The Physics of Explosive Volcanic Eruptions (Geological Society Special Publication)

The Physics of Explosive Volcanic Eruptions

Geological Society Special Publications Series Editors A. J. FLEET A. C. MORTON A. M. ROBERTS

It is recommended that reference to all or part of this book should be made in one of the following ways. GILBERT, J. S. & SPARKS, R. S. J. (eds) 1998. The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145. NAVON, O. & LYAKHOVSKY, V. 1998. Vesiculation processes in silicic magmas In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 27-50.

GEOLOGICAL SOCIETY SPECIAL PUBLICATION NO. 145

The Physics of Explosive Volcanic Eruptions

EDITED BY

J. S. GILBERT

Lancaster University, UK AND

R. S. J. SPARKS University of Bristol, UK

1998

Published by The Geological Society London

THE GEOLOGICAL SOCIETY The Society was founded in 1807 as The Geological Society of London and is the oldest geological society in the world. It received its Royal Charter in 1825 for the purpose of 'investigating the mineral structure of the Earth' The Society is Britain's national society for geology with a membership of around 8500. It has countrywide coverage and approximately 1500 members reside overseas. The Society is responsible for all aspects of the geological sciences including professional matters. The Society has its own publishing house, which produces the Society's international journals, books and maps, and which acts as the European distributor for publications of the American Association of Petroleum Geologists, SEPM and the Geological Society of America. Fellowship is open to those holding a recognized honours degree in geology or cognate subject and who have at least two years' relevant postgraduate experience, or who have not less than six years' relevant experience in geology or a cognate subject. A Fellow who has not less than five years' relevant postgraduate experience in the practice of geology may apply for validation and, subject to approval, may be able to use the designatory letters C Geol (Chartered Geologist). Further information about the Society is available from the Membership Manager, The Geological Society, Burlington House, Piccadilly, London W1V OJU, UK. The Society is a Registered Charity, No. 210161. Published by The Geological Society from: The Geological Society Publishing House Unit 7, Brassmill Enterprise Centre Brassmill Lane Bath BA1 3JN UK (Orders: Tel. 01225 445046 Fax 01225 442836) First published 1998 Reprinted 2002 The publishers make no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility for any errors or omissions that may be made. © The Geological Society 1996. All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with the provisions of the Copyright Licensing Agency, 90 Tottenham Court Road, London W1P 9HE. Users registered with the Copyright Clearance Center, 27 Congress Street, Salem, MA 01970, USA: the item-fee code for this publication is 0305-8719/98/S 10.00. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN 1-86239-020-7 ISSN 0305-8719

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V

Contents GILBERT, J. S. & SPARKS, R. S. J. Future research directions on the physics of explosive volcanic eruptions

1

DINGWELL, D. B. Recent experimental progress in the physical description of silicic magma relevant to explosive volcanism

9

NAVON, O. & LYAKHOVSKY, V. Vesiculation processes in silicic magmas

27

MADER, H. M. Conduit flow and fragmentation

51

JAUPART, C. Gas loss from magmas through conduit walls during eruption

73

WOODS, A. W. Observations and models of volcanic eruption columns

91

BURSIK, M. Tephra disposal

115

DRUITT, T. H. Pyroclastic density currents

145

Index

183

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Future research directions on the physics of explosive volcanic eruptions J. S. GILBERT1 & R. S. J. SPARKS2 1

Department of Environmental Science, Institute of Environmental and Natural Sciences, Lancaster University, Lancaster LAI 4YQ, UK 2 Department of Earth Science, University of Bristol, Bristol BS8 1RJ, UK

Scientific research can take unexpected, even counter-intuitive, directions because of technical innovation, the occasional brilliant idea that overturns conventional wisdom and new observations that provide previously unexpected insights into the way in which nature works. For these reasons no one is certain what the future holds in terms of breakthroughs. This chapter highlights some of the most recent developments in research on the physics of explosive volcanism. It pin-points cardinal areas of study poised for new research and anticipates major future developments. Advances in remote sensing and computational power are two examples of technical developments which are currently having dramatic impacts on understanding the physics of explosive volcanism. Such technical innovations, together with many good ideas and observations, are changing perceptions of the mechanisms of explosive volcanism. With an increasingly populated and ecologically stressed world, the potential effects of explosive volcanism are being exacerbated. Several megacities, e.g. Tokyo, Naples and Mexico City, now exist close to active volcanoes, and in many parts of the world economic development and population expansion have combined such that the risk of major volcanic disasters increases year by year. Volcanic activity has both local and global environmental effects. For example, fallout of volcanic ash from eruption plumes can disrupt air, sea, road and rail traffic, inhibit electrical communications, cause respiratory problems for people and animals, pollute water, damage crops, cause failure of building roofs and generally bring havoc to local communities. On a larger scale, volcanic aerosols from some events, such as the 1991 Pinatubo eruption in the Philippines, are now known to accelerate damage to the Earth's ozone layer and recent improvements in understanding volcanic gas fluxes indicate that the contributions of volcanoes to global SO2 have been underestimated. Volcanologists have a responsibility to progress their science as efficiently as possible to improve understanding and mitigation of the effects of volcanic eruptions. Therefore, prog-

nostication is worthwhile because it stimulates thought and debate even if it proves ultimately to be inaccurate. Understanding the physics of explosive volcanism is a formidable task. An explosive eruption involves a vast range of scales, material property variations, and complex interacting physical and chemical processes of extraordinary diversity. A complete description of an eruption requires an understanding of magma chambers deep in the crust, flow of viscous magma through fractures in the deformable crust, multiphase high-speed flows in conduits, vents and the atmosphere, and dispersal of ultrafine aerosols and dust by the atmosphere sometimes on a global scale. Material properties vary from brittle solids to ductile magma, which can change their viscosities by several orders of magnitude during eruption, and very low viscosity volcanic gases. A complete list of parameters that govern the entire process from the magma chamber to distant atmospheric dispersal of aerosols would exceed 100 variables. These must be incorporated into a full mathematical description of the chemistry and physics of explosive eruptions (Sparks et al 1997). The processes involved in explosive volcanic eruptions are often strongly interactive, and many are intrinsically non-linear so that the mathematical and computational studies can be difficult and results counter-intuitive. At some level of detail the dynamical character of volcanic processes makes them inherently unpredictable. Many parts of the flow system cannot be directly observed. It is, for example, impossible at the moment to envisage direct measurements in the interior of an erupting conduit. Inferences about flow conditions have to rely on observations of the products of explosive volcanism, remote methods of observing eruptions, which have their own limitations and problems, and experimental simulations. In this chapter we outline areas of science that we believe are important to improve understanding of the physics of explosive volcanism. The review does not attempt to be comprehensive, but focuses on a small number of topics

GILBERT, J. S. & SPARKS, R. S. J. 1998. Future research directions on the physics of explosive volcanic eruptions. In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 1-7.

J. S. GILBERT AND R. S. J. SPARKS stimulated by discussions at the Arthur Holmes European Research Conference, held in Santorini, Greece on 2-6 September 1996, and with many colleagues. Material properties and parameters Modelling of volcanic processes and interpretation of experiments and observations require knowledge of physical properties of materials. Dingwell (1998) summarizes some of the advances in measuring and understanding physical properties, in particular viscosity and the glass transition temperature, which determine whether magma responds as a brittle or ductile material to stresses. However, much remains to be done and there are many important properties which are not well determined or even measured. Many experiments require high pressures and temperatures in which volatile fugacities are constrained or variables such as pressure are changed in a controlled way. These experiments can be technically difficult and time-consuming. Systematic experimental determinations of properties are not always glamorous, but such studies underpin attempts to understand volcanic physics and are of vital importance. In this section we discuss a few examples of material properties for which future research is needed. Understanding of melt rheology is now very good (Dingwell 1998). Data also exist on the effects of moderate amounts of suspended crystals and bubbles on rheology (e.g. Bagdassarov, 1992; Pinkerton and Stevenson 1992; Lejeune and Richet 1995). However, explosive eruptions involve development of high vesicularities and, in lava dome explosions, extensive crystallization of the melt prior to onset of the explosion. Rheology is particularly complex if the bubble or crystal content is high, with markedly non-Newtonian behaviour (e.g. Spera et al 1988; Dingwell and Webb 1989; Pinkerton and Stevenson 1992). The rheology of high porosity foams in volcanic conduits, where acceleration achieves strain rates sufficient for fragmentation, is not well understood, but is fundamental to understanding explosive flows where foamy magma fragments into a dispersion of pumice, ash and gas. As magma degasses, and bubbles form, viscosity increases. However, intuitively, this cannot increase infinitely. A foamy melt with a very high proportion of low-viscosity gas would be expected to decrease its viscosity with increasing gas content as the proportion of the high-viscosity melt per unit volume diminishes.

High vesicularities and high crystallinities are likely not only to be non-Newtonian, but also to show hysterisis behaviour with dependence on strain role and strain history. It is also clear that such materials can deform heterogeneously (e.g. Shaw 1969; Lejeune and Richet 1995) with deformation focused in narrow zones (Lane & Phillips 1997). Such behaviour is intrinsically hard to characterize and predict. However, without a better understanding of these behaviours and properties, some important processes are likely to be misunderstood and neglected in modelling studies. Another major area of research is the kinetics of bubble formation and crystal growth. As reviewed by Navon & Lyakhovsky (1998), much attention has focused on bubble nucleation and growth, with several groups around the world developing theoretical models and carrying out experimental studies. This research has made rapid progress such that robust kinetic laws of bubble formation should emerge soon. There are still significant problems, in particular in terms of measuring and predicting nucleation rates in decompressing magma. Evidence has recently emerged on the importance of microlite crystallisation in partially degassed magma (Cashman 1992; Sparks 1997; Stix et al. 1997). Microlite crystallization may be particularly important in the pressurization of lava domes that leads to sudden vulcanian explosions and a variety of geophysical phenomena, including volcanic earthquakes. Invariably the most important microlite phase is feldspar. There are little data and limited theoretical basis for predicting microlite nucleation and growth rates, although some important progress has been made by the application of crystal size distribution theory (e.g. Cashman, 1992). The dangerous and complex behaviour of lava domes may well be governed by the kinetics of crystallization, and so prediction of lava dome behaviour will require experimentalists to determine kinetic laws. The transitions from effusive to explosive eruption (Jaupart 1998) are thought to be controlled by magma permeability. A major challenge is to understand the way in which hightemperature magmas can become permeable to gas escape. There are, however, very few data and essentially none at high temperatures. There are two main sets of permeability data on volcanic ejecta at low temperature (Westrich & Eichelberger 1994; Klug & Cashman 1996). Unfortunately these measurements disagree in that Westrich & Eichelberger (1994) found a much stronger dependence of permeability on porosity than Klug & Cashman (1996). There is the added problem that measurements of volcanic materials

FUTURE RESEARCH DIRECTIONS at low temperature may be unrepresentative of magmatic conditions, because permeability may change during cooling (for example, by microfractures related to cooling). There is an urgent need for many more permeability data on volcanic materials both at low and high temperatures. High-temperature studies are particularly important although experiments will be difficult. Permeability of magmatic foams is likely to be a rich area for research. Magmatic systems involve the three phases: solid (crystals), liquid (melt) and gas. Permeability will thus depend on the relative abundances of the phases and the strain history. In a strongly deforming foam, bubbles will change shape and interconnections between vesicles may develop, enhancing permeability. If deformation ceases, surface tension effects may close interconnections reducing permeability (Jaupart 1998; Navon & Lyakhovsky 1998), whereas Ostwald ripening may increase bubble size and permeability. Table 1 lists permeability and porosity data for samples from the on-going eruption of the Soufriere Hills Volcano, Montserrat. The data show that a sample from the lava dome, with a porosity of 33%, is more permeable than a sample of pumice with 71.5% porosity. The reason for the contrast is that the lava dome sample has a much smaller proportion of melt (glass) than the pumice due to groundmass crystallization. The glass in the lava dome sample forms a continuous phase between the feldspar microlites and has vesiculated strongly. Thus, the melt phase in the dome magma is actually more vesiculated than the melt phase in the pumice. The interconnections of bubbles in the residual melt phase allow the much denser lava dome sample to have higher permeability. Such data hint at the complexity of understanding and determining permeability in high-temperature magmatic systems. Permeability can also develop in relatively crystalline magmas by microfracturing and development of larger-scale fracture networks (e.g. Alidibirov and Dingwell 1996; Stasiuk et al 1996). Fractures can develop at magmatic temperatures and may be the cause of long-period earthquakes that precede explosive eruptions of lava domes (Chouet 1996; Sparks 1997; Stix Table 1. Porosity and permeability data for pumice and lava dome samples from the Soufriere Hills Volcano, Montserrat Rock type

Porosity (%)

Permeability (m2)

Pumice, 17 September 1996 Dome lava

71.5 33.0

4 x 10 - 1 2 5 x 10 - 1 2

et al. 1997). A much better understanding is needed of the changing mechanical properties of magma under conditions between wholly brittle and wholly ductile behaviour. Details of mechanical properties such as strength, fracture toughness and the brittle-ductile transition will be needed. The modelling of gas loss from magmas is only likely to make significant progress if permeability measurements are made of magma at high-temperature and the factors that control permeability in erupting magma are understood. Theoretical modelling and computer power The era of the supercomputer and dramatically enhanced computer power has developed the possibility of sophisticated calculations of great complexity. Sparks et al. (1997) and Woods (1998) summarized the progress of theoretical modelling on conduit flow and eruption column dynamics. Computational fluid mechanics is likely to be an increasingly important and influential approach to studying volcanic eruptions. The developments in computer technology are showing no signs of slowing down and it can be expected that the models will become increasingly complex with incorporation of more parameters and the ability to investigate three-dimensional effects. Some examples of problems which are fast becoming tractable are cited. Bubble formation in ascending silicic magmas is very complex because the nucleation rate is a highly non-linear function of supersaturation. There is interference between neighbouring bubbles in the volumes of melt from which gas is tapped, and there are strong local gradients of viscosity and diffusivity. Early numerical models (Sparks 1978), which required substantial approximations, are now being replaced by increasingly elaborate simulations (e.g. Navon & Lyakhovsky 1998), which can take account of many of the natural complexities. Complete simulations are very likely to be available during the next decade. The generation and emplacement of pyroclastic flows is a good example of the opportunities and also the problems, of computational fluid dynamics applied to explosive eruptions. There have been several studies of the motion of multiphase mixtures away from a collapsing fountain. Two rather different approaches have been used. Numerical models have considered the case of dilute particle-laden gravity currents (Bursik & Woods 1996; Dade & Huppert 1996) to calculate velocity variations, run-out distances and properties of the deposit such as thickness and grain

J. S. GILBERT AND R. S. J. SPARKS size. The other approach has been to develop supercomputer calculations solving the full Navier-Stokes equations for fountains and the flows that they feed (Valentine et al 1992; Neri & Dobran 1994). The former calculations tend to have a relatively small number of parameters and involve some conceptual simplifications which allow particular effects (such as mass flux) to be investigated by parametric studies. The latter calculations involve very elaborate computer codes and a large number of parameters, and the effects are simultaneously incorporated. To a large extent the former type of calculations have been tested against laboratory experiments, whereas the latter calculations have been presented as numerical experiments without comparison with real experiments. Recently the conventional interpretations of pyroclastic flows, drawn from field studies of the deposits, have been challenged. For example, a detailed field investigation of the Taupo ignimbrite of New Zealand has led to it having been interpreted as the result of emplacement of a predominately concentrated fluidized flow (Wilson 1985), whereas a numerical model of Dade & Huppert (1996) has suggested that the Taupo ignimbrite formed by deposition from a dilute turbulent gravity current. This debate is considered in more detail by Druitt (1998). It illustrates the future challenges for reconciliation of independent findings by modellers and field geologists. In this particular case, the field geologists need to re-examine the arguments that led them to envisaging pyroclastic flows as concentrated, and the modellers must provide convincing alternative interpretations of the geology and demonstrate that their models are robust. The supercomputer models have also revealed new features. For example, the study of Neri & Dobran (1994) revealed fluctuations in pressure and fountain height fed by a steady flux from the vent. The models, therefore, provide a stimulus for interpreting observations and recognizing new phenomena in nature. However, the models often contain many simplifications and this makes their realism open to question. For example, many of the supercomputer models of particle-laden volcanic flows involve a single particle size and, therefore, do not realistically represent nature. There is a significant gulf in culture and understanding between the supercomputer modellers, field geologists and experimentalists. The computer codes and mathematics are complex, and sometimes inaccessible. Modellers need to demonstrate that the results show phenomena identified by the calculations that can happen in the real world. The models need to

be tested against experiments and observations and the modellers need to acknowledge limitations to the models that arise from the assumptions, simplifications and numerical grid size. The assertion that the supercomputer models contain all the physics is demonstrably incorrect because many aspects of the physics of explosive eruptions are not well understood, and therefore cannot be incorporated. In addition, field-based volcanologists need to be receptive to new possibilities and ideas that the supercomputer calculations reveal, and recognize the substantial potential of this approach.

Experiments on dynamic systems Laboratory experiments on the dynamics of volcanic flows are likely to be an important component of future research. The philosophy of various experimental approaches has been outlined by Gilbert (1994) and Sparks el al (1997). Two broad types of experiments can be identified: exploratory and parametric. Exploratory experiments involve studying systems in situations which have not been investigated before and where there is little knowledge. Shock tube and explosion exploratory experiments have been a particular feature of the last few years (Mader 1998). Both analogue and high-temperature silicate systems have been investigated. These experiments were designed to examine the way in which explosive flows of degassing fluids or rapidly decompressed rock behave. The experiments seek to understand the detailed physics of flow through conduits and, although motivated by the volcanological application, the results are potentially of importance to foam-centred industries such as those involved with fire-fighting materials and chemical foaming agents. A problem with these experiments is that it is not always clear that they scale properly to the natural volcanic system. The usual strategy is to try to match the properties and dynamic forces in natural flows, but this is not always achievable. Parametric experiments seek to find a quantitative understanding of processes by studies of systems where theoretical understanding is incomplete. The experiments usually involve systematic studies to establish which parameters control a system and to recognize different regimes of behaviour. Research on bubble accumulation in magma chambers and effects on conduit flow and surface eruptive activity is a good example (Vergniolle & Jaupart 1986). The latter study demonstrated the way in which different styles of basaltic explosive activity can

FUTURE RESEARCH DIRECTIONS be controlled by the collapse of foams at the top of a magma chamber. Conduit dimensions, chamber size, magma viscosity and gas flux proved to be the main variables. Finding improved analogue systems to mimic the properties of volcanic materials is an important future task. Examples of useful systems include polyethylene glycol (PEG) which has been used to study solidification in cooling lavas (Hallworth et al 1987; Fink & Griffiths 1990), and mixtures of methanol and ethylene glycol (MEG) which have been instrumental in examining the effects of non-linear density variations that occur during mixing in eruption columns and pyroclastic flows (Huppert et al. 1986; Woods and Caulfield 1992). Gum rosin (a purified form of pine resin) has proved to be useful for laboratory experiments, because it has similar properties to rhyolite. Gum rosin is a

5

brittle amorphous solid at room temperature and changes to a liquid when it contains a small amount of dissolved acetone (Phillips et al. 1995). Solutions of gum rosin and acetone reproduce the volatile and temperature-dependent viscosity, and the phase behaviour, of hydrated magmas (Phillips et al. 1995; Lane and Phillips 1997). It would be very useful to identify an analogue system that mimics the crystallisation of degassing silicic magmas. Such a system would need to increase its viscosity at rates which are controllable in the laboratory. Seismicity during eruptions of viscous magmas From studies of Mount Redoubt, Alaska (Chouet 1996) and Galeras Volcano, Colombia

Table 2. Topics for future research in explosive volcanism Broad area of research

Details of problems to solve

Explosion triggers

• • • •

Magma chambers

• The application of more sophisticated methods of geochronological dating for understanding secular variations of explosive volcanism and residence times of magma batches

Conduit flow

• The mechanisms of magma fragmentation • The causes of pulsations in explosive eruptions

Plumes

• The fluid dynamics of explosive volcanic jets • The influence of water on explosive eruption and column stability • The development of new methods of remote sensing of volcanic plumes (e.g. using electric and magnetic plume properties) • The dispersal of small particles in plumes and development of realistic particle fallout models, including details of particle aggregation (see Bursik 1998) • The environmental effects of ashfall (particularly on human health, agriculture and ecology)

Pyroclastic flows

• The physical nature of large violent pyroclastic flows (i.e. are they dilute turbulent suspensions or concentrated dispersions, or both?)

Risk assessment

• The development of probabilistic assessments of risk from explosive phenomena

Material properties

• The rheological properties of melts containing variable amounts of crystals and bubbles • The behaviour of high-porosity foam during transport through volcanic conduits • The kinetics of crystal and bubble growth in magma • The permeability of magma • The causes of different styles of seismicity and their relationship to the onset of explosive activity

Seismicity Other planets

The governing forces of explosions of andesite domes The effects of microlite crystallization on magma pressurization The causes of volcanic earthquakes The causes of volcano instability that may trigger explosive eruptions (e.g. sector collapse)

• The implications of application of the governing equations of the physics of explosive eruptions to volcanism on other planets

6

J. S. GILBERT AND R. S. J. SPARKS

(Stix et al. 1997) it is apparent that long-period earthquakes can precede explosive eruptions and, as in the case of Galeras, the characteristics of the long-period seismicity may change after an explosive eruption. Seismicity remains the central tool for monitoring dangerous volcanoes with the potential for explosive activity. Therefore, understanding the causes of different styles of seismicity and their relationship to the onset of explosive activity must be one of the most important practical objectives of future research. Developing a robust predictive capability, as suggested by Chouet (1996), would be a very significant breakthrough. There is a general consensus that long-period and hybrid earthquakes relate to the movement of pressurized fluids along fractures. Chouet (1996) perceived the fractures to have large surface areas and that the repetitive kinds of long-period event are caused by resonation along a fracture. There is a probable link between volcano seismicity and pressurization in lava domes. Sparks (1997) identified large viscosity variations and microlite crystallization resulting from degassing as major causes of pressurization in lava domes and inferred that shallow earthquakes result. The exact causal links are not, however, well understood. The hazards posed by lava dome eruptions, which unexpectedly move into explosive activity, have become increasingly apparent with the eruptions of Galeras, Colombia (Stix et al. 1997), Lascar, Chile (Matthews et al. 1997) and the Soufriere Hills Volcano, Montserrat (Young et al. 1997). Progress in this area is likely to come from close collaboration between volcano seismologists, modellers, petrologists and physical volcanologists.

Conclusions We have selected a few out of many possible topics which have recently advanced our understanding of the physics of volcanoes, and have identified realms where future research is needed or is likely. In addition to those developed in this article, Table 2 lists important areas of volcanology poised for future research. This chapter may well be read in 20 years time when a long list of important topics missed by the authors will be readily apparent! The authors thank the Geological Society of London Conference Office, in particular Heidie Gould, for organizing the Arthur Holmes European Research Conference. R. S. J. Sparks is supported by a Leverhulme Trust Grant (No. F/182/AL).

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TAIT, S. R. 1996. Decompression of volatilesaturated rhyolitic magma in the Mule Creek vent, New Mexico, U.S.A. Bulletin of Volcanology, 58, 117-130. STIX, J., TORRES, R. C., NARVAEZ, M. L., CORTES, G. P., RAIGOSA, J. A., GOMEZ, D. M. & CASTONGUAY, R. 1997. A model of vulcanian eruptions at Galeras Volcano, Colombia. Journal of Volcanology and Geothermal Research, 77, 285-304. VALENTINE, G. A., WOHLETZ, K. H. & KEIFFER, S. W. 1992. Effects of topography on facies and compositional zonation in caldera-related ignimbrites. Geolological Society of America Bulletin, 104, 154-165. VERGNIOLLE, S. & JAUPART, C. 1986. Separated twophase flow in basaltic eruptions. Journal of Geophysical Research, 91, 12 840-12 860. WESTRICH, H. R. & EICHELBERGER, J. C. 1994. Gas transport and bubble collapse in rhyolitic magmas: an experimental approach. Bulletin of Volcanology, 56, 447-458. WILSON, C. J. N. 1985. The Taupo eruption, New Zealand II. The Taupo ignimbrite. Philosophical Transactions of the Royal Society of London Series, A3145, 229-310. WOODS, A. W. 1998. Observations and models of volcanic eruption columns. This volume. & CAULFIELD, C. P. 1992. A laboratory study of explosive volcanic eruptions. Journal of Geophysical Research, 97, 6699-6712. YOUNG, S. R., WHITE, R., WADGE, G. et al. 1997. The ongoing eruption in Montserrat. Science, 276, 371-372.

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Recent experimental progress in the physical description of silicic magma relevant to explosive volcanism DONALD B. DINGWELL Bayerisches Geoinstitut, Universitaet Bayreuth, D-95440 Bayreuth, Germany Abstract: Explosive eruptions of silicic magmas are the result of a complex interplay of physico-chemical processes (e.g. decompression, volatile saturation, bubble nucleation and growth, crystallization, foaming, fragmentation and annealing). These processes occur over a relatively wide range of temperature, pressure, stresses and time scales. Furthermore, substantial changes in the chemical and physical properties of the eruptive magmas are induced by decompression and degassing. Numerical modelling of eruptive processes presents us with a picture of a magma column where enormous vertical gradients in the physical state and properties of the continuous magma body in the conduit result from decompression and degassing. Adequate representation of the physico-chemical evolution of eruptive systems therefore requires a relatively detailed description of melt properties in order to be robust and generalizable. The source of property data for silicate melts is experimental investigation. In recent years much effort has been concentrated on rhyolitic melts under conditions relevant to explosive volcanism. Here several aspects of this work are highlighted. (1) The description of hydrous rhyolitic melt properties (density, surface tension, viscosity and thermal conductivity) has been greatly improved. (2) The solubility of water as well as its diffusivity in silicic melts has been investigated in detail at relatively low pressures. (3) Rheological and transport complexities of melts and magmas such as nonNewtonian melt rheology, viscoelasticity, crystal and bubble suspension rheology, and foam deformation, permeability and stability have been explored. (4) Studies of the mechanical strength of magma have been initiated. Some of the insights into the nature of explosive volcanism provided by these studies are also of a qualitative nature (e.g. the degree of equilibrium during degassing, the longevity of textural magmatic states and the mechanism of fragmentation). It is hoped that this description of experimental progress in melt properties will encourage the reader to conpare the assumptions, descriptions and predictions of the modelling of eruptive processes presented in this volume with the physical nature of the magmas involved that is described herein. Nomenclature M P T V X

Elastic modulus (Pa) Pressure (bar) Temperature (K) Molar volume (cm 3 mol - 1 ) Mole fraction Viscosity of melt (Pa s) Relaxation time or time scale (s)

The need for an adequate understanding of the nature and extent of physico-chemical processes involved in explosive volcanism is considerable. The growth of population centres and industry surrounding active volcanic centres means that, due to the threat to life, property and the environment, this need is increasing. An essential component of the knowledge required in order to achieve progress towards an understanding of such processes is an accurate description of the physical properties of the magma involved. A considerable literature exists on this subject and

the modern history of investigation goes back at least to the early part of this century. That literature has been amply and recently reviewed by numerous authors (rheology - Pinkerton & Stevenson 1992; Dingwell et al 1993; equation of state - Lange & Carmichael 1987; thermal conductivity - Snyder et al. 1994; surface tension - Walker & Mullins 1981; water solubility - Holtz et al. 1995; water diffusion Zhang et al. 1991) and will not be re-reviewed here. More relevant are recent advances in a number of experimental, theoretical and technical fields surrounding high-pressure and hightemperature laboratory characterization of geomaterials. Accompanying the trend towards more precise description of the physical properties of melts and magmas has been an increasingly widespread appreciation of more subtle aspects of their physical behaviour, such as mechanical relaxation, degassing kinetics, foam stability and fragmentation. Often the evaluation of these phenomena by experimental or theoretical means throws the question of the adequate characterization of magma properties

DINGWELL, D. B. 1998. Recent experimental progress in the physical description of silicic magma relevant to explosive volcanism. In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 9-26.

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back into the hands of high-temperature-highpressure experimentalists, who must improve the description of the physical property in question for further progress. The interaction between advances in the parameterization and the modelling of magmatic systems is therefore important. This chapter highlights aspects of this discussion concerning recent experimental progress in the description of the physical properties of silicic melts and magmas.

Rheology The deformation and transport history of magmas can be extraordinarily complex, involving melt segregation, ascent and decompression, foaming, fragmentation, annealing and crystallization. The link between the state of stress acting on the magma and the resultant deformation and/or transport of the magma consists of the viscous and elastic coefficients of the magma. The stresses acting on the magma can be separated into two geometric components, the volume stress and the shear stress. Together, they may be described as longitudinal stresses (e.g. Sato & Manghnani 1984; Rivers & Carmichael 1987; Dingwell & Webb 1989). In most applications involving magmas in a highly condensed state the volume term of deformation is only important for considering strains that are very much less than 1. An example of a situation where the volume component of deformation might be important is the rapid unloading of magma. The pressure drop 'felt' by a magma during rapid decompression results in a viscoelastic (see below) response of the melt phase that contains a significant volume strain component as the magma decompresses to a lower density state. When magmas contain a significant fraction of vesicles then the relatively high compressibility of the gas phase in the bubbles means that decompression events can result in quite significant volume changes. This is the one case where the volume strain term can become comparable to the shear strain term at values approaching, and perhaps greater than, 1. A good example of a volcanic product demonstrating a significant volume strain component in isolation is expanded pumice. An example of a volcanic product that has undergone combined and comparable volume and shear components of strain or longitudinal strain is tube pumice. The strain being referred to here is volumetric and shear response to stresses applied to a simple isochemical system. In decompression events in nature, depending on the time scale, the strain event may or may not be accompanied

by significant mass transfer of the volatile between phases by, for example, diffusion from the melt into the bubble phase.

The viscoelastic nature of melts and the glass transition In a viscoelastic material like magma, the stress generates three types of strain response (regardless of the tensorial properties). First, there is an instantaneous (immeasurably rapid) elastic deformation of the magma. This deformation constitutes stored energy and upon stress release it is recoverable. Secondly, there is a slower, delayed deformation that is still elastic and recoverable, and results from the rearrangement of melt structure in response to the stress. Finally, there is a viscous, non-recoverable component that is responsible for high strain transport and deformation. The instantaneous and delayed elastic components of strain are normally very small whereas the viscous component can be very large. Nevertheless, the elastic component of melt deformation can be important in magmatism because the stresses acting upon the magma may eventually be stored to levels where the strength of the magma is exceeded, resulting in brittle response of the magma (see below and Dingwell & Webb 1989). The viscoelastic nature of melts and magmas makes it essential to quantify the time scales of these types of response. They will determine whether viscous or elastic-brittle deformation is the result of the application of stress. This is easily done for the melt phase. The ratio of the viscosity to the elastic modulus gives us a time scale which is termed the relaxation time (or time scale; = . The relaxation time scale (illustrated in Fig. 1) is controlled mechanistically by the self-diffusion of Si and O in the melt, as they diffuse to new configurations in response to the applied stress (generating the delayed elasticity discussed above) (Dingwell & Webb 1989, 1990). Because this fundamental mechanism controls structural relaxation, and because the volume and shear components of viscosity and elasticity are similar, the relaxation times of volume stresses and shear stresses (and even enthalpy) are also very similar. The transition between the viscous response and the elastic response of the melt or magma is termed the glass transition. Prediction of this transition for volcanic processes using the Maxwell relation relies chiefly on adequate shear viscosity data for hydrous silicic melts. Variations in the shear modulus are relatively unimportant.

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Fig. 1. The glass transition in temperature - time space for various geological and analogue melt compositions. The curve labelled R1A is a dry calc-alkaline rhyolite (others are: AN, anorthite; AB, albite; DI, diopside; NS2, Na2Si2O5; 710, NBS standard glass 710; AN36DI64, the 1 atm anorthite-diopside eutectic composition. The nonlinear shapes of the curves reflect the non-Arrhenian temperature dependence of viscosity and structural relaxation time. The relaxation time for stresses in an individual silicate melt can change by up to 10 orders of magnitude with temperature. Various experimental techniques have been employed to define the location of the glass transition as a function of temperature including those for which the temperature and time domains are illustrated here (volume relaxation, concentric cylinder viscometry, fibre elongation, ultrasonics, shock wave propagation, scanning calorimetry, 29Si Q-species exchange and torsional deformation). Reproduced from Dingwell & Webb (1989).

The glass transition has important phenomenological consequences for the physical description of the melt phase. Figure 2 illustrates the variation of heat capacity and expansivity of a melt across the glass transition. The traces consist of low-temperature solid-like or glassy values of the derivative properties, hightemperature metastable liquid values and an intermediate transient regime, termed the glass transition, where the property is influenced by the kinetic history of the sample and the kinetics of the measurement. In Fig. 3 the variation of

the viscosity with temperature is compared with the variation of the temperature at which the glass transition is observed in heat capacity versus the cooling rate of that measurement. The apparent equivalence of the activation energy implied by the equal slope values can be used to derive a relationship between the cooling rate and the resultant viscosity observed at the glass transition temperature. A linear relationship between the cooling rate and the viscosity at the glass transition temperature, which has been quantified for several obsidians by Stevenson

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(a)

Fig. 2. The phenomenology of the glass transition expressed in scanning calorimetric and scanning dilatometric traces of the heat capacity and expansivity, respectively. The glass transition is defined as the transition from liquid-like to solid-like behaviour of the melt phase. This is reflected in the calorimetric (a) and dilatometric (b) curves as a sharp rise in the value of the heat capacity and expansivity values with increasing temperature whose origin lies in the added configurational contribution to the derivative property above the glass transition. The peak or property overshoot defining the glass transition itself is a transient phenomenon reflecting hysteresis in the glass transition interval. Data from Knoche et al. (1995).

et al. (1995), results from such a comparison. Figure 3 also leads to the inference that the processes behind enthalpy relaxation and shear stress relaxation are identical. This inference, in turn,

leads to an important simplification of relaxation in silicate melts embodied in the statement that relaxation times for the structure and the physical state of a melt are equal (cf. Siewert &

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Newtonian melt rheology

10 4 /T

(K -1 )

Fig. 3. The equivalence of the activation energy of relaxation times for the relaxation of enthalpy and of shear stress in various rhyolites. The comparison of the quench rate dependence of the glass transition temperature in calorimetric determinations and the temperature dependence of the viscosity in rheological determinations reveals equal values of the activation energy. Setting the glass transition temperature equal to the temperature of viscosity determination we obtain a relationship between the cooling rate or quench rate and the viscosity at the glass transition temperature for that cooling rate. This valuable equation (see text) provides a quantitative link between cooling rate and expected glass transition temperature where viscosity is known or can be estimated. Prediction of the glass transition temperature as a function of cooling rate allows us in turn to allot the ranges of temperature over which liquid and glassy values of the heat capacity and expansivity of the melt phase must be employed. Reproduced from Stevenson et al. (1995). Rosenhauer 1997). The glass transition reflects this thermally activated process of structural reequilibration with the result that the time scale of any process or observation must be specified to define the glass transition temperature.

The viscosities of hydrous silicic melts vary more that any other property of interest in the degassing and fragmentation of silicic magmas. A recent compilation of the experimental data and a model for the calculation of viscosity over the range of temperatures and water contents relevant in the degassing and fragmentation of silicic melts are illustrated in Fig. 4 (Hess & Dingwell 1996). Note that the decrease in melt viscosity with added water is strongly non-linear, even more so than that predicted by earlier models of hydrous melt viscosity (e.g. Shaw 1972). Secondly, the temperature dependence of the viscosity of these melts becomes significantly non-linear versus reciprocal temperature. This 'non-Arrhenian' temperature dependence of viscosity is, in fact, the general case for liquids, and dry, high-silica rhyolites belong to a very special class of melts that are almost Arrhenian. Recently, the applicability of this model (whose database of 111 viscosity determinations is dominated by synthetic system studies) to natural obsidians in the sensitive water-poor region has been confirmed (Stevenson et al. 1998). The model presented in Fig. 5 is capable of reproducing the viscosities of hydrous calcalkaline and peraluminous rhyolites very accurately, but it fails for peralkaline rhyolites which, when hydrated, have very low viscosities. A comparison of the influence of water on the viscosities of peraluminous, metaluminous and peralkaline rhyolitic melts is contained in Fig. 5 (Hess et al. 1995, Dingwell et al. 19980,b). As such, the model presented in Fig. 4 is not

Fig. 4. The viscosities of hydrous calc-alkaline rhyolitic melts. This model for the calculation of the viscosities of calc-alkaline rhyolite melts is valid for water content from 0 to 8 wt% water and temperatures corresponding to viscosities from 101 to 1011 Pa s. This range of temperature and water content effectively covers the entire range of conditions relevant to silicic magmas in explosive volcanism. Note this is a model for the Newtonian, equilibrium values of the melt viscosity and does not apply to non-Newtonian behaviour of the melt phase. Reproduced from Hess & Dingwell (1996).

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time of the melt, the melt viscosity departs from Newtonian behaviour towards lower values of apparent viscosity. This is illustrated in Fig. 6 where the decrease in melt viscosity with increasing strain rate is apparent for a range of liquid compositions from rhyolite to nephelinite (Webb & Dingwell, 1990a,b). The onset of the non-Newtonian behaviour in these liquids can be calculated to occur at 2.5 log units of deformation rate slower than the calculated structural relaxation strain rate. Thus, if the Newtonian viscosity is known, the location in temperature and strain rate of the onset of the non-Newtonian behaviour can be estimated. In other words, the liquid composition influences the onset of non-Newtonian rheology solely through its influence on the Newtonian viscosity. Fig. 5. Peralkaline, peraluminous and metaluminous hydrous haplogranitic melt viscosities expressed as the shift in temperature corresponding to a viscosity of 1011 Pas (N5, N10, A102, A105 refer to wt% excess Na2O and A12O3, respectively). The decrease in temperature (and thus in viscosity at constant temperature) with the addition of water is strong and non-linear for all three types of melt composition. As a consequence degassing of silicic magmas during volcanic ascent can be expected to generate strongly non-linear effects on the melt viscosity as a function of depth, regardless of whether the melt is peralkaline, peraluminous or metaluminous. Strongly peralkaline melts with significant water contents possess extremely low viscosities indeed, (reproduced from Dingwell et al 1998b).

a general replacement of the Shaw (1972) calculation method, which incorporates the multicomponent anhydrous basis of silicate melts. In the case of calc-alkaline dacites to rhyolites, the Hess & Dingwell (1996) model is clearly superior in being able to cope with the non-Arrhenian temperature dependence of the viscosity. Ultimately a general, multicomponent, non-Arrhenian model will be generated but many experiments are needed before this becomes feasible.

Non-newtonian melt rheology If the rate of shear deformation of the melt becomes too high the melt no longer responds in a simple Newtonian way. For the case of low strains, viscoelastic response leads to energy storage in the melt as described above. For the case of high strains, the melt responds with an effect known as 'shear thinning'. As the time scale of the deformation (the reciprocal of the strain rate) approaches the structural relaxation

Silicic suspensions The addition of crystals to a melt increases its viscosity because the deformation of the mass must be taken up entirely in the decreasing fraction of the flow cross-section of the magma occupied by liquid. Calculation of the influence of the presence of crystals at relatively low volume fraction on the viscosity of the magma is not robust or generalizable. A wide range of characteristics of the crystal population, such as crystal shape, crystal size and size distribution, can influence the relative viscosity. For crystal fractions approaching a monodisperse spherical population of crystals, such as equant phenocrysts in a flow, the Stokes-Einstein relation has been modified and applied. The Stokes-Einstein and derivative formulations have been discussed previously (see reviews by Marsh 1981; Pinkerton & Stevenson 1992; Dingwell et al. 1993; Lejeune & Richet 1995) and, in the absence of new data, will not be discussed here. StokesEinstein derivatives are adequate for such applications and relatively slight viscosity increases of up to 1 log unit for up to 25 vol.% of crystals are typically predicted. The experimental basis of that statement is, however, almost entirely based on studies of basic silicate melts and lowtemperature analogue liquids. Some complications in the approximation of Newtonian suspension viscosities from the Stokes-Einstein relation may even occur in cases of complex shapes and topologies of crystals at low crystal volume fractions. Figure 7 illustrates a recent comparison of model and experimental data for silicic lava flow obsidians (Stevenson et al. 1996). Above 25% crystal content we enter the regime of crystal-crystal interactions where

(b)

Fig. 6. The onset of non-Newtonian flow in melts expressed as log stress versus log strain rate (a) and calculated log viscosity versus log experimental strain rate (b). Four natural compositions are included (LGM, Little Glass Mountain rhyolite; CLA, Crater Lake Andesite; HTB, Hawaiian tholeiitic basalt; NEP, nephelinite). When compared at equivalent starting Newtonian viscosities, the onset of nonNewtonian viscosity occurs at the same strain rate for all four compositions. Another way of expressing this is that the offset between the relaxation strain rate (calculated using the Maxwell relation) and the strain rate corresponding to the onset of non-Newtonian viscosity, is a constant value. Reproduced from Webb & Dingwell (1990b).

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Fig. 7. Comparison of the Stokes-Einstein prediction with viscosities measured for dilute crystal suspensions in natural obsidians. Reproduced from Stevenson et al. (1996). flow will be non-Newtonian (e.g. Lejeune & Richet 1995). Here there are also very few data for silicic systems. Inferences can only be made on the basis of studies in much more basic systems and analogue materials, but these are always tenuous because of the extrapolation in either chemical composition (and therefore reactivity) or in matrix viscosity. One recent exception to this case is provided by the small strain torsional studies of Bagdasssarov and co-workers (Bagdassarov & Dingwell 1993a, b; Bagdassarov et al. 1994a) which have been summarized by Dingwell (1996). These studies of combined crystal and bubble-bearing rhyolitic suspensions reveal shear viscosity variations (Fig. 8). At 45% crystal content the samples could not fully relax, indicating rigid matrix behaviour. The most crystal-rich andesites in domes are likely to be in this regime and a significant body of work lies before experimentalists in adequately characterizing this regime (e.g. Lejeune et al. 1997). An additional complication in crystal-bearing magma rheology is that the strain embodied in the viscous flow of such magmas can rapidly lead to textural development of the flowing material. The simplest case is the flow alignment of phenocrysts and microlites in lava flows, which can easily define flow lineations or even flow foliations (relative to the flow surfaces). This strain-induced textural evolution of the crystal suspension must be modelled from the starting point of generalized constitutive relationships for the flow of the crystal suspension at all degrees of crystallinity and textural states. One might hope for a simple variation in the textural features influencing viscosity, but this is unlikely. A more likely scenario involves the progressive crystallization of phenocrysts,

microlites and, in some cases, even nanolites (Sharp et al. 1996) occurring simultaneously with the ongoing textural adjustment of older crystalline phases. Such complexities remain a challenge for the rheological modelling of crystal-rich magmas to be begun after generalizable relationships for individual textural states are found. The rheology of bubbly suspensions has been discussed in detail by Stein & Spera (1992),

Fig. 8. The influence of suspended crystalline spheres and gas bubbles on the viscosity of a rhyolitic melt. The viscosities were determined using a 1 atm torsional deformation rig at low frequencies. The addition of the crystal spheres and bubbles (1 : 1 by volume) results in the observed increase in viscosity (upper curve) whereas the gas bubbles alone generate a strong decrease in suspension viscosity (lower curve). By difference it can be inferred that the presence of crystalline spheres alone would lead to a substantially larger decrease in viscosity. Note that these are apparent Newtonian viscosities generated from viscoelasticity data. Redrawn from Bagdassarov et al. (1994).

RECENT EXPERIMENTAL PROGRESS Dingwell et al. (1993) and Bagdassarov et al (1994b). Here some key conclusions of recent work are covered. Bubble suspensions are in one sense simpler in that the shapes of bubbles at low volume fractions of vesicularity or porosity are usually spherical. Nevertheless, as complex as the situation may appear for the case of crystal suspensions, the case of bubble suspensions is, on the whole, even more complex. Intuition might tell us that the addition of bubbles to magma should reduce the viscosity as the bubbles themselves can take up deformation. The viscosity of the fluid in the bubbles is negligible compared to the silicate melt, so the deformation through both phases should reduce the viscosity of the magma. Experiments designed to measure the effects of crystals and bubbles on the viscosity of a (dry) rhyolite melt at low strain and low strain rate have indeed shown this to be the case (Bagdassarov & Dingwell 1992, 19930). Implicit here is that the bubbles in the melt can deform in response to stress. In fact, this is not always the case and under conditions of high strain rate the bubbles behave instead as rigid inclusions, similar to crystals. The essential relations used in determining whether the bubbles will be nondeformable, deformable or highly deformable (bubble breakup) are discussed by Stein & Spera (1992) and Dingwell et al. (1993). They include a characteristic shape relaxation time for individual bubbles and a critical value of the Capillary number for bubble break-up. Thus, studies of analogue liquids under such conditions demonstrate that melt viscosity in fact can be enhanced by the incorporation of bubbles (e.g. see Jaupart & Tait 1990), although this remains to be experimentally demonstrated in the low vesicularity range for rhyolites. An issue concerning the physical behaviour of magma closely related to the consideration of the viscosity of bubbly suspensions is the kinetics of foaming and foam stability. The volume fraction of bubbles in a magmatic bubble suspension can achieve very high levels. Foam may be defined as a bubble suspension with over 74 vol.% of the included phase. Overriding of this limit of close packing of spherical bubbles of uniform size is possible by bubble deformation or variable bubble size. The properties of foam depart from those of a bubbly suspension in an analogous manner to the departure of the properties of partially molten rock from those of a crystal suspension. The discussion of the rheology of such systems becomes increasingly dominated by the question of strain-induced textural evolution of the foam - i.e. the factors controlling foam

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stability. Proussevitch et al. (1993) present an analysis of the stability of foams in silicate melts. They restrict their discussion to the case of the development of foam structure where the gas pressure in the vesicles equals the fugacity of the gas component in the melt and thus no concentration gradients exist in the melt. A 'stable' foam is defined by Proussevitch et al. (1993) as one whose disruption is controlled by fluid expulsion from between the bubbles. They further term foams as subcritical when bubble walls are too thick to allow coalescence and supercritical when the critical thinness of bubble walls for disruption is reached. The foam topology is defined by the bubbles providing the porosity and the dimensions of the films and so-called 'plateau borders' between three or more bubbles. Two essential transport mechanisms lie behind the disruption of foams, the advection of melt from interbubble partitions in response to differential stresses and the diffusion of the gas phase between vesicles due to chemical potential (volatile pressure) gradients. All foams are unstable against textural ageing or maturing and eventual collapse because this process reduces the total energy of the system by decreasing the surface contribution. Nevertheless, the kinetics of foam collapse can vary greatly. Proussevitch et al. (1993) argue that for silicic magmas, the dominant mechanism for the expulsion of melt from interbubble partitions is surface tension rather than gravity at high porosity. They also point out that the thin film criterion that defines the critical thickness of films for foam disruption is likely to be the characteristic dimension of crystals in the film boundaries rather than any fundamental instabilities involving van der Waals' forces.

Permeability Static, low-temperature measurements The permeability of magmatic foam is a potentially important parameter in controlling the physics of the degassing process in the poorly understood interval between earlier rapid bubble growth and volatile exsolution and later solidstate fracturing of the cooler magma. Volatile transfer in this region may decide whether the eruptive activity of a given silicic centre is to be dominantly effusive or explosive, or whether activity will switch temporally between effusive and explosive periods (e.g. Jaupart & Allegre 1991). Magma permeability, determined on samples of rhyolite at room temperature, is illustrated in Figure 9. The permeability rises from values near

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Fig. 9. Permeability of vesicular volcanic rocks at room temperature. Two sets of data for the Mount St Helen's brown dacite stand in contrast. The permeabilities determined by Eichelberger et al. (1986) increase strongly in the porosity range of 55-60 vol. %, whereas the values determined by Klug & Cashman (1996) are relatively high over the entire range of vesicularity. This discrepancy and the relatively high values observed in both studies at high vesicularities raises the question as to whether the permeabilities measured at room temperature correspond to textures well quenched from high temperature or textures that have suffered mechanical 'damage' during cooling to room temperature. Redrawn from Klug & Cashman (1996). 10 -16 m2 at 30-60% porosity to values as high as 10 - 1 1 m 2 at porosities up to 75%. Eichelberger et al. (1986) suggested that the high permeabilities observed at high porosities should permit widespread open system behaviour with respect to volcanic degassing in silicic centres. Further investigation of the permeability of silicic pumice made by Klug & Cashman (1996) are also presented in their fig. 5 (Fig. 9 here). Both datasets agree well at high porosity but differ in the low-porosity region. Klug & Cashman argue, on the basis of scanning electron microscope (SEM) data, that their relatively high permeability values do not result from extensive cracking or other brittle deformation but rather from largely ductile coalescence' of many small bubbles. This is a process they infer as a dominant process of permeability development in the high-temperature magma. The low permeabilities observed by Eichelberger et al. (1986) could conceivably then result from postdegassing permeability reduction during flow via bubble deformation.

Dynamic, high-pressure-high-temperature measurements High-temperature in situ permeability measurements above the glass transition temperature,

where any possible remnants of brittle fracture can be healed out, would augment the present knowledge of permeability in vesicular silicic materials. In high-temperature experiments by Westrich & Eichelberger (1994) they confirmed rapid vapour transport using D2O, and interpreted their results to indicate that advective transport of the vapour rather than diffusion dominated the isotopic exchange at porosities greater than 60vol.%. They also observed rapid collapse of foams under moderate overpressures and applied their results to the modelling of open versus closed system behaviour in rhyolitic magma (Fig. 10). The decompression rates used in the experiments are, however, relatively high compared with those anticipated during magma ascent. Studies of isobaric quenching of waterrich glasses have led to the conclusion that microcracking is common (Mungall el al. 1996; Romano et al. 1996), and so brittle failure of the highly permeable samples may have occurred in the study of Westrich & Eichelberger (1994). It is important to distinguish between microcracking and thin film rupturing in generating high permeability. A complete microstructural characterization of the samples of Westrich & Eichelberger (1994), together with the identification of the mechanism of permeability development, might strengthen the application of their results to natural situations.

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metric methods (Ochs & Lange 1997), and the results modelled together with the highertemperature volumometer data of Burnham & Davis (1971). Ochs & Lange (1997) derive a partial molar volume of H2O of 27.75 ± 0.45 cm3 mol - 1 in an albitic melt at 1673 K and 1 bar, and temperature and pressure derivatives of 10.86 ± 0.46 x l O - 3 cm3 (mol-K)-1 and -3.82 ± 0.36 x 10 -4 cm 3 (mol-bar) -1 , respectively. We obtain a rough indication of the magnitude of the effect of water on melt density if we combine at 1 bar pressure those data for albitic liquids with the volume-temperature relationship from Knoche et al. (1995) for a haplogranitic liquid (VHPG = 28.12 ± 0.01 + 9.26 ± 0.06 x 10 -4 (T- 1193K)). We obtain a volume relation for the 'binary' calcalkaline rhyolite-water system of Vhydrous rhyolite (cm

Fig 10. Porosity and permeability in rhyolitic magma as a function of depth. These data represent the attempt to determine in situ high-temperature permeabilities of vesicular magma. The observed porosities generated by decompression reflect nearequilbrium values for degassed samples (given by the solid line labelled 'calculated porosity') at least for target pressures down to 8-10 MPa. Below these pressures a systematic deviation of the porosities towards values lower than the equilibrium values may be an indication of the onset of significant disequilibrium and kinetic braking of the degassing process. If disequilibrium is significant then the question arises as to the nature of the microstructure of samples and their applicability to the natural case. Reproduced from Westrich & Eichelberger (1994).

Equation of state Melt densities may be estimated using existing models for the partial molar volumes, as well as their expansivities and compressibilities using multicomponent models. Silicic melts are somewhat problematical because of the relatively low temperatures and the relatively high, but variable, water content involved in their petrogenesis. To circumvent the first difficulty, dilatometric methods have been developed to measure the expansivity and volume of silicate melts at very high viscosities and very low temperatures, just above the glass transition (Webb et al 1992). The results for partial molar volumes of major and a wide range of minor components of silicic magmas are provided by Knoche et al. (1995) together with expansivities at a reference temperature of 750°C. The second difficulty, that of the partial molar volume of water, has been recently addressed by dilato-

mol)-1

=(28.12 +9.26 x 10 -4 (T- 1193))(1 - XH2o) + (27.75 + 10.86 x 10 -3 (T- 1673))(XH2o) where X is the mole fraction of water and T is the temperature in Kelvins. At 800°C, the molar volume of the rhyolitic melt decreases from 28.01 to 27.44cm 3 mol -1 with the addition of 10 mol.% of water. Thus, the solution of 3wt% water results in a decrease in density of 1.7%/wt% H2O or 5.2% at 3wt% water; from2.30to2.18cm 3 mol - 1 ,at800°C and 1 bar. Heat capacity and transfer The relative heat contents and capacities of silicate melts has been the subject of study using various high-temperature calorimetric methods for several decades. Navrotsky (1995) provides a useful review of the types of calorimetric data available for silicate melts and the methods for obtaining them. Nevertheless, a complete picture of heats of formation and mixing in the liquid state of silicate magmas is not yet available. Silicate melts differ fundamentally from crystals and hydrous fluids in that a kinetic barrier to the equilibrium state always intercedes during cooling to room temperature. For calorimetry on magmas, the phenomenon of the glass transition has the significance that it separates a lower-temperature, disequilibrium, solid-state-like value of the heat capacity of a silicate glass from a higher-temperature, larger heat capacity of the (meta)stable silicate liquid (recall Fig. 3). Thus, extrapolations from the fairly well-described heat capacities of glasses near room temperature, although vital for reconstructing the thermal budget of cooling glassy or glass-bearing volcanic rocks cannot be

D. B. DINGWELL

20 (a)

3.5

Fig. 11. Determinations of the temperature-dependent thermal conductivities of (a) rhyolite and (b) rhyolitic foam using a thin wire source method. The thermal conductivity of the foam is less than a tenth of that of the unfoamed melt at temperatures corresponding to explosive calc-alkaline eruptions. Reproduced from Bagdassarov et al. (1994).

extrapolated to the temperatures of relevance for the viscous transport of magma in nature (i.e. above Tg, i.e. x109 kg). Heat capacity data for the liquid state are required instead. Thus, a good deal of the experimental calorimetric research conducted in the past decade has been concentrated on the high-temperature regime of metastable to stable liquid heat capacities and

heats of mixing. Most of the work has been performed on unary, binary and ternary synthetic systems (e.g. diopside, anorthite-forsterite and diopside-albite-anorthite-see Navrotsky 1995) where viscosities are low enough to ensure equilibration on reasonable time scales. Thus, silicic melts have been relatively neglected to date. The situation is changing, however, and

RECENT EXPERIMENTAL PROGRESS a recent scanning calorimetric study of haplogranitic melts up to and above the glass transition temperature (Knoche et al. unpubl. data) will soon provide a heat capacity model. The incorporation of water into such models awaits further experimental data, which are most likely to be obtained either from internally heated autoclave drop calorimetry or scanning calorimetry using slightly to substantially enhanced pressures. A good review of the thermal conductivity is included in the recent contributions by Snyder et al. (1994) and Biittner et al. (1997). Melt chemistry does not play a large role in determining the melt thermal conductivity. The exception to this rule is Fe oxide, which has a significant influence on the radiative component of thermal conductivity. Inasmuch as calcalkaline rhyolites involved in explosive volcanism can have Fe oxide contents that vary by orders of magnitude (0.05-5 wt% Fe oxide) this aspect of the thermal conductivity should be investigated systematically in the future. In the context of explosive volcanism, a much larger influence on the thermal conductivity of magmas may be that of bubbles. The thermal conductivity of rhyolite melt versus rhyolite foam (with 70-80% porosity) was determined by Bagdassarov & Dingwell (1994) using a radial heat transfer method. They showed that the conductivity of the foam is at least an order of magnitude lower than that of the equivalent unformed material atmagmatic temperatures (Fig. 11). The presence of a second phase in the form of bubbles in the magma impacts on the thermal conductivity in a potentially complex manner. The convection of gas in the bubbles can aid thermal conductivity of the bulk. This was not the case in the experiments of Bagdassarov & Dingwell (1994) due to the size of bubbles (100-200/mi). Instead the vesicles impede heat transfer due to multiple reflections on bubble walls and an effective enhancement of the optical absorption due to the increased optical path length in the bubble walls. Mechanical strength

Simplified magma strength tests The strength of what are nominally even relatively simple materials (metals, glasses) is a complex subject. The tensile strength of thick glass fibres of rhyolite has been determined using dilatometric methods by Webb & Dingwell (1990a, b) at high temperatures. Their data, included here in Fig. 12, indicate breaking strengths in the range of 108.5Pa. Interestingly,

21

Volatile Fig. 12. Estimates of the tensile strength of magma. The data labelled 'volatile free' were obtained by fibre elongation investigation of dry rhyolite melt. The data labelled H2O, CO2 and Xe were obtained using the synthetic fluid inclusion in glass decrepitation method for each saturating gas. The inclusion of bubbles of 'relatively' inert gases such as CO2 and Xe in the silicic glass leads to no noticeable effect on the tensile strength, whereas the inclusion of highly soluble H2O leads to a decrease of two orders of magnitude in the tensile strength. This contrasting behaviour may be explained by the presence of microcracks around bubbles in the latter (reproduced from Romano et al. 1996).

chemical composition seems to play no significant role in determining the breaking strength (Webb & Dingwell 1990a). The high-temperature strengths of vesicular magmas have been estimated using a technique based on the decrepitation of fluid inclusions in glasses. The results of experiments to determine the internal pressure required to fragment vesicular silicic glasses quenched isobarically from high temperature and pressure are presented in Fig. 12. The fragmentation of CO2- and Xe-saturated melts requires at least 108 Pa; similar to the values obtained for the tensile strength of the bubblefree glass fibres of Fig. 6. In contrast, the strength of the H2O-saturated vesicular melts is drastically reduced, down to about 106.3Pa. If the analogy of other anhydrous network modifying components can be used then this reduction in strength is unlikely to be due to the chemical effect of the addition of water. A more likely explanation is the presence of microcracking around bubbles that is induced by diffusive loss of water from the glass into the inclusions during cooling (Mungall et al. 1996; Romano et al. 1996).

Dynamic response of complex magmas The materials investigated above are very simplified analogues of magma. Recently, the tensile

22

D. B. DINGWELL

response of actual dome samples of magma under conditions of rapid longitudinal unloading or decompression have been determined (Alidibirov & Dingwell 1996a, b, Alidibirov et al 1998). Their preliminary data indicate that the strength of silicic magmas may be as low as a few MPa during rapid decompression. Such low strength values may be partially contributed to by the intricate geometries of vesicles making up the open porosity in certain dome magmas where crystals commonly impinge on the melt-vapour boundary as well (Mungall 1995). These seem consistent with the remarkably low strength of Unzen dome magma during edifice collapse (Sato et al. 1992). Where the solid-like or glassy response of silicic magma under high strain rates can be demonstrated to involve brittle failure, an adequate description of the failure response of the magma must include some estimate of the so-called 'fracture toughness'. The fracture toughness is a measure of the force required to generate a quantity of cracking (reported as force length - 3 / 2 ). A useful review of the fracture toughness and related properties of glasses is included in Cook & Pharr (1990). Room temperature determination of the fracture toughness of crystal-poor natural obsidians have been performed using indentation methods by Gerth & Schnapp (1996) using the formulation of Lawn & Fuller (1975). They give values in the range of 1.2-1.8MNm -3 / 2 . The temperature and pressure dependence of the fracture toughness of magma remained uninvestigated.

Surface properties The most important surface in an explosively erupting silicic system is likely to be the meltvapour interface. The evolution of this surface in an erupting silicic magma can be split into two phases; the relatively slow but accelerating foaming of the melt prior to fragmentation and the explosive production of surface during fragmentation by whatever means. Thus, surface tension data for a surface composed of a hydrous silicate melt and the saturating hydrous phase are useful starting points for the quantification of energy associated with the evolution of surface in the magmatic system. Data for the surface tension of silicate melts are provided by Khitarov et al. (1979), Proussevitch & Kutolin (1986), Walker & Mullins (1987) and Bagdassarov et al. (1994). Khitarov et al. (1979) and Bagdassarov et al. (1994b) have investigated the influence of water pressure on

the surface tension of basaltic and granitic melts, respectively. Bagdassarov et al. (1994b) performed an extensive study of the influence of various individual oxide components added to an analogue calc-alkaline rhyolite composition (HPG8) using the sessile drop method under high-temperature and high-temperature-highpressure conditions. Their results show that the surface tension of a dry metaluminous haplogranitic melt decreases significantly under increasing water pressure from 135m N m - 1 at 1 kbar pH2O to 65 mN m-1 at 4 kbar pH2O in a strongly non-linear fashion. Such a characterization of the melt-vapour surface is clearly only a first step. Nevertheless, in the foamed magmatic pumice of many Plinian eruptions it may be a useful approximation. In contrast, the products of the block-and-ash flows resulting from dome collapse are often in a far more texturally advanced state with highly deformed vesicles multiply impinged by the abundant crystals of the phenocryst, microlite and possibly nanolite populations.

Saturation and water solubility The dominant volatile controlling the explosivity of magma is water. Correspondingly, the solubility of water in silicic magmas forms a very important constraint in the modelling of volatile saturation in equilibrium magma ascent models or of volatile oversaturation in disequilibrium models of bubble nucleation and growth. The solubility of water in silicic magmas has been investigated extensively in recent years in the low-temperature and -pressure range (Holtz et al. 1992, 1993, 1995; Romano et al. 1996; Dingwell et al. 1997). There is also a considerably larger literature on the speciation and structural state of water in silicic melts, but in the present authors opinion the link between systematics of the speciation of water and the solubility limits observed has not been convincingly demonstrated. Thus, while the salient features of the recent solubility studies are presented here, the speciation studies are not treated here. Recent determinations of the pressure dependence of the solubility of water for some model rhyolitic composition melts that vary slightly in Na: K ratio confirm the earlier observations of Burnham (1981) for granitic melts in general but not in detail (Holtz et al. 1992). At pressures less than 2 kbar this composition dependence of the solubility affects the pressure dependence of the solubility very little.

RECENT EXPERIMENTAL PROGRESS

23

Fig. 13. The temperature dependence of the solubility of water in a haplogranitic melt at pressures from 0.5 to 5 kbar. The solubilities exhibit a transition from a negative temperature dependence at low pressures to a positive pressure dependence at high pressures, presumably as a result of the approach to the second critical end-point in the melt-water system. Reproduced from Holtz et al (1995). The temperature dependence of the solubility of water in silicic magma is summarized for the case of a metaluminous rhyolite composition in Fig. 13 (Holtz et al. 1995). The temperature dependence evolves from a slight negative dependence at low pressures through temperature invariance at intermediate pressure to a strong positive dependence at higher pressures. At pressures below 500 bars and at temperatures below 800°C we have very little information on the solubility of water in silicic magma. Although the temperature dependence of the solubility of water at low pressures is small compared with the pressure dependence, it is nevertheless an issue of potential relevance due to the enthalpic effects of shallow degassing of a rhyolitic magma. Although no experimental data exist on this subject, Sahagian & Proussevitch (1996) and Mastin (1997) have performed numerical simulations of the temperature drop anticipated from the enthalpic and volume effects of magma degassing at shallow depths and quasi-isobaric conditions. Mastin concludes that up to a 50°C decrease may be the consequence of degassing. Such temperature variations could develop a feedback to the solubility relation, with decreasing temperature during degassing resulting in the degassing process being confronted with a slightly higher solubility of water in the melt phase. Such effects are wholly unmodelled to date. The composition dependence of the solubility of water is illustrated in Fig. 14 for the example of a variable alkali: aluminium ratio at a pressure of 500 bars and a temperature of 1000°C. Increasingly peralkaline melts can dissolve higher levels of water at conditions of constant temperature and pressure. Mildly pera-

luminous melts appear to exhibit slightly reduced water solubilities (Dingwell et al. 1997). Interestingly, the minimum in water solubility is not found at the 1:1 alkali: aluminium ratio, a phenomenon likely to be of importance in subvolcanic plutons experiencing second boiling. Clearly, the eruption of water-rich peralkaline rhyolites may occur under conditions of much higher relative water content at shallow depths than is possible for the case of an erupting

Fig. 14. The compositional dependence of the solubility of water in haplogranitic melts as a function of the alkali: aluminium ratio of the anhydrous melt at 1000°C and 500 bar. The strongly enhanced solubility of water observed in strongly peralkaline systems, combined with the relatively low eruptive viscosities of such melts inferred from Fig. 5, are likely to mean that near-surface degassing of peralkaline rhyolites will be extremely efficient. Reproduced from Dingwell et al. (1997).

24

D. B. DINGWELL

calc-alkaline rhyolite. This fact combined with the inference of extremely low viscosities of hydrous peralkaline rhyolites (Fig. 5) at their eruption temperatures means that the efficiency of degassing of peralkaline rhyolites at very shallow depths must be very high indeed. Water diffusion Water is a highly mobile species in silicic glasses and melts. The diffusivity of water has been measured under various conditions of pressure, temperature and water content, and excellent summaries are provided by Zhang et al (1991) and Nowak & Behrens (1997). The more recent study (Nowak & Behrens 1996) demonstrates a mild positive pressure dependence of the diffusivity ( 0.3 log units for a 4 kbar pressure difference) and confirms a strong positive concentration dependence of the diffusivity ( 1.5 log units between 0.5 and 5 wt % water), and a relatively low activation energy yielding a temperature dependence of 0.5 log units

between 800°C and 1200°C. These variations have important implications for the microscopic nature of hydrous species diffusion in the melt structure but a relatively minor influence on the behaviour of water in degassing rhyolites. The most important point concerning water diffusion in rhyolites is that it is controlled by processes that are completely detached from the time scales and mechanisms associated with stress relaxation in melts (i.e. those related to Si diffusion). That this is not the case in basalts is demonstrated with the aid of Fig. 15. The reciprocal of the effective diffusive jump frequency yields a time scale that is comparable to that of viscous stress relaxation in basaltic melts, whereas in rhyolites of any volcanologically relevant water content the time scale controlling mass transfer due to water diffusion is faster than that controlling viscous stress relaxation. This fundamental point raises the possibility of bubble overpressure due to viscous retardation or the so-called 'viscosity quench' whereby bubble growth can become viscosity-controlled (see Navon & Lyakhovsky 1998).

Future perspectives The physical properties of silicic magmas remain the key link between the thermal, stress and gravity fields that provide the potential energy for explosive eruptions and the detailed response of the complex geo-material we call magma. Many gaps exist in our knowledge of these 'coefficients of magma response'. Amongst the most important are the rheology of magmatic foams and crystal-rich suspensions, the effective mechanical strength of partially degassing and crystallized magma, the PVT and thermodynamic properties of hydrous silicic melts, the stability of magmatic foam, and the thermodynamics controlling the dissolution and exsolution of water in melts at relatively low temperatures and pressures. Technical advances in the experimental synthesis and characterization of magmas should permit a vigorous experimental approach to these topics in the next years. Fig. 15. The relative time scales associated with the relaxation of shear stresses and the diffusive jump of water in the melt plotted as a function of the reciprocal temperature. The data for rhyolites exhibit a large discrepancy in the time scale that is the source of the potential role of melt viscosity in influencing late stages of bubble growth. The data for basalts exhibit no discrepancy in the time scales such that viscous 'quenching' of bubble growth in such systems is highly unlikely. Redrawn from Dingwell (1995).

Parts of this review have been presented at the Arthur Holmes European Research Conference in Santorini, August 1996, and the IAVCEI General Assembly, in Puerto Vallarta, Mexico, January 1997. Helpful reviews by anonymous, O. Navon and R. S. J. Sparks, as well as diligent editorial work by the latter, have led to substantial improvements. Much of the work of the Bayreuth group described within has been funded generously by the Deutsche Forschungsgemeinschaft. the Alexander-von-Humboldt-Stiftung, the European Commission and the Bayerisches Geoinstitut.

RECENT EXPERIMENTAL PROGRESS

References ALIDIBIROV, M. & DINGWELL, D. B. 1996a. High temperature fragmentation of magma by rapid decompression. Nature, 380, 146-149. & 1996b. An experimental facility for the investigation of high temperature magma fragmentation during rapid decompression. Bulletin of Volcanology, 58, 411-416. , & DINGWELL, D. B. 1998. Grain-size characteristics of experimental pyroclasts of 1980 Mt. St. Helens cryptodome dacite: effects of pressure drop and temperature. Bulletin of Volcano logy, in press. BAGDASSAROV, N. & DINGWELL, D. B. 1992. A rheological investigation of vesicular rhyolite. Journal of Volcanological and Geothermal Research, 50, 307-322. & 19930. Frequency-dependent rheology of vesicular rhyolite. Journal of Geophysical Research, 98,6477-6487. & 1993b. Deformation of foamed rhyolites under internal and external stresses. Bulletin of Volcanology, 55, 147-154. & 1994. Thermal properties of vesicular rhyolite. Journal of Volcanological and Geothermal Research, 60, 179-191. , & WEBB, S. L. 1994a. Viscoelasticity of crystal- and bubble-bearing rhyolite melts. Physics of the Earth and Planetary Interiors, 83, 83-99. , DORFMAN, A. M. & DINGWELL, D. B. 1998. Effect of alkalis, phosphorous and water on surface tension of haplogranitic melts. American Mineralogist, submitted. BAKER, D. R. 1996. Granitic melt viscosities: empirical and configurational entropy models for their calculation. American Mineralogist, 81, 126-134. BURNHAM, C. W. 1981. The nature of multicomponent aluminosilicate melts. Physics and Chemistry of the Earth, 13-14, 197-229. & DAVIS, N. 1971. The role of water in silicate melts: I . P-V-T relations in the system NaAlSi3O8-H2O to 10 kilobars and 1000°C. American Journal of Science, 270, 54-79. BUETTNER,

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HAGEMANN, L. 1997. Thermal properties of volcanic rock material (olivine melilite model composition) in the temperature range between 288 and 1470 K. Journal of Volcanology and Geothermal Research, 80, 293-302. COOK, R. F. & PHARR, G. M. 1990. Direct observation and analysis of indentation cracking in glasses and ceramics. Journal of the American Ceramic Society, 73, 787-817. DINGWELL, D. B. 1995. Relaxation in silicate melts: some applications. Reviews in Mineralogy, 32, 21-66. 1996. The brittle-ductile transition in high-level granitic magmas: material constraints. Journal of Petrology, 38, 1635-1644. , BAGDASSAROV, N., BUSSOD, G. & WEBB, S. L. 1993. Magma Rheology. Mineralogical Association of Canada Short Course on Experiments at

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High Pressure and Applications to the Earths Mantle, 131-196. -, HESS, K-U. & ROMANO, C. 19980. Viscosity data for hydrous peraluminous granitic melts: incorporation in a metaluminous model. American Mineralogist, 83, 236-239. -, & 1998/7. Extremely fluid behaviour of hydrous peralkaline rhyolites: Experimental viscosity data and a model. Earth and Planetary Science Letters, 158, 31-38. , HOLTZ, F. & BEHRENS, H. 1997. The solubility of water in peralkaline and peraluminous granitic melts. American Mineralogist, 82, 434-437. & WEBB, S. L. 1989. Structural relaxation in silicate melts and non-Newtonian melt rheology in igneous processes. Physics and Chemistry of Minerals, 16, 508-516. & 1990. Relaxation in silicate melts. European Journal of Mineralogy, 2, 427-449. ElCHELBERGER, J. C., CARRIGAN, C. R., WESTRICH,

H. R. & PRICE, R. H. 1986. Non-explosive silicic volcanism. Nature, 323, 598-602. GERTH, U. & SCHNAPP, J. D. 1996. Investigation of mechanical properties of natural glasses using indentation methods. Chemie der Erde, 56, 398-403. HESS, K-U. & DINGWELL, D. B. 1996. Viscosities of hydrous leucogranitic melts: a non-Arrhenian model. American Mineralogist, 81, 1297-1300. , & WEBB, S. L. 1995. The influence of excess alkalis on the viscosity of a haplogranitic melt. American Mineralogist, 80, 297-304. HOLTZ, F., BEHRENS, H. & DINGWELL, D. B. 1995. Water solubility in haplogranitic melts: compositional, pressure and temperature dependence. American Mineralogist, 80, 94-108. , , & TAYLOR, R. P. 1992. Water solubility in aluminosilicate melts of haplogranitic compositions at 2 kbar. Chemical Geology, 96, 289-302. , DINGWELL, D. B. & BEHRENS, H. 1993. The effects of fluorine, boron and phosphorus on the solubility of water in haplogranitic melts compared to natural silicate melts. Contributions to Mineralogy and Petrology, 113, 492-501. JAUPART, C. J. & ALLEGRE, C. 1991. Gas content, eruption rate and instabilities of eruption regime in silicic volcanoes. Earth and Planetary Science Letters, 102, 413-429. & TAIT, S. 1990. Physical processes in the evolution of magmas. Reviews in Mineralogy, 24, 213-238. KHITAROV, N. I., LEBEDEV, Y. B., DORFMAN, A. M. & BAGDASSAROV, N. S. 1979. Effects of temperature, pressure and volatiles on the surface tension of molten basalt. Geochemistry International, 16, 78-86. KLUG, C. & CASHMAN, K. V. 1996. Permeability development in vesiculating magmas: implications for fragmentation. Bulletin of Volcanology, 58, 87-100. KNOCHE, R., DINGWELL, D. B. & WEBB, S. L. 1995. Leucogranitic and pegmatitic melt densities: partial molar volumes for SiO2, A12O3, Na2O, K2O, Rb2O, Cs2O, Li2O, BaO, SrO, CaO, MgO, TiO2, B2O3, P2O5, F 2 O_ 1 , Ta2O5, Nb2O5, and WO3. Geochimica et Cosmochimica Ada, 59, 4645-4652.

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LANGE, R. A. & CARMICHAEL, I. S. E. 1990. Thermodynamic properties of silicate liquids with emphasis on density, thermal expansion and compressibility. Reviews in Mineralogy, 24, 64. LAWN, B. R. & FULLER, E. R. 1975. Equilibrium penny-like cracks in indentation fracture. Journal of Materials Science, 10, 2016-2024. LEJEUNE, A-M. & RICHET, P. 1995. Rheology of crystal-bearing silicate melts: an experimental study at high viscosities. Journal of Geophysical Research, 100, 4215-4229. , NEUVILLE, D. R., LINNARD, Y. et al. 1997. Rheology of the Montserrat Lavas. Terra Nova, 9, 194. MASTIN, L. 1997. Subsurface pressure in volcanic conduits and the influx of groundwater during sustained pyroclastic eruptions. EOS, Transactions of the American Geophysical Union, 78, 790. MENSCH, B. D. 1981. On the crystallinity, probability of occurrence and rheology of lava and magma. Contributions to Mineralogy and Petrology, 78, 85-98. MUNGALL, J. E. 1995. Textural controls on explosivity of lava in Merapi-type block and ash flows. Periodico di Mineralogia, 64, 233-234. , ROMANO, C, BAGDASSAROV, N. & DINGWELL, D. B. 1996. A mechanism for microfracturing of vesicle walls in glassy lava: implications for explosive volcanism. Journal of Volcanology and Geothermal Research, 73, 33-46. NAVON, O. & LYAKHOVSKY, V. 1998. Vesiculation processes in silicic magmas. This volume. NAVROTSKY, A. 1995. Energetics of silicate melts. Reviews in Mineralogy, 32, 121-143. NOWAK, M. & BEHRENS, H. 1996. An experimental investigation on diffusion of water in haplogranitic melts. Contributions to Mineralogy and Petrology, 126, 365-376. OCHS, F. A. & LANGE, R. A. 1997. The partial molar volume of H2O in NaAlSi3O8 liquid: new measurements and an internally consistent model. Contributions to Mineralogy and Petrologv, 129, 155-165. PINKERTON, H. & STEVENSON, R. J. 1992. Methods of determining the rheological properties of lavas from their physico-chemical properties. Journal of Volcanological and Geothermal Research, 53, 47-66. PROUSSEVITCH, A. A. & KUTOLIN, V. A. 1986. Surface tension of magmatic melts. Geology and Geophysics, 9, 67-69 (in Russian). , SAHAGIAN, D. & KUTOLIN, V. A. 1993. Stability of foams in silicate melts. Journal of Volcanology and Geothermal Research, 59, 161-178. RIVERS, M. L. & CARMICHAEL, I. S. E. 1987. Ultrasonic studies of silicate melts. Journal of Geophysical Research, 92, 9247-9270. ROMANO, C, MUNGALL, J., SHARP, T. & DINGWELL, D. B. 1996. Tensile strengths of hydrous vesicular glasses: an experimental study. American Mineralogist, 81, 1148-1154. SAHAGIAN, D. & PROUSSEVITCH, A. A. 1996. Thermal effects of magma degassing. Journal of Volcanology and Geothermal Research, 74, 19-38.

SATO, H. & MANGHNANI, M. 1985. Ultrasonic measurements of Vp and Qp: relaxation spectrum of complex modulus of basalt melts. Physics of the Earth and Planetary Interiors. 41, 18-33. , FUJII, T. & NAKADA, S. 1992. Crumbling of dacite domes and generation of pyroclastic flows at Unzen volcano. Nature, 360, 664-666. SHARP, T., STEVENSON, R. & DINGWELL, D. B. 1996. Microlites and "nanolites" in rhyolitic glass: microstructural and chemical characterisation. Bulletin of Volcanology, 57, 631-640. SHAW, H. R. 1972. Viscosities of magmatic silicate liquids: an empirical method of prediction. American Mineralogist, 212, 870-889. SIEWERT, R. & ROSENHAUER, M. 1997. Viscoelastic relaxation measurements in the system SiO2NaAlSiO4 by photon correlation spectroscopy. American Mineralogist, 82, 1063-1072. SNYDER, D., GIER, E. & CARMICHAEL, I. S. E. 1994. Experimental determination of the thermal conductivity of molten CaMgSi2O6 and the transport of heat through magmas. Journal of Geophysical Research, 99, 15 503-15 516. STEIN, D. J. & SPERA, F. J. 1992. Rheology and microstructure of magmatic emulsions: theory and experiments. Journal of Volcanological and Geothermal Research, 49, 157-174. STEVENSON, R., BAGDASSAROV, N., DINGWELL, D. B. & ROMANO, C. 1998. The influence of trace amounts of water on obsidian viscosity. Bulletin of Volcanology, in press. . DINGWELL, D. B., WEBB, S. L. & BAGDASSAROV. N. L. 1995. The equivalence of enthalpy and shear relaxation in rhyolitic obsidians and quantification of the liquid-glass transition in volcanic processes. Journal of Volcanology and Geothermal Research, 68, 297-306. , , & SHARP, T. 1996. Viscosity of microlite-bearing rhyolitic obsidians: an experimental study. Bulletin of Volcanology, 58, 298-309. WALKER, D. & MULLINS, O. 1981. Surface tension of natural silicate melts from 1200-1500 C and implications for melt structure. Contributions to Mineralogy and Petrology, 76. 455-462. WEBB, S. L. & DINGWELL, D. B. 1990a. The onset of nonNewtonian rheology in silicate melts. Physics and Chemistry of Minerals, 17, 125-132. & 1990b. NonNewtonian rheology of igneous melts at high stresses and strain rates: Experimental results for rhyolite, andesite, basalt and nephelinite. Journal of Geophysical Research, 95, 15 695-15 701. , KNOCHE, R. & DINGWELL, D. B. 1992. Determination of liquid expansivity using calorimetry and dilatometry. European Journal of Mineralogy, 4. 95-104. WESTRICH, H. R. & EICHELBERGER, J. C. 1994. Gas transport and bubble collapse in rhyolitic magma: an experimental approach. Bulletin of Volcanology, 56, 447-458. ZHANG, Y., STOLPER, E. M. & WASSERBURG, G. J. 1991. Diffusion of water in rhyolitic glasses. Geochimica et Cosmochimica Acta, 55, 442-456.

Vesiculation processes in silicic magmas ODED NAVON & VLADIMIR LYAKHOVSKY Institute of Earth Sciences, The Hebrew University, Jerusalem 91904, Israel Abstract: The physics of vesiculation, i.e. the process of bubble formation and evolution, controls the manner of volcanic eruptions. Vesiculation may lead to extreme rates of magma expansion and to explosive eruptions or, at the other extreme, to low rates and calm effusion of lava domes and flows. In this paper, we discuss the theory of the different stages of vesiculation and examine the results of relevant experimental studies. The following stages are discussed: (1) The development of supersaturation of volatiles in melts. Supersaturation may develop due to a decrease in equilibrium solubility following changes in ambient pressure or temperature or due to an increase in the magma volatile content (e.g. in response to crystallization of water-free or waterpoor mineral assemblages). (2) Bubble nucleation. The classical theory of homogeneous nucleation and some modern modifications, heterogeneous nucleation with emphasis on the nucleation of water bubbles in rhyolitic melts and the role of specific crystals as heterogeneous sites. (3) Bubble growth. The effect of diffusion, viscosity, surface tension, ambient pressure and inter-bubble separation on the dynamics of growth. (4) Bubble coalescence. The theory of coalescence of static foams, factors that may effect coalescence in expanding foams and shape relaxation following bubble coalescence.

Nomenclature Surface (m2) Distance between water molecules in the melt (m) Concentration (kg water/kg melt) Constants Diffusion coefficient (m2 s -I ) Free energy (J) Nucleation rate (m - 3 s - 1 ) Pre-exponential nucleation rate (m - 3 s - 1 ) Gas constant (J mol - 1 ) Boltzmann constant (JKr-1) Henry's constant (Pa -1/.2 ) Mass of gas in bubble (kg) Molecular weight of water (kg mol -1 ) Number of molecules Avogadro number (6.0225 x 1023 mol - 1 ) Bubble number density (m - 3 ) Pressure (Pa) Peclet number Radial coordinate (m) Bubble radius (m) Reynolds number Shell radius (m) Time (s) Temperature (K) Volume (m3) Radial melt velocity (ms - 1 ) Bubble growth rate (ms - 1 ) Zeldovitch factor Power in Henry's law Growth frequency (s - 1 ) Supersaturation pressure (Pa) Surface element (m2)

Vesicularity, gas volume fraction Chemical potential (Jmol - 1 ) Melt viscosity (Pa s) Geometrical factor Wetting angle (°) Density (kg m - 3 ) Surface tension (Nm - 1 ) Characteristic time (s) Power in the Avrami equation initial value crystal critical diffusive drainage and failure (of melt films) equilibrium final value gas melt (liquid silicate) magma (melt ± bubbles ± crystals) radial coordinate relaxation to spherical shape viscous viscous diffusive transition properties of the bubble-melt interface water wall rack Explosive volcanism is a high-power phenomenon. The rapid release of energy during explosive eruptions is made possible by the intimate contact between gas and melt, which results from vesiculation and fragmentation of the magma. Vesiculation is also responsible for

NAVON, O. & LYAKHOVSKY, V. 1998. Vesiculation processes in silicic magmas. In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 27-50.

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the acceleration of the melt, initially due to volume increase of the vesiculating magma and later due to the fast expansion of gas released from the bubbles during melt fragmentation and ash and pumice formation. Characterization of the processes controlling vesiculation is important for understanding and modelling the hidden part of explosive volcanism: the path from the magma chamber to the volcanic neck. Vesiculation begins with supersaturation of the volatile component in the melt and nucleation of bubbles, proceeds with bubble growth, and ends with bubble coalescence and fragmentation of bubble walls. In discussing the three processes, we begin by reviewing the theory, then describe the experimental evidence and finally make the connection to observations on natural samples. The subject has been previously reviewed by Sparks (1978), and more recently by Sparks et al (1994) and Cashman & Mangan (1994). Hence, we chose to present the current status of the subject from the point of view of our own experience, rather than thoroughly review the literature. Throughout this chapter we concentrate on the example of water bubbles in rhyolitic melt which at present is the beststudied case.

The development of supersaturation The formation of a new phase, separated from the existing one by curved interfaces, requires supersaturation of the components of the new phase in order to compensate for the energy cost of creating the new surfaces. In the case of explosive volcanism it is commonly the supersaturation of a volatile component, e.g. water or CO2, in the silicate melt. Before going into the theory of nucleation we would like to review ways by which magmas become supersaturated. The solubility of a volatile component in a melt is a function of pressure, temperature and the composition of the melt. In the case of water, which is the main volatile species in silicic melts and the example we use throughout this chapter, dissolution is complex and involves reaction with the melt to form two main species, hydroxyl and molecular water (see McMillan 1994 for review). However, at pressures up to 200 MPa (2 kbar), the effect of pressure on the solubility of water (Burnhan 1975) may be approximated by a simple square-root relation (see Nomenclature) where the Henry constant, KH, weakly depends on temperature and melt composition (Hollo-

way & Blank 1994; Dingwell 1998). The use of gas pressure rather than the fugacity of the gas requires an additional assumption of gas ideality, which is not seriously in error. Clearly, lowering the pressure is the best way to achieve supersaturation. During an eruption, pressure experienced by the magma decreases as it ascends in the conduit. In view of the typical concentrations of water in magmas, on its way up a hydrated melt eventually reaches the level where the ambient pressure equals the saturation pressure. Further ascent leads to supersaturation. Similarly, pressure may fall in response to failure of rocks enclosing the magma chamber. It also varies during convection in magma chamber. The case of vigorous convection following penetration of hot mafic magma into a silicic magma chamber was discussed by Sparks et al. (1977). The penetrating mafic magma also heats the silicic melt. The hotter melt can dissolve less water and supersaturation increases. Sparks et al. (1977) considered this effect to be secondary relative to the former one, reflecting the weak dependence of solubility on temperature. Rather than decreasing pressure, it is possible to reach supersaturation by increasing the water content of the melt. For example, during crystallization the water content of the residual melt increases as the crystallizing mineral assemblage contains little water. This mechanism is commonly referred to as Second boiling' (e.g. Burnham 1979). This process may play an important role in pressurizing magma chambers (Tail et al. 1989 and references therein) and in lava domes (Sparks 1997).

Bubble nucleation Thermodynamic equilibrium requires the formation of a separate fluid phase when the melt becomes saturated in volatiles. This process involves formation of an interface and a certain degree of supersaturation is required, so that the energy needed for the formation of the new surface is compensated by moving water from the supersaturated melt to the new gaseous phase. As the latter energy is proportional to the volume of the nucleus, there must be a critical size where the energy gained by moving a water molecule to the interior of the nucleus fully balances that required for increasing its interface with the melt. From this size up, growth is spontaneous. Classical nucleation theory allows the calculation of the required energies, the critical size of the nuclei and the rate of nucleation.

VESICULATION PROCESSES IN SILICIC MAGMAS Its formulation relies on important contributions by Gibbs, Laplace, Kelvin and many others. In the present discussion we follow the outline of Landau & Lifshitz (1980). More detailed treatments can be found in Dunning (1969) and Hirth et al (1970). All knowingly ignore some effects which may be important for nucleation of bubbles in melts; these are discussed in the section on modifications of the classical theory.

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before the formation of the nucleus, when all volume was taken by the melt and all the volatile molecules were dissolved in it, the free energy of the system was

The net change in energy upon forming a single nucleus is the difference between equations (2) and (5):

Classical nucleation theory Classical nucleation theory assumes that in spite of the microscopic size of the new phase its thermodynamic properties (e.g. energy, pressure, temperature, chemical potential and surface tension) are identical to those of macroscopic systems. That means that surface energy associated with a surface A is simply A , where a is the surface tension measured in a macroscopic system. In this case, if a spherical gas bubble of volume Vg and surface Ag is formed in melt of volume Vm, the Helmholtz free energy of the system is given by:

In a closed system of constant volume the total number of volatile molecules, nm ng, and Vm + Vg are constant. Temperature is also assumed to be constant, as heat consumption during nucleation is negligible compared with the thermal energy of the melt. The critical nucleus is in chemical equilibrium with the supersaturated melt, so that g = m. At the point of unstable equilibrium, F is at a maximum and =0

This is the Laplace (1806) equation which states that the equilibrium pressure inside a stationary bubble is greater than the pressure in the melt because of the capillary pressure, i.e. the contribution of surface tension. In the case of the critical nucleus, chemical equilibrium dictates that Pg is related to C0, the concentration in the supersaturated melt, by equation (1). It is also possible to calculate the energy needed for creating the critical nucleus. If the volume of the system, Vm + Vg, is constant, then

where we assumed that Pm and m did not change upon formation of the nucleus and that at equilibrium Ug = Um. Using equation (4) we can relate the two terms on the right-hand side of equation (6)

where R, A and Vg are the radius, surface area and volume of the critical nucleus (see also Fig. 1). P = Pg - Pm is the pressure difference acting on the bubble wall. It is also related, through the Henry constant, to the difference between the actual water content of the supersaturated melt and the equilibrium solubility atP m .

Fig. 1. The energy for formation of a nucleus as a function of its radius. The volume term depends on R3, and the surface term on R2 (see equation (7)). The total energy reaches a maximum at RCR- Further growth lowers the free energy and is spontaneous. The energy needed for creation of a nucleus of size RCR is F. In the example shown, AP = 30 MPa and a = 0.06 N m - 1 , yielding RCR = 4 nm and F = 4 x 10 - 1 8 J per nucleus.

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The second assumption made in the classical theory is that nuclei formation is due to random fluctuations and obeys statistical laws, so that the probability for formation of a cluster is, in general, proportional to exp(- F/kT). The pre-exponential term, J*, can be derived from kinetic considerations (Dunning 1969), and the rate of nucleation is

(8)

where is the concentration of potential sites taken as the number of water molecules in the melt, 3 is the growth frequency of the clusters across the critical size and Z is the Zeldovitch factor, correcting for the fact that the growth process is actually occurring at a steady state rather than at equilibrium. By relating B to the diffusion coefficient of water and simplifying Z, Hurwitz & Navon (1994) modified the expression given by Hirth et al. (1970) obtaining instead

where Vw is the volume of a water molecule in the melt and ao is the distance between neighbouring water molecules in the melt. Toramaru (1989) used a somewhat different expression

Although the two expressions are of different form, we should remember that for the critical nucleus, and is merely a geometrical factor. Calculating V w = M / p w N A , we obtain

which is trivial as the rate is controlled by the exponential term and an order of magnitude difference in J* makes very little difference to it. Figure 2 presents the rate of nucleation, J, as a function of supersaturation pressure, P. Rates below one bubble per cm3 per second (or 1 0 6 m - 3 s - 1 , dashed line in Fig. 2) are commonly taken to be negligibly slow. Very high rates are about eight orders of magnitude faster. Yet, this variation in rate corresponds to only a few MPa change in the supersaturation pressure. Similarly, varying J0 by five orders of magnitude,

Fig. 2. The effect of pressure, surface tension and the pre-exponential factor on the nucleation rate (equation (8)). Rate is very sensitive to supersaturation pressure and the transition between negligible nucleation (dashed line) to extreme rates occurs over a pressure change of only a few MPa. Surface tension is very important. For example, if a exceeds 0.1. no homogeneous nucleation is expected in melts containing less than 5% water (corresponding to maximum supersaturation 135 MPa). Five orders of magnitude uncertainties in the pre-exponential factor have small effect.

has little effect on the supersaturation pressure required to drive nucleation. The critical supersaturation is more sensitive to the choice of , whose measured values in hydrous melts vary between 0.032 N m - 1 for a Q-Ab-Or system with 3.7% water (EpeFbaum et al. 1973; EpeFbaum 1980; Bagdassarov et al. 1994) and 0.2 for dry rhyolitic melt (Murase & McBirney, 1973). Supersaturation in excess of 10 MPa is needed even for a = 0.02 N m - 1 . Extreme supersaturation, of 150 MPa or higher, is needed if >0.1Nm-1. The high supersaturation pressure predicted by classical nucleation theory for the onset of nucleation is not only a problem in silicate systems (Sparks 1978), but also in other systems as well (e.g. Bowers et al. 1995, and references therein). The overestimated supersaturations may be a result of the assumptions made in developing the theory. In the following section we discuss modifications to the classical theory.

Modifications of the classical theory Much effort has been invested in improving the classical theory of nucleation and the few examples given here illustrate some of the different approaches. It should be born in mind that many of these approaches are applicable only to simple systems, or involved parametric fits which apply only to the studied systems.

VESICULATION PROCESSES IN SILICIC MAGMAS One approach is to keep using classical thermodynamics, but to re-examine the simplifications in the expressions for the free energy of the system. Tolman (1949) examined the assumption of constant surface tension. He showed that surface tension must vary with bubble size and that, in a single component system, this effect leads to a factor of 4 reduction of the surface tension. Han & Han (1990), studying polymeric liquids, revised the assumption of constant chemical potential of the melt. However, as they fitted their results parametrically rather than using a full thermodynamical approach, the results cannot be applied to silicate systems. Modern approaches go beyond classical thermodynamics. Bottinga & Javoy (1990a) argued that the energy for formation of the new surface may be supplied by small thermal fluctuations and it is not required that all the needed energy is supplied by the supersaturated species. In their model, even in nuclei which are smaller than critical size (embryos), the chemical potential of molecules in the gas phase is low enough to establish diffusive flux and growth. The activation energy for formation of the embryos is lower than the classical value for critical nuclei, and hence the predicted nucleation rate is faster than that predicted by classical theory. The difference becomes important in systems where surface tension is high. Ruckenstein & Nowakowski (1990) described the nucleation process as assembling molecules to form clusters. They assigned different interaction potentials to molecules within the cluster and to those associated with its surface and calculated the probability for transfer of molecules across the surface layer. For large clusters, their results merge with the classical theory. However, in the case of small clusters they predict lower activation energies. This difference in activation energy may be translated to effective surface tensions that are smaller than macroscopic values by up to an order of magnitude. An interesting fact is that the distance between neighbouring water molecules in the melt is of the same order as in the gas (Fig. 3). This means that upon creation of a new nucleus, the problem may be the removal of the silicate melt out of the volume of the new nucleus rather than diffusing water molecules towards the growing cluster. This may have some important implications. For example, it means that melt viscosity and diffusivity of the silicate components, rather than water diffusivity, may be the limiting factors. Because the supersaturations predicted by classical theory are usually higher than experimental values, most works involve modification

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Fig. 3. Comparison of the number of water molecules per unit volume in a supersaturated melt and in a critical nucleus in equilibrium with it. The diagram shows that in melts with a water content of less than 2 wt% the number density of water molecules is higher than in the melt. Even in melts with a higher water content most of the water is in place before nucleus formation. which lead to lower predictions. However, there must be some ignored contributions that can lead to higher theoretical values. In the case of silicic melts we should be concerned with two important effects. One is the assumption that Pm does not change upon nucleation. Because of the low compressibility of the melt, if volume is conserved, pressure must increase upon nucleation. This effect may be neglected if is large (of the order of 1 MPa) and the volume of the nucleus is small compared with the melt volume per nucleus, but it may become important at lower pressures. Hirth et al, (1970) included this effect in their initial equations, but neglect it later in the derivation. Another important contribution to the activation energy which could become significant in melts of high viscosity is the work done by viscous deformation of the melt. Both effects should lead to higher activation energy.

Heterogeneous nucleation Most systems, and especially natural systems, are not pure and homogeneous and carry many heterogeneities. In particular, crystals present in the magma may serve as sites for heterogeneous nucleation of bubbles. If the surface energy of the crystal-gas interface is lower than that of the melt-gas, the activation energy for nucleation on the crystal face is lower compared with that required for homogeneous nucleation in the melt (Sigbee 1969). It also means that the bubble wets the crystal surface better than the melt, or that the wetting angle (Fig. 4) is larger than 90°. The

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Fig. 4. Wetting angles between bubble and crystal. The dihedral wetting angle is measured through the melt from the melt-crystal interface to the tangent to the bubble-melt interface, (a) Flat interface with crystal that is better wetted by vapour than by melt (0 > 90). (b) Bubble in a corner.

wetting angle, 0, is the angle between the crystal face and the tangent to the bubble face at the contact, measured through the melt, and is related to the three surface energies (or surface tensions) by gm = is the surface tension of the gas-melt interface, and c in the subscripts stands for the crystal. The bubble-melt interface is a truncated sphere (see for example fig. 4 of Hurwitz and Navon 1994). The surface energy of the crystalbubble-melt system can be calculated by adding the contributions of that interface and the crystal-gas contact area. It is related to the surface energy of a spherical bubble of similar radius surrounded by melt by a factor 0, where

Accordingly, the activation energy for nucleation, equation (7), is reduced by the factor 0

where A is the area of a sphere of radius R=2 P. Note that the radius of the critical truncated sphere is identical to the radius of the critical spherical nucleus in the case of homogeneous nucleation. However, the number of molecules assembled in the truncated sphere is much smaller, and hence heterogeneous nucleation is more efficient.

Fig. 5. The effect of wetting angle on nucleation measured as the ratio of the supersaturations needed for equal heterogeneous and homogeneous nucleation. Solid line - flat interface, where . Dashed line - nucleation on a corner (using the solution of Sigbee 1969).

The supersaturation required for heterogeneous nucleation may be much lower than the homogeneous limit. In the extreme case, when the vapour completely wets the crystal, — 0 and no supersaturation is needed. If vapour and melt wet the surface of the crystal equally, 0 — 902, the nucleus is a half-sphere, = 0.5 and the activation energy is reduced to half the homogeneous limit. If the melt wets the surface completely, 0 = 0. = 1 and the presence of the crystal does not help nucleation. Figure 5 shows the decrease in required for nucleation as a function of increasing 0. It can be seen that 0 in excess of 150 is needed in order to impose a substantial decrease in . Moreover, as n0 is smaller in the case of heterogeneous nucleation (nucleation sites are only around the crystals), a somewhat larger 9 is needed. A further decrease in the required supersaturation may be achieved if the crystal surface is rough. Sigbee (1969) has shown that the presence of corners, facilitates nucleation and calculated the effect of a 90 corner. If the wetting angle exceeds 135 (which means a flat interface between the bubble and the melt), no activation is necessary (Figs. 4b and 5). Thus, in addition to the gas-crystal surface energy, the roughness of the crystal faces is an important factor in determining the efficiency of the crystal as a site for heterogeneous nucleation. Experimental results in silicate systems Only a few studies have been aimed at examining the systematics of nucleation in silicate melts. Early experiments were conducted by Murase & McBirney (1973) who monitored the number

VESICULATION PROCESSES IN SILICIC MAGMAS of visible bubbles as a function of time in a 5 x 5 x 0.5 mm slab of a Newberry rhyolite obsidian. They noted that "there is an initial period of very rapid nucleation after which there is very little change". They calculated rates of nucleation from the slopes of their curves for observed number of bubbles versus time. However, as shown in the next section, the time to grow to visible size depends on melt viscosity and initial size of the nucleus. We tend to agree with Murase & McBirney that nucleation was completed with the attainment of a state of apparent equilibrium. Thus, it is possible that the calculated activation energy for nucleation (100-500 kJmol - 1 ) actually reflects the energy for viscous deformation (about 400 kJ mol - 1 , as measured separately by them), which controlled the growth to visible size and not the nucleation energy. Bagdassarov & Dingwell (1993) noted the important role that small crystals (microlites) play in nucleation of bubbles. Comparison of microlite-poor and microlite-rich obsidian revealed much faster nucleation where crystals were present. They suggested that the surface of the microlites can potentially create areas slightly enriched in water, due to water exclusion from the growing crystal. Hurwitz & Navon (1994) conducted a comprehensive study of nucleation in silicic melts. They hydrated natural rhyolitic obsidian at high pressure (150MPa, 780-850°C) to form melts with 5.3-5.5 wt% water. Then, they lowered the pressure instantaneously and allowed bubbles to nucleate and grow for various amounts of time at the new lower pressure. Finally, they quenched the experiments rapidly and examined the number density and spatial distribution of bubbles. The results demonstrate the importance of heterogeneous nucleation. Microlites of Fe-Ti oxides are very efficient as sites for bubble nucleation. In the presence of such microlites, modest nucleation was observed even after decompression by < l M P a (Fig. 6). Decompression of more than 5 MPa produced extensive nucleation (1012-1014 bubbles m-3). In the absence of microlites, no nucleation occurred at 10MPa. At >10MPa, bubbles nucleated on crystals of biotite, zircon and apatite. Modest nucleation (109-10. m - 3 ) took place even in crystal-free samples. When exceeded 80 MPa, nucleation in crystal-free samples became extensive (1011-1013 m - 3 ). Hurwitz & Navon (1994) argued that the lack of correlation of bubble density with either time or decompression suggests that nucleation was still heterogeneous. However, Toramaru (1995) has

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Fig. 6. Number density of bubbles as a function of decompression ( ) in the nucleation experiments of Hurwitz & Navon (1994). Solid circles - experiments where crystals and microlites were present; open circles - no observed crystals or microlites. Samples plotted on the bottom x-axis carry no bubbles. shown theoretically that upon a sharp drop of pressure, a single nucleation event is predicted even when nucleation is homogeneous. Hurwitz & Navon (1994) decompressed their experiments by manually opening a valve. Thus, the possibility that the random scatter of the data in Fig. 6 is the result of small variations in decompression rates cannot be ruled out. If indeed the extensive nucleation at 80 MPa marks the onset of homogeneous nucleation, then using J = lO 1 2 m- 3 s - 1 and J* = 1033 m - 3 s - 1 (Hurwitz & Navon 1994) we obtain = 0.072 N m - 1 , strikingly close to the experimental values of Epel'baum et al (1973) and Bagdassarov et al. (1994). The above experiments allowed the examination of the final products only, so that nucleation rates can only be considered as lower bounds. Nevertheless, it can be concluded that the availability of sites is an important factor in controlling nucleation rates. Rates were faster than 1 0 1 2 m - 3 s - 1 when microlites were present, and faster than 1 0 1 1 m - 3 s - 1 in the absence of microlites at > 70 MPa. The narrow size distributions in most samples (e.g. Fig. 7) suggest that nucleation took place without any significant time lag after the pressure drops.

The relative efficiency nucleation sites

of crystals as

Hurwitz & Navon (1994) noted that Fe-Ti oxides are more efficient than biotite, zircon and apatite as nucleation sites for bubble nucleation. They also noted that plagioclase appears to

34

O. NAVON AND V. LYAKHOVSKY include Toramaru (1989, 1995), Bottinga and Javoy (1990b), Proussevitch et al (1993a.), Thomas et al (1994), Manga & Stone (1994), Barclay et al (1995), Lyakhovsky et al (1996) and Proussevitch & Sahagian (1996). Here we present the general theory emphasizing only those aspects relevant for growth in viscous melts and use the water-rhyolite system as an example.

General theory Fig. 7. Bubble size distribution in nucleation experiments of Lyakhovsky et al. (1996). LGB-135: Pf = 120MPa, t = 60s; LGB-122: Pf = 35MPa, t = 5 s; LGB-116: Pf = 35 MPa, t = 35 s. All samples were initially saturated at 150 MPa.

The critical nucleus is in unstable equilibrium with the melt (Fig. 1). Chemical equilibrium ( — Mm) is commonly described by a simple Henry's law relationship

be of very low efficiency, or of no effect. Data on surface tension or wetting angles in mineralmelt-vapour systems are scarce and do not allow prediction of the efficiency of mineral phases as nucleation sites. Laporte (1992) examined the system quartz-(anorthite)-melt-vapour and reported that after 5 days at 850°C, water bubbles are enclosed in melt and do not wet quartz crystals (all anorthite was melted). Preliminary experiments in our laboratory reveal wetting of diopside, phlogopite, ilmenite and magnetite by water bubbles in rhyolite. Observations on natural samples may also reveal the wettability of different crystals. Figure 8a and b show textural relations in vesicular rhyolites from the RDO-2B drill core in the Inyo Obsidian Dome. Many vesicles include a few Fe-Ti oxide crystals that barely touch the walls or, in other words, are almost completely wetted by the bubbles. In the same sample, a feldspar is completely enclosed in melt, even where a vesicle is pressed against it (Fig. 8b) suggesting very low wettability. More observations on vesicle-mineral textural relationships may help quantify the relative efficiency of crystals as nucleation sites.

where a is a constant commonly taken as 1 for CO2 and 2 for water in rhyolite - cf. equation (1). In the case of the critical nucleus, CR = Q is the initial volatile concentration in the melt. In reality, the relations are more complex (see review by Holloway & Blank 1994), but at pressures less than 200 MPa Henry's law relation is adequate. For a given ambient pressure (Pf), and with Pg determined by equation (15), the Laplace equation (4) holds only for the critical nucleus (in either homogeneous or heterogeneous nucleation), which is at an unstable equilibrium

Bubble growth Theoretical models of bubble growth generally follow the formulation of Rayleigh (1917). In the general literature, studies are concerned mostly with growth in low-viscosity media. In the geological literature, Sparks (1978) following the growth model of a solitary bubble (Scriven 1959) estimated the role of bubble growth during eruptions. More elaborate models as well as simplified approaches presented in recent years

If a nucleus is formed with R0 < RCR* the capillary pressure closes the bubble. For nucleus with radius Ro > RCR there exists a net outward force on the interface and the bubble grows. As the tiny bubble expands, Pz goes down, equation (15) no longer holds, and water molecules evaporate from the interface in order to re-establish equilibrium. This lowers the concentration at the interface (CR) relative to the melt far away from the bubble (Co) and leads to diffusive mass flux of water towards, and into, the growing bubble. The water influx keeps Pg close to its initial value in the small bubble while the force due to surface tension decreases. This ensures continued growth of the bubble. We now proceed to describing bubble growth under constant pressure and temperature, but the general formulation presented here is also applicable for variable pressure and temperature. Examining processes that may affect growth, Sparks (1978) demonstrated that in the case of water bubbles in viscous silicate melts.

Fig. 8. Vesicle-melt-crystal relationship in JF1, a rhyolite from depth of 4m in the RDO-2B drill hole in the Inyo Obsidian Dome, (a) Reflected light photograph of two oxide crystals situated inside a bubble. The crystal-melt contact is minimal and most faces are free of melt, indicating that gc is much lower than . (b) Feldspar completely surrounded by melt and not wetted by a vesicle even where its corner impinges on the vesicle. This, as well as other, similar observations suggest that in the case of feldspar is much larger than and 9 = 0, so that no wetting occurs.

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heat transfer and evaporation of water from the interface into the expanding bubble are fast, and do not limit the process. He also demonstrated that melt acceleration may be neglected. Thus, growth is controlled by two processes: diffusion of water molecules from the bulk of the supersaturated melt towards the bubble-melt interface; and expansion of the bubble by viscous deformation of the surrounding melt. The time scale for diffusion is = R2/D and for viscous deformation = The ratio between these characteristic time scales is the non-dimensional Peclet number

When Pe 1, diffusive mass transfer is unimportant relative to the rate of expansion. This regime is not realized in viscous silicate melts. When Pe is of the order of 1 or smaller, diffusion mass transfer is the main motive for growth. This case is discussed in detail below. The mass transfer towards a spherical bubble is described by the diffusion equation

The second boundary condition is derived from conservation of water in the spherical shell of melt that surrounds the bubble (including the bubble itself). In the case of a solitary bubble, this condition is reduced to a uniform water concentration at distances much larger than the bubble radius

Multi-bubble systems may be modelled as a grid of closely packed spheres (Proussevitch et al. 1993a), each with a finite spherical shell of the melt enclosing a spherical bubble (Fig. 9). In this case, the second boundary condition is set by the water mass balance between the bubble and the melt shell around it

This solution is a reasonable approximation for

random distribution of bubbles if an average So is calculated from the number density of bubbles

The second term on the left describes the

advective flux of water due to the deformation of the melt by the growing bubble, the term on the right-hand side describes the diffusive flux. When the Peclet number is of the order of 1 or greater, equation (18) is solved in its original form, including temporal derivation of the concentration and the advective term (dynamic model). For Pe < 1, diffusion is fast enough to establish a steady-state concentration profile and the two terms on the left-hand side may be neglected, leading to the quasi-static approximation

This reduction enables a significant simplification of the mathematical procedure and enables the derivation of an asymptotic analytical solution for the growth law (Lyakhovsky et al. 1996). The following two boundary conditions are applied in solving the above equation. They are both derived from the mass conservation of water. First, the change in mass of water in the bubble (mg) is equal to the diffusive mass flux of water integrated over the whole surface of the bubble

For the case of constant final pressure, condition (20a) defines the final bubble radius that is reached when the water concentration in

Fig. 9. Schematic representation of bubbles and their melt shells. Each bubble is of radius R and is surrounded by a shell of melt of radius S which is assumed to be spherical. The shells expand during the growth of the bubbles, but the mass of melt is conserved. The volume of overlapping neighbouring shells is equal to that of the gaps, so that the shells and the bubbles exactly fill the volume of the system.

VESICULATION PROCESSES IN SILICIC MAGMAS the shell is in equilibrium with the final pressure. Substitution of the equilibrium water concentration (C = Cf) into (20a) and integration yields

In order to solve equations (18) or (18a) we still need to know the bubble growth rate and density. This is done using the continuity and momentum equations, which describe the viscous deformation of the melt around the bubble. Melt density varies slightly with water content, but is approximately constant and the continuity equation describing the velocity field in the melt surrounding a bubble with growth rate VR becomes

The relations between gas pressure in a bubble, ambient pressure, surface tension and viscous resistance of the melt are described by the Navier-Stokes equation. For an incompressible Newtonian melt with constant viscosity, low Reynolds number (Re = ) and constant ambient pressure, this equation can be integrated to give (Proussevitch et al. 1993a)

37

Pe number for a nucleus with radius close to the critical value

To estimate an upper bound of the Peclet number at the initial stage, consider a relatively low-viscosity rhyolite (105 Pas) with diffusion coefficient of 10 - 1 1 m 2 s - 1 , surface tension of O . l N m - 1 and pressure drop of only 1 MPa. Even at these relatively extreme conditions we have Pe = 4 x 10 -2 1, and diffusion is fast enough to establish a steady-state profile. The general solution of equation (18a) for water concentration in the melt around a bubble is

and analytical solutions were obtained for c1 and c2 from boundary conditions (15) and (20) (Lyakhovsky et al. 1996). For further simulation we need only the concentration gradient at the bubble-melt interface, which for a solitary bubble is of the form

and for a bubble growing from a finite shell of melt The omission of the acceleration term is justified because of the extremely low Reynolds number. Lastly, we have to relate the mass (or density) of the gas to the pressure in the bubble. This is done by the gas equation of state. Most models simply use the ideal gas equation

Taken together, equations (15), (18)-(20) and (23)-(25) define a set of equations and boundary conditions that allows a full description of the growth of bubbles, which, at present, calls for a full numerical solution. For a better understanding of the physical principles that control the process attempts have been made to obtain analytical solutions under some simplifying assumptions.

These simple solutions allow the derivation of asymptotic analytical solutions for a variety of situations. We begin with the initial stages of growth and later proceed to describing solutions for t . In the initial stages, and equation (24) is simply

Owing to the very high surface: volume ratio of the nucleating bubble, diffusion is very efficient at the initial stages of growth and succeeds in keeping Pg close to its initial value, given by equation (15). Under this approximation

Approximations under specific conditions In liquids of high viscosity (e.g. rhyolite) the Peclet number, at least at the initial stage of growth, is commonly smaller than 1. Substituting equation (16) into (17) gives the value of the

Even though growth is motivated by diffusive mass transfer, the actual growth rate is controlled by the viscous time scale, . Following Toramaru (1995), who obtained a

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O. NAVON AND V. LYAKHOVSKY

similar expression, we use the term 'viscositycontrolled' to describe this stage of exponential growth. Note that during this stage Pe < 1 and it should not be mixed with the case of Pe 1, where growth is controlled by viscous expansion, and diffusive mass transfer is negligible. Proussevitch et al (1993a) found that initial growth in their numerical experiments is slow and attributed it to the effect of surface tension. The cause of that delay was discussed by Sparks (1994) and Sahagian et al. (1994), and it was agreed that viscous resistance plays the major role. Examination of equation (29) shows that viscous resistance of the melt poses the main inhibition to growth when R > RCR, but the actual size of the nucleus and its deviation from RCR are also important. As the radius of a nucleating bubble, R0, is just slightly larger than RCR, the pre-exponential term may be smaller than either Ro or RCR, leading to slow initial growth rates of the order (Ro - RCR) In the extreme case, when Ro — RCR, the nucleus is in an unstable equilibrium and no growth takes place. Using equation (24a) we can translate the mass flux equation (19) to determine the variation in gas density

Substituting equation (29) into (30), using the ideal gas approximation (25) and ignoring surface tension, we obtain an approximation for the variation in gas pressure,

where the concentration gradient at the bubblemelt interface (28) was approximated by

Equation (31) has the following analytical solution

Fig. 10. Temporal evolution of internal pressure in a bubble growing in a viscous melt. is the difference between the initial internal pressure, P0, and the pressure at a given time, Pg. The numerical solution shows an initial drop in pressure, which quickly becomes moderate as diffusive flux increases with increasing concentration gradient. Pressure fall is very slow and for = 2 x 108 Pas, Po = I5OMPa and Pf = 120 MPa, it takes almost 104 s before the pressure falls by 5% of its initial value. The analytical curve (using equation (33)) shows close agreement with the numerical calculation.

As can be seen in Fig. 10, there is a good agreement between equation (33) and the numerical solution. The internal gas pressure in a bubble growing in the water-poor, high-viscosity melt decreases by only 5% over the first 10 4 s. Thus, in this case the constant pressure approximation that leads to exponential growth (29) is valid. Another interesting case described by equations (24) and (30) is that of negligibly low diffusion. This case was discussed by Barclay et al. (1995), who solved the problem for solitary bubbles and for bubbles in spherical shells of melt. Their solution for a solitary bubble is of the form

As there is no mass flux into the bubble, the growth rate decays with time. The important time scale is that of viscous deformation. Equation (30) can also be used to construct an approximate solution for long periods of times. When the pressure in the bubble approaches the ambient pressure, . In the case of bubble growth under relatively small

VESICULATION PROCESSES IN SILICIC MAGMAS supersaturation the Peclet number may be kept small for very large amounts of time. Under these conditions, we can substitute equation (28) into equation (30) to obtain an approximated growth law for

As the right-hand side is constant, equation (35) describes parabolic growth. Bubble radius grows as the square root of time. A more accurate approximation (allowing for a small variation in gas density, but still assuming infinite bubble separation and neglecting surface tension) was obtained by Lyakhovsky et al (1996)

square-root solution of diffusion-controlled processes. The effective diffusion coefficient is

and its value depends on the diffusion coefficient of water, initial water content and the final ambient pressure. We can now summarize the growth of a bubble at constant pressure (Fig. 11). •

•

Further terms, of the order of I/t, I/t 2 and a smaller decrease with time, are thus negligible. We see that although Peclet numbers are low and the temporal derivation of the concentration in the diffusion equation was neglected (18a), for long periods, growth is limited by the rate of diffusion and follows the common,

39

•

Immediately after nucleation, the bubble radius, Ro is somewhat larger than RCR and the bubble grows spontaneously. In these initial stages, diffusion is fast enough to supply all the needed water and prevents fast fall of pressure. Growth rate increases exponentially, see equation (29) and is controlled by viscous deformation. As the surface: volume ratio decreases, diffusive flux of water cannot maintain the internal pressure, and Pg decays to slightly above the external pressure. Growth rate is also decreasing, and strain rates are small. Growth is controlled by diffusive mass flux and follows a square-root law, equation (36). In multi-bubble systems the bubbles approach their final radius, equation (22).

As can be seen in Fig. 11, R is bounded by two analytical curves: an exponent and a square-root.

Fig. 11. Bubble growth at constant final pressure. At the initial stages the numerical simulation (thick solid line) closely follows the exponential solution (equation (29)), at later stages it is well approximated by the square-root solutions. The dashed line denotes the solution with both terms of equation (36), the solid line represent the first term only. Finally, the numerical simulation leaves the square-root solution and approaches the final radius, equation (22). The internal pressure in the bubble remains close to its initial value during the first stage and then quickly falls to slightly above the final pressure. Also shown is the time of transition between the exponential and the square-root solutions, The two thin vertical dashed lines correspond to 0.5 and 1.5 of the approximation for , equation (39). P0 = 150 MPa, Pf = 120 MPa, = 5 x 108 Pas, D = 10 -12 m2 s - 1 , C = 5.5wt%, Rf = 17/im.

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The transition from exponential to squareroot growth is marked by the crossing of the two curves. At the time of crossing ( )

We solved equation (38) for a variety of values of P0 (0.2-150 MPa), Pf (10-90% of P0) and Ro (1.01 and 2 times RCR)- Concentrations, critical radii, gas densities and viscosities were calculated using equations (15), (16), (25) and the expression of Schulze et al. (1996), respectively. Figure 12 shows that depends mainly on the Peclet number. The solution may be approximated by

As seen in Fig. 12, for bubbles of R0 close to RCR, is of the order of 20-60. Figure 11 indicates that the bubble closely follows the exponential curve for about . This means that a water-rich rhyolite at depth of a few kilometres will follow the square-root law after less than 1 s. Bubbles in a highly viscous lava flow or dome may follow the exponential law for a few hours or even days.

Fig. 12. The time scale of transition from viscositycontrolled to diffusion-controlled growth normalized to the viscous time scale . Markers - exact solutions of equation (38) for a range of conditions: Po varies from 150 to 0.2 MPa; Pf = 0.9, 0.5 and 0.1 P0; C0, RCR, pg and calculated according to equations (15), (16), (25) and the relation of Schulze et al. (1996); D = 10-11 m2 s-1 and pm = 2300kgm - 3 . Open symbols-RQ = l.OlRcR.; solid symbols - R o = 2RcR; squares - = 0.1 N m-1, T= 1000 K; triangles = 0.05Nm -1 , T=1000K; circles - a = 0.1 N m - 1 , T= 1200K. The line, equation (39), represents an approximate linear fit to the results of equation (38).

Sigmoidal shape - Avrami equation The above analysis of the different stages of bubble growth shows that rate of growth is low at the initial stage, then accelerates and finally goes to zero during approach to the final radius. This sigmoidal shape of radial growth (Fig. 11) is also conserved if bubble volume, rather than bubble radius, is considered. In the viscositycontrolled stage volume evolution should follow Vg ; during the diffusioncontrolled stage it follows ( . Avrami (1939, 1940) developed a model describing a system where one phase completely transforms into a new phase (e.g. solidification of a liquid). According to his model, the volume taken by the new phase (V) from nucleation to final volume (V f ) is

where the theoretical Avrami coefficient, , varies between 2.5 and 4 for different crystallization processes with characteristic time . The evolution of the volume fraction taken by the new phase with time follows a sigmoidal curve. Although the physics of the phase transition described by the Avrami model is different from that controlling bubble growth, the shapes of the curves describing volume fraction changes with time are of similar shape. Thus it is possible to use the Avrami equation for fitting bubble growth data. As discussed above, during the viscosity-controlled stage, V/Vf = R 3 / S 3 . This stage may be approximated by an Avrami equation with . During the diffusion-controlled stage, R and =1.5. Thus, the whole evolution of a bubble may be fitted by the Avrami equation with varying from = 3 for the initial stage to =1.5. This approximation was used by Bagdassarov et al. (1996). who found that in their experiments varies between 2.6 and 1.8, and generally decreases with increasing temperature. They attributed the lower values to the combined effect of nucleation and growth, and to the fact that some bubbles were present in the original samples. Based on the growth model presented above, we suggest that the shift in the values of reflects only a change in the growth regime with increasing temperature. At lowr temperatures (and high viscosity) most growth takes place under viscosity-controlled regime and should be close to 3. At higher temperatures (and lower viscosity) vd was shorter, more of the growth occurred at the transition from viscosity- to diffusion-controlled growth and approaches 1.5.

VESICULATION PROCESSES IN SILICIC MAGMAS

Experimental results and numerical simulations Compared with the theoretical modelling of bubble growth, experimental measurements of growth rates are scarce. Murase & McBirney (1973) studied the vesiculation of natural rhyolite at 1 bar. Their plot of radius versus time follows the typical sigmoidal shape. It rises exponentially for the first 30 min, then growth decelerates for the next 50min. Later, bubble radius declines, probably due to diffusion of water to the surface of the thin glass slabs they used. Melt viscosity in their experiments was about 107 Pas (Murase & McBirney 1973) and the vapour pressure corresponding to an initial water content of approximately 0.2 wt% was of the order of a few atmospheres. The resulting is of the order of 30-1OOs, and (using equation (39)) is of the order of 1000-4000 s, similar to the time for which exponential growth persisted in the experiments. Recently, Bagdassarov et al. (1996) and Lyakhovsky et al. (1996) published additional experimental data. Bagdassarov et al. (1996) adopted the experimental method of Murase & McBirney (1973), but used a video camera to monitor continuously the growth of individual, pre-existing bubbles in a rhyolitic obsidian at 650-925°C. The obsidian they used contained 0.14% water, corresponding to a saturation pressure of approximately 2 bars. When heated at room pressure, is 1 bar (105 Pa). The measured viscosity of their obsidian in this temperature range was 5 x 105-5 x 10 9 Pas, resulting in of 5-50 OOOs (and about an order of magnitude longer). Plotting bubble radius, R (normalized to initial radius, ), versus time, Bagdassarov et al. (1996) obtained sigmoidal growth curves. This shape fits the prediction of the growth model presented here and, indeed, replotting their data as In(R/Ro) versus time we obtain a series of straight lines for the region (Fig- 13). Comparison with equation (29), remembering that Ro >> RCR, indicates that the slope of these straight lines should correspond to and allows the estimation of melt viscosity from the bubble growth data. Indeed, Fig. 14 reveals excellent agreement between the viscosity calculated from equation (29) and that measured by Bagdassarov et al. (1996). The deviations at high temperatures reflect the larger uncertainties in the slope due to the early deviations from exponential growth. In another set of experiments, Bagdassarov et al. (1996) followed the volume expansion of cylinders of rhyolitic melt due to bubble nucleation and growth at 1 bar.

41

They fitted the results using the Avrami equation (see the discussion of their results in the previous subsection on the Avrami equations). The good agreement between the experiments of Bagdassarov et al. (1996) and the model presented above affirms the model for bubble growth in the viscosity-controlled regime. Lyakhovsky et al. (1996) used the data of Hurwitz & Navon (1994) to derive the systernatics of growth in the diffusion-controlled regime. In these experiments, the initial water content was 5.3-5.5 wt%, and at 800-850°C the corresponding viscosities were of the order of 5 x 104 Pas. Initial pressure was 150 MPa and it was dropped over a few seconds by 5-135 MPa

Fig. 13. Bubble growth in a rhyolitic obsidian at various temperatures. The data of Bagdassorov et al. (1996) were replotted in a semi-logarithmic plot, emphasizing the exponential growth in agreement with equation (29).

Fig. 14. Viscosities calculated by fitting the data of Bagdassarov et al. (1996) using equation (29) with = 105 Pas (error bars correspond to 5 x 104 and 2 x 105 Pa) compared with the best fit to the micropenetration data measured by the same authors (solid line).

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in the various experiments. Bubbles nucleated and grew at the new lower pressure. At the end of the growth period, samples were quenched in less than 2s. Under these conditions d is shorter than 1 s and growth is controlled by diffusion. In many of their experiments bubble growth was limited by neighbouring bubbles. Lyakhovsky el al. (1996) used a numerical model to simulate growth. Input parameters included the experimentally controlled, initial water content, ambient pressure, temperature and run duration, the calculated average initial separation (calculated from the observed final radius and final separation, = S3 - R3), and physical properties of the melt: viscosity, density, surface tension, Henry constant and diffusivity. The preferred values for the melt properties were: density of 2300 kg m-3 (Silver et al. 1990, using 5.4% water), surface tension of 0.06 N m - 1 (Hurwitz & Navon 1994, based on the experimental results for hydrous granitic melts of Epel'baum et al. 1973), Henry's constant of 4.4 x lO-6 Pa- 1 / 2 (5.4% water at ISOMPa), viscosity of 5 x 10 4 Pas (Persikov 1991; Hess & Dingwell 1996; Schulze et al. 1996), and a diffusion coefficient of water of 3 x 10-11 m2 s-1 (based on a subset of their data). The diffusion coefficient of water is about a factor of 4 higher than the diffusivity calculated by extrapolating Zhang et al. (1991) data to high water contents, and a factor of 2 lower than the value obtained using the equation of Nowak & Behrens (1997). The results of the numerical simulations are in very good agreement with the experimental data and are not very sensitive to viscosity. They affirm that the model well describes bubble growth controlled by diffusion. The model presented above explains well the experiments of Bagdassarov et al. (1996) and Lyakhovsky et al. (1996) for the viscositycontrolled and diffusion-controlled regimes, respectively. This close fit, as well as the agreement between the numerical results and the analytical solutions, suggest that we are now equipped with an experimentally tested numerical model for describing bubble growth at constant pressure. The model may be extrapolated to growth under variable pressure, however no experimental results for testing it are available at present.

Growth under variable pressure Analysis of growth at constant ambient pressure is important for gaining insight into the physics of growth and in modelling laboratory experi-

ments. It may be applied to modelling bubble growth in magma chambers or in lava flows and domes. Clearly, during eruption, growth takes place under falling pressure and it is important to include pressure variations in the growth model. The set of governing equations presented above is also valid for variable pressure, and numerical codes may include this effect (Toramaru 1995: Proussevitch & Sahagian 1996). Barclay et al. (1995) were able to derive analytical solution for the case of zero mass flux of water into the bubble and linear decompression. The other end-member is the case when diffusion is efficient enough to keep pace with the fall of ambient pressure. In this case, water content in the melt is uniform and determined by the equilibrium solubility at Pf (which is now a function of time). The bubble radius is given by Rf in equation (22), which grows with time and falling pressure. This simple case of equilibrium degassing has been assumed in many eruption models (e.g. Jaupart & Allegre 1991; Papale & Dobran 1993). Next we estimate the time scale, , required in order to approximate equilibrium degassing. The amount of water that has to be transferred to the bubble is equivalent to that contained in the melt shell in excess of the equilibrium solubility. Using the parabolic growth law from equations (36) and (37), we can calculate the time when the amount of water in the bubble is equal to the excess water in the shell

For

a small change in pressure, If rate of decompression is constant, e.g. = , where v is the constant ascent velocity and is the density of the conduit wall rock, we can substitute equation (37) for and equation (25) for , to obtain

Neglecting relative to , the right-hand side is well defined and for melt with approximately 4% water, ascending from = lOOMPa (about 4km), we obtain

(using D = 3 x 10- 1 1 m 2 s- 1 , . = 2200kg m-.3 and = 2800 kgm - 3 ). As actual growth during the approach to equilibrium size is slower than the square-root approximation, equations (42) and (43) may slightly underestimate the

VESICULATION PROCESSES IN SILICIC MAGMAS time needed for approaching equilibrium degassing. For a reasonable velocity of 1 m s - 1 (Klug and Cashman 1994; Sparks et al 1994) and S0 of 10/um (2.5 x 1014 bubbles per m3) the predicted equilibration time is about 30s. That means that after ascending 30m, the melt closely approaches equilibrium, so that at later times, R = Rf. If S0 = 100/um (2.5 x lO11 bubbles per m 3 ), bubbles are out of equilibrium for about 1000s, but still for most of their ascent they are very close to equilibrium. When So = 1 mm (2.5 x 108 bubbles per m 3 ), t is approximately 30 000s and bubbles are not expected to reach equilibrium. In this case, the approximations we used break down long before that time and, in reality, new bubbles must nucleate as far from existing bubbles the melt becomes highly oversaturated. Toramaru (1995) calculated the number density of bubbles produced during decompression. Assuming homogeneous nucleation and constant nucleation rates he found that typical number densities are of the order of 1010-1015m-3. His numerical calculation also predict that degassing closely approach equilibrium during most of the ascent. If melt viscosity is high, he found small deviation from equilibrium near the top of the column due to the fast decrease in water solubility with decompression at low pressure (fig. 4 of Toramaru 1995). The larger deviation from equilibrium obtained by Proussevitch & Sahagian (1996) is due to the extreme separation they used in their model (So = 1 mm, corresponding to 2.5 x 10 8 m - 3 ). Both models assumed constant melt viscosity and water diffusivity. While the variation in diffusivity is not large and should not have a significant effect, the increase in viscosity with falling water content exceeds seven orders of magnitude (Hess & Dingwell 1996; Schulze et al. 1996) and plays an important role at the final stages. The cases discussed above are illustrated in Fig. 15 that presents the results of some numerical simulations of bubble growth in a melt ascending from 4000m at 1 m s - 1 . This velocity is used by both Toramaru (1995) and Proussevitch & Sahagian (1996), and is a reasonable value for the deeper parts of the conduit (Klug & Cashman 1994; Sparks et al. 1994). If initial separation between bubbles is 1 mm (as in Proussevitch & Sahagian 1996), the water content at the outer boundary of the shell (at S) does not decrease until the bubbles grow large enough and the melt shell thins significantly. When separation is 10 times smaller, water concentration at S is higher than the equilibrium solubility for the first 1500m (or 1500 s), in agreement with equation (43). If viscosity and diffusivity are held

43

constant, at values corresponding to the waterrich melt, equilibrium is maintained all the way to the surface. If viscosity is allowed to increase with falling water content, internal pressure in the melt does not follow the external lithostatic pressure from 600 m upward, and reaches almost 2 MPa at the surface. At the same time, supersaturation builds up in the melt and the magma is not fully degassed. At this stage growth is 'viscosity-controlled', in the sense of Toramaru (1995), and, indeed, significant deviation from

Fig. 15. Bubble growth in a melt ascending from 4000 m to the surface at 1 m s - 1 . The melt is assumed to carry the equilibrium content of water at 4000 m and to nucleate with no supersaturation. (a) Water concentration in the melt away from the bubble (at S) for initial bubble separation of 1 mm and 100/um, compared with the equilibrium solubility (dotted line), (b) The last 1000m (for 100 um separation). Viscosity is allowed to grow with falling water content, deviation from equilibrium degassing is significant and internal pressure is built up in the bubbles. Melt at the surface is not fully degassed.

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equilibrium begins at approximately 1% water when melt viscosity is around 108 Pas (cf. fig. 4 of Toramaru 1995). Note that by 'viscositycontrolled' we do not mean that bubble growth at this stage can be described by equation (29), it only means that viscous resistance becomes of the same order as lithostatic pressure. Even higher supersaturation (and a high internal pressure in the bubbles) may develop if a more realistic velocity profile is used. As vesicularity increases during ascent, magma must accelerate and the decompression rate must increase (Wilson et al. 1980; Sparks et al. 1994). In this case, supersaturation away from bubbles may be large enough to trigger nucleation of a new generation of bubbles (cf. Sparks & Brazier 1982). The internal pressure within the bubbles is also expected to be higher. It may be high enough to cause microfracturing similar to that discussed by Mungall et al. (1996) and to play a significant role in fragmenting the magma. It is now possible to model accurately the growth of bubbles for a given pressure profile in a conduit. The next step is the construction of a self-consistent model of vesiculating magma where flow and growth are coupled. For that it is necessary to introduce into a flow model effective rheological properties for the vesiculating magma, e.g. average density, shear viscosity and bulk viscosity. Some of these parameters could be estimated from the bubble growth model. The magma density during equilibrium degassing is simply

Such equilibrium conditions are commonly assumed in many models of magma degassing (e.g. Wilson et al. 1980; Jaupart & Allegre 1991; Papale & Dobran 1993) and, as was shown here, are adequate for most of the ascent, but not for the last stages before fragmentation. If water mass flux into the bubbles is negligible the bulk viscosity (nbulk) of a vesiculating magma may be estimated using the continuous mixture model (Prud'homme and Bird 1978)

This approximation is adequate for describing the evolution of pumice following fragmentation (Thomas et al. 1994), however such rheology cannot be recommended for describing the slow flow in the conduit, where the contribution of diffusive mass flux of water is important and gas mass in the bubble cannot be assumed to be constant.

Bubble coalescence As bubbles grow, and the gas fraction of the magma increases, bubbles interact mechanically and may coalesce. None of the current growth models considers mechanical interactions among bubbles and their effects on growth. Similarly, theoretical and experimental studies of coalescence (at least in the geological literature) are concerned with thinning, failure and relaxation of static foams, where the internal pressure in the bubbles is in equilibrium with ambient pressure (see discussions in Vergniolle & Jaupart 1990; Proussevitch et al. 1993b; Cashman and Mangan 1994; Herd & Pinkerton 1997). Even the problem of wall thinning due to bubble expansion has not been examined in this context. Next, we describe the theory for static foams, where bubble coalescence is controlled by two important time scales: , the characteristic time of drainage and failure of a melt film separating two touching bubbles; and , the time it takes for the new bubble to regain its sphericity.

Thinning and failure of bubble walls Foam is not a stable structure and tends to collapse and reduce gravitational and surface energy. Capillary and gravity both act to drain melt from the films between touching bubbles and from the plateau borders which separate the films (Fig. 16), and both must balance the viscous resistance of the melt during drainage. Proussevitch et al. (1993b) investigated the relative contribution of the two processes. In the case of viscous melts, the failure of films, rather than plateau borders, is the important process.

Fig. 16. Schematic illustration of foam with films separating pairs of bubbles and plateau borders separating three or more bubbles. The pressure in melt enclosed in a flat film is equal to the internal gas pressure. Melt in the plateau border is under lower pressure due to the capillary pressure of the curved interface.

VESICULATION PROCESSES IN SILICIC MAGMAS The characteristic time scale of film thinning can be roughly estimated by Reynold's lubrication theory (e.g. Toramaru 1988; and note a typo in equation (1) of Klug & Cashman 1996)

(46) Integration yields the time it takes for the film to fail, T df, due to thinning to a critical thickness, h min (Toramaru 1988; Proussevitch et al 1993b) (47)

The radius of the liquid film disk, R, is of the same order as the bubble radius. AP is the pressure difference between the centre and the margin of the disk. This pressure difference may be due to gravity (for a vertical film, AP = gR) or due to capillary pressure. The capillary pressure is zero in the flat film and is

45

2 /R at the curved interface of the plateau border, where R is the radius of curvature, so that AP = 2 /R. In static foam R is comparable to the bubble radius, but note that if the foam is deformed R may be much smaller. While in some organic films hmin is of molecular size, it appears to be larger in silicate melts. This may be due to the presence of microcrystals in the melt films or because of transient stresses which may disrupt bubble films well before they have thinned to their spontaneous rupture thickness. Klug & Cashman (1996) suggested that this thickness is of the order of 1 m. Our own experience (Hilton & Navon unpubl. data, and cf. Fig. 17) is that it may be thinner, but we have not observed films thinner than 0.2 m. Using hmin = 1 m, R = 10 m and 77 = 106 Pas, equation (47) yields Tdf = 108/AP. Pressure due to the weight of the melt in the film, pmgR, is of the order of AP 0.2 Pa. This is negligible relative to surface tension that is of the order

Fig. 17. Raptured melt film retracting (from the centre to the bottom of the photograph). The film was broken during sample preparation exposing its cross-section. The film is of uniform thickness, except for the rounded and thickened tip. Thickness of wall: 0.2/mi. (Pumice from the island of Lipari. Photograph courtesy of M. Hilton.)

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O. NAVON AND V. LYAKHOVSKY

of 104Pa. The predicted characteristic time, Tdf ~ 104 s, is in agreement with the experimental observation of Bagdassarov et al. (1996), who measured the rate of bubble loss following the foaming of natural rhyolite. Proussevitch et al. (1993b) also found general agreement between theory and experiments on foamed synthetic melt. Larger driving pressures may be due to pressure difference between neighbouring bubbles, or due to bubble deformation during magma flow, but it must be remembered that equations (46) and (47) are not adequate for expanding or deforming bubbles where walls are stretched and melt is not drained through a relatively fixed geometry. For example, Barclay et al. (1995) showed that in the case of an expanding shell, the time scale is

geometry with length L and thickness /?, and if the surface of its cross-section is conserved

(49) which yields exponential decay of the thickness and (50)

The two examples discussed above demonstrate that it is not possible to apply the equation of static foam to dynamic situations. The experiments of Westrich & Eichelberger (1994) indicate that coalescence in an expanding bubbly magma occurs over short time scales of less than 1-2 min, and when vesicularity is still between 43 and 58%, long before maximum (48) packing of spherical bubbles is achieved (Klug & Cashman 1996). We observed a similar case in If Ro/h min is approximately 100, Ro/ho is approxi- our experiments. The bubbles shown in Fig. 18 mately 10, 77 = 10 6 Pas, and Pg = 1 MPa, thin- were probably quenched just before merging. ning due to growth following decompression is Accurate treatment of the mechanical interacabout 4 orders of magnitude faster compared tion between neighbouring bubbles must replace with thinning due to capillary forces. If the film the simplified examples given here, in order to is modelled as a stretching wall of rectangular gain understanding into the coalescence process.

Fig. 18. Penetration of one bubble into its neighbour (sample LGB-58 of Hurwitz & Navon 1994).

VESICULATION PROCESSES IN SILICIC MAGMAS

Shape relaxation Based on time scale analyses (Frenkel 1946), the time it takes for a coalesced pair of bubbles to relax and regain spherical shape, Trei, is generally taken as

Klug & Cashman (1996) used this relation to estimate the maximum size of bubbles that can relax in the short time available after fragmentation. For ) = 10 7 Pas, t— 10s and a — 0.2Nm-1 we expect relaxation of features whose radius of curvature is smaller than 0.2 mi. This is indeed the case in rhyolitic pumices where many coalesced walls can still be observed (Fig. 19). Note that while the sharp tip of the coalesced wall is relaxed, it will take much longer for the whole wall to be resorbed. If the film is much thinner than its length and its thickness, h, is roughly constant, then surface

47

tension acting at the vicinity of the raptured tip along the direction of the film may be approximated as ( /h)cos( y//h), where y is distance perpendicular to the film. The Navier-Stokes equations in the parallel strip with the above stress at the tip of the film have the following solution (Lyakhovsky unpubl. data)

(52)

The velocity scale predicted by the above solution is of the order , and is similar to equation (51). However, away from the tip it falls off exponentially, so that the tip thickens and becomes more blunt, exactly like the examples presented in Figs 17 and 19.

Fig. 19. Final stages of film retraction. (Pumice from the island of Lipari. Photograph courtesy of M. Hilton.)

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O. NAVON AND V. LYAKHOVSKY

Conclusions Over the last decade there has been a major advance in the understanding of the physics of vesiculation in magmatic systems. Significant contributions have been made in theoretical modelling, the experimental database has been expanded and important observations have been made on natural samples. Bubble nucleation and growth in silicic magma is now relatively well understood, but bubble coalescence and magma fragmentation, as well as some aspects of growth in dynamic systems remain poorly understood. Nucleation is controlled by the availability of heterogeneous sites, and more research is needed in order to define the nature of efficient sites and the parameters controlling their formation. Homogeneous nucleation happens only at very high supersaturation (>70MPa), if at all. Bubble growth in hydrated silicic melts (where Pe is never much larger than 1) is motivated by diffusion. During the initial stages of growth diffusion is very efficient and preserves gas pressure in the bubbles close to its initial value. Growth rates are controlled by viscosity, and follow exponential relations. After time of order /AP, gas pressure falls and approaches the ambient pressure, and growth is controlled by diffusion and follows a square-root dependence on time. Finally, the bubbles approach their final radius, which is a function of the ambient pressure, the initial water content and the separation between bubbles. Experiments at constant ambient pressure have validated the present models. The extension of these models to linear decompression (or to other decompression regimes) is straightforward, and results suggest that during ascent a typical melt, with a few wt% water and a bubble separation of 10-100 m, have enough time to equilibrate with the growing bubbles. More elaborate models call for a better understanding of the conditions and parameters which control flow of magma in the conduit. During the final stages of ascent, the water content of the melt decreases and growing melt viscosity prevents equilibrium degassing. While current models may account for this effect, mechanical interaction between neighbouring bubbles is not yet included. Experiments suggest that bubble coalescence during flow may begin long before viscosity becomes important, and even before bubbles reach the close packing limit. The theory of static foams may be applied to understanding some textural features of pumice. However, it gives little insight into the processes operating during coalescence and fragmentation in the conduit during eruption.

New experimental data and theoretical modelling are needed in this field. The new understanding of vesiculation, combined with a better understanding of bulk properties of vesiculated magma, will lead the way for a new generation of conduit flow models. We thank Steve Tait. Jenny Barclay, Don Dingwell and Steve Sparks for reviewing, Jon Fink for the Inyo Dome samples, and Matt Hilton for photographs of the pumice. The first version of this review was written during O. Navon's visit to the University of Bristol. Fellowships from the Leverhulme Foundation and the Royal Society made this visit possible and are thankfully acknowledged. Research was supported by the US-Israel Binational Science Foundation. V. Lyakhovsky is grateful for support from the Giladi program.

References AVRAMI, M. 1939. Kinetic of phase change I: general theory. Journal of Chemical Physics. 7. 1103-1112. 1940. Kinetic of phase change II: transformationtime relations for random distribution of nuclei. Journal of Chemical Physics. 8. 212-224. BAGDASSAROV. N. S. & DINGWELL. D. B. 1993. Deformation of foamed rhyolites under internal and external stresses: an experimental investigation. Bulletin of Volcanology. 55, 147-154. . & WILDING. M. C. 1996. Rhyolite magma degassing: an experimental study of melt zvesiculation. Bulletin of Volcanology. 57, 587-601. . DORFMAN, A. M. & DINGWELL. D. B. 1994. Effect of alkalies on surface tension of Haplogranitic melts. EOS, Transactions of the American Geophysical Union, 75, 724. BARCLAY. J.. RILEY, D. S. & SPARKS. R. S. J. 1995. Analytical models for bubble growth during decompression of high viscosity magmas. Bulletin of Volcanology. 57. 422-431. BOTTINGA. Y. & JAVOY, M. 1990a. Mid-ocean ridge degassing: Bubble nucleation. Journal of Geophvsical Research, 95. 5125-5131. & 1990b. MORB degassing: Bubble growth and ascent. Chemical Geology. 81. 255-270. BOWERS, P. G.. HOFSTETTER. C. LETTER. C. R. & TOOMEY. R. T. 1995. Supersaturation limit for homogeneous nucleation of oxygen bubbles in water at elevated pressure: "Superhenry's law'. Journal of Physical Chemistry. 99. 9632-9637. BURNHAM. C. W. 1975. Water and magmas; a mixing model. Geochimica et Cosmochimica Acta. 39. 1077-1084. 1979. The importance of volatile constituents. In: YODER, H. S. (ed.) The Evolution of the Igneous Rocks. Princeton University Press. Princeton. 439-482. CASHMAN. K. V. & MANGAN. M. T. 1994. Physical aspects of magmatic degassing II. Constraints on vesiculation processes from textural studies of eruptive products. Reviews in Mineralogy. 30. 447-478.

VESICULATION PROCESSES IN SILICIC MAGMAS DINGWELL, D. B. 1998. Recent experimental progress in the physical description of silicic magma relevant to explosive volcanism. This volume. DUNNING, W. J. 1969. General and theoretical introduction. In: ZETTLEMOYER, A. C. (ed.) Nucleation. Dekker, New York, 1-67. EPEL'BAUM, M. B. 1980. Silicate Melts with Volatile Components. Nauka, Moscow (in Russian). , BABASHOV, I. V. & SALOVA, T. P. 1973. Surface tension of felsic magmatic melts at high temperatures and pressures. Geokhimya, 3, 461-464 (in Russian). FRENKEL, J. 1946. The Kinetic Theory of Liquids. Clarendon, Oxford. HAN, J. H. & HAN, C. D. 1990. Bubble nucleation in polymeric liquids II. Theoretical considerations. Journal of Polymer Science, 28, 743-761. HERD, A. R. & PINKERTON, H. 1997. Bubble coalescence in basaltic lava: its impact on the evolution of bubble populations. Journal of Volcanology and Geothermal Research, 75, 137-157. HESS, K.-U. & DINGWELL, D. B. 1996. Viscosities of hydrous leucogranitic melts: a non-Arrenian model. American Mineralogist, 81, 1297-1300. HIRTH, J. P., POUND, G. M. & ST. PIERRE, G. R. 1970. Bubble nucleation. Metallurgical Transactions, 1, 939-945. HOLLOWAY, J. R. & BLANK, J. G. 1994. Application of experimental results to C O-H species in natural melts. Reviews in Mineralogy, 30, 187-230. HURWITZ, S. & NAVON, O. 1994. Bubble nucleation in rhyolitic melts: experiments at high pressure, temperature and water content. Earth and Planetary Science Letters, 122, 267-280. JAUPART, C. & ALLEGRE, C. J. 1991. Gas content, eruption rate and instabilities of eruption regime in silicic volcanoes. Earth and Planetary Science Letters, 102, 413-429. KLUG, C. & CASHMAN, K. V. 1994. Vesiculation of May 18 1980, Mount St. Helens magma. Geology, 22, 468-472. & 1996. Permeability development in vesiculating magmas: implications for fragmentation. Bulletin of Volcanology, 58, 87-100. LANDAU, L. D. & LIFSHITZ, E. M. 1980. Course of Theoretical Physics, Vol. 5, Statistical Physics, 3rd edition Pergamon, Oxford. LAPLACE, P. 1806. Traite de Mechanique Celeste, Vol. 4. Courcier, Paris. LAPORTE, D. 1992. Partial melting and melt segregation: textural constraints. Terra Abstracts, 4, 26-21. LYAKHOVSKY, V., HURWITZ, S. & NAVON, O. 1996. Bubble growth in rhyolitic melts: experimental and numerical investigation. Bulletin of Volcanology, 58, 19-32. MANGA, M. & STONE, H. A. 1994. Interactions between bubbles in magmas and lavas: effects of bubble deformation. Journal of Volcanology and Geothermal Research, 63, 267-279. MCMILLAN, P. F. 1994. Water solubility and speciation models. Reviews in Mineralogy, 30, 131-156. MUNGALL, J. E., BAGDASSAROV, N. S., ROMANO, C. & DINGWELL, D. B. 1996. Numerical modeling of stress generation and microfracturing of vesicle

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walls in glassy rocks. Journal of Volcanology and Geothermal Research, 73, 36-46. MURASE, T. & McBIRNEY, A. 1973. Properties of some common igneous rocks and their melts at high temperatures. Geological Society of America Bulletin, 84, 3536-3592. NOWAK, M. & BEHRENS, H. 1997. An experimental investigation on diffusion of water in haplogranitic melts. Contributions to Mineralogy and Petrology, 126, 365-376. PAPALE, P. & DOBRAN, F. 1993. Modeling of the ascent of magma during the plinian eruption of Vesuvius in AD 79. Journal of Volcanology and Geothermal Research, 58, 101-132. PERSIKOV, E. S. 1991. The viscosity of magmatic liquids: experiment, generalized patterns. A model for calculation and prediction. Applications. In: PERCHUK, P. L. & KUSHIRO, I. (eds) Physical Chemistry of Magmas. Springer, New York, 1-40. PROUSSEVITCH, A. A. & SAHAGIAN, D. L. 1996. Dynamics of coupled diffusive and compressive bubble growth in magmatic systems. Journal of Geophysical Research, 101, 17447-17455. , & ANDERSON, A. T. 1993a. Dynamics of diffusive bubble growth in magmas: Isothermal case. Journal of Geophysical Research, 98, 22283-22307. , & KUTOLIN, V. A. 1993b. Stability of foam in silicate melts. Journal of Volcanology and Geothermal Research, 59, 161-178. PRUD'HOMME, R. K. & BIRD, R. B. 1978. The dilatational properties of suspensions of gas bubbles in incompressible Newtonian and nonNewtonian fluids. Journal of Non-Newtonian Fluid Mechanics, 3, 261-279. RAYLEIGH, LORD 1917. On the pressure developed in a liquid during the collapse of a spherical cavity. Philosophical Magazine, 34, 94-98. RUCKENSTEIN, E. & NowAKOWSKi, B. 1990. A kinetic theory of nucleation in liquids. Journal of Colloid and Interface Science, 137, 583-592. SAHAGIAN, D. L., PROUSSEVITCH, A. A. & ANDERSON, A. T. 1994. Reply on Comment on "Dynamics of diffusive bubble growth in magmas: Isothermal case". Journal of Geophysical Research, 99, 17829-17832. SCHULZE, F., BEHRENS, H., HOLZ, F., Roux, J. & JOHANNES, W. 1996. Influence of water on the viscosity of a haplogranitic melt. American Mineralogist, 81, 1155-1165. SCRIVEN, L. E. 1959. On the dynamics of phase growth. Chemical Engineering Science, 10, 1-13. SIGBEE, R. A. 1969. Vapor to condensed phase heterogeneous nucleation. In: ZETTLEMOYER, A. C. (ed.) Nucleation. Dekker, New York, 151-224. SILVER, L. A., IHINGER, P. D. & STOLPER, E. M. 1990. The influence of bulk composition on the speciation of water in silicate glasses. Contributions to Mineralogy and Petrology, 104, 142-162. SPARKS, R. S. J. 1978. The dynamics of bubble formation and growth in magmas: a review and

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analysis. Journal of Volcanology and Geothermal Research, 3, 1-37. 1994. Comment on 'Dynamics of diffusive bubble growth in magmas: Isothermal case'. Journal of Geophysical Research, 99,17827-17828. 1997. Causes and consequences of pressurization in lava dome eruptions. Earth and Planetary Science Letters, 150, 177-189. & BRAZIER, S. 1982. New evidence for degassing processes during explosive eruptions. Nature, 295, 281-220. , BARCLAY, J., JAUPART, C., MADER, H. M. & PHILLIPS, J. C. 1994. Physical aspects of magmatic degassing I. Experimental and theoretical constraints on vesiculation. Reviews in Mineralogv, 30,413-445. , SIGURDSSON, H. & WILSON, L. 1977. Magma mixing: mechanism of triggering explosive acid eruptions. Nature, 267, 315-318. TAIT, S., JAUPART, C. & VERGNIOLLE, S. 1989. Pressure, gas content and eruption periodicity of a shallow crystallizing magma chamber. Earth and Planetary Science Letters, 92, 107-123. THOMAS, N., JAUPART, C. & VERGNIOLLE, S. 1994. On the vesicularity of pumice. Journal of Geophysical Research, 99, 15 633-15 644. TOLMAN, R. C. 1949. The effect of droplet size on surface tension. Journal of Chemical Phvsics, 17, 333-340.

TORAMARU, A. 1988. Formation of propagation pattern in two-phase flow systems with application to volcanic eruptions. Geophvsical Journal. 95, 613-623. 1989. Vesiculation process and bubble size distribution in ascending magmas with constant velocities. Journal of Geophvsical Research, 94. 17523-17542. 1995. Numerical study of nucleation and growth of bubbles in viscous magmas. Journal of Geophysical Research, 100, 1913-1931. VERGNIOLLE, S. & JAUPART, C. 1990. Dynamics of degassing at Kilauea Volcano, Hawaii. Journal of Geophysical Research, 95, 2793-2809. WESTRICH, H. R. & EICHELBERGER, J. C. 1994. Gas transport and bubble collapse in rhyolitic magma: an experimental approach. Bulletin of Volcanology, 56, 447-458. 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. Geophysical Journal of the Astronomical Societv, 63. 117-748. ZHANG, Y., STOLPER, E. M. & WASSERBURG. G. J. 1991. Diffusion of water in rhyolitic glasses. Geochimica et Cosmochimica Act a, 55. 441-456.

Conduit flow and fragmentation H. M. MADER Department of Earth Sciences, University of Bristol, Wills Memorial Building, Queens Road, Bristol BS8 1RJ, UK Abstract: This chapter provides a critical overview of syn-eruptive processes between magma chamber and vent in magmatic, sustained, Plinian-type eruptions. Phreatomagmatic effects are largely neglected. The main sources of information derive from textural studies of ejecta, theoretical models and physical experiments of eruption dynamics. Textural studies commonly find bimodal vesicle populations, indicative of several discrete nucleation events. The degree of disorder increases with explosivity. Dynamical parameters, such as nucleation density and rate and bubble growth rate, can be inferred from studies of the moments of bubble size distributions. Vesicularity in pumices is observed to vary significantly both within and between deposits, which suggests that the vesicularity at fragmentation is affected by the flow dynamics. Vesicularity variations correlate most closely with changes in magmatic composition and viscosity, but not with discharge rate. Conduit flow models can be broadly grouped into first- and second-generation models; the former generally impose a lithostatic pressure gradient and a constant Newtonian viscosity, whereas the latter include equations for the rheological changes that take place during vesiculation and solve for the pressure. Second-generation models derive highly non-lithostatic pressure gradients with the result that most of the vesiculation occurs at a high rate over a short distance just prior to fragmentation. The mechanisms of brittle and ductile fragmentation have been investigated in separate studies in non-vesiculating magmas, but which mechanism operates in explosive eruptions is not known. Dynamical laboratory experiments provide observations of the physical processes operating in conduit flows. Gas-expansion experiments have shown that it is possible to generate violent explosions by unloading in cool magmatic materials. Expanding dusty flows are found to be stable only if the bulk density increases with height. Exsolution experiments have demonstrated that acceleration precedes fragmentation and that gas evolution is enhanced by advection and bubble deformation. Deformed vesicles similar to those found in 'woody' pumice have been generated in an analogue system that has similar rheology to that found in vesiculating magmas. Large-scale exsolution experiments suggest that explosive volcanic eruptions are inherently heterogeneous; the fluctuations in discharge rate and discrete pulses and shocks commonly observed are a consequence of the large physical scale of volcanic systems. The effect of the magma chamber and conduit geometry has also been investigated. Eruption of material from a spherical flask up a narrow cylindrical tube generates quasi-steady flow conditions after an initial transient during which the discharge rate grows, as frequently observed in volcanic eruptions. The fragmentation surface does not propagate down into the magma chamber.

Nomenclature Saturation concentration of the volatile (wt%) Volatile diffusivity in the melt (m2 s -1 ) Distribution function of coordination number for polydisperse system Bubble size distribution function Acceleration due to gravity (m s-2) Rate of growth of bubble radius or shear modulus (cms-1 or Pa) Characteristic length of film dividing two bubbles (cm) Rate of nucleation of bubbles (cm - 3 s- 1 ) Solubility constant (Pa- 0.5 ) i-th moment of the distribution function

n N N0 P Pa APC APgr P0 r R R S t u

Coordination number Total number of bubbles per unit bulk volume (cm -3 ) Nucleation density of bubbles Pressure (Pa) Atmospheric pressure (Pa) Driving pressure due to capillary forces (Pa) Driving pressure due to gravity (Pa) Saturation pressure or gas pressure (Pa) Radius of curvature of liquid film (m) Bubble radius (m) Mean radius of bubbles (m) Total surface area per unit bulk volume Time (s) Velocity of gas-pyroclast mixture (ms- 1 )

MADER, H. M. 1998. Conduit flow and fragmentation. In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 51-71.

52

V Fs Ft Fv Y z zf zs

a Q cnt 7 0 p PG pL /zLo ^M p

<j <jt r

H. M. MADER Ascent velocity or velocity of fragmentation wave front ( m s - 1 ) Volume of solid (1) Total volume (1) Volume of voids (1) Water content in the melt (wt%) Height in conduit measured from reference level (m) Height of transition from singlephase flow to two-phase bubbly flow regime (m) Height of fragmentation zone, i.e. transition from bubbly flow to gas particle-droplet flow regime (m) Vesicularity Vesicularity at maximum packing Silicate-gas interfacial tension (Nm- 1 ) Volume fraction of crystals Viscosity (Pas) Viscosity of gas phase (Pa s) Viscosity of melt including effect of crystals and dissolved water (Pa s) Viscosity of anhydrous, crystal-free melt (Pa s) Viscosity of multiple-phase mixture in bubbly flow or gas particle/droplet region (Pas) Density of melt (kg m-3)

Interfacial surface free energy (N m-1) Tensile strength of magma (Pa) Average growth time of bubbles or relaxation time of linear viscoelastic material (s)

Sustained Plinian eruptions are arguably the most violent form of volcanic activity and the source of many serious hazards both to local communities and on a global scale. In a Plinian eruption a large mass of magma (1011-1014 kg, dense rock equivalent volumes 0.1-150 km3 Carey & Sigurdsson, 1989) is progressively disrupted into fragments and ejected from the volcano (discharge rates 10 3 -10 5 m 3 s -1 , exit velocities up to hundreds of m 3 s - 1 ; Wilson 1980; Wilson et al. 1980; Walker 1981; Sparks 1986). The hot gas-particle mixture that is erupted can rise in the atmosphere as a buoyant plume (column heights up to 55km; Wilson et al. 1978) from which pyroclastic material then rains down on the surrounding area or can be transported around the globe. If the bulk density of the gas-particle mixture is greater than atmospheric pressure on eruption, the column will collapse and flow down the side of the volcano as a pyroclastic flow. The duration

of a Plinian eruption is much larger than the travel time for any individual particle and so these are sustained events. Usually, after an initial growth in discharge rate, a quasi-steady eruption regime is reached (Wilson. 1980; Carey & Sigurdsson, 1987, 1989). However, discrete pulses and shocks and fluctuations in discharge rate are commonly observed (Sparks & Wilson 1982; Carey et al. 1990; Gardner el al. 1991). Accurate forecasting of these extreme events as well as the interpretation of volcanic deposits requires an understanding of the physical processes involved in their inception and evolution. This chapter focuses on the processes occurring belowground that lead to the formation of the erupted gas-particle mixture. Direct observation of these processes is not possible in the field and the flowdynamics can only be inferred from depositional features, and by using the results of scaled laboratory experiments and theoretical models. Figure 1 shows a schematic diagram of the architecture usually envisaged in plinian eruptions. Liquid magma resides at depth in a magma chamber (chamber volumes l-10 6 km 3 with their top at a few km depth or more; Koyanagi et al. 1976; Einarsson 1978; Evans & Zucca, 1988; Stauber et al. 1988; Marsh 1989) which is connected to the vent via a conduit (radius up to 100m; Eichelberger et al. 1986; Chadwick et al. 1988). The magma chamber contains a silicate melt with variable amounts of crystals and volatiles. The composition of the melt can vary widely, but in Plinian eruptions intermediate and rhyolitic compositions are most common. These systems are characterized by higher silica contents, lower temperatures and higher volatile contents, resulting in higher viscosities and lower volatile diffusivities. than more mafic compositions. The primary volatiles are H2O, CO2 and S. with water vapour usually the most dominant. Water contents are typically in the range from 2-6 wt% for intermediate and rhyolitic magmas. The lithostatic (hydrostatic) pressure in the magma increases with depth due to the weight of the overlying material and determines how much of the volatile is in solution. For example, the lithostatic pressure is about 200 MPa at 6 km depth, which is sufficient to cause 6wt% of water to be completely dissolved in the magma. The saturation condition for water in melts is to a first approximation given by Cs = where Cs is the saturation concentration of the volatile. k is the solubility constant and P is the pressure. Plinian eruptions are driven by rapid internal evolution of gas at depth within the magma as the dissolved volatiles come out of solution in the form of many bubbles. The high viscosities

CONDUIT FLOW AND FRAGMENTATION

53

VENT

MAGMA CHAMBER Fig. 1. Schematic diagram of steady-state flow conditions in Plinian eruptions. and low diffusivities of silicic magmas mean that the evolving gases cannot readily escape from the body of the magma and the result is a rapid expansion of the multiple-phase mixture. The mechanisms for triggering the exsolution of the volatiles are not fully understood. In general, however, an eruption is preceded by a build-up of pressure and an increase in the magma chamber volume. Eruption occurs when the pressure is sufficient to induce mechanical failure of the surrounding country rock and cause a decompression event. Suggestions for the cause of the pressure build-up include influx of fresh magma at depth, fractional crystallization within the magma chamber (Blake 1981, 1984; Tait et al. 1989) or external seismic excitation of entrapped bubbles (Sturtevant et aL 1996). Figure 1 shows the quasi-steady state condition. The saturation condition is reached at some depth. A pressure differential is required to cause bubbles to nucleate, so the nucleation surface will be at some height above the saturation surface. Above the nucleation surface the bubbles grow by diffusion of volatiles from the melt and by expansion due to decompression. Progressive bubble growth causes the mixture to expand and results in an upwardly accelerating flow. At some height, the liquid is fragmented into discrete particles and thereafter the flow is of a gas-particle mixture. The mechanism of the fragmentation process is contro-

versial. The eruption is sustained by the steady removal of material from the volcano which causes the saturation and nucleation surfaces to propagate downwards. These surfaces could reside either in the magma chamber, as shown in Fig. 1, or in the conduit. However, recent laboratory experiments (Mader et al. 1997) suggest that the fragmentation surface must necessarily reside in the conduit. The majority of theoretical models of conduit flow dynamics restrict themselves to a study of the quasi-steady state eruption and so neglect the problem of the initiation of the event. In many cases, a constant rise velocity of the bubble-free magma is imposed which implies a steady influx of magma at depth throughout the eruption, consistent with the first triggering mechanism proposed above. Generally, only one-dimensional flow is considered; thus, the nucleation surface resides in the conduit. The first comprehensive, quantitative model of exsolution during magma ascent to be presented was that of Sparks (1978). Many of the features of this early model have been retained in subsequent models (specifically: the presence of saturation, nucleation and fragmentation surfaces; bubble growth by diffusion and decompression; and the condition for fragmentation) and some important conclusions were reached that are still considered valid (see Sparks et al. 1994 for a recent review). However, the model

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is flawed in that bubble growth is assumed to take place in a static foam that is subject to an (imposed) lithostatic (i.e. linear) pressure gradient. Also, the mixture is assumed to be accelerated after fragmentation by bursting of bubbles due to expansion of the released high-pressure gases. This view has been corrected by subsequent models in which fewer assumptions are made and the equations describing the flow dynamics and the material parameters have been progressively refined (e.g. Wilson et al. 1980; Dobran 1992), although the models remain highly simplified in many respects and therefore incomplete. In the next section we review textural observations of geological deposits. It will be seen that the interpretation of the textures relies heavily on theoretical models of conduit flow. These will be considered in the section on Conduit Flow Models. The controversial issue of the fragmentation mechanism enters into the interpretation of textural features and conduit flow models and this will be discussed in the section on Fragmentation. Laboratory experiments represent our only means of directly observing the physical processes involved in conduit flows and arguably provide the most stringent set of observational criteria against which theoretical models must be tested. These experiments have been exploratory in nature and have had a strong impact on our view of volcanic eruption dynamics in recent years. The section on Shocktube Experiments gives an overview of the main experiments conducted.

phase or 'vesicularity', a = Vv/Vt, where Fv is the volume of voids and vt is the total volume of the sample. Researchers frequently distinguish between a 'foam' when > 0.74 and a 'bubble suspension' when < 0.74. The critical value of crit = 0.74 defines the gas volume fraction at maximum packing for undeformed, spherical bubbles of uniform radius. For a to increase beyond this critical value, given a uniform bubble size, requires the bubbles to deform to produce the polyhedral cells characteristic of highly expanded foams (Fig. 2a). crit can be greater in disordered systems that have a range of bubble sizes or in systems with non-spherical bubbles. So-called 'woody' or "tubular' pumice contains strongly strained vesicles with aspect ratios of 10 or more, and its formation is a matter of some interest. The number of nearest neighbours, n, or 'coordination number', of the vesicles in a twodimensional section provides a convenient measure of the degree of disorder in polydisperse

Textural information Textures preserved in pumice clasts reflect the complete vesiculation history of the magma from bubble nucleation and growth prior to and during an eruption through to post-eruptive processes, such as expansion and coalescence prior to quenching. A full quantitative description of the vesiculation in a pumice clast involves the determination of vesicularity and bubble number density, size, shape and wall thickness. From such data, attempts have been made to draw inferences about bubble nucleation and growth, rheology, open versus closed system degassing, explosive versus effusive eruption styles and fragmentation conditions. Vesicularity One of the most important quantitative measures used in studies of explosive eruption dynamics is the volume fraction of the gas

Fig. 2. (a) Idealized two-dimensional foams: spherical bubbles of constant size at maximum packing a = 0.74 and perfectly polyhedral foam with hexagonal cells of constant size for vesicularities a > 0.74. (b) Typical distribution functions f(n) for ordered and disordered systems using soap froth data from Glazier et al. (1990). After Mangan & Cashman (1996).

CONDUIT FLOW AND FRAGMENTATION systems (Glazier et al. 1990). In polydisperse systems, the coordination number is best described by a distribution function f(n). In highly ordered foams, f(n) has a well-defined maximum at a value of 6 (the value for monodisperse systems). As the system becomes progressively more disordered values of n = 5 become more common,

(a)

55

resulting in a broad function with a maximum around 5 (Fig. 2b). Mangan & Cashman (1996) have recently determined the distribution function f(n) for a number of basaltic scoria and reticulite samples from Kilauea volcano. The reticulite samples show a high degree of order with a pronounced peak at n = 6 in two-dimensional sections. By contrast, the scoria samples have broad distribution functions that peak at n — 5, indicative of a wide range of bubble sizes. Products of explosive eruptions tend to be polydisperse with vesicle sizes ranging over several orders of magnitude. The vesicle size distribution is often found to be distinctly bimodal. Sparks & Brazier (1982) present data for pumices, formed during Plinian-type eruptions, which show three pronounced peaks in the distribution (Fig. 3a). Two of the peaks are interpreted as distinct vesicle populations caused by bubble nucleation and growth occurring on different time scales: slow, pre-eruptive degassing within the magma chamber produces the largest vesicles (at approximately 60 m); syneruptive, explosive degassing produces vesicles in the range 5-50 m. In a later paper (Whitham and Sparks 1986), the largest peak (at approximately 1 m) is identified as being due to connections between vesicles, rather than a separate vesicle population. A further example of bimodal vesicle size distributions is shown in Fig. 3b where the largest class size was thought to be due to bubble coalescence (Orsi et al. 1992).

The fragmentation limit

(b)

Fig. 3. (a) Bubble size distribution of a pumice clast from 1980 deposit of Mount St Helens. After Whitham & Sparks (1986). Data are based on mercury porosimetry measurements, (b) Bubble size distributions for two fallout members (G7 and G8) of the Cretaio Tephra of Ischia, Italy. After Orsi et al. (1992). The distributions were determined from scanning electron micrographs of polished, ion-etched thin sections of the pumices. Bubble size is given in terms of bubble area in the two-dimensional sections.

Interpretation of vesicle size distributions hinges critically on the origin assumed for each size population, and growth times and rates. These inputs generally derive from theoretical models which are themselves based on a series of hypotheses of how an eruption might unfold. One hypothesis that enters into nearly all models of conduit flow processes concerns the condition for fragmentation. At fragmentation, the material undergoes a structural transition: prior to fragmentation the liquid phase is continuous, whereas after fragmentation the gas phase is continuous. The detailed dynamics of the disintegration of the liquid phase - the fragmentation process - is currently not understood. As a result, theoretical models fix the fragmentation limit at a specific vesicularity, with a usually somewhere in the range 0.70-0.75. This choice of fragmentation limit according to vesicularity was first proposed by Sparks (1978) with two justifications: (1) vesicularities observed in pumice clasts are commonly around

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this value; and (2) a value of a in this range represents maximum packing of spherical bubbles and therefore the point at which the growth of neighbouring bubbles starts to interfere with each other. Recent data on pumice textures have called into question both of these justifications. Vesicularities measured in pumice clasts vary significantly. Houghton & Wilson (1989), for example, find vesicularities from below a = 0.60 to more than = 0.85 in the Hatepe Plinian eruption deposit, New Zealand. This variation in a masks an even greater variation in the implied gas content, a is not proportional to gas volume, because it is defined as the ratio of the volume of gas vv to the total volume vv + Fs. In fact, as the value of the vesicularity grows, it becomes an increasingly insensitive measure of gas volume. For example, at a vesicularity of a = 0.1, a doubling of gas content increases the vesicularity to = 0.18; i.e. for a = 0.1, 2 x vv results in 1.8 x a. But, at a vesicularity of a = 0.9, a doubling of gas content increases the vesicularity to just a = 0.95; i.e. for = 0.9,2 x vv results in just 1.055 x a. The vesicularity range observed by Houghton & Wilson (1989), 0.60 < a < 0.85, implies an increase in gas volume by a factor of about 4. (2) The second justification relates to internal clast structure. The vesicularity at maximum packing crit is not a constant but depends on the bubble size distribution. It would be interesting to know how the vesicularities observed in pumice clasts relate to the packing of the bubbles within them. But, no data on this are currently available. In any case, it seems unlikely that vesicularity alone, which is a purely structural parameter, that takes no account of the rate at which the processes are occurring, should be adequate to describe the onset of the highly dynamical process of

fragmentation. We know that it is possible for magmatic materials to reach vesicularities well in excess of the maximum packing of spherical bubbles; for example, basaltic reticulite expands to > 0.95 before quenching (Mangan & Cashman 1996). Thus, it is clear that the dynamics of the flow must affect the fragmentation process and therefore the observed textures.

(1)

Several factors have been identified as possible contributors to vesicularity variations. Carey & Sigurdsson (1987) find that vesicularity variations within the AD 79 Vesuvius eruption correlate most closely with geochemical gradients, which implies an effect due to magmatic composition. Thomas et al (1994), using data from a variety of sources (Houghton & Wilson 1989; Wilson & Houghton 1990; Gardner et al 1991; Gardner 1993), consider vesicularity variations between different deposits as well as within individual deposits (Table 1). Table 1 suggests that there is a decrease in vesicularity with increasing melt viscosity but no obvious dependence on discharge rate. Unfortunately, the statistical significance of the viscosity data in particular is not strong because the decrease is not systematic and the case hinges effectively on just two data points, namely those of the Minoan Plinian and Bishop Tuff deposits which give the highest and lowest vesicularity values, respectively. The values for the bulk vesicularity of these two deposits differ by 0.07, which is just within the standard deviations of the distributions. Also, the viscosity given is that of the liquid phase with the dissolved water content at fragmentation and takes no account of effects due to vesiculation or crystallinity, which vary significantly between the deposits considered. Notwithstanding these problems, the idea that melt viscosity has a controlling effect on the vesicularity observed in deposits is supported by the most recent generation of theoretical models.

Table 1. Vesicularity data of silicic pumices. After Thomas et al. (1994) Eruption Melt viscosity (Pa) Bishop Hatepe Taupo Minoan

2x 10 8 1 x 107 2 x 107 9 x 106

Peak discharge rate Average vesicularity Average vesicularity Average vesicularity Bulk Local minimum Local maximum (kgs -1I)

7.5 1.8 1.2 2.5

x x x x

108 108 109 108

0.71 ±0.04 0.73 ±0.05 0.74 ±0.05 0.78 ±0.04

0.64* 0.681 ±0.047 0.702 ±0.038 0.731 ±0.050

0.78* 0.779 ±0.060 0.783 ±0.038 0.814 ±0.037

The bulk average is for the whole deposit. The local average is for individual units within the deposit. Maximum and minimum values for the local average are given. Data derives from Houghton & Wilson (1989), Wilson & Houghton (1990), Gardner et al. (1991) and Gardner (1993). * Average vesicularity value using eight lumped pumices, so no standard deviation is available.

CONDUIT FLOW AND FRAGMENTATION Further work by Gardner et al (1996) supports the notion that magma viscosity has a controlling influence on the fragmentation limit. They studied the ratio of gas volume to solid volume Vv/vs rather than the vesicularity a. The values of vV/vs measured for 13 different deposits vary by up to two orders of magnitude even within a given deposit. No correlations with crystallinity, initial dissolved water content or eruptive discharge rates are found. The only clear correlation of vesicularity is with the estimate of magma viscosity at the point of fragmentation. The authors propose a new fragmentation limit at vV/vs « 1.78 (a = 0.64) but emphasize that this will vary depending on the shear strain rate imposed on the material. Moreover, several authors (e.g. Thomas et al 1994; Gardner et al. 1996; Klug & Cashman 1996) point out that pumices are not quenched instantaneously at fragmentation, and that postfragmentation bubble expansion and collapse have a significant impact on the final textures. These processes also depend on magma viscosity: expansion occurs for < 10 9 Pas and collapse for < 10 5 Pas (Gardner et al. 1996).

Moments of distribution functions An alternative approach to the interpretation of vesicular textures looks at moments of the bubble size-distribution function, F(R) dR which describes the number of bubbles per unit bulk volume with radii in the interval R to R + dR. The i-th moment Mi is defined as

The zeroth, first, second and third moments are related to measurable morphological characteristics, namely: the total number of bubbles per unit bulk volume, TV; the mean radius of bubbles, R', the total surface area per unit bulk volume, S and the vesicularity, a, respectively (2) (3) (4) (5)

57

Currently, there are two different approaches taken to the interpretation of bubble sizedistribution moments. Approach 1. The first approach takes as its starting point an analogy with the theory of crystal nucleation and growth (e.g. Marsh 1988). This approach was first applied to an investigation of the vesiculation of mid-ocean ridge popping rocks by Sarda & Graham (1990). Mangan et al. (1993) and Klug & Cashman (1994) have since applied it to explosive volcanism. The bubble radii are assumed to be growing at a constant rate G which is independent of size. If the system is in steady state, i.e.

then a consideration of the balance of bubbles growing into and out of a particular size class results in the following distribution (6)

where N0 is the nucleation density, i.e. ¥(R) = N0 for R = 0, and T is a constant that represents the time available for bubble growth. Thus, for steady-state nucleation and growth, a plot of In F(R) against R is linear with a slope of (—GT) -1 and an intercept of lnN 0 . The bubble number density N can be measured and, given an independent estimate for the average growth time T, a number of dynamically interesting parameters can be calculated: bubble growth rate, G; the nucleation density of bubbles, NO = TV/Gr; and the average nucleation rate, J=N 0 G = N / T . Departures from a straight line are interpreted as due to a number of effects: vesicle coalescence or breakage; physical sorting of bubbles during withdrawal; size dependence of G; or variable nucleation rates and growth times. All the results depend crucially on the estimate of the average growth time of the bubbles T and the assumption of constant G. Klug & Cashman (1994) derive values for r between 960 and 4120s for the 1980 Mount St Helens eruption from a consideration of the Wilson et al (1980) model of conduit flow. Table 2 shows the values obtained for N (measured), G and J. The measured bubble number density, TV, is similar to that found in other andesitic to rhyolitic pumices (Heiken 1987; Toramaru 1990). The inferred nucleation rate, J, is also broadly in line with experimental observations of nucleation and growth of bubbles in silicic melts (Hurwitz & Navon 1994). However,

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Table 2. Vesicularity data and inferred dynamical parameters for Mount St Helens pumice. After Klug & Cashman (1994) Sample

Vesicularity,

Crystals (%)

Grey White

0.72* 0.86*

15 (34)f 6 (31)f

Microlites Number density (%) of bubbles, N (cm- 1 ) 7 (16)f

2.0 x 109 8.2 x 108

Growth rate, G (cms - 1 )

Nucleation rate. J

5.21 x 1Q-7 1.74x 10-6

2.09 x 106 8.55 x 105

( c m 3 s-1)

The number density of bubbles N is a measured. The growth rate G and nucleation rate J are inferred assuming a growth time T = 960s, calculated on the basis of the model of Wilson et al. (1980) and using physical parameters obtained from the literature for the Mount St Helens eruption. * Vesicularities are calculated to crystal and microlite free. f Values shown in parentheses are calculated to vesicle free.

the predicted growth rate, G, is several orders of magnitude less than average growth rates implied by bubble growth models (Sparks 1978; Proussevitch et al. 19930). This discrepancy probably arises because the assumption of constant, G, between nucleation and fragmentation is not valid. All bubble growth models derive highly non-linear growth histories (Sparks 1978; Proussevitch et al. 19930; Barclay et al. 1995). Moreover, the more recent conduit flow models (Dobran, 1992; Papale & Dobran, 1994; Sparks et al 1994), which take account of the viscosity increase of the melt as volatiles are exsolved, derive highly non-lithostatic pressure gradients with the result that most of the vesiculation occurs at a high rate over a well-defined region; for example, numerical calculations presented in Sparks et al. (1994) suggest that the Vesicularity changes from 0.2 to 0.7 over just 50m depth in times of order 10s. The study illustrates the problems associated with obtaining quantitative information of dynamical variables using an approach that necessarily relies on theoretical models, which are not sufficiently sophisticated to produce accurate and reliable results. Despite these reservations, from a qualitative point of view, the observations of Klug & Cashman (1994) support the findings of Sparks (1978) and Toramaru (1989) that the smaller diffusion coefficients and faster eruption rates of silicic magmas result in smaller bubbles (and therefore greater nucleation rates and bubble number densities for a given Vesicularity) than in more mafic magmas. Approach 2. Toramaru (1989, 1990) has proposed an alternative approach in which a theoretical framework is developed specifically for the purpose of interpreting textural parameters. A single nucleation event is assumed to take place in magma that is rising at constant speed V up the conduit. Thus, the bubble size distribu-

tion is a delta function and the i-th moment is given by where R is the mean bubble radius and N is the number density of bubbles. The crux of the approach lies in identifying the physical parameters that control the vesiculation process and hence the bubble size distribution. The controlling processes are given as nucleation facility, saturation pressure and growth processes. These processes are described by three variables: the silicate-gas interfacial tension, (nucleation); initial saturation pressure, P0 (saturation pressure); and the effective diffusivity, D/V, of the volatile, where D is the diffusivity of the volatile in the melt and V is the ascent velocity (growth processes). D/Vis a measure of the time available for bubble growth by diffusion and so the exponent £ on D/V represents the effective growth law (R t). The bubble size distribution is therefore some function of these variables and so we can write directly

(8) (9)

(10) and where (i is an integer) and depend on the physics of the process. The exponents a, b, c, /n, f and C must be determined from physical experimentation or via a theoretical model. Toramaru (1989. 1990) calculates values from a numerical model. Given values for the exponents, then values for 7, P0 and D/ V can be determined from three of the moments, provided they are independent.

CONDUIT FLOW AND FRAGMENTATION But, Toramaru (1990) finds that they are not from observations of scoria from IzuOshima and pumice from Towada volcanoes: the first and second moments (M1 and M2) vary by several orders of magnitude between eruptions and are related according to M2 the third moment (M3) shows only a narrow variation of around 10%. The variations and dependencies are assumed to be controlled primarily by the effective diffusivity D/V because the diffusivity D is expected to have the widest range of values of the various parameters (Shaw 1974). Therefore, the variation in the values of the first and second moments is interpreted as due to differences in the value of D/ V between eruptions. M2 then implies that 2z2 = z1 and hence c + 3 = 0. Similarly, the small variation in the third moment implies that M3 is independent of D/V, which also implies that z3 = c + 3 = 0. So, the conclusions are internally consistent with the condition c + 3 = 0 satisfying all of the primary observations. Bearing in mind that C is related to the implied bubble growth law, subsequent discussion in Toramaru (1990) is concerned with the significance that the constraint c + 3 = 0 has on vesiculation behaviour. This approach is unique in that it attempts to identify the physical processes involved in generating the textures and produce a theoretical model specifically to calculate the moments without drawing on analogies with crystal growth theory or using the results of bulk flow models. It is an extremely difficult problem and the model is complex. It is made tractable by a number of simplifying assumptions, such as the single nucleation event and the constant rise velocity, that do not seem strictly justified. Moreover, many inputs are required that are not well constrained at this time (notably the many exponents). For example, realistic results can only be obtained if a value for the interfacial tension 1 0 - 2 N m - 1 is chosen which is two orders of magnitude less than any of the measured values quoted in the literature (Walker & Mullins 1981; Taniguchi 1988). The model can be made more sophisticated by adding further parameters. For example, in a more recent paper Toramaru (1995) introduces a forth parameter that is the ratio of the time scale of decompression to a viscous relaxation time. But, this adds to the complexity. Conduit flow models The previous discussion has indicated the importance of theoretical models of conduit

59

flow processes in inferring dynamical parameters from vesiculation textures. The accuracy of the conclusions is limited by the accuracy with which the model describes the flow processes. The main problem that arises in modelling is that whilst information can be inferred about the starting conditions, such as temperature, pressure, composition, volatile concentration and material parameters, and direct observations above-ground can be made, direct observation of conduit flow is not possible. As a result, most models are based on a series of hypotheses about how explosive eruptions unfold. Moreover, model validation cannot be achieved by direct comparison with the natural flows. There are essentially two approaches to theoretical modelling of conduit flow. One approach is to set up a system of equations to describe the growth of an individual bubble within a defined sphere of influence. Deductions are then drawn concerning the behaviour of a large collection of such bubbles. The model of Sparks (1978) was the earliest, comprehensive model of this sort. More recent models include those of Bottinga & Javoy (1990), Proussevitch et al. (1993a), Barclay et al. (1995) and Toramaru (1995). These bubble growth models are reviewed elsewhere in this volume (Navon & Lyakhovsky 1998). The conduit flow models considered here adopt a larger-scale approach that models the bulk flow of magma up a conduit. In general, first-generation conduit flow models study one-dimensional, steady-state, isothermal flow of a homogeneous material of constant Newtonian viscosity up a vertical conduit of constant cross-section under a lithostatic pressure gradient. Different models relax one or more of these simplifying assumptions. Wilson et al. (1980), Wilson & Head (1981) and Giberti & Wilson (1990) study the effect of variations in conduit cross-section. Supersonic two-phase flow requires growth in the conduit radius. Often, this occurs at the surface, where it is marked by a flared vent. The pressure at the vent is then greater than atmospheric. Thus, these models point out the possibility that the pressure gradients in the conduit may not be lithostatic everywhere. Buresti & Casarosa (1989) studied the fragmented regime and allowed for temperature variations between the phases. Retaining an adiabatic assumption, they found that temperature changes within the two-phase flow are negligible. So, the isothermal assumption appears to be sound. The separated flow model of Vergniolle & Jaupart (1986) relaxes the homogeneous assumption by allowing for relative motion of the two phases. This model is particularly appropriate for basaltic eruptions,

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60

where the viscosity of the melt is sufficiently low that the phases can move at appreciably different speeds. Both the vesicularity and pressure drop in the separated model are found to be less than in the homogeneous model. Multiple-phase flow effects are thereby shown to be an important factor in producing non-lithostatic pressure gradients. The model of Dobran (1992) and Papale & Dobran (1994) is sufficiently different from the earlier models that we can identify it as a second generation model. This is a multiple-phase model with separate systems of equations for the different phases that thereby allows for thermal and mechanical disequilibrium between the phases. The pressure function with height is not imposed, as in the earlier work. The equations are numerically solved for pressure, vesicularity and gas and pyroclast velocities and their spatial distributions. Moreover, equations for the variation of material parameters, such as density and viscosity, as a result of vesiculation are included. The model retains the one-dimensional and steady-state flow assumptions. The relaxation of constant Newtonian rheology in this and other second-generation models (Jaupart and Allegre 1991; Sparks et al. 1994;

Woods 1995) is of particular significance. This assumption of the first-generation models is certainly wrong: the rheology of the flows is neither constant nor Newtonian. There are two factors to consider, both associated with progressive vesiculation. In the first place, pure silicate melts are considered to have a Newtonian rheology that is a strong function of dissolved gas content (Murase 1962). As volatiles come out of solution in the form of bubbles the viscosity of the melt, i.e. the pure liquid phase, increases. The increase can be as much as four to six orders of magnitude over the length of the conduit. The second factor is due to the change in the state of the fluid on vesiculation, as it goes from a single-phase liquid (or two-phase suspension of liquid and crystals) to a two-phase (or three-phase) bubbly liquid. The incorporation of gas into a liquid has an enormous impact on the bulk rheology. To use an everyday example: aerating dairy cream has the effect of turning a very runny liquid into a soft solid with no change in the viscosity of the liquid phase (i.e. the cream) itself. Whilst some research has been done on the rheological effects of crystals (e.g. Shaw 1969), the effects of bubbles are not at all well constrained. However, bubbly liquids are known to be viscoelastic and non-Newtonian with a complex rheology that is a function of bubble size distribution and strain rate. Moreover, the flow itself will affect the bubble size distribution, because it will tend to cause coalescence or tearing-up of bubbles. Thus, the rheology will be strongly coupled to the flow dynamics. In the model of Dobran (1992) and Papale & Dobran (1994) the mixture viscosity coincides with the magma (liquid + crystals) viscosity below the exsolution level (Fig. 4), whereas in the bubbly flow region zs < z < zf it is given by

(11) where

(12)

MAGMA CHAMBER

Fig. 4. Model architecture of model of Dobran (1992) and Papale & Dobran (1994). In this model the magma is accelerated from rest at some level o in the magma chamber. Exsolution begins at a height zs leading to fragmentation at a height zf. Thus, there are three different flow regimes within the conduit: o < z < rs single-phase flow; zs < z < zf two-phase bubble-foam flow; z < zf two-phase gas particle-droplet flow.

and //G is the viscosity of the gas phase; Lo is the viscosity of the anhydrous, crystal-free melt; is vesicularity; 0 is the volume fraction of crystals; and Y is the water content in the melt. Equation (12) is due to Shaw (1969) and describes the effect of the progressive loss of volatiles on melt viscosity. Equation ( 1 1 ) attempts to take account of the effect of bubbles. This equation is based on work by Ishii & Zuber (1979) and Dobran (1987), and goes some way towards achieving its goal. However, it gives the

CONDUIT FLOW AND FRAGMENTATION

Fig. 5. Pressure and vesicularity as a function of depth in the conduit for Phase IV (final plinian phase) of the 18 May 1980 Mount St Helens eruption as calculated by Papale & Dobran (1994). Note the strong departure from the lithostatic pressure distribution (indicated by the dashed line). Most of the vesiculation occurs over an extremely narrow depth range with a concomitant decrease in pressure. Fragmentation is assumed to occur at a = 0.75. z is the depth in the conduit of length L. a is the vesicularity. P is the pressure at depth z and P0 is the pressure in the magma chamber. Values for Phase IV are: mass discharge rate of 1.6 x 107 kgs -1 L = 7.2km; P0 = 220 MPa; initial dissolved water content 4.6 wt%; temperature 1193K. After Papale & Dobran (1994). viscosity simply as a function of total gas volume fraction; no account is taken of the effects of the detailed bubble size distribution, strain rate or coupling with the flow dynamics. It is not possible at present to incorporate the details of these effects because they are not known. Equations (11) and (12) adequately describe our current knowledge concerning the rheology of the two-phase bubbly flow regime and represent the most sophisticated rheological model used in conduit flow models to-date. In the gas particle-droplet region z > Zf the viscosity is given by (13)

where crit refers to the maximum packing density of spherical particles in the magmatic mixture of liquid and crystals. The mixture viscosity is the largest in the bubbly flow regime (equations (11) and (12)). It grows by up to three orders of magnitude as the gases exsolve and reaches a maximum just prior to fragmentation. After fragmentation, the mixture viscosity falls dramatically to a very low value which is close to the viscosity of the gas phase alone (equation (13)). The primary results

61

of the model are illustrated in the plot of Fig. 5. Relaxing the assumption of Newtonian rheology produces a strongly non-lithostatic pressure gradient. This occurs because the increasing resistance to flow experienced as the gases exsolve generates a very large frictional pressure drop prior to fragmentation. This result is supported by other models that allow the viscosity to vary as the gas exsolves (Jaupart & Allegre, 1991; Sparks et al 1994; Woods 1995). The model of Dobran (1992) and Papale & Dobran (1994) appears to be inconclusive on the issue of whether the fluid pressure in the conduit is higher or lower than the lithostatic pressure in the surrounding country rock. Overpressures in the conduit can lead to failure of the conduit wall and flow of gas out of the conduit and into wall cracks. Underpressures will tend to promote the flow of water into the conduit resulting in an increased probability of phreatomagmatic explosions. The original model by Dobran (1992), when applied to the May 1980 Mount St Helens eruption, produced underpressures prior to eruption, but the more detailed paper of Papale & Dobran (1994) shows overpressures. It is not clear at this stage which factors favour the development of under- or overpressures and the consequences of either occurrence have not been quantified. Fragmentation: brittle or ductile? The details of the fragmentation mechanism in explosive eruptions are currently not known. There are essentially two ways in which a liquid can fragment: brittle or ductile. A ductile fragmentation process involves the progressive thinning of liquid layers via some mechanism, such as drainage of liquid out of the layers under gravity or expansion of the gaseous phase, until the layers become so thin they necessarily rupture. A common example of this type of fragmentation is the spontaneous bursting of bubbles in soap foams. The whole process occurs in a liquid-like fashion. However, in an accelerating two-phase flow it is possible for brittle fragmentation to occur when the shear strain-rate imposed is so great that the natural relaxation time of the material is exceeded. The material then fractures in a similar way to a glass. The critical relaxation time is generally derived from the Maxwell model of linear viscoelastic materials as

where p, is the viscosity of the melt and G is the shear modulus. If we choose « 108 Pas, as a

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H. M. MADER

typical viscosity for a silicic melt at the point of fragmentation, and G 10 10 Pa (Dingwell & Webb 1989), then the critical relaxation time is T 0.01 s and the critical strain rate about 100s - 1 . This value coincides with the strain rates observed in experimental simulations of explosive eruptions (Mader et al. 1994), which suggests that the fragmentation mechanism could be either brittle or ductile. It is sometimes asked whether the fragmentation mechanism matters. Perhaps the flow is controlled by processes preceding and following fragmentation, rather than by those that occur at fragmentation. It will not be possible to decide how important the fragmentation mechanism itself is until we have understood its features. It is possible that processes occurring at fragmentation may have more of an impact on small-scale textural features than on large-scale features of an eruption, such as the discharge rate, but this is not clear at present. There are no studies available that would allow us to identify which of the two mechanisms operates in volcanic flows. However, there are separate theoretical studies of brittle (Alidibirov 1994) and ductile fragmentation (Proussevitch et al. 1993b) in magmas. Neither deals directly with liquid fragmentation due to explosive vesiculation. But a review of this research will allow us to identify the primary features associated with the two mechanisms. Alidibirov (1994) considers fragmentation occurring in discrete blasts in highly viscous magma, such as in dome explosions and in Vulcanian or Pelean types of activity. A decompression event is imposed on porous, solid magma filling a conduit (Fig. 6). The solid bubble walls are ruptured by the pressure differential created across them. The velocity of the fragmentation wave front V and the velocity of the gas-pyroclast mixture u are determined from a consideration of the energies stored in the porous body and the strength of the bubble walls using an approach due to Nikolski (1953) and Khristianovich (1953, 1979). It turns out that the energy stored in compression in the gas is much greater than the elastic energy stored in the solid. The critical condition for the bubble walls to break is given by

where a is the vesicularity, P0 is the gas pressure in the porous magma, Pa is atmospheric pressure and t is the tensile strength of the magma. The conservation equations produce a generally slow propagation velocity of the fragmentation wave V < 10ms - 1 which explains

Fig. 6. Schematic diagram of model of Alidibirov (1994). Solid, vesicular magma resides in a conduit. A decompression event causes fragmentation of the magma by brittle fracturing of bubble walls. The fragmentation surface propagates down into the magma at a speed V and the magmatic fragments are ejected upwards at a speed u.

the long duration and low rate of energy release of volcanic blasts. The theory also shows that excess pressure can be generated behind the fragmentation wave front. This changes the pressure differential across the bubble walls and thereby influences the rate of fragmentation. The excess pressure can be large enough to cause the motion of the fragmentation front to cease temporarily until the pressure differential across the bubble walls builds up and again exceeds the critical condition. In this way, several explosions can occur within a single blast. This approach cannot be applied directly to liquid systems. The critical condition in the solid system is determined simply from the tensile strength of the material. In a liquid system, flow prior to fragmentation imposes a strain rate on the material and brittle fracturing occurs when the strain rate exceeds some critical value. Proussevitch et al (1993b) present a theoretical study of the disruption of a foam by coalescence of bubbles. This is an investigation of spontaneous ductile fragmentation in a stationary foam. Bubble coalescence will always occur in a foam because it is inherently unstable; coalescence reduces the surface to volume ratio and hence the surface energy density in the foam. The films or Plateau borders (Fig. 7) bounding a bubble are assumed to rupture when they thin

CONDUIT FLOW AND FRAGMENTATION

63

Fig. 7. (a) Schematic diagram of a foam showing the location of inter-bubble films (surfaces where two bubbles meet) and Plateau borders (lines where three or more bubbles meet), (b) Generalized cross-section for Plateau borders as used by Proussevitch et al (1993b).

to some critical thickness CT — 0.1-0.05 mm, corresponding to the typical size of crystals in the melt. Proussevitch et al (19936) distinguish between three different classes of foam: in 'stable foams' the rate of disruption is limited by expulsion of fluid between bubbles; 'subcritical foams' have widely separated bubbles and so experience negligible coalescence; and 'supercritical foams' have overly thinned bubble walls which can spontaneously disrupt without further thinning. The driving pressures leading to expulsion of liquid from individual films and Plateau borders are due to gravity and capillary forces. Thinning due to gas evolution is not considered. The equations for the driving pressures are

where p is the density of the melt, g is the acceleration due to gravity, h is the characteristic length of the film dividing the two bubbles, r is the radius of curvature of the films and a is the inter facial surface free energy. A consideration of the relative size of gravity and capillary forces for the conditions typically found in natural volcanic foams and suspensions shows that expulsion of liquid from the films is controlled by capillary forces whereas the dominant driving pressure in the Plateau borders depends on the gas volume fraction; capillary forces dominate for a > 0.74 and gravity forces for a < 0.74. Proussevitch et al. (1993b) solve the NavierStokes equations for viscous flow between parallel surfaces (the films) and along triangular cylinders (the Plateau borders) separately and determine which ruptures first and so triggers a coalescence event. The results show that in silicic

melts surface tension dominates the liquid expulsion, that rupturing occurs at inter-bubble films for smaller bubbles and at Plateau borders for large bubbles (Fig. 8), and that the rate of bubble coalescence increases with time. Neither of these studies answers the problem of whether fragmentation in an explosive

Fig. 8. Results of the Proussevitch et al. (1993b) model of foam stability. The graph shows melt viscosity as a function of 'stable' bubble size in silicate froths for various vesicularities. Bubble coalescence is accomplished by the expulsion of melt along inter-bubble films (thick lines) or Plateau borders (thin line). Dashed lines are the continuation of solid lines into the field where coalescence is controlled by the other mechanism. Circles represent the initial bubble radii assumed in the model calculations. The curves for Plateau borders for vesicularities of 0.5 and 0.75 are virtually indistinguishable and are plotted as a single line. Note that smaller bubbles coalesce by film rupture, while larger bubbles coalesce by rupture of Plateau borders. Large bubbles are stable in any viscosity, but smaller bubbles can only be supported in high-viscosity melts. After Proussevitch et al (19936).

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H. M. MADER

eruption is brittle or ductile. They do, however, serve to illustrate how, in principle, one might approach this question. The approach of Proussevitch et al (1993b) would need to be extended to include thinning due to bubble growth by diffusion of volatiles from the melt. Fragmentation will occur when some critical absolute thickness is reached (ductile fragmentation) or some critical rate of thinning exceeded (brittle fragmentation) in either films or Plateau borders. The point would be to determine which critical condition is reached first, under what conditions and with what consequences. Shock-tube experiments

Rationale and approach Numerical models must be validated against observations. Conduit flow models are particularly difficult to validate because of the impossibility of observing the dynamics of the conduit flows directly. The outcome of the flow can only be observed in the form of bulk dynamical conditions at the vent (mass discharge rates, durations, flow speeds, etc.) and the resulting vesicular textures. Full validation of models cannot be achieved using only natural data. A recent approach to this problem has been to conduct dynamical laboratory experiments. The aim is not to produce a 'mini-volcano' in the laboratory but to study fundamental physical processes. A wide range of experiments investigating different aspects of gas-driven explosions have been performed and are influencing our understanding of the evolution of explosive flows. Interpretation of the experiments is complicated by the problem of scale, as discussed below. Four techniques have been used to produce explosive flows: vaporization, gas expansion, exsolution and chemical reaction. The resulting flows are generally recorded by high-speed photography and pressures are measured at several locations in the test cell. The data can provide information on bubble nucleation and growth rates, gas evolution rates, flow regimes, and interactions between flow dynamics and gas evolution. An overview of experiments to-date is provided below, grouped according to the technique, and the most significant results for conduit flow processes are summarized.

Vaporization Hill & Sturtevant (1990) studied explosive vaporization of a volatile liquid under a sudden

decompression which takes the liquid above its superheat limit. Explosive boiling occurs by homogeneous nucleation at a sharply defined interface that propagates down into the liquid. This experiment is unique in that it investigates a one-component system; the gas phase is compositionally the same as the condensed phase. In this respect, it is not a good model for an explosive eruption which is a two- (or three-) component system with compositionally distinct gaseous (H2O. CO2, etc.) and condensed (silicate melt with or without crystal inclusions) phases. There is no evidence that vaporization of the melt plays a significant role in explosive eruptions. Sugioka & Bursik (1995) studied a system in which inert particles are suspended in volatile liquid that is placed in a centrifuge. The centrifuge increases the effective gravitational acceleration up to lOOg, hence producing a vertical pressure difference up to 100pgh (approximately 105 Pa) in the shock tube, equivalent to the lithostatic pressure difference over a 5m depth of conduit. This is achieved at the expense of imposing a much greater pressure gradient in the shock tube (around 100 times greater) due to its small scale. Heterogeneous nucleation throughout the liquid occurs on the suspended solid particles. Small decompressions generate rapid bubble growth throughout the liquid and largescale breakup of the mixture. But, for large decompressions fragmentation occurs at a welldefined interface that propagates down into the liquid. The authors suggest that the growth of bubbles behind the fragmentation front is suppressed by compression of the mixture due to material being accelerated upwards from the fragmentation front. The analogy with natural volcanic systems is that the solid particles represent magmatic fragments and the vaporizing liquid the equivalent of an exsolving gaseous phase. The significance of the fragmentation surface is not clear, given that the fragments are pre-existing and not generated. Phreatomagmatic explosions are the main natural phenomenon in which vaporization is clearly a significant process. Zimanowski et al. (1991) report on experiments in which water is injected into hot melts. The experiments provide valuable information on triggering mechanisms, the effect of different water-melt mixing ratios and resulting textural features. Use of natural materials has so far been restricted to these experiments and those of Alidibirov & Dingwell (1996) (see below). Observation of the flowdynamics has not been possible, mainly due to technical difficulties associated with conducting experiments in a high-temperature furnace.

CONDUIT FLOW AND FRAGMENTATION

Gas expansion Two experiments have been reported that use gas expansion under a sudden decompression to generate two-phase explosive flows. Anilkumar et al. (1993) studied the decompression of beds of solid particles. The resulting gas particle flows are models for post-fragmentation flow processes both in the conduit and above ground. The flow dynamics depends on particle size and density stratification. Stable flows that move upwards as dense slugs are produced if there is a significant increase upwards in either the particle size or density. Where particle size or density is constant or decreases upwards the flows are unstable and become highly inhomogeneous with interpenetrating regions of high and low particle concentration. Alidibirov & Dingwell (1996) provide an experimental model for brittle fragmentation by suddenly decompressing highly viscous vesicular magma. The analogy with Pelean and Vulcanian eruption styles and lava dome collapse is close, and the laboratory pyroclasts are found to be very similar to natural samples. The experiments show that violent explosions can be produced by unloading in relatively cool magmatic materials.

Exsolution and chemical reaction Explosive volcanic eruptions are driven primarily by exsolution of dissolved volatiles within liquid magma. Therefore, experiments which use this mechanism to generate explosions are of particular interest. Mader et al. (1994) studied the decompression of carbonated water and Phillips et al. (1995) the decompression of solutions of acetone in gum rosin (natural pine resin). The exsolution experiments of Mader et al. (1994) are reported in conjunction with other experiments that exploit a chemical reaction to generate large volumes of COi internally within a liquid (Mader et al. 1994, 1996). These experiments, in common with others mentioned above, use non-magmatic or 'analogue' materials with properties such that explosions are generated at ambient room temperatures. The technical constraints are not as great as for experiments conducted using natural materials and direct observation of the flow evolution is often possible. One of the main results of these experiments is that acceleration precedes fragmentation and is a direct result of gas evolution. This observation calls into question early theoretical models (Sparks 1978) which presume that gas evolution occurs within a static foam and that acceleration

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occurs after fragmentation due to the expansion of the over-pressured gases released from the bubbles. A second important result is that the experimental gas volumes increase as t2 or r3, which is significantly greater than the t 1 . 5 growth law expected for diffusive growth of a spherical bubble at rest in an infinite fluid and frequently used in early numerical models (Sparks 1978; Toramaru 1989). The increased gas evolution rates are due to increased volatile diffusion into the bubbles by: (i) advective motions in the two-phase liquid which maintain high gas concentration gradients at the bubble walls; and (ii) bubble deformation due to high strain rates which increases the surface to volume ratio. These effects are partially counteracted by a decrease in volatile diffusion because the bubbles are not in an infinite melt; they have a finite sphere of influence and hence a finite supply of volatiles (Proussevitch et al. 1993a). The balance of these three processes in a given eruption will control the exact gas evolution rate and hence the acceleration of the two-phase mixture. The issue of bubble deformation is of particular interest because of the occurrence of 'woody' pumice, which contains strongly strained vesicles. The acetone in the gum rosin system (Phillips et al. 1995) produces solid end products that contain such strained vesicles. This is due to the rheological properties of the system. Gum rosin is an amorphous solid at room temperature. Mixtures of gum rosin and acetone are Newtonian liquids with a viscosity that is strongly dependent on the amount of acetone in solution (Fig. 9). Therefore, acetone exsolving from the mixture on decompression produces a strong viscosity gradient which closely mimics the dynamic rheology found in explosive eruptions.

The problem of scale The most significant problem associated with applying the results of the dynamical shock-tube experiments mentioned above is that of scale. A true simulation involves reproducing the flow dynamics of a large-scale system within a smallscale model by matching the ratios of important forces. Flows can be characterized by various dimensionless numbers that describe these force ratios and other relationships (e.g. void fraction, Reynolds number, mach number, relative velocities of phases, etc.). But, prediction of the flow regime (e.g. turbulent or laminar, etc.) of such high-density two-phase flows is complicated by the fact that it is affected by interactions between

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phases can act to produce the range of flow scales observed. It is tempting to think that similarity can be achieved by simply using the same materials as in the real flows and that therefore experiments on natural silicate melts are not bedevilled by the same scaling problems as those using analogue materials. This is not the case. True scaling requires scaling of material properties as well as dynamical variables. For example, consider flows in which the flow regime can be determined by a Reynolds number Re — VLjv

Fig. 9. Typical viscosity range as a function of volatile content for acetone in gum rosin solutions at temperatures of 20-40°C. The viscosities of rhyolite at 850°C and basalt at 1200°C between 0 and 10wt% are shown (Shaw 1972) for comparison. After Phillips et al. (1995).

bubbles. The map of flow regime versus dimensionless numbers has not been determined, either empirically or theoretically, for these flows. As a result, we can use the dimensionless numbers to reliably achieve similarity on the scale of individual particles but this will not necessarily result in similar bulk flow behaviour, because of the effects of bubble-bubble interaction. Recent scanning electron microscope observations of pumiceous clasts reported by Klug & Cashman (1996) emphasize that fragmentation is controlled by bubble-bubble interactions. The approach taken in the laboratory is to replicate as many of the parameters of the flow as possible. From a dynamical point of view this means that we try to replicate the velocities and accelerations. Given the same velocity, the laboratory time scale is scaled down, compared to the natural system, by the same factor as the length scale (Table 3). No requirement is placed on the absolute value of the length scale but a large range of scales must be available so that the long-range dynamical interactions between Table 3. Geometrical scaling for dynamical simulation experiments. After Anilkumar et al. (1993) Volcanic flows Laboratory flows Velocity Physical scale Time scale

100ms- 1 l-10km 10-lOOs

10-lOOms- 1 10cm l-10ms

Fig. 10. Experimental apparatus. Water that has been saturated under pressure with CO2 is fed into the highpressure Pyrex test cell TC (pressure = 1000 kPa) which is separated from the low-pressure reservoir RES (pressure approximately 4 kPa, volume 2601) by an aluminium diaphragm D. Sudden decompression and supersaturation is achieved by puncturing the diaphragm with a solenoid-driven cutter DC. The supersaturation causes CO2 bubbles to nucleate and grow progressively resulting in the expanding two-phase flow which erupts into the reservoir. Pressures are measured at transducers PT1 and PT2. The flows are recorded on video and high-speed motion photography at 1000 frames per s. (b) Flask arrangement. The test cell consists of a sphere (volume approximately 0.51, diameter 10cm) that grades smoothly into a long cylindrical neck (length 45cm, diameter 2.7 cm). Fill-depth up to 2 cm up the neck and a pressure of 760 kPa are used, (a) Constricted tube arrangement. The test cell consists of a long cylindrical tube with a metal constriction plate CP mounted at the top. The constriction is provided by circular holes of varying diameters (2.54, 1.27 and 0.64cm) drilled centrally in the metal plate. Fill-depths of approximately 50cm and 90cm (liquid volumes of between approximately 1.0 1 and 1.8 1) and pressures of 750-1040 kPa are used. After Mader et al. (1997).

CONDUIT FLOW AND FRAGMENTATION where V is the average velocity, L is the characteristic length scale and v is the kinematic viscosity of the fluid. Reducing the length scale, at the same velocity, would require a decrease in the kinematic viscosity of the liquid by the same factor to achieve similarity.

Effect of geometry and scale The effect of scale and test-cell geometry on the flow dynamics (Mader et al, 1997) has been investigated in a series of large-scale exsolution

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experiments (sample volumes up to 20 times larger and eruption durations up to 100 times longer) in two geometrically distinct test cells (Fig. 10). Two significant conclusions are reached. First, the increase in scale causes heterogeneities in the form of discrete pulses of vesicular material that move rapidly up through the test cell (Fig. llb,c). The experiments previously reported (Mader et al. 1994) produce turbulent flows with heterogeneity on a much smaller scale, relative to the test-cell dimensions. In the large-scale experiments, a larger range of

Fig. 11. Flask arrangement: photographs. The timings of important events are as follows: t = 0 diaphragm rupture; t = 9ms nucleation; t— 14ms start of flow front motion; fragmentation surface propagates down neck reaching base of neck at about t = 200ms; eruption is exhausted at about 1500 ms. Flow in (a) t = 45ms shows a well-defined fragmentation surface a height h = 8cm above the neck entrance (arrow). The fragmentation surface separates the uniformly bubbled liquid below from the fragmented spray above. Flows in (b) t = 254ms, (c) t = 264ms, and (d) t= 1278ms have the fragmentation surface at the base of the neck separating the homogeneous and virtually static foam in the sphere from the highly heterogeneous and turbulent flow in the neck. The instability at the neck entrance caused by converging streams of liquid in the sphere, coupled with the high accelerations experienced by the individual fluid particles, lead to large-scale heterogeneities and fragmentation of the liquid into small volumes of finely bubbled (i.e. dark) liquid. The finely bubbled regions have very high surface-to-volume ratios and so diffusion in these regions is high resulting in higher accelerations and velocities. The fragments start to gain speed and sweep though the neck, collecting material as they go. The process of slug growth is illustrated in (b) and (c). In (b) two diffuse slugs (arrows) of finely bubbled liquid are visible separated by a region of clear fluid. In (c) the two slugs have merged to form one dense slug (arrow). The process of the initial generation of the slugs at the neck entrance is shown in (d). Converging streams of liquid at the entrance to the neck cause oscillatory behaviour (arrows) on the scale of the neck diameter. After Mader et al. (1997).

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length scales is available and so the bubblebubble interactions have more time and space to evolve. Further scaling-up to natural dimensions will tend to increase the degree of heterogeneity to even larger scales relative to conduit dimensions. Thus, volcanic eruptions are inherently heterogeneous. The fluctuations in discharge rate (Carey etal. 1990; Gardner etal 1991) and discrete pulses and shocks (Sparks & Wilson 1982) commonly observed are an inevitable consequence of the scale of the natural phenomena. Secondly, in the flask experiments (Fig. lOb and 11), quasi-steady flow conditions are reached after an initial transient during which the discharge rate grows (Fig. 12), the pressure in the flask rises and the fragmentation surface propagates down the neck until it reaches a stable position at the exit from the sphere (Fig. 11). Thereafter the flow is characterized by a virtually static foam filling the sphere and a turbulent and fragmented flow in the neck. The pressure in the flask is at a stable value that

provides a balance between diffusion into the bubbles and flow out of the sphere. The experimental system provides a model for volcanic eruptions in which there is an evacuation of the magma chamber. In such eruptions the fragmentation surface resides at the exit from the magma chamber, after an initial transient, because it is here that the colliding fluid streams provide a source of instability within the flow, and also because this is where individual fluid particles experience the greatest accelerations. Thus, an alternative explanation is provided for the common observation of growing discharge rates at the start of eruptions, which has been previously explained in terms of steady erosion of the conduit (Wilson 1980; Carey & Sigurdsson 1987, 1989). Future perspectives This chapter has thrown up as many questions as it has answered. However, it should be clear

Fig. 12. The graph shows the velocity at the neck entrance as a function of time. The motion of individual blobs of fluid very near the neck entrance were tracked over 5 ms. The velocity at the neck entrance increases monotonically over the first 400ms of the run until a velocity of about 4 m s - 1 is reached which remains roughly constant until the eruption is exhausted. The change from increasing velocity to constant velocity at t = 400ms corresponds to a break-in-slope observed in the pressure trace measured at the top of the neck. The scatter of the data points is large because the flow is turbulent, and so there are real variations in velocity, and because the fluid blobs chosen were at variable heights and tended to change shape. The velocities measured at the neck entrance are small and the slope over the first 400ms is also small. This slope describes the Eulerian acceleration, which is the change in the velocity at a point in the flow as different fluid particles move through it. It is small compared to the Lagrangian acceleration, which is the acceleration experienced by a specific fluid particle as it moves past this point. The particles have negligible forward motion in the sphere and then reach a velocity of about 4 m s - 1 over a distance of about 2cm in the neck which corresponds to an average Lagrangian acceleration of around 200 ms-2 or 20 g. The instantaneous accelerations are likely to be several times greater than this. After Maderetal. (1997)

CONDUIT FLOW AND FRAGMENTATION that recent years have seen a tremendous leap forward in the techniques available to researchers for investigating conduit flow processes. One of the main challenges for the future is to bring together the information deriving from the three different approaches - textural studies, theoretical models and physical experimentation - to form a cohesive picture. It is perhaps possible to identify two of the most important long-term aims of research into conduit flow processes: (1) to be able to infer as much as possible (it may not all be preserved) about the dynamics of formation from pumiceous textures in the field and thereby provide the means with which we can reliably interpret the volcanic record, and (2) to produce accurate theoretical models (numerical or analytical) as forecasting tools for volcanic hazards. A full theoretical description of conduit flow requires separate equations for the different phases to allow for chemical and thermal disequilibrium and differential motion between them as well as equations for the fragmentation mechanism and the material parameters of the multiple-phase flow (such as viscosity and diffusivity) in terms of the local physical and chemical conditions and previous history of the flow. Account must also be taken of: the physical shape of the magma chamber and conduit and any erosion thereof; the triggering mechanism; loss of volatiles to the country rock; and influx of ground water from the country rock. The place of physical experimentation is to generate observations and data against which the theoretical models and textural interpretations can be constrained and validated. The author thanks J. C. Eichelberger, A. W. Woods and R. S. J. Sparks for their helpful comments on the manuscript.

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1995. Numerical study of nucleation and growth of bubbles in viscous magmas. Journal of Geophysical Research, 100, 1913-1931. VERGNIOLLE, S. & JAUPART, C. 1986. Separated twophase flow and basaltic eruptions. Journal of Geophysical Research, 91, 12 842-12 860. WALKER, D. & MULLINS, O. 1981. Surface tension of natural silicate melts from 1200°-1500°C and implications for melt structure. Contributions to Mineralogy and Petrology, 76, 455-462. WALKER, G. P. L 1981. Plinian eruptions and their products. Bulletin of Volcanology, 44, 223-240. WHITHAM, A. G. & SPARKS, R. S. J. 1986. Pumice. Bulletin of Volcanology, 48, 209-223. WILSON, C. J. N. & HOUGHTON, B. F. 1990. Eruptive mechanisms in the Minoan eruption: Evidence from pumice vesicularity. In: HARDY, D. A., DOUMAS, C. G., SAKELLARAKIS, J. A. & WARREN, P. M. (eds) Thera and the Aegean World III, Volume 2. The Thera Foundation, London, 105-113. WILSON, L. 1980. Relationships between pressure, volatile content and ejecta velocity in three types of volcanic explosion. Journal of Volcanology and Geothermal Research, 8, 297-313. & HEAD, J. W. III. 1981. Ascent and eruption of basaltic magma on the earth and moon. Journal of Geophysical Research, 86, 2971-3001. , SPARKS, R. S. J., HUANS, T. C. & WATKINS, N. D. 1978. The control of volcanic column heights by eruption energetics and dynamics. Journal of Geophysical Research, 83, 1829-1836. , & WALKER, G. P. L. 1980. Explosive volcanic eruptions - IV. The control of magma properties and conduit geometry on eruption column behaviour. Geophysical Journal of the Royal Astronomical Society, 63, 117-148. WOODS, A. W. 1995. The dynamics of explosive volcanic eruptions. Reviews of Geophysics, 33, 495-530. zZIMANOWSKI, B., FROHLICH, G. & LORENZ, V. 1991. Quantitative experiments on phreatomagmatic explosions. Journal of Volcanology and Geothermal Research, 48, 341-358.

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Gas loss from magmas through conduit walls during eruption C. JAUPART Universite Denis Diderot (Paris 7) and Institut de Physique du Globe de Paris, 4 place Jussieu, 75252 Paris Cedex 05, France Abstract: There is ample evidence that the transition from an explosive eruption regime to an effusive regime can be due to magma losing gas to fractured country rock during ascent towards the surface. This is shown by the deuterium: hydrogen isotopic ratios and dissolved water contents of erupted samples, and by various petrological observations. Field studies demonstrate that the walls of an eruption conduit are fractured and penetrated by veins infilled with vesicular pyroclasts and ash. At Mule Creek, New Mexico, USA, a fossil eruption conduit filled with lava can be studied over a height of 300m. The gas volume fraction exhibits complex variations with height and with horizontal distance from the walls, which is not compatible with closed system degassing. Lava close to the conduit walls is almost devoid of vesicles, showing that gas escape has been efficient. Gas flow through liquid magma may be achieved through fractures or through connected bubbles. Theoretical flow models which account for gas loss through conduit walls show that eruptive behaviour is very sensitive to the eruption rate and to the chamber pressure. A gradual decrease of chamber pressure, due to withdrawal of magma, leads to a transition from explosive to effusive conditions. Conversely, a gradual increase of chamber pressure, due to reinjection from a deeper source of magma, leads to a transition from effusive to explosive conditions. The initiation and propagation of gas-filled fractures during an ongoing eruption may be detected seismically. This chapter puts together several independent pieces of evidence in a coherent framework and includes a discussion of unresolved questions.

Nomenclature a Eruption conduit radius (m) GI Mass flux of magma per unit cross-section 'Effective' permeability coefficient for country rock (m 2 skg -1 ) Second coefficient of viscosity for bubbly magma (Pas) Pressure (MPa) Saturation pressure for volatile-rich magma (MPa) Country rock pressure (MPa) Mass flux of gas through permeable conduit walls (kgm -2 s-1) Radial coordinate within conduit (m) Average ascent velocity (ms - 1 ) Mass fraction of dissolved volatiles in magma Height above magma reservoir feeding an eruption (m) Volume fraction of gas in vesiculating magma First coefficient of viscosity for bubbly magma (Pas) Average density of magma-gas mixture (kgm- 3 ) Density of gas (kgm - 3 ) Density of magma (kgm - 3 )

Many volcanic eruptions exhibit an evolution from explosive phases, typically Plinian, to effusive phases where lava issues from the vent. This change of eruption regime is due to a decrease in the mass fraction of exsolved gas, such that fragmentation eventually becomes impossible. The opposite sequence is also common, and dome eruptions often lead to explosive regimes (Newhall & Melson 1983). Two explanations can be put forward for such changes of eruption regimes. One is that the magma chamber which feeds the eruption is stratified in volatile content. The other explanation is that, on its way to the surface, magma loses gas to the country rock. This second hypothesis was originally proposed by B. Taylor and J. Eichelberger in 1983 (Taylor et al 1983), and evidence in its favour has been accumulating steadily since. There now seems little doubt that it is true. The purpose of this chapter is to bring together research results from different disciplines and to discuss the implications of this phenomenon. Perhaps it is important to explain why the phenomenon deserves detailed scrutiny. First, it emphasizes the difficulty of predicting eruption conditions. Supposing that we know the magmatic conditions including volatile concentration in the chamber prior to eruption, we still are unable to calculate the gas content at the vent accounting only for pressure release of

JAUPART, C. 1998. Gas loss from magmas through conduit walls during eruption. In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 73-90.

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volatile-saturated magma. In other words, knowledge of the thermodynamic and kinetic aspects of degassing is not sufficient to assess eruptive behaviour. Conversely, as we shall see, the process of gas escape leads to important constraints on the pressures of volcanic flows and on the conditions of magma ascent. It therefore allows a useful test of flow models. Finally, changes of eruption regimes and gas contents are usually superimposed on changes of eruption rate. Such changes cannot be accounted for by shallow processes and provide a record of changing conditions in the whole volcanic system, including the magma chamber. Figure 1 illustrates the basic framework for this review and the important ingredients in the study of gas escape and changes of eruption regime. The volcanic system includes a magma reservoir whose pressure may vary with time as eruption proceeds. As magma rises towards the surface, volatiles species are exsolved from the melt and a gas phase forms. Some of this gas flows out of the conduit into country rock fractures. I shall first describe a few examples which emphasize that changes of eruption regime are not due to variations in the initial volatile content in the reservoir. I shall then review evidence for gas escape from ascending magma, using chemical and petrological analyses of eruption products as

well as large-scale studies of volcanic conduits in the field. Simple considerations on the dynamics of flow will be developed in order to define the key variables and to illustrate the relationship between the conditions of magma ascent and the eruption regime at the vent. Finally, I shall re-evaluate some observations in the light of this review and evaluate the key gaps in our understanding. This chapter focuses on eruptions involving evolved silica-rich magmas, and includes only a small discussion of basaltic eruptions. Out of necessity, evidence has been extracted from a small number of case histories. Our record is fragmentary and it is not possible to find in a single volcanic system all the complementary observations which would allow a definitive interpretation. Our aim is to emphasize that different pieces of evidence from a number of systems all point in the same direction. The conceptual model presented here is intended to provide a reference, and one must expect departures from it. Whilst the occurrence of gas escape from rising magma cannot be doubted, its importance depends on a host of variables and each eruption must be assessed individually. Changes of eruption regime and the gas budget of an eruption

A typical eruption sequence

Fig. 1. Illustration of the volcanic system studied and the various points reviewed in this chapter. The magma chamber pressure is not lithostatic and varies with time as magma is withdrawn from it or added to it. The conduit walls are fractured and the country rock is permeable, which allows gas flow out of the rising column of magma. Magma rising in the conduit experiences 'open system' degassing conditions, such that a fraction of the exsolved gas escapes.

The 1980-1986 eruption of Mount St Helens has been exceptionally well studied and offers important insights. On 18 May 1980, explosive activity was marked by a few alternations between Plinian columns and pyroclastic flow generation (Carey et al. 1990). Other minor Plinian phases followed until 12 June when a dacite dome was observed in the crater (Christiansen & Peterson 1981). A new phase of explosive activity destroyed this dome and eventually died down, leading to a second phase of dome growth. This dome was again destroyed by another explosive phase, which also subsided and made way for a third episode of dome generation. After this last episode Mount St Helens had only very minor explosive activity and issued lava for several years. Such alternations between explosive and effusive activity are not easily reconciled with the idea that the magma chamber is stratified in volatile content. At Mount St Helens, the mass discharge rate was about 107 kg s-1 in the initial explosive phases of 18 May 1980 (Carey et al. 1990) and decreased to about 10 4 kgs -1 when the eruption switched to dome growth (Swanson et al. 1987). The gas volume fraction of material erupted through the

GAS LOSS FROM MAGMAS THROUGH CONDUIT WALLS vent was much smaller in the latter, effusive, phases than in the former, explosive, phases. This indicates a correlation between mass discharge rate and gas content which is an important ingredient of the gas loss model. In quiet effusive phases which lasted for several years, the volcano discharged significant amounts of gas through fissures on the crater floor (Casadevall et al. 1981, 1983). Other features of this eruption hint at independent controls on the mass budgets of gas and magma. For example, in September 1980, the edifice went through a slow phase of swelling until strong degassing events involving almost no magma triggered deflation.

The initial volatile content of magma Mineral assemblages present in erupted melt allow the determination of the thermodynamic conditions prevailing in the chamber prior to eruption. Minerals such as amphibole are especially important because they are only stable in water-rich magmas. Yet, amphiboles may be abundant in samples taken from volatile-poor lava domes emplaced without fragmentation. Effusive and explosive products from the same volcano often contain the same population of minerals. Comparative studies of this kind have been done at Obsidian Dome, Inyo chain, California (Eichelberger et al. 1986; Westrich et al. 1988), at Mount St Helens (Hoblitt & Harmon 1993; Rutherford & Hill 1993) and in several other volcanic systems, most recently at Montagne Pelee, French Antilles (Martel 1996). Other evidence comes from melt inclusion studies. For example, peralkaline rhyolites involved in different eruption styles at Mayor Island, New Zealand, have the same initial water concentration (Barclay et al. 1996). These studies indicate that the eruptive regime may be independent of the initial volatile concentration of the melt.

Fracturing and degassing Field evidence demonstrates that, during eruption, heavily fractured volcanic edifices emit significant amounts of magmatic gas. Fissures are most active around the margins of domes, in the vicinity of the conduit system (Casadevall et al. 1981, 1983; Matthews et al. 1997; Stix etal. 1997). During dome growth at Mount St Helens, the gas flux from fumaroles and the eruption rate of magma followed the same decreasing trend as time progressed (Casadevall et al. 1981, 1983). The development of fractures in volcanic edifices and in lava can be monitored seismically. Lava dome eruptions are commonly accompa-

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nied by long-period and so-called 'hybrid' earthquakes, which have been attributed to resonating gas-filled cracks (Chouet 1996). There is a clear link between degassing and local seismic activity. For example, at Galeras volcano, gas jets breaking through the dome surface have generated long-period events (Gil Cruz & Chouet 1997). Many such events occur at shallow depth beneath domes, and hence are associated with the upper parts of the eruption conduit.

Summary: the gas budget of an eruption Many effusive eruption regimes involve magma which is initially volatile-rich, implying that significant amounts of gas do not get erupted through the main vent. The missing component is gas jetted by fissures through the volcanic edifice. Ideally, one should account for all the volatiles initially present in the magma, but this is difficult in practice. From the perspective of eruption dynamics, the important issue is whether gas is lost from relatively stagnant ponds of magma stored temporarily in the volcanic plumbing system, or from steadily rising magma. We shall see that the available data support the latter possibility.

Degassing processes during ascent in an eruption conduit D: H ratios in melts When water is exsolved from the melt and forms a gas phase, the different volatile species are not partitioned in the same way between gas and melt. This is true for the hydrogen isotopes, deuterium and hydrogen (Taylor et al. 1983; Newman et al. 1988; Dobson et al. 1989; Taylor 1991). During degassing, the melt sees its D: H ratio change as it loses its water because of mass balance constraints. The extent of contamination by meteoric water at a late stage may be assessed with the isotopic composition of oxygen. One may define a 'closed system' degassing evolutionary path in a D: H versus H2O diagram using equilibrium partitioning between gas and bulk melt (Dobson et al. 1989; Taylor 1991) (Fig. 2). This predicts a small decrease of the D: H ratio in the melt. The model is approximate because, in reality, equilibrium partitioning only applies at the gas-melt interface and because concentration gradients are set up in the melt. One may also consider a different evolutionary path, such that the gas phase is continuously extracted from the system. In this 'open' model,

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Fig. 2. Two degassing paths for magma in a D: H versus dissolved H2O diagram. D: H ratios are normalized to a standard value and are shown as 6D values. Closed system behaviour can be predicted accurately (Dobson et al. 1989). Open system behaviour is difficult to predict owing to the many processes involved (see text).

the gas phase which exsolves at any time is in equilibrium with the melt. However, this gas is lost and equilibrium partitioning is only achieved for each gas increment. This is analogous to a distillation process. The D: H ratio of the melt decreases more rapidly in this case than in the closed system one, and this produces a different evolutionary trend in a D : H versus H2O diagram (Fig. 2). Gas loss is achieved by diffusion through thin melt films or by flow through an open channel, and may not be complete. These processes do not affect H2O and HDO (water with deuterium) in the same way, and generate different D: H variations which have been documented at small water content (Anderson & Fink 1989). Open system degassing cannot be analysed quantitatively without a full physical model specifying the respective rates of exsolution, diffusion and gas escape, but it is useful to define an end member corresponding to total and instantaneous gas extraction. These ideas were applied to three young volcanic eruption sequences in California: the

Fig. 3. (a) D: H versus H2O for obsidian clasts in the AD 1340 Mono Craters eruption (from Newman et al. 1988: Dobson et al. 1989). Open circles represent samples from the pyroclastic deposit, which were erupted in explosive phases. Squares represent lava dome samples erupted in the waning stages of the eruption. The solid curve indicates the expected trend for equilibrium closed system degassing. The dashed curve shows the predictions of the instantaneous open system degassing model, (b) D : H versus H2O for various samples from Obsidian Dome. California (data from Taylor, 1991). Open circles: obsidian clasts from initial explosive phase. Open triangles: glass in pumices samples from initial explosive phase. Squares: glass samples from the dome emplaced at the end of the eruption. Stars: pyroclasts emplaced in deep fractures connected to the eruption conduit. The curve represents closed-system degassing for this system.

GAS LOSS FROM MAGMAS THROUGH CONDUIT WALLS Mono Craters, the Inyo chain and Little Glass Mountain. For these, it has been possible to determine the evolutionary path of magma for both explosive and effusive phases using obsidian clasts found in pyroclastic deposits and lava flows. Data from the Mono Craters and Little Glass Mountain eruptions have been explained by a two-stage degassing process (Fig. 3a) (Newman et al 1988; Dobson et al 1989). In these two eruptions, the initial explosive stages are characterized by closed system behaviour, contrary to the dome-building phases which exhibit open system characteristics. The transition from closed to open system degassing coincides with the change from explosive eruption to dome growth. Data from the Obsidian Dome eruption in the Inyo chain reveal more complex behaviour (Taylor 1991) (Fig. 3b). In the explosive phases, a few samples can be described exactly by a closed system model, but the majority show some degree of 'open' behaviour. All samples from effusive phases deviate significantly from closed system behaviour. The data at Obsidian Dome suggest increasingly 'open' degassing conditions as the eruption progressed. Such studies are, by definition, only possible because samples with different amounts of dissolved H2O have been preserved. This is true for both obsidian clasts and pumices. There are no magma samples from the deepest part of

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the conduit where the dissolved water content was large and where the gas volume fraction was small. In the Obsidian Dome system, the initial water content of magma prior to eruption was 4.0-4.1wt%, but no obsidian clast with more than 1.8wt% dissolved water was found (Westrich et al. 1988; Hervig et al 1989; Taylor 1991). Interestingly, in this magma, assuming closed system equilibrium degassing, the 1.8wt% dissolved water content is achieved at 60% gas volume fraction. This value is a reasonable threshold for the onset of fragmentation (Gardner et al. 1996). In other words, there seems to be no magma sample from below the fragmentation level. The same remark applies to the Mono Craters, where the maximum water concentration in obsidian clasts is 2.7 wt% and the initial water content of the magma is inferred to be about 5 wt% (Newman et al. 1988). A related study was made on Mount St Helens 'cryptodome' dacites. These are samples of relatively dense lava emplaced at shallow depth within the volcano which were rapidly decompressed when the 18 May explosion violently decapitated the edifice. Hoblitt & Harmon (1993) found that these samples had a large range of vesicularities. All have little water left in solution and had been degassed during decompression before reaching their level of emplacement. Their D: H and H2O concentrations indicate open

Fig. 4. CO2 versus H2O for obsidian samples erupted during the AD 1340 eruption of Mono Craters, California (data from Newman et al. 1988). Open circles represent clasts from the initial explosive phases and filled circles stand for dome samples. The two solid curves correspond to closed system degassing for two different initial CO2 contents (2.4 and 1.2wt% for curves 1 and 2, respectively) and the same initial water content (5.0 wt%). The solid and dashed curves, both labelled (2) and issuing from the same point, illustrate closed system and open system degassing paths for the same initial volatile contents (5.0wt% H2O and 1.2wt% CO2).

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system degassing conditions. Several samples are almost vesicle free, indicating that magma degassed and then lost its gas bubbles at depth. These magma fragments rose to their level of emplacement in non-explosive conditions, probably with small ascent velocities, and hence are analogous to obsidian clasts from effusive phases at Mono Craters and Obsidian Dome.

H2O: CO2 systematics A second method of tracking degassing conditions is to look at the H2O and CO2 concentrations of quenched melt in erupted samples. The principle is the same as before. When magma of a given initial composition of water and carbon dioxide is decompressed, a gas phase is generated which contains both species (plus some minor phases). CO2 is strongly partitioned in the vapour phase. Thus, in an open system, CO2 is degassed very efficiently and the CO 2 : H2O ratio of the melt decreases rapidly (Newman et al. 1988). In a closed system, the CO2/H2O ratio of the melt decreases slowly with decreasing H2O content. The observed trend in the pyroclastic phase of the AD 1340 Mono Crater eruption is near closed system behaviour (Fig. 4), which confirms the conclusions reached from the D : H data.

Reaction rims in hydrous phenocrysts Certain minerals are only stable in hydrous melts, and hence can only grow if pressures are sufficient to keep enough water in solution. If the melt loses its dissolved water because of decompression, those minerals are no longer stable and react out. One well-studied case is that of amphibole in Mount St Helens dacites (Rutherford & Hill 1993). At Mount St Helens, the very presence of amphibole determines the minimum amount of dissolved water in the melt in the reservoir. For this particular magma, the water concentration had to be larger than about 4wt% (Fig. 5). Amphiboles are seen both in pumices produced by explosive phases and in dome samples. A second constraint is the occurrence of reaction rims, which show that the melt surrounding the amphiboles was not in equilibrium with them when it was erupted. This indicates that the melt saw its water concentration decrease, as expected from solubility constraints. Reaction features are observed in dome samples, i.e. in magma erupted in an effusive regime. Thus, magmas from non-explosive phases initially had water content similar to those from explosive

TEMPERATURE (°C) Fig. 5. The stability field of amphibole for Mount St Helens dacite for PH2O = Ptotal and for specific oxydation conditions (log fO2 = NNO + 1) (adapted from Rutherford & Hill 1993; Geschwind & Rutherford 1995). The straight line illustrates a schematic ascent path which leads to amphibole destabilization. Also shown are the boundaries for plagioclase crystallization. Curves labelled Plag. 'An48 and An40' indicate the onset of plagioclase nucleation and the compositions of plagioclase crystals in equilibrium with the melt. These three curves indicate that the degassing of magma leads to the precipitation of plagioclase microlites.

phases. These magmas started exsolving water at depth, as demonstrated by the resorption rims, but this vapour phase is not found in the erupted magma. The logical conclusion is that this vapour phase escaped during ascent. A third constraint comes from the kinetics of amphibole resorption. Amphiboles in Plinian pumices exhibit no resorption features, indicating that not enough time was available for the reaction to proceed. Reaction rims in dome samples imply a larger time for the reaction and hence a slower rate of ascent. Reaction rims provide quantitative estimates on the timing of magma degassing at Mount St Helens. For times smaller than about 4 days, the reaction leaves no detectable trace. The thickness of reaction rims indicates that the amphiboles did not stay for more than about 10 days outside their stability field. These estimates are close to independent determinations derived from eruption rates and geophysical data, which shows that melt starts degassing at depths corresponding to the amphibole stability limit (at a pressure of about 160 MPa). This agreement also shows that magma does not get stored for any significant length of time within the volcano plumbing system.

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Micro lites Crystallization may be triggered by loss of volatiles and the implied changes of phase relationships (Fig. 5). This provides an additional constraint on the extent of degassing. In practice, one must unravel the roles of the two thermodynamic variables which act on crystallization kinetics: temperature and water content. This has been attempted for the Inyo domes (Swanson et al. 1989). Obsidian clasts found in tephra deposits and erupted during explosive phases bear few microlites, whereas samples from domes have many microlites. Once again, two different kinds of information are available. One is that melt from the dome lost large amounts of water in order to grow abundant microlites. The other is that samples erupted at large velocities got quenched before the onset of groundmass crystallization. There are therefore both kinetic and thermodynamic aspects to this problem. Similar observations were made in samples from the 1980-1980 eruption of Mount St Helens (Cashman 1992; Geschwind & Rutherford 1995). As dome building progressed with a general trend of diminishing mass flux, plagioclase microlites gradually became larger and richer in albite. These features may be reproduced by decompression experiments involving the same melt with the same initial water content (Geschwind & Rutherford 1995). The changing characteristics of the microlites may be attributed to decreasing ascent rates, and there is a good correlation between the eruption mass flux and microlite composition. This shows again that melt batches involved in effusive eruption phases at Mount St Helens degassed large amounts of water as they rose through the conduit.

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et al. 1988; Westrich et al. 1988), and thanks to erosion in the Mule Creek vent, New Mexico (Stasiuk ef a/. 1996).

Obsidian Dome, California Some observations from this system have already been discussed above. Drill holes have been made allowing study of a feeder dyke and the eruption conduit, and their relationships with the country rock. One important observation is that the conduit walls are fractured and that the fractures contain vesicular pyroclasts whose composition match those of the explosively erupted samples (Heikenet al/. 1988; Westrich et al. 1988). Furthermore, these pyroclasts are significantly degassed. At depths of 290 and 370 m, the glassy matrix of pyroclasts has about 1.1-1.2 wt% dissolved water, much less than the initial water content of this magma, 4.0-4.1 wt%. Additionally, the pyroclasts have very low vesicularity, implying that the gas phase which was present at some stage, and which is required by the low residual water contents of the glass, was lost. Heiken et al. (1988) explain this by bubble collapse. The D: H and H2O contents of these pyroclasts are consistent with open system degassing (Taylor 1991) (Fig. 3b). A final observation is that these pyroclasts have microlites, contrary to the same type of samples collected from the tuff ring at the surface. By analogy with the erupted samples studied by Swanson et al. (1989), this suggests that the 'intrusive' pyroclasts were emplaced during the late effusive regime, and provide markers for the gas escape process. The key observations are that fractures were available in country rock around the eruptive conduit, providing passageways for gas to escape. Pyroclasts found at depth have lost some of their volatile species as well as their gas bubbles.

Field studies We have summarized various lines of evidence for open system degassing which are all derived from petrological and chemical data on individual eruption products. One may argue that a few fragmented samples may not be representative of the whole volcanic system, and hence it is important to verify that the required mechanisms operate on a large-scale natural system. Two conditions must be met: one must be able to observe an eruption which went through a change from explosive to effusive conditions; and one must be able to study the conduit system at depth, preferably over a significant depth range. This has been possible through drilling in Obsidian Dome, California (Eichelberger et al. 1984, 1986;Heiken

Mule Creek vent, New Mexico In Potholes Country, New Mexico, erosion has exposed a rhyolite vent and its contact with country rock (Ratte & Brooks 1989; Stasiuk et al. 1996). An eruption produced pumice fall deposits and a lava dome, and the total vertical extent of the rhyolite outcrop is about 300m. This allows examination of the eruption products at the surface, of the final flow which fills up the conduit, of the contact between lava and country rock and of the country rock itself away from the conduit. At depth, going from the country rock to the conduit interior, one may see the following sequence (Fig. 6). The country rock is fractured

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Fig. 6. A summary of the main lithologies seen at the walls of the Mule Creek conduit (after Stasiuk el al. 1996). A - veins connect rhyolite breccia to country rock. B - pyroclastic breccia due to initial explosive phase. C - vitrophyre breccia associated with the effusive phase. D - dense vitrophyre. E - vesicular rhyolite with thin flow banding made of aligned and partially coalesced bubbles. F - tuffisite veins within the rhyolite, which may have acted as passageways for vertical gas escape. and invaded with thin veins of fine autoclastic ash. These so-called 'tuffisite' veins can be traced through the contact but do not extend far into the flow interior. They contain some small (millimetre sized) angular fragments of vesicular rhyolite. Against the country rock, there is a breccia including small, weakly vesicular, pyroclasts which can be traced into the lowermost parts of the pyroclastic deposit. This is interpreted as material emplaced during the initial and explosive phase of the eruption. Next is a vitrophyre breccia made of rather large blocks (centimetre sized) of glassy rhyolite, interpreted to be an autobreccia formed later in the eruption, presumably when the dome was being extruded. A dense vitrophyre layer is found between these thin breccia deposits at the walls and the massive lava filling up most of the conduit. This study shows that the conduit walls are far from straight and smooth. These walls were permeable and were not lined with dense and impermeable glass, which would have prevented gas leakage. Massive glassy rhyolite is found at some distance from the country rock and was emplaced late in the eruption sequence. The veins in the country rock can be likened to those described by Heiken el al. (1988) in the vicinity of the Obsidian Dome conduit. The observations at Mule Creek further demonstrate that the conditions for magma degassing are not as predicted by one-dimensional flow models. These models predict that flow pressures,

and hence the gas volume fraction, depend on height only (Jaupart & Tait, 1990). The measurements at Mule Creek show that the magma initially contained about 2.5wt% H2O. Values of gas volume fraction in the magma should have been larger than 70% over the whole outcrop if equilibrium degassing occurred. This is not observed anywhere. The massive glassy margins have lost all their gas bubbles. A key observation is that the gas volume fraction is not uniform at a given level in the exposure (Fig. 7). The bubble content is zero against the wall and increases towards the centre of the conduit. One may deduce that gas bubbles near the walls were able to escape and that a horizontal gradient of gas content, and presumably of gas pressure, was established to drive gas out of the rising magma. One is reminded of the 'cryptodome" samples from Mount St Helens, which exhibit quite a large range of vesicularities and include almost vesiclefree lava (Hoblitt & Harmon 1993). The variation of gas content with height is also peculiar. The bubble content exhibits a weak tendency, if any, to increase with decreasing pressure, i.e. with increasing height. This trend does not fit any simple closed system pressure release model. One simple interpretation is that magma lost some of its gas. If this is true, the data also show that gas escape must have occurred below the exposure level. Networks of tuffisite veins are also found in the flow interior. Some of these veins have been

GAS LOSS FROM MAGMAS THROUGH CONDUIT WALLS

VOID FRACTION Fig. 7. Void fraction as a function of depth in the Mule Creek rhyolite (from Stasiuk et al. 1996). Open circles represent samples from the glassy margins. Solid triangles represent samples from the devitrified rhyolite in the conduit interior. The void fraction increases away from the conduit walls. There is also an irregular increase of void fraction with decreasing depth. The solid curve shows the expected void fraction for closed system degassing of this rhyolite, which had an initial water content of 2.5 wt%. The dashed curve corresponds to expansion of pre-existing bubbles without new exsolution. Note that neither of these curves accounts for the measured values of void fraction.

folded by viscous flow showing that they developed whilst magma was still liquid. Such veins have also been described in samples from active domes, for example at Lascar volcano, Chile, and at the Soufriere Hills volcano, Montserrat (Sparks 1997). These veins acted as passageways for gas through the lava. Physical models of volcanic eruptions

General framework A volcanic eruption involves a complex plumbing system which connects a deep magma reservoir to the surface. Recent reviews on the fluid dynamic aspects of magma ascent are available (Jaupart & Tait 1990; Woods 1995; Mader 1998), and we shall restrict our description to a few basic and salient points. The characteristics of flow are determined by the physical properties of magma, the amount of volatiles dissolved in the melt, the kinetics of degassing and the dimensions of the plumbing system (including the reservoir and the eruption conduit). Knowledge of these variables is not sufficient for quantitative predic-

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tions of ascent conditions. One must specify the pressure in the reservoir and the exit pressure. In an explosive eruption, the exit pressure may not be atmospheric and the flow may be choked, such that the exit condition is not one of pressure, but one of velocity. Thus, the exit pressure cannot be considered as fixed and must be solved for. The same is true for the reservoir pressure, which may vary during eruption because the reservoir may be emptied or refilled. This implies that one cannot specify at once the pressure variation experienced by ascending magma. A necessary condition for rising magma to lose gas is that it must be at a higher pressure than the country rock. During eruption, the magma pressure depends on the dynamics of flow and may not be fixed a priori. This emphasizes that the feasibility of gas escape must be assessed within the framework of flow dynamics.

The dynamics of gas loss through country rock In most eruptions flow conditions vary over time scales which are much longer than the rise time from chamber to vent and hence the flow may be treated in the steady-state approximation. Consider a magma-gas mixture rising with velocity w in a cylindrical conduit of constant radius a at constant temperature. Let x denote the mass fraction of volatiles dissolved in the liquid, a and pg the volume fraction and density of exsolved gas and p\ the magma density. Quantities are horizontal averages across the conduit and conservation equations are satisfied in integral form, leaving out the finer details which cannot be specified reliably. The mass flux of liquid per unit area, G1, is equal to G\ = p\(\ - )w.

(1)

In steady state, mass conservation dictates that G\ is constant, and hence that increasing the gas volume fraction leads to a velocity increase. We assume that gas and lava rise with the same velocity and that kinetic aspects of degassing can be neglected. Magma is saturated at pressure Pi. If there is no loss of gas to the conduit walls, the gas volume fraction at pressure P is (2)

Consider now that a mass flux of gas, q, is lost through the conduit walls. At an arbitrary

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Fig. 8. Diagram illustrating the mass balance of gas for an ascending magma which loses gas to the country rock. Variables to be solved for in a dynamic flow model are average values of the ascent velocity, w, mass fraction of dissolved gas, x, volume fraction of exsolved gas, a, and flow pressure P. The mass flux of gas through the permeable walls into country rock, q, depends on permeability and the pressure difference between the flow, at pressure P, and the far-field, at pressure PL.

height z in the conduit, mass conservation for the gas molecules in the conduit reads (Fig. 8)

where allowance is made for both dissolved and exsolved gas. In this expression the horizontal mass flux of gas, q, may depend on z, for example due to increased country rock permeability towards the Earth's surface. Using equations (1) and (3), we find

(4a)

(4b) where A is positive and represents the cumulative effects of exsolution (dx/dP) and expansion (d g/dP). The volume fraction of gas is determined by two competing mechanisms: pressure release and gas loss. The gas loss term is inversely proportional to the eruption rate, and this is the key feature of this model. Note that, contrary to the equilibrium model specified by equation (2), the volume fraction of gas is not given as a function of pressure, but is involved in a differential equation which depends on the flow rate. Gas is lost to fractures in immobile country

rock, at a rate which depends on permeability and on the pressure gradient through the fractures. If magma rises fast, the amount of gas lost is small compared to the amount which flows past the fractures. If magma rises slowly, the proportion of gas lost becomes larger. The flow rate and the evolution of gas content are coupled, and this is responsible for quite complicated behaviour. This simple theory ignores one essential process, that which allows gas from the flow interior to reach the walls. There are several candidates for this process, which are discussed below and which involve different physical principles. The equations developed here are not complete and are used only to investigate basic aspects of gas escape in a dynamical framework. To proceed further, two additional equations are needed: one for the flux of gas through the conduit walls; and the other for the conservation of vertical momentum. At depth z the rising mixture is at an average pressure, P, and country rock far from the conduit is at a different pressure, PL. q, the mass flux of gas through the conduit walls, may be written as

where k is some 'effective' permeability coefficient, q is proportional to the gas volume fraction a, in order to account for the fact that only a fraction of the conduit wall is in contact with the gas phase. If a is zero, the rising mixture contains no gas and cannot lose any. An important point is that the value of the pressure contrast (P - PL) is not assumed but must be calculated as part of the solution. In other words, the occurrence of gas escape is not imposed a priori in this model. Two different assumptions have been made on the farfield pressure, PL. Jaupart & Allegre (1991) took lithostatic values, whereas Woods & Koyaguchi (1994) took hydrostatic values. The former corresponds to fissures induced by the overpressured magma. The latter implies connection to a preexisting hydrothermal system and leads to larger rates of gas flow. Several important conclusions can be gained from these schematic models. One is that the conditions required for gas escape may indeed be achieved, i.e. pressures in the rising magma may exceed pressures in adjacent country rock. A second conclusion is that the solutions are extremely sensitive to small changes in the initial conditions in the chamber. One may go from a gas-rich and rapid eruption to a gas-poor and sluggish flow. Furthermore, one may find two different solutions for given values of pressure in the chamber and at the vent. This suggests that conduit flows may be inherently unstable, and

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probably relevant to the ongoing eruption of the Soufriere Hills volcano, Montserrat (Young et al. 1997).

Bubbly liquids: flow conditions and decompression rates

AP (MPa) Fig. 9. Plot of gas volume fraction at the vent as a function of chamber pressure for the dynamic gas loss model of Jaupart & Allegre (1991). In these calculations, all variables other than the chamber pressure are kept at the same values. With decreasing chamber pressure, the amount of gas lost by ascending magma become proportionately more important and eventually leads to effusive eruption conditions characterized by a gas volume fraction below the fragmentation threshold.

may explain the observed alternations between Plinian columns, pyroclastic flows and dome eruptions. A fourth conclusion is that the chamber pressure is a key variable (the 'control' parameter in mathematical terms). One possible situation is that the chamber pressure decreases steadily with time during an eruption because the chamber is being emptied. This is shown by an analysis of several eruptions (Jaupart & Allegre 1991; Stasiuk et al 1993). The implication is that as the chamber pressure decreases the eruption rate and the eruption velocity decrease which in turn implies that the proportion of gas lost becomes progressively more important (Fig. 9). This predicts a trend of decreasing gas content at the vent, which leads to a transition from Plinian conditions to pyroclastic flows and then to dense lava eruption. Such a trend has been observed in many instances, for example at Mount St Helens in 1980. Another possibility is that the chamber pressure increases due to reinjection of magma from a deeper source. In this case one may expect the opposite sequence: an accelerating eruption with increasingly less efficient gas escape. This is

Theory and experiments on the flow of bubbly liquids are only available in the dilute limit and in the 'dry' limit where the volume of liquid is negligible. Thus, much remains to be learnt on volcanic flows at depth (Mader 1998). Here, we briefly discuss two aspects of relevance to volcanic eruptions. The picture commonly adopted for conditions below the fragmentation level is that of decompression in a laminar flow (e.g. Jaupart & Tait 1990; Woods 1995). In such models, at any level within the conduit there are no horizontal variations of pressure and the bubbles are homogeneously distributed within the flow. These two points deserve discussion because they bear directly on the measurable characteristics of eruption products, such as gas content and vesicle size. A magma parcel which rises in a conduit is subjected to shear. In these conditions, bubbles tend to migrate laterally within the flow and may not be homogeneously distributed (Herringe & Davis 1976; Cox & Hsu 1977). Furthermore, they tend to align themselves and form bubble chains parallel to the flow. Flow banding of this kind is common and has been described at Mule Creek (Stasiuk et al. 1996). This is important not only for the flow behaviour, but also for the mechanisms of gas escape. At small Reynolds numbers the flow of a vesicular liquid should be of the Poiseuille type. Assuming steady state and a vertical conduit, the decompression rate experienced by a rising parcel of magma is given by (5)

where P is the flow pressure and w the vertical velocity, which varies from zero at the wall to a maximum value in the centre. Thus, decompression rates are much larger for magma rising at the centre compared to near the walls. This has two consequences when account is taken of the kinetics of degassing, involving diffusion-limited growth of bubbles and delayed expansion due to viscous stresses (Jaupart 1992). Both nonequilibrium effects lead to the same result, which may be understood by comparing the rate of ascent with the rates of diffusion and bubble expansion. For example, a bubble growing in

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magma which flows near the conduit walls experiences a small decompression rate which allows thermodynamic equilibrium between bulk melt and gas. For magma rising near the conduit centre, diffusion may become a limiting factor. Stated differently, the importance of kinetic effects is different in different parts of the conduit. One consequence is a decrease of the mass fraction of exsolved gas, and an increase of gas pressure, towards the conduit centre. The same pressure effect can be established mathematically in a revealing manner for the other effect retarding bubble growth: viscous resistance to expansion. The proper rheological relationship between shear stress and strain rate for a bubbly liquid is (Taylor 1954) (7)

where T is the stress tensor and P the thermodynamic pressure, i.e. gas pressure, e and i are the strain rate and identity tensors, respectively. is the first coefficient of viscosity. The second viscosity coefficient K has been called dilatational viscosity and is proportional to the melt viscosity (Taylor 1954; Prud'homme & Bird 1978). Both and K vary as a function of void fraction. To investigate the behaviour of such a liquid during decompression, we consider again a cylindrical eruption conduit with constant radius a. In steady-state conditions, the equations of motion reduce to

(8a)

(8c)

where r is the radial distance from the axis of the cylinder and p is the density of the rising mixture. For the sake of argument, we assume that the viscosity coefficients are constant. Equation (8b) can be integrated directly with respect to using the boundary condition that w(a, z) = 0

(9)

This equation shows that the gas pressure varies across the conduit and reaches a maximum at the centre. This is because velocity and decompression rate are largest at the centre, implying that the retarding effect of viscosity on bubble expansion is largest there. The magnitude of the horizontal pressure variation depends on the eruption rate (Jaupart 1992). In explosive eruptions, the horizontal variation of pressure may exceed 10 6 Pa, which is significant. Both types of kinetic effects on degassing lead to the same result: a decrease of bubble pressure from the conduit axis to the walls. Such a pressure gradient is that which is needed to drive gas horizontally out of the flow. Another prediction is that the most gas-rich parts of the flow are located at the margins.

The conditionsofofmagma magma ascent The conditions ascent in ain a volcanic conduit volcanic conduit Micro lite growth In magma which is being decompressed, the loss of dissolved water acts to displace the phase boundaries. This promotes the crystallization of anhydrous minerals, for example plagioclase and pyroxene microlites in Mount St Helens dacite (Geschwind & Rutherford 1995). In turn, the growth of these anhydrous minerals acts to enrich the residual melt in dissolved water and hence to promote further degassing. Sparks (1997) has recently suggested that this acts to pressurize gas bubbles and ultimately to generate gas-filled fractures in the melt. Another effect of microlites is to modify the rheological properties of the material surrounding melt and gas bubbles. At large crystal fractions, bubble expansion is strongly impeded by the presence of crystals (Klug & Cashman 1994), which would further enhance gas pressures during decompression. A limiting case is that of gas bubbles suspended in almost rigid material. A final effect of microlites is to generate contorted melt regions, which may promote bubble connection at small gas fractions. Microlite growth must be discussed with kinetic constraints in mind. Perhaps the key observation, reported above, is that microlites are rare in Plinian samples which had large ascent velocities, and may be abundant in dome samples which rose more slowly. This has been reported for the 1980-1986 Mount St Helens eruption, the Obsidian Dome eruption and more recently for the PI Montagne Pelee eruption (Martel 1996). Thus, the onset of microlite growth is not in itself the primary cause of

GAS LOSS FROM MAGMAS THROUGH CONDUIT WALLS eruption changes. One must first explain why an eruption slows down such that sufficient time is available for microlite growth. Pumices from the late 18 May 1980 Plinian phases of Mount St Helens have interesting features (Klug & Cashman 1994). 'White' pumices have reached large vesicularity values of up to 86% and have no microlites. In contrast, 'grey' pumices, which are compositionally identical, have abundant microlites and contain fewer bubbles. This provides evidence for different degassing histories in the same magma batch. Klug & Cashman (1994) propose that microlites act to increase the magma viscosity and hence to slow down expansion. However, the key problem is to explain why there are microlites in one case and none in the other. One may first assume that all samples were subjected to the same decompression rate. As a consequence, the different microlite contents can only be attributed to different amounts of water exsolution. For the expanded microlite-free samples, one may imagine closed system behaviour with isolated bubbles within magma. In this case, the gas pressure remains larger than the bulk flow pressure (Jaupart 1992). For the less vesicular microlite-rich samples, one may imagine open system behaviour with gas being continuously lost from the expanding bubbles. This leads to gas pressures which are close to the

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bulk flow pressure, and hence to more exsolution than in closed system conditions. This would trigger microlite crystallization, and yet would not lead to more expansion because of open behaviour. This explanation seems a bit contrived, however, because it requires specific conditions for two independent processes: bubble overpressure and partial gas escape. A simpler explanation is that all the samples did not rise with the same velocity in the conduit. In this way, different pieces of the same magma batch experience different decompression rates, which lead to different exsolution rates and to different amounts of crystallization. Such ideas are consistent with the physical considerations developed above and with the deductions drawn from field studies.

Obsidian clasts in explosive eruptions Obsidian clasts found in the pyroclastic deposits of the Mono Craters and Little Glass Mountain, California, deserve special mention because they provide information on conditions in an eruptive conduit during an explosive event (Taylor et al. 1983; Newman et al 1988). Their dissolved H2O and CO2 contents follow a general decrease with increasing height in the deposit, corresponding to increasing time during eruption (Fig. 10). There is

Fig. 10. Dissolved water content of erupted samples as a function of stratigraphic level in the deposit of the AD 1340 Mono Craters eruption (data from Newman et al. 1988). Open circles stand for obsidian clasts expelled by the explosive phases and filled circles stand for dome samples. Note the range of values at a given level in the deposit and the general decreasing trend which tends towards the dome values.

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an almost continuous evolution towards dome samples, such that the dissolved volatile contents of the clasts become very small. There is not a one to one relation between dissolved volatile contents and stratigraphic height, but the overall trend is impressive and similar in both deposits studied. Obsidian clasts have been found in other pyroclastic deposits, including the Glass Creek and Obsidian Dome deposits of the Inyo volcanic chain, California and the Big Obsidian Flow vent, Newberry caldera, Oregon (Taylor et al. 1983; Taylor 1991). Two interpretations may be proposed, which attempt to account for three main features: (1) these dense clasts have not retained any gas bubbles, and yet their D : H and CO2 versus H2O systematics indicate that they degassed in almost closed system conditions; (2) their dissolved H2O and CO2 contents gradually decrease with time in the eruption sequence; and (3) their volatile contents tend towards those of dome samples. In one interpretation, obsidian clasts represent samples of a chilled margin which have been reamed out by the explosive eruption. This interpretation is not consistent with the lithologies observed along the Mule Creek conduit, where the dense glassy margins are a late feature related to the effusive phase of the eruption. Furthermore, it provides no explanation for the trend of decreasing volatile contents through the deposit. The alternative interpretation is that these clasts are fragments of the same magma batch which produced the pyroclastic deposit. In this case, one must envisage a two-stage degassing evolution. Degassing first proceeds to the preserved volatile content in essentially closed-system conditions. Second, there must be a sudden event which stops the degassing process and allows the fragment to preserve relatively large volatile contents. The clast may or may not have been a piece of vesicular magma. If it was vesicular, its network of vesicles collapsed entirely, which may occur rapidly and leave no visible trace (Westrich & Eichelberger 1994). Alternatively, the clast may have been an isolated pocket of dense melt which experienced volatile depletion through diffusion towards bubbles. In both cases, a rapid change of degassing conditions is required and a logical candidate is fragmentation itself. The gradual decrease of preserved volatile content through the deposit thus reflects a gradual rise of the fragmentation level as the eruption proceeds. This sequence implies that the fragmentation level eventually reaches the surface, at which point one predicts that fragmentation should cease and hence that the eruption should switch to dome formation. This is precisely what is observed (Newman et al. 1988).

Summary A few general facts emerge. Obsidian clasts record imperfect degassing conditions which can be described neither by closed system conditions nor by perfect open system behaviour. They also exhibit quite a large range in characteristics at any given level in a deposit, which provides evidence for heterogeneity in degassing conditions. Similar conclusions were reached from the characteristics of Mount St Helens pumices and from a discussion of the physics of decompression during conduit flow. It is likely that, at a given level in a conduit, neither gas pressure nor exsolved gas content are horizontally uniform. Gas bubbles are probably not homogeneously distributed within magma and some melt pockets may be weakly open to the flow of gas.

Discussion The mechanisms of gas loss The observations demonstrate that during eruption ascending magma loses gas before reaching the surface. Only magma which lines the conduit walls is directly connected to country rock fractures, and yet gas escape must affect the magma in bulk. We now discuss possible mechanisms, none of which has been studied in enough detail. One possibility is that magma becomes permeable. This mechanism can only be relevant if permeable bubbly magma is able to flow without fragmenting. Eichelberger et al. (1986) proposed that vesicular magma develops permeability when it reaches a gas volume fraction of about 60%, less than the 70-75% values commonly adopted for fragmentation (Woods 1995). They evaluated this hypothesis with permeability measurements on samples of vesicular lava taken from the field and with laboratory experiments under controlled conditions (Westrich & Eichelberger 1994). The specific threshold vesicularity value of 60% corresponds to random packing of spherical gas bubbles with a limited range of sizes. One difficulty is that some of the samples produced in the laboratory did not become permeable until their vesicularity values were significantly larger than 60%, indicating that there is no uniformly valid threshold. These laboratory experiments involved expansion only, without any shear. Another difficulty is that shear flow does disrupt liquid with closely packed bubbles (Li et al. 1995). One way out of this problem is that bubbles become aligned in chains due to shear, which allows bubble connection at lower values of the gas volume fraction. We have seen evidence for

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this in the Mule Creek lava. However, such bubble chains are parallel to the flow direction, and hence do not lead to cross-flow permeability which is required for gas escape at the walls. A second mechanism is the generation of fractures in overpressured magma, as observed within the Mule Creek flow and elsewhere (Stasiuk et al 1996; Sparks 1997). Such gasfilled fractures probably propagate rapidly and can transport volatiles to the walls in a short time. This mechanism would not be affected by the limitations of the permeable foam flow model. Rapid fracturing would be weakly affected by shear and would not induce bulk fragmentation. A third mechanism may operate if the flow regime is not laminar. As discussed above, the dynamics of bubbly flows are not understood with the necessary degree of generality. Such flows may develop instabilities and may generate eddies which would bring magma into contact with the conduit walls. Such an internal circulation could also be set up by wall roughness.

pyroclastic deposits and below the lowest level of exposure. A vent funnel is not really a favorable location for gas escape because it widens upwards. This acts to decrease the contact area in proportion to magma volume flux and to increase the eruption velocity (Wilson el al. 1980), both factors which act to diminish the importance of gas escape for the flow. Country rock permeability need not exist before eruption starts, and may develop because of eruption. Fracturing may be induced by overpressured magma, as suggested by the field observations and by seismic events. Another factor is wall roughness. Most flow models assume smooth conduit walls, and the contact area is directly calculated as a function of the conduit gross dimensions. In reality, it is quite likely that conduit walls are contorted, as shown by the Mule Creek exposure. This acts to increase the contact area and to generate some horizontal variations of pressure in the flow. Additionally, this leads to mixing, even in a laminar flow regime.

Textural evidence for gas escape

General significance of gas escape

One objection to the permeable foam model has been that one should find textural evidence for bubble connection. However, Westrich & Eichelberger (1994) have demonstrated in the laboratory that magmatic foams may develop permeability, lose their gas and completely collapse without leaving any trace of the bubbles they once contained. This may account for the very dense glassy margins found at Mule Creek, and for the generation of obsidian clasts in explosive eruptions. The same annealing phenomenon may obliterate the traces of gas-filled fractures in liquid magma.

The transition from Plinian column to pyroclastic flow may be due to a decrease in gas content of the erupting mixture (Sparks & Wilson 1976; Woods 1995), and hence to gas escape to the country rock, as discussed in Jaupart & Allegre (1991). At both Vesuvius in AD 79 and at Mount St Helens in 1980, pyroclastic flows were not generated immediately. According to classical degassing models and to the volatile contents of the magmas involved, predicted values for the amount of gas in the erupting mixture are too large to allow pyroclastic flows (Carey et al. 1990). Gas escape during ascent provides a solution to this discrepancy.

The conduit walls For gas to escape from ascending magma, the conduit walls must be fractured and the country rock must be permeable. Eichelberger et al. (1986) reasoned that the transition to effusive behaviour occurs late in the eruption sequence. They suggested that, at that time, magma flows through an accumulation of pyroclastic material deposited earlier. They therefore proposed that gas is lost in this shallow porous environment, in what they called the 'vent' funnel. This is not sufficient, as appreciated by Fink et al. (1992). We have seen at Mule Creek that gas loss started below the

Basaltic eruptions Most of the evidence for gas escape has been obtained for evolved silica-rich magmas, partly because these magmas are involved in the most spectacular eruptions and partly because many of them have large volatile contents. Some features of basaltic eruptions, however, may be related to the same phenomenon. For example, at Kilauea volcano, Hawaii, sustained summit eruptions are significantly richer in CO2 than flank eruptions through the rift zones (Gerlach & Graeber 1985). In the case of Hawaii, the

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effect of gas escape at depth is most visible for CO2 because it is present as a gas phase through most of the magma path, contrary to H2O which only exsolves at shallow depth. The behaviour of Kilauea has been explained by a two-stage process. According to Gerlach & Graeber (1985), magma involved in flank eruptions gets stored in the summit reservoir and loses some of its CO2 gas there before being injected into a rift zone. During eruption, however, active fumaroles are clearly visible in the rift zone and hence there is no reason to assume that gas escape is only significant during storage in the summit reservoir. One important difference between summit and rift-zone eruptions is the duration of flow beneath the surface and it is clear that near horizontal flow in the rift-zone provides plenty of opportunities for CO2 loss. Conclusion This chapter shows that many independent constraints are available on the mechanisms of magma degassing in an eruption conduit. The overall consistency of the independent observations and arguments is impressive and leaves little doubt that magmas do leak gas out to the country rock on their way to the surface. Unfortunately, the physical mechanisms which are at work and the values of such key variables as the conduit size at depth and the permeability of country rock remain poorly known. Such issues are clearly a challenge for the future, but they must be addressed if we want to make progress in our understanding of the eruptive process. In more general terms, the phenomenon of gas loss may prove very useful for properly evaluating fluid dynamical models of volcanic flows in deep eruption conduits. Such models suffer from a lack of direct constraints and in this sense temporal changes of eruption characteristics provide key informations. Many such changes occur whilst the conduit system and the magma properties remain constant. The relationship between the gas content of the erupted material and mass discharge rate (or average ascent velocity) can only be explained by a small set of dynamical models. The magnitude of the gas flux out of rising magma depends critically on the distribution of flow pressures, in both the horizontal and vertical directions. This implies that a quantitative assessment of the effect of gas loss on eruption dynamics can only be done with a physical model. It may be possible, however, to evaluate these effects during an ongoing eruption. One reason is that an eruption follows a

logical progression in time which can be accounted for and which has predictable consequences. Another reason is that gas escape probably occurs through fracturing, which can be monitored using seismic methods. I would like to thank the organizers of the wonderful Holmes Conference held in Santorini in September 1996, J. Gilbert, H. Mader and S. Sparks. Comments, criticisms, and suggestions by H. Huppert. H. Mader. P. McLeod, S. Sparks and G. Wadge, have greatly improved the manuscript.

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TAYLOR, B. E. 1991. Degassing of Obsidian Dome rhyolite, Inyo volcanic chain, California. In: TAYLOR, H. P. JR, O'NEIL, J. R. & KAPLAN, I. R. (eds) Stable Isotope Geochemistry: A Tribute to Samuel Epstein. Geochemical Society, Special Publications, 3, 339-353. , ElCHELBERGER, J. C. & WESTRICH, H. R. 1983.

Hydrogen isotopic evidence for rhyolitic magma degassing during shallow intrusion and eruption. Nature, 306, 541-545. TAYLOR, G. I. 1954. The two coefficients of viscosity for an incompressible fluid containing air bubbles. Proceedings of the Royal Society of London, A226, 34-39. WESTRICH, H. R. & EICHELBERGER, J. C. 1994. Gas transport and bubble collapse in rhyolitic magma: an experimental approach. Bulletin of Volcanology, 56, 447-458.

, STOCKMAN, H. W. & EICHELBERGER. J. C. 1988. Degassing of rhyolitic magma during ascent and emplacement. Journal of Geophvsical Research. 93,6503-6511. 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 behaviour. Geophysical Journal of the Roval Astronomical Society, 63, 117-148 WOODS, A. W. 1995. The dynamics of explosive volcanic eruptions. Reviews of Geophysics, 33, 495-530. & KOYAGUCHI, T. 1994. Transitions between explosive and effusive eruption of silicic magmas. Nature, 370, 641-645. YOUNG, S. R., WHITE, R., WADGE, G. et al. 1997. The ongoing eruption in Montserrat. Science, 276, 371-372.

Observations and models of volcanic eruption columns ANDREW W. WOODS Centre for Environmental and Geophysical Flows, School of Mathematics, University of Bristol, Bristol BS8 1TW Abstract: Direct observations, satellite imagery and field studies of air-fall deposits provide a wealth of information about eruption column dynamics. These data have stimulated a wide range of theoretical, experimental and numerical models of eruption columns to expose the fundamental physical controls. These models show that the motion of hot, turbulent, particle-laden eruption columns is sensitive to the eruption velocity and mass flux, as well as the temperature and grain size distribution of the erupting material. In turn, the eruption velocity and mass flux are sensitive to the magma volatile content, the conduit radius and the chamber pressure. The importance of magma-water interactions, the crater geometry, ambient winds and the atmospheric stratification on the motion of the erupted material are also discussed. Predictions of the models are compared and contrasted and possible avenues for future research are suggested.

Nomenclature p

Specific heat at constant pressure Energy of an explosion (J) Rotation frequency of the Earth (s - 1 ) Acceleration due to gravity (m s-2) Column height (m) Depth of the neutral cloud (m) Gas mass fraction Brunt-Vaiasala frequency for atmospheric stratification (s"-1) Mass eruption rate (m3 s-1) Flux of enthalpy produced by a steady eruption (Js - 1 ) Radius of shock front Radius of the umbrella cloud (m) Gas constant of the erupting mixture Time (s) Temperature in plume (K) Temperature of ambient air (K) Velocity of plume (m s-1 ) Height above source (m) Entrainment coefficient Dimensionless constant Ambient density (kgm - 3 ) Plume density (kgm - 3 ) Reference density (kgm - 3 )

Introduction The dynamics of volcanic eruption columns has been studied extensively over the past few years, using both detailed observations of particular eruptions and increasingly sophisticated theore-

tical models. This has led to considerable advances in our understanding of the complex phenomena involved. The purpose of this chapter is to provide a summary and synthesis of some of the key observations of eruptive phenomena and compare these with theoretical predictions. I discuss and compare the different modelling approaches and outline some of the outstanding problems. Much more detailed and comprehensive accounts of the mathematics underlying eruption column models may be found in the book Volcanic Plumes (Sparks et al 1997) and in the review by Woods (1995). I have arranged the chapter into six main sections, starting with a summary of some important historical observations of eruption columns. I then move on to describe the source conditions which control the eruption rate and style. In the following section, I describe the essential physical balances in operation and present a simple model to predict the column height. Next, I analyse the vertical structure of the plume in more detail using a horizontally averaged one-dimensional model. Finally, I describe a series of analogue laboratory experiments, which have examined specific aspects of the dynamics, and I comment on some of the sophisticated axisymmetric numerical models which have been used to study the initial stages of eruption column formation. This chapter then closes with a section on conclusions and future perspectives. Observations Over the past two decades, a number of violent explosive eruptions have been well documented. These observations have revealed many of the

WOODS, A. W. 1998. Observations and models of volcanic eruption columns. In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 91-114.

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fundamental processes involved in the formation of explosive eruption columns and provide important constraints for the predictions of theoretical models. In this section I review a number of the observations to set the scene for the subsequent modelling.

Ngauruhoe, 1975 Nairn & Self (1978) reported a series of photographs taken during the initial 1-2 min of the eruption of Ngauruhoe volcano, New Zealand, in February 1975. These photographs are particularly instructive, and Fig. 1 shows two of these photographs taken at 13 and 45s after the start of the eruption. After the eruption commenced, a cloud of ash and gas expanded rapidly above the vent, initially spreading outwards in a nearly spherical pattern. Ahead of the ejected material, a cloud of condensed atmospheric water vapour can be seen (Fig. la). This is associated with the shock wave which advanced ahead of the erupting material as a result of the initial explosion (Nairn 1976). The initial nearly spherical expansion of the ash cloud may represent the decompression of high-

pressure material just below the vent, following failure of the overlying rocks. After a few tens of seconds, when the shock had propagated several kilometres away from the vent, the continuing eruption of material evolved towards a quasisteady state, somewhat reminiscent of the decompression of an overpressured jet from a nozzle (Fig. Ib) (Thompson 1980). A distinct upward jet-like structure developed in the central core of the cloud, just above the vent. Several hundred metres above the vent this became unstable and developed a series of large convecting eddy structures. These eddies engulfed the surrounding air, mixing it with the hot ash and lowering the bulk density of the column. Eventually, the upward moving highly turbulent column became buoyant and rose more than 10km into the atmosphere. At the same time as the upward column developed, large bombs could be seen escaping from the periphery of the jet and cascading to the ground along ballistic trajectories, and a dense, ash-laden current began to develop from the outer edge of the column. This current flowed down one of the flanks of the volcano, and became progressively more turbulent in appearance as may be seen in Fig. Ib.

Fig. 1. Photograph of the developing eruption column: (a) 13s and (b) 45s after the start of the 1975 eruption of Ngurahoe, New Zealand (from Nairn & Self 1978, reproduced by permission of the Journal of Volcanology and Geothermal Research).

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(b)

Fig. 1. (continued)

Mount St Helens, 18 May 1980 The lateral blast at Mount St Helens volcano provided spectacular and very clear evidence that eruption plumes can rise from ash flows as well as developing directly above a volcanic vent. At the start of the eruption, following a catastrophic landslide and decompression of magma stored in a shallow cryptodome, a large mass of magma decompressed and generated a lateral blast flow (Kieffer 1981). This travelled at speeds of more than 100ms -1 to distances as far as 15 km from the source (Lipman & Mullineaux 1981). However, as the blast flow advanced it deposited coarse, high-density ash particles, and

it entrained and heated air. Eventually, the bulk density became smaller than that of the air, and the flow lifted off to form a cloud which rose about 25 km into the stratosphere in a period of 400-500 s (Sparks et al 1986)(Fig. 2a). After this initial explosion, the eruption at Mount St Helens continued by tapping a deeper source of magma, and a powerful Plinian phase in which an eruption column up to 15 km high ascended directly above the vent, continued for more than 8h (Fig. 2b) (Holasek & Self 1995). The material ascended through the atmosphere until the initial thermal energy was exhausted, at which stage the material intruded laterally into

(a)

Fig. 2. (a) Height as a function of time of: (i) the lateral blast cloud which formed during the initial stages of the 18 May 1980 eruption of Mount St Helens (after Sparks et al. 1986); (ii) the coignimbrite thermal which rose from the ash flow, 4-5 km from the collapsed dome during the 21 April 1990 eruption of Mount Redoubt (after Woods & Kienle 1995). (b) Height of the eruption column as a function of time during: (i) the eruption of Mount St Helens, 18 May 1980 (from Holasek & Self 1995, reproduced by permission of the Journal of Geophysical Research); and (ii) the eruption of Mount Pinatubo, 15 June 1991 (from Holasek et al. 1996, reproduced by permission of the Journal of Geophysical Research).

OBSERVATIONS AND MODELS OF ERUPTION COLUMNS the environment to form a so-called umbrella or neutral cloud. As well as the detailed data on the ascent of the cloud, satellite recordings of the cloud-top temperature provided important details of the spreading of the ash plume through the atmosphere above the eruption column. These identified that the top of the eruption column was much colder than the surrounding atmosphere (Matson 1984; Woods & Self 1993) but that, as the ash was carried away from the column by the zonal stratospheric winds, the ash temperature adjusted back to the local ambient temperature (Holasek & Self 1995; Woods et al 1996). This undercooling at the top of the column is an important signature of the cloud dynamics, and provides an important test of models of eruption columns.

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explosive activity, they identified five distinct explosions during the initial stages of the eruption. These produced five plumes, which fed an eruption column eventually rising to a height of 18 km. The ascent speeds of these plumes were in the range 8-62 m s - 1 , with angles of spread in the range 21-24°. The main eruption column gradually accelerated from 8.5 to 58 m s - 1 over the first 8 km of its ascent. The initial stages of this eruption were somewhat complex due to the inferred interaction of the magma and surface water. However, Sparks & Wilson (1982) showed that a model of a starting plume which accounted for the presence of ash particles was in fairly good accord with these data and this represented an important advance in eruption modelling. Later in this chapter we will describe a series of somewhat more refined and dynamically complete models of eruption columns which have since been developed.

Soufriere volcano, St Vincent, 1979 Sparks & Wilson (1982) reported a series of detailed observations of the eruption of Soufriere, St Vincent, on 22 April 1979. By analysing film of the first 3min of a 14-min period of

Redoubt volcano, Alaska, 1989-1990 After a large explosive eruption of Redoubt volcano, Alaska, in December 1989, a fairly

Fig. 3. Photograph of the thermal cloud which developed during the eruption of Mount Redoubt, Alaska on 21 April 1990. The picture shows the main body of material as it begins to intrude laterally into the atmosphere, with a small trailing column of ash behind (after Woods & Kienle 1995, reproduced by permission of the Journal of Volcanology and Geothermal Research).

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regular pattern of approximately weekly domecollapse events was observed during the period January-April 1990. Video recordings of these dome-collapse events provided very detailed observations of the phenomena. In each collapse event about 0.1 km3 of the lava dome broke off and fragmented as it cascaded down a narrow ice canyon below the dome. As the mixture entrained more air, the bulk density of the finegrained particles together with the air eventually became less than that of the atmosphere and, as with the lateral blast at Mount St Helens, the flow lifted off to form a coignimbrite cloud (Fig. 3). At the point of lift-off, the flow had travelled 4-5 km from the vent, again illustrating that ash clouds are not always generated directly above a volcanic vent. After about 300-400 s, the cloud had ascended more than 12 km into the atmosphere and its thermal energy was exhausted; a laterally spreading intrusion then developed in the stratosphere. Woods & Kienle (1994) developed a model of a rising coignimbrite thermal cloud which they compared with the video recordings of the ascent of the plume.

Mount Pinatubo, the Philippines, 1991 The eruption of Mount Pinatubo on 15 June 1991 was one of the largest eruptions this century, with estimates that 5-9 km3 of magma were erupted during a period of about 20 h forming an eruption column which rose more than 35km into the atmosphere (Fig. 2b) (Holasek et al. 1996). A number of video and photographic records have shown that the eruption exhibited both Plinian phases, in which the column rose directly above the vent, and coignimbrite phases, in which the column rose off ash flows some distance from the vent (Koyaguchi & Tokuno 1993; Holasek et al. 1996). Analysis of thermal AVHRR images of the cloud top again recorded that the cloud top was undercooled relative to the surrounding atmosphere (Fig. 4). The spreading umbrella cloud produced during this eruption exhibited an interesting additional feature. Although there are very limited data available, the satellite image (Fig. 4) suggests that the laterally intruding cloud may have become unstable as it spread out radially for more than 1000 km. This may signify that the effect of the Earth's rotation became important, although the mechanism is not fully understood. The instability seems to manifest itself as the wavy rippled edge of the umbrella cloud which appeared when the umbrella cloud had spread to

a radius of the order of 1000 km (Fig. 4). Note that rotational effects are unlikely to influence the motion of the ascending eruption column as they only become significant after a time of the order of 1 day, while the ascent time of material in the plume is only of the order of a few hundred seconds (Helfich & Battisti 1991).

Eruptions involving external water External water can change the character of an eruption significantly. There are two main sources of water, atmospheric vapour and surface water. Atmospheric water vapour can lead to a number of interesting effects. As well as the condensate layer which tracks the motion of the shock wave through the atmosphere just after an explosion (Nairn & Self 1978) (Fig. la), moist skirt clouds, associated with the upward displacement of the air surrounding a rising cloud, have also been observed during some eruptions (Barr 1982). As the air is displaced upwards it decompresses and cools adiabatically. This can lead to condensation of water vapour at points where the atmosphere is close to saturation conditions and where the adiabatic temperature decreases with pressure more rapidly than the liquid-vapour saturation temperature. Barr (1982) reported a series of photographs during the 17 April 1979 eruption of Soufriere, St Vincent, in which a well-defined and laterally extensive condensate layer developed around the rising ash cloud. Water vapour incorporated into the eruption column also condenses when it has been convected to greater and colder altitudes, and this can increase the thermal budget of the eruption column through the release of latent heat. In small eruptions in humid environments, for example basaltic fissure eruptions in Hawaii, the thermal energy provided by the erupting material might actually trigger moist convective clouds, similar to the clouds which form above fields of burning sugar-cane (Woods 1993). Field analysis of several deposits suggest that eruptions which involve the interaction of magma with surface water are particularly violent, producing widespread deposition of very finegrained, fragmented ash (Self & Sparks 1978). The 1965 eruption of Taal volcano, in the Philippines, involved a considerable degree of mixing of surface water and magma, and this had an important effect on the ash plume. As described by Moore el al. (1966), the eruption started with relatively little water invading the vent, and a turbulent ash column developed. However, as the eruption proceded and the vent

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Fig. 4. NOAA thermal infra-red satellite image of the Mount Pinatubo ash plume at 18:34 h on 15 June 1991. The colour-coded image of the cloud-top temperatures illustrates the formation of the rippled outer edge, which may be due to a rotational instability of the radially spreading cloud (after Holasek et al. 1996, reproduced by permission of the Journal of Geophysical Research). was eroded, more water accessed the erupting mixture, and the column became very moist, with mud rain falling from the cloud. Eventually the column collapsed and a wet, cold mud flow spread laterally from the vent. This eruption identified the crucial role which large quantities of water may have in suppressing the formation of eruption columns and in changing the character of an explosive eruption (Koyaguchi & Woods 1996). A different phenomenon occurs when basalt or basaltic-andesite comes into contact with water, as occurred for example during the 1886 eruption of Tarawara volcano, New Zealand (Walker et al. 1984). Although the interaction of water with erupting magma is a complex process, it is

thought that in some situations the water leads to enhanced fragmentation of the magma and the production of very fine-grained material so that a substantial fraction of the erupted thermal energy can be transferred to entrained air (Wohletz 1983). Such enhanced fragmentation might lead to a convective ash plume above the vent, producing a basaltic Plinian eruption, rather than fire-fountaining behaviour. A similar phenomenon might also have occurred during the 1983 eruption of Miyakejima in Japan, in which a fire-fountaining style eruption was dramatically transformed into a larger convecting cloud of ash and steam when the ascending magma intersected a shallow aquifer (Fig. 5) (Aramaki et al. 1983; Sumita 1985).

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Fig. 5. Photograph of the 'basaltic Plinian' ash plume which developed during the basaltic fissure eruption of Miyakajima, Japan, 1985 following the explosive interaction of the magma with a shallow aquifer. Earlier in this eruption, before the magma had propagated along the rift up to the shallow aquifer, the eruption was characterized by fire-fountaining activity.

Field data Even if an eruption is not observed, field data can be used to infer some of the properties of the eruption column. In particular, Bursik (1998) describes a number of models of ash dispersal from the laterally spreading umbrella cloud (e.g. Carey & Sparks 1986; Bursik et al 1992). The predictions of these models are very sensitive to the height at which ash is injected into the atmosphere, owing to variations in the strength of zonal winds with altitude. Recently, Thomas et al. (1993) have examined the vesicularity of many pumices in air-fall deposits, and this is leading to new constraints on the cooling history in the lower part of eruption columns, which may ultimately provide further important contraints on the entrainment rate and hence

dynamics of the lower part of the column (Jaupart pers. comm. 1997). A key result of these models is the prediction of the initial 'free' gas content of the erupting mixture, in contrast to the gas bubbles which remain trapped in the pumice clasts (Kaminski 1997). Unless the magma is pulverized into very fine ash, the free gas component is typically smaller than the total exsolved volatile content of the material. One important phenomenon described in the above examples is that eruption columns do not always rise above the vent of a volcano; instead coignimbrite columns may originate from massive ash flows after they have propagated some distance along the ground from the vent. This is in accordance with the interpretation of field deposits from many historic eruptions which suggest that some large explosive eruptions.

OBSERVATIONS AND MODELS OF ERUPTION COLUMNS including the 75 000 BP eruption of Toba and the 1991 eruption of Mount Pinatubo involved both Plinian and coignimbrite phases (Sparks & Walker 1977; Rose & Chesner 1987; Holasek et al. 1996). The different phases of the eruption may be distinguished because the air-fall deposits associated with coignimbrite columns are anomalously fine grained and the associated ignimbrite flow deposits tend to be fines depleted (Sparks & Walker 1977). These and a number of other observations have triggered much research modelling eruption phenomena in order to gain a better understanding of eruption column processes. Some of these models have been compared with observational data, with some success, although many more physical quantities are included in the mathematical models than have been observed or measured in the field. As a result, a number of workers have also developed analogue laboratory models to simulate the phenomena and test the theoretical models in a more controlled environment. In the following sections, I review these mathematical and laboratory models, illustrating how some of the processes described above may be quantified. Source conditions The mechanisms responsible for the ascent and eruption of magma from a reservoir 2-10 km below the surface have been described in earlier contributions to this volume (Jaupart 1998; Mader 1998). The key processes in the conduit are the decompression of the magma at depth which leads to the exsolution of volatiles and expansion of the magma-bubble mixture to form a foam. The continued decompression of the foam eventually leads to break-up of the liquid films around the volatile bubbles, and the mixture undergoes a transition from a state in which the liquid is the continuous phase to one in which the volatile gas becomes the continuous phase. Above this point, the frictional stresses exerted by the walls of the conduit become much smaller. There are two pictures of the ascent along the conduit, depending on whether the conduit is assumed to be essentially rigid, so that the pressure of the erupting mixture can differ from the lithostatic pressure, or whether the conduit is assumed to erode rapidly so that the pressure of the erupting mixture equals the lithostatic pressure at all heights (Wilson et al. 1980). The actual situation probably lies somewhere between these two extremes; however, the erosional model predicts a rather large widening

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of the conduit near the surface, but the lithic content which is predicted to accompany such erosion has not been identified in eruption deposits. In the model of a rigid, parallel-sided conduit, the pressure of the mixture is predicted to decrease slowly relative to lithostatic pressure above the fragmentation level as the density of the mixture of gas and magma fragments is considerably less than that of the surrounding rock. Therefore, in many cases, the mixture is predicted to vent with a pressure far in excess of atmospheric. In Fig. 6 I present a typical graph illustrating: (a) the vent pressure as a function of the eruption rate; and (b) the eruption rate as a function of the vent radius. The calculations follow from the model of Woods (1995) in which the conduit radius was taken to be fixed, and so the erupting mixture becomes very overpressured. Curves are given for three volatile contents, 3.6, 5 and 7wt%. The decompression process is very important because it constrains the initial velocity of the material which is supplied to the eruption column; in turn this has an important impact on whether a buoyant column or a collapsing fountain develops. The above calculations assume that all the gas is released when the magma fragments, as would occur if the majority of the magma fragments into very fine ash. However, if the fragmentation produces a more coarse-grained mixture, then some of the exsolved volatiles will be trapped in the pumices and this will lead to a smaller Tree' gas component in the erupting mixture, so that the eruption dynamics are analogous to a magma of lower gas content (Kaminski 1997). If the material is overpressured when it issues from the vent, it decompresses into the atmosphere. During the first stages of an eruption, following the initial explosion, a shock wave propagates ahead of the erupting mixture into the atmosphere. For a finite point source explosion of energy, E, propagating through an environment of density, 0, the shock front propagates a distance r = (Et2/p )1/5 over a time t (Taylor 1950). Thus, for a short-lived Vulcanian-style explosion, the explosive energy released may be estimated using observations of the motion of the initial shock wave. However, in longer lived eruptions, the effects of gravity and the continuing eruption of material become important, and this simple model cannot capture the evolving structure of the shock or of the erupted material which follows behind. However, in many maintained eruptions in which the material decompresses freely into the atmosphere or into a large U-shaped crater, the eventual quasi-steady decompression occurs in a similar fashion to that in a high-speed gas jet,

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Fig. 6. Variation of the eruption rate as a function of the vent pressure, and of the vent pressure as a function of the vent radius. Curves are given for magmatic volatile contents of 0.07, 0.05 and 0.036 (after Woods & Bower 1995, reproduced by permission of Earth and Planetary Science Letters). and the techinques developed for those phenomena may be applied (Thompson 1980; Kieffer & Sturtevant 1984). The process of decompression results in a decrease in the density and hence an increase in

the cross-sectional area and/or velocity of the flow. Wohletz el al (1984) developed a numerical model of the transient decompression based on inviscid shock tube theory for the subsurface flow and a numerical simulation of

OBSERVATIONS AND MODELS OF ERUPTION COLUMNS two-dimensional (axisymmetric) flow of the material rising into the atmosphere. These calculations illustrated the advance of the shock wave ahead of the material during the intial few tens of seconds of the eruption. More recently, Dobran et al, (1993) described further detailed numerical calculations of the decompression process above the vent, illustrating that over the first 10-30s a sequence of compression and rarefaction waves develop (Fig. 7). In the paricular calculations shown in Fig. 7 the erupting

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material eventually collapses to form an ash flow; there is a narrow region of decompression in the first 50-100 m above the vent, and above this height the Mach number decreases to values r; and (iii) entrainment coefficient e = 0.9 for all z.

OBSERVATIONS AND MODELS OF ERUPTION COLUMNS the model provides very good agreement with field data if account is made of the reduced entrainment rate at the base of a coignimbrite plume (Fig. 11). Woods & Kienle (1995) developed an analogue of this model to describe the ascent of discrete thermal clouds, and they showed that the predictions of the model were in good accordance with data on the rising coignimbrite cloud which formed during the 21 April 1990 eruption of Mount Redoubt (cf. Fig. 2). An important aspect of these models, and the associated data is that the ascent time of the material in the plumes is of the order of 200-500 s (Figs 2 and 11). The calculations and observations are in accord with dimensional arguments, which suggest that in a large buoyant convecting cloud the ascent time simply scales with 1/N, where N, approximately equal to 0.01, is the Brunt-Vaiasala frequency of the atmosphere. Experiental and numerical models of eruption columns

Experimental models There have been a number of important experimental studies of eruption columns designed to expose and provide new insight about the important physical processes. Carey et al. (1988) conducted a series of experiments to examine the effect of particle sedimentation and re-entrainment into buoyant eruption columns. A constant source of fresh, particle laden water was injected at the base of a tank filled with aqueous saline solution. By varying the injection rate and the particle loading, they identified the important control of particle sedimentation near the source on the plume dynamics. For a small particle loading the input fluid was buoyant and a plume developed directly above the source, whereas for larger particle loading the input fluid was initially dense. This formed a jet above the source which then collapsed. However, the collapse was not necessarily symmetric. As the material spread around the source, the particles sedimented from the fluid, leaving relatively fresh and hence light residual fluid. This rose from the base of the tank to form a buoyant plume above a wide source. For small particle loading the collapsed flow did not spread far from the source before becoming buoyant and rising to form a plume, whereas with larger particle loading a significant horizontal flow developed before material rose from the base as a plume. The laboratory model of the formation of buoyant plumes some distance

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from the vent is analogous to the many observations of coignimbrite eruption columns, which have been observed to rise from ash flows far from the original vent. Sparks et al (1991) and Ernst et al (1996) have developed this experimental approach further, quantifying the rate of fallout and re-entrainment of particles from the sides of the plume. These models provide new understanding of the fallout patterns in the region near the volcano, where the dispersal of ash is controlled by the large-scale transport processes. These models are described in more detail in Bursik (1998). One interesting feature of these models is the potential impact of re-entrainment on the dynamics of the plume. In a plume with small particle loading, in which the particles are dynamically passive, re-entrainment will not have a large impact on the flow. However, as shown in the experiments of Carey et al. (1988), with a larger particle loading the effect of re-entrainment is to produce instability and downward-descending particle currents at the edge of the plume. The full impact of such re-entrainment and instability has not been fully explored, yet such effects may be responsible for the simultaneous production of ash flows and eruption plumes as observed in many eruptions (Fig. 1) (Wilson & Walker 1985). In a different experimental study, Woods & Caulfield (1992) tested the one-dimensional steady models of eruption columns by comparing the predictions for collapse with a series of analogue laboratory experiments using mixtures of methanol and ethylene-glycol (MEG). MEG is less dense than water but on mixing with water the methanol reacts with the water, so that the density eventually exceeds that of water. Initially buoyant jets of MEG issued downwards into a tank filled with water and, as they descended, they became progresively more dense. With high initial momentum flux, the mixtures became denser than the water and continued to descend as a buoyant plume rather than as a momentum jet (Fig. 12a). However, with smaller initial momentum the jet speeds decreased to zero while they were still buoyant and they rose back to the source (Fig. 12b). An important feature of the experiments was that the column collapse was almost always asymmetrical. Woods & Caulfield (1992) showed that the experimental measurements of the critical flow rate required for collapse were in extremely good agreement with the predictions of an analogue model of these experimental eruption columns, based on the approach of Woods (1988), thereby supporting the relevance of the one-dimensional modelling approach.

Fig. 12. Photograph of laboratory experiments using jets of methanol and ethylene-glycol (MEG). As the mixture entrains water, the density relative to the water eventually reverses. In (a) the MEG mixes with sufficient water to become buoyant and hence forms a buoyant plume, analogous to a Plinian eruption column. In (b) the dense jet collapses before it has mixed sufficiently to become buoyant. Hence, it forms a dense flow. This eventually mixes with sufficient ambient fluid that it becomes buoyant, and forms an analogue coignimbrite plume (after Woods & Caulfield 1992, reproduced by permission of the Journal of Geophysical Researh).

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Numerical simulations of axisymmetric columns Although the one-dimensional quasi-steady models are successful at describing many of the fundamental processes in operation in an eruption column, they are unable to describe timedependent or three-dimensional phenomena. In response to this, and as a result of the increased computing power which is available today, a number of large numerical models of explosive eruptions have been developed (Valentine & Wohletz 1989; Dobran et al 1993; Neri & Dobran 1994). These are designed to model the transient development of an eruption column, following the motion of the material once ejected from the vent. The models assume that the motion is axisymmetric about the vent, and the gas and ash phases are treated as intermingled continua, accounting for mass and momentum transfer between phases. The mixing of air into the column is modelled using an eddy viscosity, based on turbulence closure models; although multiphase turbulence is not fully understood, and the eddy viscosity approach is a considerable simplification of the actual process, the qualitative effect is analogous to the entrainment coefficient used in the onedimensional models. Owing to the computational expense, many of the models account for a single or a small number of discrete particle sizes, and the models have only been run for a few hundred seconds of real time following the start of the eruption. One of the key advances of such modelling for the future will be to produce a fully three-dimensional model which can simulate the asymmetrical collapse as has been seen in analogue laboratory experiments. In many of the calculations to date, the numerical grids only extend 7-10 km above the source. As a result the models have been most successful in illustrating phenomena associated with the decompression (Fig. 7) and collapse of the erupting jet, but are unable to describe the complete motion of a Plinian eruption column. The models confirm that, apart from a highpressure region close to the vent, the main part of a steady eruption column has pressure close to that of the atmosphere. For example, many of the model calculations predict dynamic flow pressures of the order of 0.1 atm (Wohletz & Valentine 1990) once the flow has become quasisteady, and that the decompression occurs within about 100-500 m above the vent. A particularly interesting result of these models is the numerical prediction of unsteady collapsing fountains (Valentine et al 1991; Dobran et al 1993), in qualitative accord with

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laboratory observations of collapsing fountains (Carey et al 1988; Baines et al 1990; Woods & Caulfield 1992). In the numerical models, the instability appears to be connected with the reentrainment of particles and has a period of several tens of seconds. The instability is interesting because it suggests that there may be a periodic supply of material to voluminous ash flows, and if the period becomes sufficiently long this may lead to a series of discrete flows and hence depositional layering, rather than a maintained current. Conclusions and future perspectives In this chapter I have identified some of the key processes in eruption column dynamics through a description of various field observations coupled with theoretical and analogue experimental models. Although some of the basic processes are now understood, there are several important areas which merit further attention, through a combination of experimental work coupled with numerical/theoretical modelling. These include the development of: •

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•

•

an improved model of the entrainment process for use in one-dimensional models, particularly for the highly particle-laden gas thrust region; a more complete model of the effects of large dynamic pressures which arise just above the vent, in order to quantify its importance in controlling the expansion of the jet and in limiting the entrainment of air. Both of these effects are crucial for determining conditions for column collapse more accurately; a fully three-dimensional model of the behaviour of a collapsing fountain, to account for asymmetrical collapse, as has been observed in some analogue laboratory experiments (Carey et al 1988; Baines et al 1990). Understanding the controls on asymmetrical collapse is crucial for hazard analysis, because a radially spreading flow transports much less material in any one direction than a directed collapse; a more accurate method for accounting for thermal disequilibrium between ash particles and the entrained air. Again this is important for predictions of column collapse, yet to date the process has been studied only using a relatively simple parameterized model; a more accurate model of the fragmentation process and release of gas following the decompression of the magma; this will lead to more accurate constraints on the eruption evolution.

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In addition, the acquisition of more detailed field data and satellite observations of eruption columns, and of the associated dispersal of deposits in future eruptions, will provide many more stringent constraints on the assumptions included in the models, and indeed, in our understanding of the phenomena. A. W. Woods is supported by the Leverhulme foundation and the NERC. The author is grateful to Claude Jaupart, Lionel Wilson and Greg Valentine for their useful and thoughtful reviews of the paper.

References ARAMAKI, S., HAYAKAWA, Y., FUJII, T., NAKAMURA, K. & FUKUOKA, T. 1983. The October 1983 eruption of Miyakajime volcano, Japan. Journal of Volcanology and Geothermal Research, 29, 203-229. BAINES, W., TURNER, J. S. & CAMPBELL, I. 1990. Turbulent fountains in an open chamber. Journal of Fluid Mechanics, 212, 557-592. BARR, S. 1982. Skirt clouds associated with the Soufriere eruption of 17 April 1979. Science, 216, 1111-1112. BOWER, S. M. & WOODS, A. W. 1996. On the dispersal of clasts from volcanic craters during small explosive eruptions. Journal of Volcanology and Geothermal Research, 73, 19-32. BRAZIER, S. A., SPARKS, R. S. J., CAREY, S. N., SIGURDSSON, H. & WESTGATE, J. A. 1983. Bimodal grain size distribution and secondary thickening in air-fall ash layers. Nature, 301, 115-119. BURSIK, M. I. 1998. Tephra dispersal. This volume. & WOODS, A. W. 1991. Buoyant, superbuoyant and collapsing eruption columns. Journal of Volcanology and Geothermal Research, 45, 347-350. & 1996. The dynamics and thermodynamics of large ash flows. Bulletin of Volcanology, 58, 175-193. , CAREY, S. N., SPARKS, R. S. J. & GILBERT, J. 1992. Sedimentation of tephra by volcanic plumes. I, Theory and its comparison with a study of the Fogo A Plinian deposit, Sao Miguel (Azores). Bulletin of Volcanologv, 54, 329-334. CALDER, E., SPARKS, R. S. J. & WOODS, A. W. 1997. Dynamics of coignimbrite plumes generated from pyroclastic flows of Mount St Helens (August 7 1980). Bulletin of Volcanology, 58, 432-440. CAREY, S. & SPARKS, R. S. J. 1986. Quantitative models of the fallout and dispersal of tephra from volcanic eruptions. Bulletin of Volcanology, 48, 109-125. , SIGURDSSON, H. & SPARKS, R. S. J. 1988. Experimental studies of particle-laden plumes. Journal of Geophysical Research, 93, 15314-15328. CHUNG, J. N. & TROUTT, T. R. 1988. Simulation of particle dispersion in an axisymmetric jet. Journal of Fluid Mechanics, 186, 199-222. DOBRAN, F., NERI, A. & MACEDONIO, G. 1993. Numerical simulations of collapsing volcanic columns. Journal of Geophysical Research, 98, 4231-4259.

EATON, J. K. & FESSLER, J. R. 1994. Preferential concentration of particles by turbulence. International Journal of Multiphase Flows, 20, 169-209. ERNST, G., CAREY, S. N., BURSIK, M. I. & SPARKS, R. S. J. 1996. Sedimentation from turbulent jets and plumes. Journal of Geophysical Research. 101. 5575-5589. GILBERT, J. S. & LANE, S. J. 1994. The origin of accretionary lapilli. Bulletin of Volcanology. 56. 398-411. GLAZE, L. S., BALOGA, S. & WILSON, L. 1997. Transport of water vapor by volcanic eruption columns. Journal of Geophysical Research. 102. 6099-6108. HELFRICH, K. & BATTISTI, B. 1991. Experiments on baroclinic vortex shedding from hydrothermal plumes. Journal of Geophysical Research, 96. 12511-12518. HOLASEK, R. E. & SELF, S. 1995. GOES weather satellite observations and measurements of the May 18 1980 Mount St Helens eruption. Journal of Geophysical Research, 100, 8469-8487. ,' &" WOODS, A. W. 1996. Satellite observations and interpretation of the 1991 Mount Pinatubo eruption plumes. Journal of Geophysical Research, 101, 27635-27655. ISHII, R., UMEDA, Y. & Yum, M. 1989. Numerical analysis of gas-particle two phase flows. Journal of Fluid Mechanics, 203, 475-515. JAUPART, C. 1998. Gas loss from magmas through conduit walls during eruption. This volume. KAMINSKI, E. 1997. Fluid Mechanical Models of Two Geologic Processes. Ph.D. thesis, IPGP, Paris. KIEFFER, S. 1981. Fluid dynamics of the May 18 blast at Mt St Helens. In: LIPMAN. P. W. & MULLINEAUX, D. R. (eds) The 1980 Eruptions of Mount St. Helens, Washington. US Geological Survey. Professional Paper, 1250, 379-400. & STURTEVANT, B. 1984. Laboratory studies of volcanic jets. Journal of Geophysical Research. 89. 8253-8268. KOYAGUCHI, T. & TOKUNO, M. 1993. Origin of the giant eruption cloud of Pinatubo. June 15. 1991. Journal of Volcanology and Geothermal Research, 5, 85-96. & WOODS. A. W. 1996. On the explosive interaction of water and magma. Journal of Geophysical Research, 101, 5561-5574. LANE, S. J., GILBERT, J. S. & HILTON. M. 1993. The aerodynamic behaviour of volcanic aggregates. Bulletin of Volcanology, 55. 481-488. LIPMAN, P. W. & MULLINEAUX, D. R. 1981. The 1980 Eruptions of Mount St. Helens, Washington. US Geological Survey, Professional Paper. 1250. MADER, H. 1998. Conduit flow and fragmentation. This volume. MATSON, M. 1984. Satellite observations of volcanic eruption clouds. Journal of Volcanology and Geothermal Research, 23. 1-10. MCLELLAND. E. A. & THOMAS. R. E. 1990. A paleomagnetic study of Minoan age tephra from Thera. In: HARDY, D. A. (ed.) Thera and the Agean World III. Volume 2. The Thera Foundation. London, 129-138.

OBSERVATIONS AND MODELS OF ERUPTION COLUMNS MOORE, J. G. & MELSON, W. G. 1969. Nuees ardentes of the 1968 eruption of Mayon volcano, Philippines. Bulletin of Volcanology, 33, 600-620. , NAKAMURA, K. & ALCARAZ, A. 1966. The 1965 eruption of Taal volcano. Science, 151, 955-960. MORTON, B., TAYLOR, G. I. & TURNER, J. S. 1956. Turbulent gravitational convection from maintained and instantaneous sources. Proceedings of the Royal Society, A234, 1-23. NAIRN, I. A. 1976. Atmospheric shock waves and condensation clouds from Nguaruhoe explosive eruptions. Nature, 259, 190-192. & SELF, S. 1978. Explosive eruptions and pyroclastic avalanches from Ngauruhoe explosive eruptions. Journal of Volcanology and Geothermal Research, 3, 39-60. NERI, A. & DOBRAN, F. 1994. Influence of eruption parameters on the thermofluid dynamics of collapsing volcanic columns. Journal of Geophysical Research, 99, 11 833-11 857. PREVOST, F., BOREE, H. J., NUGLISCH, S. & CHARNAY, G. 1996. Measurements of fluid particle correlated motion in the far field of an axisymmetric jet. International Journal of Multiphase Flow, 22, 985-901. Ricou, F. P. & SPALDING, D. B. 1961. Measurements of entrainment by axisymmetrical turbulent jets. Journal of Fluid Mechanics, 11, 21-32. ROSE, W. & CHESNER, C. 1987. Dispersal of ash in the great Toba eruption. Geology, 16, 913-917. SARNA-WOJCICKI, A. M., SHIPLEY, S., WAITT, R. B., DZURISIN, D. & WOOD, S. H. 1981. Areal distribution, thickness, mass, volume and grain size of air-fall ash from the six major eruptions of 1980. In: LIPMAN, P. W. & MULLINEAUX, D. R. (eds) The 1980 Eruptions of Mount St. Helens, Washington. US Geological Survey, Professional Paper, 1250, 569-576. SELF, S. & SPARKS, R. S. J. 1978. Characteristics of widespread pyroclastic deposits formed by the interaction of silicic magma and water. Bulletin of Volocanology, 41, 196-212. SETTLE, M. 1978. Volcanic eruption clouds and the thermal power output of explosive eruptions. Journal of Volcanology and Geothermal Research, 3, 309-324. SPARKS, R. S. J. & WALKER, G. P. L. 1977. The significance of crystal enriched air-fall ashes associated with crystal-enriched ignimbrites. Journal of Volcanology and Geothermal Research, 2, 329-341. & WILSON, L. 1982, Explosive volcanic eruptions - V. Observations of plume dynamics during the 1979 Soufriere eruption, St Vincent. Geophysical Journal of the Royal Astronomical Society, 69, 551-570. , BURSIK, M. I., CAREY, S. N., GILBERT, J. S., GLAZE, L. S., SIGURDSSON, H. & WOODS, A. W. 1997. Volcanic Plumes. John Wiley, Chichester. , CAREY, S. N. & SIGURDSSON, H. 1991. Sedimentation from gravity currents generated by turbulent plumes. Sedimentology, 38, 839-856. , MOORE, J. & RICE, J. 1986. The initial giant umbrella cloud of the May 18 1980 explosive eruption of Mount St. Helens. Journal of Volcanology and Geothermal Research, 28, 257-274.

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SUMITA, M. 1985. Ring shaped cone formed during the 1983 Miyakejima eruption. Bulletin of the Volcanological Society of Japan, 30, 11-32. TAYLOR, G. I. 1950. The formation of a blast wave by a very intense explosion. I, Theoretical discussion. Proceedings of the Royal Society of London, A201, 159-174. THOMAS, N., TAIT, S. R. & KOYAGUCHI, T. 1993. Mixing of stratified liquids by the motion of gas bubbles: application to magma mixing. Earth and Planetary Science Letters, 115, 161-175. THOMPSON, P. A. 1980. Compressible Fluid Dynamics. McGraw-Hill, New York. TURNER, J. S. 1979. Buoyancy Effects in Fluids. Cambridge University Press, Cambridge. VALENTINE, G. & WOHLETZ, K. 1989. Numerical models of Plinian eruption columns and pyroclastic flows. Journal of Geophysical Research, 94, 1867-1887. , & KIEFFER, S. 1991. Sources of unsteady column dynamics in pyroclastic flow eruptions. Journal of Geophysical Research, 96, 2188721 892. WALKER, G. P. L., SELF, S. & WILSON, L. 1984. Tarawera 1886, New Zealand, a basaltic Plinian fissure eruption. Journal of Volcanology and Geothermal Research, 21, 61-78. WILSON, C. N. J. & WALKER, G. P. L. 1985. The Taupo Ignimbrite, New Zealand, I, General aspects. Philosophical Transactions of the Royal Society of London, A314, 199-218. WILSON, L. 1976. Explosive volcanic eruptions III. Plinian erupton columns. Geophysical Journal of the Royal Astronomical Society, 45, 543-556. & WALKER, G. P. L. 1987. Explosive volcanic eruptions - VIII. Ejecta dispersal in Plinian columns: the control of eruption conditions and atmospheric propeties. Geophysical Journal of the Royal Astronomical Society, 89, 657-679. , SPARKS, R. S. J., HUANG, T. C. & WATKINS, N. D. 1978. The control of volcanic eruption column heights by eruption energetics and dynamics. Journal of Geophysical Research, 83, 1829-1836. , & WALKER, G. P. L. 1980. Explosive volcanic eruptions - IV. The control of magma properties and conduit geometry on eruption column behaviour. Geophysical Journal of the Royal Astronomical Society, 63, 117-148. WOHLETZ, K. 1983. Mechanisms of hydrovolcanic pyroclast formation: grain size, scanning electron microscopy and experimental studies. Journal of Volcanology and Geothermal Research, 17, 31-63. , MCGETCHIN, T., SANDFORD, M. & JONES, E. 1984. Hydrodynamic aspects of caldera-forming eruptions: numerical models. Journal of Geophysical Research, 89, 8269-8285. & VALENTINE, G. 1990. Computer simulations of explosive volcanic eruptions. In: RYAN, M. P. (ed.) Magma Transport and Storage. John Wiley, Chichester, 113-135. WOODS, A. W. 1988. The dynamics and thermodynamics of eruption columns. Bulletin of Volcanology, 50, 169-191.

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-1993. Moist convection and the injection of volcanic ash into the stratosphere. Journal of Geophysical Research, 98, 17627-17636. -1995. The dynamics of explosive volcanic eruptions. Reviews of Geophysics, 33, 495-530. - & BOWER, S. M. 1995. The decompression of volcanic jets in a crater during explosive volcanic eruptions. Earth and Planetary Science Letters, 131, 189-205. - & CAULFIELD, C. P. 1992. A laboratory study of explosive volcanic eruptions. Journal of Geophysical Research, 97, 6699-6712. - & KIENLE, J. 1995. The dynamics and thermodynamics of volcanic clouds, theory and observa-

tions from the April 15 and April 21 1990 eruptions of Redoubt volcano, Alaska. Journal of Volcanologv and Geothermal Research, 62, 273299. - & SELF, S. 1993. Thermal disequilibrium at the top of volcanic clouds and its effect on estimates of column height. Nature, 350, 225-227. - & WOHLETZ, K. 1991. The dimensions and dynamics of coignimbrite eruption columns. Nature, 350, 225-227. -, HOLASEK, R. & SELF, S. 1996. Wind driven dispersal of volcanic ash plumes and its control on the thermal structure of the plume-top. Bulletin of Volcanologv, 57, 283-292.

Tephra dispersal M. BURSIK Department of Geology, 876 Natural Sciences Complex, University at SUNY, Buffalo, NY 14260, USA

Buffalo,

Abstract: Our understanding of the spread of volcanic ejecta from eruption columns has evolved significantly since the first quantitative studies. Given recent advances, it is possible to describe much of the physics of tephra dispersal, and to construct a relatively robust model for the spread of tephra in the proximal region of a volcanic eruption plume. Results from such modelling compare well with data on tephra fall deposits. More distal dispersal, driven as it is by atmospheric motions, has also been treated extensively, by atmospheric scientists as well as by volcanologists. Results from these studies have been compared with satellite data and have been found to be accurate in terms of predicting ash cloud trajectories. Comparisons of the various models with data point out several areas where research could be productively directed in future. The transition from tephra transport by the plume to that by the atmosphere is poorly understood; processes related to transport between plume and ground are poorly known; the layering of the stratosphere needs to be considered more thoroughly to understand transport processes over periods of days to months; and the total grain size distributions of fall deposits and eruption plumes have probably been incompletely characterized in the past.

Notation

Md Mi

Constant in spectral relationship between E(k) and k a, b, c Pyroclast axial lengths, a > b > c [L] ae Entrainment coefficient, dimensionless b Charactistic radius or width of plume [L] b Scale factor in turbulence spectral equation CD Drag coefficient, dimensionless Ce, Cv Specific heat of air and plume material, respectively [L 2 T - 2 Temp-1] Ci Mass concentration of particles of i-th grain size class [M L - 3 ] d Pyroclast mean diameter [L] b Characteristic increase in plume r width [L] E(k) Turbulence kinetic energy as a function of wavenumber [ML 2 T- 2 ] F Pyroclast shape factor, dimensionless F Source or sink term in advectiondiffusion equation [ M L - 3 T - 1 ] / Re-entrainment factor, dimensionless FD Drag force [M L2 T - 2 ] g Acceleration of gravity [L T - 2 ] H The Heaviside function h Thickness [L] Hb Height of base of umbrella cloud [L] Ht Height of top of umbrella cloud [L] k Wavenumber [L - 1 ]

Msi

A

msi N n p p(z) Q R R Re S s s, z

T T t Te u, U

Median grain size [L] Mass flux of pyroclasts of i-th size fraction [MT - 1 ] Total mass flux of i-th size fraction, integrated through eruption duration [MT - 1 ] Sedimentation rate per unit distance in flow direction of particles of i-th size fraction [ M L - T - 1 ] Buoyancy or Brunt-Vaisala frequency [T -1 ] Unit normal vector [L] Ree-ntrainment probability, dimensionless Pressure at height in atmosphere [ML - 1 T - 2 ] Volume flux [L 3 T -1 ] Gas constant for air [L3 T-2 Temp -1 ] Umbrella cloud radius [L] Radial coordinate in a rightcylindrical coordinate system [L] Reynolds number, wsd/v, dimensionless Plume cross-sectional area [L2] Pyroclast density [M L - 3 ] Axes of a plume centred rightcylindrical coordinate system, tangential and perpendicular to flow direction [L] Eruption duration [T] Temperature of plume material [Temp] Time [T] Temperature of air [Temp] Vector velocity [LT - 1 ]

BURSIK, M. 1998. Tephra dispersal. In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 115-144

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Components of velocity in axial directions [LT - 1 ] ue Entrainment speed [LT - 1 ] V Volume [L3] W Pyroclast immersed weight Ws Settling speed [LT - 1 ] x, y, 2 Axes of a rectangular coordinate system, z vertical [L] a Atmospheric density [M L - 3 ] 0 Bulk density of a particle-gas mixture [M L - 3 ] 7 Constant in the spectral relationship between E(k) and (k) 9 Potential temperature [Temp] k x , k y , k z Components of turbulent diffusivity in axial directions [L 2 T -1 ] A Shape factor for a gravitationally spreading plume, dimensionless v Kinematic viscosity of gas [L 2 T - 1 ] Gas density [M L - 3 ] 0 Size in 0 units [log L] Sink of source term to describe the change in particle concentration with time [ML - 3 T-1] Wavefunction u, v,

w

Subscripts 0 Initial value or value at source Hb Value at height Hb m Value of discrete wavenumber

The dispersal or spread of tephra by volcanic eruption columns within the atmosphere is critical to our understanding of global climate change (Lamb 1970) and to the safety of aircraft flying within the troposphere and lower stratosphere (Casadevall 1994). It is a complex problem, incorporating as it does aspects of both volcanology and meteorology. Nevertheless, the study of dispersal by volcanic plumes has evolved immensely over the past 30 years. The quantitative study of tephra dispersal began with the seminal work of Knox & Short (1964) to calculate the general features of a volcanic plume from point measurements in a fall deposit, Walker & Croasdale (1971) who completed the first detailed studies of the sedimentological variations within several fall deposits and Walker el al. (1971) who measured the fall speeds of moderate-sized pyroclasts. Much has been accomplished since the early works. The current state of knowledge in fact is quite evolved, and results from the relatively easy availability of data on volcanic plumes and fall deposits against which to compare models as they develop.

This chapter discusses our current understanding of tephra dispersal. It is organized along conceptual rather than historical lines, and consists of the discussion of models of tephra dispersal that have been carefully based on the physics of plumes and the atmosphere. An attempt has been made to emphasize points not treated at length in the recent book by Sparks el al. (1997), and conversely to deemphasize points more thoroughly treated therein. Where Sparks et al. (1997) include a more thorough treatment, this is pointed out. In our past efforts at understanding tephra dispersal data (e.g. Bursik el al. 1992a, \992b. 1993; Sparks et al. 1992) we relied in large part on 'inverse' modelling, i.e. determining model parameters based on least-squares fitting to the data. The current work, on the other hand, is presented from a 'forward' model approach, to emphasize the fundamental physics that underlies the phenomenology of dispersal. Two basic models are developed. One model is for 'proximal' dispersal in which the motion of the volcanic plume itself acts as the primary control of the position of dispersing particulates. The other model treats distal transport, wherein atmospheric motions determine the spread of tephra. The common starting points for both models are the advection-diffusion equation, and the dynamics of single particles moving in an infinite fluid. The models use particle, plume and atmospheric dynamics as inputs, and synthetic plume growth or fallout patterns as outputs. The hypothesis being tested in construction of the models and comparison to data is that the output should fit available volcanological data and where it does not we should gain insight into how our understanding of dispersal needs to be modified. Because one goal is to explore the first-order effects on tephra dispersal, the model is not developed directly from the multiphase Navier-Stokes equations, although it should lend insight into how these can be used to improve our current notions. Because a second goal is to validate models of ash dispersal, the results will be of some interest to those interested in the validation of ash transport forecasts for aviation (e.g. Heffter & Stunder 1993). The spread of tephra is the result of two processes: the movement of particles singly in the atmosphere; and the movement of particles within coherent structures that themselves move as coherent entities (Fig. 1). The first part of this chapter therefore treats the movement of single pyroclasts subject only to the gravitational and drag forces. The second part introduces the advection-diffusion equation, from which all

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Fig. 1. Schematic diagram showing the dispersal of tephra both within plumes and within the atmosphere. Closer to the volcano, plume dynamics dominate dispersal while more distally, atmospheric dynamics dominate.

transport relationships can be derived. The third part treats proximal transport within the gravity-driven flows - plumes, umbrella clouds and downwind plumes - whose motion is primarily or in large part derived from volcanic eruption parameters. This part of the chapter builds on the volcanic plume models outlined in Woods (1998). The fourth part treats 'passive', distal transport within coherent atmospheric structures. In each section I present the physical principles as background, then discuss the development of models based on these principles, then the results of the models compared to data. The Conclusion reviews the results of the earlier sections, points out problem areas that remain and suggests directions for the future.

Pyroclast settling speed The terminal fall speed or settling speed of a particle is an important control on its behaviour in an eruption column as well as in the atmosphere during fallout, and it is critical to characterize it well for computations. Because of the irregular shapes and inhomogeneous densities of pyroclasts, pyroclast settling speed must be determined empirically. The only volcanological datasets available for this determination are those of Walker et al. (1971) for larger pyroclasts, Wilson & Huang (1979) for smaller pyroclasts and Lane et al. (1993) for aggregates. Wilson & Huang (1979) found that the fall speed of pyroclasts of a great range of sizes could be derived from particle size, d, a shape factor, F, and the particle density, a. Particle size and shape factor are best estimated

from the three principle axial lengths of a pyroclast, a, b and c (where a > b > c) as (1) (2)

At the settling speed, ws, the balance between the immersed weight of the particle, and the drag force, FD = condition

gives the (3)

where g is acceleration due to gravity, A = is the projected area of the particle, CD is the drag coefficient, which is a dimensionless number that depends only on the particle Reynolds number Re = wsd/v (v is kinematic viscosity) and shape, and is fluid density. After substituting for A in equation (3) and solving for ws, the settling speed is found to be approximated by: (4)

The functional relationship between CD, Re and F was studied extensively by Walker et al. (1971) and Wilson & Huang (1979). After measuring the fall speeds of hundreds of submillimetre pyroclasts, Wilson & Huang (1979) found that CD is best given by: (5a)

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Suzuki (1983) proposed a modification to this relationship that seemed to provide a better fit to the experimental data, including those of Walker et al. (1971) for centimetre-sized clasts

(5b) Equations (4) and (5b) together completely characterize the vent-height settling speed of a pyroclast in the size range that typifies thick, subaerial fall deposits of several tens of micometres to tens of centimetres. The settling speed can be calculated by an iterative technique applied to initial trial guesses of drag coefficient and fall speed. The results of example calculations to fit the sea-level settling speed data of Walker et al. (1971) and Wilson & Huang (1979) are shown in Fig. 2. For all results discussed in the following sections, the density and shape factor were chosen as appropriate for the volcanic particles being considered. The most extreme value used for either density or shape factor was a value of 0.2 for the shape factor for the highly elongated felsic crystals studied in the downwind plume deposits of Mount St Helens. Pyroclasts are released from height in the atmosphere, and the variation of settling speed between sea level and a cloud-top height of 30 km is substantial. Wilson (1972) calculated the variation of settling speed with altitude and presented results in terms of the variation of fall time with height of release. His results show that a 10-fold increase in release height results in a three- to five-fold increase in fall time. This effect must be carefully considered when calculating

sedimentation patterns for clasts released from a vertical eruption column, because as clasts reach greater heights within the column their settling velocity increases accordingly, and this has ramifications for their fallout behaviour. Adjustment of the settling speed for its variation with height can be done in either of two ways. A temperature structure for the atmosphere can be assumed or measured, and from this the air density and viscosity can be directly calculated. Following this, the same iterative process outlined above for calculating the ventheight settling speed can be used for each height. On the other hand, a much simpler process has proven to yield sufficiently precise results for comparison of models with available data (Bursik et al. 1992b). In this scheme, if ws0 is the vent-height settling speed, then the settling speed at height can be given by ws, such that (6)

where z is height within the atmosphere, and N is the buoyancy frequency (discussed in a following section). Suzuki (1983) compared these two calculation methods and found that they are in rough agreement.

Atmospheric dispersal of particulates Tephra dispersal is a complexly intertwined group of processes in which a single process may predominate depending on such factors as particulate concentration, distance from source, aggregation and atmospheric wind structure. The best understood of dispersal processes can be modelled on the basis of solutions to the equation of turbulent advection-diffusion (Csanady 1980), given by

(7)

Fig. 2. Results of numerical calculation of sea-level settling speed. The line connecting the triangles follows the calculated settling speeds for particles of density 1000 kgm - 3 and shape factor of 0.8. Data from Walker et al. (1971) and Wilson & Huang (1979) as replotted by Suzuki (1983).

where t is time, x, v and z are components of a rectangular coordinate system, with corresponding velocity components u, v and w, Ci, is the concentration of particles in a given grain size class, K is an empirically determined eddy diffusivity for the atmosphere that is not necessarily isotropic (hence its separate components) and is a source or sink term that can be used to

TEPHRA DISPERSAL describe the change in particle concentration through time caused by aggregation or aerosol ripening. The second through to the fourth terms on the left-hand side represent the advection of particles by the steady wind. Depending primarily on particle size, the fourth term can represent settling. The first through to the third terms on the right-hand side can be used to model the spread of tephra by atmospheric turbulence structures, characterized by the eddy diffusivity. Equation (7), with appropriate boundary conditions, is used in the following discussion as the starting point from which dispersal characteristics for particular situations are calculated. We first treat the proximal region near the volcanic plume proper in which the dispersal of tephra is dominated by plume structure and size, and not by atmospheric motions. Proximal dispersal

Volcanic plumes In the near-vent region, the movement of tephra is intimately intertwined with the volcanic eruption plume itself. Most of the tephra in this region resides in the coherent eruption column or umbrella cloud structure. It is therefore appropriate to consider tephra dispersal based upon the concentration and velocity structure of the eruption plume itself. For the purpose of understanding the dispersal of tephra in the atmosphere, it is useful to distinguish several types of volcanic eruption plumes. The first distinction that we make is between plumes that issue directly from the vent and those that are generated from pyroclastic density currents

119

(Fig. 3). (Various names have been given to such plumes. The present work will use the term coignimbrite in a generic sense.) The recognition of coignimbrite clouds began with the observation of plumes rising from small pyroclastic density currents (e.g. Anderson & Flett 1903), eventually leading to the identification of the diagnostic features of coignimbrite ash deposits (Sparks & Walker 1977). It is now thought that many of the largest ash deposits may well be of coignimbrite origin as they are typically fine grained and are intimately associated with large ignimbrites (Woods & Wohletz 1991). In terms of tephra dispersal the critical characteristics of coignimbrite plumes are their fine grain size ( in contrast to in Plinian eruptions; Woods & Bursik 1991), their brief eruption duration relative to their rise time (these times are often subequal, making the flow closer to that of an instantaneous release of a finite (although large) parcel of gas and ash, rather than a continuous plume) and their widely distributed source region, which in the case of Mount St Helens, for example, covered an area of 600 km2 (Sisson 1995). The second important distinction is between those weak plumes that are profoundly influenced by wind as they rise, and those energetic plumes that are sufficiently vigorous to be affected relatively little by atmospheric motions (Sparks et al. 1997). Only nearly vertically directed eruption plumes issuing from the volcanic vent have been studied in sufficient detail to understand their effect on the spread of tephra. The following discussion therefore concentrates on this type of plume. The vertically directed column of a vigorous volcanic eruption is often a relatively steady

Fig. 3. Schematic illustration of eruption plumes generated from the volcanic vent and from a pyroclastic density current at distance from the vent, (a) Eruption of a vigorous plume from the central vent, also illustrating the different sections of a volcanic eruption column; (b) eruption of a coignimbrite plume from a pyroclastic flow that has travelled down the volcano flank; and (c) eruption of a weak plume with wind. The plume is rapidly bent over.

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flow, driven upward over most of its rise height by its low density relative to that of the surrounding atmosphere (Woods 1998). The density drops below atmospheric above a lowermost gasthrust region (Fig. 3), where the plume is dense and its rise is dominated by initial momentum. The low density is generated primarily by pyroclasts heating air engulfed by the turbulent eddies within the plume. In a stratified environment, such as the atmosphere, the buoyancy (which can be taken as the density of the surrounding atmosphere, a, minus the bulk density of the flow, ) is constantly changing with height, as air entrained at lower altitudes is lifted upward and decompressed, as well as heated and expanded. Eventually the buoyancy changes sign to negative at the neutral density height, Hb, and from this point, the volcanic material continues upward because of its momentum. The plume then reaches a maximum height at which its velocity approaches zero, the rise or column height, Ht. The material between Ht and some level near Hb begins to spread laterally as an intrusive gravity current to form the umbrella cloud. The bulk motion of a buoyant plume as described in the preceding paragraph is treated in the classic work of Morton et al. (1956). To a reasonable approximation, a volcanic plume can be well modelled using the approach of Morton et al. (but see Woods 1998 for a thorough treatment). Many of the scaling relationships discussed by them have proven extremely useful in predicting the behaviour of volcanic plumes (Wilson et al. 1978; Sparks et al. 1997), thus their model represents the bulk properties of a volcanic plume to within errors typically arising in field observations. Morton et al. (1956) based their work on three equations representing the values of fluid variables integrated across the entire width of a plume. These are equations for continuity, momentum flux and what they term 'density deficiency', which is derived from the more common equation for energy flux. Despite the spectacular successes in engineering as well as volcanology that have been obtained using the Morton et al. formulation of the equations of motion, it is useful to show its relationship to more exact volcanological formulations to better understand its potential pitfalls. The equations of Morton et al. for continuity and momentum flux can be derived from the more exact equations for a volcanic eruption column that were used by Woods (1988) or Glaze & Baloga (1996), by applying the Boussinesq approximation that (the air density at vent height). For comparison, part (a) below gives Glaze & Baloga's equation, and part

(b) gives the Morton et al. equation. Thus for continuity or mass flux

(8a) (8b)

and for momentum flux (9a) (9b)

For the energy flux, it is also necessary to assume that the entrainment of thermal energy from the surrounding atmosphere dominates the conversion of thermal energy to gravitational potential energy (thus the first term on the righthand side of equation (l0a) is much larger than the second term), and that the heat capacity of the material in the plume is the same as that for the atmosphere outside the plume. Thus

(l0a) can be manipulated as follows

which reduces to equation (10b) through assumption of a constant coefficient of thermal expansion and use of the Boussinesq approximation:

(l0b) In equations (8)-(10), b is characteristic plume width, w is taken as a characteristic vertical plume speed, ae is an empirical entrainment constant, equal to about 0.095 (Morton et al. 1956), g is the gravitational acceleration, Cv is the heat capacity of the erupting mixture, T its temperature, and Ce the heat capacity and Te

TEPHRA DISPERSAL the temperature of the ambient atmosphere. Note that the vertical structure of the atmosphere can be characterized either in terms of density or temperature, hence equation (10b) was simplified by assuming that

in which N is the (nearly constant) buoyancy or Brunt-Vaisala frequency of the atmosphere, and is the potential temperature given by = T e (p(0)/p(z)) R / C e , where p is atmospheric pressure and R is the gas constant for air. The buoyancy frequency represents the strength of the restoring force in the atmosphere to small adiabatic perturbations of air parcels. Its value can be found from the standard atmosphere tabulations; N2 is about in the troposphere and 4 x 10-4-5 x 1 0 - 4 s - 2 in the lower stratosphere (Pedlosky 1987). The largest errors in using the Morton et al. equations arise in the uppermost plume, where a significant fraction of thermal energy has been converted to potential energy (thus, the assumption that the potential energy can be neglected in equation (10) is violated), and in the lowermost plume, where the plume density can differ significantly from that of the atmosphere (thus, the Boussinesq approximation is not valid). However, flow in the uppermost plume is dominated by the umbrella cloud, not the vertically directed plume. The errors introduced near maximum plume height will not significantly affect the calculation of dispersal as this is handled by a separate set of equations for the umbrella cloud. Thus, we should expect the Morton et al. formulation to work best in the middle reaches of smaller eruption columns, yet it should not be too unreasonable for larger plumes. A model for a vertical eruption column. To the extent that the data contained within tephra deposits are suitable for determining eruption column characteristics from the fall deposits, equations (8b), (9b) and (l0b) have proven adequate for describing the eruption column (Bursik et al. 1992b; Ernst et al. 1996). These equations are easily solved by a Runge-Kutta routine. For initial conditions, the volume flux at the source, Q0 = , is needed. Other variables can be held constant as they either have much less direct effect on plume rise (Wilson et al. 1978; Glaze & Baloga 1996), or as they typically vary little in nature. These other variables can be taken to be the ambient temperature at the vent, temperature difference

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between erupting fluid and ambient atmosphere, and initial vertical velocity. This model has been tested extensively against plume rise height data, and against dispersal data for the Fogo A eruption, Azores (Sparks et al. 1997). The results for plume rise height are discussed at length in Sparks et al. (1997) and corroborate that the scaling relationships derived by Morton et al. from equations (8)-(10) are valid even for energetic volcanic plumes piercing the tropopause. The best results in modelling dispersal data (see section below) seem to be obtained when the buoyancy frequency is assumed to have a constant value typical of the troposphere, as most plumes rise primarily within it. Umbrella cloud In unstratified environments, such as laboratory tanks, plumes will continue upward until contacting the upper surface of their immediate environment or until they are completely diluted in the ambient fluid. This occurs because their density cannot rise above that of the ambient. In stratified environments such as the atmosphere, however, plume density eventually climbs above ambient, and the plume thereafter is subjected to a downward gravitational force (Fig. 3). The plume's momentum will carry it to Ht, from which it will slump back gravitationally toward Hb. The downflow will encounter the upflowing, trailing plume material resulting in the formation of a wedge-shaped, intrusive gravity current, the umbrella cloud. In general, these currents have an extremely sharp leading edge and a smooth outer appearance relative to the plume, suggestive of a lower degree of turbulence (see, for example, Abraham & Eysink 1969) (Fig. 4). Sparks et al. (1997) derive a relationship for the growth of an umbrella cloud fed by a steady column

(11) in which R is the radius of the outer edge of the umbrella cloud at any time, and A is an empirical constant of order unity. Therefore, the radius of the cloud may be found at any time given the initial cloud radius, bHb, and the steady flow rate into the cloud at height Hb, , and if available an initial estimate of cloud volume when the intrusion began, or Note that equation (11) can be used in the case of a finite release time by allowing the second term on the right-hand side to converge to zero at the time of eruption termination.

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Fig. 4. Photograph from the Space Shuttle mission STS-068 showing the flattening and spreading of the 1 October 1994, plume of Kliuchevskaya near the tropopause. Note also that the plume border becomes smoother with height and distance because of the weakening of the internal turbulence.

Models of the umbrella clouds of Mount St Helens, Pinatubo and Hekla. Equation (11) is an analytical expression for umbrella cloud radius as a function of time. Two possible methods for calculation of as input have been used (numerically from equations (8)-(10), and from a scaling relationship derived from Morton et al.; Bursik et al 1992b) yielding similar results. Calculations of umbrella cloud growth were compared against data for the 18 May 1980 eruption of Mount St Helens, for the 15 June 1991 eruption of Pinatubo (see also Sparks et al. 1997) and for the 1942 eruption of Hekla. The eruption of Mount St Helens on 18 May 1980 produced a vast umbrella cloud that was observed by several satellites, including the GOES-West weather satellite. Sparks et al. (1986) produced timed tracings of the cloud. The tracings can be used to plot the growth of the cloud as a function of time (Fig. 5). The data are consistent with the growth of an umbrella cloud from a continuous source with Ht 20-25 km. This height is consistent with the measurements

Fig. 5. Variation of umbrella cloud radius with time for a number of eruption parameters. Numbers next to model curves are values of Ht in km. The growth of the 29 March 1947, Hekla; 18 May 1980, Mount St Helens; and 15 June 1991, Pinatubo plumes are plotted against model curves. For Pinatubo, data from both Koyaguchi & Tokuno (1993) and Holasek el al. (1996a) are plotted. In these calculations: N = 0.017s= -1 , A = 0.8.

TEPHRA DISPERSAL from imagery of the height of the initial cloud of 28 km and of the cloud fed by the continuous eruption column following the initial cloud of 17km (Harris et al 1981; Carey et al. 1990). There is some indication of a decreasing growth rate with time (Fig. 5), consistent with the suggestion of Woods & Wohletz (1991) that the emplacement time for the initial cloud was comparable to the column rise time of about l0min, and thus that the cloud was initially fed by a column of height 28 km, thence by a column of height 17km. The sequence of gigantic eruptions of Pinatubo, in the Phillipines, on 15 June 1991 produced numerous umbrella clouds that were observed by the Japanese Geostationary Meteorological Satellite (GMS). Koyaguchi & Tokuno (1993) produced timed tracings of the largest of the clouds which was generated by the paroxysmal event. We have also independently corroborated the accuracy of the measurements. The data are consistent with the growth of an umbrella cloud from a continuous source with Ht 25-40 km. This height is consistent with the measurements from the imagery of the height of the outer edge of the cloud of at 25 km (Koyaguchi & Tokuno 1993) and with SAGE II measurements of an aerosol concentration between 20 and 25 km, and maximum altitude of 29 km (McCormick & Veiga 1992), and an eruption duration at least as long as the time of cloud growth. The estimated cloud top height from this method is also consistent with the single measurement of 34-40 km made of the central swell (Tanaka et al. 1991; Holasek et al. 1996a but it is not known how long this height was maintained. Holasek et al. (I996a) have finished a more detailed study of the Pinatubo umbrella clouds that corroborates the findings reported here.

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A sequence of timed photographs of the 1947 eruption of Hekla were taken by an amateur photographer from sufficient distance to allow a full view of the entire developed eruption cloud. These photographs were traced and scaled by Thorarinsson (1950) (Fig. 6). The plume reached a maximum height of 25-30 km only 20 min after the beginning of the eruption, but had decreased to the level of the tropopause (10 km) 1 h later. Maximum spreading rates occurred around a height of 20 km, both in the upwind and downwind directions. A prominent 'ledge' or flattening in the cloud top also occurred about 1 km above this height. The model calculations (Fig. 5) are consistent with a 20km cloud-top height, near the 'ledge' height. This may indicate that the maximum height between 25 and 30 km was so short-lived that it did not affect cloud growth, or that the eruption may be better modelled as a relatively instantaneous event, so that the contribution of the second term on the right-hand side of equation (11) contributed to cloud growth for a negligible fraction of the time that the cloud was observed.

Downwind plume With distance, wind eventually dominates plume motion. Viscous friction between the upper and lower interfaces of the downwind plume and the atmosphere results in downwind transport at the windspeed (Chen 1980). In the crosswind direction, the lenticular plume continues to spread because it is more dense at its top and less dense at its base than is the surrounding stratified atmosphere. Behaviour in this regime is thus dominated by conservation of volume flux downwind and gravity flow crosswind (Chen 1980; Bursik et al. \992a). Assuming

Fig. 6. Tracing of the Hekla 1947 plume redrawn from Thorarinsson (1950) showing the growth of the umbrella cloud with time. Times shown are Iceland Standard Time. Grid spacing is 5 km, horizontal and vertical.

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that neglible amounts of air are entrained and that crosswind plume shape can be approximated by an equivalent rectangle, the characteristic (half-) width as a function of distance can be expressed as (Sparks et al. 1997)

(12) where u can be taken as the mean windspeed at the level of the plume. Because of their corresponding greater depth, plumes from higher eruption columns spread more rapidly crosswind than do those from lower eruption columns given the same wind. Conversely, given separate plumes of the same height, those erupted into a region of higher winds will be distorted into a more elongated shape. Ernst et al. (1994) point out that under certain conditions that may be strongly dependent on the nature of the atmospheric stratification, and especially on the strength of the temperature discontinuity at the tropopause, plumes may spread due to gravity with an angle of widening much greater than that suggested by equation (12), occasionally causing the plume to split or bifurcate, as perhaps exhibited by the Rabaul plume of September 1994. However, calculations with the CANERM advection-diffusion model are consistent with the unusual spread being the result of strong wind shear effects (Fig. 7). Further downwind, the plume loses its density contrast with the atmosphere and can be advected as a lens of aerosol and gas with nearly constant width (Sarna-Wojcicki et al. 1981; Robock & Matson 1983). Only slowly does it thin, spread and disperse as shearing and smallscale atmospheric turbulence act at its margins, and as very fine ash settles out. The plume mixes with the turbulent atmosphere as it is advected. and plume width should increase with downwind distance by turbulent diffusion (Taylor 1921; Csanady 1980; Rose et al. 1988) such that in a steady mean flow (Sparks et al. 1997)

(13) where it is assumed that Fickian diffusion occurs (Bursik et al. \992a). (In both equations (12) and (13), the entire visible plume width should be about 4b, i.e. the visible plume margin lies at twice the characteristic crosswind length scale (radius) from the plume centreline; Morton et al. 1956.) Note that equation (13) should also apply in situations where a plume is just able to rise slowly into the jet stream near the tropopause and has insufficient energy to form an umbrella cloud (Fig. 4).

Models of the downwind plumes of Mount St Helens, Augustine and Kliuchevskaya Equations (12) and (13) can be solved in linear combination for the downwind spread of a volcanic plume with time and distance, assuming that the plume front advances downwind at the windspeed. For weak eruptions in which QH is relatively small, plume width increases with time and distance by atmospheric diffusion; for powerful eruptions, in which K y is relatively small, plume width increases primarily by gravity spreading. The 18 May 1980 eruption of Mount St Helens provides the best-documented example of the development of a downwind plume from a vigorous eruption column. Initially, the giant umbrella cloud of this eruption evolved into a slightly wind distorted oval pattern. As the gravity flow weakened upwind, the plume front reached a stagnation point or radius. R s (Sparks et al. 1986) at a radial distance of 15 km from the vent, where timed satellite observations of the cloud and windspeed measurements from radiosonde showed both cloud and wind travelling at 20-25 m s - 1 . Downwind, the plume spread in a parabolic pattern with vertex facing upstream consistent with equation (12) (Fig. 8). while the velocity of the downwind plume front approached the maximum stratospheric windspeed of 28 ms-1 (Fig. 9). The value of the initial volume flux, Qo. of 5 x l 0 7 m 3 s - 1 . is consistent with Q Hb of 4 10 9 m 3 s - 1 . which slightly overestimates the downwind plume spreading rate. This value for Q () is. however, consistent with observations of plume heights (calculated Ht = 23 km above sea level (ASL)). and with the sedimentation data discussed below. The other value used of 3 x 107 m3 s-1 along with a reasonable estimate of horizontal eddy diffusivity of 1 0 4 m 2 s - 1 (see Armienti et al. 1988) provides a better fit to the plume spreading data. although underestimating the width at greater distances from the vent. This underestimate may reflect the increase in the apparent diffusivity with time to a value larger than that used (Armienti et al. 1988). Frequent small eruptions of Augustine volcano in Cook Inlet. Alaska. USA. and moderate eruptions of Kliuchevskaya volcano. Kamchatka. provide examples of the dispersal of weak to moderate downwind plumes (Figs 10 and 1 1 ) . The puff-like structure of a typical Augustine plume is maintained downwind. and the plume disperses as predicted by equation ( 1 3 ) under the assumption of constant diffusivity. For the plume of 3 April 1986, that is illustrated. the effective horizontal eddy diffusivity is found to be weak at 10m 2 s - 1 , which is within the wide range

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Fig. 7. The extreme spreading of the Rabaul plume of August 1992 is shown in the satellite image (a). (b) Results of the CANERM advection-diffusion atmospheric dispersal model calculation for the plume shown in (a) are consistent with spread by strong wind-shear effects. Colours in (b) signify different concentration levels. Images from R. D'Amours.

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Fig. 8. The width of the Mount St. Helens downwind plume of 18 May 1980, 4b, as a function of distance, x, is consistent with either the model for growth by gravity spreading in the crosswind direction (equation (12); solid curve) or gravity spreading plus relative diffusion (equations (12) and (13); dashed curve). The flow rate used in the model to generate the solid curve was closer to values obtained by cloud-height measurements (e.g. Harris el al. 1981). Width was measured as the greatest north-south extent of the plume (data from Sarna-Wojcicki el al. 1981; see also Holasek & Self 1995).

Fig. 9. The distance to which a downwind plume fron travels is consistent with the assumption that it is advected at the windspeed. For the Mount St Helens eruption, maximum windspeeds near the plume base were recorded to be 29 m s -1 .

of values measured for diffusivity over brief time intervals of l 0 - l 0 4 s - 1 (Heffter 1965). The low value of diffusivity could be related to the strong wind shear noted during this eruption (Rose et al. 1988). The flow rate of material from the vent was extremely low, such that the plume was virtually sheared off at the volcano summit. In the calculations we have assumed the lowest flow rate that yields reasonable results from the eruption column model of Q0 = 2500 m3 s- 1. For

Fig. 10. For the Augustine plume the relationship for width as a function of distance is consistent with the model for downwind plume spread by atmospheric eddy diffusion (equation (13)), assuming a very low but reasonable value for atmospheric turbulent diffusivity. Wind speed is assumed to be 2 2 m s - 1 (Rose et al. 1988).

Fig. 11. The width of the Kliuchevskaya plume as a function of downwind distance is consistent with measured values of plume height and windspeed (unpublished data) - consistent with the model for spread by gravity and turbulent diffusion (equations (12) and (13)), but with a low value for horizontal eddy diffusivity. the Kliuchevskaya plume (Fig. 11) of 1 October 1994, which rose to an altitude of 15-18 km ASL, we have assumed Q0 = 3 x 106 m3 s - 1 , consistent with an eruption cloud height of 18km, and a windspeed of 25 m s - 1 , about that measured at the time of the eruption by radiosonde. The results are consistent with the data assuming negligible horizontal eddy diffusivity.

Pyroclast fallout Two methods have often been used to model the dispersal and fallout of tephra in the proximal

TEPHRA DISPERSAL region of volcanic eruption plumes. Both methods can be derived from the advection-diffusion equation by approaching it in different fashions. In the first instance, for the region closest to the plume including those areas directly under the eruption column and umbrella clouds themselves, it can be assumed that the positions of pyroclasts are controlled in large part by the plume itself (Fig. 12). Thus, a relatively robust model of the plume itself is used as discussed in the previous sections, coupled with a simplified version of the advection-diffusion equation (Bursik et al 1992a, 1992b; 1993; Sparks et al 1992; Thomas & Sparks 1992; Koyaguchi 1994). In the second method, which seems to produce results that are more reasonable at distances of 100-1000 km from vent, a simplified plume model is used along with a relatively complete version of the advection-diffusion equation. This method was first applied by Richardson & Proctor (1926) to the deposits from the December 1920 plumes of the Asama volcano, Japan. The modern version of this method for use in volcanology was first articulated by Suzuki (1983) and was later used by Armienti et al. (1988), Barberi et al (1990) and Glaze &

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Self (1991). All of the distal transport models discussed in a later section are derived fom this same formulation, but use interpolated, near real-time meteorological data to calculate plume trajectories and deposition. Most pyroclasts within a plume are retained as suspended particles because of their low fall velocities relative to plume velocity (Sparks et al. 1997). They are then released in vast quantities from the umbrella cloud, where there is no longer an upward flux of gas supporting the particles. In reality, particles within a volcanic eruption column can follow a variety of pathways once released from the plume (Fig. 12). If we simplify this picture to include only the primary particle paths - those dictated by turbulent-gravitational settling - a model for the proximal sedimentation of suspended particles from plumes and umbrella clouds can be derived from the advection-diffusion equation (equation (7)). It is assumed that the terms related to transport by diffusion processes are negligible, that the plume is maintained and long-lived (steady), and that motion of the carrying medium (eruption column, umbrella cloud or downwind plume) can be characterized by a single coordinate

Fig. 12. Schematic diagram showing the numerous trajectory that particles suspended within an eruption plume can take once they are elutriated from the plume. Most particles are ejected from the umbrella cloud, where the processes of convective sedimentation (Hoyal et al. 1998) and backflow to form a secondary intrusion (Holasek et al. \996b) may drastically alter fallout patterns.

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direction s - for which the axis is everywhere tangent to the plume centreline (e.g. Wright 1977; Hopkins & Bridgman 1985) - and speed U in that direction. Under these assumptions, equation (7) reduces to

Deposition can be measured in a number of ways. If it is measured as the mass of material deposited per unit distance in the flow direction. m si in k g ( s m ) - 1 , then (22)

(14)

If furthermore it can be assumed that

Alternatively, if deposition is measured in unit area, S i (kgs -1 m - 2 ) in the flow direction, then Si = Ciws and

(15)

(23)

where um = (U, ws) and xm = (s, z) taken in turn, and that the particles exit the volcanic flow because of a step-like concentration gradient at the flow's edge, such that

Any of these various formulations of the model can be applied to different flow geometries to calculate the amount of sediment released from a steady, moving fluid as a function of distance.

(16)

Sedimentation calculated for different plume geometries. For steady flow of the eruption column equation (20) can be written

where H(z) is the Heaviside step function, then

(24) (17)

Integrating through the depth of the flow:

(18) we obtain

(19) If we define M, = CiU x n d S , in which Mi is the mass of particles passing a given distance in unit time, n is the unit vector everywhere tangent to the flow direction and S is a surface through which plume material erupted at a single time passes, then (20)

In the case of a volcanic plume, the mass of particles that passes beyond a distance s from the plume centreline over the entire eruption duration, T, is Msi = M i T. As T is a constant, equation (20) is valid for Msi as well as for Mi. Note that Msi is also the total amount of material deposited beyond a given distance. Note that as settling speed increases or flow speed decreases, the sedimentation rate per distance increases. Equation (20) has solutions of the form (21)

where 2p is a geometric correction factor with a value of approximately 0.5 based on scaling arguments (Sparks et al. 1997). In reality, because of the strong inflow back toward the plume itself, pyroclasts up to 1 m (in the most vigorous eruptions) become re-entrained at lower heights in a plume, after falling from greater heights. The vast majority of particles in fact follow such trajectories, and therefore there can be little deposition of pyroclasts from the vertical eruption column itself. Ernst et al. (1996) have studied the re-entrainment phenomenon in detail. They have found that equation (27) can be modified to account for this by assuming that a re-entrainment region exists next to the plume such that if particles are within this region during a given time step, then the atmospheric entrainment velocity is sufficient to draw them back within the plume in time dt. Thus, the re-entrainment is proportional to the entrainment velocity, u e . and the rate at which the plume spreads or. alternatively, the characteristic increase in width. b. with time or height (25)

Equation (25) is solved using equations (8)-(10) in a numerical integration scheme to characterize the fallout from a vertically directed plume. Ernst et al. (1996) found that the re-entrainment factor, f, is relatively constant for all particles within a plume having ws/w0 < approximately 1, at a value of 0.4.

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Sedimentation from umbrella clouds in a quiescent atmosphere is modelled using the continuity condition with no entrainment, thus

If we substitute s r, U vr and h Ht - Hb (in equation (21)), and if l/(H t - Hb)vr = 2 r/QHb (by equation (24)), then we can integrate equation (21)

(27) to obtain

(28) where QHb is taken as input from the eruption column model output. In cases where grain size isopleths and isopachs are only slightly distorted by wind into ovals, a geometric correction factor can be substituted for (Bursik et al. 1992b). The method that has been successfully used assumes that the isopleths are in fact ellipses, and uses the standard calculations for ellipse perimeter based on the lengths of the semi-major and semi-minor axes. The wind modified shape of a downwind plume results in the elongated sedimentation patterns so typical of most fall deposits. If the models for the gravitational and turbulent spreading of the downwind plume (equations (12) and (13)) are adopted and applied within the relationship for sedimentation from a steady flow (equation (21)), then the mass flux of particles within a downwind plume will take the form

(29) where .xO is the distance at which particles enter the downwind plume from the spreading umbrella cloud. Proximal fallout from the Fogo A and Mount St Helens eruption plumes. For the separate parts of the volcanic flow - eruption column, umbrella cloud and downwind plume - the equations above can be used to calculate the mass of particles remaining in the flow at any distance. The initial masses of particles in different grain size fractions are determined from large sedimentological datasets wherein total grain size distributions for volcanic eruption plumes have been collated from sieve data for the major

Fig. 13. The grain size distributions in units that are the averages of available data on total grain size distributions for the eruption types shown (from Woods & Bursik 1991). The model discussed in the text breaks each of these distributions into nineteen discrete grain size fractions.

eruption types (Woods & Bursik 1991) (Fig. 13). Equation (25) has been coded into a RungeKutta routine for calculation of the changing mass of particles within the vertically directed plume, based on these initial values. Analytical solutions to equations (28) and (29) were combined into the the umbrella cloud and downwind plume spreading routines to calculate particle mass fluxes as the plume material begins to spread within the atmosphere using eruption column parameters for particle concentration and volume flux as boundary conditions. One question that might arise at this point regards the validity of equation (21) for the situation that occurs in volcanic eruption columns. Equation (21) was developed for control volumes in which the base of the volume was a solid surface and the no-slip condition necessarily held. It is not clear a priori whether this is in fact valid for plumes at height within the atmosphere. We have therefore performed experiments using two layers of fluid in tanks in the laboratory (Hoyal et al. 1998). The upper fluid contained particles and was light relative to the lower fluid, and therefore the density gradient tended to stabilize the interface between the layers. The upper fluid was then stirred to obtain a relatively uniform distribution of particles within it. The change in the concentration of particles within this upper fluid was then monitored through time using a calibrated light attenuation technique to measure the decreasing turbidity within the upper layer. The results showed quite unambiguously that the model of equation (21) was in fact valid to a relatively high degree of certainty (Fig. 14).

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Fig. 14. Results of experiments to ensure that the mathematical-physical model for loss of particles from the volcanic flows is reasonable. The fits of the model to the data are extremely good, suggesting that the simplified form of the advection-diffusion relationship (equation (21)) used in the calculations of proximal dispersal is based on sound physical principles. The mass of silicon carbide particles of different grain sizes (in grit sizes (#) from about 5 to 50 ) contained within the upper, particle-laden layer of an experimental tank decreases as an exponential function of time at a rate closely dependent on known particle fall speeds and upper layer depths (see also Hoyal et al. 1998). In terms of volcanic eruptions, the validity of the sedimentation models can be tested by relying on the relationship between the mass of particles carried within a plume and the mass that is elutriated and eventually settles to the

Fig. 15. Results from the model for deposit generated from the eruption column of the Fogo A event, for the 16-mm grain size fraction. The model yields results in excellent agreement with the data, although the model eruption duration is probably excessive, suggesting problems with the model grain size distribution. Earth's surface. Granulometric data from several volcanic deposits are in fact consistent with fallout and deposition from turbulent suspension in the vertical eruption column, the nearly radial gravity current and the windblown plume. The Fogo A deposit has provided data that can be compared with the radial umbrella cloud current and the eruption column models (Bursik et al 1992b). Results from the Fogo A eruption showed excellent agreement with both the eruption column (equations (8)-(10) and (24); Fig. 15), and the radial umbrella cloud

Fig. 16. Results of model of deposition from the umbrella cloud responsible for the Fogo A deposit. The model yields results in excellent agreement with the data.

TEPHRA DISPERSAL models (equation (28); Fig. 16). In the model run illustrated, the initial volume flux was 3 x 10 7 m 3 s - 1 , leading to an eruption cloud height of 22km. This value for cloud height is consistent with previous estimates of 21 km from inverse modelling (Bursik et al. 1992b), and 27 km from the maximum clast method of Carey & Sparks (1986). The eruption column model outputs deposition masses in kgm - 2 s - 1 . Therefore the fit to the data was accomplished by multiplying deposition at the locality closest to the vent (1.7km) by an eruption duration to fit the data at the other points. The factor (eruption duration) necessary to match the data at 1.7km from vent gives an eruption duration of approximately 6 x 105 s, or many days. As few eruptions persist for such a length of time, this result may suggest that the total grain size distribution (0 ± 3 Woods & Bursik 1991) underestimated the mass of particles in the 16-mm grain size fraction. A different total grain size distribution would yield a different eruption duration, but would not change the shape of the curve as illustrated. The laboratory work of Ernst et al. (1996) further corroborated that the eruption column model is reasonable, but also suggested that re-entrainment may sometimes be an important process. The Mount St Helens deposit from the 18 May 1980 eruption has provided data on deposition from the windblown plume (Bursik et al. \992a). The data for the downwind plume of Mount St Helens suggested that, to distances of at least 600km, the source flow characteristics rather than the atmospheric turbulence structure dominated the sedimentation pattern (equation (29); Fig. 17). The initial volume flux used in these calculations was 5 l 0 7 m 3 s - 1 , yielding an eruption column height of 23 km ASL. Thus, the plume model is consistent with data from Mount St Helens for the eruption plume height, spreading rate of the umbrella cloud, spreading rate of the downwind plume and deposition from the downwind plume. Woods et al. (1995) have shown that, after the initial gravity spreading of the downwind plume of Mount St Helens, the dispersal of fine pyroclasts of 10-100 and the bulk movement of the plume was indeed controlled by the ambient atmospheric motions. For both the Fogo A and the Mount St Helens deposits the maximum in sedimentation for many grain size classes appears to occur near where the plume begins lateral flow as an umbrella cloud or downwind plume. This suggests that the loss of vertical column speed and turbulence intensity weakens re-entrainment (f is approximately 0) and allows fallout for the grain sizes that dominate the deposits of vol-

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Fig. 17. The downwind plume deposition patterns from the Mount St Helens eruption of 18 May 1980 are compared with model results in this graph. Note the slight misestimation of the rate of fallout for the larger grain size fractions. A slightly higher volume flux would yield somewhat better results than those illustrated.

canic eruption columns, from approximately 100 to 10 cm for these intermediate-sized eruptions with column heights of 20-40 km.

Development of the downwind plume Fine ash between approximately 5 and 100 generally falls from the downwind plume at distances between 100 and 1000km of the vent. Because the ratio of surface charge to size is high for the smaller of these particles or because of the presence of moisture in the plume, and because they may remain within the plume for a long time, they can aggregate before falling out (Sorem 1982; Sparks et al. 1997). A distinct separation of gas and ash phases in the downwind plume has also been hypothesized to develop, as inferred both from electrical charge measurements (Gilbert et al. 1991) and from satellite imagery (Woods et al. 1995; Holasek et al. 1996b). Finally, the gravitational spreading of the plume can be extremely weak at these distances, hence the plume motion may be controlled in large part by ambient atmospheric motions. At these distances and for these grain sizes, therefore, it has proven important to consider the advection-diffusion equation in its full form, sometimes including the source term for particle aggregation, in modelling tephra dispersal and fallout (Armienti et al. 1988). These dispersal models are essentially the same as the distal dispersal models used in aviation warning systems, which are discussed in the next section.

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Distal transport of volcanic eruption products As an eruption comes to a close, the accumulated volcanic gas and ash ejected into the atmosphere assumes one of two typical forms (Figs 18 and 19). Smaller plumes are often markedly elongated in the downwind direction, with a prominent taper toward the upstream end. The taper is the result of either the gravity current spreading or the atmospheric relative diffusion having affected the downstream end for a longer time than the upstream end. Larger umbrella clouds are more often subequant in planform, with an elongation in the downstream direction that makes them occasionally ovoid. Because they are often the result of a complex sequence of events, larger, developed umbrella clouds are often compounded from several separate clouds. They can therefore display a complex digitate geometry. In addition, the most massive, such as that from Pinatubo, display another type of digitate margin that indicates they have reached the Rossby radius. When such a volcanic plume has persisted for a sufficiently long time, it becomes fully sub-

jected to atmospheric motions. These motions fall into a spectrum of characteristic scales, from microscopic molecular agitation to hemispheric geostrophic flow. To gain an understanding of the effects of the atmosphere on the dispersal of the finest particles generated from volcanic eruptions, the horizontal and vertical motions of the particles can be separated. The horizontal dispersal of the particles is controlled by the nearly two-dimensional nature of the largest atmospheric motions that transport ash to regional and global scales. The vertical dispersal of the particles is controlled by the smallest-scale turbulence and gravitational settling, which act on much smaller length scales. As with dispersal at distances of 100-1000 km, regional and hemispheric spread of tephra has been handled with the full advection-diffusion equation (equation (7)). Several distal transport models have been developed by national laboratories to assist in aviation hazards warning. These organizations and models include: (1) NOAA Air Resources Laboratory (ARL) Volcanic Ash Forecast Transport and Dispersion (VAFTAD) model (Heffter & Stunder 1993); (2) Lawrence Livermore National Laboratory (LLNL) Air

Fig. 18. Initial shape typical for small to moderate-sized eruption plume in the atmosphere (red), an elongated, somewhat bent lozenge of ash, gas and evolving aerosol. Other colours show a simulation of the evolution through time of the cloud as it is advected, stretched, bent and diffused in the atmospheric flow. Each colour represents a different time. The dispersing plume is represented by the centres of mass of drifting parcels of plume material. The windfield that generated the cloud spread is also shown; the field evolves through time, so the figure shows an instantaneous field.

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Fig. 19. Initial shape typical for large eruption plume developed from an umbrella cloud in the atmosphere (red), a subeqant 'droplet' of material. Other colours show a simulation of the evolution through time of the cloud as it is advected and diffused in the atmospheric flow. Each colour represents a different time. The dispersing plume is represented by the centres of mass of drifting parcels of plume material. A typical instantaneous windfield that generated the plume spread in part is also shown. Compare the size of the dispersing particle 'patch' in this figure relative to the wavelength of the windfield with that shown in Fig. 18.

Release Advisory Capability (ARAC) model (Sullivan & Ellis 1994); (3) Canadian Meteorological Centre (CMC) Canadian Emergency Response Model (CANERM) (D'Amours 1994); (4) Persistent Environment Aircraft Response Model (PEARL; described by Versteegen et al 1994); and (5) the model of Tanaka (1994). VAFTAD, CANERM and PEARL (?) are computed in an Eulerian frame, whereas the ARAC model and that of Tanaka are Lagrangian. All these models use as input interpolated data for the winds along the path of a particular plume being tracked. To obtain an understanding of the principles of distal, long-range transport from a general viewpoint, rather than for the purposes of calculating trajectories for particular eruptions, it is useful to simplify the advection-diffusion equation to neglect vertical transport (as w u, v for the quasi-two-dimensional geostrophic motions that dominate these scales of transport) and break the equation into two separate equations (e.g. Tanaka 1994), in which the advection

component is based on the spectral components of the velocities (e.g. Hopkins & Bridgman 1985; Jenkins 1985). In this formulation, the largescale advection can be characterized by the continuity equation for incompressible, twodimensional flow of the form

(30) The local dilution of the plume material by small-scale turbulence can then be satisfied by a diffusion equation of the form (e.g. Tanaka

1994)

(31) where x' and y' are local coordinates originating at the centre of mass of a given fluid parcel. The large-scale, quasi-two-dimensional atmospheric motions responsible for the distal transport of ash clouds (equation (30)) can be

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described by the turbulence energy spectrum (Gifford et al. 1988) (32)

where E is the kinetic energy of the motion at a given length scale, k is the wavenumber corresponding to the length scale, and A and 7 are constants. For the intermediate-scale atmospheric motions of wavelength 10 -3 -10 3 km (energy cascade; Gifford et al. 1988), which will dominate the spreading of most plumes, 7 has a value of 5/3. Atmospheric motions can then be simulated by using the summation or superposition of waves of progressively higher wavenumber (shorter wavelength) (e.g. Saucier 1983; Hopkins & Bridgman 1985; Pedlosky 1987) with kinetic energies (velocities) dependent on wavenumber, for example (simplifying from Behringer et al. 1991; see also Hopkins & Bridgman 1985)

(33)

where is the streamfunction, and m designates the wavenumber index. To ensure continuity (equation (30)) the velocities in the x and y directions are given by (34a)

used to approximate or a numerical smoothing kernel technique (Monaghan 1992) can be used to solve for concentration within each fluid parcel. The results of separate realizations of this procedure are illustrated in Figs 18 and 19 for small- to intermediate-scale and large-scale volcanic plumes, respectively. Only the centres of mass of the plume parcels are shown in these figures. The effect of the different scales of motion on plumes of different characteristic size is vastly different despite the fact that all atmospheric agitation has its origin in the incoming solar radiation. The largest-scale motions (greater than approxiomately 103 km) primarily draught virtually all plumes along, but cause little growth or diffusion. Observations of plumes distorted in these geostrophic and zonal winds suggest that distortion primarily occurs as simple stretching and very moderate bending. Intermediate-scale eddies are primarily responsible for complex bending and relative diffusion of small to moderate-sized ash clouds (Fig. 18). This complex bending is known as chaotic advection (Pierce & Fairlie 1993), as fluid elements are distorted in a chaotic fashion, despite the completely deterministic nature of the velocity field itself. The plume from the Hudson eruption of 1991 exhibited this phenomenon as it drifted at the edge of the polar vortex (Schoeberl et al. 1993). The largest volcanic ash clouds, such as

(34b) The energy-cascade spectrum (equation (32) can be synthesized if m are related by (Jenkins 1985)

(35) where each km — bkm+1, and b is an arbitrary scale factor. A model that demonstrates the effects of the different scales of atmospheric motion on the spread of volcanic eruption products can then be constructed by taking a few terms of the summation in equation (33), with wavenumbers separated by b = 10, for example. The centres of mass of separate plume parcels can be tracked in Lagrangian form

(36) where x = (xy), u = (uv) and t is a finite time increment, so long as u and v are related by both their continuity and spectral constraints (equations (30) or (34), and (35)). If it is assumed that the small-scale horizontal turbulence is isotropic, then an analytical solution (Tanaka 1994) can be

Fig. 20. Atmospheric sounding for Buffalo, NY. USA. from 18 November 1993, showing the layered nature of the stratosphere. Layers of positive lapse rate are separated by turbulent, isothermal layers and inversion layers. Provided by B. Coniglio, National Weather Service, Buffalo.

TEPHRA DISPERSAL that generated by the eruption of Pinatubo, seem to show almost exclusively relative diffusion by the intermediate-scale motion, with little apparent stretching or bending (Fig. 19). Vertically, a volcanic plume does disperse, but at a rate very different from the horizontal dispersion rate discussed above. This vertical dispersion occurs because of the coupled effects of the settling properties of the particles and of the

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turbulence structure of the stratosphere. Maekawa et al (1993) recently used a high-resolution VHF radar to investigate the vertical structure of the stratosphere in the 'gap region' between 30 and 60km. Their investigation showed the existence of alternating layers of 1-2 km thickness. Every other layer displayed turbulent motions with vertical windspeeds between -0.5 and 0.5ms - 1 . The other layers

Fig. 21. TOMS SO2 maps for the 4 April 1982 eruption plume of El Chichon, Mexico, showing the advection and relative diffusion of the plume through time. From D. Schneider & W. Rose.

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were relatively quiescent. These observations are consistent with the layered temperature structure observed in stratospheric soundings (Fig. 20). Because of this alternating turbulent and quiescent structure in the stratosphere, volcanic ash clouds will tend to separate into layers over long periods, lending them a distinct banded appearance (Jonsson et al. 1996). The turbulent layers will be able to retain particles longer than the quiescent layers because the turbulence will retain particles in suspension. Particles will fall relatively rapidly through the quiescent layers because of the effects of convective sedimentation (Hoyal et al. 1998).

Atmospheric transport of volcanic clouds and fine particles The effect of the different scales of atmospheric motion can be seen in the dispersal of the plumes of El Chichon, 1982, Lascar, 1986, Spurr, 1989, and Kliuchevskaya, 1994. Sparks et al. (1997) have discussed the plumes of El Chichon, 1982, Pinatubo, 1991, and Hudson, 1991, in a similar light. The zonal and geostrophic windfields affect

long-range transport, even as the smaller-scale turbulence acts to stretch, bend and diffuse the clouds. The umbrella clouds generated by El Chichon, Mexico, in March-April 1982, display the effects of the intermediate-scale atmospheric motions on a large volcanic cloud. Numerous volcanic eruptions occurred between 29 March and 4 April 1982, although the 4 April event was the largest (Sigurdsson et al. 1984). A Plinian column formed that later collapsed to generate pyroclastic flows, from which coignimbrite material was also elutriated to high in the atmosphere. For this main eruptive phase, volcanic debris was emplaced between 25 and 30 km in altitude (Hoffman 1987). The material drifted downward in altitude over the next 6 months to a height of about 20 km. Kreuger (1983) presented results of the TOMS satellite tracking and dissipation of the eruption cloud (Fig. 21), which remained confined to a relatively narrow latitudinal band for several months, although the concentrated zones of SO2 clearly underwent relative diffusion during that time. The intermediate-sized plumes from Lascar, Spurr and Kliuchevskaya all show the effects of

Fig. 22. Timed sequence of GOES imagery for Lascar eruption of 1986 showing drafting, bending and eventual dissipation of the plume over a time of 3h (each image was taken 30 min after the preceding image). Provided by L. Glaze. Copyright Springer-Verlag, reproduced with permission.

TEPHRA DISPERSAL chaotic advection in the high-latitude Rossby flow around the mid-latitude system of high- and low-pressure cells (Figs 22-24). Lascar erupted in two brief vulcanian bursts on 16 September 1986 that generated an umbrella cloud and downwind plume event, although the eruption column had dissipated within 30 min of its first appearance (Glaze et al. 1989). Photographic analysis revealed a plume height of 15 ± 1 km, which was reached within 5 min of the first burst. The GOES satellite captured a sequence of six images of the plume as it was bent, stretched and eventually dissipated in the westerly flow. Glaze et al. noted that the bend in the plume moved downwind at a slower rate than the plume head. Glaze & Self (1991) estimated the fallout pattern from the plume using an advection-diffusion model that accurately reproduced a point measurement of fallout. The Spurr eruption of 17 September 1992, generated a sub-Plinian plume that reached an altitude of 15 km with a base altitude of about 7 km (Ellis et al. 1993).

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Although the plume was initially subequant with a diameter of about 40 km in planform view, it was rapidly stretched to a southeast elongated downwind plume. The plume was then rapidly draughted by the Rossby flow to the southeast thence draughted and folded to the northeast across North America, during which time it was monitored by the AVHRR instrument (Fig. 23). Each of the ARL VAFTAD, LLNL ARAC and CMC CANERM models were used to forecast the ash trajectory using interpolated values for the winds aloft. The fits of all models to data are reasonable, although there is often a difficulty with discriminating the imaged edges and hence the relative diffusion of such a cloud in these transport codes (Fig. 25). The Kliuchevskaya eruption of 1 October 1994 shows a regional transport similar to that of both the Lascar and Spurr plumes (Heffter 1996). The downwind plume was injected into the stratospheric flow to the southeast by gravity spreading and relatively diffusion. It was then drafted, stretched and

Fig. 23. Draughting and bending of the Spurr eruption cloud as imaged by the AVHRR instrument. Note that the figure is a montage of images of one plume recorded at different times. Image has been band ratioed to emphasize the content of volcanic ash. Provided by D. Schneider & W. Rose.

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Fig. 24. Draughting and bending of the Kliuchevskaya eruption plume, also imaged by AVHRR. Note that the figure is a montage of images of one plume recorded at different times. Provided by D. Schneider & W. Rose. bent as it was carried further to the south-east, thence to the northwest in the Rossby flow. To a degree greater than the Lascar and Spurr plumes, the Kliuchevskaya plume also seems to have been disarticulated into several distinct entities by the flow. The simulation of the motion of these smaller clouds (Fig. 18) shows that many of the features of their motion can be explained by a limited number (around three?) of atmospheric wave modes. Based on a visual comparison of the simulation results with the satellite imagery, it becomes apparent that over the period of observation, each of the small to moderate sizedplumes described above exhibit three predominant modes of motion: (1) draughting and stretching in a long wavelength (A 10 3 km) flow at relatively high speed; (2) bending by

an intermediate wavelength motion (102 < A < 103 km); and (3) relative diffusion by a very short wavelength turbulence (10 - 3 < A < 1 km?) that results in slow cloud spread and dilution. Motion (1) above is the geostrophic or zonal flow that carries all air parcels in relatively narrow latitudinal bands about the Earth. Motion (2) is the Rossby, synoptic-scale motion that is induced by the regional system of high- and low-pressure cells; and motion (3) is the smallscale atmospheric turbulence that we sense in frontal systems and the spread of atmosperic clouds overhead. The predominance of three wavenumbers in determining the distal transport of volcanic plumes for observational time scales is related to the resonant triads of waves that are forced by the atmospheric dynamics (Pedlosky 1987). This lack of a smooth spectrum of

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Fig. 24. (continued) turbulence energy in the atmosphere is responsible for the breakdown of the advectiondiffusion models for volcanic debris transport (Suzuki 1983; Armienti et al 1988; Glaze & Self 1991) at scales of more than approximately 10 3 km. Simulation of the motion of a larger cloud, such as that of El Chichon (Fig. 19), shows that while such a cloud is draughted in the zonal flow, the smaller scales of motion, as they are of a characteristic size much smaller than the size of the cloud, cause the cloud to diffuse as it is draughted along, consistent with the TOMS observations.

Conclusions and future perspectives There are several possibly important paths of future research that are suggested by the results presented herein. Our understanding of basic

plume propagation rates and bulk properties appears to be reasonably evolved. Within errors of approximately 20-50% that can be expected from the quality of existing data and the varying ambient flow fields of the atmosphere, predictions of plume geometry and spread seem reasonable for plinian events, as well as coignimbrite eruptions. In general, fallout of suspended pyroclasts from vigorous Plinian-style eruptions seems reasonably well understood, in terms of total deposition at ground level. Nevertheless, studies confirming theory and laboratory work are few, and more should be performed. Excellent candidates for such studies may be the nearly radially symmetrical deposits recently described from Pululahua and Cotopaxi, Ecuador (Papale & Rosi 1993; Rosi et al 1995). Proximal dispersal from near-vent heights in the eruption column,

Fig. 25. Results of (a) the ARAC code for 1903 Z, and (b) the CANERM code for 1800 Z for atmospheric releases simulating the transport and dispersal of the Spurr eruption cloud at 18 : 53 Z. 19 September 1992. Compare with 19 September 1992 18:53 Z satellite image from Fig. 23. Figures provided by R. D'Amours & J. Ellis.

TEPHRA DISPERSAL from short-lived events (such as vulcanian blasts), wherein the rise time is longer than eruption duration, from weak plumes injected into a relatively vigorous windfield (bent-over plumes) and from coignimbrite plumes remain virtually unstudied. Because a great number of eruptions fall into these categories with serious ramifications for aviation and climate change, it is important to better characterize dispersal from such events. The mechanism of transport between plume and ground has not been investigated in much detail, and clearly effects such as backflow (Holasek et al. \996b) and convective sedimentation (Hoyal et al. 1998) need to be investigated more thoroughly. With respect to the total grain size distribution of clasts within volcanic plumes, the results from Fogo A suggest that the population of clasts within fall deposits is at present poorly characterized. Plinian eruption columns may in fact contain many more coarse particles than is currently thought, with important implications for the fallout pattern near the vent, not only from the plume itself, but also from ballistic pyroclasts. Clearly near-vent deposits should be better characterized, although this may never prove practicable because of difficulty of access. The assumption of spread of downwind plumes by gravitational spreading, thence by the wind and turbulent diffusion, seems to fit much of the available data in a general fashion as remarked above. However, the transition between these two modes of transport is as yet very poorly characterized. The results to this point suggest that our understanding of medial dispersal of ash, especially where atmospheric and plume dynamics are both important and aggregation occurs, could be developed much further. Information on this transition woud greatly aid in the further development of midrange advection-diffusion models of transport. Long-range trajectories of ash are reasonably characterized by models such as VAFTAD, CANERM and ARAC, as well as by GCMs. Nevertheless, the relative diffusion of the clouds is at times rather poorly predicted by such models, and it is important to determine more fully how the attenuation of the clouds is accomplished through turbulent diffusion, and particle dynamics and settling. The laminated or layered nature of the stratosphere also needs to be studied in terms of its effects on particle settling. Such studies will have major ramifications for the intermediate time scale (months long) development of the volcanic ash and aerosol components of eruption clouds. Given the availability and good quality of both satellite data on plume growth, and ground

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data on fallout, it seems likely that our ability to predict dispersal will continue to improve in the near term, especially as more sophisticated models for plumes and the atmosphere are developed. This work was supported by grant EAR-9316656 from the National Science Foundation and by funding from Scientific Applications International Corporation. B. Coniglio (NWS) is thanked for helping develop the author's understanding of the atmosphere. L. Glaze, W. Rose, LLNL (J. Ellis) and CMC (R. D'Amours) are thanked for allowing the use of images. I thank reviewers T. Druitt, S. Carey, L. Glaze, D. Pyle and G. Ernst for copious helpful comments.

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Pyroclastic density currents T. H. DRUITT Departement des Sciences de la Terre (CNRS-UMR 6524), Universite Blaise Pascal, 5 Rue Kessler, 63038 Clermont-Ferrand, France Abstract. High-speed, gravity-driven flows of hot particles and gas are a common and highly destructive product of explosive volcanism. They range widely in nature from expanded, turbulent suspension currents formed by lateral blasts or by the fountaining of vertical eruption columns, to highly concentrated granular avalanches formed by lava dome col-lapse or as dense underflows beneath suspension currents. The deposits from these flows, here called pyroclastic density currents, range in volume from much less than 1 km3 to thousands of cubic kilometres, and may extend over 100 km from their source. This chapter reviews the eruption, transport and deposition of pyroclastic density currents from both geological and physical perspectives, focussing on some recent advances. Nomenclature

Bi C c

g

C

P d D DEL g H k L lm

m N Pn Re Ri V V*

w V

n P

T

y

0

Bingham number Particle concentration in flow Ground drag coefficient Particle drag coefficient Particle diameter (m) Flow depth (m) Boundary layer thickness (m) Gravitational acceleration (m s - 2 ) Vertical drop of flow (m) Height of ground roughness elements (m) Horizontal runout of flow (m) Prandtl mixing length (m) Flow mass (kg) Ground-normal stress (Pa) Rouse number Reynolds number Richardson number Flow velocity (ms - 1 ) Flow shear velocity ( m s - 1 ) Particle terminal fall velocity ( m s - 1 ) Height ordinate in flow (m) Density contrast between flow and air (kgm- 3 ) Bulk viscosity of flow (Pa s) Bulk density of flow (kgm - 3 ) Particle density (kgm - 3 ) Shear stress (Pa) Yield stress (Pa) Internal angle of friction (°)

Pyroclastic density currents are rapidly moving mixtures of hot volcanic particles and gas (with or without free water) that flow across the ground under the influence of gravity. They form by the gravitational collapse of lava domes, by the fallback or continuous fountaining of vertical eruption columns or by lateral blasts

(Fig. 1). Such currents are highly destructive and have caused the deaths of more than 10000 people since 1700 (Blong 1984). During the 8 May eruption of Montagne Pelee in 1902, 2800 people were killed. Pyroclastic density currents at Mount St Helens, USA (1980), Mount Unzen, Japan (1991) and Soufriere Hills, Montserrat (1996) have each taken their toll of human life. Volcanologists conventionally recognize two end-member types of pyroclastic density current based on the textural characteristics of their deposits (Fisher & Schmincke 1984; Cas & Wright 1987). Deposits from pyroclastic surges mantle the landscape and are commonly tractionbedded. Pyroclastic surges are dilute suspension currents in which particles are carried in turbulent suspension and in a thin bed-load layer. The deposits from pyroclastic flows are generally poorly sorted, massive and thicken markedly into depressions. These characteristics are attributed to sedimentation from highly concentrated flows or from the concentrated bases of flows that are density stratified. Pyroclastic flow deposits containing pumice as a major constituent are called ignimbrites. There probably exists a continuous spectrum of density currents that encompasses what are known conventionally as pyroclastic surges and pyroclastic flows (Valentine 1987; Fisher 1991; Branney & Kokelaar 1992; Druitt 1992; Cole & Scarpati 1993). In this chapter I restrict the term pyroclastic flow to the high-concentration end member of the spectrum. The physics of pyroclastic density currents poses a considerable challenge to volcanologists. This is in part because their high velocities, high temperatures and unpredictable nature make real-time monitoring difficult, but also because the physics of high-speed multiphase flows is still in its infancy. Advances have been made in recent years by combining field studies, laboratory

DRUITT, T. H., 1998. Pyroclastic density currents. In: GILBERT, J. S. & SPARKS, R. S. J. (eds) The Physics of Explosive Volcanic Eruptions. Geological Society, London, Special Publications, 145, 145-182.

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Fig. 1. (a) Vulcanian explosion and column collapse at the Soufriere Hills Volcano, Montserrat, in August 1997. The spherical cap of the plume is about 2 km in diameter. Pyroclastic density currents can be seen travelling down drainages to the left and right of the column. Thin pumice flows that formed by the deflation of these currents travelled as far as 5km from the vent, (b) The lateral blast at Mount St Helens on 18 May 1980. The resulting density current travelled up to 25 km from the mountain. The field of view is about 4 km across.

experiments on real and analogue systems, and theoretical modelling. Pyroclastic currents have been observed at several volcanoes including Mount St Helens (e.g. Moore & Rice 1984; Hoblitt 1986; Calder et al 1997), Mount Unzen (Sato el al 1992; Yamamoto et al. 1993) and Montserrat (Cole et al. 1997), and estimates made of their flow paths and velocities. These data have been used to test physical models and to compare the physical behaviour with that of other types of geophysical flow (Kieffer 1981; McEwan & Malin 1989; Levine & Kieffer 1991). Studies of the deposits from these and other

eruptions have been valuable in developing facies models and in relating depositional features to observed flow behaviour (e.g. Fisher 1991; Druitt 1992). The eruption of Mount Pinatubo in 1991 (Newhall & Punongbayan 1996) gave volcanologists the opportunity to document a moderate-sized ignimbrite-forming eruption using modern geophysical methods and to study thin facies of the ignimbrite sheet before they were removed by erosion. Studies of both historic and prehistoric deposits have led to the development of sophisticated facies models (e.g. Wilson & Walker 1982; Walker 1985; Sohn& Chough 1989;

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Fig. 1. (continued) Cole & Scarpati 1993; Scott et al 1996). In some cases use of historical records has permitted reconstruction of prehistoric eruptions such as Vesuvius (e.g. Sigurdsson et al. 1985) and Krakatau (e.g. Carey et al. 1996) that generated pyroclastic density currents. Laboratory experiments have cast light on the role of escaping gases in the formation of ignimbrite grading (Wilson 1980, 1984). An important development has been the construction of physical models of flowing particulate suspensions, making it possible to test models against real data (Bursik & Woods 1996; Dade & Huppert 1996). Another advance has been the development of computer simulations of high-speed multiphase flows (Wohletz et al. 1984; Valentine & Wohletz 1989; Valentine et al. 1991; 1992; Dobran et al. 1993; Giordano & Dobran 1994; Neri & Dobran 1994; Straub 1996). This chapter summarizes some recent advances in our understanding of the eruption and emplacement of pyroclastic density currents. First, brief descriptions are given of pyroclastic surge and flow deposits as conventionally recognized. These are not meant to be exhaustive as many excellent reviews have already been published (Sparks 1976; Sheridan 1979; Wohletz & Sheridan 1979; Fisher & Schmincke 1984; Wilson 1986; Cas & Wright 1987; Carey 1991). The aim is to highlight features that are important in the

development of physical models, which are reviewed in the second half of the chapter. The last section includes suggestions for future lines of research. The terms dilute or low concentration are used to mean solids concentrations of several per cent or less. Dilute suspensions are maintained by fluid turbulence, and particle interactions are to a first order negligible (except in any bed-load layer). A dense or high-concentration mixture has a solid content of tens of per cent and particle interactions are important.

Pyroclastic surges and their deposits Pyroclastic surges commonly form during phreatomagmatic eruptions in which magma comes into explosive contact with water. Such base surges spread out as radial clouds across the ground or sea. Base surges occur at maars, tuff rings and tuff cones, stratovolcanoes with crater lakes, and volcanoes in marine settings. Sudden decompression of highly pressurized magma generates another type of pyroclastic surge called a lateral blast. These may be strongly directed, as when triggered by flank collapse of a stratovolcano like Mount St Helens (Kieffer

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1981) or Bezymianny (Belousov 1996), or can expand radially in all directions as at Mount Lamington (Taylor 1958) and Mount Pinatubo (Hoblitt et al. 1996). Lava domes may also produce lateral blasts (e.g. Boudon & Lajoie 1989; Fink & Kieffer 1993). Some pyroclastic surges are related directly to ignimbrites. Thin surge deposits are laid down at the leading edges of pyroclastic flows (ground surges) or by the billowing ash clouds overriding them (ash-cloud surges). Some ignimbrites have landscape-mantling veneer facies that resemble surge deposits. Pyroclastic surges travel at high velocity as turbulent suspensions. The front of the 1980 Mount St Helens lateral blast travelled at speeds up to 150ms -1 (Moore & Rice 1984), although the internal velocity may have reached 235 m s-1 (Kieffer & Sturtevant 1988). The blast had an estimated Reynolds number of 5 x 10 11 , so it was highly turbulent (Kieffer & Sturtevant 1988). The runout distance of base surges rarely exceeds a few kilometres. The Mount St Helens lateral blast travelled up to 25km before becoming buoyant and lofting to form a large umbrella cloud. Some of the 1991 preclimactic blasts at Mount Pinatubo travelled as far as 14km (Hoblitt et al. 1996). Some pyroclastic surge

deposits associated with ignimbrites are much more widespread. Possible ground surge deposits beneath the Peach Springs Tuff occur up to 100km from source (Valentine et al. 1989). If the temperature of a pyroclastic surge is less than 100oC, any steam present condenses and the surge becomes a three-phase suspension of solid particles, water droplets and gas (Wohletz & Sheridan 1979). This is called a wet surge. If the mean temperature is greater than 100 C the surge is said to be dry and free water is more likely to be absent. Dry surges form by phreatomagmatic explosions where the water:magma ratio is low or by eruptions that are purely magmatic. Temperatures as high as 250 C have been estimated for the Mount St Helens blast (Banks & Hoblitt 1981). Deposits from pyroclastic surges are well bedded on a centimetre to decimetre scale (Fig. 2). Beds commonly exhibit planar, wavy and crossstratification, while others are inversely graded, normally graded or massive. The characteristics of surge deposits have been reviewed by Wohletz & Sheridan (1979). Fisher & Schmincke (1984). Cas & Wright (1987) and Carey (1991). who provide full reference lists. Deposits from wet surges plaster vertical or overhanging surfaces and contain soft-sediment deformation

Fig. 2. Pyroclastic surge deposits of the 11 ka Upper Laacher See Tuff (Germany). The well-developed sandwave bedding is typical of surge deposits. The flow direction was from left to right.

PYROCLASTIC DENSITY CURRENTS structures. Accretionary lapilli and vesicular layers are also common. Deposits of hot, dry pyroclastic surges contain carbonized wood and are in general better sorted than those of wet surges because the segregation of particles of different sizes is more efficient in the absence of free water (Walker 1984). The deposits from some dry surges have a characteristic bipartite layering, with a coarse, fines-poor layer overlain by a thinner one richer in fine components (Schmincke et al 1973; Hoblitt et al. 1981; Fisher et al. 1983; Belousov 1996). Surge deposits exhibit a variety of bedforms (Schmincke et al. 1973; Allen 1982; Cole 1991). The most common ones are low-angle sandwaves up to 20m in wavelength and up to 2m high. Wavelengths decrease away from the vent, probably due to dynamic effects in the bed-load layer of the decelerating suspension (Valentine 1987). Sandwaves of both progressive (downstream migrating) and regressive (upstream migrating) type are observed. Allen (1982) suggested that regressive sandwaves form by the stoss-side plastering of damp ash, and are typical only of wet surges. However, it is now known that they also occur in dry surges and are in some cases closely associated with progressive forms on a local scale (Cole 1991; Druitt 1992). This implies a hydrodynamic origin rather than one related to cohesive effects, although the mechanism is not well understood. Most surge deposits exhibit systematic facies variations when traced away from the vent. Some wet surges have a proximal facies dominated by cross-stratified and massive beds, a distal planarbedded facies and a medial facies in which all three bed types occur (Wohletz & Sheridan 1979). Facies variations in dry surges are different. For example, deposits from the 1982 eruption of El Chichon in Mexico are cross-stratified throughout their extent but are intercalated in proximal regions with massive, unsorted beds that pass laterally into pyroclastic flow deposits (Sigurdsson et al. 1987). On Lipari Island, Italy, dry surge deposits of the Monte Guardia sequence exhibit systematic facies variations when traced away from the vent from disorganized, to disorganized and/or stratified, to dune-bedded, to planar bedded (Colella & Hiscott 1997). Similar variations are recorded from the Suwolbong Tuff Ring, Korea (Sohn & Chough 1989). At Mount St Helens the blast deposit is predominantly massive in depressions but becomes stratified on ridge tops and as the deposit is traced progressively further from the vent (Hoblitt et al. 1981; Fisher 1991; Druitt 1992). Flat-topped pyroclastic flow deposits occur intercalated with the proximal facies on a local scale (Fisher et al. 1987).

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Pyroclastic flows and their deposits

The spectrum of pyroclastic flow deposits The deposits from pyroclastic flow eruptions range in volume over at least six orders of magnitude (Smith 1979). Those associated with calc-alkaline stratovolcanoes such as Merapi (Indonesia) and Mount Unzen (Japan) typically have small volumes ( 1 to Ri < 1 down a slope. A hydraulic jump may occur where a current passes from supercritical to subcritical (Freundt & Schmincke 1986).

Particle transport in the suspension current Consider the behaviour of particles of different sizes in the highly polydisperse suspension current. A particle of fall velocity w is transported in turbulent suspension only if the Rouse number (2)

is less than about 2.5 (Valentine 1987). 0 varies in the range 0.12-0.65, with a commonly accepted average of 0.4. The shear velocity v* is a function of the freestream velocity v of the current, DBL the boundary layer thickness and k the height of ground roughness elements (3)

In suspension currents travelling between 100 and 300ms - 1 , only lithic clasts in the range 1-1 0cm or less can be transported efficiently by turbulence (Valentine 1987) (Fig. 20). This assumes a roughness factor (k) of 1 m and a boundary layer thickness DBL of 5m. Lower ratios of DBL:k would increase these figures. Nevertheless, given that the largest clasts to leave the vent may be several metres in size, it is clear that under many conditions only a fraction of the total population will be transported effectively by turbulence in the suspension current. Particles greater than 10cm typically constitute a few per cent of the total mass discharged during ignimbrite eruptions. There are thus two populations of particles in the initial suspension current: Pn > 2.5 and Pn < 2.5. Large particles with Pn > 2.5 are not supported by the turbulence and settle rapidly to the base of the current close to the vent. Particles for which Pn < 2.5 are carried in suspension, but to different degrees (Valentine 1987). Those with Pn