Advances in Applied Mechanics Volume 13
Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN L. HOWARTH WILLIAM ...
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Advances in Applied Mechanics Volume 13
Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN L. HOWARTH WILLIAM PRAGER T. Y. Wu HANSZIEGLER
Contributors to Volume 13 H. L. Kuo GEORGE VERONIS JOHN
V. WEHAUSEN
ADVANCES IN
APPLIED MECHANICS Edited by Chia-Shun Yih COLLEGE OF ENGINEERING THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN
VOLUME 13
1973
ACADEMIC PRESS
New York and London
COPYRIGHT 0 1973, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. N O PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY F OR M OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION I N WRITING FROM T HE PUBLISHER.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1
LIBRARY OF
CONGRESS CATALOG CARD
NUMBER:48-8503
PRINTED I N TH E UNITED STATES O F AMERICA
Contents
vii
LISTOF CONTRIBUTORS
PREFACE
ix
Large Scale Ocean Circulation George Veronis Introduction I. The Equilibrium Figure of a Self-Gravitating, Rotating, Homogeneous Mass of Fluid 11. Transformations of the Equations of Motion of a Fluid 111. The Coriolis Acceleration IV. Thermodynamic Simplifications-the Boussinesq Approximation V. Scaling of ,the Equations VI. Geostrophic Flow VII. Frictional Dissipation VIII. Modeling of Current Systems IX. The Thermohaline Circulation X. Abyssal Circulation XI. Laboratory Simulation of Large Scale Circulation (with C. C. Yang) References
2 3 6 14 18 28 33 36 42 56 72 75 90
The Wave Resistance of Ships John V. Wehausen I. Introduction 11. The Measurement of Wave Resistance 111. The Analytical Theory of Wave Resistance Bibliography References V
93 96 131 229 230
Contents
vi
Dynamics of Quasigeostrophic Flows and Instability
Theory El. L . Kuo I. Introduction 11. Tendency Toward Geostrophic Balance in Rotating Fluids 111. Simplified Hydrodynamic Equations for Large Scale Quasigeostrophic Flow IV. Permanent-Wave Solutions of the Nonlinear Potential Vorticity Equation in Spherical Coordinates V. Stability of Zonal Currents for Small Amplitude Quasigeostrophic Disturbances VI. General Stability Theory-Integral Relations and Necessary Conditions for Instability VII. Stability Characteristics of Barotropic Zonal Currents and Rossby Parameter VIII. Pure Baroclinic Disturbances IX. Finite Amplitude Unstable Disturbances X. Instability Theory of Frontal Waves XI. Concluding Remarks References
AUTHORINDEX SUBJECT INDEX
248 250 257 265 272 276 281 291 306 316 327 328
331 336
List of Contributors
Numbers in parentheses indicate the pages on which the authors’ contributions begin.
H. L. Kuo, Department of Geophysical Sciences, The University of Chicago, Chicago, Illinois (247) GEORGE VERONIS, Department of Geology and Geophysics, Yale University, New Haven, Connecticut (1)
V. WEHAUSEN, Department of Naval Architecture, University of California, Berkeley, California (93)
JOHN
vi i
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Preface
I n this volume Professor H. L. Kuo treats instabilities of the atmosphere, Professor George Veronis discusses the dynamics of the ocean, and Professor John Wehausen reviews wave resistance of ships. Thus two of the three authoritative articles are studies of our fluid environment, and the third is a testimony of one aspect of man’s successful adaptation to it. This volume should therefore appeal to meteorologists, oceanographers, and naval architects, as well as to fluid dynamicists in general. I n view of recent concerns with the environment and with the relevance of scientific work to human activities, the three articles presented herein are perhaps timely. From another point of view, it can be persuasively argued that scientific work itself is an important element of the quality of life, because it bears upon the human spirit. It is hoped that the excellence and the degree of permanence achieved in these articles will lend support to this now somewhat forgotten point of view.
CHIA-SHUN YIH
ix
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Large Scale Ocean Circulation GEORGE VERONIS Department of Geology and Geophysics Yale University. New Haven. Connecticut
Introduction
......................................................
I . The Equilibrium Figure of a Se1f.Gravitating. Rotating. Homogeneous
Mass of Fluid ................................................... I1 . Transformations of the Equations of Motion of a Fluid .................. I11. The Coriolis Acceleration ........................................... A . Conservation of Potential Vorticity ................................ B. An Angular Momentum Argument for the Neglect of 2 a cos 4 ........ IV . Thermodynamic Simplifications-the Boussinesq Approximation .......... A . The Adiabatic Hydrostatic Field .................................. B. The Boussinesq Equations ....................................... C . Use and Limitations of the Boussinesq Approximation ............... V . Scaling of the Equations ............................................ A . Small Scale Motions-the f-plane ................................. B. Motions of Intermediate Scale-the /%Plane ........................ C . Large Scale Motions ............................................ VI . Geostrophic Flow .................................................. VII . Frictional Dissipation ............................................... A . Ekman Layers .................................................. B . Ekman Layers in the Ocean ...................................... VIII . Modeling of Current Systems ........................................ A. Wind-Driven Ocean Circulation ................................... B . Extensions of Stommel’s Model ................................... C . Inertial Effects .................................................. IX . The Thermohaline Circulation ....................................... A. The Pressure Equation .......................................... B. Boundary Conditions at the Top and Bottom ....................... C . Solutions by Means of a Similarity Transformation .................. D . Generalization of the Similarity Solutions .......................... E . Solutions to the Ideal-Fluid Thermocline ........................... F. The Effect of a Barotropic Mode .................................. G . The Role of Diffusion ........................................... H . Remarks about Thermohaline Circulation Models ...................
1
2 3 6 14 15 17 18 19 21 23 28 30 31 32 33 36 36 40 42 43 49 53 56 57 58 58 60 63 65 68 71
2
George Veronis X. Abyssal Circulation.. ............................................... XI. Laboratory Simulation of Large Scale Circulation (with C. C. Yang) . . . . . . A. The Basis for the Simulation ..................................... B. Discussion of More Complete Solutions.. .......................... C. The Flow Due to a Source of Dense Water.. ....................... References ........................................................
72 75 75 78 81 90
Introduction Our theoretical picture of large scale ocean circulation has grown mostly out of the development of simple models which isolate the particular phenomenon to be analyzed. In this sense dynamical oceanography differs substantially from dynamical meteorology which has progressed hand-in-hand with the amount and types of observational data that have been accumulated. The difficulties and costs of gathering oceanographic data preclude the same type of development of oceanographic theories. The present paper contains a discussion of some of these simple theoretical models together with an attempt to extend a few of them to take into account additional features which are not normally included in the models. The presentation is necessarily selective and another author would no doubt have emphasized other models or other approaches. At the outset the plan was to discuss steady state models as well as those which include transient behavior. However, as the work progressed it became necessary to restrict attention to steady models only. The opening section includes a simple model for deriving the ellipticity of the earth. I t is followed by the derivation of the equations of fluid motion in elliptical coordinates and the approximation involved in the use of a spherical coordinate system to analyze oceanographic motions. The latter is included because the errors associated with the use of spherical coordinates are normally referred to in a casual fashion and no real attempt is made to quantify them. The approximations encountered in theoretical studies of large scale flows are then discussed and the stage is set for introducing theoretical modeling. Simple geostrophic flows and their significance are presented next because they form the basis for the remainder of the paper. The study of Ekman layers and the role that they play in large scale circulation are followed by a brief discussion of turbulent transport.
Large Scale Ocean Circulation
3
T he remaining portion of the article is devoted to a discussion of three types of studies. Th e first is wind-driven ocean circulation beginning with Stommel’s (1948) model exhibiting westward intensification of ocean circulation. Some of the difficulties in extending his model are pointed out in the subsequent development which includes an attempt to incorporate density variations into the model. Theories of the thermohaline circulation (the flows driven by sources and sinks of heat and salt) are then reviewed. T h e paper ends with a section, authored jointly with C. C. Yang, on laboratory modeling of ocean circulation. T he presentation concentrates on the evolution of theoretical oceanography by means of simple models. Because this philosophy was adopted here, the important contributions of numerical studies of ocean circulation have been referred to only in passing. It is obvious that future work in the field will probably rely more and more heavily on numerical computations, particularly as the complex phenomena associated with nonlinear interactions become the focus. Th e reason for restricting attention here to simple analytical and laboratory models is that the latter still provide the theoretician with the clearest understanding of the state of the art and what steps one must take to extend that understanding.
I. The Equilibrium Figure of a Self-Gravitating, Rotating, Homogeneous Mass of Fluid Consider a mass of incompressible, homogeneous fluid which is rotating with a constant angular velocity, S2, about a given axis. T h e velocity, v, of a fluid element at distance, R,from the axis of rotation is
and the acceleration is
and is directed inward toward the axis of rotation. At a point within the fluid the equation for the conservation of momentum is
where p is the (constant) density, p is the pressure, and F is the body force
George Veronis
4
per unit mass. Taking F = -V @ where 0 is the Newtonian gravitational potential and making use of (1.2)in (1.3) yields S2 X (S2
x R)= -V(Q2R2)/2 = -V(p/p) - V@
or
+
V[(PIP)
@ - (Q2R2/2)] = 0.
(1.4)
The first integral of (1.4) yields
( p / p )+ 0 - (Q2R2/2)= const.
(1.5)
For a finite mass such as the earth the pressure at the outer (equilibrium) surface must be a constant and (1.5) becomes
CD - (Q2R2/2)= const.
(1.6)
on the outer surface. Since the gravitational potential, @, depends on the shape of the body, the problem is an implicit one. The earth is very nearly spherical and it is convenient to express the radius as the sum of the radius, a, of the smallest inscribable sphere plus the deviation, 5, due to the bulge which results from the rotation (Fig. 1) r=a+c.
(1.7)
Then the gravitational potential, @, on the surface can be written as the sum of the potential due to the mass within the inscribed sphere and the potential due to the mass in the bulge. The potential at an external point due to the spherical mass can be expressed as the potential due to a point with all of the mass at the center. Thus, @=-
where 4 is the potential associated with the bulge and G = 6.67 x dyn cm2 gm-2 is the gravitational constant. Then (1.6) becomes 1 2
- Q2R2= const.
or, making use of (1.7),
+---4
mpa3
1
G - - Q2(a+ 5)" sin2 8 = const. 3a+5 2
where 6 is the colatitude (Fig. 1).
Large Scale Ocean Circulation
5
A first approximation to the figure of the earth is to neglect 4 in (1.9) and make use of the fact that 5 < a. Since the term involving the rotation in (1.9) is small, the lowest order expression involving 5 is $rrpa2G(5/a) - +Q2a2sin2 % = const.
(1.10)
FIG.1. A sketch of a cross section (through the polar axis) of an elliptical earth with an inner inscribed sphere of radius a. The bulge due to the rotation is denoted by 5.
The difference between ( / a evaluated at the equator (0 = ~ 1 2 )and the pole (0 = 0) yields the ellipticity (C/a)eq.-
( 5 / a ) p o l e = +Q2/9,~G
= iCPa/g M
1/580,
(1.11)
where g = 4rrGpa/3 = 980 cm sec-2 is the gravitational acceleration at the earth’s surface. The value of the ellipticity given by (1.11) is about half the known value. Higher order approximations in ( / a cannot alter the result substantially and the only correction that may yield an improved estimate must come from 4. A second approximation may be obtained by retaining 4 in (1.9) while still retaining the approximation 5 < a. Then (1.9) becomes
4 + Q rrpaG[ - ;Q2a2 sin2 % = const.
(1.12)
Now the term involving 5 incorporates the geometrical effect of the bulge. On the surface the potential due to the mass of material in the inscribed sphere will be smaller where 5 > 0 and larger where M < 0. However, it is more convenient to choose an alternative manner of accounting for this geometrical effect. The potential is evaluated on the spherical surface Y = a and the geometrical effect is taken into account by assuming a surface distribution of mass at Y = a with negative mass where 5 > 0 and positive mass where 5 < 0. The surface distribution of mass at Y = a due to the bulge associated with 5 can be written in terms of a jump condition at Y = a in the radial gradient of the potential (Ramsey, 1964) as
(1.13)
6
George Veronis
where a + and a - correspond to the value of a approached, respectively, from outside and from inside the spherical boundary. Substituting (1.13) into (1.12) yields
$
+ Qa[a4/ar]:zg? = @Pa2
sin2 0
+ const.
(1.14)
The potential due to the spherical mass is included in the constant term and will not contribute to the ellipticity. If the constant term is neglected, the problem reduces to one in which there is no mass except for the surface distribution on r = a so that Laplace's equation is satisfied inside and outside the surface, i.e.
V+=O
for r < a
and r > a .
(1.15)
Equation (1.15) can be solved by separation of variables, and the pertinent solution for present purposes is
+
= K(r/a)2(1 - 3
cos2 0) = K ( ~ / Y1)~ (3 C O S ~0)
Y
< a,
r > a.
(1.16)
The application of condition (1.14) yields
K = -R2a2/4,
(1.17)
and (1.13) then gives
[a$/a~];rgT = - 5Q2a(l - 3 cos2 0)/4=
--47TGp5.
(1.18)
Hence,
C/U = 5Q2(1- 3 COS'
0)/1&Gp = 5Q2a(1- 3 cos2 0)/12g.
(1.19)
The ellipticity is therefore evaluated as
(1.20) [(/a]& = 5Q2a/4gM 1/232, i.e. the gravitational attraction of the material in the bulge serves to increase the ellipticity by a factor of 2.5. The result given in (1.20) was derived by Newton by another method. Much more exhaustive treatments of the problem have been presented by various authors (e.g., Jeffreys, 1962). The most accurate estimate of the ellipticity (1/298) is based on measurements of orbits of artificial satellites.
II. Transformations of the Equations of Motion of a Fluid T o a very good approximation the shape of the earth can be taken to be an oblate spheroid (sometimes called a planetary ellipsoid) with the minor axis of the ellipsoid along the axis of rotation. The equations of motion
7
Large Scale Ocean Circulation
can be written in terms of oblate spheroidal coordinates with gravity taken as a constant on the surface of the ellipsoid. However, oblate, spheroidal coordinates are never used for the analysis of oceanic or atmospheric currents. T he usual procedure is to work with the equations on a sphere. We shall go through the derivation of the equations in oblate spheroidal coordinates and then show the approximations involved in the use of spherical coordinates. T h e vector form of the equations for the conservation of momentum is aV
at
+V
1 vv f 2 a x v = - - vp - v@, P
(2.1)
where the centripetal acceleration terms are incorporated into the gravitational potential 0.Th e conservation of mass can be written as
5 + v . v p f p v. v = at
0.
T o write the equations in terms of any curvilinear coordinate system we introduce the generalized coordinates (ql, q 2 , q3)where the qi can be related to the rectangular Cartesian coordinates by the relations qi
= qi(X1,
~2
9
~
3
)
or
Xi
= xi(q1,
q2
43).
(2.3)
Then (Margenau and Murphy, 1949) the derivative along any direction, si , can be written as
where the Q i are defined by 3
Qi2=
C
j=l
ax (2)
In general, the following relations will be used 3
v= xvjij,
(2.6a)
j=1
V=
C
l a ij--,
]=I
Q j a q j
(2.6b) (2 .6 ~ )
8
George Veronis
where ii are the unit vectors along the three coordinate directions and the last relation follows from the first two. From ( 2 . 6 ~ )we have
where the last term in (2.8) is necessary because the unit vectors can change direction. Now the surface of the earth is, to a good approximation, an oblate spheroid for which the orthogonal surfaces are (Fig. 2) (1) oblate spheroids,
FIG.2. Oblate spheroidalcoordinates with orthogonal surfaces given by oblate spheroids with q1 = constant, hyperboloids of one sheet with q2 = constant, and planes (one is the plane of the figure) through the xp axis with q 3 = constant. Also shown are the polar coordinates in the equatorial plane.
q1 =const., (2) hyperboloids of one sheet, q2=const., and (3) planes through the x3 axis, q3 = const., where the x3 axis is taken as the minor axis of the ellipse and the xl, x2 plane is orthogonal to the x3 axis. I n the following the coordinates (x,y , z ) will replace (xl, x 2 , x3). In addition, it is useful to define the polar coordinates R, X in the xy plane. Note that A-q3. T h e intersection of the surface of the earth and a plane through the x axis is the ellipse given by
where the values of a and q1 on the surface of the earth are determined by noting that at z = 0, re = R = a cosh ql, where re is the earth’s equatorial
9
Large Scale Ocean Circulation
radius, and at R = 0, rp = z = a sinh ql, where rp is the earth’s polar radius. The curves orthogonal to the ellipses are the hyperbolas R2 a2 sin2 q2
-
2 2
a2 cos2 q2
= 1.
(2.10)
The relations between the coordinates of the different systems are x = a cosh q1 sin q2 cos q 3 ,
y = a cosh q1 sin q2 sin q 3 , z = a sinh q1 cos q 2 ,
(2.11)
R = a cosh q1 sin q2 , y/x = tan X = tan q 3 , x 2 + y 2 = R2.
(2.12)
The values of the Q, in (2.5) take the form
Q1 = Q2= a(sinh2 q1 + cos2 q2)1/2, Q3 = a cosh q1 sin q2 .
(2.13)
Furthermore, the derivatives of the unit vectors can be calculated.
.
ai, - a2 cos q2 sin q2 -_ 12, 841
ail
- tc2 sinh
-392
Qi2
-ail _ - a sinh q1 sin q2 1.3 ,
Qi2
ai,
8 1
ai, -Go,
-ai,_ -0,
-ai,_ -
841
842
a43
.,
12
-a91 _-
393
q1 - cosh q1
M
u2 cos
q2 sin q2 .
(2.14)
11,
Qi2
sinh q1 sin q2 Qi
.
u
11 -
cosh q1 cos q2 . 12 * Qi
The equations of motion expressed in terms of oblate spheroidal coordinates become dv, v1v2 - _ _ _ _ a cos q2 sin q2 -
dt
Qi
VZ2
81
u2 sinh
2Qa . 1 ap 1 a@ sinh q1 sin q2 v3 = - --- --, Qi PQi Qi 841
-__
v z
q1 cosh q1 - 3a sinh q1 sin q2
81Q3
(2.15)
George Veronis
10
v32
91--
a cosh q1 cos q2
Q1Q3
(2.16)
and the conservation of mass equation is
(2.18) where
d dt
- -= _
a +I-3 v i a
at
j = l
(2.19)
Q j aqj
The gravity potential @ appears only in Eq. (2.15) since we have assumed that the equipotential surface is an ellipsoid. Next suppose that the equipotential surface of the earth is taken to be a spherical surface. For spherical coordinates we have (91, q z , 43) = (r, 8, A),
Qi =
sin 8,
(2.20)
z = r sin 8.
(2.21)
1, Q z = r, Q3
=r
and x = r cos 8 cos A,
y = Y cos 8 sin A,
The equations of motion take the form
11
Large Scale Ocean Circulation
where d/dt is defined by (2.19) and the velocity components (v,., v a , vA) are along the directions (r, 8, A). Now, what error is involved in replacing the set (2.15) to (2.18) by the set (2.22) to (2.25)? We answer this question by relating: (a) the velocities (vI, v a , vA) in spherical coordinates to (vl, c u 2 , v 3 ) in oblate spheroidal coordinates; (b) the total time derivatives in the two systems; (c) the respective pressure gradients in the two systems; and (d) the velocity divergences in the two systems. It has already been assumed that @ is constant on a particular q1 surface in the oblate spheroidal system and on the spherical surface (r = a, say) in the spherical system. Furthermore, we make the following identification i.e. the radius, u, of our spherical earth is the mean of the polar and equatorial radii for the ellipsoid. With the foregoing relations the only difference between the two systems lies in the different metric terms associated with the variations of the unit vectors. From Eq. (2.11) we derive r = (x2 + y 2
+ z2)112= a(cosh2 q1 sin2q2 + sinh2 q1 cos2 q 2 ) 1 / 2 = a(sinh2 ql
+ sin2 q2)1/2
= a(Cosh2 q1 - cos2 q2)1/2,
(2.27)
and from (2.12) and (2.27) sin 6' = R/r = cosh q1 sin q2/(sinh2q1
+ sin2q,)lI2.
(2.28)
Equation (2.28) can be solved for sin q2 in terms of sin 8 sin q2 = sin 6'
(2.29)
1 - sin2 8/cosh2 q1
Now Eq. (2.9) for the ellipse may be rewritten as
R2 a2 cosh2 q1
+ 2 cosh2 ql[lz2-l/cosh2
-
ql] -
(2.30)
But the general expression for an ellipse with ellipticity e is written as 2 2 R2 = 1, b2 + b2(1 - e)2
(2.31)
where b is the major radius. Thus from (2.30) and (2.31) we can make the following identification between cosh q1 and the ellipticity l/cosh2 q1 = 2e - e2.
(2.32)
12
George Veronis
At the earth's surface the ellipticity is 1/298 so that a good approximation to (2.32) is 1/cosh2 q1 w 2e. (2.33) Making use of (2.32) yields (2.29) in the form sin q2 = sin 8[l - e cos2 O
+ - -1,
(2.34)
where terms of order e2 are omitted. In a similar fashion we may write cos q2 = cos 8[l
+ e sin2 8 + .
sinh q1 = -(1 (2e)u2
5
(2et'!2+):
cash q1 = -(1
.I,
.)
+ .. ' ,
(2.35)
Q1 = Q2= u cosh ql[l - e sin2 O + * * .I, Q3= u cosh q1 sin 8[l - e cos2 0 + . -1. Straightforward substitution of the foregoing into the metric terms of Eqs. (2.15) to (2.17) yields, to first order in e, a2 cos
q2 sin q2 - e cos 8 sin 8 +... -
a
Q13
a2sinh q1 cosh Q13
q1 - 1 - $e cos 28 + a
u sinh q1 sin q2 - 1 - e(1 -
a
+ e(1 - 4 cos 28) +
81 u cosh q1 cos
=sin O[l - e(1 - cos 20)
q2
= cos
Qi
Qi
*
8[l
+
.], (2.36)
*
+ e(1 + cos 28) + - - .],
2 - e(1- 2 cos 28) + . . . , a
Qi
q1 - u2 sin qz cos q2 Q13
QlQ3
-
cot 0[l
. -1
a
u sinh q1 sin q2
u cos q2 cosh
9
*.
QlQ3
+
--
+ + cos 28) + -
Q1Q3
u cosh q1 cos q2 - cot 8[1
3
+ e(l a
cos 26)
+ . . -1- _e _ sin 28 _ 2a
+..a.
13
Large Scale Ocean Circulation Hence, (2.15) to (2.18) become dt
1 - 2 e cos 20) -
a
$ [I
- e(1
-y)] 5 1 ap
dv2 -+ v1v2 (I - e cos 28) dt a 2 ~
- 252
cos 8 v3[l
e sin 20
-
1
a@
(2.37)
e vI2 e sin 28 a [ l + e(1 - cos 28)] + 2a v32
1 aP , + e( 1 - cos 28)] = - -
(2.38)
PQZ 842
+ 252 cos 8 v2[1 + e(1 - cos 28)] + 252 sin 8 v,[l
- e(l
+ cos 28)] (2.39)
3 1 avj dP -dt+ P I Cj = 1 Q j a q j
--+
~
a
e
2-41 - 2 ~ 0 ~ 2 8 ) Vl a
[l + e (1-- co;2e)1
v2-e-
2a
(2.40)
The set (2.37) to (2.40) differs from the set (2.22) to (2.25) only by terms of order e, once the identifications between velocity components, gradients, divergences, and total rates of change have been made. I t is instructive to observe that the largest discrepancies are 3e/2 so that neglecting the metric terms arising from the elliptical correction involves an error of 1/200. This error is substantially larger than the error made in oceanographic studies when the spherical radius r is replaced by its mean value, r = a, in the spherical equations of motion. Furthermore, it is worth noting that the metric terms proportional to sin28 in Eqs. (2.37), (2.38), and (2.40) have no counterpart in the spherical system. Since the first two are associated with the radial velocity, it is difficult to visualize a physical situation when they could contribute materially. The last term in Eq. (2.40) would simply add a small contribution to the preceding term. And finally, one should note that differentiated forms of the equations, such as the vorticity equation or the divergence equation, will involve larger errors when the elliptical metric terms are neglected.
14
George Veronis
I n oceanographic studies the equations are more often written in terms of spherical coordinates with latitude instead of colatitude and with the mean radius, a, replacing r in the coefficients. These equations take the form du dt
uw
uvtan+
a
a
-+---
+2Q cos
+ w - 2Q sin q5 v = - pa cos + ax’ ~
wv u2tan+ -+a++2!2 sin + u =
dv dt
a
-dw _u 2 + v~2 dt
a
--+--+ 1
1 dp pdt
- cos 2!2
au
acosq581
+ u=
1 av a
a+
a
--1 aP Pa a+ ’
1 aP
-- --g, P
(2.42) (2.43)
at-
t an
(2.41)
aw 2w + ++= 0, ar a
(2.44)
where
d --_a _
dt-at
u a +--+--+w-
acosq5 ah
v a
a
a+
a7
(2.45)
and gravitational acceleration has been assumed constant. T h e velocities u, v, w are now in the directions of increasing h, r.
+,
111. The Coriolis Acceleration In almost all studies of oceanographic phenomena the horizontal component of the Coriolis acceleration can be neglected. A crude but simple way of showing that this assumption is justified for large scale flows is to make use of the fact that the hydrostatic equation is known to be a very good approximation in this case. Then (2.43) becomes
+
and the term 2Q cos u is neglected. Now the kinetic energy equation of the system does not contain a term involving the rotation Q because v (2S2 x v) vanishes. Hence, for consistency, if 2Q cos u is neglected in (2.43), it is necessary to neglect 2!2 cos w in (2.41) since the latter term would otherwise contribute to the kinetic energy. Two additional justifications of the neglect of 2Q cos in the equations are given in this section. First the conservation of potential vorticity is
+
+
+
15
Large Scale Ocean Circulation
derived and from scale analysis the neglect of 2 Q c o s 4 is shown to be justified except in the immediate vicinity of the equator. Then the same conclusion is drawn from an angular momentum argument proposed by Phillips (1966). I t is important to observe that the neglect of the horizontal component of the Coriolis acceleration means that the direction determined by gravity, i.e., the local vertical direction, is singled out as a dominant direction in large scale flows. If seawater were homogeneous, the direction parallel to the axis of rotation would be dominant. Hence, the neglect of 2Q cos 4 automatically includes the strong constraint of stratification. In homogeneous models of ocean circulation only the vertical component of the Coriolis acceleration is kept. Therefore, the effect of stratification is included even though it may never appear in the actual model. This point will be mentioned again later.
A. CONSERVATION OF POTENTIAL VORTICITY I n large scale oceanic flows the vorticity of the fluid is an important variable because of the inherent vorticity which each fluid particle has as a result of the earth’s rotation. A quantity, called potential vorticity, is conserved in the absence of dissipation. Its derivation follows. Consider the equation of motion in vector form av -+(VXV)XV+V at
1 +252~~=--Vp-V@,
(3.1)
P
9 +pv dt
v = 0,
and suppose that some state variable, s, is conserved so that dv dt
-=-
as
at
+v.
vs=o,
(3.3)
where s = s(p, p). T he curl of (3.1) yields do
_- o.vv+wv.v=-v dt
where 0
= 2s2
+ v x v.
(3 xvp, -
16
George Veronis
Eliminating V . v from (3.2) and (3.4) yields
T h e gradient of (3.3) yields d
- vs f V v . vs= 0. dt
(3.7)
Then the scalar product of (3.6) with Vs plus the scalar product o f o / p with (3.7) leads to the expression
f
(Y) =O,
where use has been made of the fact that s is a state variable so that Vs- Vp x V p vanishes. T h e quantity, Vs-w/p, which is conserved along a trajectory, is the potential vorticity. Th e derivation given here is that of Ertel (1942). T h e principal use of conservation of potential vorticity for present purposes is to note that in spherical coordinates (Vs. w)/p takes the form
and that the component of the Coriolis term parallel to the earth's surface, 2!2 cos 4, is negligible compared to the normal component, 2!2 sin 4, if
(3.10) Denoting characteristic vertical and horizontal scales of variation of s by D and L , respectively, we can rewrite (3.10) in terms of D and L as
DIL 4 tan 4.
(3.11)
For large scale oceanic motions (i.e. when potential vorticity is a useful variable) the ratio D / L is not larger than Hence, the horizontal component of the earth's rotation is negligible at latitudes beyond a fraction of a degree from the equator.
17
Large Scale Ocean Circulation
B. AN ANGULAR MOMENTUM ARGUMENT FOR
THE
NEGLECT
OF 2 Q C O S 4
A more general argument for the neglect of the horizontal component of the Coriolis term has been proposed by Phillips (1966), who writes the set (2.41)-(2.43)before the “ shallow ” approximation is invoked as
du dt
-= F ,
+ (2Q +A)(. sin + - w cos +) r cos +
dv - = F B - (2Q+ 5 ) u sin dt Y cos 4 dw dt
2Q +-)u Y
U
cos
+
+
(3.12)
wv
-
-
(3.13)
Y
cos
+ +-
V2
(3.14)
Y
where
dh ~=h,-, dt h, = Y cos 4,
d4 ~=h,;Zi, h,
=I ,
dr ~ = h , dt ’ h, = 1 ,
and F,, F , , F, include pressure gradients and frictional forces per unit mass. These equations satisfy the angular momentum balance d -dM __ -
dt
dt
[Y
cos
+ (u + Qr cos +)]
=Y
cos+F,
.
(3.15)
If the “ shallow ” approximation is made in the form h, = a cos +,
h, = a
(3.16)
then (3.12) to (3.14) are written with a replacing Y, i.e., the set (2.41) to (2.43) is obtained. However, as Phillips points out, the set (2.41) to (2.43) does not have the angular momentum balance which is obtained by replacing r in (3.15) by a. In fact, there is no function A(+,Y) such that A F , = dM/dt, in general. If the terms 2Q cos w + (uw/a) were absent from (2.41), the equation would give the proper (approximate) angular momentum balance. If the momentum equations are recast into the vector invariant form
+
av _ F +g - V ( i at -
v2)
+ v x curl(v +Rr
cos
+ i,)
(3.17)
18
George Veronis
and if the relations (3.16) are now used in all curvilinear operators, the final set of component equations is du
-dt= F A
+ (22 +La cos) 4 v sin 4
(3.18)
--
) u sin 4
(3.19)
dt
dw
-=
dt
F, -g .
(3.20)
Hence, the horizontal component of the Coriolis parameter as well as the metric terms which gave rise to the angular momentum difficulty are now absent. Consistent with this approximation,the term 2w/a in (2.44) should be neglected. Although Phillips’ argument is correct, there are obvious cases where the final set (3.18) to (3.20) is inapplicable. Even with the “shallow” approximation, flow between two concentric spheres in a laboratory cannot be analyzed by this set of equations. In other words, some justification for the use of the simplified set of equations must be provided by the particular physical system under consideration. For large scale oceanic flows, the set (3.18) to (3.20) is consistent and represents a good approximation.
IV. Thermodynamic Simplifications-The Boussinesq Approximation The equations of motion inherently contain descriptions of a wide variety of processes ranging from high frequency, small scale phenomena such as sound waves, to low frequency, large scale motions which describe the general circulation. For the study of a particular class of phenomena it is very helpful to “filter ” the equations so that a simplified set of equation is available. For example, for investigations of large scale flows it is common practice to consider seawater as essentially incompressible. The ensuing simplification is an enormous help for it reduces the equations to a much more tractable form. In the present section the analysis for deriving this so-called Boussinesq set of equations is given and some of the limitations of the simplified equations are pointed out. The starting point is a brief discussion of the thermodynamic properties of seawater. Seawater contains many dissolved salts (Sverdrup et al., 1942) the most abundant of which is sodium chloride. Because these salts appear in ap-
Large Scale Ocean Circulation
19
proximately the same relative concentrations in seawater,* for dynamical purposes it suffices to lump them together and to define the salinity, s, as a composite measure of the concentrations of the salts. T h e density of seawater is, therefore, a function of salinity as well as of temperature and pressure. p = p(s,
T , PI.
Empirical formulas for determining p in terms of s, T , and p are summarized by Sverdrup et al. (1942) and by Fofonoff (1962a). Surface water in regions of low salinity and high temperature has a density as low as 1.02 gm/cm3. Water in the deepest trenches is subjected to the highest pressures and has a density as high as 1.07 gm/cm3. Hence, the maximum range of density is only about 5% so that where p appears as a coefficient it can be replaced by its mean value, pm, and a maximum error of only 2.5% is incurred. However, when variations in density are important as driving terms for the motion field, replacing p by pm means that the dynamic effects of density variations are ignored. T h e Boussinesq approximation, which is developed below, allows one to ignore the density variation when it contributes only a small quantitative correction and to keep the variation where it is dynamically significant.
A. THEADIABATIC HYDROSTATIC FIELD I n the absence of motion the conservation of momentum reduces to the hydrostatic equation
vp = -gp. Hence, surfaces of constant density and pressure are level (horizontal) and the only admissible variations in p and p are in the direction of gravity,t z. To complete the description of the motionless state it is necessary to specify the equation of state, the thermodynamic process which controls the state variables, and known values of the state variables at some level. Even in this case, however, there will be an infinite number of systems which can satisfy these constraints because seawater is a multicomponent fluid (salt and temperature can vary). I n order to specify a unique static state the salinity is here taken to be constant, with the value s = 34.85%. Hence,
* Carritt and Carpenter (1958) discuss the variability of relative concentrations. t If diffusion of momentum and heat are included, there can be no relatively static state for a rotating system because the equations describing eyuipotential surfaces are, in general, not solutions to the steady diffusion equations. The slow flow which results is called Sweet-Eddington flow. The implicit assumption here is that it is negligible compared to the motions generated by driving forces of the system.
20
George Veronis
variations in s will occur only via interactions with the bounding media (the atmosphere, the bottom etc.) and will normally be associated with motions of the fluid. With s a known constant the density is a function of T and p only. Now, the first law of thermodynamics is
6 g = T dq = C , dT
+ T(aq/ap),dp,
(4.3)
where q is specific entropy, c, is specific heat at constant pressure and Sq is the specific quantity of heat added. For an adiabatic process S q = 0 and it follows from (4.3) that the vertical variations of the state variables are related by
where the hydrostatic equation has been used to give the right-hand term. T h e thermodynamic relation (%IaP)T = - ( a v / a T ) ~ 7
(4.5)
where v is specific volume, can be used to write (4.4)in the form
( a T / a ~= ) , -g a TIC, (4.6) where o! = I/v(av/aT), is the coefficient of thermal expansion. Values of o! and c, as functions of s, T , and p are tabulated by Fofonoff (1962a) who also gives an empirical formula for (aT/ap),as determined by Fofonoff and Froese (1958). For present purposes it suffices to note that (aT/az),for s = 34.85 % has values ranging from O.O16"C/1000m at the surface with T = -2°C to 0.209"C/1000m at 10,000 m depth with T = 4°C. A typical range for the adiabatic temperature is 0.6"Cfrom the surface to 4000 m depth. Hence, the adiabatic temperature gradient is small relative to observed vertical temperature gradients in the upper layers of the ocean but the two are comparable in deep water. Formulas for the calculation of density as a function of s, T , and p are summarized by Fofonoff (1962a). Hence, from the hydrostatic relation (4.2), the adiabatic temperature gradient (4.6), and the formula for p as a function of T and p (s = 34.85 yo),one can calculate the adiabatic density field given p and T at one level. As stated earlier, the observed range of values of p is from 1.02 gm/cm3 at the surface to 1.07 gm/cm3 at a depth of 10,000 m. T h e range of adiabatic density variations is therefore comparable to the observed range. Hence, in contrast to temperature variations the density variations do not differ much from the adiabatic density variations, i.e. the principal contribution to changes in density comes from the pressure field. T h e adiabatic, hydrostatic state will be denoted by variables with subscript a. 9
21
Large Scale Ocean Circulation
B. THEBOUSSINESQ EQUATIONS T he conservation of momentum for a fluid is now expressed as
dv dt
p-+2pQ
x
v = -0j-gj5,
(4.7)
where the pressure and density have each been divided into an adiabatic part and a perturbed (tilde) part, and p = p a ri; and p = p a +fi. T h e adiabatic, hydrostatic field has been subtracted from the right-hand side. On the left-hand side the density appears as a coefficient. Using the fact that the total variation of density is small relative to the mean value enables one to approximate (4.7) by
+
dv -++a )( v = - -1 v p - g - ,P (4.8) dt Pm Pm where pm rn 1.035 is a mean value of the density for the ocean. T h e formal requirement for (4.8) to be valid is
< 1.
(4.9) T h e equation of state is linearized about the adiabatic state, as discussed below. Thus, (IP-Pm(/Pm)=6
P = Pa
+
(g)*,s T + ($) b + ($) T. s
P.T
5
+-
*
..
(4.10)
Hence, the perturbation density can be written as P/pa M P/Pm = - aT
+K$ +Y?,
(4.11)
where
If (4.11) is substituted into the right-hand side of (4.Q the vertical component of (4.8) has the terms
(4.13) on the right-hand side. For oceanic motions the vertical scale of variation,
H , is lo5 cm so that the first term has magnitude 1) I / H . T h e isothermal compressibility, K , has a magnitude of 4.5 x 10-l1 in cgs units (Sverdrup et al., 1942) so that the t e r m g K j has magnitude 1fi1 / H swhere H , = l/gK is the scale height for seawater. Its value is 2 x lo7cm, i.e., substantially larger
22
George Veronis
than the vertical scale H , so that the pressurefluctuation term in the equation of state is dynamically insigniJcant. This means that the equation of state (4.11) can be approximated by
+
p"/pm Fz: 03 ys".
(4.14)
T h e equation for the conservation of mass takes the form (4.15)
T h e first and third terms are O(6)compared to the last term so that both can be neglected. T h e second term involves a time scale which must be specified. If velocity and length scales, V and L, respectively, are chosen, and if a/& is taken to be O( V / L )(i.e., local changes are due to convection), the second term is the same order as the third. An alternative procedure is to specify the time scale on the basis of some other physical process, such as buoyancy oscillations. I n either case the second term can be neglected. T h e reason why +/at must be considered separately is that it is the term which is associated with acoustic phenomena and one must specify the physical process in order to filter out acoustic waves. Hence, to lowest order (4.15) takes the form v*v=o. (4.16) T h e equation for the conservation of salinity may be used in its primitive form dfldt = 0. (4.17) T h e first law of thermodynamics is p (deldt)= -pV
v,
(4.18)
where e is specific internal energy. For the hydrostatic, adiabatic state the first law was expressed in the form (4.3) to give the adiabatic temperature gradient (4.6). It is convenient at this point to express the change in internal energy in terms of increments in specific volume, temperature, and salinity for a process at constant pressure as de dt
dF dt
p - = c, --p v
.v +
ds" c, -, dt
(4.19)
where c, is the specific heat per unit change of salinity. Even though the actual value of c, is not available for seawater, the form (4.19) suffices for the purpose of this development. Substituting (4.19) into (4.18) yields dp dt
ds" dt
c -+cc,--0.
(4.20)
Large Scale Ocean Circulation
23
In view of the conservation of salinity (4.17), Eq. (4.20) reduces to
dF/dt = 0.
(4.21)
Since variations in salinity occur only in the perturbed state and since the adiabatic temperature gradient (4.6)has already been evaluated, Eqs. (4.17) and (4.21) can be used for the perturbation salinity and temperature. In summary, the equations take the following form when the Boussinesq approximation is employed. For the adiabatic, hydrostatic state,
S = const.,
aT/az = -gaT/c,, For the perturbed state
V p = -gp, p = p( T , p , S).
dv 1 -+2Q x v = - - v p - g e , dt Pm V . v 0, Ps/pmz= -a8 =z
dsldt = 0,
d8ldt = 0,
(4.22)
(4.23) Pm
+ YS,
dps/dt = 0,
where 8 is the potential temperature, the adiabatic temperature gradient having been accounted for in the last of Eqs. (4.22);s is the deviation of the salinity from some constant value; and is the potential density, expressed as a linear combination of 8 and s.
C. USE AND LIMITATIONS OF THE BOUSSINESQ APPROXIMATION The set of equations (4.22) and (4.23) is sometimes called a Boussinesq set because it was derived with the use of the Boussinesq approximation. In laboratory experiments and other very shallow layers of fluid the Boussinesq set is a very good approximation to the primitive set of equations for certain classes of motions. For shallow layers, molecular processes (conduction, diffusion, etc.) can be added to the system in a straightforward fashion. For oceanic flows of even moderate scale it is necessary to treat mixing processes by some approximate technique (eddy coefficients, mixing length theory, or some analogously crude method). The great advantages of the Boussinesq system are (a) the filtering out of acoustic waves and other phenomena associated directly with compressibility, (b) the linearization of the equation of state for the perturbation variables, and (c) the relatively simple form of the energy equation. Potential density is the dynamically significant part of the density field, the remaining part being assumed to adjust adiabatically when an element is displaced vertically. The widespread use of the Boussinesq approximation
24
George Veronis
in analyses for single-component fluids and the straightforward interpretation of the adiabatic temperature and density for such fluids would lead one to expect an equivalent simplification for the oceanic case. However, there are limitations on the use of the Boussinesq system for large scale ocean circulation problems. T h e problem is immediately apparent upon examination of observed Auid properties in the deep ocean. Table 1 shows typical values of T and S TABLE 1 CALCULATED PROPERTIES OF TWO SAMPLES OF SEAWATER PRESSURE OF 4000 dB WITH THE SAME DENSITY BUT DIFFERENT TEMPERATURES AND SALINITIES
AT A
T("C) ~ ~ O i o o )
u(gm/liter) &"C) u@(gm/liter)
Sample 1
Sample 2
2.30 34.90 45.93 1.95 27.92
2.09 34.85 45.93 1.74 27.89
and calculated properties for two elements of seawater at 4400 m depth. The two elements have a common value of density" of 1.04593 but different temperatures and salinities. Observe that the two elements have different valuest of a, but the same value of a. Since the in situ density determination states that the two elements have the same density, study of the potential density alone would lead one to the incorrect conclusion that sample 1 is heavier than sample 2 and that it would, therefore, seek a lower level according to inviscid gravitational stability theory. Table 2 contains typical values of T and S and calculated properties for two elements of water, also at 4000 m depth, but with slightly different densities and substantially different values of T and S. In this case, sample 2 is actually heavier than sample 1 according to the in situ density determination but sample 1 appears heavier according to a, values. On the basis of
* To avoid unnecessary decimal digits oceanographers customarily use the quantity sigma, defined by u = 100(p - 1) where p is expressed in cgs units. For example, seawater with a density of 1.04593 has a value of u equal to 45.93. t T h e quantities u, 8, and were calculated with the computer subroutines of the Woods Hole Oceanographic Institution. Density (or a) is obtained from Ekman's (1908) empirical formula. Potential temperature (8) is calculated from the polynomial expression by Fofonoff and Froese (1958). Potential density (no)is derived from the Knudsen (1901) formula for u with p = 0 and with the temperature replaced by 8.
25
Large Scale Ocean Circulation
in situ density, inviscid gravitational stability theory predicts that sample 2 should sink and sample 1 should rise (relative motions) whereas the same criterion, based on potential density, indicates a sinking of sample 1 and a rising of sample 2. The difference between the densities of the fluid parcels in this example is small (a difference of 0.01 in ue is close to the limit of reliability) but it is typical of observed differences in abyssal waters. TABLE 2
CALCULATED PROPERTIES OF T w o SAMPLES OF SEAWATER PRESSURE OF 4000 dB WITH DIFFERENT DENSITIES, TEMPERATURES, AND SALINITIES
AT A
WC) S(O/OO)
u(gm/liter) wc) ae(gm/liter)
Sample 1
Sample 2
2.21 34.90 45.95 1.86 27.92
1.89 34.85 45.97 1.55 27.91
Figure 3 shows the distribution of 0 0 for a vertical section in the western Atlantic Ocean as calculated by Lynn and Reid (1968) from observed data. According to the distribution of potential density the bottom kilometer or so of water is unstably stratified over a large range of latitudes south of the equator. I n situ stability calculations show that the water is, in fact, gravitationally stable. One must conclude, therefore, that the use of potential density for determining the gravitational stability of water in the deep ocean can lead to incorrect results. This apparently paradoxical instability of abyssal water can be explained as follows. In Fig. 4 isopycnals for seawater are shown for depths of 0, 2000, and 4000m. The curves slope more steeply upward to the right as the depth (or pressure) is decreased, i.e. the entire pattern of isopycnals rotates counterclockwise as p is decreased. Hence, a layer of fluid which is neutrally stable or even slightly stable near 4000 m depth appears to be unstable if the reference surface is the top surface as it is for potential density. It should be observed that the apparent instability is present even when the equation of state is highly nonlinear (the complete expression for u at each level was used to generate the curves shown in Fig. 4).T h e difficulty can be avoided only by allowing the various coefficients, whether they be the coefficients in the complete expression for (T or in the linearized Boussinesq form, to be functions of pressure. Allowing a and y in (4.14) to be
8p0S
I
60° I
I
40' I
I
2po
I
00 I
I
2p0
I
4P0
I
I
8p0N
FIG.3. A plot of potential density, ug, as a function of latitude and depth in a longitudinal section west of the Mid-Atlantic Ridge, according to Lynn and Reid (1968). (Courtesy of Deep-sea Research.)
Large Scale Ocean Circulation
27
functions of pressure would make the equations analytically intractable. However, with negligible error, one could approximate the pressure effect by taking u and y to be functions of depth. T h e equation of state would then be more complicated than the form with u and y constant but it would at least be linear.
S FIG.4. Curves of constant density plotted in the potential temperature-salinity plane for depths of 0 (solid curves), 2000 m (dashed curves), and 4000 m (dash-dot curves).
The issue discussed in this section is not important for phenomena such as internal waves where the particle motions are restricted to local regions. However, for large scale circulation problems, where fluid at the surface in polar regions may sink and eventually make its way to the bottom, the use of the Boussinesq approximation in its simplest form would preclude the possibility of direct verification of particle trajectories by comparing theory with observation. For example, water mass analysis shows that the water near the bottom between 30"s and the equator in Fig. 3 originates in the Antarctic Ocean (Weddell Sea) and the tongue of water with ua > 27.9 comes from the Greenland Sea. At the points of origin, water from the Greenland Sea is denser than the water from the Antarctic. This feature is reflected in the values of a,. In the abyss these waters stratify with Antarctic water lying below the water from the Greenland Sea. This inversion is a direct consequence of the effect of pressure on the densities of water of different temperatures and salinities. T h e simple Boussinesq system could not lead to the observed distribution because, according to the Boussinesq system, water which is densest at the point of origin would, in the absence of mixing, end up in the deepest part of the ocean.
28
George Veronis
V. Scaling of the Equations The equations of motion can be scaled to exhibit important balances for large scale flows. By “large scale” we mean flows whose characteristic horizontal scales are substantially larger than the vertical scale (or depth). The scaling procedure makes use of observed (or perhaps only plausible) magnitudes of quantities to reduce the equations to a simpler form. Results from subsequent analyses of these equations can be compared to observations of appropriate phenomena or features in order to obtain an a posteriori check on the validity of the model equations. For this purpose the variables and operators in the Boussinesq equations of motion will be represented as follows :
a
i ----a cos
-=TQ6,, at
u = VU’,
a
i
l a
+ aA - L aa,
v = VV’,
a
w = WW’,
a
1
1 H”,
a+-
-=-
p = (Ap)p’,
p = (AP)p’,
ar
where the prime quantities and the 6 operators are nondimensional and of order unity and the scales T , L, H , V , W, Ap, and AP are to be determined by restricting attention to certain chosen magnitudes which reflect processes of interest. T h e Boussinesq equations on a sphere then take the following nondimensional form : 7St u
T
+ R[u 6, u + v 6, u +(p/s)w 6, u + vpuw - ~ u tan v $1
6,v
+2p cos 4 w - 2 sin 4 v = - PS,p,
(5.1)
+ R[u6,v + z, 6, w + ( ~ / E ) 6,v w + qpvw +7u2 tan 41 + 2 sin 4 u = - P 6,p,
(54
+
+
6 , ~R(u 6 , ~ Z, 6 , +~( P I & ) 6,w - r p 2 - qv2)]- 2 s cos = -P 6,p - (Fs2/R)p,
F ~ [ T
7atp
S,u
+ R[u a a p + v +
+ S,v
6
- qv tan
6
~ ( ~ 1 s S,P]= ) ~
0,
+ + 2rpw + ( p / s ) 6,w = 0,
+u (5.3)
(5.4) (5.5)
where the primes have been dropped and the following nondimensional parameters appear
R = V/QL, p = W/V, 7 = L/a, E =H / L , F =g HQ2 Ap/pm N 2 / Q 2 , P = AP/QVLpm.
(5.6)
T h e parameter R is called the Rossby number and F is a ratio of frequencies.
Large Scale Ocean Circulation
29
Formal expansions of the variables in powers of each of the small parameters in (5.6) would be messy. It is possible to make some simplification by means of two assertions which lead to balances of primary interest. The first assertion is that the horizontal scale is always much larger than the vertical scale so that E < 1. This assertion is taken to be a dominant one in the sense that all conclusions which result from imposing it as a restriction are valid for the flows to be considered. Since the parameter p = W/V is divided by E in several places, it is necessary to make a statement about the amplitude of p. Hence, the second assertion is that the vertical divergence is always upper bounded by individual horizontal divergence so that p 5 E or, equivalently, W < VH/L. Several important consequences follow from these two assertions. First, in the vertical equation of motion the Coriolis term is O ( E )and the remaining acceleration terms are O(2). The bouyancy term is ([Fe2/R]).I n the ocean F 9 1 and R < 1 almost everywhere so that the parameter F@/R may be O(1). Since large scale flow is is known to be essentially hydrostatic, the parameters P and Fe2/Rmust be of the same order. In the following, the parameter P will be set equal to one and F 2 / R must then be of order unity also. An important consequence of this argument is that the lowest order $ow will be hydrostatic and geographic. T h e latter balance follows directly when P = 1 and both T and R are small. Second, the vertical convection term in each of the first four equations is at most of the same magnitude as the horizontal convection terms. Third, the metric terms involving the radial velocity, w, are at most O ( E )and hence can be neglected to lowest order. Fourth, consistent with the hydrostatic relation and the second assertion ( p 5 E ) above, the horizontal component of the Coriolis terms, 2p cos 4 w, in (5.1) can be neglected to lowest order. Therefore, the equations simplify to the following set 7
6,u
+ R(u 6,u + v 6,u + w 6,u
T
6,v
+ R(u 6,v + v 6,v + w 6 , v +qu2 tan 4) -+ 2 sin 4 u = -6,p, 7- 6,p
+ R ( u 6,p
6,u
+ 6,v
-
~ u tan v 4) - 2 sin 4 v = -S,p,
+ v 6,P
- ~v tan
+ w 6,P) = 0.
4 + 6,w = 0,
(5.7) (5.8)
(5.10) (5.11)
where p has been set equal to its upper bound, e, and Q = F 9 / R . Even though Q is order unity it will be retained as a tracer. The last set of equations is the conventional starting point for almost all studies of large scale flows (with frictional and diffusive processes normally introduced by means of parameterizations of small scale phenomena).
30
George Veronis
Further simplification of the set (5.7)-(5.11) requires additional restrictions imposed by the particular phenomenon under investigation. For example, for flows whose horizontal length scale is much smaller than the radius of the earth the condition 7 < 1 leads to a simplification because some of the geometrical distortion terms associated with the spherical geometry can be ignored. Another type of simplification results for relatively weak flows for which the condition R < 1 is valid. For flows which satisfy both of these latter conditions, it is necessary to compare the relative magnitude of R and q in order to derive the appropriate ordering of the equations. Large scale flows in the ocean can be divided into several types, depending on the horizontal scales which are involved. Phillips (1963) has designated two of these by analogy to atmospheric motions as: (a) motions of type 1, in which q 5 R and R < 1, and (b) motions of type 2, in which R < 1 and q 1. T h e specific conditions and simplifications are developed below and a set of equations is presented for three different scales of motion.
-
A. SMALLSCALEMOTIONS-THEPLANE T h e Gulf Stream meanders and the eddies observed by Swallow and Hamon (1960) are examples of type 1 motion. The horizontal length scale, L, of these motions is of the order of 100 km; hence, it is large compared to the depth ( E < 1) but small compared to the scale of the oceanic basins, or equivalently, of the radius of the earth (6 1). Other typical magnitudes for Gulf Stream meanders are
H-105 cm,
Ap/pm-10-3,
V-102 cm sec-'.
(5.12)
Hence, the following magnitudes obtain for the parameters in Eqs. (5.6) ~=10-',
Q-1,
P-1,
R-IO-l,
~-10-',
(5.13)
where AP, and therefore P, is determined by the hydrostatic balance condition. An additional simplification is obtained by making use of the fact that q is small and expanding the trigonometric coefficients about the latitude near which the motion is to be studied. Thus choose a (mid-) latitude, and write 4 = d o 4' = d o ( Y l 4 = d o qy', (5.14)
+
+
where the linear distance, y , has its origin at yields sin 4 and cos 4 about sin 4 = sin do
+
+ = 4,.
Then expanding
+ qy' cos do + . . . , cos 4 = cos do- qy' sin do4 . . . . (5.15)
31
Large Scale Ocean Circulation From the original east-west derivative we can write
1
a
cos40
a
1
-
cos4,
a (5.16)
so that the nondimensional rectangular Cartesian coordinates, defined to replace A. Analogously,
XI,
has been
(5.17) so that x, y , and z form a local rectangular coordinate system. T o lowest order in 7 the equations are those for a fluid in a uniformly rotating system
au + RV
r-
at
T
aP VU-f v = -ax ’
(5.18)
av + RV VV f f u = --,aP -
(5.19)
aPlaz = -Qp,
(5.20)
at
aY
a, + RV -
(5.21)
- + +-
(5.22)
Vp = 0, at au av aw = 0. ax ay az Here, the primes have been dropped T
.
a ax
a +w ii az
v V -siU - f v ay
and f = 2 s i n + , = c o n s t . T h e set (5.18) to (5.22) is sometimes called an f-plane system and is appropriate for the study of smaller scale properties of large scale phenomena but it is inadequate to describe larger scale flows where the sphericity of the earth is important.
B. MOTIONS OF INTERMEDIATE SCALE-THEP-PLANE Theoretical studies of wind-driven ocean circulation of intermediate scale or larger must take into account the nonuniform vertical component of the Coriolis parameter. A set of equations for ocean basins which lie on the equatorial side of 45”latitude can be derived with geometrical considerations. Again choose a latitude 4, as in (5.15) and assume that 7 = L / a 1 so that sin 4,cos 4,and 8, can be approximated as in Eqs. (5.15) and (5.16).
+
George Veronis
32
Now compare the terms 6,v and r]v tan q5 in the continuity equation (5.11) 6,V=-
av
7vtan+=qtanq5,v
"'
sin q50 cos q50
aY '
+-
-1
. (5.23)
T he conditions 7 < 1 and do< 45" require 7 tan q5, < 1 and the metric term can be neglected to lowest order. Similarly the metric terms in the substantial derivatives can be neglected. Th e Coriolis term in (5.7) can be written as 2 sin q5 v = 2 sinq4, v(l qy' cot q5, + -). (5.24)
+
-
Now even though one assumes 7 < 1, the term qy'cot 4, cannot be neglected because q50 < 45" and cot q5, > 1. Indeed, with decreasing q50 the term 7y' cot 4,may dominate. Th e equations in this case reduce to the 2q set (5.18)-(5.22) with the Coriolis parameter written as 2 sin q5, cos d o y -fo 8 y and they are called the 8-plane equations. They are used quite extensively in theoretical oceanography and almost exclusively in studies of wind-driven ocean circulation (Stommel, 1948; Munk, 1950; Fofonoff, 1954; Bryan, 1963). In cases where the 8-plane is used for basins poleward of 45" latitude an obvious error is incurred in neglecting the metric term in the continuity equation while the 8 term is retained.
+
+
C. LARGE SCALE MOTIONS When the motion to be investigated has truly large length scale, e.g., the global circulation of the oceans brought about by differential incident solar radiation between equatorial and polar regions, the equations take on a different form. Here, L = a = 6 x1O8crn, V-1 cmsec-', 7-1, and R If gravity waves or other high frequency phenomena are to be incorporated into the analysis, the time scale is small and the time derivatives must be retained, i.e., T 1. For long period quasi-steady motions, T < 1 and the use of R < 1 simplifies the equations at lowest order to the form 2 sin 4v = (l/cos q5) ap/aA, (5.25a)
-
-
2 sin q5 u = -ap/+,
(5.25b) (5.25~)
dpldt 1 cos q5
= 0,
aua/\+ a42 (v cos q5) ] + -
:
- = 0.
(5.26a) (5.26b)
Large Scale Ocean Circulation
33
When dissipation processes are included, either via parameterization of boundary layer effects or heuristic treatments of internal dissipation processes, Eqs. (5.25) and (5.26) are used for general circulation models of the ocean. Many of the so-called thermohaline circulation studies in which large scale circulation is driven by imposed surface density conditions (Welander, 1959, Robinson and Welander, 1963 ; Needler, 1967; Kozlov, 1966; Veronis, 1969) make use of amodifiedor extended formof (5.25)and
(5.26). Although the three sets of equations discussed above are applicable for the.study of suitably restricted phenomena, it is obvious that the phenomena are not isolated from each other and that some interaction will occur and indeed may be of great importance. Thus, even though some properties of the motions of type 1 may be studied by the equations (5.18) to (5.22) the generation of such motions depends on the overall circulation of the ocean. For example, the Gulf Stream is a necessary part of the overall circulation of the oceans and the dynamical balances which describe this smaller scale feature must be appended to the equations (5.25) and (5.26) via either a boundary layer or some other asymptotic procedure when large scale circulation is studied.
VI. Geostrophic Flow Steady, linear, frictionless flow in a rotating system is geostrophic and hydrostatic. Because of the fundamental nature of geostrophic flow in rotating fluid theory, a simple example of geostrophic balance is given here. This example is essentially that which was first presented by Phillips (1963) but a simple geometrical argument is used. Consider a right cylindrical basin partially filled with water and rotating about its axis of symmetry with constant angular velocity, R. The parboloidal free surface (Fig. 5a) represents a constant pressure surface and can be calculated as a consequence of the hydrostatic relation and the exact balance between the radial pressure force and the centripetal acceleration
( l h ) ap/az= -g ,
(1/p) ap/ar = n2r.
(6.1)
The solution, subject to the condition that p = 0 at z = h, is
where h is the height of the free surface above the bottom of the vessel and h, is the height at Y = 0. If the container were rotating with constant
34
George Veronis
FIG.5 . The free surface height, h, is shown as a function of the radial coordinate, Y, for rotation rates of (a) R, (b) R IARl, and (c) R everywhere except 0 lhnl between the dashed lines.
+
+
+
angular velocity Q AQ (Fig. 5b) the same equations with Q replaced by Q As2 would hold. The free surface would have a steeper slope for AQ >O. Now consider a thought experiment in which the container rotates with angular velocity Q but a portion of the fluid, shown by the region between dashed lines in Fig. 5c, rotates with angular velocity Q AQ. The free surface would have the shape given by (6.2) everywhere except above the special region where S2 must be replaced by AQ in (6.2). Now subtract Eqs. (6.1) from the corresponding equations with Q replaced by Q AQ to obtain the following balance of forces of the second system (the fluid in the special region) relative to the rotating frame of the first system
+
+
+
+
(l/p) 8 A p / & = 2Q AQr +(AQ)%.
(6.3)
Large Scale Ocean Circulation
35
0. Third, the net transport of fluid is to the right of the applied surface stress. The latter result can be verified by integrating Eq. (7.3) from z = - 00 to z = 0 but it is simpler and more straightforward to integrate Eqs. (7.1) directly. The result is (7.4a) (7.4b) and it is easily seen that the transport is to the right of the surface stress. Fourth, the transport given by (7.4) is independent of the details of the flow. I n anticipation of the situation for oceanic flow (7.1) may be written in the more general form (7.5a)
a
(7.5b)
2ii= - ( ETY), az
where E may be a function of z (because of V) and T is the vertical stress in the surface layer. Th e result (7.4) follows directly, the only requirement being that cc vanish at great depth. An important consequence of the foregoing considerations emerges when the prescribed horizontal velocities (or stresses) have horizontal variations of a scale much larger than the Ekman layer depth (Charney, 1955a). Then the Ekman layer solutions are locally valid near the surface and a vertical flow out of the Ekman layer results when the continuity equation is integrated from z = - 00 to z = 0 with the boundary condition zZo=Oatz=O. Thus
or t%-m=
[-
i a (&$) 2 ax
-
a
--
aY
(ET,')]= 1k . v 2
X
(ET,),
(7.6)
where Eqs. (7.4) have been used to evaluate the vertical integrals of u and v. T he parameter E is included with T~ within the del operator in anticipation of oceanic considerations.
Large Scale Ocean Circulation
39
An extension of the theory of pure Ekman layers is necessary in order to make the theory useful for oceanic (and laboratory) flows. T h e interior region, i.e. the fluid outside the Ekman layer, is normally in motion. T h e usual procedure for linear theory (Greenspan, 1968) is to write the dependent variables as the sum of an Ekman layer contribution (denoted by a tilde) and an interior contribution (denoted by subscript I) which does not decay at the edge of the Ekman layer. This approach, or an analogous one, is necessary for treating the flow in the vicinity of a rigid boundary where the total velocity must vanish. For example, consider the flow in the vicinity of a bottom boundary, z = - 1, where the total velocity must vanish. As noted earlier the solution (7.2) is valid there with t; = -( 1 z)E- 112 but the boundary values of the Ekman layer components are (7.7a)
+ -=
,uO
-%b
1
zjo = -q,, ,
(7.7b)
where subscript b corresponds to interior velocities evaluated at the bottom. Hence, the solution in the bottom Ekman layer can be written in terms of the interior velocity as
ii = -(uIb cos 5 - q,,sin t;)et,
(7.8a)
5 + GI,, sin 0 and c 4 - 1 if u, < 0. T h e transport P can now be evaluated from (8.26) as
P=/
0
6dzmd6,(1+c/k).
(8.37)
-1
Then Eq. (8.35) becomes
y6,
= -2
where y = 2/3 cos 4 (1
a.;,/aA,
+
(8.38)
C/K)/fE”2,
and the solution is
6,= 6,,e-yi,
(8.39)
53
Large Scale Ocean Circulation
where Gbo( =6,) o) can be evaluated as before so as to yield zero net meridional flux across the entire basin. For 6, > 0 the thickness of the boundary layer i ~ f E l ’ ~ / cos2 ( 2 q5( 1 c/K)), i.e. it is thinner than the corresponding boundary layer of the homogeneous system. The reason for this is that the entire vorticity of the interior must be dissipated in the bottom Ekman layer by means of a relatively weaker meridional flow, 6,. Hence, the longitudinal gradient of the meridional flow must be larger, or equivalently, the boundary layer must be thinner. It is interesting to note that the thinner the western boundary layer is near the bottom the more likely it is that some other process such as lateral friction or nonlinearity may become more important and eventually dominate the flow. For example, it was pointed out above that 6,= 0 (this is equivalent to zero boundary layer thickness) requires that lateral friction be introduced. Of course, effects of bottom topography or of nonlinearity or of a more realistic density distribution could alter the argument dramatically. For 6, < 0 the present solution is not valid because it grows exponentially with increasing longitude. T h e reason is that negative bottom velocity cannot lead to the required dissipation of negative vorticity unless 6, becomes even more negative; hence the solution does not boundary layer. Observational evidence (Swallow and Worthington, 1961) indicates that south of Cape Hatteras the deep western boundary layer flow is indeed southward. This southward flow appears to be related to the thermally forced circulation (Stommel, 1965) and because of bottom topography it may be displaced eastward relative to the Gulf Stream. Hence, it is possible that the bottom frictionalmodel may not beinapplicable even in this case. T h e issue is by no means clear, therefore, and will probably not be resolved until success is achieved with models which incorporate both bottom topography and stratification. If wI does not vanish with depth in the interior, effects of bottom topography and stratification must be incorporated everywhere and the winddriven circulation cannot really be separated from the general circulation of the oceans.
+
C. INERTIAL EFFECTS Several different methods have been employed to study nonlinear effects in wind-driven ocean circulation. These include perturbation analyses (Munk et al., 1950; Veronis, 1966), numerical studies (e.g., Bryan, 1963 ; Veronis, 1966), and studies which accept the interior flow as given and which treat nonlinear effects in the boundary layers (Fofonoff, 1954; Charney, 1955b; Morgan, 1956; Carrier and Robinson, 1962).
54
George Veronis
Perhaps the principal benefit from these investigations has been an increased understanding of the nonlinear process rather than an advance in our knowledge of the structure of the wind-driven gyres. The reason is that the models tend to be too idealized to admit incorporation of additional fcatures ; consequently, one cannot build an ordered hierarchy of successively more complicated models. T h e present discussion will be restricted to nonlinear effects in Stommel’s original model on the ,&plane although some remarks will be made about variations on the model. T h e nondimensional vorticity equation for the vertically averaged mass field is
R v . V C $ / ~ V + E C = ~ *XV T,
(8.40)
where
or, in terms of the stream function,
T h e nonlinear terms here are only crude approximations to the actual nonlinear terms because the vertical average of the product v . V( is not generally given by the product of the vertically averaged quantities. The Rossby number, R , can be defined in terms of the amplitude of the windstress since the velocity amplitudes are determined by 7. However, for present purposes it suffices to note that R is so small that nonlinear effects in the interior of the ocean are negligible unless the boundary current penetrates the interior. T h e effect of the inertial terms can be treated by perturbation methods (Veronis, 1966) in a straightforward manner. However, it is easy to see the qualitative effect of nonlinearity by rewriting (8.40) for the western boundary layer region as v * V(R5
+f)=
-E
0) the radial velocity, uI, is negative, i.e. the interior flow is toward the apex (irrespective of the location of the source). For a sink-driven flow the radial velocity is toward the rim of the basin. Hence, the apex models the northern boundary (pole) and the rim the equator. There are two points to be made in connection with (11.9). T h e first is that for source-driven flow the free surface rises locally because a column of fluid moves radially inward without changing its height. Since the equilibrium height increases with radius, such an inward flow raises the level of the fluid at any location. There must be boundary layers near the rim and near the apex since the radial flow must vanish at those extremes. For sink-driven flow a column of fluid moves radially outward thereby decreasing the level of the fluid. The second point has to do with how the change of height affects the local vorticity. Conservation of mass requires that a local increase of height of a column of fluid be accompanied by a horizontal convergence. For an inviscid fluid the angular velocity of inward-moving particles must increase in order to conserve angular momentum. T h e result is an increase in the magnitude of the vorticity. Thus, when S2 is positive (counterclockwise rotation) a source-driven flow will cause the relative vorticity to increase locally throughout the interior. Hence, the overall vorticity pattern must exhibit positive vorticity, i.e., there must be a flow toward the rim in a boundary layer on the left side looking downward. In an analogous manner it is easy to verify that a boundary current on the left side and toward the
78
George Veronis
apex is associated with sink-driven flow. Hence, a western boundary current is generated in the pie-shaped basin when the flow is driven by a source or a sink.
B. DISCUSSION OF MORECOMPLETE SOLUTIONS A complete linear analysis of the source-sink flows is not difficult to carry out (Kuo and Veronis, 1971). However, the details require an analysis making use of Stewartson shear layers in a rotating fluid and such a treatment would take us too far afield. Consequently, the present discussion will be confined to a qualitative description of the flow and the reader is referred to the Kuo-Veronis paper for details. Figure 14 illustrates the circulation driven by a source located at the junction between the western boundary and the rim for a pie-shaped basin with a semicircular cross section. T h e fluid in the experiment is a thymal blue solution and the flow is visualized by dye streaks generated by pulsing wire electrodes in the middle layers of the fluid (Baker, 1966). This example is especially instructive because both the eastern and western boundaries lie along the diameter, the western boundary along the left half of the diameter and the eastern boundary along the right half. Hence, the eastern and western boundaries meet at the apex. T h e flow is essentially radially inward everywhere in the interior so that the apex serves as a sink for the interior flow, irrespective of where the actual source is. T h e intense western boundary layer flow is obvious in the figure. Flow along the eastern boundary is slow, consistent with the partial analysis given in the previous section. Hence, at the apex there is a divergent flow parallel to the boundary because the slower moving eastern boundary flow cannot supply the amount of fluid required by the intense western boundary layer so fluid is sucked into the vicinity of the apex from the interior. This convergent flow from the interior is evident in the figure. I n a linear flow the structure of the western boundary layer is controlled by frictional processes. Consequently Stewartson. layers of thickness Ell3 and Ell4 ( E = v/fiu2is the Ekman number introduced earlier) serve to redistribute the fluid from the source and also to satisfy the conditions of zero flow at the solid side boundaries. T h e simplest equations for the western boundary layer admit a solution which satisfies only the condition of zero normal flow at the wall. It is an exact laboratory analog of Stommel’s theoretical model for wind-driven ocean circulation. When a more complete mathematical model is formulated so that tangential velocities also vanish at the side walls, the model is the analog of the HidakaMunk lateral friction model of wind-driven ocean circulation.
Large Scale Ocean Circulation
79
FIG. 14. T h e circulation generated by a source at the southwest corner of a p'ie-shaped basin with semicircular cross section (radius = 45 cm, R = 1.9 cm sec-l, Reyiiolds numbc:r = 1.33, mean height = 9 cm).
80
George Veronis
Near Y = a a boundary layer must be included in order to adjust the radial flow to zero at the rim and also to bring the tangential flow to zero there. This boundary layer turns out to be thicker than that near the western boundary. T h e Ekman layer near the bottom serves no function in the lowest order equations except within the boundary layers at the sides and at the rim. Hence, the solution found in the previous section is the exact lowest order solution for the interior flow. T h e analysis for the more complete problem requires including nonlinear as well as frictional terms in the equations. The nondimensional form of the momentum equations for this steady flow is
Rv* VV + 2 k
XV=
-Vp
f
EV'V,
and the free surface condition takes the form w =[
+ Fru +F R v . V[.
Here, the velocity scale is chosen as V = Q/AE1/',where Q is the volume flux per unit time from the source and A is the area of the basin ; the length scale is a, the radius of the basin; E = v / Q a 2 ;the dimensional a[/at is set equal to V [ = Q / A so tha [ = Ell2; the [ scale is chosen as QVa/g= RFa; and the Rossby number is defined as R = V / Q a= Q/QaAE'/'. T h e specific choice [ = E l l 2 is made because for the linear problem the velocities are all O(1) as a result. When the Rossby number is E1/', inertial effects are important. T h e ratio R/E1l2is the Reynolds number, denoted here by y , so linear flow occurs for y < 1 and nonlinear effects are important for y 2 1. A study of nonlinear flows by perturbation methods is possible for y-1 and theory and experiment agree for these moderate Reynolds numbers (Veronis and Yang, 1972). A more direct simulation of the North Atlantic gyre including the Gulf Stream is obtained when the flow is driven by a sink, because the flow in the western boundary layer is toward the north (apex). T h e laboratory Gulf Stream is more intense than the flow some distance from the western boundary and, as a result, the apex serves as a point of convergent flow parallel to the boundary. Hence, the boundarycurrent must fanout at the apexto provide fluid for theinterior flow. As the strength of the sink is increased, the flow in the western boundary layer intensifies. T h e perturbation analysis for y-1 can no longer be 1 and it is used to study the nonlinear flow patterns that occur for y necessary to consider a numerical analysis for the latter cases. A numerical study has been made by Beardsley (1972) for the nonlinear flow in the sliced cylinder model of oceanic circulation. Several features emerge in our experiments with the flows driven by a sink at the southwest corner of the basin. T h e western boundary layer is
Large Scale Ocean Circulation
81
formed near the rim and then flows to the apex before it couples again to the interior flow. Between the rim and the apex the western boundary layer flow is extremely stable. In none of the experiments, even with Reynolds numbers as high as 70, has the western boundary layer shown an indication of becoming unstable. A second feature is that as the flow becomes nonlinear a high pressure region forms in the interior near the apex. Th e western boundary jet leaves the boundary in the vicinity of the apex and much of the fluid is swept around the high pressure region and flows toward the rim on the offshore side of the western boundary layer. Hence, a recirculating gyre is generated near the western boundary. Such a feature appears also in numerical analyses of wind-driven ocean circulation (Veronis, 1966) as well as in other experimental studies (Beardsley, 1969). A photograph of the flow pattern for an experiment with y = 5 is shown in Fig. 15 and the high pressure region at the center of the recirculating gyre is clearly present just to the southwest of the apex. A third feature is that sufficiently intense flows become unstable. As noted earlier, the western boundary layer remains stable. However, as the western boundary jet leaves the vicinity of the apex and penetrates into the interior, swirls formon either side and the flow becomes transient. Figure 16 exhibits both the stability of the western boundery layer and the instability of the jet after it leaves the apex and flows into the interior. For this flow the value of the Reynolds number is 20.
C. THEFLOWDUE TO
A
SOURCE OF DENSEWATER
The experiments reported above have all been carried out with working fluids of constant density. When the density of the fluid varies, it is not possible to simulate the beta effect in the simple manner outlined above. Nevertheless, since abyssal circulation is driven by sinking of dense water in polar regions, it is of some interest to determine what the effect is when the source fluid has higher density than the ambient fluid of the basin. Such experiments are basically transient because the relative amounts of water of different densities change monotonically with time. Even so, however, we were surprised to observe the magnitude of the change from the homogeneous case to the two-fluid case. Photographs from three experiments are shown below. I n all three cases the flow is in the linear range (Reynolds number = 0.75), the rotation rate is 1.9 rad sec-l, the mean depth of the fluid initially is 8 cm, and the ~ . the first density of the fluid in the basin initially is 0.998 gm ~ r n - In experiment the density of the source water is also 0.998gmcm-3. For
82
George Veronis
FIG.15. Same as Fig. 14 except that the flow is sink-driven and Reynolds number = 5. For the nonlinear flow shown here a high pressure region forms near the western boundary south of the apex.
Large Scale Ocean Circulation
83
FIG.16. Same as Fig. 15 except that Reynolds number = 20. T h e jet bifurcates as it leaves the apex and forms swirls on either side. These swirls eventually separate and become self-contained eddies which are swept around the high pressure region southwest of the apex.
84
George Veronis
the second and third experiments the densities of the source water are 1.001 and 1.075 gm ~ m - respectively. ~ , The motion was observed by two different methods. In the first, illustrated in Figs. 17-19, the source fluid is colored and the penetration of the colored fluid is shown shortly after the experiment was started and then toward the final stage of the experiment. Figures 17a and 17b show the penetration of homogeneous source fluid after 2 min and after 90 min, respectively. The ball of fluid at the apex reflects the disturbance created by the source itself. Since the transport in the bottom Ekman layer is to the left of the flow above the Ekman layer, the increased thickness of the dark band of fluid around the sides and rim of the tank in Fig. 17b shows
FIG.17. Dyed fluid with the same density, p = 0.998, as the ambient fluid is introduced through a source at the apex and makes its way around the sides of the basin. T h e penetration of the dyed fluid is shown (a) 2 min and (b) 90 min after source was turned on.
Large Scale Ocean Circulation
85
the inward penetration associated with the Ekman layer. Note especially that the ball of fluid near the apex penetrates very slowly into the interior and even after 90 min it is confined to a sector within 13 cm of the apex. Figures 18a and 18b illustrate the change that takes place when the source fluid is slightly more dense than the fluid in the basin. Here, it is important to note that the flow pattern in Fig. 18b was photographed after a delay of only 21 min. The denser fluid was generally observed to make the circuit around the sides in a time substantially shorter than that required by the homogeneous fluid. The most striking difference between Figs. 17b and 18b is in the penetration of the ball of fluid from the source near the apex. The penetration into the interior is much greater for the dense fluid and, in fact, it works its way to the rim boundary current near the western
FIG.18. Same as Fig. 17 except source fluid has density 1.001. (a) After 2 min and (b) after 21 min.
86
George Veronis
side. This penetrative character of the dense fluid is important in connection with the observed high concentrations of tritium in the bottom 100 m of the ocean at mid-latitudes (Rooth, 1971). Tritium is a bomb-produced radioactive tracer which has a relatively high concentration in the surface waters and throughout the depth in polar regions but its presence is barely observable in deep water at mid-latitude. The present experimental result suggests that the high concentration near the bottom is associated with the Ekman layer flux of dense water which sinks to the bottom in the Greenland Sea and then spreads southward in the Ekman layer. The source fluid in the experiment stays close to the bottom whereas in the homogeneous case it tends to distribute vertically. When the density of the source water is large, as in Figs. 19a and 19b, the circuit around the sides of the tank is even faster (there is a ten-minute
FIG.19. Same as Fig. 17 except source fluid has density 1.075. (a) After 2 min and (b) after 10 min.
Large Scale Ocean Circulation
87
FIG.20. Experiment of Fig. 17 repeated. The flow of fluid originally in the basin is visualized by the thymol blue technique. Shows flow after 5 min.
interval between the two photographs). The penetration of the ball of fluid from the source is now erratic but very strong. Earlier photographs show that the ball extends to the rim within four minutes after the beginning of the experiment. Figure 19b shows another feature of the highdensity source flow. There is a strong flux of fluid out of the western
88
George Veronis
boundary current about a third of the distance from the apex. This feature was observed in all of the experiments when the density of the source fluid was 1.075. The source fluid in this experiment also tends to stay close to the bottom. The second method of observation is shown in Figs. 20-22, where the thymol blue technique mentioned earlier was used to mark the fluid which is initially in the basin before the experiment was started. Figure 20 shows a flow similar to that of Fig. 14 although the source is located at the apex in the former and at the junction between the western boundary and the rim in Fig. 14. The flow here is steady and consistent with what is expected from linear theory. The ball of fluid resulting from the source near the apex is visible in Fig. 20 and a steady eddy occurs near the rim as the intense western jet impinges on the boundary.
FIG. 21. Experiment of Fig. 18 repeated. The flow of fluid originally in the basin is visualized by the thyrnol blue technique. (a) After 5 min and (b) after 7 min.
Large Scale Ocean Circulation
89
The startling contrast introduced when the source fluid is slightly denser ( p = 1.001 gm cm-3) is shown in Fig. 21. The photographs of the fluid pattern in the homogeneous fluid were taken two minutes apart, and the disordered character of the flow is apparent in each photograph. The western boundary current is toward the rim only in the southern half of the region in Fig. 21a but even there it has reversed and exhibits eddy-like structure in Fig. 21b. Near the rim the eastward jet of homogeneous fluid reverses as the source flow works its way around the rim. The latter behavior is seen in the marked fluid near the rim on the westernmost radial marker. Later photographs show that this reversal progresses eastward with the source fluid. The chaotic behavior associated with a source of dense fluid is exhibited in Fig. 22a where just after the start of the experiment even the
FIG.22. Experiment of Fig. 19 repeated. The flow of fluid originally in the basin is visualized by the thymol blue technique. (a) After 2 min and (b) after 12 rnin.
90
George Veronis
interior flow no longer exhibits behavior expected for the homogeneous experiment. T he boundary layer near the rim is completely reversed and there is no longer a southward western boundary current. T h e flow pattern after 12 min is shown in Fig. 22b. Th e intense flow out of the western boundary is again apparent and the interior flow is a collection of completely disordered eddies. There is not much hope of obtaining a complete analysis of the flows generated by a dense fluid source. However, some features, such as the rapid penetration of the dense source fluid via the Ekman layer and the diverging fluid from the western boundary layer, can be obtained in a model more suited for the particular purpose.
ACKNOWLEDGMENT Support from the National Science Foundation under Grant GA 25723 is gratefully acknowledged. REFERENCES BAKER, D. J. (1966). A technique for the precise measurement of small fluid velocities. J. Fluid Mech. 26, 573-575. BAKER,D. J. and ROBINSON, A. R. (1969). A laboratory model for the general ocean circulation. Phil. Trans. Roy. SOC.London Ser. A 265, 533-565. R. C. (1969). A laboratory model of the wind-driven ocean circulation. J. Fluid BEARDSLEY, Mech. 38(2), 255-271. BEARDSLEY, R. C. (1972). A numerical model of the wind-driven ocean circulation in a circular basin. J. Geophys. Fluid Dynam. (in press). BLANDFORD, R. R. (1965). Notes on the theory of the thermocline. J. Mar. Res. 23, 18-29. BLANDFORD, R. R. (1971). Boundary conditions in homogeneous ocean models. Deep-sea Res. 18, 739-751. BRYAN, K. (1963). A numerical investigation of a non-linear model of a wind-driven ocean. J . Atmos. Sci. 20, 594-606. BRYAN, K. (1969). Climate and the ocean circulation. 111. The ocean model. Mon. Weather Rev. 97, 806-827. BRYAN,K., and Cox, M. D. (1967). A numerical investigation of the oceanic general circulation. Tellus 19, 54-80. CARRIER, G. F., and ROBINSON, A. R. (1962). On the theory of the wind-driven ocean circulation. J . Fluid Mech. 12, 49-80. CARRITT, D. E., and CARPENTER, J. H. (1958). The composition of sea water and the salinity-chlorinity-density problems. Physical and chemical properties of sea water. Nut. Acad. Sci.-Nut. Res. Counc., Publ. 600, 67-86. CHARNEY, J. G. (1955a). The generation of ocean currents by wind. J . Mar. Res. 14, 477-498. CHARNEY, J. G. (1955b). The Gulf Stream as an inertial boundary layer. Proc. Nut. Acad. Sci. U S . 41, 731-740. CRAIG,H., SCHLATER, J. G., CHUNG, Y., EDMOND, J. M., KROPNICK, P. M., and WEISS, R. F. (1970). Geochemical and temperature profiles in equatorial Pacific bottom water. Trans. Amer. Geophys. Union 51, 326 (abstr.).
Large Scale Ocean Circulation
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DEARDORFF, J. W. (1972). Numercial investigation of neutral and unstable planetary boundary layers. J. Atmos. Sci. (in press). V. W. (1905). On the influence of the earth’s rotation on ocean currents. Ark. Mat., EKMAN, Astron. Fys. 2, 1-53. EKMAN, V . W. (1908). Die Zusammendruckbarkeit des Meerwassers. Publ. Circ. Cons. Explor. Mer 43, 1-47. ERTEL,A. (1942). Ein neuer hydrodynamischer Wirbelsatz. Meteorol. Z. 59, 277-281. FOFONOFF, N. P. (1954). Steady flow in a frictionless homogeneous ocean. J. Mar. Res. 13, 254-262. FOFONOFF, N. P. (1962a). Physical Properties of sea water. “ T h e Sea,” Vol. 1, pp. 3-30. Wiley (Interscience), New York. FOFONOFF, N. P. (196213). Dynamics of ocean currents. “ T h e Sea,” Vol. 1 , (Interscience) pp. 323-395. Wiley FOFONOFF, N. P., and FROESE, C. (1958). Program for oceanographic computations and data processing on the electronic digital computer ALWAC 111-E. PSW-1 Programs for properties of sea water. Fish. Res. Ed. Can., Manuscript Rep. Ser. No. 27 (unpublished manuscript). H. (1968). “ T h e Theory of Rotating Fluids.” Cambridge Univ. Press, GREENSPAN, London and New York. L. N. (1963). On a time-dependent motion of a rotating GREENSPAN, H. P. and HOWARD, fluid. J. Fluid Mech. 17, 385-404. HIDAKA, K. (1949). Mass transport in ocean currents and lateral mixing. J . Mar. Res. 8, 1 932-1 936. Holland, W. (1967). On the wind-driven circulation in an ocean with bottom topography. Tellus 19, 582-600. K. (1966). Ekman drift currents in the Arctic Ocean. Deep-sea Res. 13, 607HUNKINS, 620. JEFFREYS, H. (1962). “ T h e Earth.” Cambridge Univ. Press, London and New York. Knudsen, M. (1901). “ Hydrographical Tables.” Williams & Norgate, London. Kozlov, V. F. (1966). Certain exact solutions of the non-linear equation for density advection in the ocean. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okean 2, 1205-1207; see Atmos. Oceanic Phys. 2, 742-744 (1 100). Kuo, H. H. and VERONIS, G. (1970). Distribution of tracers in the deep oceans of the world. Deep-sea Res. 17, 29-46. Kuo, H. H., and VERONIS, G. (1971). The source-sink flow in a rotating system and its oceanic analogy. J. Fluid Mech. 45, 441-464. LYNN,R. J . , and REID,J. L. (1968). Characteristic and circulation of deep and abyssal waters. Deep-sea Res. 15, 577-598. MARGENAU, H. and MURPHY, G. M. (1949). “ T h e Mathematics of Physics and Chemistry.” Van Nostrand-Reinhold, Princeton, New Jersey. MORGAN, G. W. (1956). On the wind-driven ocean circulation. Tellus 8, 301-320. MUNK,W. H. (1950). On the wind-driven ocean circulation. J. Meteorol. 7 , 79-93. MUNK,W. H. (1966). Abyssal recipes. Deep-sea Res. 13, 707-730. MUNK,W. H., GROVES,G., and CARRIER, G. F. (1950). Note on the dynamics of the Gulf Stream. J . Mar. Res. 9, 218-238. NEEDLER, G. T . (1967). A model for thermohaline circulation in an ocean of finite depth. J . Mar. Res. 25, 329-342. NEEDLER, G. T. (1972). Thermocline models with arbitrary barotropic flow. Deep-sea Res. 18, 895-903. NIILER,P. A,, ROBINSON, A. R., and SPIEGEL,S. L. (1965). On thermally maintained circulation in a closed ocean basin. J. Mar. Res. 23, 222-230. PEDLOSKY, J. and GREENSPAN, H. P. (1967). A simple laboratory model for the ocean circulation. J. Fluid Mech. 27, 291-304.
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PHILLIPS,N. A. (1963). Geostrophic motion. Rew. Geophys. 1, 123-176. PHILLIPS,N. A. (1966). The equations of motion for a shallow rotating atmosphere and the “ traditional approximation.” J . Atrnos. Sci. 23, 626-628. RAMSEY,A. S. (1964). “ Newtonian Gravitation.” Cambridge Univ. Press, London and New York. REID, J. L. (1965). “ Intermediate Waters of the Pacific Ocean.” Johns Hopkins Press, Baltimore, Maryland. ROBINSON, A. R., and WELANDER, P. (1963). Thermal circulation on a rotating sphere; with application to the oceanic thermocline. J. Mar. Res. 21, 25-38. ROOTH,C. (1971). Private communication. ROOTH,C. and ~ S T L U N D ,G. (1972). Penetration of tritium into the Atlantic thermocline. Deep-sea Res. 19,481-492. STOMMEL, H. (1948). The westward intensification of wind-driven ocean currents. Trans. Amer. Geophys. Union 29, 202-206. STOMMEL, H. (1955). Lateral eddy viscosity in the Gulf Stream Systems. Deep-sea Res. 3,88-90. STOMMEL, H. (1957). A survey of ocean current theory. Deep-sea Res. 4, 149-184. STOMMEL, H. (1965). “ The Gulf Stream.” Univ. of California Press, Berkeley. STOMMEL, H., and ARONS, A. B. (1959). On the abyssal circulation of the world ocean. I. Stationary flow patterns on a sphere. Deep-sea Res. 6, 140-154. STOMMEL, H., ARONS,A. B.,and FALLER,A. J. (1958). Some examples of stationary planetary flows. Tellus 10, 179-187. SVEFUIRUP,H. U. (1947). Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Nut. Acad. Sci. U S . 33, 318326. SVERDRUP, H. U., JOHNSON, M. W., and FLEMING, R. H. (1942). “ T h e Oceans.” PrenticeHall, Englewood Cliffs, New Jersey. J. C., and HAMON, B. V. (1960). Some measurements of deep currents in the SWALLOW, eastern North Atlantic. Deep-sea Res. 6, 155-168. SWALLOW, J. C., and WORTHINGTON, L. V. (1961). An observation of a deep countercurrent in the western North Atlantic. Deep-sea Res. 8, 1-19. VERONIS, G. (1963). On inertially-controlled flow patterns in a ,%plane ocean. Tellus 15, 59-66. VERONIS, G. (1966). Wind-driven ocean circulation. Deep-sea Res. 17, 17-55. VERONIS,G. (1969). On theoretical models of the therrnohaline circulation. Deep-sea Res. 16, Suppl. 301-323. VERONIS, G., and YANG,C. C. (1972). Non-linear source-sink flow in a rotating pie-shaped basin. J. Fluid Mech. 51, 513-528. VONARX,W. S. (1952). A laboratory study of the wind-driven ocean circulation. Tellus 4, 311-318. WARREN, B. (1963). Topographic influences on the Gulf Stream. Tellus 15, 167-183. WEBSTER, F. (1961). The effect of meanders on the kinetic energy balance of the Gulf Stream. Tellus 13, 392-401. WELANDER, P. (1959). An advective model of the ocean thermocline. Tellus 11, 309-318. WELANDER, P. (1968). Wind-driven circulation in one- and two-layer oceans of variable depth. Tellus 20, 1-16. WELANDER, P. (1971a). Some exact solutions to the equations describing an ideal-fluid thermocline. J . Mar. Res. 29, 60-68. WELANDER, P. (1971b). The thermocline problem. Phil. Trans. Roy. SOC.London Ser. A 270.69-73.
The Wave Resistance of Ships JOHN V. WEHAUSEN Department of Naval Architecture University of California. Berkeley. California
I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1 . The Measurement of Wave Resistance . . . . . . . . . . . A . Residuary Resistance and Refinements-Froude’s Method . B. Momentum Considerations . . . . . . . . . . . . . . C. Wave-Pattern Analysis . . . . . . . . . . . . . . . . 111. The Analytical Theory of Wave Resistance . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . B. The Exact Problem in an Inviscid Fluid . . . . . . . . . C . Perturbation Expansions . . . . . . . . . . . . . . . . D . Methods of Solution . . . . . . . . . . . . . . . . . E. Further Results, Variations. and Extensions . . . . . . . F‘. Numerical Methods . . . . . . . . . . . . . . . . . . G . Comparison of Theory and Experiment . . . . . . . . . H . Applications of the Theory . . . . . . . . . . . . . . I . Higher-Order Theories . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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93 96 96 100 109 131 131 134 137 142 152 174 177 198 214 229 230
For Georg Weinblum for his 75th birthday
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I Introduction A newcomer to the subject would probably ask the following questions . What is wave resistance? How do you measure it? How do you compute it? We shall discuss in the following pages all three questions and try to describe how they have been answered up to now . The answer to the question. “What is wave resistance?”. is not as elementary as it might seem at first glance. One observes that waves follow a moving ship. supposes that some expenditure of energy is required for 93
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John V . Wehausen
their generation, and probably recognizes that the propagation of these waves is associated with the presence of a gravitational field. However, wave formation is not the only expense of energy. One also knows that water is endowed with viscosity and that any body moving through a viscous fluid experiences a resistance, partly due to tangentially acting stresses on the body, but partly also because boundary-layer growth and, of course, separation if it occurs yield a resistance resulting from integrating normal components of the stress over the body. A new question then is, “Can we clearly separate resistances due to gravity alone and to viscosity alone?” That is, is there a “gravitational resistance ” and a “ viscous resistance ”? A moment’s thought shows that this is not likely. T h e frictional resistance, i.e., the part due to tangential stress, will certainly depend upon the wave profile along the ship. On the other hand, the wave pattern itself is going to depend in some fashion upon the ship’s boundary layer and wake. T h e effects of gravity and viscosity interact in essential ways. There seems to be no neat practicable definition of gravitational or wave resistance without introducing assumptions or approximations. Since any discussion of measurement or calculation of wave resistance necessarily touches on its definition, we shall have to return to this question subsequently. Finally, one may ask, “Why make any decomposition at all?” The answer is primarily economic. One would like to be able to predict the power requirements of a full-scale ship from its designed lines. T o do this by calculation from the Navier-Stokes equations is beyond our powers. T o do it by model tests leads to dilemmas that can be resolved, at least in part, if we can find the components of resistance, especially as they depend upon the effects of gravity and viscosity. This has usually been simplified further into trying to separate the resistance into parts due to wave making and to friction. As we shall see below, the separation has been moderately successful, in fact, enough so that in this account of the subject we shall be able to avoid dealing with viscous resistance except in a superficial way. A further reason for trying to separate wave resistance and to understand its relationship with hull shape is ship design. For conventionally shaped ship hulls one cannot reduce the frictional resistance significantly by redesign of the hull. Eddy resistance can be kept small by properly designing the afterbody. Added resistance resulting from bilge vortices can also be avoided by proper design. (Wave-breaking resistance, although it shows up in certain experiments as viscous resistance, will be lumped here with wave resistance). This still leaves great freedom in designing the hull, and experience and observation have shown that optimum designs (from thestandpoint of resistance) are those that make the smallest waves. Rational
The W a v e Resistance of Ships
95
hull design requires an understanding of how a suitably defined wave resistance is connected with hull geometry. Although apparently not the first person to try to examine in a systematic and scientific way the resistance of a ship, William Froude (1810-1879) seems to have been the first to appreciate fully the differing roles played by friction and wave making in ship resistance and the significance of this difference in trying to project data from model tests to full-scale size. This is already evident in a memorandum of December 1868 from Froude to the Chief Constructor of the Navy in which he outlines the advantages of model tests and proposes construction of a model tank. In a later paper (1876) Froude discusses the components of ship resistance and their implications for ship design with such insight that this paper can still be read with profit. Froude’s analysis laid solid foundations for future tests of ship hulls and his ideas still dominate the subject. However, Froude’s method, which will be briefly described later on, has one important disadvantage, explicitly recognized by him, namely, that it did not allow one to predict by purely analytical means the resistance of a ship. T h e first significant step in this direction was taken by J. H. Michell(l863-1940) who in a paper published in 1898 derived an analytic expression for the wave resistance of a ship moving in a calm, inviscid fluid. I n much the same way that Froude’s ideas have dominated the field of ship model testing, Michell’s results have dominated much of the subsequent analytical study of wave resistance. T h e present account of the subject is divided as follows. T h e first part treats the measurement of wave resistance. This in turn is divided into a brief exposition of Froude’s method followed by a more lengthy treatment of the determination of wave resistance from wave-pattern measurements. The second part treats analytical methods for calculating wave resistance and some applications. There exist already a number of expository accounts of the separate parts of this article. For a discussion of the problem of separation of wave and viscous resistance one can hardly do better than to read Sharma’s (1964, 1965) account of this subject. For questions concerning the determination of wave resistance from direct wave measurements a paper by Eggers, Sharma, and Ward (1967) gives not only an excellent survey of the subject but also a valuable bibliography. For the analytical theory of wave resistance there are a number of expository accounts directed at different audiences. There is a fairly comprehensive treatise by Kostyukov (1959, 1968). Of the others listed below perhaps the most comprehensive are by Wigley (1949), Lunde (1951a), and Sabuncu (1962b), the last not being really very accessible because of language. T h e others are more restricted
John V . Wehausen
96
in scope or in prospective audience. See the following references: Bessho (1957b), Gadd (1968), Havelock (1926b, 1951), Inui (1954,1957), Kostyukov (1959, 1968), Lunde (1951a, 1957, 1969), Maruo (1957), Sabuncu (1962b), Wehausen (1956), Weinblum (1950, 1959, 1963), Wigley (1930b, 1935, 1949).
11. The Measurement of Wave Resistance A. RESIDUARY RESISTANCE AND REFINEMENTSFROUDE’S METHOD Determination of the wave resistance by experiment has proceeded along two rather different paths. I n the older one, to be described here, wave resistance is usually identified with a component extracted from the total resistance by a method similar to that originally devised by Froude. This component is the “residuary resistance,” defined below, or some part of it. It is easy to criticize the method and to propose modifications, and many persons have done so. Although some of the analyses that have been proposed in recent years are ingenious and throw considerable light on the components of ship resistance, I believe that it will be clear from the brief description below of Froude’s method that, in fact, the effects of gravity and viscosity are “ inextricably interwoven,” to borrow a phrase from Froude himself. The purpose of this section is then not to expound all the recent developments in this field but rather to give some perspective concerning this method of trying to assess the contribution of wave making to the resistance of a ship. Consider a ship moving steadily on a constant course in a calm sea. Let its length, say a t the waterline, be L and its velocity U , and let the density of the water be p and its kinematic viscosity v. T h e ship moves at some expense of energy. Let E be the rate of energy expenditure. We may call R = E / U the resistance of the ship. If we neglect certain other physical parameters, especially the vapor pressure of the water and surface tension, and consider our ship as one of a family of geometrically and dynamically similar ships, we may write R = F ( L , U , p, v, g). We must include g, for we know that the gravitational force plays an important role in the formation of the wave pattern behind the ship. We need not dwell upon the ways of making this dimensionless, but write down immediately
C, =
R zp U2L2
=f( U/(Lg)1‘2, UL/v)= f ( F n , Rn),
The Wave Resistance of Ships
97
where F n is the Froude number and Rn the Reynolds number. C, is evidently a function of the two variables Fn and Rn and can be represented by a surface over the (Fn , Rn)-plane that will be characteristic for this family of ship hulls. If one experiments with just one of these hulls, how much of the surface can one construct! Experiments at one place and during a restricted time interval will allow variation only in U, i.e., variation along a ray in the (Fn , &)-plane defined by
With a given hull very little variation in A is possible in practice. T h e value of g is practically constant, although one can make thought experiments with dropping elevators, merry-go-rounds, or towing tanks on the moon ; and v can be varied rather little for practically useful fluids, perhaps by a factor of two. We shall call this result the “ ship-model tester’s dilemma ” (although it is only one of several), for it shows why tests on a model cannot in principle give information about the prototype (which we assume to be much larger). Since Ap $ A,, the model experiments simply explore the wrong part of the surface C,(Fn , Rn), as shown schematically on Fig. 1. If one wishes to allow for some variation in v for the model test, one may replace the ray labeled with A, by a narrow sector. It is not customary to use three-dimensional representations. Instead, one projects the curves cut out of the surface C,(Fn, Rn) onto the (Rn, C,)-plane, and, in order to bring the curves conveniently onto one graph, uses a logarithmic scale for
FIG.1. Sketch showing parts of surface CT(Rn,Fn) determined by a model test and needed for prototype ship.
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John V . Wehausen
0 0
30.00 30.00 24.00
6.350 6.350 7.938
+ (1
X 0
12.00 9.00 6.00 4.00
15.875 21.167 31.750 47.625
REYNOLDS NUMBER
FIG.2. Values of resistance coefficient CT obtained from tests with models of Lucy Ashton. [From Hinterthan, 1956 (Fig. 18).]
R n . Figure 2 shows a typical plot of such resistance curves for a family of models. T h e constant-A curves are labeled by the model lengths, which may not be consistent in a dimensionless plot but is hardly confusing since in practice v changes but little. T h e curves cut out of the surface C,(Fn, Rn) by constant-Fn planes are also labeled, but again not dimensionlessly. T h e prototype in this case, Lucy Ashton, is only 190 f t long. I n order to give some idea of the magnitudes of F n and Rn , we remark that usually 0.1 < F n < 0.5 and that for the usual merchant ship 0.15 < F n (0.3. T h e values of Rn depend then upon ship size. For five-foot models (the smallest usually tested because of laminar-flow difficulties) the Rn range corresponding to 0.15 < F n < 0.3 is approximately 1 x lo6 < Rn < 2 x lo6, for a 20-foot model, 7.5 x lo6 < Rn < 1.5 x lo7, and for a 500-foot ship, 1 x lo9 < Rn < 2 x lo9. T h e dilemma mentioned above was resolved after a fashion by a bold assumption of William Froude. T h e assumption as we shall state it below is not really in the form given by Froude. Froude did not know about Froude or Reynolds numbers, or about dimensional analysis. He discovered were by experimenting with geometrically similar models that, if UL kept constant, the wave patterns of models of different sizes were similar.
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However, the equation given below represents (almost) in our notation his assumption, and in any case is usually called Froude’s hypothesis.
C,(F?Z, Rn) = C,(Fn)
+ CF(R7.2).
It follows immediately from this assumption that the curves Fn = const. are all at a constant vertical distance from one another in the ( C , , Rn)-plane and that the curves A = const. are all congruent, one being obtained from the other by sliding along the curves Fn = const. Hence, in applying Froude’s assumption to model testing, one tests the model at the same Froude number as the prototype, which in any case is more practical than testing at the same Reynolds number, and then finds the total resistance coefficient of the prototype from cTp
cT?n
- cFm -k
cFp
*
What is necessary now, of course, is to know the curve C,(Rn) for the given hull form In fact, it would be sufficient to know the function CT(Fn, Rn) for any fixed value of Fn . Any such curve is known as an “ extrapolator.” One could consider Froude’s assumption as purely an empirical one to be supported by the examination of data like those shown in Fig. 2. However, there was a rationale behind Froude’s assumption. Let us decompose C , into parts obtained by integrating over the ship’s wetted surface the components in the direction of motion of the normal and tangential components of the stress vector:
CT(Fn Rn) == cnoi-m(Fn> Rn) f Ctang(Fn Rn). This decomposition is, of course, precisely defined. Now one argues that the tangential force is primarily determined by the viscosity of the water and the normal force primarily by the effect of gravity as a result of the = waves produced by the ship, or, in dimensionless variables, that Cnorm Cnor,(Fn)and C,,,, = Ctang(Rn).In a final bold step one replaces the tangential force by the tangential force on a flat plate of the same water-line length and wetted-surface area as the ship at rest. If S is this area, one defines a frictional-resistance coefficient by C,(Rn) = (flat-plate resistance)/ipUU2S. (Note that L2has been replaced by S . Other coefficients will now be similarly redefined). Th e difference C, - C, = C, is now assumed to be a function of Fn only and is called the “ residuary-resistance ’’ coefficient. The advantages and weaknesses of Froude’s assumption are nearly obvious. T he choice of C, has yielded a function of Rn that is independent of hull form. T he only problem is to measure it with adequate accuracy over a sufficiently wide range of Reynolds numbers. This problem, for years a lively topic among naval architects, and still not a dead one, will not be discussed here. The curve in Fig. 2 labeled “ Schoenherr Line ” is one of
100
John V . Wehausen
the standard formulas for C, , but there are others. T h e residuary resistance contains everything else : wave resistance, eddy resistance, curvature effects, effects of trim and sinkage, the difference between the true wetted area and the wetted area at rest, etc. T h e weakness of the assumption is, of course, to assume that C , is a function of Fn alone. A frequent refinement is to estimate an eddy or form resistance as a function of the form: C, = a bC, , where a and b are empirical coefficients depending upon the hull form. I t is not uncommon to find a = 0 and b g 0.16. T h e constants are chosen so that at the lower end of the measured total-resistance curve, where the C , curve becomes approximately parallel to C,, the curves C , and C, a bCF run together. T h e curve C , -- CF- C , is often identified with the wave resistance, although not without awareness on the part of naval architects of the shortcomings of the definition. A more sophisticated version of Froude’s method, but also one requiring more effort in application, is to define the viscous resistance as half the resistance of a double model tested at appropriate Reynolds numbers in a wind tunnel or deeply submerged in a towing tank. If this is denoted by C,(Rn), then C , - C, will still depend upon Rn as well as Fn inasmuch as the double-model experiment does not include the effect of varying wave profile and sinkage and trim upon frictional resistance, or even upon such phenomena as separation. Even at low Froude numbers when there are almost no waves, the wakes of the single and double model may be different because of the lack of turbulent interchange across the free surface with the single model. On the other hand, C, comes much closer to including all the effects of viscosity associated with a given hull form than do the methods based upon flat-plate frictional resistance. Many further refinements of Froude’s assumption are possible and, since any improvement makes more accurate the prediction of prototype performance, there are frequent proposals. Since it is not our main purpose to discuss such matters, we cite only a few recent papers concerning this topic: Lackenby (1965), Shearer and Cross (1965), Weinblum (1970), Shearer and Steele (1970). As mentioned earlier, the purpose of this somewhat lengthy but still superficial introduction to the phenomenological theory of ship resistance is to try to provide some perspective about the nature of the resistance derived from such experiments, for a theoretically computed wave resistance is often compared with them.
+
+ +
B. MOMENTUM CONSIDERATIONS In the following paragraphs we shall derive some general expressions, in terms of certain surface integrals, for the force acting upon a body in steady translational motion. The derivation will be carried out for a viscous fluid
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satisfying the Navier-Stokes equations and the boundary conditions appropriate to them. However, the final formulas are also correct for an inviscid fluid if one sets v = 0. It is not obvious that this will be so, for the boundary conditions appropriate to an inviscid fluid are different from those for a viscous fluid. However, one may verify the correctness of the formulas for an inviscid fluid by examining, as one proceeds through the proof, the effect of altering the boundary conditions on the body. We begin with a coordinate system fixed in the fluid, 07 directed upward (against gravity), & in the direction of motion, and 0 5 to the starboard; &% coincides with the undisturbed water surface. We recall that for an incompressible Navier-Stokes fluid the stress tensor is given by Tij
+p(ui, +
== -@ij
j
(2.1)
uj. i),
where (&, Z 2 , 2,) = (3,7, Z), the velocity components of the absolute . shall use the conmotion are (u, w , w ) = (ul, u2, u,) and ui, = a ~ , / & ~We vention that repeated indices denote summation. The velocity components must then satisfy the equations
pauilat
+
ui, i = 0, puk u i , k
= -psi - pg6i2
+
pu, k k
i=
i
3*
2i
(2.2) (2*3)
If the velocity at any point on the surface S of the body is V = (U,, U,, U3) = ( U , V , W ) ,then the boundary condition on the body is u i = Ui on S. For an inviscid fluid it is only the kinematic condition u 1. n1.= U.n. I I'
(2.4)
(2.5)
Since we shall not be considering the effects of cavitation, there is no loss in generality in taking the atmospheric pressure to be zero. We shall also neglect surface tension, for it plays practically no role for bodies of the size of concern to us. The dynamical boundary condition on the free surface is then simply Ti,
n, = -pni
+ p ( u i ,,+ u,,i)nk= 0,
i = 1, 2, 3.
(2.6)
If the fluid is inviscid, one need only set p = 0 above, to obtain the usual
p
= 0.
(2.7)
In addition, the kinematic condition must be satisfied. If we express the free surface by jj = Y(2,5,t ) , this takes the well-known form
Y$,jj, t)u - v
+ Yrw + Y t = 0.
(2.8)
John V . Wehausen
102
Consider now the fluid bounded by the wetted surface S of the body and a (possibly moving) control surface t; enclosing the body. Part of t; may consist of free surface but the combined surface S and C must bound only fluid. Figure 3 shows schematically two possibilities. The total momentum of the fluid in the volume V is Qi
pui dV,
=
and the rate of change of momentum is
If we now use the Navier-Stokes equations to replace aui/at and use the continuity equation ui,{= 0, we obtain
=
s,
bui(Uknk-Uknk) -tTiknk]dS-pgl VIsiZ,
(2.10)
where I VI is the volume of V. On any part of I; that is the free surface both terms in the integrand vanish, so that the integral over Z extends only over its submerged portions. In the integration over S the first part of the integrand vanishes because of the kinematic condition (2.5). The remainder is just the force acting on the body:
Fi = - J j T i k nk dS. S
If we let Zc stand for the submerged part of C, we finally obtain
FIG.3.
Sketches illustrating surfaces S, C, , and Z;,
(2.11)
The Wave Resistance of Ships
103
We now add a further restriction to the surface C. I t is to move together with the body. If we now assume that the motion has continued for such a long time that the mean motion of the velocity field within V no longer changes with time, then dQi/dt = 0. (Note that we are allowing turbulent motion in the wake as long as the mean motion is steady.) This now gives us a formula for the mean force acting on the body in terms of an integral over the control surface Zc on which
U, nk = Un,,
(2.13)
where U is the velocity of the body in the direction of 62. I n finding the mean value of (2.12) one must take especial care with the quadratic term pui uk nk . If we write temporarily u i = G i uit, where C i gives the mean velocity field, then
+
pui uknk = pGi zik nk
+pui’uk’nk.
We now drop the bars over the mean velocity and denote it by ui . We then find for the mean resistance R = -Fl, the only component of Fi in which we are interested here, the following expression :
R =-
1
[pul(un, - ukn k )-pn, - p u i n k
+
ZC
+
dS. (2.14) T h e quantities u i ,p , R are now all mean quantities, and -Pzdltz&‘nk is a Reynolds stress, which may be much more than the viscous stress. Although we have used only one component from (2.12), one should be aware that much of the ensuing analysis could also be carried through for side forces and sinkage, and, with a further extension of the calculations, for trim. This is an exact formula in the sense that no mathematical approximations have been made in deriving it from the original assumptions. As was mentioned earlier, this is also an exact formula for an inviscid fluid if one sets p = 0. Even if the viscosity of water is taken into account, the term with p as coefficient is usually quite small compared with the other terms. This does not mean, however, that the values of the integral will be nearly the same in either case, for the velocity uk and the pressure p satisfy different differential equations and boundary conditions in the two cases. I t will be convenient to make here a slight change in notation. Since the motion has been assumed steady in the mean in the frame of reference of the ship (taken here as Oxyz), we may write p(ul,k
uk,l)nk]
7,z, t ) = Ci(R - ut,7, 2 ) = C i ( X , y, z), P(%7,%t ) =$(a - ut,7 , 2 )= y , z),
Ui(X,
m,
Y(2,x, t ) = F(X - ut,X) = F(x, z),
(2.15)
John V . Wehausen
104 so that au, -=-u-
at
as,
au, -=-
azk
ax
asi,
etc.
ax,
With this understanding we shall drop the tildes. Formula (2.14) is not altered by this. It is convenient to make some special choices for 6,. Let us first suppose that the ship is moving down a rectangular canal and let V consist of the fluid bounded by the side walls, bottom, and planes ahead of and behind the body perpendicular to the walls and bottom. Of the plane areas bounding this volume we denote the one on the starboard wall by C,, on the port wall by X p ,on the bottom by X H ,ahead of the body by C A ,and behind by C B. Taking account of boundary conditions on the solid surfaces and of the simple form of the normal vectors (always directed out of the fluid) to C,, we find
R=-J
puz dx dy
+ J puZdx dy + J =P
ZS
-
LA
[pu(U-~)-~-pU'2+2pu,]
+J
[pu( U - U) - p
pu,, dx dx
ZH
- pU12
dydx
+2 p , ] dy dz.
(2.16)
We now modify this formula by taking X A far enough ahead of the body so that there is no disturbance of the fluid, a possibility that we assume as one of our boundary conditions. Then only p = pgy remains in the integrand and the upper limit of the y-integral is 0. We may imbed this integral in the Z A integral if we take account of the fact that the upper limit in the y-integral over Z B is Y (xB, x). We then have R=-J
p u, dx dy ZS
+J ZP
pu, dx dy
+J
pug dx dx
ZH
(2.17) where xBis the x-coordinate of the moving plane C B, and z p and xs are the x-coordinates of the walls X P and Cs, respectively. Finally we make one more change suggested by Landweber and Jin Wu
The Wave Resistance of Ships
105
(1963) and, in the form used below, by Sharma (1965). Define a “total head ” as follows : P g H h y , z ) =P
+pgy +
*PP[(U
+ +4
- y2 v2
(2.18)
and let pgH0 = &pU2.
I t follows from Euler’s integral that for irrotational motion of an inviscid fluid H = Ho everywhere. Hence H - H,, represents a kind of measure of the effect of viscosity. A straightforward manipulation allows one to put (2.17) into the form
R
=-
J
p ~ dx ,
dy
+J
pu,
dx dy
ZP
ZS
+21 p J
(- u2 + v2
+J
pug dx
dz
ZH
+ w2) dy dz
ZB
(2.19) I n the case of irrotational flow of an inviscid fluid, one is left with only the last two terms:
1 R = - p J ( - u2 v2 w 2 )dy dz
+ +
ZB
+ -21 pg J
eS
Y2(xB, Z ) dZ.
(2.20)
ZP
This is, of course, a genuine “ wave resistance,” but, as mentioned earlier, its value will not be the same as that of the corresponding terms in (2.19). T h e integrals over C s , C p ,and C H and the contribution of the term 2pu, in the integral over X B are generally negligible in comparison with the other terms. However, this is not the same as neglecting viscosity. If the body is moving in an infinitely deep fluid unbounded horizontally, one may choose as the control surface Xc a rectangular box as we did above for the canal, noting, however, that Xs, C p, and C, are no longer physical surfaces, so that the integrands in the integrals over these surfaces are more complicated :
Z’ H
[puv
+PUT - p(uY+ v,)] dx dx.
(2.21)
John V . Wehausen
106
If one lets the planes, Xs, X p , X H recede to infinity, these integrals converge to zero and one is left with the last three integrals of (2.19) with zp=-a, xs =00. If one sets p = 0, one obtains again (2.20). On the other hand, if one lets X B recede to infinity and sets p = 0, one obtains R = J puw dx dy - J puw dx dy - J puv dx dx.
(2.22)
LY
BP
&S
Whenever we have set p = 0 above, we have also supposed the Reynolds stress to vanish.
1. Separation of Wave and Viscous Resistance Since the last two terms of (2.19) coincide in appearance with (2.20), a true wave resistance in an inviscid fluid, one might be tempted to simply define these two terms as the “ wave resistance ” and
as the “viscous resistance.” However, if one applies (2.19) to a flow without a free surface and with no gravity, one still has the term 1 2
-p
”!
(-u 2 + v 2
+w”)dydz,
EB
even though one now has a true viscous resistance. This integral must evidently play a role in each component in any attempt to separate viscous from wave resistance. The first attempt to base a separation of viscous and wave resistance upon momentum considerations goes back to Tulin (1951). Since then the ideas have been further developed and refined by Jin Wu (1962), Landweber and Wu (1963), Sharma (1964, 1965), Tzou and Landweber (1968), Baba (1969a), Brard (1970a,b), and others. The main idea in these attempts is that there is a boundary-layer plus wake region (hereafter BLW) where the velocity field is rotational, but that outside this region it is irrotational. Then, from a well-known theorem concerning vector fields, we may decompose v as follows : v=v,+v,
(2.23)
The Wave Resistance of Ships
107
where
1 v --curl
S,,
~-+rr
curl v -dv, r
curl v = curl vR, curl vI= 0,
v I ~ v - v R ,
div vR= 0,
(2.24)
div vI= 0.
Since vI is irrotational, it can be generated from a velocity potential vI such that grad rpl = v I . T h e function v1 is by its definition defined everywhere in the fluid, and in the region exterior to BLW. However, v R is also irrotational exterior to BLW and can be generated there by a velocity potential v R .Hence, exterior to BLW = grad(vl
+
YR).
Within BLW this representation does not hold, for v, is not irrotational there. However, v1 within BLW is a harmonic extension of cpI outside BLW, and since it seems reasonable to assume that the field v is continuous with continuous first and second derivatives, the field vRand hence vI may be presumed to have the same properties. I n particular, the function y I will be continuous everywhere in the fluid. Now we should like to assume that vRcan be extended harmonically into BLW. However, we cannot expect that this extension can be made without there being a discontinuity in v R .It will be convenient to define the extended q R so that the discontinuity is on the plane x = 0. Let us now define a potential flow vP
= grad(?l
+
P)R)
= grad Y
P>
which is defined everywhere, but with a possible discontinuity within BLW on z = 0. We further define a wake velocity field by
vw = v - vp. Although we know the boundary condition satisfied by v on the ship's wetted surface, namely v = ( U , 0,O), we cannot easily deduce the boundary behavior there of v, or vw . In particular, we cannot expect that vp.n= U n l , the boundary condition that would ordinarily be imposed for an irrotational flow in an inviscid fluid, for, to speak loosely, the function ' p p exterior to BLW does not know that there is a solid boundary at S. Presumably there does exist a stream surface S' starting at the ship's bow, perhaps not closed, upon which this condition will be satisfied. We now substitute this decomposition into (2.19), discarding the terms in p as being negligible, but retaining the Reynolds stress. T h e possible
John V. Wehausen
108
discontinuity of y p on x = 0 requires an easy modification of (2.19), for the control surface C, must now include the two sides of the surface of discontinuity in order to exclude any discontinuity within the fluid volume V . After some small manipulations one obtains the following formula:
1
I-zpS
[vw2+ww2 t 2 v ~ v w +2wPww] dy dx
ZW
1 + j p g / [ Y w z +2YPYw] dz, where C, is that part of C, within BLW. We now define a "wave resistance " by the first three integrals in (2.25):
1 R w = 2 p Jz,[-upz
+ up2 + wpz]dy dz (2.26)
Corresponding to this we define a
"
viscous resistance by "
R,=R-Rw
(2.27)
The definition of R, is sometimes simplified by discarding the last two integrals of (2.25), although it has not been really established that they are negligible. T h e definitions are precise. Whether they are useful depends to some extent upon how well they lend themselves to independent measurement. Measurement of R, will not be discussed, but various practical approximation procedures are discussed by Sharma (1964, 1965). Measurement of Rw will be discussed in the next section, but the underlying assumption
The Wave Resistance of Ships
109
of the measurement methods excludes the integral over the surface of discontinuity. An empirical proof of the usefulness of the definitions would consist in showing that the sum of the independently measured resistances adds up to the measured total resistance. We note that it is not true that, as defined above, Rw is a function of Fn alone and R , of Rn alone. Consequently we do not anticipate that measurement of these two components will necessarily be helpful in predicting full-scale resistance from model tests. We emphasize that Rw as defined here is not the same as the wave resistance that would be obtained if one were to find the velocity potential that satisfies rp, = Un, on S. As mentioned above, there presumably exists a surface S‘, containing S in its interior and coinciding with S at the bow, upon which rpPn = Un,. If S’ is not closed, but has a ‘ I wake ” trailing off to infinity, one must exclude from the integrals over C, that part occupied by the tail of the body. Th e surface S‘ is, however, irrelevant to the proposed definition of R , for S. On the other hand, it is useful to be aware of S’, for it suggests methods of making numerical experiments for estimating the effect of the wake upon wave resistance. 2. Wave-Breaking Resistance Recently Baba (196913) has pointed out that the rate of energy loss in breaking of ship-generated waves can be treated as a separate component of the total resistance. Th e occurrence of wave breaking is clearly a phenomenon in which gravity, i.e., the Froude number, plays the dominant role. On the other hand, energy lost in wave breaking will not be recorded in the “ wave-resistance ” measurements as defined above, but rather in the measurement of the “ viscous resistance.” This has been confirmed by Baba’s experiments. C. WAVE-PATTERN ANALYSIS Formulas like (2.17) and (2.19) can obviously serve as the basis for an experimental method of determining the resistance of a steadily moving body. However, the measurements would clearly be tedious, even after neglecting the terms in p, for measurements of H , u, v , w are necessary over the whole plane Z B . If one assumes irrotational flow, however, and hence (2.20), Eggers (1962, 1963) has shown that one may calculate R from surface-profile measurements alone. There are two versions of the method. I n one, transverse profiles are measured and R is derived from (2.20), in the other, longitudinal profiles are measured and R is calculated from (2.22). Both methods rely upon linearization of the boundary conditions at the free surface and hence upon measuring profiles far enough
John V. Wehausen
110
behind or to the side of the ship (or other moving body) that no serious error is introduced by this approximation. On the other hand, it puts no restriction upon the form of the ship or the cause of the disturbance, as will be done in the next section when wave resistance is calculated analytically. T he effect of the neglect of viscosity will be discussed later.
1. Transverse Profiles in a Canal We begin with the case of transverse profiles. Suppose, as we did in deriving (2.17),that the ship is moving steadily in a canal. Let its bottom be in the plane y =-h and its side walls in the planes z = &b. The undisturbed free surface is contained in y=O. As before, the coordinate system is moving with the velocity U of the ship. It will be convenient to suppose that the ( y , z)-plane intersects the ship amidships. Let x = x B be a plane sufficiently far behind the ship so that for x < x B it will be an acceptable approximation to replace the exact free-surface boundary conditions for irrotational flow, namely,
by the linearized conditions
+g Y(x,). =o, + UY,(x, z ) 0,
- Uyz(x, 0, 2)
yy(x, 0, ).
(2.29)
=
or, after eliminating Y , by yzz(x,0, z)
+ Kyy
= 0,
K
=g / U 2 .
(2.30)
In addition to Laplace’s equation, y must also satisfy the boundary conditions
(2.31) A complete set of solutions of Laplace’s equation that satisfy these three boundary conditions is given by the functions
I
cos k,x sin k,x cash pn(Y
nr + h)cos 2b ( z - b),
p,
> 0,
k n > 0,
(2.32)
111
The Wave Resistance of Ships where kn2= p n 2- (nT/2b)2, =pzp
(2.33)
+ (nrr/26)2.
Th e p,, must satisfy (2.34) and the p n pmust satisfy (2.35)
If n 2 1, there is a single solution for p, for each value of U2/gh. If n =0, there is one positive solution for po if U2/gh < 1, but none if U2/gh > 1. There are infinitely many solutions for p n p ,which we shall number with p = 0, 1, 2, . . . in order of increasing value. If n = 0, there is no solution pooif U2/gh < 1, but pooexists if U2/gh > 1. There is no steady-state solution if U2/gh = 1. The functions {cos(n~r/26)(z- 6)) form an orthogonal set of functions. However, the families (cosh p n ( y h), cos p n p ( y h), p = 0, 1, 2, . . .} do not, contrary to the behavior of such families in some similar situations. Since these form a complete set of functions for 'p in the region x 5 x B , we may express 'p as follows:
+
x (-a, cos k,x g
+& uk,,x
+
nrr + 6 , sin k,x)cosh p,(y + h)cos (z 2b
1 h
+
-
b)
nrr 2b
cnpexp k,, x cos p n p ( y h)cos - ( z - b).
(2.36) T he associated free surface is than given according to the linearized boundary condition by
Y(x,z ) = C (a, sin k , x n
+ b, cos k,x)cos nrr -( z - b ) 26
nnexp k,, x cos - ( z +b). (2.37) P 2b Th e first summation starts with n = 0 or n = 1 according as U2/gh < 1 or U2/gh> 1, respectively. I n the second summation the term corresponding to n = p = 0 is absent in the first case, but present in the second. f
1 c,, n.
112
John V . Wehausen
The coefficients a,, b, , cnPhave not as yet been determined, but they are uniquely defined by q in the region x 2 x B. Before substituting the expression for q into (2.20), the latter may be simplified a little by taking advantage of linearization in the double integral and integrating up to the surface y = 0 instead of y = Y(x, x). After a tedious but straightforward calculation one obtains the following expression for R. m 1 R = -pgb C e,-l(an2 2 n=Oorl
+ bn2)
where e0 = 4, E , = 1 for n 2 1. Note that the coefficients cnP do not enter into the determination of R, as indeed they should not inasmuch as (2.20) must be valid for any plane behind the ship and the contribution from the second summation in the expression for q must vanish as x B-+ -a.The first summation in q will be said to be the “ free-wave potential ” and the second to represent the “ local disturbance.” There now remains the problem of determining the coefficients a,,, b, needed in R. Multiply the expression for Y ( x , x ) by b-’ cos(mrr/2b)(x- b) and integrate from -b to b. This yields the equation
+ b, cos k,x + C cmPexp k,, x
a , sin k , x
P
=-Ib
mrr
Em
-b
Y(x,X)COS -(Z- b) d X 2b
= Y,(x).
(2.39)
Since x < 0, it is evident that the exponential terms will become negligible if I X I is large enough. Let us suppose that profiles have been measured at N locations x l , . . . , x N where this is the case. We then obtain N equations to determine the two unknowns a , , b, :
a , sin k m x i
+ b, cos k m x i= Y,(x,),
i = 1, . . . , N .
(2.40)
Any two of them will suffice provided sin k,(x, - x,) # 0. However, if N > 2, it is appropriate to use the method of least squares to obtain a more reliable estimate of a , and b, . T h e necessary formulas are as follows:
a, = - A - l
C Y,(x,)cos k,xi
sin k,(xi - xi),
i. j
b, = A - l
C Y,(x,)sin
k m x isin k,(xi - xi),
i. i
1 A = - C sin2 km(xi- x,). 2 i.j
(2.41)
The Wave Resistance of Ships
113
I n planning an experiment it is obviously advantageous to maximize A in order not to amplify measurement errors in Y ( x i , z). For N = 2, A = 4 sin2 k,(xl - x z ) is obviously a maximum for k , 1 x1 - x21 = 7r/2. However, since k , varies with m, an optimum spacing for one value of k , may be poor for another one. Since one usually plans to determine all the necessary values of a , and b, from one set of measurements, Y(xi,x), i= 1, . . . , N , this is a disadvantage of taking N = 2. This difficulty is ameliorated by taking a greater number of cuts and using least squares, for the function
A,(k,S)
1 N-1
1
C sin2 k,(xi - x j ) = C 2
=-
i,i
q=
1
( N - q)sin2 qk,6,
(2.42)
where 6 = xi + 1 - x i has an increasingly wider flat part about k,6 = 7712 as N increases. Figure 4 shows 2 W 2 A , ( k , 6 ) for N = 2, 4, 6 , 8, and 10.
FIG.4.
Graph of function showing effect of spacing and number of cuts upon A.
It is possible to avoid this difficulty with A if one can measure the slope Y , ( x i ,z) at the same time as Y ( x i ,z). T h e double measurement on a single profile then suffices in principle to determine the coefficients, for the equations become a, sin k,x
+ b, cos k,x = Y,(x),
a,,,k , cos k , x - b, k , sin k, x = Y,'(x),
(2.43)
114
John V . Wehausen
with solution
a,=-
1
[Y,(x)k, sin k,x
+ YA(x)cos k,x],
km
(2.44)
1
b,
=
knl
[Y,(x)k, cos k,x
-
YA(x)sin k,x].
If double measurements are made on N cuts x,, . . . , x N , a least-squares fit gives simply the average of the values obtained from each single profile. There remains the question of how far behind the ship the profiles must be measured before we can neglect the exponential terms. Since k,, < k,, and k,, < k, ,, it suffices to examine k,, , KO,, and k,, . It is not difficult to establish that
,-,
k,, > rr[@/h)2
+ (n/2b)2]1’z.
(2.45)
Hence k,, > r / h and k,, >.rr/2b. We recall that k,, does not exist if the speed is subcritical, U2/gh< 1. We assume this for the moment. If XI > # max(h, Zb), then both exp k,,x and exp k,,x are less than 0.009. This is an unsatisfactory result, for it does not take account of the size of the c,, and is overconservative for large h or b. Th e most satisfactory investigations have been numerical experiments by Kobus (1967) and Landweber and Tzou (1968), both carried out for channels of infinite depth but finite width. By introducing a mathematical disturbance approximating that caused by a ship, they are able to estimate the effect of the local disturbance upon Eggers’ method. Th e results of Landweber and Tzou are too elaborate to summarize briefly. However, the conclusion of both papers for shiplike forms is that profiles taken more than a ship length behind the ship will result in errors of less than 1 o/o for the range of practical Froude numbers. This problem has also been discussed in Eggers (1966, p. 667), where a computed curve displays for various distances behind the ship the ratio of the resistance computed from transverse-cut analysis neglecting the effect of the local disturbance to the true resistance. The Froude number was 0.36. Th e results conform to those of Landweber and Tzou. If U2/gh> 1, KO, appears. This may be very small if the flow is only slightly supercritical and is bounded above by .rr/2h. It seems likely that this exponential term may have to be taken into account if measurements are made at the usual distances behind a ship model.
I
2 . Transverse Profiles in Unbounded Fluid Several situations can be analyzed in a manner similar to that above: motion in an infinitely deep canal, motion in horizontally unbounded fluid of finite depth, and in horizontally unbounded fluid of infinite depth.
The Wave Resistance of Ships
115
We choose the last case, for it is most different from that above. However, we shall treat it in approximately the same way. Consider a region x < X, sufficiently far behind the ship that linearized free-surface boundary conditions can be used. Let a(k)= (KP)1/2,
P(k) = &[K+ ( K 2 + 4R2)112],
+ k2.
P2 = a'
(2.46)
One may show that the most general potential function satisfying the linearized free-surface condition and vanishing as y + - 03 is given by g w 1 p,(x, y , z ) = - dk -exp ~(k)y{[-F,(k)cos ax
U o
a(k)
+ [-F2(k)cos
ax
+ Gl(k)sin axlcos kz
+ Gz(k)sin axlsin kz}
+
x { [ K pcos py - ( k 2 p2)sin pyI x [F(k,p)cos kz G(R,p)sin kz]),
+
x < x, ,
(2.47)
where the F i , G i , F(R, p), G(R, p ) are general, but must satisfy certain conditions to ensure proper behavior of the integrals. T h e free surface is then given by ~ ( xz,) = Jo
m
~A{[F, sin ax + G, cos altos k z
+[F,sin ax + G, cos axlsin kz} + JmdpImdkexp(k2 +
p2)1/2x[~(~, p)cos
0
~z
+~
( kp)sin , kz].
0
(2.48) Formula (2.20) for the resistance takes the following form :
1 0 m R =p dyJ dz(-qx2 2 .--m -m
1
+ qy2+ y a 2 )+ 21 pg J m Y 2dz.
(2.49)
-m
Substitution of q into this expression for any fixed x 5 x B is simplified by the fact that the second double integral can be discarded and by use of the following Fourier-integral indentities. If
f n ( x )= Sm[/3,(k)cos kx 0
+ y,(k)sin kx] dk,
n = 1, 2,
(2.50)
John V . Wehausen
116 then
(2.51) m
Jm -m
n, m = l , 2.
fnfrndx=nJo (pnBrn+YnYrn)dk,
After simplification one eventually finds
(2.52) There now remains the problem of determining the Fi and G i , the analogs of the an and b, in the canal. We take Fourier transforms of Y(x,z):
1 *
57
j
Y(x,x)cos kz dz = Y,(x, k)
-m
+
1
7,
m
+ 1 d p ~ ( kp)exp(k2 , + W
= F,(k)sin a ( ~ ) x ~ , ( ~ ) c a(k)x os
p2)1/2x
0
(2.53)
I-,
Y(x,z)sin kz dz E Y2(x,k) = F,(k)sin
a(k)x
+ G, ( ~ ) c oa(k)x s +j
W
+G(K, p)exp(k2
+
p2)1/2x.
If profile measurements Y(xi, z) are made far enough aft that one can neglect the integrals in p, one obtains essentially the same formulas as were found for the a, and b,: N
F,(k) = - A - l
Y , ( x , , k)cos a(k)x, sin a(k)(xi- x i ) , i. i = 1 N
G,(k) = A - l
1 Y,(xj,k)sin a(k)x, sin a(k)(xi
- xj),
(2.54)
i.1=1
1
A = - 1 sin2 a(k)(xi- x j ) , 2 i. i
p = 1, 2.
The remarks about choosing 1 xi -xi 1 made for the canal carry over here without change. If one has simultaneously measured slope Yz(x,,z ) and amplitude Y ( x i ,z), then as before one may derive estimates of F , and G, for each x i and average for greater accuracy:
G,(k)= 2[ Y,(x,k)a(k)cos ax - YDz sin a x ] .
The Wave Resistance of Ships
117
In practice one will, of course, compute the F, G for a discrete set of k's sufficiently close to one another to allow calculation of R with the required accuracy. Under what conditions may the local disturbance be neglected? T h e only numerical experiments that are relevant are those of Kobus (1967) and Landweber and Tzou (1968) discussed earlier, and these were carried out for infinitely deep canals. It seems reasonable to suppose that the same results hold here. In fact, as pointed out by Sharma (1966), a numerical quadrature of (2.52) in which one takes Ak = r/2b and k, = nr/2b yields exactly (2.38) with appropriate correspondences between the a,, b, and the functions F, , G, , F,,G,. Consequently one may expect the two formulas to be valid under the same circumstances.
3. Longitudinal Profiles in Unbounded Fluid From the viewpoint of convenience the measurement of longitudinal profiles has important advantages over transverse profiles, the most important being that a wave gauge can be fixed in position while a ship or model passes by it and that measurements do not have to be made in the ship's own wake. We suppose that there exist values x, > 0 and z p < 0 such that when x 2 z , or z 5 z p it is allowable to use the linearized free-surface boundary condition. If zs' 2 z, and zp' 5 z p , then from (2.22)
-
R(S)+ R(P).
(2.56)
The conceivable contribution from a closing portion of C at, say, x = x B < 0 vanishes as x B-f - co in an unbounded fluid because the disturbance decreases as I x I -1/2, as we shall show later. We consider first only the part R"). In the region z 2 z,, y 5 0 any potential q P ) ( x , y , z) satisfying the linearized free-surface condition and vanishing as y-f - 00 can be represented as follows: (~(s)(x,y,
x) =
k2 f d k exp[--y(k)z]exp - y[-f(k)cos K
UO
kx +g(k)sin kx]
+ 5 Jwdkexp Kk2 - y{[-f,(k)cos kx +g,(k)sin kxlcos y(k)z + [-f,(K)cos kx +g,(k)sin kxlsin y(k)z} k2 +5 JwdpSadk exp[-(k2 +p2)1~2x][cospy + sin py] KP K
0
0
John V . Wehausen
118 where
k K
y(k) = - ( k 2 - KZ)l’Z,
k 2K ,
k y ( k ) = - ( K 2- kZ)l”,
O zs was large enough so that the double integral could be neglected. Sharma (1966) has made a rather thorough study of this question as well as of the effect of truncation by means of numerical experiments [the results are also summarized in Eggers, Sharma, and Ward (1967)l. Briefly, he produces a disturbance by means of a known distribution of sources and sinks satisfying the linearized free-surface condition and computes the values of Y(x, x) and of F,(k),G,(K),and R. He then uses the computed values of Y in the various formulas above in order to test the effect of varying z , of using finite lengths of Y(x, z), and of applying the truncation correction mentioned above. I n the case considered the optimum value of z was about SUz/g. It may seem strange that there is an optimum value of 2. Suppose that 20,
I
I I
EXACT THEORETICAL VALUE
9
FROMY -CUT WITH
k
@ TRUNCATlON CORRECTION FROM Yz-CUT WITHOUT
v)
905 4
T
T 2
2
F
d
'
@ TRUNCATION CORRECTION FROM Y -CUT WITHOUT @TRUNCATION CORRECTION
The Wave Resistance of Ships
123
measurements terminate at x, and that L(z) is the useful length of profile at z. Then because of the nature of ship wave patterns, which will be discussed later,
xt?- L(z)N tan 70"32' =2.8. z
Hence increasing z shortens L(z), while decreasing x increases the unwanted effect of the local disturbance. Figure 5 from Sharma (1966, p. 772) shows the effect of truncation upon R , the importance of a truncation correction and the superiority of measurements of Y , instead of Y if no correction is made. Figure 6
THEORETICAL VALUE K2G ( k k O
-"tiL -20
I
I
1 3 TRANSVERSE WAVE NUMBER k / K +
1
I 4
FIG.6. Comparison of free-wave spectra derived directly from knowledge of disturbance and from theoretically computed Y. [From Eggers, Sharma, and Ward, 1967 (Fig. 6, p. 128), by permission of the Society of Naval Architects and Marine Engineers.]
John V . Wehausen
124
from Eggers, Sharma, and Ward (1967) shows the results of another numerical experiment where F(k) and G(k) are computed directly and also from the theoretical values of Y . This is a more discriminating test than a computation of R, and the agreement is impressive. For further discussion of such numerical experiments one should consult the cited papers.
4. Longitudinal Profiles in a Canal It is not possible in a canal to carry through a development analogous to that for an unbounded fluid. First of all, since all the wave energy is channeled down the canal, no pair of longitudinal cuts on each side of the ship can be long enough so that the contribution from a closing cut at x = x B < 0 can be neglected in computing R. Also, since all the energy passing through a longitudinal cut is (ideally) reflected from the walls, there is no net flux. However, as shown by Eggers (1962), it is in principle possible to evaluate the coefficients a, and b, from longitudinal profile data by means of the following formulas: mrr +2 sec (zs2b
a,
=
b,
= 2 sec - (z, - b) lim
mrr
2b
T-m
Y(x,y)sin k,x dx, 1
-
(2.72) 50
Y(x,z)cos k, x dx,
TJx0-T
where z = z, is the plane in which the profile is measured. In practice this procedure has not been successful, partly because tank walls are not perfectly reflecting, partly because it is not possible to get a long enough profile to approximate the necessary limits. Recently, however, Moran and Landweber (1971, 1972), using a modified procedure, have made both numerical and physical experiments with such profiles. T h e former show the theoretical feasibility of the procedure. Results of the latter are consistent with data obtained independently by other means. Perfect reflection is assumed. There are other ways of making use of longitudinal-profile measurements in a towing tank. In most tanks model dimensions are usually limited to a size that avoids the necessity of making wall corrections. Consequently, if a measured longitudinal profile is truncated before the first reflected waves from the tank walls affect it, that section of the profile should approximate well the profile that would be measured in an unbounded fluid. Th e truncated tail must, however, be supplied by a theoretical extrapolation, as proposed by Newman (1963) and Sharma (1963) and described in the last section. One should note that the difficulty of making a reliable
The Wave Resistance of Ships
125
extrapolation increases as the Froude number, and hence the wavelength ( s 2 r U 2 / g )increases, for one has fewer and fewer wavelengths upon which to base an extrapolation. Figure 7 shows two typical records up to the first reflection. However, the procedure seems to work satisfactorily in practice, as will be seen when we turn to experimental results.
FIG.7. Two typical records of longitudinal cuts, including the first reflected wave from the side wall.
5. Other Methods of Direct Measurement When the transverse-cut method was used to determine a,,, and b, for the waves in a canal, this led to a separate calculation for each value of m, preceded by taking a Fourier transform. These calculations required a knowledge of Y ( x i ,zj)= Y i jfor i = 1, . . . , P ; j = 1, . . ., Q. I t seems reasonable to ask why one should not bypass the Fourier transform and determine all the a,, b,, n = 0, . . . , N , in one calculation. If we may neglect the local disturbance, each measured value Y i , gives rise to an equation satisfied by the a, and b, : nr
C (a, sin k n x i + b, cos k,xi)cos (z,2b N
h=O
+
b) = Yij,
i = 1, . . ., P,
j = 1, ..., Q. (2.73)
If PQ 2 2 N 2, there are in principle enough measurements to determine the desired a , , b,. If PQ > 2 N 2, one may exploit the method of least squares to find a “best” solution. This procedure was first suggested by Hogben (1964) and further developed in Gadd and Hogben (1965) and Hogben (1970). Obviously there must be imposed further requirements on the grid of measurement points for the method to be applicable (for example, P > 2 N + 2 , Q = 1 will not work). These problems are discussed in the referenced reports. Another ingenious method due to Ward (1963, 1964) called by him the “XY-method,” is based upon measurement of the force acting on
+
126
John V. Wehausen
a long vertical cylinder situated in a position that would also be suitable for a wave probe measuring longitudinal wave profiles. The fundamental assumption underlying the method is that the force exerted on the cylinder by a plane oncoming wave is proportional to the wave amplitude, a prediction of the linearized theory of water-wave diffraction in an inviscid fluid. Ward’s experimental studies of this assumption are reported in Ward and Snyder (1968). It appears to be adequately substantiated for the intended use. The method relies upon combination of the formulas (2.20) and (2.22), the force components on the cylinder providing the data for the integral along the longitudinal cut and a separate wave gauge providing the data for an approximation to the closing transverse cut. For details one should consult the cited references. A recent modification of Ward’s idea has been given by Roy and Millard (1971).
6 . Eflect of the Wake The underlying theories for the determination of wave resistance from profile measurements all require irrotational flow. When the methods are applied practically, what is measured? If there is indeed a boundary-layerplus-wake region BLW surrounded by a region in which the flow differs only insignificantly from an irrotational one, then a longitudinal profile taken far enough to the side so that the wake region is not intersected should determine the velocity potential y p used in (2.26) to define R , in a viscous fluid. On the other hand, a determination of y p from a set of transverse profiles that intersect the wake, which may not be avoidable, may be expected to show an effect of the fact that part of the data used to determine y p is being taken in a region where the flow is not irrotational. This may be an explanation of some evident discrepancies between quantities deduced from the two sorts of measurements. The most important attempt to study the effect of the wake by means of numerical experiments is in a paper by Tatinclaux (1970). He takes as body an infinitely long vertical strut of ogival section in unbounded inviscid fluid. A wake is artificially produced by giving the form of the vorticity o in the fluid behind the midsection of the ogive. This is equivalent to assigning H,-H inasmuch as the Navier-Stokes equations for steady motion in the coordinate system Oxyx can be written in the form
pgV(Ho - H ) = W x
(V-
V) - ~ A fpv’ v
VV’
(2.74)
Let L be the length of the ogive and 2bo its beam. Then Tatinclaux makes the following special choice for o :
U
(2.75)
The Wave Resistance of Sh;Ps
127
for x aft of the ogive. For the region between the midsection and the after end x/L is deleted. For the computations h = 0.1, c = 0.25. A solution of the equations of motion for an inviscid fluid with linearized free-surface conditions is then constructed. The velocity is the gradient of a harmonic function outside the wake to which must be added a term V x A inside the wake. (Outside the wake V x A contributes VpR to the velocity.) T h e form of the surface Y(x,z) can then be calculated. For Fn = 0.25 and for sections X B at one, two, and four model lengths behind the model, Tatinclaux calculates the exact wave profile, an I ‘ asymptotic ” profile that neglects the local effect of thevorticity, and the profile that would exist if there were no wake at all. Differences are substantial even after four model lengths. For the resistance it makes no difference whether the exact or the O, for x (0.
(3.10)
Various modifications of this problem are possible, depending upon the physical situation. I n particular, in a towing tank a model can be restrained from trimming and squatting so that u = e = 0. Such modifications are not difficult to make, and this one actually simplifies the problems. Modification of the boundary conditions in order to accommodate a submerged vessel is also not difficult. Since U has been fixed in the problem formulated above, the unknown quantities are y , Y , T , a, and e . It is obvious from the boundary conditions that they are hopelessly intertwined with one another and that some approximations will be necessary. Before proceeding to the discussion of approximation, we shall formulate one more “ exact ” problem, that of a moving pressure distribution. Here the pressure above the water surface is given, but the surface profile itself is unknown. T h e kinematic condition (2.2813) remains the same, but the dynamic condition (2.28a) must be replaced by
where P(x, X) is a given function, which may be chosen to vanish outside a certain region in order to model a particular physical situation like a moving hovercraft. Although much of the complexity of the ship problem has disappeared, this is still a nonlinear problem. As in (3.5), one can com-
The Wave Resistance of Ships
137
pute the x-component of the force exerted by the pressure distribution upon the water:
s
F, = /P(x, z)n, ds = P(x, z) Y,(x, x ) dx dz.
(3.12)
C. PERTURBATION EXPANSIONS In order to obtain any solutions of the exact problems formulated above, or of others that one may formulate, some method of approximation is necessary. T he standard procedure has been by way of either regular or singular perturbation expansions. In either case one introduces a dimensionless parameter E > 0 connected with the problem in such a way that as E -+ 0 the disturbance near the free surface becomes smaller and smaller, except possibly at certain isolated points or lines. Each value of E labels one of a family of flows. Many perturbation approximations involving moving bodies have a different aim from this one. For example, in thin-wing theory in an unbounded fluid one wishes primarily to avoid the tedious computations associated with conformal mapping of the given wing section onto a circle. However, it still seems fair to say that in problems involving motion near a free surface the nonlinear boundary conditions at the free surface seem to present the greatest obstacle. Hence one chooses a perturbation parameter associated with vanishing of the surface disturbance. It will still be true, however, that some of the classical perturbation approximations also fulfill this requirement. There seem to be two main ideas behind free-surface perturbation schemes for moving bodies. One is to have the body sufficiently deeply submerged that the free-surface disturbance resulting from its motion is not very great, and, of course, must become smaller the deeper the submergence. I n the other, one introduces a family of bodies whose members can be made to approximate more and more closely to a body whose motion will not disturb the fluid. Th e most familiar example of the latter in fluid mechanics is thin-wing theory. For a moving pressure distribution one simply assumes that the imposed pressure P is " small." For moving submerged bodies the two approaches can be schematized as follows. Let L be the length in the direction of motion, d a typical length perpendicular to this direction (say a vertical or horizontal dimension, or perhaps the square root of the maximum cross-section area), and ( a , b, c ) a fixed point in the body. There are four relevant lengths: U2/g,b, d, L. In a typical deep-submergence approximation one keeps U21gLand d / L fixed and takes E = d/ I b I . In a typical thin- or slender-body approximation one keeps U2/gLand b/L fixed and chooses ~ = d l L I. n the former case,
John V. Wehausen
138
letting 1 b ( -+co entails E+ 0 and ever deeper submergence of the given body without any distortion of the body. In the latter case, letting 8 --f 0 requires changing the shape of the body so that it approaches a flat disk or a spindle, depending upon how one has chosen d. The mathematical problem is to formalize these ideas into a systematic approximation procedure. An extensive and thorough study of this problem has just been made by Ogilvie (1970), especially with regard to singular perturbation problems. Ogilvie’s paper is also complemented by one by Newman (1970) expounding recent results in the slender-body approximation. Consequently, we shall restrict ourselves here to a token presentation of the thin-ship approximation and to a statement of results for some other approximations.
1. Thin-Ship Approximation In the case of a surface ship there is, of course, no possibility of a deepsubmergence type of approximation. Consequently one is forced to fall back upon one involving the geometry of the ship. There are several ways of approaching the problem and perhaps the greatest recommendation of the one chosen here is that the resulting problem can be solved analytically. We begin by imbedding the hull form (3.2) in a family of hulls derivable from a standard hull f ( l ) ( x ‘ ,y’), i.e., we shall consider hulls of the form
z = &&f(l)(X’,y’),
&
> 0.
(3.13)
Evidently, as E --f 0 the hulls approach a flat disk, the centerplane section of the ship. Corresponding to each value of E there will be a velocity potential ~ ( xy,, z ; E ) , a free surface Y ( x , x ; E ) , and a trim a(&),sinkage e(&), and thrust T(E).Although it is possible, as shown by Ogilvie (1970), to treat the approximation by the method of inner and outer expansions, the results are the same in the first order as one obtains by assuming a regular perturbation expansion, and in the second order seem a little dubious. The problem of approximation has also been treated by Wehausen (1963) by use of Green’s theorem and Green functions and again (1969) by use of Lagrangian coordinates. Both are lengthy to expound, and we shall simply assume here, following Peters and Stoker (1957), that q(x, y , z ; &) = & I p ( X , y , z )
Y ( x , z ; &)
= &Y‘l’(X, z)
+
+
&2,(2)
+ . . ., + . . .,
&(2)Y(2)
+ + . .., e(&)= &e(l)+ 2 e ( 2 ) + . . . , T (E= ) E ~ T+ ‘ ~c ’3 F 2 )+ . . . . a(&)= &&)
&%(2)
(3.14)
The Wave Resistance of Ships
139
Starting the last expansion with 2 anticipates what will be forced upon us, and is not a real restriction. T h e underlying assumption that ~ c p ( l )and & Y ( lare ) small cannot be uniformly valid for a ship that does not have an appropriately shaped bow and stern, for there will be stagnation points on the stem and sternpost, where the velocity will be U , which is not small no matter how small E > 0 is. T h e expansions (3.14) are now substituted into the equations (2.28), (3.3), (3.7), (3.8), or (3.9), and (3.10), expanded in powers of E , further expanded in Taylor series where necessary, and all terms assembled according to powers of E . We omit the rather tedious details, but give the first-order results. The velocity potential ~ ( l must ’ satisfy the following equations :
Acp‘l) = 0, l?g)(x, 0, x)
+
Y 1
and r2
= [(x
The expression for sistance R = - T:
-
o2+ (y + 2h + d2+ (2
-
5) 11/2* 2
v may now be substituted into (3.17) to find the re-
11". d . JJ@d q f l ( x ,y)fe(t,v)G,(x, y , 0 ; t,
PU2 R = -7T
so
7, 0).
(3.52)
so
Only the last integral in G above contributes because of symmetry considerations. One finds again (3.39) with the function M ( x , y ) given in a slightly different form, obtained by letting h = sec 8. Substitution of y into (3.16) and (3.18) gives expressions for Y and a pair of linear equations for finding cc and e. This is a more convenient expression for analyzing Y than the one obtained from (3.32). I n particular, one may first determine the surface generated by a moving source and then by superposition find the one associated with the ship.
151
The Wave Resistance of Ships
Not all problems can be solved with more or less equal ease by Fourier’s method and by the method of Green functions. Fourier’s method will generally have required a coordinate system in which variables can be separated and then boundary conditions imposed on one of the coordinate surfaces. I n order to illustrate the greater power of the method of Green functions, let us consider the problem of the motion of a “deeply submerged” body. We recall that the body boundary condition is then (3.20). In order to find y , we begin as we did for the thin-ship problem, using Green’s formula. However, the first integral does not simplify, but remains
(3.53) T he other integrals all vanish as they did there. If we now use the boundary condition (3.20), we may rewrite Green’s formula as follows:
+ S v(Q)Gy(P;Q)dS
kv(P)
=
u JSn,(Q)G(P;Q)dS.
(3.54)
If in the left-hand integral we let P+Po on S, then by a well known theorem the integral converges to - 2T(Po)
+J dQ)G”(PO; Q) dS.
We then have the following integral equation for y(P),where P is a point of s:
If this equation can be solved for y on S by numerical methods, then y is determined everywhere by (3.54). (This is the same integral equation that is to be solved in the inconsistent surface-ship problem mentioned above, except that one must also take account of the line integral.) Instead of starting with Green’s formula (3.48), one may also assume that y can be represented by a source distribution:
dP)= J U(Q)G(P> Q) dS.
(3.56)
This yields immediately an integral equation for u:
2no(P) + a(Q)G,(P, Q) d S = Un,(P), S
This is essentially the same as (3.55).
PeS.
(3.57)
152
John V. Wehausen
Another common procedure is to try to expand y ( P ) in a series of singularities of all orders, all located at some fixed point within S. The boundary condition on S then gives an equation to be used in determining the coefficients of the series. Not all bodies can be treated by this method.
E. FURTHER RESULTS, VARIATIONS, AND EXTENSIONS I n the preceding section several problems of steady motion with a free surface were formulated and solved, but the solutions were not analyzed in any way, nor were any numerical evaluations shown. In this section we shall try to fill this gap. In addition, there are many problems that can be solved that are variations of the ones already solved. Although the results of these variations are important for the subject, the methods of solution are usually not different in nature from those already illustrated, although working out the details may require a very elaborate analysis. Consequently, we shall chiefly restrict ourselves here to giving a census of known results, a guide to recent literature and some computed results that seem to illustrate important behavior. Since many of the classical results of wave-resistance theory are already expounded in Lunde (1951a), Kostyukov (1959, 1968), and Wehausen and Laitone (1960), there seems to be no need to reproduce formulas that can easily be found in these places. Consequently, for results already available in these sources only references will be given. In a later section we shall discuss comparison between theory and measurement. Many computational results will be postponed to that section. T he ones reproduced here will be computations with no associated measurements. Although the emphasis in Section I1 and also in this one has been on three-dimensional motion and ships, most of the considerations can be carried over to two-dimensional motion. In fact, the possibility of using analytic functions of a complex variable allows an elegance of treatment not available in the three-dimensional problems. We shall not, however, discuss such problems except where the method or result throws light upon threedimensional problems. A fairly complete summary of results up to about 1959 can be found in Wehausen and Laitone (1960, Sections 20 and 218). Asymptotic analysis of wave patterns behind moving singularities or pressure points is only marginally relevant to the purposes of this article, principally in connection with light thrown upon the determination of wave resistance from wave-pattern measurements. The most relevant result for this purpose is already given in (3.25). However, at the end of this section various recent papers on this subject will be listed.
153
The Wave Resistance of Ships 1. Kochin’s H Function
Consider a deeply submerged (not necessarily symmetric) body with boundary S. We have showed in Section II1,D in (3.53) that
d P >=
1
f [YV(Q)G(P,Q) - vGVI dS(Q)
The form of H will depend upon the boundary conditions imposed upon G. In (3.51) we have given an example for h = a.For this case Kochin’s i@ function is defined as follows: S ( k , 8) =
1
{rp,, - kv[ny
S
+ i(n, cos 8 + n, sin B)]}exp kwdS,
(3.59)
where w =y
s i x cos B + i z sin 8.
(3.60)
Kochin (1936) derived the following representation in terms of % for the last integral in (3.58):
-n
iK
n12
K
n
4772
-n
- Re -[ S ( K sec2 8, O)exp(Kw sec2 B)sec2 B d8 2~ -n12
1
1
- Re -J’ d8 sec2 8
S ( K sec2 8( 1 - A), 0)
-m
x exp[Kw sec2 O ( 1 - A)]
ax x
(3.61)
-
where the integral with respect to A is to be interpreted as a Cauchy principal value. Other forms for the X function are possible. Kochin shows that
J
+
X ( k ,8) = {y,, i cos O[q,n2- y 3n,]
+ i sin 8 [ ~n,y-
‘pun.]}exp
kw dS.
(3.62) Furthermore, if cp is representable in the form
?(P)=
1
1
u(Q)G(P,$2)dS’
(3.63)
John V . Wehausen
154 then
1
%(A, 8) = - a(P)exp kw dS.
(3.64)
S
T h e real usefulness of the A? function is in the information that it contains concerning the force acting on the body. The following formulas are from Kochin (1936):
F,
2
-R
= --
&(K sec2 8, 8) I sec3 8 d8,
(3.65)
+-!pK2 47.f F 2 --
d8sec48
A?(Ksec28(1-h,8)12--
1-A h dh,
-n
p ( K sec2 8, 8) I sin 8 sec4 8 d8.
It is evident that the & function is closely related to the pair of functions P , Q in (3.37). Indeed, Havelock (1932~)introduced functions playing a very similar role. There is also a close relationship between the % function and the free surface Y(x,z). This is more or less evident from (3.61) and is explicitly stated and exploited in Eggers, Sharma, and Ward (1967). T h e definition and exploitation of the % function for other boundary conditions have been carried through by others, notably by Haskind (1945a,b), who has used it to treat two- and three-dimensional problems with finite depth and acceleration. (It is also useful in problems with oscillating boundaries but these are not being treated here.) 2. Properties of Michell’s Integral Several properties of Michell’s integral can be deduced immediately either from (3.15) or from inspection of (3.38) or (3.39): (1) R is proportional to B 2 ;( 2 ) R is independent of the direction of motion of the ship; (3) R, and the wave pattern, are the same whether the ship is free to trim and squat or fixed in position. Although the derivation of these properties has been given only for h = co and horizontally unbounded fluid, they are also consequences of thin-ship approximations in various other situations that will be considered later. It is evident that in any comparison of theory with experiment these should be the first predictions to be tested, for they require no prior numerical calculation.
155
The Wave Resistance of Ships
It is possible to derive still other information without numerical calculation by examining the behavior of Michell’s integral at high and low Froude numbers. There may be some question about the usefulness of such approximations, for the accuracy of the linearized theory decreases as the Froude number becomes either very small or very large, or put in another way, the ship must become ever thinner in these two limits to maintain the same accuracy. However, the low-Froude-number expansion has proved useful in calculation and both bring out interesting properties, one of them surprising, of the behavior of Michell’s integral. A low-Froude-number expansion seems to have been given first by Wigley (1942) and later further developed by Inui (1957) and exploited by him and still later by Bhattacharyya (1970) in calculations for certain mathematical ship forms representable by polynomials. A more elaborate study was made by Kotik (ca. 1956) in a paper that unfortunately had only a small circulation in manuscript. The first term in the Iow-Froudenumber expansion is
Formula (3.66) assumes that fz exists for all x in BL). If there are corners in the waterline, there will be further terms corresponding to these. One can find some formulas for a ship of the form f(x,y ) = X ( x )Y ( y ) in Wehausen (1956). One property of the Michell resistance is immediately clear from (3.66) and that is that there are an infinite number of maxima and minima as F,,approaches zero. Equation (3.66) shows also that the most important wave-making property of a smooth ship at slow speed is the tangent angle at stem and stern at the water surface. Michell (1898) himself gave a proof of the fact that R, --f 0 as Fn -+co. I n a discussion to Wigley (1942) Havelock stated the correct form of the asymptotic expansion at large Froude number, namely, R , R , log Fn , where R1 and R 2 are power series in F n - , starting with Fn-,.Newman (1964) has derived the first term in each series. We give here only the most important term :
+
R,
=
~-lpgL-~A$,Fn-~ log Fn + O(Fn-,),
(3.67)
where A,, is the waterplane area. The result is surprising, for one would expect the most important term in the wave resistance to be related to the displacement. In fact, if the body is completely submerged, the leading
156
John V . Wehausen
term is proportional to VzFn-2,as was first pointed out by Weinblum (1936a). One can find a somewhat different approach to the high-speed limit in Michelsen (1966). Havelock, in the discussion to Wigley’s paper cited above, points out that, if the ship form has been enlarged to take account of boundary layer and wake in such a way that there is a tail of finite cross section extending to infinity, then Rn will not converge to zero as Fn-t m but to some finite value. It. may seem like a trivial remark to note that R, 2 0 for all F,, for Michell’s integral. However, the wave resistance given by the slender-body approximation does not have this property and becomes negative at low and high Froude numbers. M. G. Krein, in work reported by Kostyukov (1959, 1968, Section 40), has shown that in fact R, > 0 for any thin ship of finite length, draft, and displacement, but also constructs a form of infinite length, but finite draft and displacement, that has R , = 0. Krein also shows an easy way to construct functions S(x, y) vanishing on the underwater profile of a hull z = if(., y) such that z = +(f S) has the same displacement and Michell wave resistance as f itself. It is not clear that f S 2 0, as it must be for a real ship. However, even this can be achieved by proper choice of 6. A study of mathematical properties of Michell’s integral has also been made by Birkhoff and Kotik (1954b). From this paper we give two further representations of R, . Let the domain of definition of f(x, y) be extended to the whole (x, y) plane by settingf= 0 outside S o . Define
+
+
where
W(x, y ) = ( 4 q - 1 ’ 2 exp(--x2/4y). Then
where M is defined in (3.39) and Y ois the Bessel function usually denoted by this letter. An obvious advantage of the first form is that all information about the hull is isolated in the function H .
The Wave Resistance of Ships
157
3. Moving Pressure Distributions Recent investigations of moving pressure distributions have all been made in order to clarify some aspect of hovercraft behavior. Although some of these contain computations for infinitely deep unbounded fluid, the computations are mostly for comparison with computations with finite depth or canal walls, or both. These results will be mentioned in subsequent sections devoted to these cases. Several others will be considered still later when we deal with comparison between theory and experiment.
4. Finite Depth Replacing the boundary condition vu+0 as y + co by yY(x, -h, y ) = 0 leads to somewhat more tedious computations than for infinite depth but to no real conceptual difficulties, although subcritical and supercritical speeds play a role. Th e same methods are applicable, and indeed it is evident from Michell’s paper that he knew how to carry through Fourier’s method for a thin ship in finitely deep water even though he gave no formulas. If one uses Green functions, formulas (3.16), (3.17), (3.18), (3.50) are still valid with a Green function for finite depth. T h e same Green function can be used in the integral equation (3.55). The Green function, first derived by Sretenskii (1937), has already been given in (3.51b). Th e wave resistance formula analogous to (3.36), also first given by Sretenskii (1937), can be found in Lunde (1951a, p. 51ff), Wehausen and Laitone (1960, p. 581), and Kostyukov (1968, Section 28). T he velocity potential and wave resistance for a pressure distribution moving over water of finite depth were first given by Havelock (1922) in a special case and in general by Lunde (1951b). They may also be found in Wehausen and Laitone (1960, p. 599). Calculations of the wave resistance for distributions of rectangular and elliptical planform have been made by Barratt (1965). Included are results for infinite depth. Yim (1971) has made calculations for planforms with parallel sides but pointed ends. Huang and Wong (1970) have calculated the surface displacements for rectangular planforms. Doctors and Sharma (1972) have calculated the resistance for rectangular planforms with the pressure distribution making a continuous transition at the edges from its maximum value to zero. Calculations are for finite and infinite depth, various beam/length ratios and various transition behaviors. Some calculations showing the effect of finite depth for a thin ship are given in Fig. 12 together with other calculations for resistance in rectangular canals. The behavior for pressure distributions is similar. Both show a marked maximum near Fh = U/(gh)’”= 1.
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John V. Wehausen
5.0
4.O
3.O
2.0
1 .o
0.2
0.3
0.4 0 :4
FIG.12. Resistance coefficient R" = RW/8.rr-lpgB2TzL-' for BIT = 3 and the following cases: L / B = 5 , L / H = 3.75, h/T = 4: (a) 2b/B = 1 3 . 3 3 , (b) 2b/B = 6.66; L / B = 10, L/h = 5 , h/T = 6: (a) 2b/B = 20, (b) 2b/B = 10, (c) 2b/B = 6.66, (d) 2b/B = 5 . (2b = canal width, h = water depth) [From Kirsch, 1966 (Fig. 8b, p. 175) by permission of the Society of Naval Architects and Marine Engineers.]
5 . Motion in Rectangular Canals The velocity potential and wave resistance for a thin ship were first worked out by Sretenskii (1936, 1937) and by Keldysh and Sedov (1936). The results may be found in Lunde (1951a, p. 57ff) and Kostyukov (1968, Sections 11 and 29). The analogous problem for a moving pressure distribution has been solved by Newman and Poole (1962). Calculations have been given by Kirsch (1962, 1966) for a thin ship in a canal and earlier for a smaller range of variables by Voitkunskii (see Apukhtin and Voitkunskii, 1953, Chapter 7). Calculations similar to Kirsch's have also been made by Ueno and Nagamatsu (1971). The chief difference is that they try to satisfy the body boundary condition more accurately by solving an integral equation for centerplane source strength. The results differ noticeably but not significantly. Newman and Poole (1962) g'ive extensive calculations for moving pressure distributions of various shapes. Figure 12 is one of the figures from Kirsch's paper. The figure shows R* = R , / ~ T - I ~ ~ B ~plotted T ~ L against -~ Froude number for infinitely
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159
for unbounded fluid of finite depth (Sw)and deep unbounded fluid (a), for canals with several ratios of canal width 26 to ship beam B . The computations were made for a model of rectangular sections and parabolic waterlines. One should note especially the discontinuity in the resistance that = 1 when canal walls are present. Finite depth occurs at Fh= U/(gh)1’2 produces a sharp maximum in the neighborhood of F h = 1, but no discontinuity unless the walls are present. Kolberg (1963, 1966) has treated a generalization of the problem of ship motion in rectangular canals in which the canal bottom and walls are allowed to be rough, but with only “ small ’’ roughness protuberances, so that boundary conditions on the bottom and walls can be referred to smooth reference planes. Th e analysis becomes exceedingly complex in detail if not in principle. A similar investigation has been made by Biktimirov (1967). There are no computations.
6 . Motion with Acceleration T h e only kind of motion considered u p to now has been steady straightline motion that has been going on long enough so that the fluid motion is also steady in the ship coordinate system. However, it is obviouslyof interest to consider motion on curved paths or rectilinear motion started from rest and accelerated to some final velocity. In dealing with such problems one can proceed pretty much as we have up to now, formulating first exact problems and then linearized ones. We shall avoid here the interesting problem of combined oscillatory and forward motion, but shall mention some results on motion started from rest and along circular paths (see below). I n solving such problems it is useful to have a Green function for a singularity of variable strength moving on an arbitrary path. Such a Green function has been given by Haskind (1946) and Brard (1948) and may be found in Wehausen and Laitone (1960, pp. 490-495) for finite and infinite depth and for two and three dimensions. T he motion of a circular cylinder under a free surface starting from rest has been considered in two papers by Havelock (1949a, b) and also by Maruo (1957). This work and some computations are briefly described in Wehausen and Laitone (1960, pp. 610-617). T h e wave resistance of a thin ship in accelerated motion was first derived by Sretenskii (1939) and his result rederived by Havelock (1949a) in a different way, and also by Shebalov (1966). Th e theory is given by Lunde (1951a, p. 40ff, 55ff, 59ff), who also extends it to finite depth and canals. A different approach to the problem may be found in Wehausen (1963). Wehausen (1964) has used Sretenskii’s formula to investigate the asymptotic behavior as t -+00 of the
John V . Wehausen
160
wave resistance of a ship model started from rest and accelerated to constant speed. The problem is of obvious importance in model testing. Some computations for a mathematical ship form are included. Calculations of Sretenskii's formula for a constant acceleration have been carried out for a mathematical ship form by Efimov, Chernin, and Shebalov (1967). The result is then compared with the Michell wave resistance at each instantaneous Froude number. They have also compared different rates of acceleration and find that for 0.3 < Fn < 0.6 the wave resistance increases as the acceleration decreases. Shebalov (1962) considers a submerged body moving with variable velocity in a fixed horizontal direction, carries through the steps analogous to Kochin's in deriving his %' function and finds expressions for the force acting on the body in terms of the %' function. The essential step is to have available the time-dependent Green function. Havelock (1917a) found the wave resistance of a two-dimensional pressure distribution suddenly brought into being at t = 0 and moving with constant velocity. D'yachenko (1966) solves the analogous three-dimensional problem, but without requiring the impulsive start. As an example he treats a two-dimensional distribution moving with constant acceleration I
I
I
I
I
Curve I : Curve 2: curve 3:
I
I
o/g = o v/g = 0.05 u/g = 0.1
E/L = 0.5 h / L = 0.5
I
2
3
4 I
FIG. 13.
/
5
6
7
8
2Fn2
ResistancecoefficientR, = RWpg/4po2LB for rectangular pressure distribution
( p o )in acceleratedand steady motion. [From Doctors and Sharma, 1972 (Fig. 11, p. 258), by permission of the Society of Naval Architects and Marine Engineers.]
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161
from rest. Doctors and Sharma (1972) have rederived these results, including the effect of finite depth, made numerous calculations, and exploited the results in various ways. Here we reproduce in Fig. 13, one of their computed curves showing a comparison of steady wave resistance with two steady accelerations. The pressure distribution is rectangular, but the pressure does not drop discontinuously to zero. Finally we note that Warren and MacKinnon. (1968) have calculated the wave resistance for a “thin” disk moving in its own plane along an arbitrary, not necessarily horizontal plane.
7. Circular Path Havelock (1950) has computed the tangential and radial forces acting on a submerged sphere moving at constant angular velocity along a circular
path. We reproduce the resistance formula: If the radius of the sphere is a, the radius of the path r, the depth of the center d, and the angular velocity 0, then
R=
4X2pa6Q4m
gr
C n5Jn2(n2Q2r/g)exp(-2n2Q2
d/g).
(3.69)
1
He also computes the same quantities for a spheroid and in addition the moment acting on it. Sretenskii (1957a) independently calculated the force components acting on a sphere, but with a factor 8/3 instead of 4. In a note added in proof he attributes this to Havelock‘s having satisfied the boundary conditions on the sphere more accurately. Some results for a more general body have been stated by Perzhnyanko (1960). Havelock shows comparative graphs of R against Qr (both made appropriately dimensionless) for the sphere (r = d, 4d, and m) and for the spheroid. Shkurkina (1966) has considered the problem of variable angular velocity along a circular path and in particular the case when the sphere is suddenly set into motion along the path. 8. Stratified Fluids
It has been known at least since some experiments of Ekman (1906) that, if a ship is moving in a layer of fresh water over a layer of salt water, there is an extra resistance associated with waves generated at the interface. In particular, a large resistance maximum can occur at speeds well below those at which surface-wave resistance becomes important. One will find some discussion of the phenomenon with references to earlier literature in Wehausen and Laitone (1960, pp. 503-505).
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John V . Wehausen
The first ones to carry over the thin-ship theory to this situation were Sretenskii (1959) and Hudimac (1961). Essentially what is required is deriving the Green function for the problem, now complicated by an additional set of interfacial boundary conditions. Hudimac gives the Green function only if the singularity is moving in the upper fluid. Sretenskii gives it also when the singularity is in the lower (infinitely deep) fluid. They then give equivalent generalizations of Michell's integral to the case of a thin ship moving in the upper fluid. Sretenskii gives an additional formula for a " thin submarine " in the lower fluid. Sretenskii's results were extended to finite depth by Uspenskii (1959). Later Sabuncu (1961) rederived these results and completed them by finding also the Green function when the lower fluid was of finite depth, the upper one bounded by a horizontal plane (a rigid ice sheet), and both simultaneously. The associated resistance is also given. Later Sabuncu (1962a) made some calculations of the resistance for a body generated by a source and sink of equal strengths that
FIG.14. Resistance coefficient 2 5 0 R / 7 r ~ U ~ Vfor ~ ' ~'' Rankine solid " moving in upper layer of a stratified fluid. [From Sabuncu, 1962a (Fig. 4).]
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163
would have generated a Rankine ovoid of length/diameter 10.5 in an unbounded homogeneous fluid. It is not easy to state exactly what sort of body is generated, but presumably something like a Rankine ovoid. I n any case the computations show clearly the effect of the internal wave at low Froude numbers. Figure 14is reproduced from Sabuncu's report and shows clearly the low-speed " dead water " resistance. He calculated the resistance also for other relative depths of the upper fluid, with the " Rankine ovoid" always resting on the interface. He also calculated the moment acting on the body as a result of the two sorts of wave. 9. Nonuniform Current I n all considerations up to now we have assumed the ship to be moving in still water, which is equivalent, of course, to having the ship held still and a uniform stream of water flow past it. In rivers, however, the flow is known not to be uniform, so that it is of special interest to those concerned with inland waterways to study the effect of nonuniform flow upon the resistance. T he theoretical investigations have been carried out chiefly by Kolberg (1959b, 1961) and Cremer and Kolberg (1964). Since the flow is no longer irrotational, there is no longer a velocity potential and this initial hurdle must be gotten over. Let the velocity of the undisturbed stream relative to the ship be U ( y )and let V = (u, v, w ) be the disturbance velocity caused by the ship. Then Euler's equations for an inviscid fluid, linearized by neglecting second-order terms in (u, v, w ) , become uu,+ uv= -p-'pz, (3.70) u v , = -p - 'p, -g ,
uw, = -p -'p2. By using the continuity equation u,
+ v, + w , = 0,
one can eliminate
u, v, w and obtain the following partial differential equation inp'
Ap' - 2 u ' U - l ~ ;= 0.
=p
+pgy:
(3.71)
T he boundary conditions can also be reformulated in terms of p'. If there is an external pressure being applied (see 3.23), then the free-surface boundary condition becomes n
(3.72)
and the surface is given by 1 Y(x,2) = - [pyx, 092) +PI. Pg
(3.73)
John V. Wehauserz
164
Bottom and side-wall conditions (if necessary) become
p;(% -4 y ) = 0,
P;(% y , 9) = 0.
(3.74)
An equation analogous to the fourth one of (3.15) is also necessary. It is evident that one may try to carry through a program similar to that already completed for, say, the “ thin ship ” and the moving pressure distribution. T he first step for the former will be to construct a Green function for the equation and boundary conditions above satisfied by p‘. Kolberg is able to do this in the usual way for both h = 00 and h < co if he assumes
U ( y )= v exp Py,
P > 0.
(3.75)
The construction is extended to canals in Cremer and Kolberg (1964). Kolberg then shows that the linearized body boundary condition (see 3.15, Eq. 3). (3.76) w(x, y , f0) = F w x x , y ) can be satisfied by a distribution of his Green functions with strength proportional to U(y)fz(x,y ) . Th e analog of (3.17) follows immediately from (3.5):
R =2
J P’(% y, 0 ) f h y ) dx dy.
(3.77)
SQ
T he solution for a moving pressure distribution can also be carried through, and this is done by Kolberg (1961) and Cremer and Kolberg (1964). Finally, in the latter paper there is also an investigation of the asymptotic behavior of the wave pattern behind a moving singularity: They use this pattern to establish some rules of “ equivalent velocities ” allowing one to pass from nonuniform to uniform flows.
10. Viscosity EfJects Within the context of this chapter the effect of viscosity can be taken into account only by some sort of ad hoc procedure. T h e most straightforward idea of this sort is to increase the ordinates of the ship hull by the amount of the displacement thickness at each point and then to complete the afterbody by either an infinite or finite wake. Such ideas were put forward by Havelock (1926a), together with calculations, and then subjected to a more thorough and critical analysis later (1948) in a paper with many valuable insights. Havelock’s ideas were formalized somewhat by Lavrent’ev (1951) and computations based on Lavrent’ev’s formulas have been made by Wigley (1962, 1963, 1967). In comparing the wave resistances of Havelock’s modified ship forms with the originals, it is clear that the modifications have produced resistance curves conforming more closely to the
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165
behavior of observed ones than do those of the originals. Rather than reproduce some of these results, we prefer to show a recent calculation of Milgram (1969). Milgram’s “ ship ’’ has cosine waterlines and triangular sections. The “ wake ” consists of a triangular prism joined to the hull by a parabolic transition. The modified ship has the following equation :
@?(1 +y/T)cos r x / L ,
5 +L,
+ (.rrl2L)(x xo)”(*L +xo)], x 5 BL. @( 1 +y/T)(+L)(BL + x*),
B q1 +y/T)[cos z=[
xo 5 x
7
.
4
-
-*L F x <xo,
-
Computations of the Michell wave resistance were made for LIT = 10 and for xo= -0.40L and -0.45L. The wave resistance coefficient R,/ ;pU2($B)2is shown in Fig. 15. The effect of the wake evidently is more important at the smaller Froude numbers, as had been observed by Havelock.
0’40F-----l 0.36 0.32
-
0.28
-
NJ 0.24m
-
-LN0.20 -
N
3
U
-
0.08
-
NO SEPARATION SEPARATION AT 0.95 L SEPARATION AT 0.90 L
-
-
-
Fn
FIG.15. Effect of wake upon wave resistance. [From Milgram, 1969 (Fig. 2, p. 71), by permission of the Society of Naval Architects and Marine Engineers.]
Wigley (1938) follows a more empirical method in trying to correct for viscosity. He observes that the afterbody of a ship seems not to play as large a role in wave making as predicted in the Michell theory. He therefore proposes an empirical correction factor in which the effect of the afterbody is reduced by multiplication by an empirical factor depending upon the Froude number. The corrections seem generally to deform curves in the right direction, although they do not fully account for
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John V . Wehausen
discrepancies. It should be noted that this correction may be considered also as a correction to the thin-ship boundary condition. Similar but more extensive correction methods of Inui will be mentioned later. There have also been attempts to consider the effects of viscosity in a more fundamental way. The first of these was by Sretenskii (1957b) who used a linearized form of the Navier-Stokes equation (Oseen’s approximation) to treat a moving pressure distribution. Tangential forces are neglected. Formulas for resistance are derived that are not dissimilar in structure to those for an inviscid fluid. If P(x, z) represents the imposed pressure, then
(3.78) where
m2 = k2
+ la,
D = (2vm2 + ikU)2+ mg - 4vam3(m2- ikU/v)l12, and
R=-
1
k 2 P Im
jj
mk
( I 2+J 2 )dk dl,
(3.79)
-m
where m
I
+ iJ= jj exp i(kx + Zz)P(x, x) dx dx. -m
Kolberg (1959a) has extended the results to water of finite depth. Essentially the same problem is studied by Gruntfest and Nikitin (1966) except that they treat an initial-value problem in which the pressure distribution is suddenly brought into existence at t = 0. They examine the asymptotic behavior as t --f co for a pressure distribution degenerated to a singular line distribution of length L in the direction of motion. No reference is made to Sretenskii’s work and it is not clear that the two results are consistent with one another. A different sort of approach to the effect of viscosity is made by Lurye (1968). He also assumes small disturbances with respect to U and uses Oseen’s equations. However, he starts with a solution containing a singularity and wake-like flow following it. (In the neighborhood of the singularity the disturbance is, of course, not small.) T o this he adds another velocity, also small, chosen so that the sum still satisfies Oseen’s equations
The W a v e Resistance of Ships
167
but also the linearized free-surface conditions for a viscous fluid. This second flow is constructed in the form of double Fourier integrals and is not very perspicuous. T. Y. Wu (1963) has followed still another path in trying to take account of viscosity in a more fundamental way. He starts from the initial ad hoc idea of increasing the ship by its boundary layer plus wake, but instead of simply taking a nominal displacement thickness and wake or using that for a flat plate in an infinite fluid, he has considered the interaction between the waves and the boundary layer. For the boundary-layer calculation he uses von Kdrmhn’s momentum-integral equation and similarity methods associated with this equation. For the exterior potential flow he uses the thin-ship potential. In the author’s words, the work is “ a very preliminary effort.” I n a paper by Webster and Huang (1970) equations for a turbulent boundary layer are solved by an approximate method. However, the emphasis here is on prediction of separation and the influence of the Froude number upon this rather than calculation of the effect of the boundary layer upon the wave resistance. Brard’s (1970a,b) investigations are more ambitious but too complex to summarize in any detail. He assumes linearized Navier-Stokes equations and linearized free-surface conditions, as did Sretenskii and Lurye. In addition, he assumes that the position of the boundary layer and wake as well as the vorticity distribution within this region are known. He wishes to satisfy the exact boundary conditions on the hull and does this by expressing the velocity as the sum of four vector fields, three of them irrotational, the fourth determined by the known vorticity distribution. A Green function appropriate to the free-surface condition is constructed and used to derive integral equations for the irrotational vectors. Although no solutions are obtained, the decomposition of the velocity is used as a basis for a decomposition of the resistance. He especially addresses himself to the problem of a rational definition of wave resistance in a viscous fluid. Intermediate between the attempts to account for viscosity by artificially adding wakes and by using the Navier-Stokes equations are papers by Tatinclaux (1970) and Beck (1971). Th e former has been discussed near the end of Section I1 and is concerned chiefly with the effect of the wake upon direct measurement of wave resistance from wave profiles. Beck couples the thin-ship approximation with a rotational wake of known vorticity distribution and position, but does not assume a viscous fluid. Th e rotational wake is eventually taken in the form of a U-shaped vortex sheet trailing aft from the ship with draft and breadth as parameters. Th e resistance is then expressed as a sum of the Michell resistance, the wave resistance of the wake itself, the interaction of the two and an
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John V . Wehausen
" equivalent viscous resistance " resulting from a head loss associated with the wake. Computations of the wake and interaction resistance were carried out for a hull with parabolic sections and waterlines, one extensively tested by Lackenby (1965) and for which measured values of the equivalent viscous resistance, the total resistance and the Michell resistance were available. Variation of the draft of the vortex sheet showed only a small effect, variation of the ratio of the wake breadth to the ship beam showed a greater effect, but still a small one compared with the total measured resistance. However, the interaction resistance shows substantial variation as a function of the attachment point. Beck uses the attachment point (for a fixed breadth and draft of the wake) as a parameter to be selected so as to make the measured total resistance agree with the sum of the four component resistances. He is able to make the selection only for the two largest Froude numbers. However, one should remember that the model of the wake is a very simplified one, and a more elaborate one would presumably allow fitting at any Froude number. Beck's procedure may seem artificial in that he must start with experiment data to determine the vortex strength in his wake model and then must again use experiment data. However, one must keep his aim in mind. He is trying to find out whether it is possible to discover a wake model that will allow one to explain the smoothing out of the humps and hollows in the Michell resistance that are observed in measurement. Once this is established, one might hope to proceed to more elaborate models, perhaps in the manner of Brard (1970a,b). We mention finally a paper by Dugan (1969) in which the plane problem of motion of a submerged cylinder normal to its generators is considered. Oseen's equations and linearized free-surface boundary conditions are used. A Green tensor is derived and the problem reduced to solution of a pair of coupled integral equations. For the special case of a flat plate of finite length moving edgewise beneath the free surface Dugan is able to derive an approximate solution (assuming among other things small Reynolds number). Although the result is still rather remote from most ship problems, it displays a resistance component due entirely to the interaction of viscosity and gravity.
11. Surface- Tension Effects Once one has found the appropriate Green function for a fluid with surface tension, one can repeat the calculations already made for thin ships, submerged bodies, etc. T h e Green functions for both unsteady and steady motion are given in Wehausen and Laitone (1960, p. 636ff). The thin-ship theory has been carried through by Webster (1966). Computations
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169
are given for ship models of lengths 2, 5, and 10 ft. For the last two the effect of surface tension is negligible, for the first substantial. I t appears, however, that 5 ft is about the lowest one should go in model tests. Other interesting things turn up. For example, certain kinds of singular behavior present in the case of gravity alone disappear when surface tension is included. 12. Slender Ships Derivations of the wave resistance that make use of the slender-body approximation may be found in papers by Maruo (1962), Joosen (1963), and Tuck (1963ab, 1964ab) each from a somewhat different point of view. The slender-body expression for the wave resistance is expressed in terms of the section-area curve (3.80) The following formula requires S’(x) to be continuous except at the ends, where it must, however, remain finite:
L/2
- S’(-+L)
+
S”(X)Yo(K(4L x)) dx
J-L/2
+ S’(@)/L’ZS”(x)Y,(K(+L-x)) dx + S’(+L)S’(-+L)Yo(Fn-2) - LI Z
2 - - JO
-T
2
-
jodY drljz(-%
s”, p,.
;
Y)fz(-%
rl)ln(irKlr
-T
drlfZ(SL Ylf,(*-G rl)ln(iyKb
+?ID
+ rl I ),
(3.81)
where In y=O.5772.. . is Euler’s constant. If the ship is such that S‘(f1;/2) = 0, then all terms after the first vanish. This would be the case with a raked stem, for example, but not a vertical one with finite entrance angle. This form for the resistance was first given by Maruo. A somewhat different form, expressed in Stieltjes integrals, is known as Vosser’s integral :
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John V . Wehausen
It is not possible, however, to obtain the last two integrals in the first form directly from this one. [In fact, (3.82) predicts infinite resistance if S’( fL/2) # 0.1 Maruo pointed out that (3.81) could be derived from Michell’s integral as a limit when the draft becomes small. As he remarks, this in itself does not mean that Michell’s integral is more accurate than Maruo’s expression. However, Wehausen (1963) has argued that Michell’s integral does indeed carry the most important information about wave resistance independently of the beam/draft ratio. In any case, the slender-body approximation has turned out to be a disappointment for wave resistance. In some sense it ought to have been better than thin-ship theory, for ships have beam/draft ratios usually between 2 and 4 and would seem offhand to be better suited to this approximation than the thin-ship one. However, for both low and high Froude numbers the resistance becomes negative and its behavior at moderate Froude numbers usually has more extreme humps and hollows than the Michell resistance. The difficulties are discussed by Kotik and Thomsen (1963), where some computed curves are also shown.
13. “Flat” Ships Inasmuch as a ship’s beam is usually two to four times its draft, one naturally asks why a counterpart to the Michell theory has not been developed in which the ship boundary condition is satisfied on the waterplane section rather than the centerplane section. There is, in fact no difficulty in formulating the problem. The difficulty is in solving it. I t leads to an integral equation with a particularly troublesome kernel and in particular to no explicit formula. In the two-dimensional case one can solve the problem, and it has been done by several persons. References and a description of one method may be found in Wehausen and Laitone (1960, pp. 587-592). The three-dimensional problem has been studied by Maruo (1967). The problem is formulated and an integral equation derived. He proceeds then to develop two approximations. First he works out a high-aspect ratio theory (an analog of the lifting-line approximation in wing theory). This can be completely worked out. Next he treats the low-aspect ratio case. This leads to another integral equation, simpler than the original one but still too difficult. By next considering a high-Froude-number limit, he is able to find some explicit results. Three-dimensional planing at high Froude numbers has also been treated by Wang and Rispin (1971). However, they have been able to solve their integral equation for aspect ratios that are O( 1). They give computations for rectangular plates with aspect ratios between 0.5 and 2.0. Comparison
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171
with experimental data of Sottorf shows fairly satisfactory agreement. Hogner in a series of papers (1932a, b, 1936, 1954) has proposed a flatship theory further elaborated into an “ interpolation formula ” designed to interpolate between thin and flat ships. The formulas must, however, be regarded as conjectured useful expressions rather than proved ones. They have not been sufficiently explored numerically so that one can judge them from this point of view.
14. ‘‘Shallow” Water When the depth of water becomes small compared with the wave length of typical waves, the equations and linearized boundary conditions that we have been using are replaced with another set of approximating equations, the so-called “ shallow-water ’’ equations. We shall not introduce the equations here [they are nonlinear and can be found in Stoker (1957, Chapter 10) or Wehausen and Laitone (1960, Chapter E)], but remark that these equations can be further linearized. These linearized shallow-water equations can be used to examine the flow about a steadily moving ship. T h e problem was attacked by Michell in his original paper on thin-ship theory (1898), and solved for a wall-sided thin ship touching the bottom, i.e., for a “thin ” vertical strut in a steady stream. T h e result was rediscovered by Joukowksi (1909) in 1903. T h e problem has been reconsidered by Tuck (1966), who does not require the ship to extend wall-sidely to the bottom. Furthermore, Tuck does not start from the linearized shallow-water equations, but from the exact equations and derives his results directly in connection with a slender-body approximation for the ship. T h e result for resistance turns out to be the same as Michell’s but he also finds expressions for vertical force and moment. T h e character of the answer depends upon whether the ship speed is sub- or supercritical, i.e., whether F h = U(gh)-1’2is (1 or > l . T h e result for resistance follows : Fh 0.316, so that the applicability of Egger's calculations to ships free to trim and heave becomes doubtful in this region. T h e smallness of sinkage and trim for Fn < 0.316 lends further support to the usefulness of the inconsistent problem in this region. On the other hand, the experiments of Shearer (1951, see Fig. 20) and others show a fairly substantial difference in measured resistance between models free to trim and ones fixed, even in the region Fn < 0.3 16. Experiments in which the wave resistance is directly measured as well as further calculations would clarify the situation.
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A report by Eng (1969) also gives an almost complete second-order theory for a vertical strut with parabolic waterlines. The theory is somewhat deficient in that certain free-surface corrections were simplified in order to reduce computer time. In view of Egger’s results this was perhaps not serious. In any case he found fairly satisfactory agreement with values from experiments especially designed to measure wave resistance. Eng has also modified (3.117) by retaining certain terms in e4 that guarantee that R will not be negative, an event not excluded by (3.117). One should note, however, that a situation in which the term in e3 can overpower the one in ea does not seem appropriate for application of the theory. The inconsistent approximation has been the subject of a number of investigations in recent years. Here we may distinguish between at least two approaches. In each, one assumes the representation (3.56). However, in one approach, instead of solving the integral equation (3.57) with the proper Green function, say (3.51), one simplifies G for the purpose of easier numerical solution by replacing G by the Green function appropriate to a rigid surface at y = 0, e.g., in the case (3.51) by r - l + r ; l . However, in using the solution in (3.56) the complete expression for G is used. This is often called the “ zero-Froude-number approximation ” or the “ doublebody approximation.” In the other approach one attempts to solve (3.57) with the proper Green function. This means, of course, a different source density distribution for each Froude number. A modification of this approach consists in satisfying the body boundary condition to the second order and the free-surface condition to the first order. In another modification one distributes sources on the centerplane section but determines (if possible) the source strength from the exact body boundary condition. The double-body approximation was apparently first used by Inui (e.g., 1957) to construct hull shapes from given source distributions by the inverse streamline-tracing method. However, he traced the streamlines assuming a rigid surface at y = 0. This approximation has been used either in a direct or an inverse method by Breslin and Eng (1963), Pien and Moore (1963), Yokoyama (1963), Ikehata (1965), Ogiwara, Maruo, and Ikehata (1969) (here combined with a slender-body approximation), and Bhattacharyya (1970). Several of these show comparisons with experiment and it seems fair to say that the calculated curves show no better agreement than does the Michell resistance (see, e.g., the first cited paper), in fact, usually much worse. Special experiments by Gadd (1966) seem to bear this out. This approximation has also been investigated theoretically by Kotik and Morgan (1969). They point out that a velocity potential satisfying the body boundary condition and the rigid free-surface condition can be generated by infinitely many sorts of source and/or dipole distributions, and that it is not clear that all will lead to the same wave resistance. In fact, Kotik
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and Mangulis (1962) had already observed that a vertical circular cylinder could be generated by either a vertical line of dipoles or by a surface distribution and that the former would yield an infinite wave resistance, the latter a finite one. Eggers in his discussion of Pien (1964) has made a similar point. Kotik and Morgan restrict themselves to source and dipole distributions on the surface of a body. They show that if the body is completely submerged the wave resistance is unique no matter how the distribution of sources and dipoles on its surface is chosen. I n the case of a body intersecting the surface one must allow an additional dipole distribution over the waterplane area in order to obtain uniqueness of the wave resistance. Numerical calculations in which the complete Green function is used have been carried out by Kajitani (1965), Nakatake and Fukuchi (1967), Gadd (1969, 1970) and Kobayashi and Ikehata (1970). Of these we note that Kajitani uses an inverse (streamline-tracing) method, Nakatake and Fukuchi distribute the sources on the centerplane section, Gadd (1969) satisfies the body boundary condition only to second order, but Gadd (1970) distributes sources over the surface, as do Kobayashi and Ikehata, to satisfy this condition exactly (except for the error inherent in the numerical methods). Several of these papers give comparisons with Michell resistance, the resistance computed by the double-model approximation, and experiment. Although we shall not reproduce any results here, it is evident that in comparison with the Michell resistance agreement with experiment has been improved for Fn < 0.35, which was not the case with the doublemodel approximation. However, there also seems to be evidence of difficulties associated with the numerical calculations. These will certainly be overcome in the near future. Brard (1971, 1972) has made a thorough study of the potential-theory problem arising in the inconsistent problem and calls especial attention to the necessity of a line integral around the intersection of the ship and the plane y = 0 in the integral equation [see (3.49) and the remarks following (3.55)]. None of the above authors has included this. Finally we mention briefly a nonsystematic correction procedure of Guilloton (1964). His idea is to start from the velocity field given by the Michell theory and to map it into a new velocity field that will provide a better approximation to the “ real ” flow. Since the procedure starts with the Michell potential, the associated source distribution is on the centerplane and Guilloton’s procedure is a kind of inverse streamline-tracing method, but with the hull forced to be a streamsurface. I n some respects it is similar to satisfying the hull boundary condition to the second order. There is also a small free-surface correction. The method has been studied further by Guilloton (1965) and has been applied by Emerson (1967, 1971). I n spite of the method’s unsystematic character Emerson’s calculations
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show good agreement with experimental measurements for Froude numbers below those for which sinkage and trim (neglected in the theory) become substantial.
3 . Other Higher-Order Calculations I n the approximation schemes that we have been using it has been assumed that for a given velocity U the moving body was either sufficiently deeply submerged or so constituted geometrically, say thin enough, that it did not disturb the free surface much, except possibly near the stem and stern of a surface vessel. Keeping U fixed is equivalent to keeping Fn fixed. If U is quite large or quite small, it may be necessary for the body to be correspondingly quite deeply submerged or quite thin for the approximation to be useful. Even the second- and higher-order approximations treated above are subject to this. One might, however, wish to change the conditions of the approximation and have, for example, EIFn = constant as E +0, or even E = constant as while Fn -+ 0. Not all conceivable combinations will lead to useful approximation schemes, but some have been examined. T he analysis is almost always by way of matched asymptotic expansions. T he following papers treat some aspect of such problems for low Froude numbers: Ogilvie (1968), and Dagan (1971b). Some aspect of the problem for high Froude numbers is considered in the following papers : T . Y. Wu (1967), Ogilvie (1967), Dagan (1971a). T he flow in the immediate neighborhood of the bow of a surface ship will be badly represented in any approximation scheme that assumes the flow to be a small perturbation of a uniform flow. It is evident that some different method of approximation is required there. This problem has been investigated by Dagan and Tulin (1969, 1970a,b). We omit any detailed discussion of these various results, for they have been dealt with by Ogilvie (1970). We close with mention of a paper by Newman (1971) in which the usual perturbation expansion is carried to the third order in the neighborhood of the cusp line. He finds that the solution diverges on the cusp line, or put differently, that no steady third-order solution exists. If this result can be confirmed, it will have an important effect upon analytic approaches to the theory of ship wave-resistance as well as waves. ACKNOWLEDGMENTS T h e author wishes to express his gratitude to several colleagues who have read parts of a preliminary draft of this paper for their suggested improvements and for errors pointed out. In particular, his thanks go to K. Eggers, L. Landweber, J. N. Newman, and W. C. Webster.
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In addition, he gratefully acknowledges the support of the Fluid Dynamics Branch, Office of Naval Research, during the summer of 1971 when most of the foregoing was written. BIBLIOGRAPHY For the most part the papers listed below are those cited in the text. An exception is formed by the papers on wave-resistance theory published before 1951. For this period I have tried to give a fairly complete bibliography, although some papers will certainly have been overlooked either through inadvertance or ignorance. In addition, because few extra titles were involved, I have included, as well as I was able to determine it, a complete list of the late W. C. S. Wigley’s papers, not including reports. Papers on wave resistance published in the USSR are not covered as thoroughly as would have been desirable. Fortunately, this lack is compensated by the bibliography in Kostyukov’s (1959, 1968) book and also by Palladina’s (1957) guide to Russian literature on the theory of ships. For more recent papers no attempt at completeness has been made, and many substantial papers are not here for no other reason than that their content did not fit easily into the exposition. For example, there are many more papers on the determination of wave resistance by means of measurement of wave patterns than occur here. And this is not the only example. Fortunately, papers on wave-resistance theory, and on ship hydrodynamics in general, appear in a relatively limited number of journals or conference proceedings, in pleasant contrast to other fields. Consequently, it is not difficult to put together an almost complete bibliography on any special topic. First there are the publications of several societies of naval architects : Bulletin de l’dssociation Technique Maritime et Akonautique (Park), Jahrbuch der Schiffbautechnischen Gesellschaft (Hamburg), Journal of the Zosen Kiokai (through vol. 122; thereafter the following), Journal of the Society of Naval Architects of Japan (Tokyo), Transactions of the Royal Institution of Naval Architects (London), Transactions of the North-East Coast Institution of Engineers and Shipbuilders (Durham), Transactions of the Institution of Engineers and Shipbuilders in Scotland (Glasgow), Transactions of the Society of Naval Architects and Marine Engineers (New York), Trudy Tsentral’nogo Nauchno-Issledovatel’skogo Instituta imeni A . N . Krylova (Leningrad), and Trudy Leningradskogo Korablestroitel’nogo Instituta (Leningrad). In addition there are several journals devoted to topics in ship research: International Shipbuilding Progress, Journal of Ship Research, Schiffstechnik. Of the journals devoted to fluid dynamics in general, one will occasionally find papers in the following: Journal of Fluid Mechanics, Physics of Fluids, Prikladnayn Matematika i Mekhanika, and Izvestiya Akademii Nauk S S S R . Mekhanika Zhidkosti i Gaza. In addition to the journal literature important papers have appeared in the proceedings of the Symposia on Naval Hydrodynamics (every two years since 1956), the International Towing Tank Conferences (every three years), and the International Seminar on Theoretical Wave Resistance held in Ann Arbor in 1963. There is also a rather extensive report literature, some of it rather informal and of limited distribution, some of it intended as a permanent record. The latter is especially true of the reports of several of the large ship research laboratories. This nearly exhausts the sources of current literature, but of course not completely, as one may easily determine from the papers below. Papers have been identified in the text by author and year. If more than one paper has been published in one year, they are further distinguished in the text by letters, e.g., 1951a, 1951b, the first listed paper being 1951a, the second 1951b. The corresponding letters have also been added to the dates in the bibliographical data below. They are not, of course, properly a part of the data, but have been included for ease of reference.
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ALLEN,R. F. The effects of interference and viscosity in the Kelvin ship-wave problem. J . Fluid Mech. 34 (1968), 417-421. P. A.; VOITKUNSKII, Ya. I. Soprotivlenie vody dvizheniyu sudov. (The resisAPUKHTIN, tance of water to the motion of ships.) Gos. Nauch.-Tekhn. Izdat. Mashinostroi. Sudostroi. Lit., Moscow-Leningrad, 1953, 356 pp. BABA,E. Study on separation of ship resistance components. J . SOC.Nav. Architects Jap. 125 (1969a), 9-22. Also in Selec. Pap. 7, 12-16. BABA,E. A new component of viscous resistance of ships. J . Soc. Nav. Architects l a p . 125 (1969b), 23-34. Also in Selec. Pap. 7, 17-21. BARRATT, M. J. The wave drag of a Hovercraft. J . Fluid Mech. 22 (1965), 39-47. BARRILLON, E. G. Calcul de la part de rksistance due aux vagues formCes par le navire. Rev. Gen. Mec. 17 (1933), 136-140, 200-203, 292-294. BECK,R. F. The wave resistance of a thin ship with a rotational wake. J . Ship Res. 15 (1971), 196-216. BESSHO,MASATOSHI. On the wave resistance of a submerged circular cylinder (consideration of the finite water surface disturbance). J . Zosen Kiokai 100 (1957a), 1-6. BESSHO,MASATOSHI. On the wave resistance theory of a submerged body. SOC.Nav. Architects Jap. 60th Anniv. Ser. 2 (1957b). 135-172. BESSHO,MASATOSHI. On the wave-making resistance of submerged prolate spheroids. J . Zosen Kiokai 109 (1961), 59-72. BESSHO,MASATOSHI. A new approach to the problem of ship wave. Mern. Def. Acad. Math. Phys. Chem. Eng., Yokosuka,Jap. 2, no. 2, 161-174 (1963b). BESSHO,MASATOSHI. On the problem of the minimum wave making resistance of ships. Mem. Def. Acad. Math. Phys. Chem. Eng., Yokosuka, Jap. 2, no. 4, 1-30 (196213). BESSHO,MASATOSHI. Wave-free distributions and their applications. Int. S a . Theoret. Wave Resistance, Ann Arbor, 1963a, 891-906. BESSHO,MASATOSHI. On the minimum wave resistance of ships with infinite draft. Int. Sem. Theoret. Wave Resistance, Ann Arbor, 196313, pp. 985-1010. BESSHO, MASATOSHI. Solutions of minimum problems of the wave-making resistance of the doublet distribution on the line and over the area perpendicular to the uniform flow. Mern. Def. Acad. Math. Phys. Chem. Eng., Yokosuka, Jap. 4 (1964), 25-42. BESSHO,MASATOSHI. The minimum problem of the wave resistance of the surface pressure distribution. 6th Symp. Nav. Hydrodyn., Washington, D.C., 1966, pp. 775 -790; disc. pp. 790-792. BHATTACHARYYA, ~ F S W A RUber . die Berechnung des Wellenwiderstandes nach verschiedenen Verfahren und Vergleich mit einigen experimentellen Ergebnissen. SchifYstechnik 17 (1 970), 51-64. BIKTIMIROV, Yu. K. Theoretical investigation of the hydrodynamic reactions of wave nature by the method of spatial distribution of dipoles. (Russian.) Tsent. Nauch.-Issled. Inst. Krylov. Tr. Sb. Statei P O Gidromekhanike Dinamike Sudna, pp. 147-157. Izdat. “ Sudostroenie,” 1967. BIRKHOFF, G.; KOTIK,J. Theory of the wave resistance of ships. 11. The calculation of Michell’s integral. Trans. SOC.Naw. Architects Mar. Eng. 62 (1954a), 372-385; disc. 385-396. BIRKHOFF, G. ; KOTIK,J. Some transformations of Michell’s integral. Publ. Nut. Tech. Univ. Athens 10 (1954b), 26 pp. G.; KORVIN-KROUKOVSKY, B. V.; KOTIK,J. Theory of the wave resistance of BIRKHOFF, ships. I. The significanceof Michell’s integral. Trans. SOC.Nav. Architects Mar. Eng. 62 (1954), 359-371; disc. pp. 385-396.
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BRARD,R. Vagues engendrkes par une source pulsatoire en mouvement horizontal rectiligne uniforme. Application au tangage en marche. C . R. Acad. Sci. 226 (1948), 2124-21 25. BRARD, R. Composantes rbelles et apparentes de la rtsistance B la marche. Bull. Ass. Tech. M a r . Aeronaut. 70 (1970a), 229-256; disc. pp. 257-260. BRARD,R. Viscosity, wake, and ship waves. J . Ship Res. 14 (1970b), 207-240. BRARD,R. Thkorie semi-1inCarisi.e des vagues d’accompagnement d’un navire de surface. Bull. Ass. Tech. M a r . Adron. 71 (1971), 255-269; disc. 270-275. BRARD,R. T h e representation of a given ship form by singularity distributions when the boundary condition on the free surface is linearized. J . Ship Res. 16 (1972), 79-92. BRESLIN,J. P.; ENG, KING. Calculation of the wave resistance of a ship represented by sources distributed over the hull surface. Int. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1081-1110. CALISAL, S.; Effect of wake on wave resistance. J . Ship Res. 16 (1972), 93-112. S.; MOFFITT,F. H.; WEHAUSEN, J. V. Measurement by transverse wave profiles CALISAL, of the wave resistance of three forms of “minimum” resistance and of Series 60, Block 0.60. Univ. Calif., Berkeley, Coll. Eng. Rep. NA-68-1 (July 1968), i 46 pp. CHERKESOV, L. V. T h e development and decay of ship waves. (Russian.) Prikl. M a t . Mekh. 27 (1963a), 725-730. CHERKESOV,L. V. Ship Waves in a viscous fluid. (Russian.) Dokl. A k a d . Nauk. SSSR 153 (1963b), 1288-1290. L. V. T h e development of ship waves in a fluid of finite depth. (Russian.) CHERKESOV, I z v . A k a d . Nauk. SSSR, Mekh. Zhidk. Gaza 1968, no. 4, 70-76. CHEY,YOUNG,H . T h e consistent second-order wave theory and its application to a submerged spheroid. J . Ship Res. 14 (1970), 23-51. G. D. Surface waves generated by a travelling pressure point. Proc. R o y . SOC. CRAPPER, Ser. A. 282 (1964), 547-558. CRAPPER,G . D. Ship waves in a stratified ocean. J . Fluid Mech. 29 (1967), 667-672. F. Der Stromungseinfluss auf den Wellenwiderstand von Schiffen. CREMER, H. ; KOLBERG, Forschungsber. Landes Nordrhein- Westfalen 1264 (1964), 73 pp. CUMBERBATCH, E. Effects of viscosity on ship waves. J . Fluid Mech. 23 (1965), 471-479. G. Free-surface gravity flow past a submerged cylinder. J . Fluid Mech. 49 (1971a), DAGAN, 179-192. DAGAN, G. Nonlinear effects for two-dimensional flows past submerged bodies moving at low Froude numbers. Hydronautics, Inc. Tech. Rep. 7103-1 (June 1971b). vi 45 6 pp. DAGAX, G.; TULIN, M. P. Bow waves before blunt ships. Hydronautics, Inc. Tech. Rep. 117-14 (Dec. 1969), iv 4- 45 + 6 pp. DAGA~S, G.; TULIN,M. P. T h e free-surface bow drag of a two-dimensional blunt body. Hydronautics, Inc. Tech. Rep. 117-17 (Aug. 1970a), v 31 3 pp. DAGAN,G . ; TULIN, M. P. Nonlinear free-surface effects in the vicinity of blunt bows. 8th Symp. N a v . Hydrodyn., Pasadena, CaliJ., 1970b, pp. 607-622 ; disc. 622-626. S. D. T h e wave resistance of an air-cushion vehicle in steady DOCTORS,L. J.; SHARMA, and accelerated motion. J . Ship Res. 16 (1972), 248-260. DUGAN,J. P. Viscous drag of bodies near a free surface. Phys. Fluids, 12 (1969), 1-10, D’YACHENKO, V. K. Wave resistance of a system of surface pressures in unsteady motion. (Russian.) T r . Leningrad, Korablestroi. Inst. 52 (1966), 83-91. S. D. Bugwulste fur langsame, vollige Schiffe. Jahrb. Schiflbautech. ECKERT,E. ; SHARMA, Ges. 64 (1970), 129-158; Erort, 159-171. EFIMOV,Yu. N.; CHERNIN,K. E.; SHEBALOV, A. N. Calculation of the wave resistance of a thin ship in unsteady motion. (Russian.) T r . Leningrad. Korablestroi. Inst. 58 (1967), 47-55.
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EGGERS, K. W. H. Uber die Ermittlung des Wellenwiderstandes eines Schiffsmodells durch Analyse seines Wellensystems. I, 11. Schzflstechnik 9 (1962), 79-84; Disk. 85; 10 (1963), 93-106. EGGERS, K. W. H. Second-order contributions to ship waves and wave resistance. 6th Symp. N a v . Hydrodyn. Washington, D.C., 1966, pp. 649-672; disc. 673-679. EGGERS, K. W. H. An evaluation of the wave flow around ship forms, with application to second-order wave resistance calculations. Stevens Inst. Tech., Davidson Lab. Rep. SIT-DL-70-1423 (June 1970), viii A- 32 pp. EGGERS,K. W. H.; KAJITANI, H. A comment concerning local-wave influence on longitudinal-cut wave analysis. Proc. 12th Int. Towing Tank Conf., Rome, 1969, pp. 151-1 54. EGGERS, K. ; WETTERLING. W. Berechnungen zum Wellenwiderstand an der elektronischen Rechenmaschine G2 in Gottingen. [Erganzung zuin T M B Rep. 886 (1955): G. P. Weinblum : A systematic evaluation of Michell’s integral.] Institut fur Schiffbau der Universitat Hamburg, 1956, 19 pp. 19 pp. S. D.; WARD,L. W. An assessment of some experimental EGGERS, K. W. H.; SHARMA, methods for determining the wavemaking characteristics of a ship form. Trans. SOC. N a v . Architects Mar. Eng. 75 (1967), 112-144; disc. 144-157. EKMAN,V. W. On dead water. The Norwegian North Polar Expedition, 1893-1896. Scientific Results, vol. 5, no. 15, viii 152 pp. 17 plates. Christiania, 1906. EMERSON, A. The application of wave resistance calculations to ship hull design. Trans. Inst. N a v . Architects 96 (1954), 268-275; disc. 275-283. EMERSON, A. The calculation of ship resistance : an application of Guilloton’s method. Trans. Inst. N a v . Architects 109 (1967), 241-248; disc. 269-281. EMERSON, A. Hull form and ship resistance. North-East Coast Inst. Eng. Shipbuilders Trans. 87 (1971), 139-150; disc. D27-D30. ENG,KING. Development and evaluation of a second-order wave-resistance theory. Stevens Inst. Tech., Davidson Lab. Rep. 1400 (Aug. 1969), ix 70 pp. EVEREST, J. T . ; HOGBEN,N. Research on hovercraft over calm water. Trans. Inst. N a v . Architects 109 (1967), 311-322; disc. 322-326. J. T . ; HOGBEN, N. A theoretical and experimental study of the wavemaking of EVEREST, hovercraft of arbitrary planform and angle of yaw. Trans. Inst. N a v . Architects 111 (1969), 343-357; disc. 357-365. EVEREST, J. T.; HOGBEN,N. An experimental study of the effect of beam variation and shallow water on ‘thin ship’ wave predictions. Trans. Inst. N a v . Architects 112 (1970), 319-329; disc. 330-333. V. S.; CHERKESOV, L. V. Development of ship waves in a nonhomogeneous FEDOSENKO, fluid. (Russian.) I z v . Akad. Nauk S S S R . , Mekh. Zhid. G a z a 1970, no. 4, 137-146. FROUDE, W. Observations and suggestions on the subject of determining by experiment the resistance of ships. (Memorandum sent to Mr. E. J. Reed, Chief Constructor of the Navy in December 1968). T h e Papers of William Froude, pp. 120-127. The Institution of Naval Architects, London, 1955. FROUDE, W. The fundamental principles of the resistance of ships. Proc. Roy. Inst. G t . Brit. 8 (1875-1878), 188-213 (1876)= T h e Papers of William Froude, pp. 298-310. The Institution of Naval Architects, London, 1955. GADD,G. E. An approach to the design of low-resistance hull forms. 6th Symp. N a v . Hydrodyn., Washington, D.C., 1966, pp. 705-729. GADD,G. E. On understanding ship resistance mathematically. J . Inst. Math. Appl. 4 (1968), 43-57. GADD,G. E. Ship wavemaking in theory and practice. Trans. Inst. N a v . Architects 111 (1969), 487-498; disc. 498-505.
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GADD,G. E. A method for calculating the flow over ship hulls. Trans. Inst. Nav. Architects 112 (1970), 335-345; disc. 345-351. GADD,G. E. ; HOGBEN, N. The determination of wave resistance from measurements of the wave pattern. Nut. Phys. Lab., Ship Div. Ship Rep. 70 (Nov. 1965), 40 pp. 10 figs. GERTLER, M. A reanalysis of the original test data for the Taylor Standard Series. David Taylor Model Basin Rep. 806 (1954), xiv 45 226 pp. GIESING, J. P. ; SMITH, A. M. 0. Potential flow about two-dimensional hydrofoils. J . Fluid Mech. 28 (1967), 113-129. GRAFF.W . ; KRACHT, A.; WEINBLUM, G. Some extensions of D. W. Taylor’s Standard Series. Trans. SOC.Nav. Architects Mar. Eng. 72 (1964). 374-396; disc. 3 9 6 4 0 3 . GRUNTFEST,R. A. Ship waves in uniformly accelerated motion. (Russian.) Prikl. Mat. Mekh. 29 (1965), 192-196. GRUNTFEST, R. A. ; NIKITIN, A. K. On wave resistance of a viscous fluid. (Russian.) I z v . Akad. Nauk SSSR, Mekh. Zhid. Gaza 1966, no. 4, 118-126. GUILLOTON, R . Contribution B l’ttude des carknes minces. Science et Industrie, Paris, 1939, vii 117 pp. GUILLOTON, R. A new method of calculating wave profiles and wave resistance of ships. Trans. Inst. Nav. Architects 82 (1940); 69-90; disc. 90-93. GUILLOTON, R. Evaluations approximatives concernant les profiles minces. Science et Industrie, Paris, 1946a, 20 pp. GUILLOTON, R. Further notes on the theoretical calculation of wave profiles and of the resistance of hulls. Trans. Inst. Nav. Architects 88 (1946b), 308-320; disc. 321-327. GUILLOTON, R. Streamlines on fine hulls. Trans. Inst. Nav. Architects 90 (1948), 48-60; disc. 60-63. GUILLOTON, R. Potential theory of wave resistance of ships with tables for its calculation. Trans. SOC.Nav. Architects Mar. Eng. 59 (1951a), 86-123; disc. 123-128. GUILLOTON, R. Rtflexions sur l’ttude thkorique des carhes. 4th Congr. Int. Mer, Ostende, 1951(b), Rapp. 1, 596-624. GUILLOTON, R. A note on the experimental determination of wave resistance. Trans. Inst. Nnv. Architects, 94 (1952), 343-356; disc. 356-362. GUILLOTON, R. Reflections on the theoretical study of ship hulls. SOC.Nav. Architects Mar. Eng. Tech. Res. Bull. 1-15 (1953), 21 pp. GUILLOTON, R. Compltments sur le potentiel lintarise avec surface libre appliquk a l’ktude des carknes. Bull. Ass. Tech. Mar. Aeronaut. 55 (1956), 337-376; disc. 377-383. GUILLOTON, R. The waves generated by a moving body. Trans. Inst. Nav. Architects 102 (1960a), 157-172; disc. 172-173. GUILLOTON, R. Les vagues de sillage. Bull. Ass. Tech. Mar. Aeronaut. 60 (1960b), 1-19. GUILLOTON, R. Examen critique des mCthodes d’Ctude thtorique des carknes de surface. Schiflstechnik 9(1962), 3-12. GUILLOTON, R. Mouvements liquides produits par les bateaux. Schiflstechnik 1 0 (1963), 8-16. GUILLOTON, R. L‘ttude thCorique du bateau en fluide parfait. Bull. Ass. Tech. Mar. Aeronaut. 64 (1964), 538-561 ; disc. 562-574. GUILLOTON, R. La pratique du calcul des isobares sur une c a r h e lintariste. Bull. Ass. Tech. Mar. Aeronaut. 65 (1965), 379-394; disc. 395-400. GUILLOTON, R. RCflexions sur les carknes de rCsistance minimum. Bull. Ass. Tech. Mar. Aeronaut. 66 (1966), 223-239; disc. 240-251. HASKIND, M. D. Translation of bodies under the free surface of a heavy fluid of finite depth. (Russian.) Prikl. Mat. Mekh. 9 (1945a), 67-78. Translated in N A C A Tech. Memo. 1345 (1952), 20 pp. HASKIND, M. D. Wave resistance of a solid in motion through a fluid of finite depth. (Russian.) Prikl. Mat. Mekh. 9 (1945b), 257-264.
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HASKIND, M. D. The oscillation of a ship in still water. (Russian.) Izv. Akad. Nauk S S S R , Otd. Tekh. Nauk 1946, 23-34. Translated in SOC.Nav. Architects Mar. Eng. Tech. Res. Bull. 1-12 (1953), pp. 45-60. HAVELOCK, T. H. (Havelock’s papers on hydrodynamics have been edited by C. Wigley and published under the title “ T h e Collected Papers of Sir Thomas Havelock on Hydrodynamics” by the Office of Naval Research, Washington, D.C., 1966. They will be referred to below under the abbreviation “ Coll. Papers.”) HAVELOCK, T . H. The propagation of groups of waves in dispersive media, with application to waves on water produced by a traveling disturbance. Proc. Roy. SOC.Ser. A 81 (1908), 398-430. HAVELOCK, T. H. The wave making resistance of ships: a theoretical and practical analysis. Proc. Roy. SOC.Ser. A 82 (1909), 276-300 = Coll. Papers, pp. 34-58. HAVELOCK, T. H. Ship resistance: a numerical analysis of the distribution of effective horsepower. Proc. Univ. Durham Phil. SOC.3 (1910a), 215-224 = Coll. Papers, pp. 59-68. HAVELOCK, T. H. The wave-making resistance of ships: a study of certain series of model experiments. Proc. Roy. SOC.Ser.A . 84 (1910b), 197-208 = Coll. Papers, pp. 69-80. HAVELOCK, T. H. Ship resistance: the wave making properties of certain travelling pressure disturbances. Proc. Roy. SOC.Ser.A. 89 (1914), 489-499 = Coll. Papers, pp. 94-104. HAVELOCK, T. H. The initial wave resistance of a moving surface pressure. Proc. Roy. SOC. Ser. A . 93 (1917a), 240-253 = Coll. Papers, pp. 105-118. HAVELOCK, T. H. Some cases of wave motion due to a submerged obstacle. Proc. Roy. SOC. Ser. A . 93 (1917b), 520-532 = Coll. Papers, pp. 119-131. HAVELOCK, T. H. Wave resistance: some cases of three-dimensional fluid motion. Proc. Roy. SOC.Ser. A . 95 (1919), 354-365 = Coll. Papers, pp. 146-157. HAVELOCK, T. H. The effect of shallow water on wave resistance. Proc. Roy. SOC.Ser. A 100 (1922), 499-505 = Coll. Papers, pp. 192-198. HAVELOCK, T . H. Studies in wave resistance: influence of the form of the water-plane section of the ship. Proc. Roy. SOC.Ser. A . 103 (1923), 571-585 = Coll. Papers, pp. 199-213. HAVELOCK, T . H. Studies in wave resistance: the effect of parallel middle body. Proc. Roy. SOC.Ser. A . 108 (1925a), 77-92 = Coll. Papers, pp. 214-229. HAVELOCK, T. H. Wave resistance: the effect of varying draught. Proc. Roy. SOC.Ser. A 108 (1925b), 582-591 = Coll. Papers, 230-239. HAVELOCK, T . H. Wave resistance: some cases of unsymmetrical forms. Proc. Roy SOC. Ser. A 110 (1926a), 233-241 = Coll. Papers, pp. 24C248. HAVELOCK, T. H. Some aspects of the theory of ship waves and wave resistance. North-East Coast Inst. Eng. Shipbuilders Trans. 42 (1926b), 71-86 = Coll. Papers, pp. 249-264. HAVELOCK, T . H., The method of images in some problems of surface waves. Proc. Roy. SOC.Ser A . 115 (1927), 268-280 = Coll. Papers, pp. 265-277. HAVELOCK, T. H. Wave resistance. Proc. Roy. SOC.Ser. A . 118 (1928a), 24-33 = Coll. Papers, pp. 278-287. HAVELOCK, T. H. The wave pattern of a doublet in a stream. Proc. Roy. SOC.Ser. A 121 (1928b), 515-523 = Coll. Papers, pp. 288-296. HAVELOCK, T. H. The vertical force on a cylinder submerged in a uniform stream. Proc. Roy. SOC.Ser. A 122 (1929a), 387-393 = Coll. Papers, pp. 297-303. HAVELOCK, T. H. Forced surface-waves on water. Phil. Mag. [7] 8 (1929b), 569-576 = Coll. Papers, pp. 304-311. HAVELOCK, T. H. The wave resistance of a spheroid. Proc. Roy. SOC.Ser. A 131 (1931a), 275-285 = Coll. Papers, pp. 312-322. HAVELOCK, T. H. The wave resistance of an ellipsoid. Proc. Roy. SOC.Ser. A 132 (1931b), 480-486 = Coll. Papers pp. 323-329.
The Wave Resistance of Ships
235
HAVELOCK, T. H. Ship waves: the calculation of wave profiles. Proc. Roy. SOC.Ser. A 135 (1932a), 1-13
=
Coll. Papers, pp. 347-359.
HAVELOCK, T. H. Ship waves: their variation with certain systematic changes of form. Proc. Roy. SOC.Ser. A 136 (1932b), 465-471 = Coll. Papers, pp. 360-366. HAVELOCK, T . H. The theory of wave resistance. Proc. Roy. SOC.Ser. A 138 (1932c), 339-348
= Coll.
Papers, pp. 367-376.
HAVELOCK, T. H. Wave patterns and wave resistance. Trans. Inst. Nav. Architects 76 (1934a), 430-442; disc. 442-446
= Coll.
Papers, pp. 377-389.
HAVELOCK, T H. The calculation of wave resistance. Proc. Roy. SOC.Ser. A . 144 (1934b), 514-521
= Coll.
Papers, pp. 390-397.
HAVELOCK, T. H. Ship waves: the relative efficiency of bow and stern. Proc. Roy. Soc. 149 (1935), 417-426
= Coll.
papers, pp. 398407.
HAVELOCK, T . H. Wave resistance: the mutual action of two bodies. Proc. Roy. Sac. Ser. A . 155 (1936a), 460-471 = Coll. Papers, pp. 408-419. HAVELOCK, T. H. The forces on a circular cylinder submerged in a uniform stream. Proc. Roy. SOC.Ser. A 157 (1936b), 526-534
= Coll.
Papers, pp. 420-428.
HAVELOCK, T. H. Note on the sinkage of a ship at low speeds. Z . Angezu. Math. Mech. 19 (1939), 202-205
= Coll.
Papers, pp. 458-461.
HAVELOCK, T. H. The approximate calculation of wave resistance at high speed. NorthEast Coast Inst. Eng. Shipbuilders Trans. 60 (1943), 47-58
= Coll.
Papers, pp. 500-511.
HAVELOCK, T. H. Some calculations of ship trim at high speeds. Int. Congr. Appl. Mech., Paris, 1946 = Coll. Papers, pp. 520-527.
HAVELOCK, T. H. Calculations illustrating the effect of boundary layer on wave resistance. Trans. Inst. Nav. Architects 90 (1948), 259-266; disc. 266-271 = Coll. Papers, pp. 528-535.
HAVELOCK, T. H. The wave resistance of a cylinder started from rest. Quart. J . Mech. Appl. Math. 2 (1949a), 325-334
= Coll.
Papers, pp. 536-544.
HAVELOCK, T . H. The resistance of a submerged, cylinder in accelerated motion. Quart. J . Mech. Appl. Math. 2 (1949b), 419-427 = Coll. Papers, pp. 545-553. HAVELOCK, T. H. The forces on a submerged spheroid moving in a circular path. Proc. Roy. SOC.Ser. A 201 (1950), 297-305
= Coll.
Papers, pp. 554-562.
HAVELOCK, T. H. Wave resistance theory and its application to ship problems. Trans. SOC. Nav. Architects Mar. Eng. 59 (1951), 13-24
=
Coll. Papers, pp. 563-574.
HAVELOCK, T. H. The moment on a submerged solid of revolution moving horizontally. Quart. J . Mech. Appl. Math. 5 (1952), 129-136 = Coll. Papers, pp. 575-582.
HAVELOCK, T . H. A note on wave resistance theory: transverse and diverging waves. Schiflstechnik 4 (1957), 64-65
=
Coll. Papers, pp. 615-616.
HAVELOCK, T. H. T h e forces on a submerged body moving under waves. Trans. Inst. Nav. Architects 96 (1954), 77-78.
HINTERTHAN, W. B. Report on geosim analysis according to Schoenherr line. David Taylor Model Basin Rep. 1064 (1956), 88 pp.
HOGBEN, N. The computing of wave resistance from a wave pattern by a matrix method. Nut. Phys. Lab., Ship Div. Ship Rep. 56 (Oct. 1964), 6 pp.
HOGBEN, N. An investigation of hovercraft wave-making. J . Roy. Aeronaut. Soc. 70 (1966), 32 1-329.
HOGBEN, N. Derivation of source arrays from measured wave patterns. 7th Symp. Nav. Hydrodyn., Rome, 1968, pp. 1557-1560.
HOGBEN, N. Automated analysis of wave patterns behind towed models. N u t . Phys. Lab., Ship Div. Ship Rep. 143 (July 1970), 27 pp.
HOGBEX, N. “Equivalent source arrays ” from wave patterns behind trawler type models. Trans. Inst. Nav. Arch. 113 (1971), 345-360; disc. 360-363.
236
John V . Wehausen
HOGBEN, N. Automated recording and analysis of wave patterns behind towed models. Trans. Inst. N a o . Architects 114 (1972), 127-150; disc. 150-153. HOGNER, E. Ueber die Theorie der von einem Schiff erzeugten Wellen und des Wellenwiderstandes. Proc. 1st Int. Congr. Appl. Mech., Delft, 1924, pp. 146-160 (plates XIII, XIV). HOGNER, E. On the theory of ship wave resistance. A r k . M a t . Astron. Fys. 21A, no. 7 (1928), 11 pp. HOGNER, E. Schiffsform und Wellenwiderstand. Hydromechanische Probleme des Schiffsantriebs, Hamburg, (1932a), pp. 99-1 14. HOGNER, E. Eine Interpolationsformel fur den Wellenwiderstand von Schiffen. Juhrb. Schiffbautech. Ges. 33 (1932b), 4 5 2 4 5 6 . HOGNER, E. Influence lines for the wave resistance of ships. I. Proc. R o y . SOC.Ser. A 155 (1936), 292-301. HOGNER, E. A complementary method for evaluating ship wave resistance. 7th Int. Conf. Ship Hydrodyn., Oslo, 1954, 9 pp. HSIUNG,C.-C.; WEHAUSEN, J. V. Michell resistance of Taylor’s Standard Series. 12th Int. Towing Tank Conf., Rome, 1969, Proc., pp. 176-180. HUANG, T . T . ; WONG,K. K. Disturbance induced by a pressure distribution moving over a free surface. J . Ship Res. 14 (1970), 195-203. HUDIMAC, A. A. Ship waves in a stratified ocean. J . Fluid Mech. 11 (1961), 229-243. IKEHATA, MITSUHISA. The second order theory of wave-making resistance. J . Zosen Kiokai 117 (1965), 39-57. IKEHATA, MITSUHISA. On experimental determination of wave-making resistance of a ship. Jap. Shipbld. Marine Eng. 4, no. 5, (1969), 5-14. IKEHATA, MITSUHISA; NOZAWA, KAZUO. Determination of wave-making resistance of a ship by the method of wave analysis. I, 11. (Japanese)J. SOCN a v . ArchitectsJup. 121 (1967), 62-71 ; 124 (1968), 37-49. Translated in Selec. Pup. 4, 1-27. INUI,TAKAO. Japanese developments on the theory of wave-making and wave resistance. 7th Int. Conf. Ship Hydrodyn., Oslo, 1954, 60 pp.; disc. 61-70. INUI,TAKAO. Wave-making resistance of ships travelling on a shallow water. Proc. 6th Jup. N u t . Congr. A p p l . Mech., 1956, pp. 357-360. INUI,TAKAO. Study on wave-making resistance of ships. SOC.N a v . Architects Jap., 60th Anniv. Ser. 2 (1957), 173-355. INUI,TAKAO, Wave-making resistance of ships. Trans. SOC.N a v . Architects M a r . Eng. 70 (1962), 282-326; disc. 326-353. ISAY,W. H. ; MUSCHNER, W. Zur nichtlinearisierten Theorie der Unterwassertragflugel. Schiflstechnik 13 (1966), 107-112. ISHII,MASAO.Semi-submerged ship form with minimum wave making resistance. (Japanese.) J . SOC.N a v . ArchitectsJap. 123 (1968), 1-12. JOOSEN, W. P. A. The velocity potential and wave resistance arising from the motion of a slender ship. Int. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 711-742; disc. 743-745. JOUKOWSKI (ZHUKOVSKII), N. E. On the concomitant wave. (Russian.) T r . O t d . F i z . Nauk Obshchest. Lyubitelei Estestvoznanzya, 14, vyp. 1, 3-13 (1909) = Sobranie Sochinenii, Vol. 11, pp. 712-725. Gos. Izdat. Tekh.-Teor. Lit., Moscow, 1949. KAJITANI, HISASHI.The second order treatment of ship surface condition in the theory of wave-making resistance of ships. (Japanese). J . Zosen Kiokai 118 (1965), 84-107. KARP,S.; KOTIK,J.; LURYE,J. On the problem of minimum wave resistance for struts and strut-like dipole distributions. 3rd Symp. N a v . Hydrodyn., Scheveningen, 1960, pp. 56-110; disc. 110-119.
The Wave Resistance of Ships
237
KELDYSH, M. V.; SEDOV,L. I., The theory of wave resistance in a channel of finite depth. (Russian.) T r . Konf. Teorii Volnovogo Soprotivleniya, Moscow, 1936, pp. 143-152. See Thomson, Sir William. KELVIN,LORD. KHASKIND, M. D. See Haskind. KIM, W. D. Nonlinear freee-surface effects on a submerged sphere. J. Hydronaut. 3 (1969), 29-37. KIRSCH,M. Ein Beitrag zur Berechnung des Wellenwiderstandes im Kanal. Schzflstechnik 9 (1962), 123-127. KIRSCH,M. Shallow water and channel effects on wave resistance. J. Ship Res. 10 (1966), 164-181. ; IKEHATA, MITSUHISA. On the source distribution representing KOBAYASHI, MASANORI ship hull form. J . SOC.N a v . ArchitectsJap. 128 (1970), 1-9. KOBUS,H. E. Examination of Egger’s relationship between transverse wave profiles and wave resistance. J. Ship Res. 11 (1967), 240-256. KOCHIN,N. E. On the wave resistance and lift of bodies submerged in a fluid. (Russian.) T r . Konf. Teorii Volnovogo Soprotivleniya, Moscow, 1936, pp. 65-134= Sobranie Sochinenii,Vol. 11, pp. 105-182, Akad. Nauk SSSR, 1949. Translated in SOC.Nav. Architects M a r . Eng. Tech. Res. Bull. 1-8 (1951), 126 pp. KOLBERG, F. Der Wellenwiderstand von Schiffen auf flachem Wasser. Ing.-Arch. 27 (1959a), 268-275. F. Untersuchung des Wellenwiderstandes von Schiffen auf flachem Wasser bei KOLBERG, gleichformig scherender Grundstromung. 2. Angew. Math. Mech. 39 (1959b), 253-279. KOLBERG, F. Der Stromungseinfluss auf den Wellenwiderstand von Schiffen. Ing. Arch. 30 (1961), 123-140. KOLBERG, F. The motion of a ship in restricted water. Int. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 617-696. KOLBERG, F. Zur Theorie der Bewegung eines Schiffes bei begrenzten Fahrwasserverhaltnissen. Forschungsber. Landes Nordrhein- Westfalen 1596 (1966), 45 pp. W. R. Calculation of the wave profile and wave KORVIN-KROUKOVSKY, B, V.; JACOBS, making resistance of ships of normal commercial form by Guilloton’s method and comparison with experimental data. Soc. N a v . Architects Mar. Eng. Tech. Res. Bull. 1-16 (Dec. 1954), 63 21 70 pp. A. A. Teoriya korabel’nykh voln i volnovogo soprotivleniya. (The theory of KOSTYUKOV, ship waves and wave resistance.) Gos. Soyuz. Izdat. Sudostr. Promyshl., Leningrad, 1959. 312 pp. KOSTYUKOV, A. A. “ Theory of Ship Waves and Wave Resistance.” Effective Communications Inc., Iowa City. 1968, vi 400 pp. KOTIK,J. Some aspects of the problem of minimum wave-resistance. Int. Sem. Theoret. Wave Resistance, A n n Arbor, 1963, pp. 955-983. KOTIK,J. ;Mangulis, V. Comparison of two approximate dipole distributions for alenticular cylinder in a semi-infinite fluid, J . Math. Phys. (Cambridge, Mass.) 41 (1962), 280-290. KOTIK,J. ; MORGAN, R. The uniqueness problem for wave resistance calculated from singularity distributions which are exact at zero Froude number. J. Ship Res. 13 (1969), 61-68. KOTIK,J.; NEWMAN, D. J. A sequence of submerged dipole distributions whose wave resistance tends to zer0.J. Math. Mech. 13 (1964), 693-700; erratum 14 (1965), 141. KOTIK,J. ; THOMSEN, P. Various wave resistance theories for slender ships. Schiflstechnik 10 (1963), 178-186.
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John V . Wehausen
KRACHT, A. Lineartheoretische Abhandlung iiber die optimale Verringerung des Wellenwiderstandes gegebener Schiffsformen durch einen Wulst in symmetrischer oder asymmetrischer Anordnung. Schiflstechnik 15 (1968), 39-44, 61-70. KRACHT, A. A theoretical contribution to the wave resistance problem of ship-bulb combinations: verification of the negativeness of the interaction term. J . Ship Res. 14 (1970), 1-7. LACKENBY, H. An investigation into the nature and interdependence of the components of ship resistance. (With Appendices by E. F. Relf and J. P. Shearer.) Trans. Inst. N a v . Architects 107 (1965), 474-486; disc. 486-501. LANDWEBER, L. ; Tzou, K. T. S. Study of Eggers’ method for the determination of wavemaking resistance. J . Ship Res. 12 (1968), 213-230. LANDWEBER, L. ; Wu, JIN.The determination of the viscous drag of submerged and floating bodies by wake surveys. J . Ship Res. 7 , no. 1, 1-6 (1963). LAVRENT’EV, V. M. Influence of the boundary layer on the wave resistance of a ship. Dokl. Akad. Nuuk S S S R [N.S.] 80 (1951), 857-860. Translated in Taylor Model Basin Transl. 245 (1952), 4, pp. LEE,A. Y . C. Source generated ships of minimum theoretical wave resistance. Trans. Roy. Inst. N a v . Arch. 111 (1969), 449-476; disc. 476-485. LIGHTHILL,M. J. A new approach to thin aerofoil theory. Aeronaut. Quart. 3 (1951), 193-210. LIN, WEN-CHIN;PAULLING, J. R.; WEHAUSEN, J. V. Experiment data for two ships of ‘‘minimum ” resistance. 5th Symp. N a v . Hydrodyn., Bergen,1964, pp. 1047-1060; disc. 1060-1064. W. C. ; WEHAUSEN, J. V. Ships of minimum total resistance. LIN, WEN-CHIN;WEBSTER, Int. Sem. Theoret. Wave Resistance, Ann Arbor, 1963, pp. 907-948; disc. 949-953. LUNDE, J. K. Wave resistance calculations at high speeds. Trans. Inst. N a v . Architects 91 (1949), 182-190; disc. 190-196. LUNDE, J. K. On the linearized theory of wave resistance for displacement ships in steady and accelerated motion. Trans. SOC.N a v . Architects Mar. Eng. 59 (1951a), 25-76; disc. 76-85. LUNDE,J. K. On the linearized theory of wave resistance for a pressure distribution moving at constant speed of advance on the surface of deep or shallow water. Norg. Tek. Hegskole, Trondheim, Skipsmodelltankens Medd. 8 (1951b), 48 pp. LUNDE,J. K. The linearized theory of wave resistance and its application to ship-shaped bodies in motion on the surface of a deep previously undisturbed fluid. SOC.N a v . Architects Mar. Eng. Tech. Res. Bull. 1-18 (1957), 70 pp. LUNDE,J. K. Wave resistance. Proc. 12th Int. Towing Tank Conf., Rome, 1969, pp. 66-79. LURYE,J. R. Interaction of free-surface waves with viscous wakes. Phys. Fluids 11 (1968), 261-265. MAKOTO, OHKUSU. Wave analysis of simple hull form-effect of B/L. J . SOC.N a v . Architects Jap. 126 (1969), 25-34. MARUO, HAJIME.Modern developments of the theory of wave-making resistance in the non-uniform motion. SOC.N a v . Architects Jap. 60th Anniv. Ser. 2 (1957), 1-82. MARUO, HAJIME. Calculation of the wave resistance of ships, the draught of which is as small as the beam. J . Zosen Kiokai 112 (1962), 21-37. MARUO, HAJIME.Experiments on theoretical ship forms of least wave resistance. Int. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1011-1030. MARUO, HAJIME.Problems relating to the ship form of minimum wave resistance. 5th Symp. N a v . Hydrodyn., Bergen, 1964, pp. 1019-1046.
The W a v e Resistance of Ships
239
MARUO, HAJIME. A note on the higher-order theory of thin ships. Bull. Fac. Eng. Yokohama Nut. Univ. 15 (1966), 1-21. MARUO, HAJIME.High- and low-aspect ratio approximation of planing surfaces. Scht'stechnik 14 (1967), 57-64. MARUO, HAJIME. Theory and application of semi-submerged ships of minimum resistance. Jap. Shipbld. Mar. Eng. 4, no. 1, 5-15 (1969). MARUO, HAJIME.Application of the wave resistance theory to the ship form design. Korea-Japan Seminar Ship Hydrodyn., Seoul, 1970, pp. 2-1-2-22 14 figs. MARUO, H A J I M E ; BESSHO, MASATOSHI. Ships of minimum wave resistance. J . Zosen Kiokai 114 (1963), 9-23. Translated in Selec. Pap. 3, 1-18. MARUO,HAJIME;IKEHATA, MITSUHISA. Determination of wave-making resistance of a ship by the method of wave analysis. 111. (Japanese.) J . SOC. N a v . ArchitectsJap. 125
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(1969), 1-8. MARUO, HAJIME,; ISHII, MASAO.Semi-submerged ship with minimum wave making resistance. (Japanese.) J . Zosen Kiokai 116 (1964), 22-29. MICHELL, J. H. The wave resistance of a ship. Phil. Mag. [5] 45 (1898), 106-123="The Collected Mathematical Works of J. H. and A. G. M. Michell," pp. 124-141. Noordhoff, Groningen, 1964. MICHELSEN, F. C. Expressions for the evaluation of wave-resistance for polynomial center plane singularity distributions. Int. Sem. Theoret. Wave Resistance, Ann Arbor, 1963, pp. 857-889. MICHELSEN, F. C. Asymptotic approximations of Michell's integral for high and low speeds. Schi'stechnik 13 (1966), 33-38. MICHELSEN, F. C. The application of Gegenbauer polynomials to the Michell integral. J . Ship Res. 16 (1972), 6@65. MICHELSEN, F. C. ; KIM,HUNCHOL.On the wave resistance of a thin ship. Schi'stechnik 14 (1967), 46-49. MILGRAM, J. H. The effect of a wake on the wave resistance of a ship. J . Ship Res. 13
(1969), 69-71. MORAN, D. D. ; LANDWEBER, L. A longitudinal-cut method for computing the wave resistance of a ship model in a towing tank. 16th Amer. Towing Tank Conf., Sao Paulo,
1971. MORAN, D. D.; LANDWEBER, L. A longitudinal-cut method for determining wave resistance. J . Ship Res. 16 (1972), 21-40. NAKATAKE, KUNIHARU ; FUKUCHI, NOBUYOSHI. On the source distribution which represents ship form. (Japanese.)J. SOC. N a v . Architects West Jap. (Seibu Zosen K a i ) 34 (1967),
1-24. NEWMAN, J. N. The determination of wave resistance from wave measurements along a parallel cut. Int. Sem. Theoret. Wave Resistance,'Ann Arbor, 1963, pp. 351-376; disc.
377-379. NEWMAN, J. N. The asymptotic approximation of Michell's integral for high speed. J . Ship Res. 8, no. 1, 10-14 (1964). NEWMAN, J. N. Applications of slender-body theory in ship hydrodynamics. Annu. Rev. Fluid Mech. 2 (1970), 67-94. NEWMAN, J. N. Third-order interactions in Kelvin ship-wave systems. J . Ship Res. 15
(1971), 1-10. NEWMAN, J. N.; POOLE,F. A. P. The wave resistance of a moving pressure distribution in a canal. Schi'stechnik 9 (1962), 21-26. NIKITIN,A. K. On ship waves on the surface of a viscous fluid of infinite depth. (Russian.) Prikl. Mat. Mekh. 29 (1965). 186-191.
John V . Wehausen
240
OGILVIE, T . F. Nonlinear high-Froude-number free-surface problems. J . Eng. Math. 1 (1967), 215-235. OGILVIE, T . F. Wave resistance: the low speed limit. Univ. Mich. Dept. Naval Architecture Mar. Eng. Rep. 002 (Aug. 1968), iii 29 pp. OGILVIE, T . F. Singular perturbation problems in ship hydrodynamics. 8th Symp. N a v . Hydrodyn., Pasadena, 1970, pp. 663-806. OGIWARA, S E I K; MARUO, ~ HAJIME; IKEHATA, MITSUHISA. On the method for calculating the approximate solution of source distribution over the hull surface. (Japanese.) J . SOC.N a v . ArchitectsJap. 126 (1969), 1-10. Translated in Selec. Pap. 7 , 1-11. PALLADINA, 0. M. Teoriya korablya. Ukazatel' literatury na russkom yazyke za 17741954 gg. (Theory of the Ship. Guide to the literature in the Russian language for the years 1774-1954). Gos. Soyuz, Izdat. Sudostroi. Promysh., Leningrad, 1957. PAVLENKO, G. E. The ship of least resistance. (Russian.) T r . Vsesoyuz. Nauch. Znzh.Tekh. Obshch. Sudostroen. 2, no. 3 (1937), 28-62. PERZHNYANKO, E. A. The problem of the wave resistance of a body moving in a circle. (Russian.) Dokl. Akad. Nauk SSSR 130 (1960), 514-516. PETERS, A. S.; STOKER, J. J. The motion of a ship, as a floating rigid body, in a seaway. Comm. Pure Appl. Math. 10 (1957), 399-490. PIEN,PAOC. The application of wavemaking resistance theory to the design of ship hulls with low total resistance. 5th Symp. N a v . Hydrodyn., Bergen, 1964, pp. 1109-1137; disc. 1173-1153. PIEN,PAOC.; MOORE, W. L. Theoretical and experimental study of wave-making resistance of ships. Znt. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 131-182; disc. 183-188. PLESSET,M. S.; WU, T . YAO-TSU. Water waves generated by thin ships. J . Ship Res. 4, no. 2, 25-36 (1960). ROY,J.-F.; MILLARD, A. Une mkthode de mesure de la rksistance des vagues. Bull. Ass. Tech. M a r . Aeronaut. 71 (1971), 481-495; disc. 496-501. SABUNCU, T. The theoretical wave resistance of a ship travelling under interfacial wave conditions. Norw. Ship. Model Exp. Tank, Trondheim, Publ. 63 (1961), ii 124 pp. T. Some predictions of the value of the wave resistance and moment concerning SABUNCU, the Rankine solid under interfacial wave conditions. Norw. Ship Model Exp. Tank, Trondheim, Publ. 65 (1962a), ii ii 38 pp. SABUNCU, T. Gemilerin dalga direnci teorisi. (The theory of wave resistance of ships.) Istanbul Tek. Univ. Gemi Enstitiisii Biil. 12 (1962b), viii 77 7 pp. SALVESEN, N. Second-order wave theory for submerged two-dimensional bodies. 6th Symp. N a v . Hydrodyn., Washington, D . C . , 1966, pp. 595-628; disc. 629-636. SALVESEN, N. On higher-order wave theory for submerged two-dimensional bodies. J . Fluid Mech. 38 (1969), 415-432. SHARMA, S. D. Uber Dipolverteilungen fur getauchte Rotationskorper geringsten Wellenwiderstandes. Schiffstechnik 9 (1962), 86-96 ; Disk. 96. SHARMA, S. D. A comparison of the calculated and measured free-wave spectrum of an inuid in steady motion, Znt. Sem. T?ieoret. Wave Resistance, Ann Arbor, 1963, pp. 201-257; disc. 259-270. SHARMA, S. D. Untersuchungen uber den Zahigkeits- und Wellenwiderstand mit besonderer Beriicksichtigung ihrer Wechselwirkung. Inst. Schiffbau Univ. Hamburg, Ber. 138 (Dec. 1964), 491 pp. S. D. Zur Problematik der Aufteilung des Schiffswiderstandes in zahigkeitsSHARMA, und wellenbedingte Anteile. Jahrb. Schiffbautech. Ges. 59 (1965), 458-504; Erort.,
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SHARMA, S. D. An attempted application of wave analysis techniques to achieve bow-wave reduction, 6th Symp. Nav. Hydrodyn., Washington, D.C., 1966, pp. 731-766; disc. 767-773. SHARMA, S. D. Der Wellenwiderstand eines flach getauchten Korpers und seine Beeinflussung durch einen aus dem Wasser herausragenden Turmaufbau. Schiflstechnik 15 (1968), 88-98. SHARMA, S. D. Some results concerning the wavemaking of a thin ship. J . Ship Res. 13 (1969), 72-81. SHARMA, S. D.; NAEGLE, J. N. Optimization of bow bulb configurations on the basis of model wave profile measurements. Univ. Mich., Dept. Nav. Architecture Rep. 104 (Dec. 1970), ii 55 pp. SHEARER, J. R. A preliminary investigation of the discrepancies between the calculated and measured wavemaking of hull forms. North-East Coast Znst. Eng. Shipbld., Trans. 67 (1951), 43-68; disc. D21-D34. SHEARER, J. R.; CROSS, J. J. The experimental determination of the components of ship resistance for a mathematical model. Trans. Znst. Nav. Architects 107 (1965), 4 5 9 4 7 3 ; disc. 486-561. SHEARER, J. R.; STEELE, B. N. Some aspects of the resistance of full form ships. Trans. Znst. Nac. Architects 112 (1970), 4 6 5 4 7 9 ; disc. 4 7 9 4 8 6 . SHEBALOV, A. N. On the forces acting on a body of arbitrary form in unsteady motion under a free surface. (Russian.) Zzv. Akad. Nauk SSSR, Mekh. Mashinostr. 1962, no. 2, 38-47. SHEBALOV, A. N. Theory of wave resistance of a ship in unsteady motion on calm water. (Russian.) Tr. Leningrad. Korablestroi. Znst. 52 (1966), 209-220. SHKURKINA, 2. M. Determination of forces acting on a sphere under nonsteady motion along a circular path. (Russian.) Vestn. Mosk. Univ. Ser. Z . Mat. Mekh. 21 (1966), no. 3, 98-109. SHOR,S. W. W. Trial calculation of a hull form of decreased wave resistance by the method of steepest descent. Int. Sem. Theoret. Wave Resistance, Ann Arbor, 1963, pp. 453-464. SIZOV, V. G. On the theory of the wave resistance of a ship on calm water. (Russian.) Z z z . . Akad. Nauk SSSR. Otd. Tekhn. Nauk. Mekh. Mashinostr. 1961, no. 1, 75-85. SMORODIN, A. I. On the application of an asymptotic method for the analysis of waves in the unsteady motion of a source. (Russian.) Prikl. Mat. Mekh. 29 (1965), 62-69. SRETENSKII, L. N. Sur un probkme de minimum dans la thkorie du navire. C. R . (Dokl.) Acad. Sci. URSS [N.S.]3 (1935), 247-248. SRETENSKII, L. N. On the wave-making resistance of a ship moving along in a canal. Phil. Mag. [7] 22 (1936), 1005-1013. SRETENSKII, L. N. A theoretical investigation on wave resistance. (Russian.) Tr. Tsent. Aero-Gidrodinam. Znst., vyp 319 (1937), 56 pp. SRETENSKII, L. N. On the theory of wave resistance. (Russian.) Tr. Tsent. Aero-Gidrodinam. Znst., vyp. 458 (1939), 28 pp. SRETENSKII, L. N. Computation of the tangential forces of wave resistance of a sphere moving on a circular path. (Russian.) Akad. Nauk SSSR, Tr. Morsk. Gidrofiz. Znst. 11 (1957a), 3-17. SRETENSKII, L. N. Sur la resistance due aux vagues d’un fluide visqueux. Proc. Symp. Behaviour Ships Seaway, Wugeningen, 1957b, pp. 729-733. SRETENSKII, L. N. Wave resistance of a ship in the presence of internal waves. (Russian.) Izv. Akad. Nauk SSSR, Otd. Tekhn. Nauk, Mekh. Mashinostr. 1959, no. 1, 56-63.
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STOKER, J. J. Water Waves. The Mathematical Theory with Applications. Wiley (Interscience), New York, 1957, xwiii 567 pp. TANAKA, HIRAKU,; ADACHIHIROYUKI; OMATA,SADAO. Study on wave analysis by use of INUID geosims. (Japanese.) J . SOC.N a v . Architects Jap. 128 (1970), 19-29. TANAKA, HIRAKU; YAMAZAKI, YOSHITSUGU; IENAGA, ITSUO;ADACHI,HIROWKI ; OGURA, MICHIHITO; OMATA, SADAO. Some application of the wave analysis on the geosim-models and their actual ship. (Japanese.) J. SOC.N a v . Architects l a p . 126 (1969). 11-24. Translated in Selec. Pap. 7 , 22-38.
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TATINCLAUX, J.-C. Effect of a rotational wake on the wavemaking resistance of an ogive. J. Ship Res. 14 (1970), 84-99. THOMSEN, CI.-PETER. Ein Ansatz zum Entwurf von Vorschiffen geringen Wellenwiderstandes. Schiffsstechnik 17 (1970), 80-90. THOMSON, SIR WILLIAM(Baron Kelvin). On stationary waves in flowing water. 111. Phil. Mag. [S] 22 (1886), 517-530=Math. Phys. Pap. 4, 282-295. TIMMAN, R. The wave pattern of a moving ship. Simon Stevin 35 (1961/62), 53-67. TIMMAN, R. ; VOSSERS, G. A solution of the minimum wave resistance problem. 3rd Symp. N a v . Hydrodyn., Scheveningen, 1960, pp. 48-55. TUCK, E. 0. The steady motion of a slender ship. Dissertation, Univ. of Cambridge, 1963a, 190 pp. TUCK, E. 0. On Vossers’ integral. Int. Sem. Theoret. Wave Resistance, Ann Arbor, 1963b, pp. 697-706; disc. 707-710. TUCK, E. 0. A systematic asymptotic expansion procedure for slender ships. J. Ship Res. 8, no. 1 , 15-23 (1964a). TUCK,E. 0 . On line distributions of Kelvin sources. J . Ship. Res. 8, no. 2, 45-52 (1964). TUCK, E. 0. The effect of non-linearity at the free surface on flow past a submerged cylinder. J . Fluid Mech. 22 (1965), 401-414. TUCK, E. 0. Shallow-water flows past slender bodies. J. Fluid Mech. 26 (1966), 81-95. TUCK, E. 0. Sinkage and trim in shallow water of finite width. Schifstechnik 14 (1967), 92-94. TUCK, E. 0.; COLLINS,J. I . ; WELLS,W. H. On ship wave patterns and their spectra. J . Ship Res. 15 (1971), 11-21. TULIN, M. P. The separation of viscous drag and wave drag by means of the wake survey. David W . Taylor Model Basin Rep.772 (1951), 6 pp. Tzou, K. T. S.; LANDWEBER, L. Determination of the viscous drag of a ship model. J . Ship Res. 12 (1968), 105-115. UENO,KEIZO; NAGAMATSU, TETSURO. Effect of restricted water on wave-making resistance. J. SOC.N a v . Architects West Jap. 41 (Mar. 1971), 1-18. URSELL,F. On Kelvin’s ship-wave pattern. J . Fluid Mech. 8 (1960), 418-431. USPENSKII, P. N. On the wave resistance of a ship in the presence of internal waves (under conditions of finite depth). (Russian.) Akad. Nauk S S S R , Tr. Morsk. Wrofiz. Inst. 18 (1959), 68-84. VOSSERS, G. Some applications of the slender body theory in ship hydrodynamics. Dissertation (proefschrift), Technische Hogeschool te Delft, 1962a, v 96 pp. VOSSERS. G. Wave resistance of a slender ship. Schiffstechnik 9 (1962b), 73-78. WANG,D. P. ; RISPIN,P. Three-dimensional planing at high Froude number. J . Ship Res. 15 (1971), 221-230. WARD,L. W. The XY method of determination of ship wave resistance from the wave pattern. Int. Sem. Theoret. Wave Resistance, Ann Arbor, 1963, pp. 381410; disc. 411-414. WARD,L. W. Experimental Determination of Ship Wave Resistance from the Wave Pattern, Webb Inst. Naval Arch., Glen Cove, New York, 1964, viii 68 pp.
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WARD,L. W.; SNYDER, J. D . 111. Forces due to Gravity Water Waves on a Long Vertical Circular Cylinder. Webb Inst. Naval Arch., Glen Cove, New York, 1968. v -t 36 pp. WARREN,F. W. G . A stationary-phase approximation to the ship-wave pattern. J . Fluid Mech. 10 (1961), 584-592. WARREN,F. W. G.; MacKINNON, R. F. A problem of gravity wave drag at an interface. J . Fluid Mech. 34 (1968), 263-272. WEBSTER, W. C. T h e effect of surface tension on ship wave resistance. Dissertation, Univ. of Calif., Berkeley, 1966. ii 114 pp. T. T. Study of the boundary layer on ship forms. J . Ship Res. WEBSTER, W. C . ; HUANG, 14 (1970), 153-167. WEBSTER, W. C . ; WEHAUSEN, J. V. Schiffe geringsten Wellenwiderstandes mit vorgegebenem Hinterschiff. Schtffstechnik 9 (1962), 62-67; Disk. 67-68. WEHAUSEN, JOHNV. Wave resistance of thin ships. Symp. N a v . Hydrodyn., Washington, D . C . , 1956, pp. 109-133; disc. 133-137. WEHAUSEN, JOHNV. An approach to thin-ship theory. Int. Sem. Theoret. Wave Resistance, Ann Arbor, 1963, pp. 819-852; disc. 853-855. WEHAUSEN, JOHN V. Effect of the initial acceleration upon the wave resistance of ship models. J . Ship Res. 7,no. 3, 38-50 (1964). WEHAUSEN, JOHN,V. Use of Lagrangian coordinates for ship wave resistance (first- and second-order thin-ship theory). J . Ship Res. 13 (1969), 12-22. WEHAUSEN, J. V.; LAITONE, E. V. Surface waves. “Encyclopedia of Physics,” Vol. IX, pp. 446-778. Springer-Verlag, Berlin, 1960. WEINBLUM, G. Schiffe geringsten Widerstands. Proc. 3rd Int. Congr. Appl. Mech., Stockholm, 1930a, pp. 449-458. WEINBLUM, G. Anwendungen der Michellschen Widerstandstheorie. Jahrb. Schzzbautech. Ges. 31 (1930b), 389-436; Erort. 436-440. WEINBLUM, G. Uber die Berechnung des wellenbildenden Widerstandes von Schiffen, insbesondere die Hognersche Formel. 2. Angew. Math. Mech. 10 (1930c), 453466. G., Hohle oder gerade Wasserlinien? Hydromechanische Probleme des SchiffsWEINBLUM antriebs, Hamburg, 1932a, pp. 115-131, 417419. WEINBLUM, G. Schiffsform und Wellenwiderstand. Jahrb. Schiffbautech. Ges. 33 (1932b), 419451 ; Erort. 456460. WEINBLUM, G. Untersuchungen uber den Wellenwiderstand volliger Schiffsformen. Jahrb. Schiffbautech Ges. 35 (1934), 164-192. WEINBLUM, G. Widerstandsuntersuchungen an scharfen Schiffsformen. Schiffbau 36 (1935), 355-359, 408-3 14. WEINBLUM, G. Rotationskorper geringsten Wellenwiderstandes. 1ng.-Arch. 7 (1936a), 104-1 17. G.Die Theorie der Wulstschiffe. Schiffbau 37 (1936b), 55-65. WEINBLUM, WEINBLUM, G. Beitrag zur Ausbildung volligerer Schiffsformen. Schzybau 37 (1936c), 285-292. WEINBLUM, G . Wellenwiderstand auf beschranktem Wasser. Jahrb. Schzffbautech. Ges. 39 (1938). 266-289; Erort. 289-291. WEINBLUM, G. Schiffsform und Widerstand. Schzffbau 40 (1939), 27-23, 46-51, 66-70. WEINBLUM, G. Analysis of wave resistance. David W . Taylor Model Basin Rep. 710 (1950), 102 pp. WEINBLUM, G. T h e wave resistance of bodies of revolution. (Appendix I1 by J. Blum.) David W . Taylor Model Basin Rep. 758 (1951), 58 pp. WEINBLUM, G. A systematic evaluation of Michell’s integral. David W . Taylor Model Basin Rep. 886 (1955), 59 pp.
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WEINBLUM, G. Ein Verfahren zur Auswertung des Wellenwiderstandes vereinfachter Schiffsformen. Schzflstechnik 3, 278-287 (1956). WEINBLUM, G. Applications of wave resistance theory to problems of ship design. Trans. Inst. Eng. Shipbuilders Scotland 102, 119-152 (1959); disc. 153-163. WEINBLUM, G. On problems of wave resistance research. Int. Sem. Theoret. W a v e Resistance, A n n Arbor, 1963, pp. 1-44; disc. 4 5 4 9 . WEINBLUM, G. Schiffe geringsten Wellenwiderstandes. Schiflstechnik 12 (1965), 131-136. G. Uber die Unterteilung des Schiffswiderstandes. Schifl Hafen 22 (1970), WEINBLUM, 807-81 2. WEINBLUM, G. ;AMTSBERG, H. ; BOCK,W. Versuche iiber den Wellenwiderstand getauchter Rotationskorper. Schzffbau 37 (1936), 411419. Translated in David Taylor Model Basin Rep. T-234 (1950), 22 pp. G. P.; KENDRICK, J. J. ; TODD, M. A. Investigation of wave effects produced by WEINBLUM, a thin body-TMB Model 4125. David Taylor W. Model Basin Rep. 840 (1952), 14 pp. WEINBLUM, G. ; SCHUSTER, S. ; BOES,CHR.; BHATTACHARYYA, R. Untersuchungen iiber den Widerstand einer systematisch entwickelten Modellfamilie. Jahrb. Schiflbautech. Ges. 56 (1962), 296-319; Erort. 320-324. G. ; WUSTRAU, D. ; VOSSERS,G. Schiffe geringsten Widerstandes. Jahrb. WEINBLUM, Schiflbautech. Ges. 51 (1957), 175-204; Erort. 205-214. WIGLEY, W. C. S. Ship wave resistance. A comparison of mathematical theory with experimental results. I , 11. Trans. Inst. N a v . Architects 68 (1926), 124-137 (plates X, XI); disc. 137-141 ; 69 (1927), 191-196 (plate XVIII); disc. 196-210. WIGLEY,W. C. S. Ship wave resistance. Some further comparisons of mathematical theory and experiment result. Trans. Inst. N a v . Architects 72 (1930a), 216-224 (plates XXIV,XXV); disc. 224-228. WIGLEY,W. C. S. Ship wave resistance. Proc. 3rd Znt. Congr. Appl. Mech., Stockholm, 1930b, vol. 1, pp. 58-73; disc. 73. WIGLEY,W. C. S. Ship wave resistance. An examination of the speeds of maximum and minimum resistance in practice and in theory. North-East Coast Inst. Eng. Shipbuilders Trans. 47 (1931), 153-180 (plates 11-VI); disc. 181-196 (pl. VII). WIGLEY,W. C. S. A note on ship wave resistance. Hydromechanische Probleme des Schiffsantriebs, Hamburg, 1932, pp. 132-138. WIGLEV, W. C. S., A comparison of experiment and calculated wave-profiles and waveresistances for a form having parabolic waterlines. Proc. Roy. SOC.Ser. A 144 (1934), 144-159 (4 plates). W I G L E Y ,C. ~ . S. Ship wave-resistance. Progress since 1930. Trans. Inst. N a v . Architects 77 (1935), 223-236 (plates XXVI, XXVII); disc. 237-244. WIGLEY, W. C. S. The theory of the bulbous bow and its practical application. North-East Coast Inst. Eng. Shipbuilders, Trans. 52 (1936), 65-88 (plate I). WIGLEY,W. C . S. Effects of viscosity on the wave-making of ships. Trans. Inst. Engr. Shipbuilders Scotland 81 (1938), 187-208 (1 plate); disc. 208-215. WIGLEY,W. C. S . The wave resistance of ships: a comparison between calculation and measurement for a series of forms. Congr2s Znt. Zng. N a v . , Li2ge, 1939, pp. 174-190. WIGLEY, W. C. S. The analysis of ship wave resistance into components depending on features of the form. Trans. Liverpool Eng. SOC.61 (1940), 2-25; disc. 26-35. WIGLEY, W. C. S. Calculated and measured wave resistance of a series of forms defined algebraically, the prismatic coefficient and angle of entrance being varied independently. Trans. Inst. N a v . Architects 84 (1942), 52-71 ; disc. 72-74. WIGLEY,W. C. S. Comparison of calculated and measured wave resistance for a series of forms not symmetrical fore and aft. Trans. Inst. N a v . Architects 86 (1944), 41-56; disc. 57-60.
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WIGLEY, W. C. S. L’Ctat actuel des calculs de rCsistance de vagues. Bull. Ass. Tech. M a r . Aeronaut. 48 (1949), 533-564; disc. 565-587. WIGLET,W. C. S. Water forces on submerged bodies in motion. Trans. Znst. N a v . Architects 95 (1953), 268-274; disc. 274-279. WIGLEY,W. C. S. Possible developments in calculation of wave resistance of ships. Schifstechnik 3, 17-18 (1955). WIGLEY,W. C. S. The effective virtual mass of a spheroid moving near the free surface of a fluid. Actes 92me Congr. Znt. Mec. A p p l . , Bruxelles, 1957, Vol. 1, pp. 203-206= Schifstechnik 4 (1957), 65-67. WIGLEY,W. C. S. The effect of viscosity on wave resistance. Schiffstechnik 9 (1962), 69-71 ; disc 71-72. WIGLEY,W. C. S. Effects of viscosity on wave resistance. Znt. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1293-1310. WIGLEY, W. C. S. A note on wave resistance in a viscous fluid.Schifstechnik 14 (1967), 10. WIGLEY, W. C. S.; LUNDE,J. K. Calculated and observed wave resistances for a series of forms of fuller midsection. Trans. Znst. N a v . Architects 90 (1948), 92-104; disc. 104-110. Wu, J I N . The separation of viscous from wave-making drag of ship forms. J . Ship Res. 6, no. 1 , 26-39 (1962). Wu, T . YAO-TSU. Interaction between ship waves and boundary layer. Znt. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1261-1287; disc. 1288-1291. Wu, T. YAO-TSU. A singular perturbation theory for nonlinear free surface flow problems. Znt. Shipbuilding Progr. 14 (1967) 88-97. YEUNG,R. W. Sinkage and trim in first-order thin-ship theory. J . Ship Res. 16 (1972), 47-59. YIM,BOHYUN.On ships with zero and small wave resistance. Znt. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1031-1075 ; disc. 1076-1079. YIM,BOHYUN.Some recent developments in theory of bulbous ships. 5th S y m p . Naw. Hydrodyn., Bergen, 1964, pp. 1065-1098. YIM,BOHYUN. Analyses on bow waves and stern waves and some small-wave-ship singularity systems. 6th Symp. N a v . Hydrodyn., Washington, D.C., 1966, pp. 681-698; disc. 699-701. YIM,BOHYUN.Higher order wave theory of ships. J . Ship Res. 12 (1968), 237-245. YIM,BOHYUN. On the wave resistance of surface effect ships. J . Ship Res. 15 (1971), 22-32. YOKOYAMA, NOBUTATSUO. On the relations between a practical ship-hull form and an attempted singularity distribution. Znt. S e n . Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1111-1128. ZHUKOVSKII, N. E. See Joukowski.
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Dynamics of Quasigeostrophic Flows and Instability Theory H . L . KUO Department of Geophysical Sciences The University of Chicago. Chicago. Illinois
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . I1. Tendency Toward Geostrophic Balance in Rotating Fluids . . . . A . Adjustment of Pressure and Nondivergent Flow Fields Toward Geostrophic Balance . . . . . . . . . . . . . . . . . . . . B. Solution of the Wave Equation and the Adjustment Process . . . 111. Simplified Hydrodynamic Equations for Large Scale Quasigeostrophic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Estimates of the Magnitudes of the Thermodynamic Variables in Quasigeostrophic Flows . . . . . . . . . . . . . . . . . . B. Scaling of the Hydrodynamic Equations . . . . . . . . . . . C . Expansion of the Flow Variables in Powers of Ro and the FirstOrder Potential Vorticity Equation . . . . . . . . . . . . . D . The Boundary Conditions in Terms of $I . . . . . . . . . . . IV . Permanent-Wave Solutions of Nonlinear Potential Vorticity Equation in Spherical Coordinates . . . . . . . . . . . . . . . . . . . A . Development of the General Permanent-Wave Solution . . . . B. The Vertical. Function and the Eigenvalues . . . . . . . . . . V . Stability of Zonal Currents for Small Amplitude Quasigeostrophic Disturbances . . . . . . . . . . . . . . . . . . . . . . . . VI . General Stability Theory-Integral Relations and Necessary Conditions for Instability . . . . . . . . . . . . . . . . . . . . . A . Stability Conditions for Pure Barotropic flow . . . . . . . . . B. The Semicircle Theorem for Three-Dimensional Baroclinic Disturbances . . . . . . . . . . . . . . . . . . . . . . . VII . Stability Characteristics of Barotropic Zonal Currents and Rossby Parameter . . . . . . . . . . . . . . . . . . . . . . . . . A . Stability of the Sinus Profile U = (1 cos y)/2 . . . . . . . . B. Stability of the Bickley Jet . . . . . . . . . . . . . . . . . C . Disturbances in a Hyperbolic-Tangent Zonal Wind Profile . . . VIII . Pure Baroclinic Disturbances . . . . . . . . . . . . . . . . . A . The Constant f Model and Boussinesq Approximation . . . . . B. Approximate Solutions of Equation (8.1) for a Nonzero b . . . . C . The General Baroclinic System . . . . . . . . . . . . . . . D . Laboratory Experiments on Baroclinic Instability . . . . . . .
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IX. Finite Amplitude Unstable Disturbances . . . . . . . . . . . . A. Method of Solution . . . . . . . . . . . . . . . . . . . . B. General Equations for Wave Perturbations in a Two-Level or Two-Layer System . . . . . . . . . . . . . . . . . . . . C. Inviscid Finite Amplitude Disturbance, @ # 0, r = 0 . . . . . . D. Viscous Equilibration for /3 = 0, Y # 0 . . . . . . . . . . . . X. Instability Theory of Frontal Waves . . . . . . . . . . . . . . A. The Basic State . . . . . . . . . . . . . . . . . . . . . B. Perturbation Equations and Boundary Conditions . . . . . . . C. Frontal Wave Solution. . . . . . . . . . . . . . . . . . . D. Nonlinear Development of Frontal Wave. . . . . . . . . . . XI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction T h e earth’s atmosphere is a mixture of gases and is under the influences of earth’s gravity and rotation, hence the disturbances set up in it can behave either as sound waves, surface gravity waves, internal gravity waves, inertial waves, Rossby or other kinds of vorticity waves, or some combination of these different types of motions. This is also true of the oceans. The actual nature of the disturbance depends strongly on its frequency or wave length, which are usually determined by the way the disturbance is produced. I n this respect it is useful to classify the disturbances into two different categories, namely, (1) forced motions and (2) free motions. By forced motion we mean those motions which can be attributed directly to some known forces, such as the oceanic and atmospheric gravitational and thermal tides, monsoons, mountain and sea breezes, and motions set u p by the varying topography and differential heating. On the other hand, free motions are those which cannot be attributed to any given force directly, but are the results of some intrinsic instability of the system. Within this category we have the long Rossby waves in the upper troposphere, the low level cyclone waves, and the disturbances created by convective and shear instabilities. I n this paper we shall limit our discussions to the large scale, low frequency flows only, that is, the motion systems whose aspect-ratio H / L is much smaller than unity and whose period is longer than one day, where H is the vertical scale of variation and L is the horizontal scale of variation. It can be shown that for such large scale flows the
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vertical acceleration can be neglected in the equation of motion and the influence of the local density variation can be disregarded in the continuity equation. We then have the hydrostatic and the anelastic approximations of the hydrodynamic equations. For the oceans, the latter can be replaced by the Boussinesq approximation, which is to take the density as constant except when it is associated with the gravity where a buoyancy force is introduced due to the density difference. These approximations have the effect of eliminating the sound waves from t h e system but leaving the low frequency disturbances represented quite accurately. A very important property of the low frequency (large scale) disturbances in a rotating fluid is the tendency toward geostrophic balance, in which the Coriolis force in the horizontal plane (i.e., normal to the gravity g) is balanced, or nearly balanced, by the horizontal pressure gradient force, so that the equation of motion in the horizontal plane is approximately given by where f = 2w sin is commonly referred to as the Coriolis parameter even though it is only the vertical component, w(=7.293 x s-l) is earth's rate of rotation, V is the velocity measured in a frame fixed on the earth, k is the unit vector along the vertical, p is density, p is pressure, and V h is the horizontal grad operator. When disturbances are produced in the atmosphere and in the oceans, they may contain many components which are not in geostrophic balance. Such flows are usually of the inertial-gravity-wave type with prominent horizontal convergences and divergences, and hence are accompanied by large cross-isobar components. These flows usually act as the agents in carrying away the unbalanced field from the source and leave a predominantly geostrophically balanced field behind. This adjustment toward geostrophic balance will be discussed in Section 11. T he flow is said to be quasigeostrophic when (1.1) is satisfied approximately, but the departure from this balance is of importance for the determination of changes of the flow fields. In such flows the velocity V is predominantly rotational, i.e., the vorticity 5 = vh X is relatively large, while the horizontal divergence vh * V is small. Such flows, when established, are governed by the nonlinear quasigeostrophic potential vorticity equation, which we shall derive in Section 111. T he inviscid nonlinear potential vorticity equation permits two- and three-dimensional barotropic permanent wave solutions containing many components with arbitrary coefficients both in Cartesian and in spherical coordinates. Specific combination of these solutions can be used to represent the observed mean flow patterns in the atmosphere.
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The potential vorticity equation is linearized with respect to a basic zonal current U(y, 2) and the general stability problem is formulated in Section V, while in Section VI the general stability theory is represented by two integral relations and an extended circle theorem. The stability properties of barotropic flows are discussed in some detail in Section VII, especially with regard to the eigenvalues for the cos2(y/2d), sech2(y/d), and tanh y / d profiles. It is found that, for the same profile, an easterly current is more unstable than a westerly current under the influence of a positive Rossby parameter p, while the westerly current is more unstable under the influence of a negative /3. The linear theories of pure baroclinic disturbances without and with the influence of Rossby parameter are discussed, and the solutions related to the baroclinic disturbances in the rotating annulus experiments are reported briefly. In Section I X a general method of obtaining finite amplitude solutions of the baroclinic potential vorticity equation is formulated, and an inviscid, oscillatory solution and a viscous, equilibrium solution obtained by Pedlosky for the two-level model are presented. Finally, the instability theory of frontal cyclones is presented in Section X as a separate problem, even though it is closely related to the baroclinic wave theory for quasigeostrophic Aow. As is unavoidable in a paper like this, only a few papers on each subject have been mentioned. Additional references can be found in the papers cited, and so extensive bibliography is not included here
II. Tendency Toward Geostrophic Balance in Rotating Fluids One very important character of the motion of a rotating fluid is the tendency toward geostrophic equilibrium, in which the Coriolis force of deflection is balanced by the pressure gradient perpendicular to the direction of motion. This equilibrium is brought about by a mutual adjustment between the mass (pressure) and the momentum distributions toward the geostrophic condition whenever an imbalance exists, such as when certain momentum is suddenly imparted to part of the fluid without an accompanying pressure gradient, or when a pressure gradient is produced by extraction or addition of mass in a certain region. This process was first discussed by Rossby (1938), and later on by many others. For example Cahn (1945), Obukhov (1949), and Raethjen (1950), have examined the adjustment problem for a homogeneous rotating fluid, while Bolin (1953) and Veronis (1956) have investigated the stratified fluid problem, and Kibel’ (1955) analyzed the three-dimensional flows. In this
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section we shall demonstrate this process with Obukhov's (1949) simple, vertically averaged barotropic model, by introducing m
p V dx,
(2.la)
(Y =
:),
(2.lb)
as the dependent variables, where p , is the surface pressure, p o is its mean value, H = p o / g p o is the scale height of the atmosphere, and P is the potential energy of the air column which is equal to the vertically integrated pressure, viz., (2.lc) The vertically integrated and linearized equations of motion and continuity equation are then given by ( a V / a t ) + f k x V=-gHVm,
(2.2)
anpt = -v v = -vzV. (2.3) For convenience we decompose V into its nondivergent part V , and irrotational part V , and introduce a stream function for V* and a velocity potential for V , , viz. V = V,
+ V , = k x V$ + V V .
(2.4) On applying the operators V x and V . to (2.2) we then obtain the following vorticity and divergence equations, respectively : V 2 [ h+fvl= 0, V2[vt-f* +@TI = 0. Combining (2.5) with (2.3) we find the potential vorticity equation V2$bt-fTr, = 0.
(2.51 (2.6) (2.7)
This equation shows that the potential vorticity Q, given by = v2* -f T ,
(2.7a)
is independent of t and hence is a function of x , y only. We can use this equation to determine the final distributions of and m from their initial distributions.
H . L. Kuo
25 2
For wave perturbations we can remove the operator V2 from (2.5) and (2.6) and obtain (2.5a) (2.6a) On applying a/at to (2.6a) and substituting $t and rt from (2.5a) and (2.3) we then obtain the following wave equation in p): Co2=gH.
f f 2 y - Co2V2p)= 0,
(2.8) We assume that the initial values y o , $o, and ro are given. Equation (2.6a) then furnishes the initial y t , so that (2.8) can be solved with the initial conditions p)tt
yo = el(% y ) , yto
=f$o
(2.8~)
- gHr0 = ez(x,y ) .
(2.8b)
Note that if p) is decomposed into its Fourier wave components exp i(k,x k 2 y - ut), we then find that u is given by
+
u2
=f2 +
(K,2
+
K22)C02.
(2.8~)
This relation shows that these waves are dispersive and hence an initially localized distribution of the unbalanced wave energy represented by ez(x,y ) will be carried away by these surface waves and spread out over the whole domain, thereby also altering the mass distribution. Thus, the wave system represented by (2.8) and (2.8a,b) actually furnishes the mechanism through which the adjustment toward geostrophic equilibrium is established, and the rate at which the equilibrium is reached is also determined by this system of equations in the present model.
A. ADJUSTMENT OF PRESSURE AND NONDIVERGENT FLOWFIELDS TOWARD GEOSTROPHIC BALANCE T o obtain the solutions of (2.7) and (2.8) as an initial value problem, we first decompose I+$r,and p into a steady field and an unsteady or wave field, viz.,
$=*+*‘,
(2.9) where 4, 7j, p) are functions of x, y only and $’, r’,and y’ are functions of x,y , and t. From (2.5a) and (2.6a) we then find 7T=++r’,
p=q+p)’,
(2.10)
(2.11)
Quasigeostrophic Flows and Instability Theory
25 3
Therefore the F field has no steady part so that the steady flow is nondivergent, while ii and $ satisfy the geostrophic relation. Further, since the potential vorticity is independent of t, Q must be given by the steady field and also equal to its initial value. Thus we have
n = V2$- f 77 = no =
v2*o-fro
(2.12)
>
Q' = VZ*' -fT' = 0.
(2.13)
Substituting ii from (2.11) in (2.12) we then obtain the following equation for $: (2.14) where h = (gH)li2/f is called the radius of deformation or the radius of influence of the system. For any given initial distribution of Q, the solution of this equation can be obtained from the following general formula *(Xi
y) = -2T
SSn,(E,
'I)Ko(P) dt
4,
(2.15)
where Ko(p)is the zeroth-order Bessel function of imaginary argument, also known as Kelvin's function, whose asymptotic expressions are
+ log(2/p)
K,(p) =-0.5772
=(42p)
112
e
--P 7
and p is given by P2 = [(x - ElZ
+ (Y
for p < 1 for p 9 1
(2.15a)
'I)21/h2. (2.15b) When the disturbance depends only on one space coordinate, the solutions of the steady and unsteady systems (2.14) and (2.7) and (2.7a,b) can be obtained rather easily. For example, with an initial unbalanced vortex motion given by -
(2.16a) T o = To = 0,
we find that the final solutions of the potential vorticity equation (2.14) are
(2.16b)
H . L. Kuo
254
Here R is a horizontal scale length and AIR is the strength of the flow, and
5 = r/R,
r2 = x2
+y2,
p = R2/h2.
These solutions show that, when p is small, the difference between 6 and coo is insignificant, while for relatively large values of p, the change of z, is appreciable. For example, the curves in Fig. l a represent the function 5, v o , and ii for AIR = 5 m s - l and p = 0.0516, which corresponds to R = 500 km and A = 2200 km. It is remarkable that in this example z, changes so little while the change of pressure is so drastic, amounting to a reduction (A < 0) of 20 mb at the center. This situation is characteristic of the disturbances whose horizontal dimension R is small compared with A. For large values of AIR, the change of z, becomes significant, as can be seen from the values of 6 / v 0 in Table I, which are based on h = 2700 km.
2.0
l 1.2 . b y ;
0.0
m s'
0.6 0.4
0.4
0
it@.!
0.2
-
c
Hours
I I I I I
0 500 1000 1500 2000 km r
0 500 lo00 1500 km
r
FIG.1. Adjustment between pressure and velocity distributions in rotating fluid. (a) Adjustment of pressure to given initial velocity field; (b) diminution of unbalanced pressure field; (c) -pressure change at the center of an unbalanced vortex.
TABLE I
VALUES OF U/v0
500 3000
5000
I
1000
2000
3000
4000
5000
0.99 0.75 0.52
0.99 0.74
0.99 0.71 0.47
0.98 0.64 0.39
0.98 0.51 0.27
0.51
Quasigeostrophic Flows and Instability Theory
255
If, on the other hand, the initial disturbance is in the form of an unbalanced pressure, such as the sudden elevation of the free surface or localized distribution of potential energy, we shall find that most of it will be carried away by the gravity waves and only a very small fraction will be left behind to be balanced by a wind system, and the more so the smaller the initial horizontal scale of the perturbation. This case is demonstrated by the following example.
Initial jield:
IIIo=o, n=o= A [ 2
yo=(),
+ p - (4+ p)x + x2]e-.?.
(2.17a)
Final steady jield:
Afr
= - [x - 2]e-",
2
=
ARZf
( I - x)e-*,
(2.17b)
ii = Ap(1 - x)e-'.
where x = r2/2R2. Thus, when p < 1 the final pressure perturbation ii is very small compared with its initial value n o ,as is illustrated in Fig. lb. Thus we conclude that when p is small, the pressure always adapts to the velocity. Since p is proportional t o p , we expect pressure adjustment to take place on larger scales at low latitude. T h e physical reason for the behavior discussed above is that when an imbalance beween the pressure gradient and Coriolis force is present in a region, adjustment of mass distribution can be accomplished rapidly near the edge of the region but only very slowly far away from the boundary. Therefore unbalanced pressure gradient in a small region can readily be obliterated by a mass flow in the direction of the pressure gradient or Coriolis force, whereas far away from the boundary the velocity has to adjust to the pressure gradient. Another point worth mentioning is that large scale variations can only be established gradually and hence the pressure and velocity distributions have ample time to adjust to each other, therefore we do not expect strong imbalances to occur over a large area. On the other hand, large velocity or pressure gradient concentrations can easily develop over small regions. According to the adjustment theory discussed above, only the small scale velocity concentration can persist while the unbalanced pressure gradient will soon be obliterated. Thus, the adjustment mechanism seems to be responsible for the streakiness of the velocity distributions in the atmosphere and in the oceans.
H . L. Kuo
256
B. SOLUTION OF THE WAVEEQUATION AND THE ADJUSTMENT PROCESS As has been pointed out already, the adjustment of the pressure and the velocity distributions toward geostrophic equilibrium is actually accomplished through the divergent flows. For the problem under consideration, these flows are governed by the dispersive wave equations (2.8) and (2.8a,b). T h e solution of this system can be obtained more readily by using
4 x 9 Y,
5, t ) = COS(f5/Co)d~,y , t )
as the dependent variable, so that (2.8) is transformed into the following simple three-dimensional wave equation Utt = V,%.
(2.18)
The initial conditions (2.8a,b) are then given by
u(x, Y , 5, 0) = 4% y)cos(f5/co),
(2.18a)
4%y, 5, 0) = 4% y)cos(f5/co).
(2.18b)
The solution of this system can readily be obtained by Cauchy’s method and the function ~ ( xy , t ) is simply given by u(x,y, 0, t). Thus we have
where the argument offi and 5 ct, and
f2
are x
+ p cos 6, y + p sin 6, the limit for
p is p
.q = ( C 2 t 2
-p
y .
T h e pressure variation can be obtained from (2.19) by first applying V2 to it and then integrating over t. For the initial disturbance represented by (2.16a), the change of the pressure at the center with time is represented in Fig. lc. It is seen that the geostrophic value of the pressure at the center is established within three to four hours. Many works on the adjustment problem, such as those of Cahn (1945), Bolin (1953), and Veronis (1956), are centered on the solution of the dispersive wave equation. However, from the point of view of the large scale flows, it seems that the solution (2.15) is more important and interesting. I t is evident that in a stratified medium, the adjustment process works essentially within individual layers bounded by isentropic surfaces, even though adjacent layers influence each other to a certain degree. T h e equation that governs the adjusted state is the general potential vorticity equation
QuasigeostrophicFlows and Instability Theory
257
( 3 . 1 3 ~ )to be derived in the next section, while the mass adjustment will be accomplished both by surface waves and by the internal gravity waves.
111. Simplified Hydrodynamic Equations for Large Scale QuasigeostrophicFlow We consider that the motion system under consideration is characterized by a horizontal scale length L, a vertical scale length D, a velocity U, and a time scale T . When the ratio LID is much larger than unity, the motion will be referred to as of large scale. For the large scale motions in a stably stratified and rotating fluid, the following parameters are of paramount importance in determining the nature: of the motion, (i) (a) T h e thermal Rossby number
gD vh8 gD Ah8 = -ROT= fo2L 8, fo2L2 0, ' (b) The mechanical Rossby number
(ii) The planetary Richardson number
gD A,0, &=--gD2 38, - -fO2L2t7,az -fO2L2 0, ' (iii) 'The planetary Froude number P=-*
fo2L2 gD
(iv) The Ekman number
E = vifo D2, where fo is an appropriate value of the Coriolis parameter for the region under consideration, 0 is the potential temperature and 0, is its normal value, which is taken as a function of z only, 0, is a vertical average of 8, for the level considered, A,0 is the horizontal difference of 0 within the horizontal distance L , and A20, is the vertical difference of 8, within the vertical distance D.The motions are considered as the results of the horizontal temperature gradient AhO, so that U may be identified with the thermal wind gAhO/ffls. We then have ROT= Ro.
25 8
H . L. Kuo
Because of the existence of the mean stratification in the earth's atmosphere, the vertical scale length D of the large scale motions is of the order of the depth of the troposphere, which is about lo4 m, while the typical , horizontal wave length) is of the order of horizontal scale L( ~ ! x / 4A= lo6 m, essentially 1/10 of the distance from the equator to the pole. Thus, if we use the equator to pole difference of 8 [ %SO "C] as the representative horizontal temperature gradient, we shall have V,8 m 5.0 x "C m-l. On using fo = 10-4s-1 we then find
ROTM 0.15. On the other hand, the normal vertical stratification of 8, gives aO,/az= 6.0 x "C/m, and hence S, M 2.0. The planetary Froude number p depends only on fo , D, and L and is of the order of 0.1 for the values of fo, L, and D used above. Thus there is an important class of motions for which R, and p are much smaller than 1 while S, is of the order of unity. I n what follows we shall derive the simplified versions of the hydrodynamic equations that are adequate for this class of motion.
A. ESTIMATES OF THE MAGNITUDES OF THE THERMODYNAMIC VARIABLES IN QUASIGEOSTROPHIC FLOWS
As will be shown later in this section, the large scale motion of a stably stratified fluid is mainly geostrophic. That is to say, the horizontal velocity V and the pressure perturbation p' =p -ps(z)are related approximately by the geostrophic wind equation
fopsk x V= 0 , ~ ' . (3.1) On replacing V by U and Vhp' by p'/L for order of magnitude consideration we then find
[ P I = [fopsLu1= LfoLU/RT,lp, = [(pQ'gH)ROlp,,
(3.la)
where H = RTJg is the equivalent depth of a homogeneous atmosphere with total pressure p , and uniform density p, , and is of the order of 7.5 km, which is nearly equal to D.Thus we have [p'lp,] = E = p R o M 0.015. Since, according to the equation of state, p'/ps and T'/Tsare of the same order of magnitude as p ' / p s , we have
x/xs = EX*
(34
for all the thermodynamic variables, where X ' represents either p', p', T', or 8' and 2,represents the vertical average of the corresponding basic state thermodynamic variable, and X * represents the nondimensionalized perturbation of these quantities. In this form X * is of order unity.
Quasigeostrophic Flows and Instability Theory
259
B. SCALING OF THE HYDRODYNAMIC EQUATIONS The nonsteady quasigeostrophic flows usually propagate with speeds comparable to the mean current velocity U and hence the quarter period of the time variation is of order TI = L/U. I n addition, the stable stratification of the atmosphere renders w' to be of order UR, D/L instead of UDIL. Therefore the appropriate scalings of this class of motion are
x' = Lx, u' = uu,
pi +AmpsQ,
Y' = LY, v'
=
uv,
= Epsp*,
Z' = Dz,
t' = tL/U,
W' = wUDR,/L,
el = &eSs,
(3-3)
f'=E,
where the primed quantities are dimensional and the unprimed quantities are dimensionless, and fi is an appropriate mean value off' for the area under consideration. Substituting these transformations in the horizontal and the vertical equations of motion, the continuity and the heat equations we then obtain the following nondimensionalized equations
dV Ra - +fk x V = -V@ dt
+ EV3'V,
(3-4)
where
and V' = V, is the dimensionless horizontal velocity, u, = -( l / p s ) (a p , / a z ) , Q = Q'/CpTsp,Q' being the rate of accession of heat per unit mass, V = V, is the horizontal gradient, and V32 is the three-dimensional Laplacian. We mention that in the scalings in (3.3) both the dependent variables and their space and time derivatives are assumed to be of order unity. Thus, the terms on the right-hand side of (3.5) can be neglected whenever the ratio D2/L2is much smaller than unity, and hence for such large scale motions the pressure distribution is hydrostatic even in the disturbed state. Further, the viscous term in (3.4) can also be neglected except close to the surface where it becomes important in creating the Ekman flow.
H . L. Kuo
260
Further, since E is much smaller than R, for almost all large flows under consideration, the last term in (3.6) can also be neglected. This approximation to the continuity equation is called the anelastic approximation. Thus for E = p R < R, (3.4)-(3.7) can be simplified to the following: d V +fk x
V = -V@,
(3.4a)
dt
a@
- - a,@
ax
+p"
a@
-s =
= - +s,@
ax
apsw = 0, v v+-Ro -
(3.5a) (3.6a)
*
ps
0,
ax
(3.7a) where
1
ae,
e,
ax
s, = - -= ps,.
With R, set to unity, these equations are called the primitive equations in meteorology and they represent the proper simplification of the original equations (3.4)-(3.6), involving only the neglecting of the viscous dissipation, the vertical acceleration in ( 3 4 , and the local density change in the continuity equation (3.6a). The results of the hydrostatic and the anelastic approximation are that high frequency internal gravity waves and acoustic waves are excluded from the system, but otherwise the system is still able to carry the relatively low frequency internal gravity waves as well as inertial waves and vorticity waves.
C. EXPANSION OF THE FLOW VARIABLES IN POWERS OF R, AND THE FIRST-ORDER POTENTIAL VORTICITY EQUATION For a Rossby number R, of the order of 0.15 or smaller, the series expansions of the flow variables in powers of R, can be expected to converge rapidly so that the first few terms should give sufficiently accurate results for many problems. We shall therefore develop the dynamic equations from a formal expansion in R, by setting
X=
C ROmXm,
(3.8a)
m=O
w = w1+ R o w ,
+
* * *
,
(3.8b)
Quas&eostrophic Flows and Instability Theory
261
where X stands for any one of the variables u,cu, @, or s. Further we write
f =J’K = (1 +ROPY),
(3.8~)
where h=pS,R;l is of order 1, s,= S z p = h R o , P=/3’L2/LT, which is taken as of order unity, where p‘( =dJ’/dy‘) is the dimensional Rossby parameter. Substituting these expansions in (3.4a)-(3.7a) and equating to zero the coefficient of Rooand Ro, we then obtain the following systems of zeroth- and first-order equations :
k x Vo=-V@o,
(3.9a)
a(Do/az= so,
(3.9b)
v.vo=o;
(3.9c)
8% - s, - m0, --
(3.10b)
ax
(3.10~) as0
-
dt
+ Vo V S ,+ Szwl= Q1. *
(3.10d)
This set of equations is essentially the same as those obtained by Charney and Stern (1962) and by Pedlosky (1964a). The higher order hydrostatic and continuity relations are of the same forms as (3.10a) and (3.10b) while the higher order equations of motion and heat equation are given by the following for m 2 1
(3.11a) (3.11b) Thus the zeroth-order equations (3.9a-c) signify that V o , @, , and so are connected by the geostrophic and the hydrostatic relations, and that V , is nondivergent to the extent that a constant fo is used in the geostrophic relation. Notice that these relations do not contain the time rate of change explicitly and hence they are not able to reveal the time changes of the flow variables, even though time variations may be implied implicitly.
262
H . L. Kuo
T o obtain the time changes we must make use of the higher order equations (3.10a-d) and (3.11a,b). Since (3.10a) contains Vl and Q1in addition to aVo/at,it is difficult to use them directly even though (3.4a) and (3.7a) form a closed system. However, by applying V x to (3.10a) and making use of (3.10~)we can eliminate O1and Vl from this equation and obtain
where
Here we have included a first-order frictional term even though this term is small. Note that the geostrophic motions possess vorticity because the @-field forms closed contours. Eliminating w1 between (3.12) and (3.10d) and including the first-order dissipative terms for generality, we obtain the following first-order nonlinear quasigeostrophic potential vorticity equation in terms of a single dependent variable Q 0 : (3.13) where q is the potential relative vorticity, given by (3.13a) The dimensional form of q is
(3.13b) where t,h =p’/fopsis the dimensional stream function. Equation (3.13) can also be obtained from Ertel’s potential vorticity equation by the application of the geostrophic relation (see Ertel, 1942; Kuo, 1972). For a homogeneous incompressible rotating fluid of variable depth H( =Ho + h), the quasigeostrophic potential vorticity equation takes the form
d (f3) = 0. dt
H
(3.14)
Quasigeostrophic Flows and Instability Theory When f and H , are functions of y only, and h written as
263
< H , , this equation can be ay ax
=0,
(3.14a)
where
a*
The second part of represents the influence of the variation of the undisturbed depth on the flow. It can often be used in laboratory experiment to simulate the Rossby parameter. Because of the change of depth, it is no longer possible for the particles to move along the contours of constant H and hence steady geostrophic flow is replaced by wave motions, just as under a variable f.
D. THEBOUNDARY CONDITIONS IN TERMS OF 4 For flows in a region bounded by rigid side boundaries, the kinematic condition to be satisfied is the vanishing of the normal velocity. Therefore we have = 0. (3.15a)
vo,
On the other hand, the conditions for w at the bottom surface and aloft must be expressed in terms of so through (3.10d) and then in terms of Q, through (3.10b). For example, when the surface is not horizontal but has a topography given by h(x, y ) , we shall have W, =
V, * Vh
T h e condition at an upper level can be specified in terms of Qo or psQo in a similar manner. It is rather remarkable that the complete set of the first-order hydrodynamic and thermodynamic flow fields in nonsteady quasigeostrophic flow can be determined by the solution of the single potential vorticity equation together with the associated boundary conditions alone. We note that the internal gravity waves have been excluded from the present system by the condition of geostrophy, as is evidenced by the fact that (3.13) is of first order in t and hence it cannot represent gravity waves. One shortcoming of the formal expansion used above is the approximation off by the mean value fo in both the geostrophic relation and in the
H . L. Kuo
264
coefficient of the divergence term in the vorticity equation (3.12), resulting in the appearance of f o 2 in the expression (3.13b) for q. Even though the approximation o f f by f o is justifiable in middle and higher latitudes, it can hardly be valid in lower latitudes. Further, the neglectingof the variable part off is based on the assumption that L < ~ a / 2and , hence is not valid for the very long waves. Therefore, for some problems it is desirable to relax the ordering consistency in the R,-expansion since higher order equations are not being used anyway. Thus, we may replace fo2 by f 2 in (3.13b) so that the potential vorticity is given by (3.13~) Since the hydrostatic relation is valid for all the large scale flows, it is often convenient to employ p as the vertical coordinate and use w = dp/dt as the measure of the vertical velocity. T h e horizontal pressure gradient is then measured by the gradient of the geopotential @ =gz of the isobaric surface. T h e primitive equations in dimensional forms are then given by
dV
-++fk dt
x
v=
-V@,
(3.4b) (3.5b)
am v . v+--0,
aP
(3.6b) (3.7b)
where
r = -T-ae= -(yd),
e
aP gP yd and y being the dry adiabatic lapse rate and the actual lapse rate, respectively, and Q the rate of accession of heat. It is seen that the continuity equation (3.6b) is formally like that for a homogeneous and incompressible fluid, even though inhomogeneity and compressibility are included implicitly. T h e quasigeostrophic potential vorticity in pressure coordinate is given by (3.13~") where S = - R r l p and ah,
= @/f.
Quasigeostrophic Flows and Instability Theory
265
On multiplying (3.10a) scalarly by ps V oand (3.10d) by psso7integrating over the entire volume T , and making use of (3.10b,c) and (3.15a), we then obtain the following equations for the changes of the kinetic energy K = 4j r p s V o V , dr and the available potential energy E =g/2 Jz pssgdr/s2: (3.16a)
-aE_ - _ 2t
jrgPsWls0 + D*
(3.16b)
9
where D, is the rate of viscous dissipation of K and D , is the rate of generation or destruction of E by diabatic heating. These equations are of the same forms as that given by the primitive equations (3.4a)-(3.7a). It is seen that the sum of K and E is conserved for inviscid and adiabatic changes and hence the quasigeostrophic system of equations represents an energetically consistent system.
IV. Permanent-Wave Solutions of the Nonlinear Potential Vorticity Equation in Spherical Coordinates
A. DEVELOPMENT OF THE GENERAL PERMANENT-WAVE SOLUTION For inviscid and adiabatic flows the three-dimensional quasigeostrophic potential vorticity equation (3.13) in spherical coordinates reduces to
a%,
+
*A
411 - *11 q A
+ 2Q*A
=0
9
(4.1)
where a is the radius of the earth, the subscripts denote partial differentiations, and h is the longitude, q(= sin @) is the sine of the latitude, and q is the relative potential vorticity, which is given by
(4.la) where S = -p-la log e/ap, and Vs2 is the horizontal Laplacian operator in spherical coordinates, viz., (4.lb)
H . L. Kuo
266
In this section we shall seek permanent-wave solutions of (4.1), namely, solutions of the type #(A, 77, P, 4 = *(A - at, 7, PI?
(4.2) where CL is the constant angular phase-velocity of the perturbation. For such disturbances (4.1) can be written as
'FAG, -yP, G, where"" and G are given by
= 0,
(4.3)
+a 2 q , G = 2(!2 + a ) + ~ A2Y. Y =$
(4.3a) (4.3b)
The first integral of (4.3) is
G = F(Y), (4.4) where F is an arbitrary function ofY. In this investigation we shall restrict ourselves to the case where F ( Y ) is a linear function of Y, that is, G = -(p/a2)Y, (4.4a) where p is a constant which can be chosen to fit the specific situation. In this case $ is given by
A'$
P +$ = -(ZQ a2
+p~)q.
(4.5)
We write the solution of this equation as
where Pz(v) is the associated Legendre function and Niis the vertical amplitude function, given, respectively, by
where n j is a positive integer, including zero, 5 =plp, ,p , being the pressure at the surface, and S* and 1," are given by
T a log 8, S"=----..--
TC
RT
ac
+
'
ljz = -[nj(nj 1)- p]. fo2aZ
(4.8a) (4.8b)
QuasigeostrophicFlows and Instability Theory
267
Since the solution (4.6) satisfies the finiteness condition forI/,I over the entire globe, we only need to impose boundary conditions at the top (5 = 0) and at the ground surface, which we shall assume to be flat and rigid, so that the condition at the bottom surface is the vanishing of the vertical velocity. According to the hydrostatic and the quasigeostrophic approximations, we have w = - +apw - = p ap
at
ax
;( ). --
Substituting this expression in the adiabatic heat equation (3.10d) and setting w to zero we then obtain the following relation for 5 = 1 : a2(*ct
+ + s*t)
*A
*tv - *v
$4, = 0,
(4.9)
where s = -8 log t9,/8 P, .
(6.7a) (6.7b)
For antisymmetric profiles extending to infinity, such as the U = tanh by profile, the condition (6.7a) alone appears to be sufficient for the existence of unstable disturbances. We pointed out that the stability characteristics of an easterly current differ from that of the corresponding westerly current on account of the influence of p. I n fact, the stability characteristics of an easterly current under the influence of /3 are exactly the same as that of the westerly current under the influence of -/3, as can readily be seen by a change of the direction of the x-coordinate. Hence we shall use a negative /3 to characterize the flow properties of an easterly current.
B. THESEMICIRCLE THEOREM FOR THREE-DIMENSIONAL BAROCLINIC DISTURBANCES
It has been shown by Howard (1961) that for the two-dimensional inviscid disturbances in a nonratating system, the upper bounds of c, and ci are given by the half of the difference between the maximum value (U,) and the minimum value (Urn)of the basic current. A similar result can also be derived for the three-dimensional quasigeostrophic and baroclinic disturbances under the influence of p. For this purpose we shall at first nondimensionalize the potential vorticity equation (5.49 in pressure coordinate by setting u'= u,
u, a=kL,
c' =
u, c, y' = LT, p = (fo L / J S)5 , / 3 = p f L 2 / U M , $'=LU,$,
where the primed quantities are dimensional. For simplicity, we assume that the stability factor S is constant. In terms of these dimensionless variables, (5.4) becomes
Th e boundary conditions for $ are
H . L. Kuo
280
Here we limit ourselves to the unstable disturbances only, so that we have ci# 0 and hence c # U. We transform (6.8) further by setting
$ = ( U - c)F(r, 5).
(6.10)
Equation (7.9) and the boundary conditions (5.5a,b) then become d
( U - c)' F=O F,=O
"1
-
d5
- a'(
U - c)'F
+p( U - c)F = 0. (6.11)
at 7 l = % , % , at 5=51,52.
(6.11a) (6.1 lb)
For the unstable disturbances both c and F are complex. Thus on multiplying (6.11) by the conjugate complex F" of F and integrating over 7 and 5 from T~ to q 2 and from to C2, respectively, and making use of the conditions (6.11a,b) we then find
J"JU- c)'Q dA = /3
J" ( U -
c)l
A
+
FI ' d A ,
(6.12)
+
where Q ='a I F I ' 1 F,, I ' 1 F, 1 ', which is positive definite, A = (yz -q1)(12 - 5,). This equation can be separated into its real part and its imaginary part, which are given by
J [( U - c,)'
-
ci']31Q d A = p
J(U
- c,)
I F I ' dA,
(6.12a)
1
ci[J(U- cr)Q d A - P- I FI d A ] = 0. (6.12b) 2 Since ci differs from zero for the unstable modes under consideration, the quantity in the bracket must vanish. On substituting (6.12b) in (6.12a) we find
s,
[U'-(C,' + c i 2 ) ] Q d A = p J " U I F I ' d A .
(6.13)
A
Now we have
( U - Urn)(U- U M ) = U 2 - ( U m + U M ) U + U m U M 1 0 ,
10- UI I A U ,
(6.14a) (6.14b)
where UM and U , are the maximum and the minimum of U and (Urn UM)/2, AU = ( U , - Urn)/2.Hence (6.13) yields
+
(c,-
o=
Quasigeostrophic Flows and Instability Theory
281
where P2 and Q are the area averages of I FI and Q. Thus c, - 0 and ci are bounded by the square root of the right-hand side and hence are bounded by the absolute value of ( U , - Urn+pF2/Q]/2.Equation (6.15) becomes identical with the relation derived by Pedlosky (1964a) when F2/Qis taken as equal to (k2 kI2)- l. Another limit of c,2 has been obtained by Miles (1964b) for the baroclinic system, viz.
+
+
4ci2< { ( p / ~ s ) ~ U " ~K( T a ~ / Y-Tl } )m a x .
(6.16)
VII. Stability Characteristics of Barotropic Zonal Currents and Rossby Parameter For the nondivergent perturbations in a barotropic zonal current (5.4) reduces to
T he appropriate boundary conditions for
$=O
t,h
are
at y1,yz.
(7.la)
For convenience we nondimensionalize this system by setting
Y"
u=- u m a x
Y'd' $=-
U"
P ,
,
c=-,
C"
umax
b = - ,Pd2
(7 4
~=kd, umax duma, where the starred quantities are dimensional, Urn,, is the maximum value of the basic current and d is a measure of the horizontal scale of U. Expressed in terms of these dimensionless variables, (7.1) becomes
We take b positive for westerly current and negative for easterly current, corresponding to a positive and a negative Urn,, , respectively.
A. STABILITY OF THE SINUS PROFILE U = (1
+ cos y ) / 2
T he solutions corresponding to this sinusoidal current have been discussed by Kuo (1949). It is readily seen that, for this basic current the condition for instability is simply b < 1/2. When this condition is satisfied,
H . L. Kuo
282
the flow is unstable and the upper transition from stability to instability is given by the neutral solution = cos y/2 = U1l2,
= J3/2,
=
= 112 - b.
(7.4) For 0 5 b 5 1/2, the lower transition from the unstable waves to the stable modified Rossby-Haurwitz type waves is given by the neutral solution $k
ffk
= cos2"y/2) =
$0
ao2=
UA,
u k
c, = Umin(=O),
[4h = 1
1 - ha,
ck
+ (9 - 16b)1'2].
(7.5)
This neutral solution does not exist for the easterly current ( b < 0). Besides these symmetric neutral solutions, there exists an antisymmetric neutral solution given by y, aka= 0, c k 2 = 1/2 - b. (7.6) Since this velocity profile is symmetric about y = 0 and since the boundaries are symmetrically located, the most unstable solution of (7.3) is also symmetric about y = 0. Therefore the appropriate boundary conditions are $k2
= sin
$'(O) = 0, $(r)= 0. (7.7) The solutions for this profile have been obtained by an iterative method, both for positive and for negative b values, which is to integrate (7.3) for a given a and a given c numerically, and then to correct c by reducing the discrepancy at the boundary to zero. The dimensionless phase velocity c, and growth rate S = a c i so obtained are illustrated in Figs. 6a and 6b,
umaxTh umax~rlx I
l
l
,
,
0.6
c,
Cr/
00
812
2
,
( 0 )
4
6
8
10
I
1214
L/ D
0.8
a
FIG.6 . Dimensionless eigenvalues for barotropic disturbances in a sinus profile. (a) c,; (b) growth rate a t i .
Quasigeostrophic Flows and Instability Theory
283
respectively. Of particular interest is the result that the easterly current (i.e., b < 0 ) is made more unstable by the /3 effect within the range 0 > b 2 -0.25 (approximately), while the westerly current is made more stable. In addition, the most unstable wave length is slightly longer in the Easterlies than in the Westerlies under the influence of a positive b. Numerical results have also been obtained by Yanai and Nitta (1968, 1969), who also found that when the horizontal shear exceeds certain critical values, the neutral, Rossby-Haurwitz type waves change into singular wave solutions of the continuum type, with some discontinuities in $’, while regular neutral solutions cease to exist. We point out that for this velocity profile the actual half width is d’ = Td. Thus, for d’= lo6 m, Urn,, = 10 m s-l, fl= 2.29 x 10-l1 m - l s-l, b is of the order of 0.23. For this case, the most unstable disturbance in the easterly current corresponds to a = 0.5. Hence the most unstable wave length is L = k d = 4000 km. Before we discuss the other stability problems, it appears worthwhile to mention briefly the destabilization of the easterly sinus profile U=-siny, O < ~ < T (7.8) by the influence of /3. Since the inflection points of this profile are either on the boundaries or outside the range of y, this velocity distribution is stable when /3 is absent. However, with b(=fld2/Um,,) < 1, the absolute vorticity gradient QOy = b - sin y changes its sign in (0, T) and hence the flow is made unstable by b. The same result applies also to a westerly current when a negative b is present.
B. STABILITY OF THE BICKLEY JET For the jet with the profile U = U+/Umax=sech2y,
-co 0. This latter solution represents the lower transition for stability to instability in the westerlies. Another symmetric solution, given by
c=l, J! C ,
(9
&“=[3-
+4 ~ 2 1 ,
= (sech ~ ) “ ‘ / ~ ( t a n y)2-a2!3 h
(7.15)
(7.16)
exists in the Easterlies ( b < 0). This solution represents a submode of the stable modified Rossby wave and is not a stability boundary for small -b. However, for b < -1, this solution does closely represent the lower transition from instability to stability. Another neutral solution is given by
2 = 2[1 - (3b/2)I1I2- 1,
+ 3)/6,
c = (2
$ = sechy . tanh y. (7.17)
This is an antisymmetric solution and it exists in the b < 1/2 region but it does not represent transition from stability to instability. T h e solutions of the system (7.11) and (7.12) have been obtained by an iterative method; the dimensionless eigenvalues c, and 6 = aci so obtained are represented in Figs. 7a and 7b. The numbers attached to the curves in Fig. 7a are the values of b. It is seen that the phase velocity of the unstable disturbance is always within the range of the basic current and for positive b its value decreases as the wavelength increases, while for negative b, c, decreases to a minimum and then increases with the wavelength. Here again, the most interesting results are the fact that the easterly jet is made more unstable by the b influence within the range b 5 0.84, while the westerly jet is made more stable by b. In contrast to the case of the sinus profile, here the most unstable wavelength is reduced by the b effect both for the Westerlies and for the Easterlies, and this reduction is very promi-
Quasigeostrophic Flows and Instability Theory
1.5
2.0
285
2.5
b FIG.7.
Eigenvalues for U = sech2y. (a) c,; (b)
OLC,
nent. Notice that the growth rates in Fig. 7b are of the same order of magnitude as that in Fig. 6b. For an easterly current with a halfwidth = lo6m = 1.76 d, U,, = 10 m s-l, we find b = 0.74. The most unstable disturbance for this b corresponds to cc =: 2n-d/L = 1.28, therefore the most unstable wavelength is L = 2790 km.
H. L. Kuo
286
C. DISTURBANCES IN A HYPERBOLIC-TANGENT ZONALWIND PROFILE The intertropical convergence zone (ITCZ) is a region in the tropics roughly parallel to the equator, containing deep, intense cumulus convection. The convectively active part of the zone is usually very narrow, of the order of 100 km. Its latitudinal position varies from 3" or so to 20" or more, with the average of about 10" throughout the year. But what is most astonishing is the fact that it is almost never found at the equator. Satellite pictures of the ITCZ usually reveal it not as a truly parallel zone, but often in a disturbed state with wavelike disturbances superposed on it. Wind observations also show the regular appearances of wave disturbances and vortices along the ITCZ, and many of these disturbances develop into hurricanes and typhoons. The mean wind in the intertropical convergence zone is usually characterized by a nearly uniform current on one side and a different uniform current on the other side, with a rapid change of direction across a relatively narrow zone of transition. Such zonal velocity distributions are illustrated in Fig. 8a by the mean zonal wind profiles observed over the Pacific and the Atlantic, given by Yanai (1961) and by Riehl(1969), respectively. Since the ITCZ is a zone of active convection and upward motion, such a zonal wind distribution can surely be expected theoretically because the convergence toward the ITCZ will definitely create such a wind system under the influence of the Coriolis deflection, provided the ITCZ is not located at the equator. These observed mean wind distributions in the intertropical convergence zone can be represented analytically by a hyperbolic tangent profile, viz.
where U* is the dimensional mean zonal current, 0 is its mean value, and 2U, is the total shear. It can be seen that, with d - 150 km, the profile (7.18) can fit the observed Atlantic profile in Fig. 8 quite well. The dimensionless gradient of the absolute vorticity of this wind distribution is given by Qo,=b-
U,,=b-2~(1-~'),
(7.18a)
where x = tanh 77. Thus, for this mean wind (7.3) becomes
4m-
b - 2 4 1 - 9) [.2+
Z+c
1*=o.
(7.19)
287
Quastgeostrophic Flows and Instability Theory -20
-30 0 '
- I Om/s
' I
I
IOOmb
300 500 700
050
20'N.
I
U 8SOmb
Majuro
EQ
I
1
(b)
A*€
he+
Torowo
I
lo"
Ae+
20'
. I Om/s
0
F I G . 8. Observed mean zonal wind profiles and structure of perturbations in the tropics. (a) Vertical distributions; (b) horizontal distributions over the Pacific; (c) mean velocity profile over the Atlantic.
H . L. Kuo
288
The boundary conditions for a,b is
+-to
as
(7.19a)
7 p - f ~ .
The velocity profile (7.18) and the vorticity gradient (7.18a) are represented in Fig. 9. From (7.18a) we find that the necessary condition for instability is
I bl < b, = 4 x 3 -3"
(7.20)
= 0.7698.
From Fig. 9 we see that, when (7.20) is satisfied, QOnchanges its sign at two values of 7,viz., 7 = r)cl and 7 = vC2.Thus two neutral solutions exist when 161 < b, . The phase speeds of these neutral disturbances are equal to the current velocities at vCland 7 c 2 ,which are given by the roots of the following cubic equation 2zC3- 2xc
+ b = 0.
Hence zCjis given by
+
cos[(% 2Trj)/3],
z,j = ($)1/'
j
= 1, 2,
3;
% = ~ 0 ~ - ~ ( - 2 2 7 ~ ' ~ b / 4 )7, ~ / 2 5 % < 7 ~ .
(7.21)
For b = 0 we have zC1= - 1, zc2= 0, zc3= 1; hence all three values are allowed. However, for b > 0 the magnitude of zC1becomes larger than 1 and hence is outside the velocity range and must be excluded, while the other two roots zCzand zcg lie within the range 0 5 zC25 3 -11' and zC35 1 and hence each one gives rise to a neutral solution. These 3-lI2 solutions are given by (7.22).
0, the stable waves are damped.
H . L. Kuo
310
C. INVISCID FINITE AMPLITUDE DISTURBANCE, p # 0, r = 0
It has been shown in Section VII1,B that, for u2 0, hence we also expect to find a second-order zonal mean created by the nonlinear transport in (9.2). Therefore we set
$,(y, T ) = r)2&2)
+ r)3&3) +
According to (9.14) and (9.10a) we have
*
.
(9.19)
Quasigeostrophic Flows and Instability Theory
311
Substituting these expansions in (9.5) and equating to zero the coefficients of 7 and 7 2 to zero we find the following system of equations:
+ K 2vL1 = 0, K3yL1’+ K4vL1)= 0,
K1vL1)
(9.2 1a)
)
(9.2 1b)
K,vL2)+ K2 vk2’= - k q‘(1) z
(9.22a)
9
K3v(22)+K4vL2)=-K441), i 2 Klv(23)+ K 2 v 4( 3 ) = _ _ 42 (2) k
K , vL3)+ K , ~ $ 3 = ) -
2
q,C 2 )
(9.22b)
+ qL1’+ FvP’,
(9.23a)
+ qL1’- Fcpi”,
(9.2313)
where q y ) = dq(*)/dTand
+
+
+
K1= ( U , U, - C ) ( L Y ~ F ) - /3 - FUc , K2 = - F( U4 U, - c), K , = - F( U , - c), K , = ( U4- c ) ( d F ) - /3 F U , , (9.24a) F ( r ) - ( F ct2)vr’. 4“’ = Fyr’ - (a2 F)v$’, 44(r) - ~2 (9.24b)
+
+
+
+
Eqs. (9.21a,b) simply yield the relations for the marginal solution obtained already (9.25a)
---_K3 -
F
(9.25b) * K, Notice that the left-hand sides of the inhomogeneous equations (9.22a,b) and (9.23a,b) are the same as in (9.21a,b), therefore their right-side member must bear definite relations to have a solution. Dividing (9.22a,b) by K , and K , , expressing q$l)in terms of vj through (9.24b) and making use of the relations in (9.2513) we then find u2
+F
-
(p - FU,)/( U , - C )
H . L. Kuo
312
where A=&). It can readily be shown that the two coefficients of A are equal, so that these two relations are identical. Hence we have (9.27) This relation shows that a phase different between i,h2 and i,h4 exists when A differs from zero. For convenience, we set q$) to zero so that we have
i,h2 = Re(qA ~ , b= ~ Re[yqA - iq2 -
+ q3yL3)+
(B+FUc)
k F ( U g + UC-c)"
* *
.)eik(t-Ct) sin my,
A+,,3p)i3)+
(9.28a)
...]eik(z-ct)sinmy. (9.28b)
The mean stream functions $2") and I&$) can now be determined. T o simplify the real forms of i,h2 and i,h4 we set
A = Reie,
t, = 5 + 0.
f = k(x - ct),
(9.29)
We then find
i,h& = [qR cos i,hk = {(y
c1+ q3yL3)+
* *
. ]sin my,
+ C17 d)qR cos t1+ Clq2R sin tl+ .
-
(*L 4 2 ) Y
= - (*L 4 ; ) Y = -
* *
(9.3Oa) }sin my, (9.30b)
kmFCl
2 RR sin 2my,
(9.30~)
where
(B + FUC) c, = kF(U4+ U,-C)~'
(9.30d)
Thus, the solution of (9.2) and (9.2a) gives
1-
sinh(2F)lI2Cy- (y2/2)] m c o ~ h ( F / 2 ) ~ ' ~ y , (F/2)1/2
(9.31)
where Ro is the initial value of R and
c -- (FkmC, +2m2)
*
(9.31a)
Qumigeostrophic Flows and Instability Theory
313
The corresponding change of the vertical shear is given by
Similarly, when (9.23a) is divided by K , and (9.23b) divided by K 4 , we find that the left sides of these two equations are the same and therefore their right-hand members must be equal. On making use of the results obtained above in the various functions involved and equating these righthand members one finds the following equations for R and 8:
R = Co2R- N oR(Rz- Ro2) + L2/R3,
(9.33a)
R28= L,
(9.33b)
where L is a constant of integration and co2
No
2k2/YF2 + 2F)2UC’
(9.35c)
= a4(a2
k2mC2 8(F + 2m2)(a2 2F) ((2F-
+
8m2 (2m2 F ) tanh(F/2)1’2y2 x (F/2)l
[
+
a”) a’+
] +2m2(2a2- F)].
The first integral of (9.35) is given by
1
- R2-
2
1
+
+N
(COz N oRO2)R2 - R4 = 2E, 4
(9.34)
where E stands for the initial amount of the total energy. Just as in the truncated nonlinear two-level baroclinic wave problem discussed by Lorenz (1963), R(T) is given by the elliptic function R( T )= R,,, dn[kR,,,(N/2)1/2(
T - TO)].
(9.34a)
The most interesting result revealed by this solution is that the disturbance and the basic flow change together rhythmically, keeping the total energy constant at every moment. The amplitude of the wave oscillates between the maximum R,,, and the minimum Rmin,which depends on the initial amplitude R,. The variation is such that the wave extracts energy from the basic state when R is increasing, but as R grows beyond its equilibrium value R e , the environment becomes increasingly more stable and finally the direction of the energy transfer reverses and the amplitude of the disturbance diminishes.
H . L. Kuo
3 14
D. VISCOUS EQUILIBRATION FOR / I = 0, r # 0 In nondimensional form r is the ratio between the square root of the Ekman number and the Rossby number. Here we take r = O(1). For this case we find from (9.13) that the critical vertical shear UC=AUmi,is given by
U, =
2ra k(2F - a2)1/2*
(9.35a)
We consider again that I AUI is slightly above U, such that
AU= Uc+A, ci =
A < U,
(A>O),
(2F - a2)l” aA. 2(a2 F )
+
(9.35b)
For sufficiently small A, the behavior of the disturbance is again determined by its interaction with the mean flow. However, (9.35b) indicates that for this case the slow time scale should be
T = At = q2t,
(9.36)
7 = All2.
Using the same expansions for & and qj as given by (9.17) and (9.21) we then find from (9.10) that the 7- to q3-order equations are given by
K,q$)
+ K2q$) = G$),
(9.37a) s=
+
1, 2, 3 (9.3713)
K3cp(a) K49)(4S) = Gf),
where K, , K 2 ,K3, K , are given by (9.24a) except j3 is replaced by ira2/k and G$) and Gc,“)are given by GL1)= G(1)= 0 (9.38a) (32)
(33)
=J63)
(33)
where
JP)
=J&2),
+
Gi2) =Ji2);
+ $’&I)
=Ji3) - F&)
(9.3813)
2
-- pL1)
k
- -z pi 1 ) k
’
(9.38~)
’
are the s-order contributions from the nonlinear transport
J’ of (9.1”) or (9.4). The two homogeneous equations for s = 1 yield the phase velocity c = U‘I +(U$).
(9.39)
Quasigeostrophic Flows and Instability Theory
315
Using this relation in the coefficients K , in (9.24a) we find
uc (a' Kl = 2
-F ) -
ira2 k '
K , = -K,
UC K --(F - a') 4 2
= - FUC
-
2 '
ira2 - -, k
(9.40a)
Notice that the amplitude ratio y is complex in this solution and that the upper wave is lagging. Therefore there is an energy transfer from the mean flow to the perturbation for this marginal solution, which is needed for the wave to be maintained against the viscous dissipation. This spatial structure of the wave accounts for the +order contribution of the nonlinear vorticity transfer J'. Thus, the equations (9.38a,b) for s = 2 give (9.41) When J is taken as J' =J -J, this equation is satisfied identically. The solution of the 'q order part of (9.5) gives
R2ak(2F' - -$p-
$W-
8rm
Similarly, the equations for s = 3 require the equality of 1/K2 times the right side of (9.37a) and l / K 4 times the right side of (9.37b). The result is the following first-order equation for the amplitude R of z,&): d dT
-R' = 2R2[kCoi- k2NR2],
(9.43)
where R 2 is the square of the amplitude and coi =
(2F - a2))"a , 2(a2 F )
+
N=
F %(a2
+F ) [4m2(a2- F ) + 3a2(2F-
a')].
(9.43a) The solution is therefore of the same form as that given by Stuart and Watson (1960), viz. (9.44) Thus, the amplitude of the wave approaches asymptotically a steady value
R' = Co,/kN.
(9.44a)
H . L. Kuo
316
There is also a linear phase change
e=
( k / 2 ) ~ (A= Re 1. (9.44b) This is simply a reflection of the fact that the phase velocity of the wave is equal to the mean current velocity 0 = U , ( U , A)/2, which differs from that of the marginal wave speed given by (9.39). T he cases with smaller Y , e.g., Y 4000 km) have their maximum intensities at upper levels while the shorter waves ( L < 2000 km) are limited to the lower part of the troposphere, therefore we may identify these shorter baroclinic disturbances with the cyclone waves and attribute the origin of the cyclones to the general baroclinic instability. However, surface extratropical cyclones usually form on a frontal surface, wherein a major part of the temperature or density contrasts between neighboring air masses is concentrated into a narrow transition layer which, on the scale of the large scale flow, amounts essentially to a surface of discontinuity in temperature or density. Thus, treating the cylone wave as a disturbance on the front will definitely bring the theoretical result closer to reality. T he frontal cyclone theory goes back to the Bjerknes-Solberg (1922) cyclone model, which depicts the extratropical cyclone as an unstable wave which develops from a small perturbation on a quasistationary front characterized by a cyclonic shear. A mathematical model was first formulated by Solberg (1928; cf. also V. Bjerknes et al., 1933, Chapter 14; J. Bjerknes and Godske, 1936), with two planes parallel to the frontal surface serving as boundaries for mathematical expediency. A physically sound and mathematically tractable model was later on formulated by Kotschin (1932), who also obtained a neutral solution of the system. A significant advance on the instability theory of frontal waves has been made by Eliasen (1960), who obtained solutions of Kotschin’s equations
Quasigeostrophic Flows and Instability Theory
317
for a range of values of the important parameters relevant to cyclone waves and demonstrated that the flow pattern given by the unstable solutions are very similar to the observed flow pattern in developing cyclones. The Kotschin equations have also been integrated numerically by Orlanski (1968), who also covered other ranges of values of the parameters and showed that the Margules type front is unstable for inviscid disturbances of all wave lengths except a number of isolated neutral disturbances, just as in Charney's continuous model of the baroclinic problem.
A. THEBASICSTATE The basic state is characterized by a balanced stationary front which separates two homogeneous fluids moving with constant velocities U, and U , in the x-direction in a rotating system, as illustrated in Fig. 15. The pressure distribution of this state satisfies the hydrostatic and the geostrophic relations (10.la) (10.lb) where j = 1 refers to the layer below and j the front. From (10.la) we find
Pdz, y ) =pz"(y)+gp@ P1@,
- x)
Y ) =Pz(h0)+gp1(ho - .) =Pl"(y) + g p l ( H - x)
=2
refers to the layer above
0I ho I 2I H, 0I
2
I ho < H ,
0I zI H,
X
FIG.15.
The frontal surface model.
y
y I 0, (10.2a)
a
0I y I
2 D,
(10.2b)
H. L. Kuo
318
where H is the total depth, D is the width of the frontal belt and p*(y) is the pressure on the top boundary. From these relations we find aj2
ap2*
aY
aY
dH dY
-= -+gp2 -=
u,,
-fpz
( 10.3a)
Therefore the slope of the front is given by
-dh0 _ -tancr dY
f fP --[pzU2-p1U1]=-(U2O -gAP gAP
Ul),
(10.4)
where Ap=pl-p2. We assume that the stratification is stable so that Ap > 0. We also take Ap as much smaller than either p1 or p 2 , so that p1 = p 2 = p can be used when they occur individually.
B. PERTURBATION EQUATIONS AND BOUNDARY CONDITIONS We assume that every flow variable is composed of an undisturbed part and a small departure, viz.,
Vj = (U,
+ u’)i + vj’j + wj’k,
p , = p j +p,’,
h = ho(yj
+ c‘,
j = 1, 2. (10.5)
The pertubation pressure p i is also taken as hydrostatic, hence we have PAX,
Pl(X,
y , z 2 , t)=P2+p2’(x,y, zz, t)=p”+gp,(H--z,)
y , z, 4 =I1 + P l ’ ( X , y , z1,t ) =P+ +gpz(H-h) +gpl(h-zz,),
hIz25H (10.6a)
0 i z 1 i h iH. (10-6b)
The linearized equations of motion and the continuity equation are
;( + uj $i.
1
- fv; = - -piz, P
1 +fUj’ = - -piy, P u;$+ vjy + w;, = 0. Vj‘
(10.7a) (10.7b) (10.8)
Quasigeostrophic Flows and Instability Theory
319
From the continuity of pressure across the interface we find
Pl'
-P 2 '
=g(Pl - P 2 ) 5 ' ( X , y ,
t),
a p j p Z=o
(10.9a) (10.9b)
so that the perturbation pressure is independent of x. From the equations of motion we also conclude that the horizontal velocities uj' and vj' are also independent of height within each individual layer, viz.,
(10.9~) Therefore the continuity equation gives w j as a linear function of x , viz., j
au
avji
(ax
ay)
-L+- ( x - x o j )
w.'--
(10.9d)
where xol = 0 and xO2= H . In addition, we have at the interface x = h ( 10.10)
Combining this equation with (10.9~)for z = h , we then obtain ah
(10.11)
where d1= ho i
d2 = - ( H -
ho).
(10.1 la)
The equations (10.71, (10.9a), and (10.11) constitute a closed system for the variables p j ' , u j ' , vj', and ['. As in other stability problems, we take the perturbations as represented by the product of a wave factor and their amplitude, viz., (u' v'
p' 5') = ( 0 9 , v*, p , [)ei',
(10.12)
where 6 = (kx + w t ) is the phase of the wave disturbance and w is the frequency. It is understood that only the real parts of these complex expressions are to be taken to represent the real variables. Substituting this representation in (10.7a,b) and solving for ujDand vj* in terms of p j we then obtain (10.13a) uj8 = (kwj*Pj -fPjv)lpFj > vj* wj#=
= i(kfPj - wj*Pupju)lPF~9
w
+k U j ,
Fj =f
-my2.
(10.13b) (10.13~)
H . L. Kuo
320
Substitutions of (10.12) in (10.9a) and (10.11) result in the following: Pl
-
Pz =d P 1 -
(10.14)
P2)L
+ v jdho - + d,(iku,* + vry) = 0.
(10.15) dY Inserting u,", v,*, and 5 from (10.13) and (10.14) in (10.15) we then find the two following equations for p , and p , : iw,*