclastic sediments is identified by analysis of grain-size distributions and settling velocities, combined with field observation of mudflow deposits of the Pliocene Souri tephra bed within the fluvial Tokai Group, central Japan. The tephra consists predominantly of volcanic glass particles and pumice grains. Subaerial liquefied flow has previously been considered impossible, but this study shows it may occur when grains have relatively low settling velocities due to their surface roughness and/or low density. Experimentally determined settling velocities for these tephra grains show that the transition from laminar to turbulent flow is promoted at a lower grain Reynolds number than for smooth spherical particles of equivalent volume. The typical flow unit is divided into subunits A, B and C, in ascending order. Grains in subunit A were supported under hydraulic settling equivalence, and hence both glass particles and pumice grains having the same settling velocity were deposited simultaneously from suspended sediment dispersion. The grain-support mechanism of subunit B is autofluidization, because the superficial escape velocity of fluid released from subunit A can be greater than superficial fluid velocity at minimum fluidization for grains in subunit B. Subunit C consists of comparatively finer grains, which are elutriated from the fluidized part, and deposited as suspension fallout. This is the first quantitative and qualitative description of subaerial liquefied flow, and the succession described can be considered representative of such deposits.

INTRODUCTION This paper examines the Pliocene Souri tephra bed in Japan, and determines that some parts of the tephra bed are subaerial liquefied flow deposits. This conclusion is based on qualitative outcrop observation and quantitative physical properties. Lowe (1976) concluded that autofluidization of uniformly sized sediment is impossible, because the fluid escape velocities are never greater than minimum fluidization velocities. His conclusion was drawn from non-volcaniclastic grains. The results discussed in this paper, however, indicate that autofluidization can occur in the case of particular volcaniclastic grains. With respect to the terminology in this paper, hydraulic settling equivalence exists when grains having the same terminal settling velocity are deposited simultaneously (Rubey, 1933; Komar et al., 1984). Dispersive pressure equivalence indicates that grains are supported by the same intergranular dispersive pressure. These terms are used in the same sense

Volcaniclastic deposits are formed by a variety of sedimentary processes and mechanisms (e.g. Nakayama & Yoshikawa, 1997), and several texts provide comprehensive descriptions of such sediments (e.g. Fisher & Schmincke, 1984; Cas & Wright, 1987). However, many problems and controversies remain in volcaniclastic sedimentology. Among studies of volcaniclastic sedimentology, the studies based on grain-size distribution and grain density are fundamental (e.g. Walker, 1971), because they provide some quantitative physical data. Cole and Stanley (1994), Nakayama et al. (1996a) and Nakayama (1997) discuss the relevance to the grain-support mechanism of hydraulic settling equivalence and grain dispersive pressure equivalence. These criteria are important, but are insufficient for fully analysing grain-support mechanisms. Grain-support mechanisms also include fluidization, liquefaction, matrix buoyancy and other mechanisms (Lowe, 1982).


233

PGCC16 9/12/08 3:24 PM Page 234

234

K. Nakayama

as Cole and Stanley (1994) and Nakayama (1997). Liquefaction and fluidization are used as defined by Lowe (1976). Liquefaction occurs when a loosely packed sediment collapses, the grains temporarily losing contact with each other, and settling within their own pore fluid. The particles fall a short distance, fluid is displaced upward, and a more tightly packed grainsupported framework is established. Fluidization occurs when a fluid forced vertically through loose sediment exerts an upward drag force on the grains equal to the downward force of gravity. Autofluidization can occur only when escaping pore fluids resulting from liquefaction can fully support grains in the flow.

SOURI TEPHRA BED The Souri tephra bed occurs within the fluvial MioPleistocene Tokai Group (Nakayama, 1994, 1996). In some outcrops, the tephra bed directly overlies flood deposits with immature paleosols. The tephra bed is a marker-tephra, distributed over a distance of more than 40 km (Fig. 1). The source volcano of the tephra is inferred to lie 50 to 100 km north of the most proximal Souri tephra bed (Nakayama et al., 1996b; Kurokawa et al., 1998). Correlative tephra beds are recognized in areas on the Japan Sea side of Honshu, suggesting that the Souri tephra and its correlatives are traceable over an area of 8 × 104 km2. Nakayama and Furusawa (1988) give outcrop descriptions and petrologic characteristics of the tephra. The Souri tephra bed is white to pinkish white in colour, and varies from 0.3 to 3.8 m in thickness. More than 98% of the tephra grains consist of volcanic glass particles and pumice grains, and less than 2% are mineral fragments. The glass particles consist of fibrous shards with a refractive index of 1.4979 to 1.4997. Among the mineral fragments, biotite, amphibole, quartz and feldspar particles constitute more than 70%, with subordinate pyroxene, zircon, and opaque mineral particles comprising the remainder. The depositional processes of the Souri tephra bed have been described by Nakayama et al. (1996b), using facies analysis. Six volcaniclastic sedimentary facies are recognized, which are interpreted as pyroclastic fall and reworked (mudflow, channel fill, flood flow, swamp, and lacustrine) deposits. Nakayama et al. (1996b) concluded that the depositional processes comprised six stages: first and second eruptions of pyroclastic fall constitute stages I and II, mudflow in stage III, fluvial reworking in stage IV, a third eruption in stage V, and final fluvial reworking in stage VI.

The pyroclastics and volcaniclastics in all stages were deposited under subaerial conditions. The stage III ‘mudflow’ deposits of Nakayama et al. (1996b) are the subject of this chapter (Fig. 1). According to Nakayama et al. (1996b) and additional observation, ‘mudflow’ deposits are confined to a small area less than 4 km2. These deposits consist of multiple units of poorly sorted ash and lapilli-ash. The lapilli-grade material consists of pumice grains. Each unit of the facies can be divided into subunits A, B and C in ascending order (Figs 1 and 2). Subunit A is a poorly sorted massive unit of comparatively coarse grains, ranging from 19 to 56 cm in thickness. Subunit A frequently displays coarse-tail grading. Subunit B varies in thickness from 15 to 51 cm, and is similar to subunit A, but consists of slightly finer grains and contains dish structures. Subunit C consists of markedly finer grains, and is less than 6-cm thick. In the lowermost flow unit of stage III in Fig. 1, the thickness of subunit A increases westward, while that of subunit B decreases in the same direction. Despite these variations in subunits A and B, the total thickness of each ‘mudflow’ unit (subunits A + B + C) is roughly constant at 66 to 68 cm. Correlation of all units between four sections is difficult, because two to five flow units occur at individual outcrops. However, the lowermost flow unit in each section may be correlative. Although the basal position of this unit (directly overlying the fallout unit) is not in itself good evidence for correlation, its thickness is constant in all sections. This, and gradual and consistent lateral change in subunit thickness within it, support correlation of the lowermost flow unit. Sedimentary facies of the Tokai Group in this small area indicate subaerial deposition from a fluvial system (gravelly to sandy low sinuosity streams), with no evidence for lacustrine environments (Nakayama, 1994, 1996). The fluvial system flowed west throughout deposition (Nakayama, 1994).

ANALYTICAL METHODS AND RESULTS Grain density, grain settling velocity, and grain-size distributions were analysed in an effort to quantify the depositional mechanism of the ‘mudflow’ deposits. Sampling horizons are shown in Fig. 1. The analytical procedure is the same as that of Nakayama (1997). Grain density Samples for measurement of grain density were collected as 102 to 103 cm3 blocks from each sample

Fig. 1. Left: Distribution of the Pliocene Souri tephra bed (thick solid line). Stippled area shows the distribution of the Tokai Group. Right: Representative columnar sections of the Souri tephra bed. Sample numbers with arrows indicate sampling horizons. Stage divisions after Nakayama (1996b). Stages V and VI of Nakayama (1996b) are not recognized at these localities.

PGCC16 9/12/08 3:24 PM Page 235

Liquefied flow of volcaniclastic sedimentation 235

PGCC16 9/12/08 3:24 PM Page 236

236

K. Nakayama

Fig. 3. Correlation between grain size and grain density. Measured grains larger than 0 phi (1 mm) are pumice grains, and those less than 0 phi are glass particles.

Fig. 2. Typical succession of subaerial liquefied flow deposits at locality 2 in Fig. 1, showing lowermost massive subunit A, overlying subunit B containing dish structures, and their fine-grained subunit at top.

horizon. All samples from locality 2 were combined as one mixed sample, which was sieved into 0.5 phi classes. These size fractions were then immersed in water for one month. A few pumice grains still floated after this immersion. They were discarded and not used for measurement. Grain density was measured with an electric specific gravimeter (Masuda ED-120T) on clasts in the 3.5 to −4.0 phi (0.088 to 16 mm) diameter classes. Immersed volcanic glass particles were recognized by their grey colour, whereas pumice grains were white. Clasts smaller than 2 phi (0.25 mm) were solely glass particles, and clasts larger than −1 phi (2 mm) pumice grains. Clasts in the range between −1 and 2 phi were mixed populations of glass particles and pumice grains. Grains larger than 2 phi and smaller than −1 phi were separated by handpicking into pumice grain and glass particle populations for separate measurement.

Results showed that glass particle densities ranged from 1.80 to 2.60 g cm−3, with an average of 2.10 g cm−3. This average density is slightly less than the published representative density of 2.3 g cm−3 for rhyolitic to dacitic bubble-wall shards (Fisher, 1965; Fisher & Schmincke, 1984). Pumice grain densities were less than 1.80 g cm−3, with an average of 1.50 g cm−3, concordant with the empirical pumice density range of less than 2.0 g cm−3 cited by Fisher (1965), and Smith and Smith (1985). Larger pumice grains tend to have lower densities, however, correlations between grain density and grain size within the pumice and glass populations are weak (Fig. 3). Consequently, to simplify the quantitative analysis in the following sections, the mean densities of pumice grains and glass particles are used in the calculations. Settling velocity Terminal settling velocities of the grains were obtained by settling in a tube 18 cm in diameter and 190-cm long, filled with distilled water. Times of descent were measured over a 100-cm length, the upper line being 65 cm from the top of the tube, which ensured that grains had reached terminal velocity. Baba and Komar (1981) found that a 65-cm length was sufficient for terminal velocity to be reached. They used a tube 6-m long for accurate measurements; however, 100-cm length is adequate for this experiment, because the maximum settling velocity is less than 10 cm s−1, whereas

PGCC16 9/12/08 3:24 PM Page 237

Liquefied flow of volcaniclastic sedimentation the maximum observed by Baba and Komar (1981) was over 20 cm s−1. Grains between 3.5 to −4.0 phi (0.088–16 mm) separated into 0.5 phi classes were used for the measurements. The samples used were the same as those for which grain density was determined. Grain settling terminal velocity (W [cm s−1]) is determined by equating forces due to particle immersed weight and fluid drag: W2 =

8a(σ − ρ )g 3C d ρ

(1)

where a is the grain radius in cm, σ and ρ are the grain and fluid densities in g cm−3 respectively, g is the acceleration due to gravity in cm s−2, and Cd is a non-dimensional drag coefficient. If grains are small enough, settling velocity conforms to Stokes’ law, which is strictly valid only at very small grain Reynolds numbers (Re). Rubey (1933) and Gibbs et al. (1971) suggest formulas valid for all grain Reynolds numbers. Cd can be determined by the relation to grain Reynolds number; for example, Cd = 24/Re in Stokes’ law, or Cd = (24/Re) + 2 according to Rubey (1933). In the present experiments, assuming Cd = (24/Re) + X, X can be obtained through regression of the measurement data assuming Eq. (1). ⎛ 24 ⎞ Cd = ⎜ ⎟ + 3.4 ⎝ Re ⎠ Re =

2aW η

(2) (3)

in which η is the fluid dynamic viscosity in cP. Grain settling velocities for the Souri tephra grains can be obtained by the equation (see Fig. 5C): W=

−18η +

324η 2 + 81.6 ρa 3 (σ − ρ )g 39ρ a

(4)

Equation (4) shows that the transition to turbulence is promoted at a lower grain Reynolds number than is the case for smooth spherical particles of equivalent volume. This is due to the surface roughness of bubble-wall shards, a result in agreement with Graf (1971). Similar settling velocity characteristics can be read from the glass shard settling velocity curve of Fisher (1965). Grain-size distribution Samples for grain-size distribution were collected as 103 to 104 cm3 blocks from the sampling sites and horizons shown in Fig. 1. Samples were hand crushed in the laboratory, and then immersed in water for

237

one month. Grain distributions were measured by wet sieving combined with a SK Laser Micron Sizer (Shinsei PRO-7000). As more than half of the samples contain pumice grains, hand sieving was used to minimize abrasion. Grains smaller than 3 phi (0.125 mm) were measured using the SK Laser Micron Sizer. Statistical parameters were calculated after Folk and Ward (1957). Glass particle and pumice grain populations were separated using the methods of Visher (1969) and Inokuchi and Mesaki (1974), and mean diameters of both pumice grains and glass particles were calculated. Grain-size distribution parameters are given in Table 1 and Fig. 4.

DISCUSSION The deposits studied are mainly massive, poorly sorted facies lacking large clasts. Nakayama et al. (1996b) qualitatively interpreted them as ‘mudflow’ deposits, but did not examine the grain-support mechanism quantitatively. Grain-support mechanisms of mudflows are considered to be mainly matrix strength and buoyancy (Lowe, 1979). Grain-support mechanisms of other types of sediment gravity flow, including grain dispersive pressure, fluidization and turbulence must have played some role in the grain support mechanism of the deposits in the Souri tephra bed. Hydraulic settling equivalence and dispersive pressure equivalence are examined below, followed by consideration of the possibility of fluidization. Hydraulic settling and dispersive pressure equivalence Hydraulic settling equivalence and dispersive pressure equivalence during grain transport have been examined using the methods of Cole and Stanley (1994), Nakayama et al. (1996a), and Nakayama (1997). This method is applicable only to the Souri samples which contain mixed pumice grain and glass particle populations (all samples of subunit A and some from subunit B). Hydraulic settling equivalence of pumice grains and glass particles can be tested by comparing the settling velocities of each grain type calculated from Eq. (4). The calculated settling velocities of pumice grains (WL [cm s−1]) and glass particles (WH [cm s−1]) in samples from subunit A show excellent agreement, whereas those from subunit B do not (Fig. 5A). It has been recognized that hydraulic settling equivalence is the key grain-support mechanism in the Bouma Ta divisions of turbidites (Middleton & Humpton, 1973). Coarse-tail grading, which is frequently recognized

PGCC16 9/12/08 3:24 PM Page 238

238

K. Nakayama

Table 1. Statistical parameters of the grain-size distributions and settling velocities for pumice grains and glass particles from the Souri tephra bed Sample number Subunit C 42C01 41C01 33C01 32C01 31C01 23C01 22C01 21C01 12C01 11C01 Subunit B 42B03 42B02 42B01 41B02 41B01 33B02 33B01 32B02 32B01 31B02 31B01 23B02 23B01 22B03 22B02 22B01 21B02 21B01 12B03 12B02 12B01 11B02 11B01 Subunit A 42A02 42A01 41A03 41A02 41A01 33A01 32A03 32A02 32A01 31A02 31A01 23A02 23A01 22A02 22A01 21A02 21A01 12A02 12A01 11A01

Dm

HDm

LDm

Pred HD m

phi

cm

phi

cm

WH cm s−1

1.27 1.43 0.96 1.05 1.02 2.13 1.45 1.00 1.39 0.98 0.96 1.41 1.14 0.97 1.36 1.00 1.01 0.93 0.98 1.03 0.85 1.02 0.97

4.50 4.20

0.0044 0.0054

0.20 0.80

0.0871 0.0574

0.393 0.577

5.557 4.361

0.44 1.04

0.0736 0.0485

4.30 3.50 3.10

0.0051 0.0088 0.0117

0.70 −0.20 0.60

0.0616 0.1149 0.0660

0.508 1.280 1.857

4.548 6.471 4.739

0.94 0.04 0.84

0.0520 0.0971 0.0558

4.00

0.0063

0.60

0.0660

0.736

4.739

0.84

0.0558

4.50 3.40 3.30

0.0044 0.0095 0.0102

0.90 −0.10 0.20

0.0536 0.1072 0.0871

0.393 1.414 1.555

4.178 6.233 5.557

1.14 0.14 0.44

0.0453 0.0906 0.0736

4.20

0.0054

1.10

0.0467

0.577

3.825

1.34

0.0394

4.50 3.80

0.0044 0.0072

1.40 0.10

0.0379 0.0933

0.393 0.928

3.322 5.776

1.64 0.34

0.0320 0.0789

3.90 4.10 3.90

0.0067 0.0058 0.0067

0.40 0.00 −0.10

0.0758 0.1000 0.1072

0.828 0.652 0.828

5.138 6.001 6.233

0.64 0.24 0.14

0.0641 0.0845 0.0906

0.86 0.85 0.83 0.86 0.86 1.01 0.90 0.85 1.10 0.96 0.99 0.90 0.91 0.90 0.68 0.79 0.87 1.04 0.87 0.91

2.90 2.70 2.20 2.30 2.50 2.50 2.70 2.80 2.50 2.50 1.90 2.80 2.80 2.40 1.80 1.90 2.40 1.80 2.30 2.20

0.0134 0.0154 0.0218 0.0203 0.0177 0.0177 0.0154 0.0144 0.0177 0.0177 0.0268 0.0144 0.0144 0.0189 0.0287 0.0268 0.0189 0.0287 0.0203 0.0218

1.60 0.90 1.20 1.40 1.80 1.30 1.30 1.00 1.80 1.10 0.70 1.70 1.90 1.60 1.10 1.30 1.00 1.30 1.80 1.60

0.0330 0.0536 0.0435 0.0379 0.0287 0.0406 0.0406 0.0500 0.0287 0.0467 0.0616 0.0308 0.0268 0.0330 0.0467 0.0406 0.0500 0.0406 0.0287 0.0330

2.183 2.529 3.463 3.269 2.891 2.891 2.529 2.354 2.891 2.891 4.067 2.354 2.354 3.078 4.276 4.067 3.078 4.276 3.269 3.463

3.002 4.178 3.654 3.322 2.694 3.486 3.486 3.999 2.694 3.825 4.548 2.847 2.544 3.002 3.825 3.486 3.999 3.486 2.694 3.002

1.84 1.14 1.44 1.64 2.04 1.54 1.54 1.24 2.04 1.34 0.94 1.94 2.14 1.84 1.34 1.54 1.24 1.54 2.04 1.84

0.0279 0.0453 0.0368 0.0320 0.0243 0.0343 0.0343 0.0423 0.0243 0.0394 0.0520 0.0260 0.0226 0.0279 0.0394 0.0343 0.0423 0.0343 0.0243 0.0279

phi

cm

SD

Sk

Kr

5.77 4.90 5.80 5.70 4.97 5.93 5.53 5.80 5.83 5.53

0.0018 0.0033 0.0018 0.0019 0.0032 0.0016 0.0022 0.0018 0.0018 0.0022

1.95 1.50 1.55 1.70 1.75 1.55 1.45 1.70 1.90 1.65

−0.01 0.06 0.01 0.11 0.09 0.10 0.18 0.11 0.19 0.12

1.28 1.34 1.51 1.19 1.17 1.45 1.42 1.19 1.29 1.12

3.90 3.57 4.50 4.37 4.23 3.33 2.90 4.62 3.67 4.47 4.33 3.30 3.13 4.53 3.98 4.43 4.33 3.67 4.33 4.43 3.83 4.00 3.80

0.0067 0.0084 0.0044 0.0048 0.0053 0.0099 0.0134 0.0041 0.0079 0.0045 0.0050 0.0102 0.0114 0.0043 0.0063 0.0046 0.0050 0.0079 0.0050 0.0046 0.0070 0.0063 0.0072

1.90 1.70 1.55 1.70 1.85 1.20 1.70 1.58 1.20 1.35 1.80 1.45 1.70 1.48 1.00 1.45 1.75 2.10 1.45 1.35 2.15 1.95 2.00

0.34 0.49 −0.16 −0.03 −0.09 0.46 0.15 −0.10 0.23 −0.29 −0.11 0.36 0.34 −0.01 0.22 −0.21 −0.05 0.08 −0.21 −0.23 −0.04 −0.01 0.06

2.57 2.33 1.47 1.47 1.70 1.80 2.23 2.33 2.00 1.93 1.43 2.53 2.63 2.07 1.57 1.43 1.87 1.17 2.10 2.00

0.0169 0.0198 0.0362 0.0362 0.0308 0.0287 0.0213 0.0198 0.0250 0.0262 0.0370 0.0173 0.0161 0.0239 0.0338 0.0370 0.0274 0.0445 0.0233 0.0250

3.70 3.85 3.80 4.05 3.80 4.00 3.60 3.10 3.80 3.70 3.45 3.65 3.30 3.80 4.65 4.25 3.90 3.90 3.75 3.55

0.06 0.00 −0.08 0.02 −0.04 −0.01 −0.15 −0.16 −0.24 −0.07 0.03 −0.01 0.03 −0.28 −0.09 −0.08 −0.34 −0.32 −0.28 0.06

WL cm s−1

phi

cm

Note: Statical parameters were calculated from cumulative curves after Folk and Ward (1957). Separation of glass particle populations from pumice grain populations based on the methods of Visher (1969) and Inokuchi and Mesaki (1974). Abbreviations: Dm = mean diameter, SD = standard deviation, Sk = skewness, Kr = kurtosis, HD m = mean diameter of heavier grain (glass particle) population, LD m = mean diameter of lighter grain (pumice grain) population, WH = settling velocity of HD m, WL = settling velocity of LD m. Settling velocities are calculated based on Eq. (4). Pred HD m = predicted mean diameter of glass particles on dispersive equivalence according to Eq. (5). Diameters are shown in both phi units and cm.

PGCC16 9/12/08 3:24 PM Page 239

239

Liquefied flow of volcaniclastic sedimentation

Fig. 4. Cumulative grain size by subunit. Bends in the curves at 20–25% in subunit A and around 15% in subunit B distributions reflect transitions from pumice grain populations to glass particle populations. Statistical parameters for each curve are given in Table 1.

in subunit A, is also found in the Ta division of turbidites (Middleton & Hampton, 1973). There are thus similarities in sedimentary facies and interpreted grain-support mechanism between subunit A of the subaerial deposits and Bouma Ta division of subaqueous deposits. Grains under dispersive pressure equivalence are tested by Eq. (5) because grains that experience equal dispersive pressure will be deposited together along a given horizon (Bagnold, 1954, 1956; Sallenger, 1979). HD m = LD m

ρL ρH

(5)

where HD m is the diameter of a heavier grain of density ρH, and LD m is the diameter of a lighter grain of density ρL. Diameter is in cm, and density is in g cm−3. The heavier grains in the samples are glass particles with mean density of 2.10 g cm−3, whereas the lighter grains are pumice grains with mean density of 1.50 g cm−3. Measured mean diameters of the glass particle and pumice grain components are used for HD m and LD m. Predicted diameters of glass particles for ideal dispersive pressure equivalence (Pred HD m) can be calculated from Eq. (5) with known LD m, ρL and ρH. Figure 5B compares the measured glass particle mean diameter (HD m) with predicted diameter (Pred HD m) for dispersive pressure equivalence. Samples from both subunits

A and B plot well away from the ideal relationship line, suggesting that dispersive pressure equivalence did not prevail. From the above, it can be concluded that subunit A was deposited under hydraulic settling equivalence, but the grain-support mechanism of subunit B was neither hydraulic settling equivalence nor dispersive pressure equivalence. Autofluidization Fluidization in a liquefied flow is examined by comparison between superficial fluid velocity at minimum fluidization (Vm f ) and superficial fluid escape velocity from the liquefied bed (Vesc ). Fluidization in a fluidized flow is considered to exist when the upwardmoving fluid is absorbed from subjacent fluid external to the flow. However, in these examples of the Souri tephra, external fluid supply is not possible, because of their deposition under subaerial conditions. Superficial fluid velocity at minimum fluidization (Vm f ) in cm s−1 is presented as a generalized equation by Kunii and Levenspiel (1969). Vm f = −

150βη + 3.5α Dρ

2

⎛ 150βη ⎞ Dg (σ − ρ ) (6) ⎜ ⎟ + 1.75αρ ⎝ 3.5α Dρ ⎠

where η, ρ, σ, and g are as in Eqs (1) to (4). D is the grain diameter in cm, and the constants α and β are

PGCC16 9/12/08 3:24 PM Page 240

240

K. Nakayama functions of particle shape and voidage at minimum fluidization. Wen and Yu (1966) suggest mean values of α = 14 and β = 11 for a wide variety of systems. Their data also indicate that α = 10.3 and β = 7.83 for spherical particles. Lowe (1976) considered that the general values (α = 14, β = 11) of Wen and Yu (1966) are reasonable for natural sediments, and thus they are used here. Equation (6) is effective only for cohesionless grains. Generally, grains less than 0.001 cm in diameter are cohesive. The actual Vm f on such small grains is in fact greater than that predicted by Eq. (6). Superficial fluid escape velocity (Vesc ) from the basal high-concentration part within a flow is calculated using the equation of Maude and Whitmore (1958). Vesc = CoVdisp Vdisp = W(1 − Co )

(7) n

(8)

Combining Eqs (7) and (8) Vesc = CoW(1 − Co ) n

(9)

in which Vdisp is the aggregate fall velocity of the grain dispersion part within a flow (see Fig. 5). W is the grain settling terminal velocity indicated by Eq. (4). All velocities are in cm s−1. The exponent n is empirically determined to vary between 2.4 and 4.7 for fully inertial and fully viscous systems, respectively, i.e. n depends on grain Reynolds number; n = 2.4 for Re ≤ 1, and n = 4.7 for Re ≥ 103. In this paper, n is estimated assuming a linear relation between n value (0.4 < n < 4.7) and grain Reynolds number (1 < Re < 103). Co is the value of initial concentration (see Fig. 6) in a flow. Co can be estimated to vary approximately between 0.4 and 0.6, because the requisite porosity for minimum fluidization generally ranges between 40 and 60% (Leva, 1959). The value of Vesc is therefore calculated Fig. 5. (left) Souri tephra Grain properties. A: Settling velocities of glass particles (WH ) versus settling velocities of pumice grains (WL). Settling velocities are calculated based on Eq. (4). Line with slope = 1 indicates ideal hydraulic settling equivalence (HSE). B: Measured glass particle size (HDm ) versus predicted glass particle size (Pred HDm ) for dispersive pressure equivalence calculated using Eq. (5) (Sallenger, 1979). Line DPE indicates ideal dispersive pressure equivalence. C: Potential for fluidization. Curve W indicates terminal settling velocity. Curve Vm f indicates the superficial fluid velocity at minimum fluidization using Eq. (6) (Kunii and Levenspiel, 1969). Vesc40 is the superficial fluid escape velocity from the liquefied bed at the initial concentration of 40%, Vesc60 is that for 60% initial concentration. The grain range in which Vesc is greater than Vm f has potential for autofluidization (shaded zone). The zone with densest shading has the highest potential for autofluidization.

PGCC16 9/12/08 3:24 PM Page 241

Liquefied flow of volcaniclastic sedimentation based on both 40% (Vesc40) and 60% (Vesc60) concentrations. Velocity curves for Vesc40, Vesc60, Vm f , and W are drawn in Fig. 5C. Theoretically, the grain range in which Vesc is greater than Vm f has potential for autofluidization (Fig. 5C). This shows that grains less than 0.04 or 0.07 cm in diameter can autofluidize depend on the concentration value of Vesc . However, since grains finer than 0.001 cm in diameter are cohesive, only the grain range between 0.001 and approximately 0.05 cm has potential for autofluidization. Furthermore, the line Vm f in Fig. 5C can be expected to level out around less than 0.001 cm because of grain cohesion. The largest difference between Vm f and Vesc and hence the highest potential for autofluidization is located in the grain size range between approximately 0.004 to 0.01 cm. Mean diameter (Dm ) values of subunit B samples lie around this grain range (Table 1), and thus their grains can be supported by fluidization. The velocity curves in Fig. 5 differ significantly from those of non-volcaniclastics. Figure 7 of Lowe (1976) shows the velocity curves for quartz spheres, for which Vesc is never greater than Vm f . Lowe (1976) thus concluded that autofluidization of uniformly sized grains is not possible. However, the results of this study show that autofluidization is indeed possible in volcaniclastic sediments such as the Souri tephra. Druitt (1995) demonstrated experimentally that ‘hindered settling’ may occur within concentrated and poorly sorted particle dispersions, as the sedimentation of large particles causes a vertical fluid flux strong enough to fluidize or elutriate smaller particles. The hindered settling of Druitt (1995) is almost equivalent to the autofluidization of Lowe (1976). Vesc of large particles is greater than Vm f of smaller ones, even if the Vesc curve is never greater than the Vm f curve. Experiments by Druitt (1995) indicate that any particle dispersion has some potential for autofluidization, depending on the particle concentration and sorting. In this study, the tephra grains with relatively low settling velocities have potential for autofluidization, independent of their sorting characteristics. Over the grain size range in which Vesc is greater than Vm f , the curves in Fig. 5C suggest that once grains have begun to settle, ongoing fluid escape could support following grains of the same size. The curves imply that perpetual motion for grain support could thus be possible. However, this is not tenable as the curves in Fig. 5C were drawn using Eqs (6) and (9), assuming constant grain concentration or retention of initial grain concentration. In reality, grain concentration (Co ) changes with time, and thus, the concentration ratio decreases. Vesc calculated with Eq. (9) will

241

also change, depending on the Co values. When Vesc becomes smaller than Vm f , autofluidization is terminated. Furthermore, Eq. (9) is not meaningful outside the range assumed for fluidization (Co ≈ 40–60%). For very small values of Co, we can calculate values of Vesc with Eq. (9), but results are meaningless, because very small values of Co indicate that the grain dispersion is fully scattered, and such dispersion cannot produce upward-escaping fluid. Conversely, high values of Co are also invalid in Eq. (9). In this case, grain dispersive pressure by grain collision must be a more effective grain support mechanism than fluidization. With respect to settling velocities, it is peculiar that settling velocities of pumice grains in subunit B are slightly larger than those in subunit A (Fig. 5A). This results from assumption of a constant density value for the pumice grains. In fact, larger pumice grains tend to have lower densities (Fig. 3), because they contain more closed vesicles. Pumice grain content also needs to be considered. Pumice grain content in subunit B is much less than that in subunit A (Fig. 4). However, fluidized potential estimation is not directly related to these features, because it is discussed based on whole grain populations. Beside the problem of the assumption of constant grain concentration and variation of grain density, further consideration of other factors, including size sorting, are necessary for fully understanding autofluidization. However, it remains significant that the Souri tephra exhibits the grain size range in which Vesc is greater than Vm f . Depositional succession of subaerial liquefied flow Grain-support mechanism can also be assessed qualitatively. In subunit A, grains were apparently supported by hydraulic settling equivalence, as discussed above. Subunit A was thus formed by deposition of a hydraulically equivalent bed load, or by rapid deposition from suspended load. Subunit A is massive and commonly graded, and there is no evidence of the influence of shear stress on grains; grain imbrication and stratification are lacking. Rapid deposition of subunit A from suspended load resulted in subsequent upward squeezing of fluid. Therefore, fluidization in subunit B must have been caused by fluid squeezed from the basal high-concentration layer which formed subunit A. Subunit C, consists of finer deposits, which are likely to have been elutriated from the fluidized part of the flow. The selective winnowing of finer grains by escaping pore fluids has been described (e.g. Lowe, 1975; Sparks, 1976). Mean grain diameters of subunit

PGCC16 9/12/08 3:24 PM Page 242

242

K. Nakayama

Fig. 6. Model for a subaerial liquefied flow, including an autofluidized horizon. Vesc is the superficial fluid escape velocity from the (most) concentrated part (subunit A). Vdisp is the aggregate fall velocity of the dispersion part (subunit B). Co is the initial concentration. The flow is divided into three subunits, and their corresponding deposits are shown at left. The columnar section at left is a representative succession of a subaerial liquefied flow.

B ranges between 0.0041 and 0.0134 cm. Corresponding upward fluid velocity in fluidization ranges between 0.013 and 0.078 cm s−1 in the case of initial concentration of 40%, and between 0.008 and 0.046 cm s−1 in the case of 60% concentration. The mean grain diameters of subunit C range between 0.0016 and 0.0033 cm, but are mainly from 0.0016 to 0.0022 cm (42C01 and 31C01 samples are exceptional large in Table 1). Terminal settling velocities for the grain diameter between 0.0016 and 0.0022 cm can be calculated as from 0.054 to 0.101 cm s−1, which are similar to the upward fluid velocities from subunit B. From the above discussion, it is concluded that the ‘mudflow’ deposits of Nakayama et al. (1996b) in the Souri tephra were subaerial liquefied flows. Single flow units can be divided into three subunits, A to C. Basal subunit A is massive, and was rapidly deposited from a suspended load under conditions of hydraulic settling equivalence. The overlying subunit B was formed by autofluidization, which was produced within the flow by upward fluid flow from subunit A. The uppermost subunit C is the finest grained, and is elutriated from the autofluidized part of subunit B. This succession of subunits may comprise a representative succession of a typical subaerial liquefied flow. The mechanisms are summarized in Fig. 6. Implications of travel distance and flow depth Travel distance and flow depth are two factors which could significantly influence the subaerial flow origin interpretation for these beds. Firstly, as noted above, the source volcano of the Souri tephra is considered to lie 50 to 100 km north of the most proximal outcrop of the tephra bed. It seems unlikely, however, that laminar flow characteristic of liquefied flows (Lowe,

1976), could be maintained over such distances, and the possibility that the mudflows represent the terminal deposits of longer turbulent flows from the Souri source must be considered. Secondly, thickness of the beds and settling velocities of the grains suggest that travel distance could only be several kilometres, even if settling velocities are lowered as a result of hindered settling or particle concentration. This also seems incompatible with direct derivation from the distant Souri source, solely as liquefied flows. Estimation of travel distance is thus important to the interpretation. As noted above, outcrops of the mudflows are confined to a small area. Lateral change in thickness of subunits A and B in the lowermost flow unit of stage III (Fig. 1), and general flow direction to the west result in the hydraulic equivalent bed load part (subunit A) increasing in thickness along the flow direction because of gravitational settling. Localities 1 and 4 are separated by a distance of 1.5 km, over which subunit B decreases from 34 to 23 cm in thickness. Total travel distance of the basal flow unit can be estimated by assuming that subunit B thins at a constant rate from the maximum thickness (66–68 cm) of the entire basal unit at the proximal end, to zero at the distal end. Thicknesses and relative distances of localities 1 and 4 give a thinning rate as 11/150 000 (the thickness change from 34 to 23 cm/locality distance of 1.5 km). A total travel distance of the flow can be calculated as 9.0 (to 9.3) km (lowermost flow thickness of 66–68 cm/thinning rate of 11/150 000). Although the Souri tephra bed is exposed over a distance of 40 km, the liquefied flow deposits can only be traced about 2 km. This and the thinning calculations show that the travel distances of the flows must be comparatively short from a few to 10 km. A short-lived subaerial liquefied flow in laminar flow mode is thus tenable, even though

PGCC16 9/12/08 3:24 PM Page 243

Liquefied flow of volcaniclastic sedimentation an origin as a terminal facies of a longer, turbulent flow, cannot be excluded. Original flow depth can be roughly estimated as two times the deposited unit thickness, because grain concentration (Co ) of 40–60% is realistic (Leva, 1954). Observed velocities of modern active mudflows range between 1.4 to 50 m s−1 (Janda et al., 1981; Blong, 1984; Fisher et al., 1997). Assuming a realistic flow velocity of 10 m s−1, a flow would take 150 s to travel the 1.5 km separating localities 1 and 4. The combination of assumed flow depth and flow travel time of 150 s indicates that the concentrated particle settling velocity of subunit A can be calculated as 0.15 cm s−1 (twice the thickness change in subunit B (34 –22 cm) / flow travelling time of 150 s). Single-particle settling velocities of mean diameter particles in subunit A are greater than 2 cm s−1 (Table 1). The concentrated particle settling velocity of subunit A must therefore be significantly reduced due to grain concentration. The calculated value of 0.15 cm s−1 is probably realistic. Short travel distance, concentrated particle settling velocities, and lateral thickness change are all compatible with origin as a localized subaerial liquefied flow.

CONCLUSIONS Subaerial liquefied flow deposits have been recognized in reworked volcaniclastic sediments of the Pliocene Souri tephra bed in central Japan. Single flow units can be divided into three characteristic subunits A, B and C in ascending order. Grain density, settling velocity, and grain-size distributions of the deposits were measured to quantitatively examine depositional mechanism. Hydraulic settling equivalence, dispersive pressure equivalence, and fluidization have been tested, based on these data. In the massive basal subunit A, grains were deposited rapidly from suspended load under hydraulic settling equivalence. The overlying subunit B was produced by autofluidization, where grains were supported by fluid squeezed upward from subunit A. Autofluidization can occur when the grain diameter ranges between 0.001 and 0.05 cm, but is most likely in the range from 0.005 to 0.01 cm, as in the case of the Souri tephra. Upward fluid velocity in subunit B is estimated to range from 0.008 to 0.078 cm s−1. The uppermost subunit C consists of finer grains, which were elutriated from the autofluidized part of subunit B. Travel distance of these flows is estimated to be less than 10 km. The succession of subunits A to C as described is considered to be a representative succession of a typical subaerial liquefied flow.

243

ACKNOWLEDGEMENTS My sincere thanks to B.P. Roser for his critical reading and helpful comments on the manuscript, to B.C. Kneller for the comments on an early version, and to D. Lowe, A. Freundt, and T.H. Druitt for their constructive reviews and comments on the submitted manuscript. This study was supported by a Japanese Ministry of Education grant-in-aid (10440142; K. Nakayama).

REFERENCES Baba, J. & Komar, P.D. (1981) Measurements and analysis of settling velocities of natural quarts sand grains. J. sediment. Petrol. 51, 631–640. Bagnold, R.A. (1954) Experiments on the gravity-free dispersion of large spheres in a Newtonian fluid under shear. Proc. R. Soc. London, Ser. A 225, 49–63. Bagnold, R.A. (1956) The flow of cohesionless grains in fluids. Phil. Trans. R. S. London 249, 49–63. Blong, R.J. (1984) Volcanic Hazards: A Sourcebook on the Effects of Eruption. Academic Press, Sydney, 424 pp. Cas, R.A.F. & Wright, J.V. (1987) Volcanic Successions: Modern and Ancient. Chapman and Hall, London, 528 pp. Cole, R.B. & Stanley, R.G. (1994) Sedimentology of subaqueous volcaniclastic sediment gravity flows in the Neogene Santa Maria Basin, California. Sedimentology 41, 37–54. Druitt, T.H. (1995) Settling behavior of concentrated dispersions and some volcanological applications. J. Volcanol. geothermal Res. 65, 27–39. Fisher, R.V. (1965) Settling velocity of glass shards. DeepSea Res. 12, 345–353. Fisher, R.V. & Schmincke, H.-U. (1984) Pyroclastic Rocks. Springer-Verlag, New York. Fisher, R.V., Heiken, G. & Hulen, J.B. (1997) Volcanoes. Princeton University Press, Princeton. Folk, R.L. & Ward, W. (1957) Brazons river bar; a study in the significance of grain size parameters. J. sediment. Petrol. 27, 3–26. Gibbs, R.J., Matthews, M.D. & Link, D.A. (1971) The relationship between sphere size and settling velocity. J. sediment. Petrol. 41, 7–18. Graf, W.H. (1971) Hydraulics of Sediment Transport. McGraw-Hill, New York. Inokuchi, M. & Mesaki, S. (1974) Analysis of the grain size distribution of bed material in alluvial rivers. Geograph. Rev. Japan 47, 545–556 (in Japanese with English abstract). Janda, R.J., Scott, K.M., Nolan, K.M. & Martinson, H.A. (1981) Lahar movement, effects, and deposits. In: The 1980 Eruptions of Mount St. Helens, Washington (Eds Lipman, P.W. & Mullineaux, D.R.), U.S. Geol. Surv. Prof. Pap. 1250, 461– 478. Komar, P.D., Baba, J. & Cui, B. (1984) Grain-size analyses of mica within sediments and the hydraulic equivalence of mica and quartz. J. sediment. Petrol. 54, 1379–1391.

PGCC16 9/12/08 3:24 PM Page 244

244

K. Nakayama

Kunii, D. & Levenspiel, O. (1969) Fluidization Engineering. John Wiley, New York. Kurokawa, K., Tomita, Y. & Kaneko, A. (1998) Correlation of the Itayama-Nym ash layer in Niigata region, the YT3 ash layer in the Himi Group and the Souri ash layer in the Tokai Group: Detection of the Pliocene widespread volcanic ash layer in central Japan. Earth Sci. (Chikyu Kagaku) 52, 292–300 (in Japanese with English abstract). Leva, M. (1959) Fluidization. McGraw-Hill, New York. Lowe, D.R. (1975) Water escape structures in coarse-grained sediments. Sedimentology 22, 157–204. Lowe, D.R. (1976) Subaqueous liquefied and fluidized sediment flows and their deposits. Sedimentology 23, 285–308. Lowe, D.R. (1979) Sediment gravity flows: their classification and some problems of application to natural flows and deposits. Soc. Econ. Paleontol. Mineral. Spec. Publ. 27, 75–82. Lowe, D.R. (1982) Sediment gravity flows: II, Depositional models with special reference to the deposits of highdensity turbidity currents. J. sediment. Petrol. 52, 279–297. Maude, A.D. & Whitmore, R.L. (1958) A generalized theory of sedimentation. Br. J. Appl. Physics 9, 477– 482. Middleton, G.V. & Hampton, M.A. (1973) Sedimentary gravity flow: mechanics of flow and deposition. In: Turbidites and Deep-Water Sedimentation. Soc. Econ. Paleontol. Mineral., Pacific Sec. Short Course Lecture Notes, Los Angeles. Nakayama, K. (1994) Stratigraphy and paleogeography of the Upper Cenozoic Tokai Group around the east cost of Ise Bay, Central Japan. J. Geosci. Osaka City Univ. 37, 77–143. Nakayama, K. (1996) Depositional processes of fluvial sediments in an intra-arc basin: an example from the Upper Cenozoic Tokai Group in Japan. Sediment. Geol. 101, 193–211. Nakayama, K. (1997) Reworked volcaniclastics of Ohta tephra in central Japan: depositional mechanism with

reference to hydraulic settling and dispersive pressure equivalence. J. geol. Soc. Japan 103, 897–907. Nakayama, K. & Furusawa, A. (1988) Volcanic ash layers in the Seto and Tokoname Groups, Aichi Prefecture, Japan. J. geol. Soc. Japan 95, 189–208 (in Japanese with English abstract). Nakayama, K. & Yoshikawa, S. (1997) Depositional processes of primary to reworked volcaniclastics on an alluvial plain; an example from the Lower Pliocene Ohta tephra bed of Tokai Group, central Japan. Sediment. Geol. 107, 211–229. Nakayama, K., Kono, K. & Masumoto, A. (1996a) A preliminary report on volcaniclastic sedimentology with reference to grain size; an example from the Pliocene Ohta Tephra Bed. J. sediment. Soc. Japan 43, 27–38 (in Japanese with English abstract). Nakayama, K., Masumoto, A. & Hosoyama, M. (1996b) Depositional processes of Pliocene Souri tephra bed, central Japan. Geosci. Rep. Shimane Univ. 15, 63–73 (in Japanese with English abstract). Rubey, W.W. (1933) Settling velocities of gravel, sand, and silt. Am. J. Sci. 25, 325–338. Sallenger, A.H. Jr. (1979) Inverse grading and hydraulic equivalence in grain flow deposits. J. sediment. Petrol. 49, 553–562. Smith, G.A. & Smith, R.D. (1985) Specific gravity characteristics of recent volcaniclastic sediment: implications for sorting and grain size analysis. J. Geol. 93, 619–622. Sparks, R.S.J. (1976) Grain size variation in ignimbrites and implications for the transport of pyroclastic flows. Sedimentology 23, 147–188. Visher, G.S. (1969) Grain size distribution and depositional processes. J. sediment. Petrol. 39, 1074 –1106. Walker, G.P.L. (1971) Grain-size characteristics of pyroclastic deposits. J. Geol. 38, 696 –714. Wen, C.Y. & Yu, Y.H. (1966) A generalized method for predicting the minimum fluidization velocity. Am. Instit. Chem. Engineer. J. 12, 610–612.

PGCC17 9/12/08 3:23 PM Page 245


Depositional and eruptive mechanisms of density current deposits from a submarine vent at the Otago Peninsula, New Zealand U . M A R T I N * and J . D . L . W H I T E Geology Department, Otago University, PO Box 56, Dunedin, New Zealand

ABSTRACT The remnant of a small Miocene volcano, consisting largely of phonolitic pumice, is preserved on the Otago Peninsula at Allans Beach. Its 80-m thick deposits record a variety of particulate gravity flows from a subaqueous eruption. Lack of wave-generated structures suggests deposition below wave base, and for the eruptive centre a depth of about 200 m is estimated, based on the present day wave climate along the Otago Peninsula and an overlying pillow/hyaloclastite unit. A thick unit of phonolitic pumice represents the main phase of the Allans Beach eruption. A lower, crudely bedded zone at the base is characterized by alignment bedding defined by elongated pumice fragments. The overlying part of the unit is well bedded, crudely doubly graded and locally shows weak crossstratification. A thin-bedded upper zone is fine grained and lacks clear current indicators. A 20-m thick pillow/hyaloclastite unit overlies the pumice deposits. This phonolitic pumice sequence indicates deposition from dense suspension out of a series of closelyspaced high-particle concentration granular flows, followed by high- and low-concentration turbidity currents, shed directly from a subsiding subaqueous eruption column during the waning of an initially higher flux eruption. Development of the eruption column was a result of the high gas content of the magma. The column formed and collapsed entirely beneath seawater. Pumice shed from the column chilled rapidly and the grains mixed with water to form aqueous density currents from which deposition occurred.

INTRODUCTION deposits (e.g. Cashman & Fiske, 1991; White, 1996a; Fiske et al., 1998), in large part because of the inaccessibility of most young, subaqueously-formed volcanoes. This chapter examines a succession of deposits that are compositionally distinctive, sufficiently vesicular to imply high magmatic gas content, and well bedded. They are inferred to have formed from a single small subaqueous eruption, and we specifically address the relation between style of deposition, the nature and evolution of the eruption, and the changing role of magma–water interaction during its course.

Small-volume explosive volcanic eruptions form cones, rings, or mounds consisting of bedded tephra that is deposited by suspension fall or density currents. Additional bedded deposits may form by subsequent redeposition of pyroclastic debris from primary deposits. It is essential that the role of transport and deposition in producing the deposits be understood in order to allow ‘removal’ of these effects, in order to interpret eruptive processes from pyroclastic deposits. For subaerial eruptions, the nature of sedimentation from density currents and from suspension has been extensively studied, and quantitative models are available linking many deposit characteristics with eruptive processes (e.g. Sparks et al., 1997 and references therein). Far less has been accomplished for subaqueous

SETTING Volcanic rocks exposed at Allans Beach belong to the Dunedin Volcanic Complex (DVC), which makes up the Otago Peninsula near Dunedin, New Zealand. The

* Present address: Westfälische Wilhelms-Universität, Geologisch-Paläontologisches Institut, U. museum, 48149 Münster, Corrensstraße 24.


245

WV

MV

magma mingling peperite 1 peperite 2 peperite 3 beach sand

lithofacies B lithofacies A1 A1 near-vent lithofacies A2 lithofacies LT

younger volcanics

dip and strike low tide line proposed fault

phonolite dike

dikes

Allans Beach

Otago Peninsula, basalt, dikes (partly pillows and hyaloclastites Lithofacies C vent breccia (pumice)

Belmont

92

0

10

Fig. 1. Map of field area. AB = Allans Beach, WB = Waipuna Bay, WR = Wellers Rock, MP = Maori Pa.

AB

WR

N

Wharekakahu

200m

246

MP

WB

Dunedin Volcanic Complex Otago Peninsula New Zealand

PGCC17 9/12/08 3:23 PM Page 246

U. Martin and J.D.L. White

PGCC17 9/12/08 3:23 PM Page 247

Density current deposits from a submarine vent DVC formed from 13–10 Ma and rests on Cretaceous and Tertiary sedimentary rocks which are underlain by pre-Cretaceous basement of quartzofeldspathic schist (Haast Schist Group) (Coombs et al., 1960). The sedimentary rocks comprise marine sandstones, mudstones and limestones which were deposited on the continental shelf, and a non-marine to marginal marine basal unit. Miocene marine sediments on the Peninsula give evidence of a submarine setting for the early phases of the volcanic activity (Scott, 1991). No dramatic sea level change is reported during Miocene time, although Haq et al. (1987) calculated a global marine highstand during the Miocene. Coombs et al. (1960) concluded that at the early stage of the First Main Eruptive Phase of the DVC, volcanism took place in a submarine setting, due to the presence of pillow basalts and peperites, which they described as ‘pillows separated by a matrix of hard sandstone’, at Maori Pa Point (Fig. 1). Pillow basalt, hyaloclastites and peperites exposed along almost the whole southern coast of the Otago Peninsula further support this idea. Interbedded tuffs within marine sandstones and limestones at Varleys Hill (c. 3 km NW of Allans Beach), Waipuna Bay (Fig. 1; Coombs et al., 1960) and along the Otago Harbour (Wellers Rock; U. Martin, unpublished data) also indicate a submarine environment for the early stages of volcanism, as do depositional features of the Allans Beach succession itself.

247

ALLANS BEACH VOLCANIC STRATIGRAPHY All the rocks at Allans Beach were mapped by Benson (1969) as products of the first main eruptive phase (agglomerates, tuffs and basalts), indicating eruption after a lesser initial trachytic eruptive phase of the DVC. In this study we recognize five pyroclastic units, which are cut by a variety of penecontemporaneous and younger dikes (Figs 1 and 2). The pyroclastic units are listed below: 1 Unit LT is the lowest stratigraphic unit exposed at Allans Beach (Fig. 2), and consists of fine-grained tuff deposits (Table 1) derived from an early eruption from a different vent (MV; Martin & White, 1997). 2 Unit A1 consists of basanitic, very poorly bedded and poorly sorted juvenile scoria, non-juvenile volcanic clasts, pyroxene and feldspar crystals and, locally, sandstone and schist fragments (Table 1). It is interpreted as a high concentration mass-flow deposit, from eruption at a western vent WV (Fig. 2; Martin & White, 1997). 3 Unit A2 consists of schist fragments, a variety of non-juvenile volcanic clasts, quartz pebbles, basanitic clasts and scoria (Table 1). It is interpreted as a nearvent high-concentration density current deposit, from an eruption at the small vent MV in the middle of the field area (Figs 1 and 2). 4 Unit B is a lithic-rich basanitic lapilli tuff (Table 1) that locally overlies lithofacies A1 (Fig. 2).

Fig. 2. Interpretive cross-section along line WV–MV, showing inferred relationships between the lithofacies exposed at Allans Beach.

PGCC17 9/12/08 3:23 PM Page 248

248


Table 1. Summary of sedimentological features and inferred depositional processes of older volcanic units at Allans Beach Unit

Composition

Texture

Structures

Inferred depositional process

Older volcanics LT from different source, at least 2-m thick

Trachytic

Well-sorted, coarse ash to fine lapilli, subangular grey

Well-bedded (3–7 cm) and laminated, graded beds, cross-bedded, local scour fillings

Turbidity currents

Initial deposits A2 about 2-m thick

Basanitic, scoria, tuff clasts from older volc. up to 30 cm, large schist clasts, quartz pebbles, sandstone clasts, different volcanic clasts

Poorly sorted, subrounded schist and sandstone clasts

Bedded, beds 10–30 cm, lensoidal patches with smaller and rounded clasts and less matrix, larger clasts slightly inverse graded, large scale cross-bedding

Near-vent highdensity current

Initial deposits A, at least 5-m thick

Basanitic, juvenile scoria, non-juvenile volcanic clasts, pyroxene, locally schist and sandstone clasts, grainsize between 0.2 and 1.5 cm, occasionally large bombs scattered in dense ash matrix, glass in matrix altered to clay

Not sorted, subangular to subrounded, green–grey blue

Not bedded, massive character, becoming slightly bedded with distance to the vent, no block sags

High concentration, mass flow, probably cohesionless debris flow

Lithic-rich lapilli tuff B, c. 3-m thick

Basanitic, juvenile scoria, non-juvenile volcanic clasts, quartz pebbles, schist fragments, feldspars, matrix altered to clay

Poorly sorted, clast size Cross-stratification, beds 2–6 cm, locally 12 cm defined by grain size, occasional scour-filling structures, feldspars concentrated in thin graded beds

5 Unit C comprises phonolitic pumice deposits of the major eruptive phase at Allans Beach, probably also erupted from vent WV (Figs 1 and 2). The phonolitic pumice deposits are the main focus of this paper. Unit C is inferred to represent the major eruptive phase of the Allans Beach succession. The pumice deposits consist of more than 95% pumice, so in this case the term ‘pumiceous’ is not applied because it would imply a significant proportion of non-pumice components (cf. Fisher, 1961; Schmid, 1981). The phonolitic pumice beds of this unit dip radially away from the western massive breccia (WV), suggesting that the breccia marks the vent site (Figs 1 and 2).

PHONOLITIC PUMICE DEPOSITS (UNIT C) The pumice succession comprises about 80 m of phonolitic pumice, which can be subdivided into three distinct subunits based on observed volcanic textures, shapes of pyroclasts, and sedimentary structures character-

High-density turbidity currents

istic of each unit (Fig. 3): (i) a structureless sequence of about 5–10 m, representing the base of the succession; (ii) a 50–60-m thick bedded lapilli tuff; (iii) a thinly laminated tuff at the top. The original thickness of the upper tuff is unknown because of the eroded surface. Subunit 1 The lower 5–10 m of the deposit is unstratified or poorly stratified and consists almost entirely of coarse lapilli tuff. It is here called the massive division of the pumice sequence (Fig. 3a). Massive beds are defined by the appearance of discrete fine and thin (0.5–2 cm) beds or laminae, which occur at various levels in the subunit, usually at intervals of 0.5 to 2 m. The crudely developed beds sometimes are defined by elongated subhorizontally aligned pumice fragments (alignment bedding), which have 2–5-mm thick yellowish vesicle-free rims. The original glass forming the rim is altered to clay. Clasts consist mainly of pumice, and are subangular to subrounded. Usually the lapilli size clasts are sub-

0

10

20

30

40

60

70

80

90cm

-bedding thickness changes from 3-8 cm -slightly normal graded,some larger clasts inverse graded -largest clasts are about 5 cm long and aligned -clasts sub-angular to sub-rounded -open framework structure -poorly sorted -undulating bottom and top

-bedding thickness changes from 3-5 cm -slightly reverse graded -average grain size about 3 mm -largest clasts about 1 cm -undulating top and bottom -trains of coarser clast interupt this sequence -trains usually one clast thick -clasts in trains are aligned

-bedding thickness change from 15-20 cm -matrix supported -grey ash tuff matrix -slight normal grading -contains coarse trains, one grain size thick -clasts in trains are aligned in bedding plane -trains are clast supported

-lapilli-size framework clasts -large irregular clasts up to 30cm are aligned subhorizontally -large clasts concentrate in the middle part -no bomb sags -sequence weakly normal-graded

b. transition zone

0

50

70cm

-undulating top and bottom -beds often show load casts and flame structures -both matrix-rich and matrix-poor beds present -both bed types slightly fine upwards -slightly normal to inverse graded -first fining upward cycle is about 50cm thick -clasts of matrix-rich beds about 3-5mm -clasts of matrix-poor beds 2cm -largest clasts about 15 cm -matrix-rich layers contain 1cm thick trains -trains formed by coarse-grained clasts -trains usually 1 clast thick -some clasts are a(p) a(i) imbricated -bedded part generally about 40-50m

Density current deposits from a submarine vent

Fig. 3. Stratigraphic column of unit C. (a) Representative features of massive subunit 1. (b) Transition zone between subunit 1 and 2. (c) Bedded subunit 2.

0

10

20

30

40

50cm

-bedding defined by the appearance of fine beds or laminae -poorly defined beds 0.5m thick or more -ungraded -average grain size about 2cm -largest clasts about 30cm -undulating top and bottom

a. subunit 1

c. subunit 2

PGCC17 9/12/08 3:23 PM Page 249

249

PGCC17 9/12/08 3:23 PM Page 250

250

U. Martin and J.D.L. White Stretched vesicles are usually found along concave– convex contacts. Occasionally there are some brown, amorphous patches, which may represent relicts of original fine-ash matrix. Interpretation

Fig. 4. Cauliflower bombs in the massive subunit 1, with very irregular and chilled margins, indicating contact with water during fragmentation. Hammer length c. 40 cm.

angular, whereas the larger ones are subrounded. Larger clasts are up to 30 cm and lie subhorizontally within the unit without any impact sags. Some larger pumice fragments show chilled, cracked and irregular margins (Fig. 4) surrounding a more highly vesiculated interior (‘cauliflower bombs’; Lorenz, 1973). In specific regions near the base of the lower part of the massive pumice division there are lithic fragments, mainly basanitic clasts inferred to have been picked up in the vent. The fragments are subrounded and usually not larger than 1 cm. Only very rare fragments of Tertiary sedimentary strata or the basement schist and tuff rip-up clasts occur in the pumice deposit. There are no accretionary lapilli. Vesicles in the pumice clasts are tube-like, which gives the pumice a woody appearance. Some pumice clasts are irregular in shape. Vesicularity is visually estimated at 70–80%. Microprobe analysis of locally preserved pumice-clast glass and whole-rock X-ray fluourescence analysis, each indicate a phonolitic composition. In many places the pumice glass is strongly altered to zeolites, shows devitrification structures and a pale creamy colour in thin section. The tubular vesicles of the less vesicular pumice clasts are often widely separated, which gives the pumice a thick-walled glassy texture. Individual pumice fragments differ widely in crystallinity, with some wholly glassy and others having a subtrachytic texture defined by the feldspar crystallites. Very rare feldspar phenocrysts up to 0.1 mm, are also present. The pumice lapilli are tightly packed with only a minor (1–2%), largely devitrified, vitric ash matrix. Two-dimensional clastcontact geometries range from point contacts to conformable concave–convex grain–grain contacts.

The abundance of highly vesiculated pumice clasts records expansion of magmatic volatiles (e.g. Sparks, 1978; Fisher & Schmincke, 1984). The apparently uniform vesiculation of about 70% suggests that there was a high degree of magmatic fragmentation. High vesicularity of the pumice clasts at Allans Beach reflects a high gas content, which is consistent with the generally high (3–6%) water content of phonolitic melts (Schmincke, 1988) in comparison to basaltic magma (0.2–1.5%). The vesicularity index of magmatic fragmentation given by Houghton and Wilson (1989) lies uniformly in the range 70–80% regardless of magma viscosity. The presence of cauliflower bombs, in contrast, indicates contact with water during fragmentation (Lorenz, 1973). Woody pumice forms in response to shear of vesiculating magma against the walls of the vent (Fisher & Schmincke, 1984; McPhie et al., 1993). The elongated and subrounded clasts could have been bent and deformed in the hot centre of a submarine eruption column and chilled at the vapour–water margin before deposition, a possibility also supported by the fact that no real primary welding features have been identified, suggesting that the bulk of the glassy pyroclasts cooled before deposition. Some of the large clasts may have retained heat better than the small ones, thereby becoming bent and deformed during transport. The rounding of the larger clasts is not a result of abrasion but instead resulted from surface tension shaping of fluidal clasts. The absence of reworked and abraded clasts or of any evidence of time gaps, suggests ongoing deposition directly from a submarine eruption column. The massive division records sedimentation of a substantial influx of phonolitic pumice. The ungraded character of this division is evidence for rapid high concentration suspension sedimentation (Chough & Sohn, 1990) such as occurs from high particle concentration flows or flow bases (Lowe, 1982). Alignment bedding suggests deposition from currents rather than from direct fall-out (Hiscott & Middleton, 1980). Deposits from direct water-settled fall would be graded and would not contain any rip up clasts. The lack of well-developed internal stratification in thick currentemplaced beds, together with alignment bedding, may indicate deposition by progressive aggradation from one or more gravity currents during quasi-steady flow (Kneller & Branney, 1995). The fine beds or laminae,

PGCC17 9/12/08 3:23 PM Page 251

Density current deposits from a submarine vent which occur at various levels within the massive part, are interpreted to have been deposited during irregularities in the quasi-steady flow. Irregularities are dependent on temporal fluctuations in the downward grain flux and the clast concentration. The concentration fluctuation may arise from (1) flow unsteadiness (e.g. as a result of pulsing flow), (2) heterogeneities in grain-size population as provided to flows from the eruption source or (3) concentration variations within the current as it passes (Kneller & Branney, 1995). Slower deposition by fallout from current tails during pauses between distinct flows is also possible. The presence of small amounts of interstitial fine ash in the lapilli-supported deposits can be interpreted at least in two ways: 1 The proportion of fines to coarse clasts is representative of the eruption products at that time in the eruption. Although there is little fine ash, the common presence of large clasts within the fine-to-medium lapilli tuff shows that sorting overall is relatively poor, which in this interpretation is taken to indicate deposition by overcapacity currents (e.g. Hiscott, 1994), in which there was little separation of fines from the body of the currents. 2 The small proportion of vitric ash particles in the matrix was deposited by being captured in the irregular margins of larger clasts prior to deposition. This interpretation does not require sedimentation from an overcapacity current, and the ash : lapilli ratio may be unrepresentative of the eruptive mixture. If deposition occurred by progressive aggradation from an ongoing current, then it is most likely that deposition was from overcapacity currents (Lowe, 1988). A certain amount of compaction after deposition is indicated by the concave–convex conformable contacts. The presence of concave–convex contacts at all levels in the massive unit, not only in the centre of beds where maximum heat retention typically occurs in hot pyroclastic deposits, suggests that the contacts are the result of compaction following alteration of pumice glass to deformable clay–zeolite aggregates (Branney & Sparks, 1990). Glass shards are subangular and not flattened, indicating that lapilli were not deformed while hot, but as altered glass much later. A small amount of unaltered fine ash is present between the larger pumice clasts, even when alteration of pumice glass is extensive. Subunit 2 Well-developed stratification distinguishes this unit from subunit 1. Lapilli tuff beds of this bedded subunit consist of thin to medium (5–10 cm) bedded pumice

251

Fig. 5. Overview of subunit C2. Approximately 10-m high outcrop of bedded subunit 2 and part of transition zone in the lower part of the outcrop. Photograph shows typical bedding features (undulating bedding, coarser-clast trains in the matrix-rich layers (CT), repetition of matrix-poor (MP) and matrix-rich (MR) layers). Note the different resistance to weathering in the alternating layers, due to the presence of calcitic cement in matrix-rich layers and absence of cement in matrix-poor layers and faint cross-stratification (CS). Outcrops are locally cut by normal faults (stippled lines), which offset the bedding planes. Hammer is c. 30 cm.

(Fig. 5) with undulating bedding contacts (Fig. 6) and common shallow scouring. No significant channels were observed in the area. Beds are defined by the alternation of matrix-rich and matrix-poor layers (Fig. 3c), which also are different in colour, the vitric ash matrix-rich beds being grey and matrix-poor beds yellowish and which overall show a slight thinning and fining upward. At least three fining upward cycles are present at Allans Beach. Individual beds are generally ungraded or inversely to normally graded, locally with low angle crossstratification (Fig. 5). Many beds show diffuse load- or flame-like structures (Fig. 6). Beds generally fine and

PGCC17 9/12/08 3:23 PM Page 252

252


thin upward. In a c. 5-m thick transition zone (Fig. 3b) between the bedded subunit 2 and the massive subunit 1, beds are much thicker (30–50 cm) but with the same sedimentological features as the bedded unit (Fig. 3c). Some irregular clasts are elongated, lying with long axes sub-parallel to stratification. Other clasts show a long-axis parallel [a(p)a(i)] imbrication (Fig. 7) of 90–120°/15–20°, which is consistent with bedding dip in indicating transport from W–WNW. Common in the matrix-rich part are local concentrations of scattered larger clasts, or even lapilli tuff, in diffuse lenses or irregular clusters (Fig. 3b). These coarser horizons are usually only one or few clasts thick, with gradational boundaries at a millimetre to centimetre scale. The phenocryst and microlite content of clasts in both matrix-rich and matrix-poor parts is comparable with that of clasts in subunit 1, and shows no apparent change throughout the sequence. The coarse-grained beds have a compact-framework supported texture, devoid of fine ash, and are dominated by subangular to subrounded clasts (see comments for subunit 1).

Clasts are closely packed and interstitial space is occupied by carbonate cement. Larger pumice clasts appear locally in clusters. As in subunit 1 pumice clasts are locally aligned and have a ‘woody’ appearance resulting from strongly stretched vesicles. The vesicles have very irregular shapes and vary greatly in size. Vesicularity is 60–80%, but generally nearer 60%. Interpretation The well-bedded character of subunit 2, together with imbrication and low-angle cross-stratification records deposition from turbidity currents. Inverse grading of smaller pumice clasts which is not related to clast density appears in some beds. The variation in clast vesicularity reflects complexities in the relative timing of vesiculation (Houghton & Wilson, 1989), and waterinduced fragmentation. Magma–water interaction at early stages significantly reduces the vesicularity indices and broadens the ranges, whereas late-stage interaction has only a minor effect on the index (Houghton & Wilson, 1989). Since the vesicularity of the pumice in subunit C2 has not decreased dramatically compared with subunit C1, and does not vary greatly, a late-stage interaction is indicated. The considerably higher proportion of fines in the beds reflects a change in the material supplied at the source, which may have resulted from a change in fragmentation style during the eruption (Wohletz & Sheridan, 1983; Zimanowski et al., 1997). Several fining upward cycles suggest that the eruption rate increased and decreased repeatedly. The repetition of the graded coarse and fine beds indicates that the pumice clasts in each bed were transported and deposited by turbidity currents, where,

(a)

(b)

Fig. 6. Photograph (a) and line drawing (b) showing a close-up of undulating bedding. A load structure (L) is visible in the middle of the image, caused by shaking of the loosely packed, low-density beds during deposition. The thicker bed at the base belongs to the transition zone. Note the clusters of large chilled clasts (C) and undulating bedding panes (UB). Scale (little card) is 10 cm.

PGCC17 9/12/08 3:23 PM Page 253


253

currents (S1) followed by traction carpet sedimentation (S2; Lowe, 1982). The load-like structures, usually formed beneath the coarse-grained beds, are interpreted to have formed by shaking of the beds (Johnson, 1985; Scott & Price, 1988), during deposition of subsequent layers. Such soft-state deformation is facilitated by the overall low bulk density of the pumice, and loose packing of coarse-grained layers. There are three possibilities to explain the change from high- to low-density currents: 1 the change in particle concentration could have been caused by an increasing magma flux; 2 the dilute currents were derived from different parts of the eruption column (convective region) and were mixed with larger amounts of water before transportation; 3 the low concentration currents could also be residual currents decoupled from coeval high concentration precursors, or low-concentration ‘clouds’ elutriated from a high concentration underflow. Subunit 3

Fig. 7. Imbricated clasts within subunit 2 in the upper middle of the image (see arrows) are an indicator of lateral flow. Larger clasts at the bottom are stretched, aligned and occur in clusters (see arrow). Hammer is c. 30 cm.

from each current, the coarse-grained material was deposited from the body or head and the fine-grained material from the tail. The fining upward cycles are interpreted as a ‘doubly graded’ turbidite series formed in response to gradual decrease in eruption rate (Fiske & Matsuda, 1964). The normally graded to ungraded matrix-rich beds together with the appearance of low-angle crossstratification record traction deposition from dilute turbidity currents (Ta,c ), while the inverse-graded, coarse-grained beds were deposited from traction carpets (Lowe, 1982). The crudely stratified matrix-rich layers with coarse-grained pods and trains result from rapid fall-out of dense suspension with subsequent tractional transport from the body, as indicated by discontinuous trains of coarse clasts (Sohn & Chough, 1989). Cross-stratification in the matrix-poor layers, followed by coarse-grained inverse graded beds, suggests traction deposition from the base of high-density

The topmost unit, a fine-grained tuff (Fig. 2), is poorly exposed and lacks clear current indicators. It is laminated, graded, and in places gives an impression of slight cross-lamination. The coarser parts contain small pumice clasts (1 mm) and a large amount of crystals, mainly feldspars and, locally, pyroxene and very rare biotite. The finer-grained laminae at the top consist entirely of fine cuspate glass shards, and broken feldspar and pyroxene crystals. The contact with the underlying subunit 2 is not exposed. The top of the fine-grained subunit 3 is mainly eroded, but in places a 20-m thick pillow/hyaloclastite unit partly overlies it. Interpretation Lack of clear current indicators makes it difficult to determine whether this unit was deposited by many small volume aqueous fall events, as a series of Td,e ash turbidites or both. The latter could have formed either as eruption-fed turbidity currents during the last stages of eruption, or by episodic remobilization afterwards. As discussed earlier, subunits 1 and 2 contain few or no phenocrysts, whereas subunit 3 contains a large amount of crystals. This trend is common in eruptions of highly differentiated magmas, which first erupt crystal-poor or crystal-free magma (Schmincke, 1988). The concentration of feldspar and pyroxene crystalrich beds in subunit is thus consistent with it representing the latest stage of the Allans Beach eruption.

PGCC17 9/12/08 3:23 PM Page 254

254


SUBSEQUENT ACTIVITY The deposits subsequently were invaded by a number of dykes, some in the form of irregular lobes, along which unusual margins have formed, as reported by Martin (1998). The contact with the pumice sediment is irregular, with metres-long apophyses extending from the dykes. The irregular form and fluidal margins of the dykes suggest that they intruded into the pumice while it was still unconsolidated and watersaturated (Busby-Spera & White, 1987; Kano, 1989; McPhie, 1993). In thin section, flow-banding of glass along the contact is apparent, and is interpreted to result from shearing of heat-softened glassy pumice clasts due to the intrusion of the dykes. It has been inferred that water vapour under pressure (Sparks et al., 1980) and salt in the seawater entered the glass, promoting softening and therefore shearing. Thermal modelling for the Allans Beach peperite (Martin, 1998 and unpublished data) demonstrates that welding is most likely to have occurred subaqueously.

SUBMARINE ERUPTION AND DEPOSITION OF THE ALLANS BEACH MOUND A model for emplacement of unit C at Allans Beach must account for: 1 an absence of significant compositional changes within the unit; 2 The fact that much of the unit consists of highly vesiculated phonolite-glass pumice; 3 a change in clast vesicularity from subunit 1 to subunit 2; 4 the presence of vitric ash as a minor component in subunit C1; 5 absence or rarity of lithics; 6 lack of evidence for significant pauses during deposition; 7 the shallow dips of bedding. In addition, there are several lines of evidence that, taken together, strongly indicate a subaqueous eruptive and depositional setting. In terms of regional and local paleogeographical setting, a subaqueous setting is indicated by the presence of pillow basalts, peperites, and hyaloclastites as well as interbedded tuffs in marine sandstones at the Otago Peninsula, and the presence of fused peperite. In addition, sev-

eral features of the deposits suggest a subaqueous setting. Interaction with water is indicated by the presence of cauliflower bombs. The pumice beds (subunit 1) at Allans Beach contain only a small amount of fines and no, or very rare, accidental lithic fragments. This indicates a less vigorous fragmentation and a shallower fragmentation locus compared to subaerially formed successions (e.g. Laacher See, Germany; Bogaard & Schmicke, 1984), which are together suggestive of interaction with standing water. High instantaneous depositional rates are suggested for subunit 1 at Allans Beach by massive, thick, beds with pumice trains (Sohn & Chough, 1989). Beds in subunit 2 are laterally continuous, have diffuse boundaries and show only subtle scours and grading. Such beds are typical of deposits from unconfined sediment gravity flows. Subtle softstate deformation also suggests deposition onto a water-saturated sediment (Johnson, 1985; Scott & Price, 1988). Continuous, finely laminated fine ash beds of subunit 3 represent suspension deposition. These features are interpreted to have formed by rapid deposition onto wet sediments from sedimentgravity flows, followed by suspension settling in the absence of other currents. The Allans Beach mound dips gently radially outward from a vent breccia. The vent breccia is surrounded by the deposits at the same or slightly higher elevations, strongly suggesting that the vent environment was the same as that of the deposits; either both were subaerial or both were subaqueous. Gas-escape pipes, carbonized vegetation, which are indicative of hot emplacement, and accretionary lapilli are missing. Thus positive evidence for subaerial deposition, nor evidence of water droplets which are essential to form accretionary lapilli, is lacking. There are no gullies, rills, or sedimentary features indicative of stream or wave erosion or reworking. There are only two internally consistent interpretations of these data: 1 All features formed fully in a subaqueous setting, as is implicit in the preceding subunit interpretation. 2 All features formed in a fully subaerial environment, in which there was with no condensation of water in eruption plumes or bursts, no ballistic transport, no disruption of the substrate, and no subsequent erosion and reworking. The site must have intersected a very shallow water table to allow formation of peperites. We think (1) is more likely for eruption and deposition of the Allans Beach mound. Taking into account the lack of any wave-generated structures, a deposi-

PGCC17 9/12/08 3:23 PM Page 255

255


6 fig. 8b warm water

5 4 water ingestion into column

cold water

Older volcanics

2

Pre-volcanic sediments

1

3 Intensive fragmentation

Intensive vesiculation clasts not in scale

(a) Fig. 8. Schematic diagram illustrating the interpreted processes of eruption and deposition of Allans Beach succession. (a) Cross-section through the vent region. Magma was strongly vesiculated as it rose into the vent (1). Fragmentation took place as the melt exited the vent (2). Fragmentation in the core of the column was wholly magmatic (2). At the column margin gas condensation brought clasts into contact with seawater and hydroclastic fragmentation occurred (5) together with chilling of pumice–bomb margins. Whole column at exit had mix of large and small clasts. Some large clasts are fragmented into smaller clasts (6), while others were chilled to form cauliflower bombs. Gas–pumice mix has been transported upwards by high-speed gas-vapour stream (straight arrows). Curved arrows indicate turbulent movement.

tion below wave-base is indicated. These data together suggest that unit C formed from a submarine eruption, driven largely by magmatic fragmentation, that fed tephra into an evolving series of mobile aqueous sediment gravity flows (Fig. 8 a–d). This interpretation is further elaborated below.

GAS THRUST TO AQUEOUS FLOWS This section provides an interpretation of other features in terms of subaqueous setting, particularly the result of transforming a gas-supported stream of hot

clasts into aqueous density currents depositing cold clasts. Large clasts best retain an imprint of this twophase history. The larger chilled cauliflower margins (Fig. 4) are inferred to have chilled abruptly as clasts left the gas-thrust column, initially at the column margin and continuing during aqueous transport for some of the larger clasts. The high vesicularity of the pumice in subunit 1 at Allans Beach suggests an energetic and gas-rich eruption, that probably formed a sustained eruption column and water exclusion zone (Kokelaar & Busby, 1992; White, 1996) in which vapour and pumice were transported upwards as a high-speed gas-particle stream (Wilson et al., 1980; Fig. 8b). The

PGCC17 9/12/08 3:23 PM Page 256

256


hot magmatic gases and steam steam, hot water

1

2

cold sea water

3

Fig. 8b. Close up of the margin of the eruption column. Arrows indicate turbulent movement, different shading shows temperature variation among clasts (1 = hot, 3 = cold). Due to the large temperature difference, pumice clasts ingested water (2), which aided rapid cooling of the pumice.

gas-thrust part of the column was probably not very high due to drastic deceleration resulting from the dense overlying water column (Cashman & Fiske, 1991). Lack of any evidence for significant pauses during deposition suggests an uninterupted depositional sequence. As hot pumice passed from the gas-thrust region into the aqueous convective column, the large temperature difference between pumice and seawater caused the pumice to ingest water as it was rapidly cooled under hydrostatic pressure (Whitham & Sparks, 1986). This caused the aqueous convective column’s margin to become unstable, an effect also promoted by the greater drag effect of water, which enhanced turbulent mixing with seawater (Fisher & Schmincke, 1984). A pumice-laden high-concentration density current was formed as the convective column collapsed.

The vesicularity of the pumice clasts in subunit 2 is lower and more variable than in subunit 1 and the abundance of cauliflower bombs is lower indicating a stronger interaction with water. The fewer large cauli-flower bombs and the appearance of more ash in subunit 2 suggest a change in eruption style from more magmatic to more hydromagmatic (Wohletz & Sheridan, 1983; Wohletz & McQueen, 1984), because hydromagmatic fragmentation was probably more efficient. Interaction of magma and water most likely occurred at the transition from the gas-thrust region of the eruption column during the Allans Beach eruption, as hot clasts entered the convective part of the column (Fig. 8c). Some large clasts may have been disrupted to form small glass shards. Interaction in the vent is unlikely because deposits contain very little wall rock clasts.

PGCC17 9/12/08 3:23 PM Page 257

257


high flux stage

1 collapsing column at water/pumice margin water

3 high density currents

2

top mixing with surrounding water

older volcanics

pre-volcanic sediments

high density current particles + water

particles + steam + magmatic gas

magmatic gas + particles

particles + steam + water

particle movement

Fig. 8c. Collapse of the margin of the column (1) is promoted by the turbulent mixing of water with pumice to form high-concentration aqueous density currents (2). Different shading and patterns (3) indicate turbulence, differences in temperature and concentration of gas, vapour and water.

The vapour forming the gas thrust column condensed, and the column disintegrated to form a series of dilute turbidity currents from which the top beds of the sequence were deposited (Fig. 8d).

CONCLUSION AND IMPLICATIONS The pumice deposits at Allans Beach records deposition from laterally-flowing eruption-fed density currents. The emplacement of high- and low-concentration pyroclastic density currents in subaqueous settings has been documented widely (e.g. Fiske & Matsuda, 1964; Kokelaar, 1983; Kokelaar & Durant, 1983; Cashman & Fiske, 1991; Nishimura et al., 1992; White, 1996; Smellie & Hole, 1997). Most cases discussed are from basaltic or rhyolitic eruptions. Submarine eruption and deposition of highly vesiculated phonolitic pumice has not been previously described.

Compared generally to basaltic or rhyolitic products of subaqueous eruptions (e.g. Kokelaar & Durant, 1983; Cashman & Fiske, 1991; Mueller & White, 1992) those preserved at Allans Beach show transitional characteristics. The higher degree of vesiculation of the phonolitic pumice clasts at Allans Beach indicates a stronger magmatic role than for basaltic scoria. Conversely, bedding is better developed than in high-flux rhyolitic pyroclastic flow deposits. This may indicate that the primary controls on the depositional process of subaqueous eruptions are the eruption flux and magmatic gas content, which directly control column dynamics, rather than the mode of fragmentation, which controls grain size, shape and distribution.

ACKNOWLEDGEMENTS This chapter reports results from the PhD study of

PGCC17 9/12/08 3:23 PM Page 258

258


floating pumice

2

fall out

water

water

1

ash turbidites

pumice beds older volcanics

pre-volcanic sediments legend see fig. 8c Fig. 8d. As eruption waned, more water was ingested into the column (1) and more vapour condensed (2); the eruption column disintegrated and formed a series of dilute turbidity currents in the final stage of eruption.

the first author, supported by a University of Otago Postgraduate Scholarship. Special thanks to Karoly Nemeth for valuable field assistance and helpful discussions in the field. Thanks also to John Smellie, Chuck Landis and Tony Reay for helpful suggestions in the field. Two anonymous referees are thanked for their constructive reviews.

REFERENCES Benson, W.N. (1969) Geological Map of New Zealand, Dunedin District. 1 : 50000. New Zealand Geological Survey. Department of Scientific and Industrial Research. Bogaard, P. & Schmicke, H.U. (1984) The eruptive center of the Late Quaternary Laacher See Tephra. Geol. Rdsch. 73, 933–980. Branney, M.J. & Sparks, R.S.J. (1990) Fiamme formed by diagenesis and burial-compaction in soils and subaqueous sediments. J. geol. Soc. London 147, 919–922. Busby-Spera, C.J. & White, J.D.L. (1987) Variation in peperite textures associated with differing host-sediment properties. Bull. volcanol. 49, 765 –775.

Cashman, K.V. & Fiske, R.S. (1991) Fallout of pyroclastic debris from submarine volcanic eruptions. Science 253, 275–280. Coombs, D.S., White, A.J.R. & Hamilton, D. (1960) Age relations of the Dunedin Volcanic Complex and some paleogeographic implications. Part 2. N.Z. J. Geol. Geophys. 3(4), 572–579. Chough, S.K. & Sohn, Y.K. (1990) Depositional mechanics and sequences of base surges, Songaksan tuff ring, Cheju Island, Korea. Sedimentology 37, 1115–1135. Fisher, R.V. (1961) Proposed classification of volcani-clastic sediments and rocks. Geol. Soc. Am. Bull. 72, 1395–1408. Fisher, R.V. & Schmincke, H-U. (1984) Pyroclastic Rocks. Springer, Heidelberg, 474 pp. Fiske, R.S. & Matsuda, T. (1964) Submarine equivalents of ash flows in the Tokiwa Formation, Japan. Am. J. Sci. 262, 76–106. Fiske, R.S., Cashman, K.V., Shibata, A. & Watanabe, K. (1998) Tephra dispersal from Myojinsho, Japan, during its shallow submarine eruption of 1952–1953. Bull. Volcanol. 59, 262–275. Haq, B.U., Hardenbol, J. & Vail, P.R. (1987) Chronology of fluctuating sea levels since the Triassic. Science 235, 1136–1167. Heiken, G. & Wohletz, K. (1991) Fragmentation processes in explosive volcanic eruptions. In: Volcaniclastic

PGCC17 9/12/08 3:23 PM Page 259

Density current deposits from a submarine vent Sedimentation (Eds Fisher, R.V. & Sinth, G.A.) SEPM Special Publication, 45, 20–26. Hiscott, R.N. & Middleton, G.V. (1980) Fabric of coarse deep-water sandstones Tourelle formation, Quebec, Canada. J. sediment Petrol. 50(3), 703–722. Hiscott, R.N. (1994) Traction-carpet stratification in turbidites – fact or fiction? J. sediment Res. A64(2), 204 –208. Hiscott, R.N. (1994) Loss of capacity, not competence, as the fundamental process governing deposition from turbidity currents. J. sediment Res. A64(2), 209–214. Houghton, B.F. & Wilson, C.J.N. (1989) A vesicularity index for pyroclastic deposits. Bull. Volcanol. 51, 451– 462. Johnson, S.Y. (1985) Water-escape structures in coarsegrained, volcaniclastic, fluvial deposits of the Ellensburg Formation, South-Central Washington. J. sediment Petrol. 56(6), 905–910. Kano, K.-I. (1989) Interactions between andesitic magma and poorly consolidated sediments: Examples in the Neogene Shiraham Group, South Izu, Japan, J. Volcanol. geothermal Res. 37, 59–75. Kneller, B.C. & Branney, M.J. (1995) Sustained highdensity turbidity currents and deposition of thick massive sands. Sedimentology 42, 607–616. Kokelaar, B.P. & Durant, G.P. (1983) The submarine eruption of Surtla (Surtsey, Iceland). J. Volcanol. geothermal Res. 19, 239–246. Kokelaar, B.P. (1983) The mechanism of Surtseyan volcanism. J. geol. Soc. London 140, 939–944. Kokelaar, B.P. & Busby, C. (1992) Subaqueous explosive eruption and welding of pyroclastic deposits. Science 257, 196–200. Kurszlaukis, S., Buettner, R., Zimanowski, B. & Lorenz, V. (1998) On the first experimental phreatomagmatic explosion of a kimberlite melt. J. Volcanol. geothermal Res. 80, 323 –326. Lorenz, V. (1973) On the formation of maars. Bull. Volcanol. 37(2), 183 –204. Lorenz, V. (1975) Formation of phreatomagmatic maardiatreme volcanoes and its relevance to kimberlite diatremes. Phys. Chem. Earth 9, 17–27. Lowe, D.R. (1982) Sediment gravity flows: II. Depositional models with special reference to the deposits of highdensity turbidity currents. J. sediment Petrol. 52(1), 279–296. Lowe, D.R. (1988) Suspended-load fallout rate as an independent variable in the analysis of current structures. Sedimentology 35, 765–776. Martin, U. & White, J. (1997) Evolution of the Dunedin Volcanic Complex. In: Abstracts of the International Association of Sedimentologists, 18th European Meeting, Heidelberg, Germany, pp. 228–229. GAEA heidelbergenesis, Institute of Geology and Palaeontology, Auprecht-karls University, Heidelberg, Germany. Martin, U. (1998) Melting and mingling of phonolitic pumice deposits with intruding basaltic dikes: An example from the Otago Peninsula, New Zealand. In: Abstracts of the International Volcanological Conference (IAVCEC), Cape Town, South Africa, p. 40. University of Cape Town, South Africa. Mueller, W. & White, J.D.L. (1992) Felsic fire-fountaining beneath Archean seas: Pyroclastic deposits of the 2730 Ma

259

Hunter Mine Group, Quebec, Canada. J. Volcanol. geothermal Res. 54, 117–134. McPhie, J. (1993) The Tennant Creek porphyry revisited: A synsedimentary sill with peperite margins, Early Proterozoic, Northern Territory. Aust. J. Earth Sci. 40, 545 –558. Nishimura, A., Rudolfo, K.S., Koizumi, A., Gill, J. & Fujioka, K. (1992) Episodic deposition of Pliocene– Pleistocene pumice from the Izu-Bonoin arc, leg. 126. Proc. Ocean Drilling Prog. Sci. Results 126, 3 –21. Schmid, R. (1981) Descriptive nomenclature and classification of pyroclastic deposits and fragments: recommendations of the IUGS Subcommission on the Systematics of Igneous rock. Geology 9, 41– 43. Schmincke, H.-U. (1988) Pyroklastische Gesteine. In: Sedimente und Sedimentgesteine (Ed. Fuechtbauer, H.). Schweitzbarter, Stuttgart, 1141 pp. Scott, B. & Price, S. (1988) Earthquake-induced structures in young sediments. Tectonophysics 147, 165 –170. Scott, G.H. (1991) Revision of the boundary between Lillburian and Waiauan stages (middle Miocene, New Zealand), NZ J. Geol. Geophys. 34, 397– 406. Smellie, J.L. & Hole, M.J. (1997) Products and processes in Pliocene-Recent, subaqueous to emergent volcanism in the Antarctic Peninsula: examples of englacial Surtseyan volcano construction. Bull. Volcanol. 58, 628– 646. Sohn, Y.K. & Chough, S.K. (1989) Depositional processes of the Suwolbong tuff ring, Cheju Island (Korea). Sedimentology 36, 837–855. Sparks, R.S.J. (1978) The dynamics of bubble formation and growth in magmas: A review and analysis. J. Volcanol. geothermal Res. 3, 1–37. Sparks, R.S.J., Sigurdsson, H. & Carey, S.N. (1980) The entrance of pyroclastic welding. J. Volcanol. geothermal Res. 7, 97–105. Sparks, R.S.J., Gardeweg, M.C., Calder, E.S. & Matthews, S.J. (1997) Erosion by pyroclastic flows on Lascar Volcano, Chile. Bull. Volcanol. 58, 557–565. White, J.D.L. (1996a) Pre-emergent construction of a lacustrine basaltic volcano, Pahvant Butte, Utah (USA). Bull. Volcanol. 58, 249 –262. White, J.D.L. (1996b) Impure coolants and interaction dynamics of phreato-magmatic eruptions. J. Volcanol. geothermal Res. 65, 1–17. Whitham, A.G. & Sparks, R.S.J. (1986) Pumice. Bull. Volcanol. 48, 209 –223. Wilson, L., Sparks, R.S.J. & Walker, G.P.L. (1980) Explosive volcanic eruptions – IV. The control of magma properties and conduit geometry on eruption column behavior. Geophys. J. R. astron Soc. 63, 117–148. Wohletz, K.H. & Sheridan, M.F. (1983) Hydrovolcanic explosions. II. Evolution of basaltic tuff rings and tuff cones. Am. J. Sci. 283, 385 – 413. Wohletz, K.H. & McQueen, R.G. (1984) Experimental studies in hydromagmatic volcanism. In: Studies in Geophysics: Explosive Volcanism: Inception, Evolution and Hazards. National Academy Press, Washington, pp. 158–169. Zimanowski, B., Buttner, R. & Lorenz, V. (1997) Premixing of magma and water in MFCI experiments. Bull. Volcanol. 58, 491– 495.

PGCC18 9/12/08 3:23 PM Page 261


Deltaic density currents and turbidity deposits related to maar crater rims and their importance for palaeogeographic reconstruction of the Bakony–Balaton Highland Volcanic Field, Hungary K. NÉMETH University of Otago, Geology Department, P.O. Box 56., Dunedin, New Zealand

ABSTRACT The Bakony–Balaton Highland Volcanic Field (BBHVF), active in the late Miocene, is located in the Central Pannonian Basin of Hungary and consists of around 100 mostly alkaline basaltic eruptive centres. After volcanism, deposition took place in lakes inside the maar craters. Above the primary volcaniclastic deposits, thick maar-lake volcaniclastic sediments occur. The steeply dipping (25–35°), 25–30-cm thick, coarse-grained, inverse-to-normal graded beds of reworked tuff represent the foresets of large Gilbert-delta fronts built into the maar crater lakes of the BBHVF. The coarse-grained beds were deposited by lowdensity granule debris flows and grain flows. Interstratified 10–15-cm thick beds of fine-grained, cross-bedded, reworked volcaniclastic sandstone and mudstone were probably deposited by turbulent sediment gravity flows. The delta fronts generally indicate transportation from north to south, suggesting a strong N–S trending fluvial system, active during or, shortly after volcanism in the BBHVF. The juvenile fragments in the deltaic sediments are often highly vesiculated, rounded/semirounded glassy lapilli. These suggest that the maar volcanism was related to widespread Strombolian-type explosive volcanism that followed the maarforming phreatomagmatic events. Deposits derived from scoria cones were easily washed into the steepwalled maar basins and deposited by debris flows.

INTRODUCTION (main karst aquifer), Permian Red Sandstones and Silurian schist formations fuelled the initial phreatomagmatic eruptions and produced unusual ‘Tihanytype’ maars (Németh & Martin, 1998, 1999a,b) with unusually deep maar basins. Similar structures were reported from Mexico by Aranda-Gomez and Luhr (1996). This paper gives a short summary of the basic characteristics of maar lake deposits and especially the Gilbert-type deltas of the BBHVF, mostly on the Tihany peninsula.

In the Bakony–Balaton Highland Volcanic Field (BBHVF ), Hungary (Figs 1, 2 and 3), the formation of large maars and subsequent lake sedimentation (Németh & Martin, 1998) occurred between 7.56 and 2.8 Ma (Balogh et al., 1986; Borsy et al., 1986). The distribution of large hydrovolcanic centres strongly depends on the occurrence of the Pannonian Sandstone Formation in the pre-volcanic stratigraphy. In the west of the area, the Pannonian Sandstone Formation reaches a thickness of 600 m, decreasing eastwards. Where thick Pannonian Sandstone beds are present, normal maar volcanic structures developed, due to the abundant water content of the unconsolidated sand, which led to phreatomagmatic explosions (Lorenz, 1986; White, 1996). With the progressive drying-out of the sand, large lava lakes and Strombolian scoria cones developed in the maar basins, due to the onset of magmatic explosive activity. At the eastern side, the groundwater content of the Mesozoic carbonates

GEOMORPHOLOGY The BBHVF is a region of low elevation bordered by Mesozoic and Palaeozoic fault ridges with an average height of 250–400 m. Between these ridges are Pannonian lacustrine clastic-filled basins (Fig. 1). The individual eruptive centres show a strong correlation


261

N

Slo

e Lak

n

ato l a B

?

?

? sin ? ba

li-

Ka

?

?

e Lak

10 km

Tihany peninsula (on Figs 2 & 3)

Maar lake volcaniclastics (Gilbert-type deltas) in original position

Maar lake volcaniclastic fragments from Holocene debris flanks

?

on t a l Ba

120

200

Phreatomagmatic sequence Late Miocene (Pannonian) fluvio/lacustrine sediments (mostly covered) Mesozoic or older rocks on the surface

Strombolian scoria cones

Lava or intrusive body

Topographic contours at 20 m intervals between 120 and 200 m then at 100 m intervals

120

?

?

?

Fig. 1. Simplified geological map of the Bakony–Balaton Highland Volcanic Field (BBHVF), Hungary.

N

120

200 300

?

0

120

Yu Hr Calculated transportation direction of tephra into the maar basins Transportation direction (sure) T (unsure)

Ro 100 km

Hungary

Budapest

Lake Balaton

LHPVF BBHVF

300

0

Sk

500

30

A

200

0

UA

400

0 30 0

0 30 20 20

0

0

0

20

30

40

4

120

300

200 300

00

120

262

12

Bakony- Balaton Highland Volcanic Field (BBHVF)

PGCC18 9/12/08 3:23 PM Page 262

K. Németh

PGCC18 9/12/08 3:23 PM Page 263

263

Palaeogeographic reconstruction of the BBHVF

Bay of Füred

Lake Balaton

05

15

110 25 00

22

20

Kiserdõ-tetõ 35 120

15

Lake Külsõ (swamp)

25

35

12 0

Lake Belsõ 20 130 25

30

23

45

160

140

Hawaiian spatter deposits

140

Csúcs-hegy

25

15 0

20

N

18 08 1 40

12 0

10

18

Str ombolian fall deposits Hot spring cemented maar -lake sediments

180

Maar-lake carbonate sediments Maar-lake reworked volcaniclastics Primary phreatomagmatic volcaniclastics

Lake Balaton

Holocene volcanic debris flank Pannonian Sandstone Formation and Holocene Formations

Reconstruction of the possible positions of the maar crater rims 20

Dip direction of primary phreatomagmatic deposits

10 Dip direction of reworked volcaniclastics

1 km

Topographic contours at 10 m intervals Fig. 2. Geological map of the Tihany peninsula showing the distribution of primary and reworked volcaniclastic deposits in the area.

PGCC18 9/12/08 3:23 PM Page 264

264

K. Németh log CFU/cm–2

110

Number of subjects

14

Csúcs-hegy

0 TNF-a (pg/ml)

Delta fr Delta fr Delta fr Tephra lobe of Dip dir Possible

Fig. 3. Reconstruction of large Gilbert-type delta fronts on the Tihany peninsula. Different patterns represent different lobes of large Gilbert-type delta fronts. Delta fronts with similar pattern belong to the same maar crater.

PGCC18 9/12/08 3:23 PM Page 265


Fig. 4. Overview (E–W) of a hillside of Kiserdõ-tetõ (Tihany peninsula) formed by a remnant of a Gilbert-type delta front. Dashed lines represent dipping of beds (LK, Lake Kûlsõ; LB, Lake Belsõ).

265

Fig. 5. Overview of the Csúcs-hegy outcrop at Tihany. Note the steeply dipping beds of Gilbert-type delta front (GD) covered by silicified maar lake carbonates (ML).

with the occurrence of the Pannonian clastic sediments, indicating the importance of these beds as a water source for the phreatomagmatic explosive activity during the volcanism. The hydrovolcanic landforms were maars with low rims. Strombolian scoria cones and shield volcanoes formed on the more elevated ridges (Fig. 1). In those areas, where the karst aquifer was present in the basement rock, unusual maar volcanic structures (Tihany-type maar volcanoes) developed (Németh & Martin, 1998, 1999a,b). Where thick Pannonian sediments are present, normal maar volcanoes built up. Both types of maar volcanic centres functioned as local basins within which thick maar lake deposits accumulated. The more steeply dipping beds have better preservation potential, due to the strong carbonate and silica cementation, eroding as steep ridges, usually higher than the flat-topped primary volcaniclastic remnants (Fig. 4). The erosional remnants of former Gilbert-type delta fronts are probably the most commonly preserved volcanic landforms in the BBHVF (Figs 4 – 6).

PRIMARY AND REWORKED VOLCANICLASTIC DEPOSITS RELATED TO MAARS: TEXTURAL DIFFERENCES Maars are monogenetic volcanic craters, cut into preeruptive country rocks and surrounded by a low ring wall (tephra /tuff ring) of pyroclastic material (e.g. Fisher & Schmincke, 1984; Cas & Wright, 1987). Originally the term ‘maar’ described a topographic feature, consisting of a crater and a tephra /tuff rim. This term incorporates the ring wall, the crater sediments, the diatreme, and the feeder dyke. The syn-eruptive processes

Fig. 6. Overview of the Kiserdõ-tetõ outcrop at Tihany. The lower part of the outcrop represents the crater fill (lower diatreme) deposits (LD), covered by steeply dipping Gilberttype delta front beds (GD). The dashed line represents the position of a chute structure above the lower diatreme deposits.

PGCC18 9/12/08 3:23 PM Page 266

266

K. Németh

are driven by magma /groundwater, Fuel- (Impure) Coolant Interaction (FCI) and produce mostly base surge and phreatomagmatic fallout beds up to a few tens of metres’ thick around the excavated maar crater (e.g. Wohletz, 1986; White, 1996). During the posteruptive processes subsequent to the formation of the maar basin, the undercutting of the groundwater level lead to formation of a lake. The filling of this lake was usually controlled by (i) mass movements (mass flows of any type from the inner crater wall), (ii) delta deposits, (iii) atmospheric loads, mostly ash fall from nearby eruptive sites, (iv) production of organic matter in the lake and (v) intensive mineral-rich spring activity (Büchel, 1993). Especially in ancient maars and their deposits, it is crucial to distinguish primary and secondary volcaniclastic deposits. Primary volcaniclastic deposits of maar volcanoes are mostly generated by turbulent pyroclastic density currents (base and /or pyroclastic surges). The tephra forming the low rims of tuff rings and maars is usually fine grained, and tends to be consolidated (Moore, 1967; Sheridan & Wohletz, 1983). Abundant finegrained ash contains a high proportion of fine to medium-grained blocky pyroclasts (Wohletz, 1986). In general the volcanic glass fragments from these primary deposits are non- or weakly vesiculated, and they do not show significant geochemical variation. Maar ejecta contains abundant non-juvenile debris, torn from the pre-eruptive substrate (Lorenz, 1986). Near-vent explosion breccias may show large impact sags but many large blocks, however, are carried by the surge currents and are matrix-supported with no underlying depressions. Beds with abundant accretionary lapilli antidunes and scour-fills are common (e.g. Sheridan & Wohletz, 1983). Volcaniclastic rocks from reworked volcaniclastic beds, such as the beds of a delta front building into a maar crater, can be separated from primary volcaniclastics by their textural characteristics as well as by the presence of different volcanic glass clast populations in the same beds (tachylitic and sideromelane glasses). Usually the scoriaceous clasts are better rounded and relatively well sorted. Remnants of altered, palagonitized, irregular rims around larger clasts are significant signs that reworking processes produced the deposits. The relatively high percentage of free, broken crystals in the individual rock fragments, especially those rimmed by palagonite, also represent post-eruptive reworking. Importantly the typical bedding characteristics (bomb sags, accretionary lapilli) of individual units are useful features that distinguish primary from reworked deposits in an ancient maar sequence.

GILBERT-TYPE DELTAS Gilbert-type deltas are produced by progradation of alluvial or fluvial systems into a standing body of water, either lacustrine or marine, usually in a basin with a steeply inclined margin (Gilbert, 1885). They have a very steep depositional surface, i.e. foreset slopes that are inclined near the angle of repose and dominated by gravity-driven processes (e.g. Massari, 1996; Sohn et al., 1997). Tephra deltas occur in most of the BBHVF maar craters. In several places, exceptionally good exposures show the structure of the steeply dipping foreset of the individual delta fronts (Figs 4–6). Continuous transitions between the flat-lying deltaic topset and the steep foreset have not yet been described from the BBHVF. Deltaic topset strata have probably been eroded, or are just not well exposed in the studied areas. In several places (mostly on the Tihany Peninsula, Figs 1 and 3) complex geometrical structures of steeply bedded, scoriaceous, coarse-grained lapilli beds intercalated with finer-grained, gently dipping, cross-bedded reworked tuff layers, suggest complex transition zones between topset and foreset zones of Gilbert-type delta fronts, as similar characteristics were reported from the Hopi Butte eruptive centres (White, 1992). The best exposures of deltaic foresets are located in the Tihany Peninsula (Kiserdõ-tetõ, Csúcshegy), with at least four delta fronts (Figs 1–3). The beds of the delta front foresets are usually not covered by other post-maar beds. They have also been eroded in most of the topset settings. Locally, late maar-lake carbonate beds overlie the original foreset beds, representing the late central lacustrine facies in the maar basin. Volcaniclastic beds, interpreted as foresets of the BBHVF delta fronts, are generally 5–25-cm thick, and dip steeply (20–30°) towards the former crater position, forming a concave geometry controlled by the steep inner morphology of the former maar basin (Figs 5 and 6). The total thickness of the foreset sequences is at least 40 m at Tihany peninsula (Kiserdõ-tetõ and Csúcs-hegy). The individual beds usually have irregular tops and bases (Fig. 7). Coarsegrained beds are usually laterally discontinuous and they grade into fine-grained cross-beds. Scoriaceous, fragment-rich scour-fill structures are common. Locally, carbonate (calcite) enrichment is also common, in several cases forming faint bedding. In several places, the large scoriaceous clasts contain a large amount of probably secondary carbonate (calcite) and the

PGCC18 9/12/08 3:23 PM Page 267


Fig. 7. Irregular base and top in a coarse-grained, grainsupported bed sets from the Kiserdõ-tetõ locality (Tihany).

clasts are carbonate-cemented. Fine carbonate-rich beds and laminae are present mostly in the bedded reworked tuff sequences. Bed-specific carbonate alteration probably resulted from both influx of detrital carbonate across the pre-volcanic surface, and alteration of tephra beds by carbonate saturated lake waters during periods of flood or periods of high net evaporation respectively, as White (1989, 1992) described in the Hopi Buttes maar craters. This interpretation for the origin of the carbonate would require revision of our view of the paleoclimatology during the eruptive history of the BBHVF, which is usually considered to be a dry, cold subarctic climate. The evaporative original the carbonate in this interpretation suggests a relatively warm, dry summer and therefore strong seasonality of the region during the period of volcanism. However, the high carbonate content of maarlake water could also have been derived from runoff from the large Mesozoic carbonate hills or from

267

Fig. 8. Inverse-to-normally graded, coarse-grained beds of Lithofacies 1 (L1) from the Csúcs-hegy locality (Tihany). Note the interbedded fine-grained, low-angle crosslaminated Lithofacies 2 (L2) in the middle of the photo. ML is a maar-lake silicified carbonate lamina set above L1. The coin is 2.5 cm in diameter.

subsurface water via new fractures which developed immediately after the phreatomagmatic activity. The carbonate content of these beds could also be explained by substantial hot spring water entering the maar lakes. Coarse-grained beds show characteristic inverseto-normal grading (Fig. 8). Isolated outsized basaltic clasts (without any impact craters or folds), usually 5–10 cm in diameter, lie within many beds. Between the coarse-grained foreset beds there are a few finegrained primary volcaniclastic beds, usually with fallout characteristics. They usually contain numerous accretionary lapilli, with a maximum diameter of 1 cm. These interbedded primary volcaniclastic beds suggest ongoing volcanic eruptions adjacent to the

PGCC18 9/12/08 3:23 PM Page 268

268

K. Németh MEAN GRAIN SIZE MAX. CLAST SIZE FACIES (CM) Sand Gravel (ash) (lapilli) F MC F M C 0 10 20 30 40 L1 L2 CH MC PF

~9 fig. 16

fig. 8

3

2

THICKNESS (METRES)

fig. 11

1

0 outsized accidental lithics outsized basaltic lithics scoria gravels scour fill carbonate lamina primary volcaniclastic lamina

modified grain flow turbidity currents FACIES INTERPRETATION chutes filled by debris flow suspension deposited carbonate primary volcaniclastic (fall out) Simplified measured log from Csúcs-hegy Gilbert-type delta (Tihany peninsula - west maar)

earlier-developed maar basins which were already accumulating clastic sediments.

LITHOFACIES OF GILBERT-TYPE DELTA FRONTS In general, two major lithofacies groups can be identified on BBHVF delta fronts, similar to those described by Massari (1996) on coarse-grained, Gilberttype progradational wedges in marine environments: (1) massive or graded beds, generally clast-supported; and (2) units with various types of internal lamination, including flat, low-angle, broadly convex-up, and

Fig. 9. Simplified log from the Csúcs-hegy (Tihany) locality. Compare with Fig. 5.

wavy lamination, and backset cross-lamination. The two lithofacies groups were described from two type localities at Tihany peninsula from the Csúcs-hegy (Fig. 9) and Kiserdõ-tetõ (Fig. 10). Lithofacies 1 (L1) The first facies group consists of massive or graded beds, usually with regular, sheet-like geometries. They generally show extensive lateral persistence, although very gradual pinch-outs are common. They are generally a few decimetres thick. Coarse-grained beds consist of pebble and sand (lapilli)-size scoriaceous clasts of average diameter 0.5–1 cm. These beds are

PGCC18 9/12/08 3:23 PM Page 269


269

MEAN GRAIN SIZE MAX. CLAST SIZE FACIES (CM) Sand Gravel (ash) (lapilli) F MC F M C 0 10 20 30 40 L1 L2 CH MC PF

4 fig. 7

3

2 THICKNESS (METRES)

fig. 12

1

0 outsized accidental lithics outsized basaltic lithics scoria gravels bedded tuff fragment

Fig. 10. Simplified log from the Kiserdõ-tetõ (Tihany) locality. Compare with Fig. 6.

scour fill

predominantly clast-supported (Fig. 11). Most beds are inversely-graded or structureless. Normal grading is always related to inversely graded bottom zones of beds (Figs 8 and 11). Some layers show an upward increase in the packing of the larger clasts, with a transition from matrix-supported (sand matrix) to clast-supported texture. The carbonate filling is characteristic of the upper clast-supported parts of the beds. Imbrication of elongated, platy clasts (mostly schist clasts, Tihany Peninsula) is also common as well as slump scars, and small (1–2-m long, 15–20-cm deep) elongated downslope-running scours (Fig. 12). From general studies of normal marine and lacustrine environments, it is considered that massive or

modified grain flow turbidity currents FACIES INTERPRETATION chutes filled by debris flow suspension deposited carbonate primary volcaniclastic (lower diatreme) Simplified measured log from Kiserdõ-tetõ Gilbert-type delta (Tihany peninsula - east maar)

Fig. 11. Hand specimen of a coarse-grained, grainsupported lapilli tuff from Csúcs-hegy (Tihany).

PGCC18 9/12/08 3:23 PM Page 270

270

K. Németh suspension’. This mechanism (‘modified grain flow’) could be applied to model the transportation and depositional environment of the coarse-grained lithofacies group on the slope of the inner side of the crater wall. Slump scars, and small elongated downsloperunning scours (chutes) are common features (Fig. 12), and abound in many modern (Prior et al., 1981; Kostaschuk & McCann, 1987; Prior & Bornhold, 1988; Syvitski & Farrow, 1989) and ancient (Postma, 1984; Postma & Roep, 1985; Postma & Cruickshank, 1988; Colella et al., 1987) deltas. Possibly due to lack of exposure, large-scale chutes have not yet been satisfactorily identified in the BBHVF. Despite its significant depth (up to 300 m), the maar lake itself is not a large basin, and its length is not adequate to develop large chute structures as is common in normal marine environments (e.g. width from 10 to 200 m; depths from 2 to 20 m). The only large chute-type channel identified from the Tihany peninsula was at the Kiserdõ-tetõ outcrop, where a feature measuring 7–10-m wide and 1.5–2-m deep is filled with matrixrich debris flow deposits. However the chutes on deltafronts in the maar basin are generally infilled either by deposits of smaller, less-robust turbidity currents, or by debris flows spawned by slumping of the chute walls, or by suspension settling related to the stream channel mouths, from which the chutes often emanate (Nemec, 1990).

Fig. 12. Small chutes and scours (Ch and arrows with dashed lines) on the lower bed surface of a coarse grained bed (Lithofacies 1) from the Kiserdõ-tetõ (Tihany) outcrop. Their elongation is identical with the bed dip direction. The coin is 2.5 cm in diameter.

graded layers are usually deposited by flows in which a critical value of concentration was exceeded, so that the ability of the bed-load layer to move and sort the sediment particles was suppressed (Lowe, 1988). While normally-graded beds may be the result of the collapse of turbulent, high-concentration suspended sediment clouds, layers showing inverse grading throughout, or within the basal division, and predominant a-axis imbrication, may have been generated by highly concentrated, cohesionless sediment gravity flows dominated by frictional/inertial effects. Such flows are often termed ‘modified grain flows’ (Lowe, 1976, 1982). Waning flows may also produce inverseto-normal grading as argued by Hiscott (1994, 1996). In fact, if any traction structures are present, it must be waning flow rather than the result of a ‘collapsing

Lithofacies 2 (L2) The second facies group shows diffuse, planar, crude to very regular lamination with an average grain size of 1 mm (Fig. 13). Individual beds are few centimetres thick (up to 20 cm). The laminae may grade laterally into flat lenses or pebble-sized scoria clasts (average of 1 cm in diameter). Outsized pebbles (up to 15 cm in diameter) mostly of scoria or juvenile lithics are relatively common. Often, large (mostly) pyroxene crystals (up to 1 cm in diameter) form small lenses (5–10-cm long). Individual laminae are ungraded, but faint normal grading can be identified in thicker (>3 cm) beds. This lithofacies contains a high proportion of lenticular units, with 10–25-cm thick planar or shallow concave-up scoured bases and broadly convex-up tops. This lithofacies group is well represented in sequences in Tihany (Kiserdõ-tetõ), but in other places they represent only a small proportion of the total thickness (Káli Basin) (Fig. 1). Diffuse planar lamination on the foreset slopes of Gilbert-type deltas has commonly been modelled as

PGCC18 9/12/08 3:23 PM Page 271


271

likely to occur in the BBHVF maar-lake sediments. High-sphericity rollable clasts are concentrated in the lenses probably because they are more mobile than clasts of other shapes (Massari, 1996).

SCORIA-CONE-DERIVED CLASTS RELATED TO LARGE HYDROMAGMATIC VOLCANIC FIELD

Fig. 13. Lithofacies 2 (L2) from Csúcs-hegy (Tihany). Dashed line represents the border between Gilbert-type delta front beds and silicified maar-lake carbonates (ML). Note the light-coloured, carbonate-rich lamina (arrow) in the fine-grained lithofacies (L2).

resulting from freezing of successive traction carpets at the bases of highly concentrated flows (Colella et al., 1987; Postma & Cruickshank, 1988; Nemec, 1990). Lowe (1982) argued that traction carpets may develop under the influence of dispersive pressure and under upper-stage plane bed conditions, in relatively coarsegrained materials under high-density turbidity currents during their evolution from relatively steady, non-depositing flows to highly unsteady, rapidly sedimenting flows. It is suggested that, even in the case of bed materials coarser than sand, lamination may reflect temporal and spatial variations in bed shear stress and lift forces acting on sediment particles in motion over the bed, related to strongly fluctuating conditions such as burst and sweep cycles (Hiscott, 1994, 1996; Massari, 1996). In the planar bedded, mostly fine-grained volcaniclastic facies, large lenticular units with highly rounded, well-sorted scoriaceous fragments, averaging 1–2 cm in diameter are common. These lenses are elongated and have a faint inner structure with diffuse bedding. Lenticular units similar in geometry and internal structure to those described in the BBHVF delta fronts have been reported from (1) the foreset, toeset and bottomsets of Gilbert-type deltas and other delta fronts in marine or lacustrine environments (Colella et al., 1987; Postma & Cruikshank, 1988), (2) coarsegrained deposits of high-density turbidity currents (Hendry & Middleton, 1982; Surlyk, 1984), (3) deposits of hyperconcentrated flood flows at points of flow expansion in subglacial eskers (Brennand, 1994) and (4) alluvial-fan deposits related to unconfined sheetfloods (Blair, 1987). The first two of these are most

While studying the texturally distinctive beds of the Gilbert-type delta fronts, a large proportion of scoriaceous lapilli was found within individual beds (Figs 14 and 15). The lapilli grains are usually tachylite and are microvesiculated (10–50%—visual estimation), with elongated, often calcite- or zeolite-filled vesicles. Usually the beds are well sorted and the lapilli are subrounded to well rounded. Around the individual lapilli there are often thin, altered, usually palagonitized rims (Fig. 14). These kinds of rims do not always entirely cover the lapilli. Sideromelane clasts are usually rare, small and broken, with a wide range of grain sizes within the same bed. Large crystals, usually pyroxenes, olivines or quartz crystals of metamorphic origin, are also represented in the beds but in relatively low proportions. A wide range of chemical compositions of the sideromelane glasses are present, even in the same beds. All of these characters suggest that the the clasts originated from Strombolian eruptions. In several beds, thick algae rims on individual grains indicate repeated algal blooms in the maar lake (Fig. 15). Most of the reworked tephra which makes up the delta fronts in the maar basins was not supplied

Fig. 14. Reworked tuff from fine-grained lithofacies (L2, Kiserdõ-tetõ, Tihany) in thin section. Note the roundness of the grains as well as their dark colour rim (mostly altered glass). S, sideromelane; T, tachylite, Q, quartz. The shorter side of the picture is 4 mm.

PGCC18 9/12/08 3:23 PM Page 272

272

K. Németh

Fig. 15. Algae-rimmed scoria lapilli from reworked volcaniclastic deposits (L1, Csúcs-hegy, Tihany). The shorter side of the picture is 4 mm.

from the phreatomagmatic ejecta wall of that crater. The maar volcanoes own ejecta are usually of base surge and phreatomagmatic fallout origin, with a large amount of disrupted accidental lithics from the pre-eruptive rock formations. In contrast, the deltaic tephra in almost all locations in the BBHVF is dominantly composed of variably vesiculated tachylitic juvenile glass and lithic fragments. Sideromelane glasses of phreatomagmatic origin are usually rare and small. Thus, the Gilbert-type delta fronts represent reworked remnants of a widespread Strombolian scoria cone field on the BBHVF.

MAAR LAKE CARBONATES

Fig. 16. Silicified maar-lake carbonate beds (ML, Csúcshegy, Tihany). Note the soft-sediment slumping in the lower part of the photo.

In a few places in the BBHVF a thick, silicified, freshwater carbonate succession covers the volcaniclastic beds. In the Tihany Peninsula the most extensive silicified freshwater carbonate unit was deposited in a maar lake. The carbonate beds are very fine grained and well bedded (Fig. 16). The beds contain a low proportion of volcanic ash which forms dark-coloured laminae. The individual laminae are 0.1–0.7-mm thick, with each lamina representing approximately half-year deposition length, assuming that dark (autumn, winter) and light (spring, summer) laminae couplets represent a full-year depositional cycle due to effects of strong seasonality on lake deposition, similar to varvic (rhythmite) deposits in glacier lakes or other young maar lakes (e.g. Heinz et al., 1993; Talbot & Allen, 1996). The total thickness of carbonate beds in Tihany reaches 10 m, which suggests a minimum of 7150 years and a maximum of 50 000 years of relatively quiet lake sedimentation. Since this calculation is

based on the current (post-erosion) thickness of the maar-lake carbonates, these estimates probably represent minimum time spans since it is unknown what thickness of sediment has been eroded in the last few million years. It is, however, noteworthy that this calculation is consistent with other calculations based on alginite and oil shale deposits which fill maar craters of the nearby Late Miocene volcanic field at the Little Hungarian Plain Volcanic Field (140 000 years— Gérce) and in the central part of the BBHVF (50 000 years—Pula) (Jámbor & Solti, 1976). The thinly laminated carbonate deposits formed in isolated crater lakes. The laminite beds that represent lacustrine sequences were deposited by fallout from suspension, and consist of biogenic and chemically produced carbonate, together with a small amount of quartzo-feldspathic silt. The thinly laminated beds lack any signs of bioturbation, which suggests that

PGCC18 9/12/08 3:23 PM Page 273

Palaeogeographic reconstruction of the BBHVF the bottom water was unsuitable for aquatic life (e.g. White, 1989, 1992). Thermally stratified water with only seasonal overturn, and alkali-rich crater lake water may together have been responsible for the lack of benthic organisms. The carbonate beds contain a large number of soft-sediment structures such as slumps, folds, water escape and fluidization structures (Fig. 16). The large number of hot spring pipes suggest that they played an important role in forming these sedimentary structures. Another possibility, especially in the Tihany region, is that maar eruptive centers and late Strombolian scoria cones were still active, while maar-lake sedimentation was occurring in nearby maar craters. Thus each explosion could have caused an earthquake in the region and shaken the unconsolidated sediment in the maar lake bottom nearby. Large (1-m high, 30–50 cm in diameter), ovoid shaped structures could also be interpreted as rhizoliths, as reported by Jones et al. (1998) from recent hot spring pools from thermal areas of New Zealand.

CONCLUSION: GENERAL MODEL The relative abundance of the sedimentary structures and textures described above suggests that maarlake sedimentation was an important process at the BBHVF. The higher preservation potential of maarlake deposits as Gilbert-type delta fronts or maar-lake carbonates is probably the reason why this type of sediment is one of the most common in the BBHVF. The steeply-dipping Gilbert-type delta front beds form characteristic hills, in contrast to the usually flattopped primary volcaniclastics. The large proportion of scoriaceous fragments in the Gilbert-type delta foreset beds is a sign of widespread Strombolian activity in the BBHVF during volcanism and maar-lake sedimentation. In recent times these eruptive centers were deeply eroded, and are now only represented by a few feeder dikes or isolated, scoriaceous outcrops. The usually N–S elongation of the individual eruptive centres and the development of the Gilbert-type deltas mainly on the north sides of the maars, suggest that the maar-forming explosions took place in a N–Soriented fluvial system (Fig. 17). Processes generally claimed in the literature to be responsible for sediment dispersal on the foresets and toesets of Gilbert-type bodies in normal sedimentary environments are as follows: 1 direct underflows generated by river floods; 2 basinward-flowing river plumes carrying suspended load;

273

3 flows related to slope failures; and (4) storm-driven seaward-directed flows, which may remove sediments from the nearshore zone and contribute gravity-driven sediment surges to the foreset slope (Massari & Parea, 1990). Of these possibilities, (2) is not relevant to freshwater deposits and (4) is unlikely since the maar basin itself is usually no more than a few kilometres (0.5– 3 km) in diameter, and therefore real storm-driven flows cannot develop. Features thought to be characteristic of fluvialdominated Gilbert-type deltas, such as radiating, steep-walled chutes cutting foreset slopes, and the frequency of debris flows (e.g. Prior & Bornhold, 1988), have not been clearly identified on the delta fronts of crater lake deposits of the BBHVF. However small downslope-running scours (chutes) are abundant, suggesting intensive small-scale sudden mass movements on the delta flanks, which were probably generated by continuous small stream-induced underflows. Thus large fluvial systems probably did not operate during the formation of the Gilbert-type deltas, but small streams were probably common. Small debris flow deposits and erosion of channels are described. Slope failures played an exceptionally important role in the mass transfer of sediments, as is usually the case in other steep crater lakes (e.g. Büchel & Lorenz, 1993). A general model is proposed, invoking small stream systems through the former volcanic field as was modelled on the Hopi Buttes in Arizona (White, 1992) but due to lack of exposure no better reconstruction can be suggested. In other Gilbert-type delta fronts from lacustrine or marine environments, sand-dominated foreset deposits are generally thin-bedded, graded to planar-laminated or ripple-cross-laminated, and are commonly intercalated with silt layers similarly to Lithofacies 2 (L2) in the BBHVF. These features suggest deposition by dilute underflows or turbidity currents (Jopling & Walker, 1968; Gustavson et al., 1975; Stanley & Surdam, 1978; Dunne & Hempton, 1984; Flores, 1990). These processes were probably responsible for producing L2 deposits in the BBHVF. In general, gravel-dominated foreset deposits, like Lithofacies 1 (L1) of the BBHVF, are crudely stratified, and their individual beds are difficult to discern because of abrupt lateral changes in bed thickness, grading patterns and grain fabrics. Although these features have hampered detailed facies analysis, several characteristics have been recognized in gravelly foreset deposits, such as a predominance of inverse grading, lack of mud matrix, abundance of well-sorted or openwork gravel layers and lenses, and coarsening

PGCC18 9/12/08 3:23 PM Page 274

274

A.

K. Németh Pre-volcanism

Lacustrine sediments Fluvial sediments

Pannonian fluvial sedim ents

N

Pannonian lacustrine sediments

B.

Phreatomagmatic eruptions, maar-forming process

River

N

Base surge

C.

Strombolian activity adjacent to maar-craters

Maar

N

eatomagmatic

D.

Erosion of the scoria cones into maar-craters. Gilbert-type delta front building stage

Base surge

Scoria cones

Maar

N

Eroded scoria cones

Gilbert-type delta fronts Gilbert-type delta fronts

E.

M a a r- l a ke s e d i m e n t a t i o n interacting with products of nearby active phr eruptive centres

Active phreatomagmatic eruptive center Base surge Maar-lake

N of clast size towards the down-dip margin of a bed or the base of foreset slopes. Most of these features have been interpreted mainly in terms of the grainflow theory described by Bagnold (1954), i.e. as cohesionless debris flows, modified grain flows and grain

Fig. 17. Phreatomagmatic explosive activity and maar-lake sedimentation in a small stream fluvial system with Strombolian-type scoria cones, and its relation with maar-lake sedimentation. (A) Pre-volcanic shallow lacustrine and fluvial sedimentation. (B) Phreatomagmatic explosive maar-forming volcanic activity in active hydrogeological zones. (C) Strombolian activity due to the lack of water supply. (D) Intensive erosion of the maar-crater walls as well as the scoria cones into the maar basins. (E) Base surge and phreatomagmatic fallout tephra from nearby active vents depositing into a maar crater, intercalated with different type of maar-lake deposits.

avalanches (Postma, 1984; Postma & Roep, 1985; Colella et al., 1987; Massari & Parea, 1990; Mastalerz, 1990; Nemec, 1990; White, 1992). Other workers have inferred turbulent flow processes (e.g. Postma & Cruickshank, 1988).

PGCC18 9/12/08 3:23 PM Page 275

Palaeogeographic reconstruction of the BBHVF The observed cyclic facies changes from L1 to L2 (i.e. from graded or massive (mostly coarse) beds to some kind of laminated facies), suggests an increase in flow turbulence which was probably due to a decrease in flow concentration (e.g. Postma, 1984). Highly concentrated flows dominated by frictional forces, existing as such, or segregated near the flow base by a process of ‘gravity transformation’ (Fisher, 1983), can ‘freeze’ on relatively steep slopes such as a maar-crater inner slope (20–70°), whereas more dilute flows may accelerate downslope, incorporate water, and became more mobile and turbulent (Postma & Roep, 1985). In this way the relative abundance of a particular lithofacies (L1 or L2) is related to the position of the remnant of a delta front. In the west at Tihany peninsula (Csúcs-hegy), L1 is dominant, thus this area must represent a medial position on the delta slope. The eastern delta front remnants contain progressively increasing amounts of L2, and thus represent medial to distal position on the delta slope. A generalized schematic model is shown in Fig. 17, showing the evolution of the depositional environment of the BBHVF maar craters. Figure 17(A) and (B) shows the hydrovolcanic activity in the fluvial basin, and in Fig. 17(C) the development of the widespread Strombolian scoria cone field is shown. In Fig. 17(D) the mostly N–S Gilbert-type delta front development is reconstructed. Figure 17(E) shows the possibility of intercalated primary volcaniclastics with the ongoing maar-lake deposition. The asymmetric nature of the preserved Gilbert-type delta fronts, suggest that stream systems entrained the tephra from the northern side of the lakes. The sediment load was probably transported and deposited in the maar lake by underflows depending on the density difference between river/stream and lake water, as was reported in a recent analogue from the Eifel District, Germany (Drohman & Negendank, 1993). Sediment-laden stream water flowed down the delta-front from N–NW to S–SE, derived probably from spring runoffs. Another possibility is that delivery of fluvial sediments to the BBHVF maar craters could have resulted in simple slope instability on the steep inner side of the maar craters, a process which is also common and well reported in the recent maar craters e.g. Ukinrek Maar, Alaska (Büchel & Lorenz, 1993). In general, the nature of the reworked volcaniclastic beds of Gilbert-type delta fronts from the BBHVF suggest that the processes that formed these beds are similar to those described for fjord deltas and other large, coarse grained, deep water deltaic systems (e.g.

275

Prior & Bornhold, 1988; Kazanci, 1990; Kazanci & Varol, 1990). The striking difference between these and maar lake-evolved Gilbert-type delta fronts is that maar craters of the BBHVF were probably fed by a freely migrating small stream system transporting mostly scoriaceous tephra to the entire northern halves of the semicircular lakes. The result in most cases was a crescent-shape delta foreset that prograded directly inward from the crater wall, rather than building outward from a single, constricted point source (e.g. as in fjords), such as that described by White (1992) from maar lakes of Hopi Buttes, Arizona. The fact that all the deltaic foresets studied at Tihany are onlapped by all other crater-filling sediments (silicified maar-lake carbonates), indicates that the deltas developed relatively early in the craters’ history, which is similar to what White (1992) reported from the maar lakes of Hopi Buttes, Arizona. Another striking similarity to the Hopi Buttes maar craters is that deposits within these craters show very limited input of their own ejecta (e.g. phreatomagmatic ash—sideromelane). This emphasizes that maarcrater deposits are largely insensitive to their own eruptive processes, but are more sensitive to the sedimentary environment into which the volcanoes erupted and the post-eruptive sedimentary environment of the surrounding area. According to the reconstructed N–S small stream systems during the volcanism, the paleogeomorphology of the region was probably very similar to that seen at the present day, with small stream systems and large swampy areas between gentle hill sides, as it has been since the Pleistocene (e.g. Cserny, 1997).

ACKNOWLEDGEMENTS I would like to thank Ulrike Martin (University of Otago, New Zealand), Nizamattin Kazanci (University of Ankara, Turkey), László Fodor (Eötvös University, Hungary), for suggestions and discussions during the field work. Thanks also due to James D.L. White (University of Otago, New Zealand) for discussions about phreatomagmatism and crater lake sedimentation and to Gábor Csillag (Geological Institute of Hungary) for showing me key outcrops in the field. The manuscript benefited from the suggestions and comments of journal reviewers, Ray Cas and Tim Druitt. Volume editor Ben Kneller is also thanked for his constructive suggestions. Thanks also due to Andrei van Dusschoten (University of Otago, New Zealand) for careful language corrections.

PGCC18 9/12/08 3:23 PM Page 276

276

K. Németh

REFERENCES Aranda-Gomez, J.J. & Luhr, J.F. (1996) Origin of the Joya Honda maar, San Luis Potosi, México. J. Volcanol. geothermal. Res. 74, 1–18. Bagnold, R.A. (1954) Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. S. London, Ser. A 225, 49–63. Balogh, K., Árva-Sós, E. & Pécskay, Z. (1986) K/Ar dating of post-Sarmatian alkali basaltic rocks in Hungary. Acta Miner. Petrol., Szeged 28, 75 –93. Blair, T.C. (1987) Sedimentary processes, vertical stratification sequences, and sedimentology of the Roaring River alluvial fan, Rocky Mountain National Park, Colorado. J. sediment. Petrol. 57, 1–18. Borsy, Z., Balogh, K., Kozák, M. & Pécskay, Z. (1986) Újabb adatok a Tapolcai-medence fejlõdéstörténetéhez. [New data to the evolution of Tapolca Basin] (in Hungarian with English abstract) Acta Geogr. Debrecina 23, 79 –104. Brennand, T.A. (1994) Macroforms, large bedforms and rhytmic sedimentary sequences in subglacial eskers, southcentral Ontario: implications for esker genesis and meltwater regime. Sediment. Geol. 91, 9–55. Büchel, G. (1993) Maars of the Westeifel, Germany. In: Lecture Notes in Earth Sciences: Paleolimnology of European Maar Lakes (Eds Negendank, J.F. & Zolitschka, B.), Vol. 49. Springer-Verlag, Berlin, pp. 1–13. Büchel, G. & Lorenz, V. (1993) Syn and post-eruptive mechanism of the Alaskan Ukinrek Maars in 1977. In: Lecture Notes in Earth Sciences: Paleolimnology of European Maar Lakes (Eds Negendank, J.F. & Zolitschka, B.), Vol. 49. Springer-Verlag, Berlin, pp. 15– 60. Cas, R.A.F. & Wright, J.V. (1987) Volcanic Succesions. Modern and Ancient. Allen and Unwin, London, 528 pp. Colella, A., De Boer, P.L. & Nio, S.D. (1987) Sedimentology of a marine intermontane Pleistocene Gilbert-type fan-delta complex in the Crati Basin, Calabria, southern Italy, Sedimentology 34, 721–736. Cserny, T. (1997) Environmental geological research in the Lake Balaton region. Z. Geomorph. Suppl. Bd. NF 110, 137–144. Drohman, D. & Negendank, J.F.W. (1993) Turbidites in the sediments of Lake Meerfelder Maar (Germany) and the explanation of suspension sediments. In: Lecture Notes in Earth Sciences: Paleolimnology of European Maar Lakes (Eds Negendank, J.F. & Zolitschka, B.), Vol. 49 SpringerVerlag, Berlin, pp. 195 –208. Dunne, L.A. & Hempton, M.R. (1984) Deltaic sedimentation in the Lake Hazar pull-apart basin, southeastern Turkey. Sedimentology 31, 401– 412. Fisher, R.V. (1983) Flow transformations in sediment gravity flows. Geology 11, 273–274. Fisher, R.V. & Schmincke, H-U. (1984) Pyroclastic Rocks. Springer-Verlag, New York, 472 pp. Flores, R.M. (1990) Transverse and longitudinal Gilberttype deltas, Tertiary Coalmont Formation, North Park Basin, Colorado, USA. In: Coarse Grained Deltas (Eds Colella, A. & Prior, B.), Vol. 10. International Association of Sedimentologists, Special Publication, pp. 223 –233. Gilbert, G.K. (1885) The topographic features of lake shores. U.S. Geol. Surv. Ann. Rep. 5, 69–123.

Gustavson, T.C., Ashley, G.M. & Boothroyd, J.C. (1975) Depositional sequences in glaciolacustrine deltas. In: Glaciofluvial and Glaciolacustrine Sedimentation (Eds Jopling, A.V. & McDonald, B.C.), Vol. 23, Society of Economic Paleontologists and Mineralogists, Special Publication, pp. 264 –280. Heinz, T., Rein, B. & Negendank, J.F.W. (1993) Sediments and basin analysis of Lake Schalkenmehrener maar. In: Lecture Notes in Earth Sciences: Paleolimnology of European Maar Lakes (Eds Negendank, J.F. & Zolitschka, B.), Vol. 49, Springer-Verlag, Berlin, pp. 149 –161. Hendry, H.E. & Middleton, G.V. (1982) Grosses Roches Conglomerate. In: Excursion 7B: Paleozoic Continental Margin Sedimentation in the Quebec Appalachians, Field Excursion Guide Book (Eds Hesse, R., Middleton, G.V. & Rust, B.R.), Eleventh International Sedimentological Congress, McMaster University, Hamilton, Ontario, Canada, 22–27 August 1982, pp. 153 –160. Hiscott, R.N. (1994) Traction-carpet stratification in turbidites – fact or fiction? J. sediment. Res. A64, 204 –208. Hiscott, R.N. (1996) Traction-carpet stratification in turbidites – fact or fiction? – Reply. J. sediment. Res. A65, 704 –705. Jámbor, A. & Solti, G. (1976) A Balaton-felvidéken és Kemenesháton felkutatott felsõpannóniai olajpalaelõfordulás földtani viszonyai. [Geological conditions of the Upper Pannian oil-shale deposit recovered in the Balaton Highland and at Kemeneshát] (in Hungarian with English abstract) MAFI Jel. 1974-rõl, 193 –219. Jones, B., Renaut, R.W., Rosen, M.R. & Klyen, L. (1998) Primary siliceous rhyzolithes from Loop Road hot springs, North Island, New Zealand. J. sediment. Res. 68(1), 115 – 123. Jopling, A.V. & Walker, R.G. (1968) Morphology and origin of ripple-drift cross-lamination, with examples from the Pleistocene of Massachusetts. J. sediment. Petrol. 38, 971–984. Kazanci, N. (1990) Fan-delta sequences in the Pleistocene and Holocene Burdur Basin, Turkey: the role of basinmargin configuration in sediment entrapment and differential facies development. In: Coarse Grained Deltas (Eds Colella, A. & Prior, B.), Vol. 10. International Association of Sedimentologists, Special Publication, pp. 185 –198. Kazanci, N. & Varol, B. (1990) Development of a mass flow-dominated fan-delta complex and associated carbonate reefs within a stransgressive Palaeocene succession, central Anatolia, Turkey. Sediment. Geol. 68, 261–278. Kostaschuk, R.A. & McCann, S.B. (1987) Subaqueous morphology and slope processes in a fiord delta, Bella Coola, British Columbia. Can. J. Earth Sci. 24, 52–59. Lorenz, V. (1986) On the growth of maars and diatremes and its relevance to the formation of tuff rings. Bull. Volcanol. 48, 265–274. Lowe, D.R. (1976) Grain flow and grain flow deposits. J. sediment. Petrol. 46, 188–199. Lowe, D.R. (1982) Sediment gravity flows: II. Depositional models with special reference to the deposits of highdensity turbidity currents. J. sediment. Petrol. 52, 279–297. Lowe, D.R. (1988) Suspended-load fallout rate as an independent variable in the analysis of current structures. Sedimentology 35, 765–776. Massari, F. (1996) Upper-flow-regime stratification types on steep-face, coarse-grained, Gilbert-type progradational

PGCC18 9/12/08 3:23 PM Page 277

Palaeogeographic reconstruction of the BBHVF wedges (Pleistocene, Southern Italy). J. sediment. Res. 66(2), 364–375. Massari, F. & Parea, G.C. (1990) Wave-dominated Gilberttype gravel deltas in the hinterland of the Gulf of Taranto. In: Coarse Grained Deltas (Eds Colella, A. & Prior, B.), Vol. 10. International Association of Sedimentologists, Special Publication, pp. 311–331. Mastalerz, K. (1990) Diurnally and seasonally controlled sedimentation on a glaciolacustrine foreset slope: an example from the Pleistocene of eastern Poland. In: Coarse Grained Deltas (Eds Colella, A. & Prior, B.), Vol. 10. International Association of Sedimentologists, Special Publication, pp. 297–309. Moore, J.G. (1967) Base surge in recent volcanic eruptions. Bull. Volcanol. 30, 337–363. Németh, K. & Martin, U. (1998) Groundwater and gas rich magma controlled phreatomagmatic (maar/diatreme) volcanism in the Balaton Highland Volcanic Field. Pannonian Basin, Hungary. IVC 98, Abstract Book, Cape Town, RSA, p. 43. Németh, K. & Martin, U. (1999a) Volcanic glass study of volcaniclastic deposits of Tihany-type and normal maar volcanoes of the Bakony–Balaton Highland Volcanic Field (BBHVF), Hungary. IUGG 99 – IAVCEI 99, Abstract Book, Birmingham, UK, p. B171. Németh, K. & Martin, U. (1999b) Late Miocene paleogeomorphology of the Bakony–Balaton Highland Volcanic Field (Hungary) using physical volcanology data. Z. Geomorph. NF 43(4), 417– 438. Nemec, W. (1990) Aspects of sediment movement on steep delta slopes. In: Coarse Grained Deltas (Eds Colella, A. & Prior, B.), Vol. 10. International Association of Sedimentologists, Special Publication, pp. 29 –73. Postma, G. (1984) Mass–flow conglomerate in a submarine canyon: Abrioja fan delta, Pliocene, southeast Spain. In: Sedimentology of Gravels and Conglomerates (Eds Kostern, E.H. & Steel, R.J.), Vol. 10. Canadian Society of Petroleum Geologists, Memoir, pp. 237–258. Postma, G. & Cruickshank, C. (1988) Sedimentology of a Late Weichselian to Holocene terraced fan delta, Varangerfjord, northern Norway. In: Fan Deltas: Sedimentology and Tectonic Settings (Eds Nemec, W. & Steel, R.J.). London, Blackie, pp. 144 –157. Postma, G. & Roep, T.B. (1985) Resedimented conglomerates in the bottomsets of Gilbert-type deltas. J. sediment. Petrol. 55, 874 – 885.

277

Prior, D.B. & Bornhold, B.D. (1988) Submarine morphology and processes of fjord fan deltas and related high-gradient systems: modern examples from British Columbia. In: Fan Deltas: Sedimentology and Tectonic Settings (Eds Nemec, W. & Steel, R.J.). London, Blackie, pp. 125 –143. Prior, D.B., Wiseman, W.J., Jr. & Bryant, W.R. (1981) Submarine chutes on the slopes of fiord deltas. Nature 290, 326 –328. Sheridan, M.F. & Wohletz, K.H. (1983) Hydrovolcanism: basic considerations and review. J. Volcanol. geothermal. Res. 17, 1–29. Sohn, Y.K., Kim, S.B., Hwang, I.G., Bahk, J.J., Choe, M.Y. & Chough, S.K. (1997) Characteristics and depositional processes of large-scale gravelly Gilbert-type foresets in the Miocene Doumsan fan delta, Phoang Basin, SE Korea. J. sediment. Res. 67(1), 130 –141. Stanley, K.O. & Surdam, R.C. (1978) Sedimentation on the front of Eocene Gilbert-type deltas, Washakie Basin, Wyoming. J. sediment. Petrol. 48, 557–574. Surlyk, F. (1984) Fan-delta to submarine fan conglomerates of the Volgian-Valanginian Wollaston Group, east Greenland. In: Sedimentology of Gravels and Conglomerates (Eds Koster, E.H. & Steel, R.J.), Vol. 10. Canadian Society of Petroleum Geologists, Memoir, pp. 359–382. Syvitski, J.P.M. & Farrow, G.E. (1989) Fjord sedimentation as an analogue for small hydrocarbon-bearing fan deltas. In: Deltas: Sites and Traps for Fossil Fuels (Eds Whateley, M.K.G. & Pickering, K.T.), Vol. 41, Geological Society, London, pp. 21– 43. Talbot, M.R. & Allen, P.A. (1996) Lakes. Rhytmites. In: Sedimentary Environments: Processes, Facies and Stratigraphy, 3rd edn (Ed. Reading, H.G.) Blackwell Science, Oxford, pp. 100–102. White, J.D.L. (1989) Basic elements of maar-crater deposits in the Hopi Buttes Volcanic Field, northeastern Arizona, USA. J. Geol. 97, 117–125. White, J.D.L. (1992) Pliocene subaqueous fans and Giberttype deltas in maar crater lakes, Hopi Buttes, Navajo Nation (Arizona), USA. Sedimentology 39, 931–946. White, J.D.L. (1996) Impure coolants and interaction dynamics of phreatomagmatic eruptions. J. Volcanol. geothermal Res. 74, 155 –170. Wohletz, K.H. (1986) Explosive magma–water interactions: thermodynamics, explosion mechanisms and field studies. Bull. Volcanol. 48, 245–264.

PGCC19 9/12/08 3:21 PM Page 279


Synsedimentary deformation in the Lower Muschelkalk of the Germanic Basin K . F Ö H L I S C H and T . V O I G T Institut für Geowissenschaften der Friedrich-Schiller-Universität Jena, Burgweg 11, 07749 Jena, Germany

ABSTRACT The Lower Muschelkalk (Anisian, Triassic) of the Germanic Basin represents an epicontinental carbonate depositional system in which synsedimentary deformation structures occur very commonly. Several types of synsedimentary deformation developed: load structures are most abundant in the marginal sandy facies, whereas in the marly limestones of the basin centre, subaqueous mass movements like debrites and slumps prevail. The rheological behaviour of carbonate mass flows is significantly controlled by diagenesis (degree of carbonate redistribution). Micritic intraclasts of debris flows were formed in situ by shearing of early diagenetic carbonate nodules from their marly envelope during transport. Debrites and slide deposits are accompanied by the development of vein structures and by ball-and-pillow structures, which are thought to be of seismic origin. The wide regional distribution of individual sediment gravity flow deposits imply that the mass flows were triggered by earthquakes. Orientation of vein structures suggest that the epicentres were distributed at major graben structures near the early Triassic Tethys margin.

INTRODUCTION depositional system which was established during the Anisian. The Germanic Basin was situated at 20° N and was connected to the open marine Tethys across the Moravo-Silesian and East Carpathian Gateways (Ziegler, 1982) (Fig. 1). Open marine environments prevailed in the eastern part of the Germanic Basin as reflected by rich faunas, calcareous algae and formation of small reefs (Szulc, 1993). To the west, decreasing faunal diversity and the occurrence of magnesium-rich carbonates point to a more restricted environment with higher salinities. A retrogradational clastic margin of fine-grained shallow marine sandstones in southern Germany and western France reflects the late Anisian transgression of the Muschelkalk sea (Schwarz, 1970). To the north-west, a transition to laminated cryptalgal limestones marks an extended coastal sabkha. The main part of the basin is characterized by micritic nodular limestones, the so-called Wellenkalk Facies. In contrast to Schwarz (1970), who inferred an intertidal depositional environment for the Wellenkalk, most other authors (e.g. Seilacher 1993; Klotz 1990; Voigt & Linnemann, 1996) interpret the depositional environment of the nodular limestones as shallow marine

This chapter proposes a model for the formation of a range of synsedimentary deformation structures in marine carbonate mudstones. The mass flows described are accompanied by a broad variety of particulate mass movements implying a close relationship of synsedimentary deformation and redeposition of cohesive and non-cohesive material. Deformation of cohesive material may represent one end member of particulate gravity current movement. Both in situ deformation and mass movements are confined to distinct stratigraphic horizons and indicate that major events (earthquakes or catastrophic storms) may cause gravity particulate currents, soft sediment deformation structures and cohesive mass flows contemporaneously. They can be used as a stratigraphic tool which defines precise time lines within sedimentary basins.

GEOLOGICAL SETTING The Lower Muschelkalk (Middle Triassic) of the Germanic Basin represents an epicontinental carbonate


279

PGCC19 9/12/08 3:21 PM Page 280

280

K. Föhlisch and T. Voigt

Fig. 1. Area of study with outcrops of the Lower Muschelkalk. The detailed map covers the southern, south-eastern and central part of the Germanic Basin and is bordered in the south-east by the Vindelician Massif.

(subtidal) on the basis of fossils and sedimentary structures (intercalated storm layers). The Wellenkalk Facies of the Lower Muschelkalk is subdivided into three nodular mudstone units which are separated by bioclastic limestone horizons of varying thickness and composition (Fig. 2). The occurrence of hardgrounds, ooids and rich faunal assemblages of crinoids and brachiopods, which are lacking in the micritic limestones of the Wellenkalk, as well as sedimentary structures such as current ripples, trough cross-bedding and low-angle bedding, point to highly energetic environments during the deposition of those units. The rather uniform-thickness distribution of the Lower Muschelkalk (80 –110 m) is slightly modified by an internal, N–S-striking swell–basin morphology. The 50–100-km broad depressions are dominated by marly limestones and thin biomicrites, whereas the swell areas display an increased abundance of bioclastic storm layers and generally decreased thicknesses. An indistinct cyclicity of the Lower Muschelkalk was recognized by Fiege (1938) and later used for regional correlation (e.g. Schulz, 1972; Kramm, 1986, 1997; Götz, 1996). The cycles are not yet verified by geochemical or geophysical data but were supported by investigation of the trace fossil assemblages (Knaust, 1998). The continuity of facies and thickness indicate only minor fluctuations of sea level, water chemistry

and carbonate production for the central part of the Germanic Basin during the deposition of the Lower Muschelkalk. Sediments of the Lower Muschelkalk Micritic limestones are the dominant lithofacies of the Lower Muschelkalk. The carbonate content varies between 90 and 95% by weight (As-Saruri & Langbein, 1987). In general the carbonate content is represented by calcite, although in the southern Germanic Basin a gradual transition to dolomitic micrites occurs (Figs 3 and 4). The Wellenkalk Facies is characterized by diffusely defined nodules of micritic carbonate which are surrounded by marl seams. As-Saruri and Langbein (1987) identified equigranular isometric crystals of up to 5–10 µm diameter, which were probably formed by recrystallization during diagenesis. The prevailing diagenetic structures depend on carbonate content, degree of bioturbation and primary porosity. Early cementation of burrow systems is indicated by pressure shadows around horizontal 0.5–1.0-cm thick tubular traces of Rhizocorallium and Teichichnus. The preservation of complete undeformed biomorphs within the nodules and circular cross-sections of burrows point to early diagenetic

PGCC19 9/12/08 3:21 PM Page 281

281

Synsedimentary deformation

Fig. 2. Stratigraphic units of the Lower Muschelkalk of the Thuringian Depression compared with the South German Basin. The Lower Muschelkalk sediments of the southern Germanic Basin show decreased thickness (about 50 m) and are dominated by fine-grained sandstones at basin margins. The calcareous Wellenkalk Facies of the central Germanic Basin and the Thuringian Depression has nearly constant thickness of about 100–120 m. Marly nodular limestones with intercalations of bioclastic arenites prevail.

cementation of concretions. Compaction affected the remaining marls and was accompanied by strong pressure solution reflected by microstylolitic structures. Platy limestone beds and underbed cementation of bioclastic arenites are very common and represent features of early diagenetic carbonate redistribution (Ricken, 1986). The terrigenous component of the Lower Muschelkalk is low and consists of quartz, feldspar and mica. The clay mineral spectrum is dominated by illite and smectite (Füchtbauer, 1950; Kruck, 1974). A minor amount of kaolinite and mixed-layer minerals was identified by Backhaus and Flügel (1971). A gradual transition from limestones to marls occurs towards the basin margin (Figs 3 and 4), accompanied by a

gradual increase in grain size of the material, and the passing of marls into sandstones (‘Muschelsandstein’) which plane-bedded mica-rich quartz sand intercalations.

terrigenous fine-grained consists of with marly

Synsedimentary deformation structures of the Lower Muschelkalk Conspicuous synsedimentary deformation the Lower Muschelkalk succession was observed early this century (Reis, 1910; Börner, 1936; Haltenhof, 1962) and was mostly attributed to tidal action or channel-margin slumping (Schwarz, 1970). Conglomeratic limestone beds (debrites) and slump

PGCC19 9/12/08 3:21 PM Page 282

282


Fig. 3. Facies distribution of the Lower Muschelkalk in the southern Germanic Basin (modified from Schwarz, 1975). Marginal parts of the Basin are dominated by fine-grained marine and fluvial sandstones derived from the surrounding massifs. A gradual transition to marls and limestones occurs towards the basin centre.

structures have been used for regional stratigraphic correlation (Wagner, 1897; Reis, 1910; Mägdefrau, 1957). Deformation of particulate material dominates the shallow-marine, sandy facies at the western and southern basin margin. Typical structures of synsedimentary deformation in that area are load structures (Richter, 1962). In situ deformation together with gravitational mass movements are typical features of synsedimentary deformation, both of the central and the south–eastern part of the Germanic

Basin (Schwarz, 1970, 1975; Vossmerbäumer, 1973; Simon, 1977). Increasing abundance of such features is detectable at the base of Muschelkalk, beneath the Oolithbank Member, in the Middle Wellenkalk, around the Terebratelbank Member and below the Schaumkalk Member (Puff, 1997; Föhlisch & Voigt, 1998). The intensity of deformation increases towards the Moravo-Silesian Gateway at the northern margin of the Tethys, where Szulc (1993) described debrites containing megaclasts up to 4–5 m in diameter.

PGCC19 9/12/08 3:21 PM Page 283


283

Fig. 4. Cross-sections through the Lower Muschelkalk of the Germanic Basin (after Schwarz, 1975). Limestones of the central south German Basin pass into marls and fine-grained sandstones to basin margin.

TERMINOLOGY AND CLASSIFICATION In this chapter we follow the classification of mass movements established by Nemec (1990). The scheme is based on the rheologic behaviour of mass movements and uses the earlier investigations of Rupke (1978) and Stow (1986). Nemec (1990) subdivides mass movements according to their transport mode into falls, flows with fluidal behaviour, flows with plastic behaviour, slumps, slides and creeps. Flows with plastic behaviour include debris flows, grain flows and liquefied flows. The characteristic feature of all plastic flows is their distinct shear strength. Deformation and redeposition of cohesive material In contrast to particulate material, cohesive material is able to absorb a distinct shear rate by elastic deformation. Irreversible deformation starts if the ‘yield point’ is exceeded (so-called Bingham behaviour) (Leeder,

1987; Martinsen, 1994; Stow et al., 1996). The maximum stress which can be supported without irreversible deformation is defined as the yield strength (Hampton, 1979). The yield strength depends on the internal friction and the cohesion of the material. Unlithified, weak sediments require much less stress than lithified materials to reach the same state of strain (Jones, 1994). Deformation and redeposition of particulate material The rheological behaviour of turbiditic and fluidal flows is approximately that of Newtonian fluids. Lacking a shear strength, they show nonlinear behaviour under the influence of shear stress (Uriel & Serrano, 1973). Under stress conditions, grain–grain sliding occurs because the frictional contacts between the grains are far weaker than the atomic bond strength within the individual minerals (Jones, 1994), and deformation starts immediately with the onset of shear stress.

PGCC19 9/12/08 3:21 PM Page 284

284


In general, pore-fluid content controls the rheologic behaviour before attaining the liquid limit (Einsele, 1989). After reaching the liquid limit, fluidal flows can take up water due to their turbulent movement. Fluidal flows move downslope as long as shear occurs until all the sediment load is dropped (Stow et al., 1996).

GRAVITATIONAL MASS FLOWS AND DEFORMATION STRUCTURES OF THE LOWER MUSCHELKALK The following section contains description and interpretation of a broad variety of deformational structures which are explained by the occurrence of both cohesive and non-cohesive materials in varying thicknesses of the Lower Muschelkalk succession. Debris flow deposits and slump structures Debris flow deposits of the Lower Muschelkalk typically consist of mud-supported conglomerates which reach average thicknesses on the order of 1–2 m. The maximum measured thickness is 3.7 m. Large outcrops allow reconstruction of sheet-like sediment bodies with very wide distribution (in few cases at least 120 km2) and only minor, irregular thickness variations. The amount of marly matrix fluctuates between 65 and 95% (estimated after the method of Terry & Chilingar, 1995) and varies strongly both vertically and laterally (Fig. 5). Most clasts (90%) consist of micritic carbonate (Fig. 6). The silicon content of the clasts is about seven times lower than the silicon content of the matrix (X-ray fluorescence analysis), reflecting the different clay content (Fig. 7). The size of the spherical pebbles ranges from 0.1 cm to 15.0 cm (average 2–3 cm), roundness is between 2 and 4 (estimated after Pilkey et al., 1967). About 10% of the clasts consist of bioclastic, intraclast-bearing arenites and broken hardgrounds. The size of clasts varies irregularly both in vertical and in lateral direction. Large noduleshaped or sheet-like, clast-enriched subunits commonly occur within the debrites. In extreme cases, clast-supported fabrics with densely packed intraclasts were formed. They are concentrated at the bases of the flows, pointing to separation of layers with different densities (depending on clast content) during transport. The process of separation was accompanied by

Fig. 5. View of a part of a matrix-supported debris flow, 10 m beneath Oolithbank Member, Kunitz (near Jena, South-eastern Germanic Basin). Debris flow is 2.3 m thick, area of figure is 0.7 m from the base. Extensional cracks are mainly orientated vertically and were caused by postdepositional compaction. Fewer fractures are orientated oblique to the bed surface. They probably developed prior to gravitational redeposition under conditions of stress amplification due to great differences in Poisson numbers of matrix and nodules.

internal loading and plastic deformation. The debris flow deposits commonly show indistinct stratification reflecting the primary bedding of undisturbed autochthonous deposits. The upper surfaces of the debris flow deposits are usual flat. Internal listric shear planes occur with decreasing frequency from bottom to top. Clayey material is concentrated on the shear planes. The basal parts of the debrites occasionally contain bioclastic blocks up to 30 cm in diameter, which were entrained from the underlying partly-eroded succession. The depth to the basal detachment of the mobilized layers does not exceed 4 m. Slump structures are widely distributed in the Lower Muschelkalk of the Wellenkalk Facies and reach thicknesses of about 0.3 m. Maximum thicknesses are on the order of 1.5 m. Their lateral extent exceeds several square kilometres, and they can be traced in a regional scale. Slumping is generally caused by the influence of shear stress on anisotropic, layered materials, so that they are restricted to a distinct facies type of the Lower Muschelkalk which is characterized by thin-bedded marlstones with 1–2 cm thick inter-

PGCC19 9/12/08 3:21 PM Page 285


285

Fig. 6. Thin section of a debris flow, 7 m beneath Oolithbank Member, Steudnitz, south-eastern Germanic Basin. During burial the marly matrix of the debris flow became more compacted than the early diagenetically cemented intraclasts which show extension fractures.

Fig. 7. Chemical composition (X-ray fluourescence analysis) of intraclasts and matrix of a debris flow of the Lower Muschelkalk compared to that of marly limestones (Lower Muschelkalk). Matrix of debrites shows the highest and intraclasts the lowest content of SiO2, Al2O3 and K2O, reflecting the clay mineral component. Content of clay and carbonate of nodular limestone is generally between that of debris flow matrix and intraclasts. Corresponding to results of energy dispersive x-ray investigations on thin sections it points to genesis of debris flow intraclasts from cores of early-diagenetic concretions.

calations of platy limestone beds. A prevalence within units of higher clay content (

Particulate gravity currents

An Introduction to Gravity Currents and Intrusions

Currents

Currents

Particulate Colloids

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Particulate Emissions from Vehicles (RSP)

Deadly Currents

Cross Currents

Cross Currents

Cross Currents

Deadly Currents

Dark Currents

Hidden Currents

Under Currents

Hidden Currents

Particulate gravity currents

An Introduction to Gravity Currents and Intrusions

Currents

Currents

Particulate Colloids

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Gravity

Particulate Emissions from Vehicles (RSP)

Deadly Currents

Cross Currents

Cross Currents

Cross Currents

Deadly Currents

Dark Currents

Hidden Currents

Under Currents

Hidden Currents

Recommend Documents