BOWOM TURBULENCE
BOWOM TURBULENCE
FURTHER TITLES IN THIS SERIES 1 J.L. MERO THE MINERAL RESOURCES O F THE SEA
2
L.M. FOMIN
THE DYNAMIC METHOD IN OCEANOGRAPHY
3 E.J.F. WOOD MICROBIOLOGY OF OCEANS AND ESTUARIES
4
G.NEUMANN
OCEAN CURRENTS
5
N.G. JERLOV
OPTICAL OCEANOGRAPHY
6
V. VACQUIER
GEOMAGNETISM IN MARINE GEOLOGY
I W.J. WALLACE THE DEVELOPMENT O F THE CHLORINITY/SALINITY CONCEPT IN OCEANOGRAPHY
8
E. LISITZIN
SEA-LEVEL CHANGES
9
R.H.PARKER
THE STUDY O F BENTHIC COMMUNITIES
1 0 J.C.J. NIHOUL MODELLING O F MARINE SYSTEMS
11 0.1. MAMAYEV TEMPERATURE-SALINITY ANALYSIS O F WORLD OCEAN WATERS
1 2 E.J. FERGUSON WOOD and R.E. JOHANNES TROPICAL MARINE POLLUTION
13 E. STEEMANN NIELSEN MARINE PHOTOSYNTHESIS
1 4 N.G. JERLOV MARINE OPTICS
15 G.P. GLASBY MARINE MANGANESE DEPOSITS
16 V.M. KAMENKOVICH FUNDAMENTALS OF OCEAN DYNAMICS
17
R.A. GEYER
SUBMERSIBLES AND THEIR USE IN OCEANOGRAPHY AND OCEAN ENGINEERING
18 J.W. CARUTHERS FUNDAMENTALS O F MARINE ACOUSTICS
Elsevier Oceanography Series, 19
BOTTOM TURBULENCE PROCEEDINGS OF THE 8th INTERNATIONAL LIEGE COLLOQUIUM ON OCEAN HYDRODYNAMICS Edited by VJAQUES C.J. NIHOUL Rofessor o f Ocean Hydrodynamics, University of Lihge, Likge, Belgium
ELSEVIER SCIENTIFIC PUBLISHING COMPANY Amsterdam - Oxford - New York 1977
ELSEVIER SCIENTIFIC PUBLISHING COMPANY 335 Jan van Galenstraat P.O.Box 211, Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER NORTH-HOLLAND INC. 52, Vanderbilt Avenue New York, N.Y. 10017
Library of Congress Cataloging in Publication Data
Liege Colloquium on Ocean Hydrodynamics, 8 t h , Bottom t u r b u l e n c e .
1976.
( E l s e v i e r oceanography s e r i e s ; 1 9 ) Bibliography: p. I n c l u d e s index. 1. Turbulence--Cmgresses. 2 . Turbulent boundary layer--Congresses.* 3. Ocean bottom--Congresses. I. Nihoul, Jacques, C. J. 11. T i t l e . GC203.L53 1976 551.4'7 77-3546 ISBN 0-444-41574-2
Elsevier Scientific Publishing Company, 1977. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Scientific Publishing Company, P.O. Box 330,Amsterdam, The Netherlands
0
Printed in The Netherlands
V FOREITORD
I'hile
the atmospheric boundary layer has been extensively
investigated, the marine boundary layer above the sea floor although very similar in character
-
-
was, until recently, much
less well k n o w n ; the difficulty of making measurements in the sea, near the bottom, and the cost in equipment and human effort of any single experiment, reflecting on the calibration and the quality of the models. Bottom turbulence is however a determinant factor in such important problems as bottom friction and energy dissipation in marine circulation, sedimentation, bottom erosion, recycling of nutrients, trapping and release of pollutants, etc.. Understanding bottom turbulence is prerequisite for the development of accurate forecasting models of the marine systems which, nowadays, the extensive exploitation of the sea requires. In the recent years, the perfection of advanced techniques and the extension of the research effort have brought n e w interesting results an1 a more comprehensive insight into the characteristics of marine turbulence in the bottom boundary layer. Furthermore, the detection, in the bottom layer of the sea, of semi-coherent structures and the simultaneous study of the effects of the suspended sediments load have contributed, beyond the simple investigation of marine turbulence, to a better understanding of the general features of turbulence and such phenomena
-
still much debated
-
as drag reduction by
additives. The International LiSge Colloquia o n Ocean Hydrodynamics are organized annually.
Their topics differ from one year to
another and try to address, as much as possible, recent problems and incentive n e w subjects i n physical oceanography. Assembling a group of active and eminent scientists from different countries and often different disciplines, they provide a forum for discussion and foster a mutually beneficial exchange of information opening o n to a survey of major recent discoveries, essential mechanisms, impelling question-marks
VI and valuable suggestions for future research. The Scientific Organizing Committee of the Eighth Colloquium saw the desirability of bringing together, on the important topic of bottom turbulence, specialists from different fields, experimentalists and modellers, hydrodynamicists and sedimentologists. T h e present book which m a y be regarded a s the outcome of the colloquium comprises the proceedings of the meeting and specially commissioned contributions o n observations, parameterization and modelling of turbulence in the bottom boundary layer of the sea.
Jacques C.J. N I F O U L
VII
The Scientific Organizing Committee
of
the Eighth International
LiSge Colloquium onocean Hydrodynamics and all t h e p a r t i c i p a n t s w i s h to e x p r e s s t h e i r g r a t i t u d e to the B e l g i a n V i n i s t e r o f E d u c a t i o n , the N a t i o n a l S c i e n c e F o u n -
dation
of B e l g i u m , t h e
University
of
L i a g e and t h e O f f i c e of N a v a l R e s e a r c h for their m o s t v a l u a h l e support.
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IX LIST OF PARTICIPANTS
.
Mr
Y. ADAM, Institut d e MathEmatique, UniversitE de LiPge, LiPge, BELGTUM.
Dr
.
L. ARMI, l!oods Hole Oceanographic Institution, \.roodsHole Massachusetts, U.S.A.
Mr
.
A. BAH, Institut d e MathEmatique, Universitd de LiPge, LiPge, BELGIUM.
Prof. J. BOWMAN, State University of New York, Stony Brook, New York, U.S.A. Prof. G. CHABERT D'HIERES, Institut d e MEcanique, UniversitE Scientifique et MEdicale de Grenoble, St. Martin d'Heres, FRANCE. G.S. COOK, Systems Oceanography Branch, Naval Underwater
Dr.
Systems Center, Newport, Rhode Island, U.S.A. Prof
I
W.O.
CRIMINALE, Department of Oceanography and Geophysics
Program, University of Washington, Seattle, Washington, U.S.A. Dr.
A.M. DAVIES, Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merseyside, ENGLAND.
Prof. A. DISTECHE, Institut d e Zoologie, Universit6 de Lisge, LiSge, BELGIUM. Dr.
A. EDVARDS, Scot. Marine Biological Association, SCOTLAND.
Dr.
R.D. FLOOD, Woods Hole Oceanographic Institution, Wood s Ho 1e , Mas sac hu s e t t s , U
Mr.
. S .A.
J. FONT, Instituto d e Investigaciones Pesqueras, Barcelona, SPAIN.
Dr.
C 1 . FRANKIGNOUL, Max-Planck Institut fsr Meteorologie,
Hamburg, GERMANY. Dr. nr
.
C.M. GORDON, Naval Research Laboratory, Washington,U.S.A. P.K. KUNDU, School of Oceanography, Oregon State University, Corvallis, Oregon, U.S.A.
Miss
H. LAVAL, Institut d e MathEmatique, UniversitE de LiPge, Lisge, BELGIUM.
X Dr.
G. LEBON, Institut d e MathQmatique, UniversitE de Liege, LiBge, BELGIUM.
Prof
C. LE PROVOST, Institut d e Mgcanique, Centre National d e Recherches Scientifiques, Grenoble, FRANCE.
Mr.
A. LOFFET, Tnstitut de MathEmatique, Universitg de Liege, Liege, BELGIUM.
Dr.
F. MADELAIN, Centre National pour 1'Exploitation des OcEans, Centre OcEanologique de Bretagne, P,rest, FRANCE
Prof. J.C.J.
NTHOUL, lnstitut d e MathEmatique, Universitg d e
Litge, LiSge, BELGIUM. Prof. J . J . PETERS, Waterbouwkundig Laboratorium. Borgerhout, BELGIUM. Mr.
G. PICHOT, Institut d e MathEmatique, Universit6 d e LiBge, LiPge, BELCIUM.
Dr.
R.D. PINGREE, T h e Laboratory, Plymouth, Devon., U . K .
Prof. RAMMING, Universitat Hamburg, Institut fiir Veereskunde, Hamburg, GrRMANY. Dr. Dr.
H.W. RIEPMA, K.N.M.I., J.
D e Bilt, THE NETHFRLANDS.
RODRF., Oceanografiska Institutionen, Universitv of
Got henhur g , G 6 t ebo rg , SWEDEN. Dr.
F. RONDAY, Institut de MathEmatique, Universitg de L i e g e , LiGge, BELGIUM.
Mr.
Y. RUNFOLA, Institut de MathEmatique, UniversitE de LiPEe, LiSge, BE1,GIUM.
Mr.
U.J.
SALAT, Instituto de Investigaciones Pesqueras,
Barcelona, SPAIN. Prof. J . I . . SARMIENTO, Lamont-Doherty Geological Observatory, Columbia University, Palisades, N.Y., U . S . A . Prof. J.D. SVITI1, Department of Oceanography, University of Washington, Seattle, U.S.A. Mr.
J. S M I T Z ,
Institut d e MathEmatique, Universit6 de Liege,
Liege, DELGIUM.
XI Mr.
R.L.
SOULSBY, Institute of Oceanographic Sciences,
Taunton Somerset, U . K . Dr.
J.S. TOCHKO, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, U.S.A.
Dr.
VANDEF.BORGHT, Laboratoire de Chimie Industrielle, Universitd Libre de Bruxelles, BELGIUM.
Prof. G.L. WEATHERLY, Florida State University, Department of Oceanography, Tallahassee, U.S.A. Dr.
B. WILLIAMS, N.A.T.O.
Saclant A.S.W.
Centre, La Spezia,
ITALY. Dr.
A.J. WILLIAMS 3rd. Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, U.S.A.
Prof. M. WIMBUSCH, Nova University, Oceanographic Laboratory, Dania. Florida, U.S.A. Prof. R. WOLLAST, Laboratoire de Chimie Industrielle, Universitd Litre de Bruxelles, Bruxelles, BELGIUM.
This Page Intentionally Left Blank
CONTENTS
. . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . . LIST OF PARTICIPANTS . . . . . . . . . . . . . . . . . FOREWORD
V VII IX
A.M. DAVIES : The numerical solution of the threedimensional hydrodynamic equations, using a . B-spline representation of the vertical current profile
. . . . . . . . . . . . . . . . . . . . .
1
A.M. DAVIES : Three-dimensional model with depthvarying eddy viscosity I.D. LOZOVATSKY, R.V.
. . . . . . . . . . . . .
OZMIDOV, J.C.J.
27
NIHOUL :
. .
49
. . . . . . . . . . . . .
59
Bottom turbulence in stratified enclosed seas C.M. GORDON & J. WITTING : Turbulent structure in a benthic boundary layer A.J. WILLIAMS 3 r d & J . S . of
TOCHKO : An acoustic sensor
.
a3
. . . . . . . . . . . .
99
velocity for benthic boundary layer studies
J.C.J. NIHOUL : Turbulent boundary layer bearing silt in suspension (abstract)
R.D. PINGREE & P.K. GRIFFITHS : The bottom mixed layer of the continental shelf (abstract)
. . . .
101
G.L. WEATHERLY & J.C. VAN LEER : O n the importance of stable stratification to the structure of the bottom boundary layer o n the Western Florida shelf. J.D.
. . . . . . . . . . . . . . . . . . . . .
103
SMITH & S.R. McLEAN : Boundary layer adjustments to bottom topography and suspended sediment
. . .
123
L. ARM1 : The dynamics of the bottom boundary layer of the deep ocean W.O.
. . . . . . . . . . . . . . . .
153
CRIMINALE Jr. : Mass driven fluctuations within the Ekman boundary layer
. . . . . . . . . . . .
165
P.K. KUNDU : On the importance o f friction in two typical continental waters : off OreRon and Spanish Sahara
. . . . . . . . . . . . . . . . .
187
XIV J.P. VANDERBORGHT & R. WOLLAST : Mass transfer properties in sediments near the benthic boundary layer..
. . . . . . . . . . . . . . . . . . .
J.J. PETERS
:
209
Sediment transport phenomena in the
Zaire River
. . . . . . . . . . . . . . . . . .
22 I
G.L. WEATHERLY : Bottom boundary layer observations in the Florida current M.J.
BOWMAN & W . E .
. . . . . . . . . . . .
237
ESAIAS : Coastal j e t s , fronts,
and phytoplankton patchiness
. . . . . . . . .
255
J. SALAT & J. FONT : Internal waves in the N.-W. Africa upwelling G.S.
. . . . . . . . . . . . . . .
COOK, R.W. MORTON & A.T.
269
MASSEY : A report on
environmental studies of dredge spoil disposal sites
.....................
275
Part I : An investigation of a dredge spoil disposal site. Part 11: Development and u s e of a bottom boundary layer probe. SUBJECT INDEX
. . . . . . . . . . . . . . . . . . .
30 I
1 THE NUMERICAL SOLIJTION OF THE THREE-9IMENSIONAT. HYDRODYNAMIC EQUATIONS, USING A B-SPLINE REPRESELlTATION OF THE VERTICAL C IIRRENT
PROF IL E
A.M. DAVIES Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merseyside L 4 3 7 R A , England. ABSTRACT A numerical model
is described in which vertical current
structure may be determined using a n e w method involving expansion through the depth in terms of B-splines.
By way of a first
test, wind induced motion in a simple rectangular basin is computed, yielding surface elevations and vertical current profiles in good agreement with those obtained by Heaps ( 1 9 7 1 ) using an integral transform method,
The effect of varying eddy viscosi-
ty is investigated, considering the changes thereby produced in the wind induced vertical and horizontal circulations and in the surface and bottom currents. INTRODUCTION Two-dimensional finite difference models, based on the vertically-integrated equations of continuity and motion, have been used extensively in recent years to calculate tides and storm surges.
This approach is satisfactory for problems where
the primary aim is to calculate changes in sea surface elevation, but for problems involving water circulation, and particularly in engineering the calculation of the forces exerted by the sea on off-shore structures, a knowledge of vertical current profile
is required.
The use of a Laplace transform method to recover
the vertical current structure from a two-dimensional vertically integrated model has been proposed by Jelesnianski ( 1 9 7 0 ) and applied by Forristall ( 1 9 7 4 ) to the calculation of current profiles generated by a hurricane in the Gulf of Mexico.
This me-
thod is particularly suitable for determining the depth distribution of currents at a specific position for a given moment in
2
time, but for circulation studies the size of the computations would make i t less convenient. Finite difference models with grid boxes in both the horizontal and the vertical have been used recently in circulation studies (e.g. Leendertse 1 9 7 3 ) .
This model involves vertical
integration over each layer, and the use of a coefficient of interfacial friction.
The bottom stress, however, is expressed
in terms of the current in the bottom layer, a physically more realistic assumption than that employed in many two-dimensional models where the bottom stress is related to the depth mean current.
However, solutions in the vertical are only available
at discrete points, and the determination of a continuous velocity profile is not possible. Heaps ( 1 9 7 1 ,
1976)
has overcome this latter problem for
both the linear and non-linear hydrodynamic equations by expanding the two components of horizontal current in terms of depthdependent eigenfunctions with time-dependent, horizontallydependent coefficients.
Both surface and bottom boundary condi-
tions are satisfied in the limit as the number of terms in the
In practice, Heaps shows that the expansion tend to infinity. expansion converges very rapidly, yielding a technique which is particularly economic in computer time. In this paper a method is proposed in which the two components of horizontal current are expanded in terms of the product of depth dependent functions (B-splines), vary with time and horizontal position.
and coefficients which The determination of
the coefficients is accomplished by substituting these expan
-
sions into the two equations of motion and minimizing the resulting residual with respect to each coefficient in a least squares sense.
The surface and bottom boundary conditions are sa-
tisfied exactly by using linear combinations of B-splines. The application of the present method to the solution of the linear three-dimensional hydrodynamic equations, assuming a rectangular basin of constant depth with a constant eddy viscosity and a constant bottom friction coefficient, yield nearly identical solutions for wind induced motion to those obtained by Heaps ( 1 9 7 1 ) ,
providing an initial confirmation of the
3
accuracy and stability of the method.
The time variation of
both horizontal and vertical circulation induced by the wind is calculated for a number of cases having different eddy viscosi.ty, and the influence of eddy viscosity u p o n surface and bottom currents together with the induced circulation is examined. SOLUTIO'J
OF THE BASIC EQUATIOhTS U S I Y G AM EXPAMSIPN OF R-SPLINES
For a homogeneous fluid, neglecting shear stress i n the horizontal, the advcctive terms, and the equilibrium tide, the equations of continuity and motion may be written
where denotes time,
t
x,y,z C a r t e s i a m co-ordinates, form
2ft han :d set,
with x and y in the horizontal plane of the undistorbed sea surface, and z measuring depth below that surface, h
undisturbed depth of water,
E;
elevation of the sea surface ahove the undisturbed level,
u,v
components of the current at depth z
,
in the direc-
tions of increasing x,y respectively, P
the density of the water,
Y
the geostrophic coefficient, uniform and constant,
g
the acceleration due to gravity.
Also,
F,G denote internal shear stresses at depth z
,
in the
4
x,y directions respectively, given by au
-,
F = - p N - - az
av G = - P N - az
(4)
where N is a coefficient of eddy viscosity,-i general varying
,
with x,y and z lysis.
av at
but taken as a constant in the following ana-
Substituting ( 4 ) into ( 2 )
+ yu =
-
g
ac + a aY
(N
and ( 3 ) gives
av z )
(6)
To solve these equations i t is necessary to specify both
surface and bottom boundary conditions.
At the surface,
where F s , Gs denote the components of wind stress over the water surface in the x and y directions, suffix o denoting evaluation at z = 0. Similarly at the sea bed, z = h
,
where G B, F B denote the components of bottom friction in the x and y directions. Assuming a slip condition at the sea bcd :
where k is
a
constant coefficient, ( E )
gives
,
A n o slip bottom boundary condition, namely uh = vh = 0 , when employed with a coefficient of eddy viscosity which varies near the sea bed, is used in an extension of the present paper (Davies 137Ga).
However, for constant eddy viscosity, the rela-
tionships given by ( 1 0 ) are appropriate. Expanding the two components of velocity in terms of depth dependent functions Mr(z)
(4'th order B-splines) gives
The B-splines have a number of particularly useful features which make them a good choice as a set of basis functions.
They
have been used extensively for the accurate fitting of numerical data (Powell 1970), and yield very accurate solutions when used in solving linear hydrodynamic equations (Davies 1976b) and n o n linear partial differential equations (Davies 1 9 7 6 ~ ) . The incorporation of boundary conditions is particularly easy due to the piecewise nature of the functions. Points along the z a x i s , at which the E-spline changes from a z e r o - t o a non-zero function are termed k n o t s , Xr.
A fourth
CX
order B-spline Mr being non-zero over the interval X though at the points Xr-4
and X r
,
< z r-4 provided these knots are
single, Mr and its derivatives vanish. shows the region 0
F o r example, Fig. I
z < h divided into ten interior k n o t segand ( 1 2 ) , with
ments, corresponding to m = 1 3 i n equations ( 1 1 ) knots at 0 = X
<
XI
< X2
...
<
hg < XIo
= h.
In order to sup-
port the fourth order B-splines additional k n o t s are required <
at
0 and h , < A l l
<
AI2
<
< XI3
.
From this dia-
gram i t is obvious that only the first three B-splines, V 1 and V 3 and the last three M I ,
,
MI2
,
,
M2
M I 3 are non-zero at the
boundaries z = 0 and z = h respectively, enabling boundary conditions to be readily incorporated.
_ _
0
+-*I. W.CT
FIG,I.
~~
.
~
~~
~>-
h
1 .z11
sf. *D
: Distribution of B-splines and associated knots with depth.
6
F o r constant eddy viscosity N , the surface boundary condition for the u component of current becomes using (7) and ( I I ) ,
where
The positions of the k n o t s may be chosen arbitrarily.
A
uniform distribution w a s in fact used, and i n this case V 2 is zero.
Thus only the derivatives of M I and
M3
are non-zero at
the surface boundary giving, T,
= AIVl
+
A3V3
Rearranging ( 1 4 ) gives
The bottom boundary condition for the u component of current
I n this case M m-2 the boundarv z
?
Mm-l
?
M
m
'
- are
m'd dMm-2 and d z d z
non-zero at
= h Riving (16.2)
Rearranging ( 1 6 . 2 )
gives
where
c1
= v3/v
and C3 = k Urn I / (NV,
+ k LTm)
This can be written a s
A similar expression to ( 1 9 )
can be derived for the v component
o f current namely,
Substituting ( 1 9 )
and (20) into equations (5) and ( 6 ) , dropping
the bar on the 3 , m-2 and m-1 Iindependent o f z
+ g % ax
and
-
T
terms gives, f o r eddy viscosity
,
d2M1(z) +m-l Z Ar d2Mr(z)~ dz2
r=2
dz2
=
RI
8
The residuals R l and R
2
arise because expansions ( 1 9 )
and (20)
are only approximate solutions to equations (5) and ( 6 ) . However R , and R 2 can be minimized in a least squares sense with respect to the coefficients Ar and B, by making the integral of their product with each basis function zero over the region 0
2
z
2 h yielding the set of coupled differential equa-
t ions,
and
NT
1
where k
I
m- 1
h
MY(z)Mk(z)dz 0
= 2,3,
...
-
N
Br r=2 Z
I
h
o
M"(z)Mk(z)dz
'
= 0
(24)
m-I.
Fxpressing ( 2 3 )
and ( 2 4 ) in matrix form and rearranging
gives
and
which c a n be solved for vectors (a') and(b'),
the time
9
where
derivatives of A r and B r ( C
h
1
is a square matrix, with element r,k given by
(a)
is a column vector, r element given by A,
(a')
i s a column vector, r element given by
(e')
is a column vector, k element given by
I 0M$z)%(z)dz
dA r dt h
1 MI(z)Mk(z)dz 0
( f ' ) is a column vector, k element g i v e n by
h lIk(z)dz
1
0
(b)
is a column vector, r element is Br
(b')
is a column vector, r element is
(9)
is a column vector, k element given by
'5, dt
I
h VY(z)Mk(z)dz 0
h
( D ) i s a matrix, element r, k given by and
:T
=
dT
dTx , dt
T; =
$ ,
1 Y:(z)Mk(z)dz 0
d2Mr(z) M;(z)
=
dz2
FINITE DIFFERENCE REPRESENTATION I n order to solve equations ( 2 5 ) steps the vectors ference form. N(D)(b)
a'
and
b'
and ( 2 6 ) at discrete tine
have to be expressed i n finite dif-
I n this formulation the terms N( D ) ( a )
were centred in time giving for ( 2 5 ) ,
rence form : At
Multiplying ( 2 7 )
At
by
At
and rearranging gives,
and
i n finite diffe-
10
+
[
(C)
+
(D)
-
Let (CC) =[[C)
I
(s'~)) (.)]-I
Then multiplying ( 2 8 ) by (CC) and rearranging gives
Similarly equation ( 2 6 )
gives
Substituting expansions ( 1 9 )
and ( 2 0 ) into equation ( I ) ,
using
forward differencing for at;/at gives in matrix form the continuity equation
S(t+At)
=
-
6(t)-
At(h)T(f)
(31) dA2
where
(T)T is
the r o w vector
and
(h)T is
the r o w vector
dA3 dx
dB2
dB3
,...
1-
dBm- I dY
Using (29), (30) and (31) sequentially the coefficients A r , Br and elevation t; may be calculated through time using a time stepping procedure.
Then, from expansions ( 1 9 )
and (20). the
two components of current u and v may be calculated at any depth z
.
11
For a computationally economic solution, it is necessary to discretize in the horizontal plane,(x,y coordinates), though still retaining a continuous solution in the vertical.
A num-
ber of spatial finite difference schemes exist in the literature, any one of which c a n be used to solve these equations.
A
convenient method using a staggered grid system is described in NO((TH
G
+
x
r x - x - - x ~ x ~ x ~
O
+
x Q
O
Q
+
+
J Q
+
Q
+
Q
+
Q
I
X
+
Q
X
J
X
Q
+
Q
x
/
x
Q
+
Q
+
x Q
+
~ Q
+
+
+
Q
+
+
+
+
x C
+
+
Q
+
+
x
Q
+
+
Q
+
x
Q
+
+
Q
+
Q
+
Q
Q
+
Q
+
Q
Q
+
+
Q
+
Q
+
Q
x
Q
+
Q
+
Q
+
Q
+
x
Q
+
:+:I
+
Q
x
+
x
x
Q
1+
:+:I
+
x
G
+
x +
x +
x
G
x
+
Q
x
Q
+
Q
x +
Q
x
Q
+
x
Q
+
+
x
+
x
Q
+
Q
x
x
Q
Q
+
Q
x
Q
+
--x--x-.xpx
x
Q
x
Q
Q
Q
x
Q
x
Q
+
x
x
Q
x
x
Q
Q
x
x +
+
x x
X
I
+
x
+
Q
x
O
+
Q
x
+
x
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+
x
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x Q
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% x-L ~ x-x-
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.
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.
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WIND
x
+
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x +
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x +
+
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x Q
+
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x
+
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STRESS 0
x +
Q
x .
+
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+
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+
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SOUTH
F I G . 2 : The distribution of grid points within the rectangular basin, 0 a 6-point ; + a u-point ; x a v-point ; land boundary.
-
12 detail by I!eaps
(1971),
and an indication of its application to
the present method is given here,
Using a staggered grid sche-
me in the x,y) -plane, as indicated in Fig.2, with grid lines parallel to the co-ordinate axes, forming a rectangular mesh of sides Ax12 in the x direction, and Ay/2 in the y direction. Points on the mesh are of three types, a circle indicating an elevation point (S-point), a + sign, points at which the vector ( 5 ) is evaluated (u-points), and x sign, points at which the vector (IJ)
is evaluated (v-points).
The equations of continuity and motion ( 3 1 ) , ( 2 9 ) and (30) when discretized using this grid system given for continuity,
where
and
(zi)T
is a finite difference row vector having elements
(FilThas
elements
n being the number of columns in the grid.
depends i upon the x component of wind stress evaluated at u point i , and TYi
,
Where T,
y component of wind stress at v point i ,
For the u component of current,
of vectors. For v component of current,
13
(t+A t) - 5 (t+A t)
4y i+n
-gAt{ci
}(CC)
(f)-Aty(CC)
(C) ( $ j t ) ) + y
I
(4)
TY;(CC)
(34) = 0,25
where ^(t) T,
i
Using ( 3 2 ) ,
(33)
{Tit) + T,(t) + T,(t)+ Txjri} i+n+l i+n i
starting from a n initial state of
and ( 3 4
rest in which 5 , and b . ) are z e r o , for every grid point i , solutions for the elevation 5 , and coefficients a and b r ,
(ai)
describing the state of the sea, under the influence of wind stresses T, and Ty c a n b e calculated at progressive time steps Currents at any depth being calculated from equations ( 1 9 )
At,
and (20).
Incorporation of various boundary conditions being
accomplished in a n analogous manner to that described by Heaps (1971).
,
The vertical component of current w
at any depth z
,
be-
low the undisturbed surface, c a n be calculated from
Substituting equations ( 1 9 )
and ( 2 0 ) into ( 3 5 ) , and rewri-
ting in vector form gives,
where
(pT(s)
=
(?IT(@
+
(a)T
and
(F)T have
w(z)
(36)
been defined previously, and ( g ) h Mk(z)dz.
vector k'th element given by
is a
I
2
Evaluating w at elevation points i
,
gives for the finite
difference representation of ( 3 6 ) Wi(Z)
where
=
(ai) T
(9)
+
(a.) and (z.)are -1
-1
(hi)T ( g ) as defined previously.
(37)
14 EVALUATION OF B-SPLINES AND ASSOCIATED INTEGRALS The spline fonctions involved in the calculat on can read ly be evaluated using a numerically stable method formulated by Cox ( 1 9 7 2 ) . using the recurrence relationship
for s = 2 . 3 ,
...
n
starting with
- Xi-l)
when A 2
i-1
-c
z
> hi-]
c
'i
,
X .1
c -
2
n, the order of the spline. is four for all B-splines used in this paper, with the exception of the above and has been omitted from the notation. In order to solve the system of equations ( 3 2 ) . ( 3 3 ) , ( 3 4 ) and ( 3 7 ) . it is necessary to evaluate a number of integrals involving B-splines, which form the component elements of the various matrices and vectors, namely integrals of the form
,
Although these integrals can be calculated numerically using an appropriate quadrature formula, this method is particularly time consuming and inaccurate.
A better method is to
expand the splines in terms of Chebyshev polynomials. each knot interval X j of degree n-1
,
2 z 2
'j+l
Over
the B-spline i s a polynomial
and can be expressed as
n- 1 ..
and Ti(Z)
is a Chebyshev polynomial of the first kind.
The
double prime indicates that the first and last y are to be halved when the sum is evaluated. and Parker 1 9 6 8 ) :
The yji is given by (see Fox
15
with
Zk
=
cos
kn n- I
zk(A.+lz
and
k
=
A.)+
A.+ J
A.+I
2
D u e to the piecewise nature of the B-splines, many of the integrals which occur will be zero, and expressing these as sums of integrals evaluated between k n o t s , gives further zero integrals.
Using the transformation given i n ( 3 9 ) , the inte-
grals c a n be expressed in terms of integrals of Chebyshev polynomials.
Namely
and
Making the substitution Z
=
c o s 0 and using
these integrals cgn be readily evaluated, and all necessary matrices and vectors formed. APPLICATION T O WIND INDUCED MOTION IM A SI)fPLE RECTANGULAR SEA
In order to test the stability and accuracy of the method, the wind-induced surface elevations and currents in a simple rectangular sea having dimensions and rotation representative of the North Sea have been calculated.
Heaps ( 1 9 7 1 ) has applied
an integral transform method to this problem and presented results for the time variation of u and v components of current and sea surface elevation, together with steady state current profiles.
Results from this work are compared with his.
This
is also a good problem f o r studying the circulation induced by a sudden wind field, and the manner in which current profiles develop as energy is transferred by shear stress from the upper
16 water layers, to the lower layers, finally being dissipated by friction at the sea bed.
Three cases of varying eddy viscosi-
ty are considered here, and the influence of this parameter u p o n the initial horizontal and vertical circulation of the water, together with the development of the current profile at two points within the basin are considered in d e t a i l , Fig.2. tion.
shows the rectangular closed basin under c0nsider.a-
The water, initially at r e s t , is subjected to a uniform
wind stress of magnitude 1 5 dyn/cm2 in the direction of decreasing y Sea.
,
corresponding to a north wind blowing o v e r the North
Parameters used i n the calculation are Ay = 8 0 0 1 1 7 k m
Ax = 40019 k m y
= 0.44
h-'
P
P = O
h = 6 5 m 2
= 1.025 glcm
Q = -
g = 98/cm/s
1 5 dynlcm
2
Since the components of wind stress are constant, TX= P I p N and T = Q / P N at all grid points ; their spatial and time deriY vatives are zero. The coefficient of bottom friction k is a constant at 0.2 cmlsec.
In all the numerical calculations a time step A t
=
3 mins
was used and thirty five B-spline functions w e r e required i n the expansions to give a n accurate result.
Heaps (1971) only
required ten eigenfunctions to model accurately the motion in the basin.
The difference in the number of functions required
is understandable, in that the eigenfunctions used by Feaps are physically meaningful functions, representing vertical modes of the basin.
T h e B-splines, however, have n o physical
significance, being chosen because of the ease by which they make possible the incorporation of boundary conditions and their excellent numerical properties. i n the depth domain
no
Since at any given point
more than three B-splines are non-zero,
it is obviously necessary to have a high density of these functions (i.e. a large number of terms i n the expansion)
in
the
depth domain, in problems involving rapid changes of current with depth.
In solving a number of cases involving flow i n a
one-dimensional channel, for which analytical solutions, c o n sisting of low frequency harmonics exist, Davies (1976b) found
17 that fifteen to twenty B-splines were requ red to accurately model the motion.
In the present problem, particularly imme-
diately following the onset of the wind,
a
rapid change in cur-
rent occurs near the water surface ; const tuting a more complex flow structure, than in the problems solved by Davies using a smaller expansion.
Heaps ( 1 9 7 1 )
shows that in this
layer the contribution from higher modes is quite significant, hence accurate representation of these higher modes requires a large number of B-splines.
A good method for reducing the num-
ber of B-splines required in the expansion, but still retaining a high density of functions close to the surface would be to increase the knot spacing in the central depth region.
Al-
though this has not been investigated, i t should reduce the number of terms required in the expansion. The tiFe variation in surface elevation at point B for
N
= 650
2
cm lsec is shewn i n Pig. 3 . 10.
10.
lII1LI”R(II 10. SO. so.
70.
The first maximum in the
80.
“l++++-++ +’ -10
N.650 FIG.3. : Time variation of surface elevation 5 at the corner point B : !I = 6 5 0 crn2lsec.
18 elevation is approximately I0 cms higher than that calculated by Heaps, and for the other cases (N=130, 2 6 0 0 cm 2 /sec) a similar difference is obtained.
This can be attributed to the dif-
ferent methods of incorporating boundary conditions and to any small consistent differences in current profile which when integrated produce a significant effect upon the surface elevation.
The development through time of the two components of
current is shown in Fig. 4 (N that given by Heaps (1971).
=
6 5 0 ) and is nearly identical to
Figure 5 shows current profiles at
position A for the three cases after the establishment of a steady wind-driven circulation.
The numerical variation of the
current with depth is almost identical to that obtained by Heaps (1971) except that for the case P? = 1 3 0 cm 2 /sec, the magnitude of the u and v components of current at the surface are appraximately 5 cm/sec greater.
As
explained previously, in
cases where the current varies very rapidly with depth representing the current profile with B-splines becomes more inaccuThe present method yields surface elevations for the
rate.
steady state solution of 98 cms, 105 cms and 9 4 cms at point B for EJ = 130, 6 5 0 and 2 6 0 0 cm 2 /sec respectively, results within a
few centimetres of those obtained by Heaps (1971). Figures 6 and 7 illustrate the change in circulation pat-
terns which occur after the onset of the wind.
Five horizontal
sections through the basin are shown in terms of a normalised depth
S
=
Z/h ranging from the surface
(S
= 0.0)
of a quarter of the depth, to the bottom ( S
=
in intervals
1.0).
A vertical
North-South cross section through the centre of the basin is also shown, in which the vertical component of current w is multiplied by 100. Current vectors at nine points from surface to bottom are shown. The circulation produced by the wind field for a uniform eddy viscosity of 130 cm 2 /see is eiven in Fig. 6.
After 5 hrs
the u and v components of current have just passed their maximum values (see Figs. 4a and 4b) and away from the basin walls, at the surface, the current distribution in the horizontal is fairly uniform.
The magnitude of the current diminishes with
depth, and the current vectors rotate, illustrating a return flow in the positive u direction near the sea bed.
19 T I HE[ HRB 1 10.
20.- 0.
20- 30.
40.
60.
60.
70.
80.
DEPTH
N = 650 F I G , 4a. TInEl nR6 I 10.
20.
30.
40-
SO.
60.
70.
00.
DEPTH
-70
N = 650 FIG, 4b.
: Current c o m p o n e n t s u, v at t h e c e n t r a l position A showing the time variation of each component from surface to b o t t o m , s = 0 (0.25) 1.0 f o r k = 0 . 2 c m / s e c , 17 = 6 5 0 c m 2 / s e c .
FIG.4.
20
-19mi
n
FIG. 5a.
CI V W I A
FTG. 5 b .
FTG.5. : V e r t i c a l d i s t r i b u t i o n o f c u r r e n t at t h e c e n t r a l p o s i t i o n A at t = 300 h o u r s , s h o w i n g t h e e f f e c t u p o n u a n d v of 130, 6 5 0 , 2 6 0 0 c m Z / s e c , c h a n g e s in e d d y v i s c o s i t y , N k = 0.2 cmlsec.
-
21 ..
.. .. .. .. .. ..
......... ......... ......... .......... ........ ......... .........
... ... ... ... ...
......... ......... ......... .........
...........
. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....I.I I . .
I
0.60
.oo
rl/11.,:
1
......... ......... ......... ......... ......... ......... , WIND ......... ......... ......... .............. ......... ............. ......... ............. ......... ............. ....... ......... ......... , .. .. .. .. .. .. .. .. .. .. .. .. .. . ......... . . ......... ......... . . . . . . . . . . . . . . . . .
......... ......... ....... \.
......... ......... ......... ......... ......... ,..-..-. ......... ......... ......... ......... ......... ~
......... ......... .........
:,.i'[ I
0.25
. "" I
11°C
i
10
......... ......... ......... .........
......... ..... ,\......... ::::::::, :::::::::
,\ \, ,i.,,\.\,
......... ......... ......... ......... .......... ........
.\.>
::::::::I
::::::::: .....
,\
1 : : : : : : : : : : : : :
,.. ... ,,,
.........
, , , , , I _ _ .
0.00
pm?z?q[
...
_.-.\,
,\.,.\.\, , , .........
\,
I 0.00
I
U.26 - i
.oo
TIM
.
7K
I00 CM/S
N = I30
FIG.6. : Current vectors at various levels from surface to bottom, s = O ( 0 . 2 5 ) 1 . 0 and circulation in a vertical NorthSouth section throiigh the centre of the hasin (vertical component of current scaled by 100, origin of vectors shown by a cross) for N = 130 cm2/sec, k = 0.2 cm/sec.
The vertical cross section illustrates the very large surface current produced b the wind with some return flow near the edges of the basin
Two small vertical gyres at the northern
and southern end of the basin have also developed. The next two plots show the diminution of the surface current, at its energy passes through the viscosity effect deeper into the basin.
A vertical circulation pattern is gradually
established, in which there is
a
strong surface flow to the
south, and a return flow in the lower three quarters of the basin, the return flow having its maximum in the region 2 0 to 2 5 m from the surface.
Fig. 7 illustrates the time variation in circulation for
22
.......
__
.. ............. .. .
.. ... .. ,... ,,.. I , . . ,
I
,
,
,
,
* .
. I
I
1
o . ,
,, ,. .. .. ,
.....
.. ,. ,.,. .,. . . I
I
I
........
I 0.26 0.00
,.-----.
I \ \ \ \ \ . \
I I l I I l r I I I I I I )
l l l l l l ~
I 1 1 1
l l l l l r 111111 111111 111111
:::::::::I
,
,
,
,
*
I
I
.......
........ ........ ........ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . .,..... ....... ,,,,, , , , . 1 I
,
,
....... r e
--
. - - - - - I
I
I I
.........
6
8
, , , , I
. ........ ........ ......... ....... .. .. .. .. .. .. .. .. ......... . . . . . . . ............... ,.,,. ,.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..II ........ ........ . ,,,,,, * . , ,,,I ,,
~
I.oo
I!--:::::::::::::::) ............. 1 ................ I
llnE
.... . . . . .... ......... .... .... ..,. .... .... .,.. ,,,, .... .,.. .... ..... .........
......... .........
i
10
~-
.......... ........
......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... .........
=
......... .........
0.so
0.26
.oo
. _ - - - - _ ......... .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. ........ ...... ,.I_*_.
*
.. , ., ,, ....... .. .. .. .. .. ....
,
.
#
.
.
,
,
,
.
,
.
.
,
,
*
.
,
.
.
,
.
,
I
*
I
I
*
*
, , < a I
,
.
.
, . I .
... ......... ......... ......... L
:::::::::I 0.60
N = 650
I , , , . , # * *
0.76
- = 100 C W S
.
.
I
,.,*
WIND
,,,+
___....____-------. ............... . .. .. .. .. .. .................. .......... . , . , * . . < ................ ............... ...............I ......... ......... ................. . . . . . . . . . ................
,,,*
0.26
0.00
.........
....... 11ni
.. .. .. .. .. .. .. ... ... ... ... ... ... ...
, ,, , , , I , , , ,,,
1111111 1111111 I I I I I I I I I I I I I I
.......... ........
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. WIND .. .. .. .. .. .. .. .. .. ..\\\\\ .. .. .. .. .. .\,\\.\.-5:;:::< ,\\\\.\\-,,,,,,I .. .. .. .. .. .. .. .. .. J . . . . . \......... .......I .. .. .. .. .. .. .. .. .. . .. .. .. .. ................... .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
0.75
......... _____-........ .......
*
1111111l
.. .. .. .. .. .. .. .. ..
0.50
, , , a > , *
1111111l 1 1 I I
.. .. .. .. .. .. .. ........ ...... ,, , .. ,,,,*..
::::I
.. .. .. .. .. . . . .
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
, , , a
‘I
;I
4
I.oo
1lllE
i
7s
FIG.7. : ICurrent vectors at various levels from surface to bottom, s = 0 (0.25) 1.0 and circulation in a vertical NorthSouth section through the centre of the basin (vertical component of current scaled by 100, origin of vectors shown by a cross) for N 650 cmzfsec , k 0 . 2 cm/sec. 2 an eddy viscosity of 6 5 0 cm /sec. Again at the surface away
-
-
from the constraining effect of the basin walls, the current distribution is fairly uniform, though reduced in magnitude, compared with the previous case.
Due to the higher eddy visco-
sity of the system the currents‘ direction and magnitude do not change
so
rapidly wi-th depth.
The vertical cross section
through the basin illustrates the much smaller surface current, and the g-radual establishment of a return flow in the bottom half of the basin, a different situation to that occurring previously.
The small vertical gyres are present initially,
though the vertical component of current has been reduced by nearly a half.
23 A similar circulation pattern to this is obtained with an
eddy viscosity of 2 6 0 0 cm 2 /sec, though with much reduced current amplitudes and only slight variations of current in the vertical. These figures, together with Figs. 5a and 5b illustrate the considerable variation, particularly in surface current which occurs with change in the eddy viscosity parameter,
The depth
at which most of the return flow occurs, and the magnitude and direction of the bottom current vary significantly with eddy viscosity which obviously has a pronounced effect not only upon the steady state solution, but upon the time variation of the circulation pattern produced by the wind. This variation in the circulation pattern of the system, with eddy viscosity is particularly inlpbrtant in the calcolation of currents generated by storm surges, a situation where a steady state is never obtained. MATHENATICAL EXTENSION TO A BASIN OF V A R Y T N G DEPTH From the mathematical analysis of the preceding sections the formulation is such that for a basin having a variable depth, each grid point in the model would require different matrices (C),
(D) and vectors
(e) ,
( f ) and ( $ )
these involve integrals from 0 to h
since elements of
, h varying with position
(x,y) within the basin. As shown previously, these matrices are composed of integrals of the form,
Defining a variable 0
2
z
2 h , knots X
tegrals become
.i
s
such that
s
=
z/h where 0 5
being specified in the
s
s
2
I
as
domain, these in-
Their numerical evaluation being as described previously.
24
The equations of motion ( 2 5 ) , ( 2 6 ) and the continuity equation are now given by
where (C),
(D),
( e ) , (E)
over the region 0 to I .
and
(d)
involve integrals of splines
The representation in finite differen-
ce form being analogous to that described previously, h being represented by the depth at the grid point. CONCLUDING REMARKS A three-dimensional numerical model has been described in-
volving the representation of the vertical profile of current in terms of an expansion of B-splines.
Results from the mode?
agree satisfactorily with those obtained by Beaps ( 1 9 7 1 )
expres-
sing vertical current structure in terms of a series of eigenfunctions.
In this way, the accuracy and stability of the pre-
sent approach has been tested. Application has here been restricted to a simple rectangular sea of constant depth.
However, extension to the physi-
cally more realistic situation of a varying bottom topography has been described.
Bottom friction and eddy viscosity may be
varied in an arbitrary manner in the horizontal.
Thus here is
a considerable degree of flexibility in the choice of these parameters which marks a n advance on earlier formulations. Future work involving the calculation of current profiles in terms of B-splines, with a depth-varying eddy viscosity, is
25
presently in progress, and an extension to the case of eddy viscosity specified i n terms of vertical gradients of horizontal currents is being considered. ACKNOULEDGMENTS The author is indebted to Dr. N.S. Heaps for many valuable discussions and useful suggestions concerning this work. care with which Mr. R.A.
The
Smith annotated the diagrams is much
appreciated. REFERENCES Cox, M.G.,
1972. J. Inst. Maths. Applics.
lo,
134-149.
Davies, A.M., pub1 i she'd.
1976(a).
M6m. SOC. R. Sci. LiSge, Sbr.8, to be
Davies, A.M.,
1976(b).
Submitted J. Comp. Phys.
Davies, A.M.,
1976(c).
I n preparation.
Forristall, G.Z., 2721-2729.
1974. Journal of Geophysical Research,
2,
Fox, L. and Parker, I . B . , 1968. "Chebyshev Polynomials in Numerical Analysis", Oxford University Press. Heaps, N.S.,
1971. Mbm. SOC. R. Sci. LiSge, S6r. 6,
2,
143-180.
Heaps, N.S., 1976. Second International Meeting on Computing Methods in Applied Science and Engineering, Paris, D6c. 1975, Springer-Verlag, in press. Jelesnianski, C.P.,
1970. Mon. Weather R e v , , =(6),
Leendertse, J.J., Alexander, R.C. Report R-1417-OWRR.
and Liu,
S..
462-478.
1973. Rand
Powell, M.J.D., 1970. "Numerical Approximations to Functions and Data", Athlone Press, London, 65-83.
This Page Intentionally Left Blank
27
THREE-DIMENSIONAL MODEL WITH DEPTH-VARYING EDDY VISCOSITY A.M. DAVIES Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Nerseyside L43 7RA, England. ABSTRACT This paper extends previous work by Davies (1976a),
solving
the three-dimensional hydrodynamic equations for tides and storm surges using a n expansion of B-splines, by allowing depth variation of vertical eddy viscosity and studying the influence of such variation on vertical current structure.
The effect of
changes in the surface and the bottom eddy viscosities upon current profile, for wind-driven motion in a rectangular basin, is examined in detail. A no-slip bottom boundary condition is used, a logarithmic current profile then being pbtained close to the sea bed.
The
dependence of vertical apd horizontal circulations on eddy viscosity is also considered. INTR O D U C T IO?l Over the past twenty years a number of papers have been published on the variation of eddy viscosity with depth in tidal flows.
The importance of this viscosity in determining the
vertical profile of horizontal current particularly in the surface and bottom boundary layers, has been discussed. The experimental work of Lesser (1951), Bowden, Fairbairn and Hughes (1959),
Charnock (1959) and
on the structure and dis-
tribution of shearing stress in a tidal current has confirmed that the current has a logarithmic profile near the sea bed. To obtain this logarithmic profile from theory i t is necessary to postulate an eddy viscosity which changes with depth, having a low value close to the sea bed.
At the sea bed itself
a
boundary condition of no slip is assumed to apply, a condition associated with laminar flow there. Away from the bed, the
28
flow becomes turbulent.
Bowden, Pairbairn and Nughes (1959)
have assumed that the coefficient of eddy viscosity increases linearly with increasing height from the bed through a distance 0.14h (where h denotes the total depth of water), above which it remains uniform to the surface.
Kagan (1966) uses a
similar model in calculating the structure of the tidal current in the Southern Bight of the North Sea, obtaining good agreement between calculated and observed values. The change in eddy viscosity brought about by wind stress has been considered by MunK and Anderson (1948).
According to
their findings, the surface eddy viscosity rises rapidly from between 4 and 6 cm
2
corresponding to wind speeds below
/see
2
lmfsec to above 9 5 0 em /see for winds greater than 1 5 mlsec. The increase in tRe depth of the surface turbulent boundary layer for the higher wind speeds has been investigated by Hansen (1975) for a stratified lake, and, by Nihoul (1973) for the upper ocean.
Bind driven currents in shallow water, using
a linear variation of eddy viscosity, diminishing with depth, have been calculated analytically by Thomas (1975).
A three-
dimensional numerical method has been used by Liggett (1970) to determine the sensitivity of the wind driven circulation in a lake to changes in both horizontal and vertical eddy viscosity. Johns (1966) has modelled analytically the tidal profile in a river estuary, using a depth variation of eddy viscosity, d i minishing with depth.
Bowden and Hamilton (1975) use a numeri-
cal model to calculate circulation and mixing in a tidal estuary, relating the eddy viscosity linearly to depth and depth mean horizontal current.
In a previous paper (Davies 1976a) a method was proposed for the solution of the three-dimensional hydrodynamic equations by expanding the two components of horizontal current in terms of depth dependent functions (R-splines),
and coeffi-
cients which vary with time and horizontal position. This method yielded a continuous current profile in the vertical. A slip bottom boundary condition was used and the coefficient of eddy viscosity was assumed to be uniform through the
29
vertical.
Here the method is extended to permit an arbitrary
variation of eddy viscosity in the vertical employing a noslip bottom boundary condition,
With this new formulation i t
i s possible to model the wind-induced turbulent surface layer and the bottom Boundary layer, The method is applied here to the motion set up by wind in a closed rectangular basin of constant depth.
A simple varia-
tion of eddy viscosity with depth is used, and the effect on the not on of changes in the magnitude of this viscosity in the surf ace and bottom boundary layers is investigated. The dimensions of the basin and the wind strength are identical w th those used By Davies (1976a). permitting direct compar i son of the present results with those obtained with constant eddy viscosity and a slip bottom boundary condition. SOLUTION OF THE HYDRODYNAMIC EQUATIONS The equations of continuity and notion for a homogeneous fluid, neglecting non-linear terns, and shear in the horizontal, may be written as
loud^ h
a
a h + j---JO~dz
ac -
+ at = 0
where t
denotes time,
x,y,z
a left handed set of Cartesian coordin tes, rith z
the depth below the undisturbed surface and
x,y coordinates in the horizontal plane, h
undisturbed depth of water,
5
elevation above the undisturbed depth,
30
u,v
x and y components of current at depth z
P
the density of water,
Y
geostrophic coefficient,
g
the acceleration due to gravity.
,
denote internal shear stresses at depth z
A l s o F,G
,
in the
x,y directions respectively, given by
with N ( z ) ,
the coefficient of vertical eddy viscosity, general-
ly varying with x,y and z depth dependent (i.e.
,
although here taken as solely
a function of
Substituting (4) into ( 2 )
aa ut -
yv
=
-
g
ac + a ax az
2).
and ( 3 ) gives
(N(z)
x) au
and
The surface boundary condition is given by
where F s
,
G s denote the components of wind s ress
cting o n
the water surface in the x and y directions, suffix 0 denoting evaluation at z =
0.
Similarly, suffix h will denote evalua-
tion at the sea bed z = h
.
The method described here is sufficiently general to allow for a n arbitrary depth variatior, of eddy viscosity, thus enabling turbulence characteristics of the bottom layer to be modelled with the use of a no-slip boundary condition (8) h = o Expanding the two components of velocity, and the eddy viscosiU h = V
ty in terms of m depth dependent 4'th order B-splines gives m
Mr(z)
31
The advantages of using B-splines a s basis functions have been described in detail in Davies (1976a) and references there cited. For the case of eddy viscosity varying with depth, given by the function N ( z ) ,
the coefficients Er can readily be obtained
as a , s o l u t i o n to the matrix problem (C)(E)
=
(f)
(12)
h where (C) is a matrix, element i,j given by
(E)
a column vector j
(f)
a column vector j th element
M.M.dz
o l J
th element E j
h
f
N(z)Mj (z)dz 0
The integrals involved in matrix (C) can be readily evaluated (Davies 1976a) and the matrix inverted by one of a number of standard numerical techniques.
Integrals in vector ( _ f )
,
for continuous, smooth functions can be accurately evaluated using Gauss quadra.ture
if analytic solutions cannot be found.
The surface boundary condition for the u component of current becomes using (7) and (9)
where
being the derivative of the r th B-spline at z = o
In practice a uniform distribution of knots was used throughout the region of support and from sea surface to sea bed. With this distribution all the B-splines except the first
32 and the third Rave zero derivatives on the boundary giving : Tx = A,Vl + A3V3
(14)
The bottom boundary condition for the u component of current using (8) and (9) gives m E Ar(x.y,t)Mr(z) r=l
I
(15)
- 0
z=h
Since Mm-2 , Mm-] , Mm are the only non-zero B-splines at z = R , equation ( 1 5 ) gives
where
Using ( 1 4 ) and ( 1 6 )
to eliminate A 1 and Am from (9) yields
in which
In more compact notation this gives u(~,y,Z,t) =
TX M](z)
m-3 + A2M2(z)
+ A3M3(z)
“1
-
+
Am-2Mm-2(z)
+
+
1 A M r=4
( 2 )
(18)
Am-IMm-](2)
where
and
A similar express on to ( 1 8 ) be derived.
Thus
for the v component of current may
33
where T
= - -
Y
GS
NO
Substituting ( I I ) ,
(18)
and ( 1 9 )
into equations ( 5 )
and ( 6 )
and minimizing resulting residuals in a least squares sense. dropping the bar from 3rd , m-2 and m-I terms, gives
-
!"[0
'
j=l
dM. (z)
Ej
m
'{% T
dM1(z)
m-I
7
d2M1(z)
+
Mk dz ' y}] dz
r=2Ar
m-1
+ dz2
d2Mr(z) Ar -} M r=2 dz2
(z)
Z
where k = 2,3,...
k (z)dz = 0 m-1
A similar equation may be derived from the v-equation of motion, and writing these equations in matrix form gives : for the u-equation of motion :
and for the v-equation of motion :
m
where
34
h
(C) is
a
matrix, with element r,k given b y
I Mr(z)Mk(z)dz 0
( 5 ) is a column vector k th element given b y
I
h MI(z)Mk(z)dz 0
(z) is
a column vector k th element given b y
1h Mk(z)dz 0
(a), Ar
(g'),
,
(p)
and
(!I),
column vectors of coefficients
B r and their derivatives (denoted b y dashes) with res-
pect to time,
(cj)
[:)=
is a column vector, elements given b y h dM1 dM1 h dMm d M 1 M2dz d z d z M 2 dz, dz dz
...
I --
...I - -
dm-2
I
h
d2MI MI 7 M2 d z 0 dz
,...
...I
h
dLMl Mm
7M2dz dz
0
+
. . .I
h
d2Ml
Mm 0
and the i th r o w of matrix (D)is
-.
7 Mm-ldz dz
-
given b y
h dMI d M I =Midz
h d V l dV
1
0
...I -m Y . d z d z dz
h dwm dM I rMidz
. ,.
I0
I
h dMm dMm M.dz
35
-
-
h d2Ml ioMl 7 M i d z
...!
h
dLM,
h
d2Mm M l y Midz 0 dz h
dLMm
I Mm Midz.. . I Vid: Mm dz 0 dz 0
Writing equations (21) and ( 2 2 )
(a)
the terms (D)
and (D)
(b)
i n finite difference form, with
centred i n time, gives, with some
rearrangement,
where
The terms [D) (a) and ( D )
(b)
initially were evaluated at
the lower time level, however this scheme proved to be unstable with a time step greater than 3 0 secs.
Centring these terms in
time enhanced the stability of the scheme and a time step of 180 secs became possible. The continuity equation i n matrix form,using forward differencing for aS/at i s given by
where
( -a )T
and
(h)T is
-
dTx 1 is a r o w vector [dx Vl
' dx dA2
' * "
1-
dAm- I dx
a r o w vector
Using equations ( 2 3 ) , ( 2 4 ) and ( 2 5 ) stepping procedure the vectors (a(t+At))
sequentially in a time and (b(t+At) ) and
36 elevation 5 may be generated numerically. vectors ( 5 ) and
(b)
The components of
namely coefficients Ar and Br are substitu-
ted in (9) and ( 1 0 ) to yield components of current at depth z. The finite difference representations in ( 2 3 ) .
( 2 4 ) and
(25) are identical to those used for the case of uniform eddy viscosity (Davies 1976a).
Calculation of the vertical compo-
nent of current, and the numerical algorithms used ,to evaluate the various integrals, have been described i n detail previously (Davies l976a). EFFECT OF THE DEPTH VARIATION OF EDDY VISCOSITY UPON CURRENT PROFILE The wind induced surface elevations and currents i n a simple rectangular sea having dimensions and rotation representative of the North Sea have been calculated previously (Davies 1976a) assuming a uniform vertical eddy viscosity and a slip boundary condition a t the sea bottom.
Using a n expansion of
3 5 B-splines and a time step of 3 minutes, f o r different values
of the viscosity, calculated current profiles through depth were found to be in good agreement with corresponding profiles determined by Heaps (1971) using a n integral transform method. From these calculations i t w a s evident that a sufficiently large number of spline functions had been incorporated to represent accurately the steep gradients in current profile near the surface occurring with low values of eddy viscosity (130cm2 /sec). For more rapidly changing profiles the method would require a larger number of terms in the expansion.
The numerical experi-
ments described in the present paper again deal with wind induced motion in a rectangular basin but consider a depthdependent eddy viscosity.
A time step of 3 minutes and a n ex-
pansion of 35 B-splines are again employed. The shorter sides of the sea basin are x directed and of length 4 0 0 k m , subdivided by 9 mesh lengths ; the longer sides are y directed of length 800 k m and subdivided b y 17 mesh lengths.
The basin initially at rest is subjected to a sudden
wind stress in the direction of decreasing y of magnitude
37 15 dyn/cm
2
.
This represents a north wind blowing over the sea.
Parameters used in the calculation are A x = 400/9 k m ,
by = 8 0 0 / 1 7 k m ,
0.44 h-’,
y
p
= 1.025 g/cm
h 3
,
=
6 5 m,
2 g = 9 8 1 cm/sec
F s = O , The components of wind stress F s eddy viscositv at the surface, N o fore so also are
T,
,
No) and
( = F,/p
,
G s and the value of
are constants and there-
Ty ( = G , / p
No)
.
The assumed law governing depth variation of eddy viscosity is shown in Fig. I .
This consists of linear variations i n the
surface and bottom layers connected with a uniform middle value.
The depth of the surface layer, dl
tant at I I metres.
For N 1
,
,
was maintained cons-
the coefficient of eddy viscosity 2
in the central depth region, a constant value of 6 5 0 c m /sec was used.
In tidal f l o w , Bowden, Fairbairn and Hughes (1959) the bottom boundary layer, d 2 ,
suggested that the thickness o f should take the value 0.14h(9.1
metres i n this case),
To main-
tain symmetry with the surface layer w e , i n fact, took d2= Ilm.
SURFACE
Z
SEA BED
NFIG.].
: Depth variation of coefficient of eddy viscosity.
By choosing different values for No and N 2
,
the surface and
bottom values of eddy viscosity, the importance of these values in determining the current structure in the basin w a s calculated. Numerical experiments were carried o u t , each generating the wind induced motion in the basin from a state of rest for a prescribed viscosity variation of the system shown in F i g . 1 .
In these experiments, d l
,
d 2 and N I were held constant while
No and N 2 were varied as shown i n Table I. Initially the coefficient of surface eddy viscosity N o was 2 varied, N I and N 2 being k e p t constant at 6 5 0 c m /sec and 1 3 0 2
c m /sec respectively. Expt. (i) With N o
=
2
2 c m lsec, a very low value, the motion of the
water reached a near steady state condition after approximately 40 hours.
Elevation at the south-west corner of the basin,
at Point B in Fig. 2 of Davies (1976a), ut cnis I -20. -10. 0 .
10.
attained a maximum of U( cn/s 1
-10. 0.
TIHE
N
= 76 a 190.86O.lSO
FIG. 2a.
10.
TInE = 7s N = Zr860.130 FIG. 2b.
39 U( CW8 I -10. 0 .
U(W8I -LO. -10. 0 .
10.
10.
TIM s 76 W D 660~660~130
VICN8)
-so. -LO.
-10. 0 .
TIM W
=
10.
to.
76 2600~8600190
D
F I G . 2c.
ui cn/8 ) -to.
-10.
0.
TInE
D
7s
= 6SO.6SO.13O F I G . 2d. W
U(CW6) 0.
10.
- t o . -10.
10.
V(CIV8)
-so. -LO.
vicn/81 -10. 0 .
-30. -40. -10. 0 .
10.
T TInE W
8
=
10.
to.
T
to.
TIRE = 7s I = 2600.660.3.t6
76 t800.660.20
FIG.2e. FIG.2f. FIG.2. : T h e v e r t i c a l p r o f i l e s o f u a n d v a t t h e c e n t r a l p o s i tion A at t = 75 h o u r s , f o r v a r i o u s e d d y v i s c o s i t y d i s t r i b u tions N O , N I , 7J2
.
40
23 c n and a steady state value of 17 cm.
T h e diminution of
the u and v components of current with depth w a s particularly rapid, especially for the v component of current, as illustrated in Figure 2a for Point A at the centre of the basin.
The
motion of the water consisted of a strong surface flow (approximating to 9 0 cm/sec) i n the top two metres, in the direction
of the wind, with a small return f l o w in the bottom half o f the basin.
Although probably not physically meaningful, the flow
pattern i s one to be expected with this particular eddy viscosity distribution.
T h e low viscosity in the upper layer per-
mits this layer to slip freely over the lower layers, with little downwards transfer of energy, Expt. (ii) Increasing N o
2 to 1 3 0 c m /sec permits the transfer of energy
from the wind to the lower water layers to increase,
The ele-
vation at B reaches a maximum of 198 cm, but the system L s still more heavily damped than the results obtained using a 2 uniform coefficient of eddy viscosity N = 6 5 0 c m /sec and a bottom slip condition with coefficient of friction k = 0.2 cm/sec The depth variation of u and v
,
at Point A after 7 5 hours is
shown in Fig.2b. illustrating a high surface shear in the v component of current, and a logarithmic profile for both c o m p o nents of current in the bottom boundary layer. Expt. (iii) 2
Taking N o = 6 5 0 c m /sec is particularly interesting since the results can justifiably be compared directly with those ob2 tained using a uniform eddy viscosity N = 6 5 0 c m /sec with a bottom slip condition (Davies 1976a).
The time variation of
surface elevation at Point B is nearly identical to that of the previous case, the height of the first maximum being reduced by something less than 2 cms.
T h e damping of the sea surface ele-
vation, Fig. 3, is very similar to that obtained by Heaps 2 (1971) using a n eddy viscosity N = 6 5 0 c m /sec and a bottom coefficient k = 0 . 4 cmlsec. The depth variation of the two horizontal components of
41
N=650,650,130 FIG.3. : Time variation of surface elevation 5 at the corner point 3 : N o = 6 5 0 , M I = 6 5 0 , PI2 = 1 3 0 cm2/sec. current are illustrated for Point A i n Figure 2c.
The compo-
nent of surface current is reduced by approximately 2 0 cm/sec, with a comparable reduction i n the u component of surface current, from the u,v obtained in Expt. (ii).
Comparing Fig. 2c.
with Fig. 2b., i t is evident that the higher value of N o has
42
reduced the shear in the upper boundary layer, though there is no
observable change in the current structure in the lower
layer.
The current profile within the upper three quarters of
the depth is very similar to that obtained with uniform eddy viscosity (N = 6 5 0 cm 2 /sec, k = 0.2 cmfsec), (Davies 1976a), although within the bottom boundary layer the results are markedly different.
The current profile within this layer, calcu-
lated with a uniform eddy viscosity distribution, was not logarithmic, and changed in magnitude by less than 1 cm/sec through the layer.
Fig. 2c. clearly demonstrates the logarithmic va-
riation calculated assuming a linear variation of viscosity within this Layer, illustrating the importance of the bottom boundary condition and eddy viscosity distribution in determining the bottom profile. Expt. (iv) 2 Increasing the surface eddy viscosity to 2600 cm /sec a physically realistic value for a wind stress of 1 5 dyneslcm (Munk and Anderson 1 9 4 8 , Neumann and Pierson 1966) produced a less than I cm change in sea surface elevation throughout the whole period at B , although the magnitude of the v component of surface current at A is reduced by approximately as illustrated in Fig. 2d.
10
cm/sec
The current profile in the bottom
half of the basin is nearly identical to those obtained previously, illustrating the slight effect that changes in surface eddy viscosity have upon bottom currents. T o examine the effect of varying bottom eddy viscosity a series of numerical experiments was performed in which the coefficient of bottom eddy viscosity was changed, the surface and central region eddy viscosity remaining at 2600 cm 2 /sec and 6 5 0 cmL/sec respectively. Expts. (v) and (vi) 2 2 Reducing N2 to a value of 6 5 cm /sec from one of 1 3 0 c m / s e c produced no noticeable change in results, although a further reduction to 2 0 cm 2 /sec produced a visible change in the current profile close to the sea bed, Fig. 2e, characterised by a
43
high shear layer of rapidly changing current within the bottom three of four metres.
The damping of the sea surface elevation
was also reduced, giving a time variation at Point B very simi2 lar to that obtained with uniform eddy viscosity ( N = 6 5 0 cm/sec
k
=
0.2 cm/sec).
The damping of current at Point A is similar
to that obtained for the uniform eddy viscosity case, and therefore i t is evident that this value of bottom eddy viscosity dissipates energy at a similar rate to the constant eddy viscosity model with a linear bottom friction coefficient k = 0.2 cm/sec.
The variation of horizontal current distribution, 75 hours from the onset of the wind, with depth is illustrated in Fig.4
-TI
.. .. .. .. .. .. .....
-l
. .. .. ....... .. ..
,.... ,
I
.
,
,
.
,
I
.
I
,
I
.
0.75
I .on
.. .. .. .. .. .. .. .. ..
,
.
.. .. .. .. .. .. .. .. .. ... ... ... ... ... ... .........
.. .. .. .. .. .. .. .. .. ..
__
I
0.
0.25
0.00
.........
.. .. .. .. .. .. .. .. ..
.
,
,. .* .. .. .. ,
......... .. .. .. .. .. .. .. .. .. ......... ......... .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. ......... ......... .........
.........
.
,,... ,
.. .. .. .. .. .. .. .. .. .........
~~
TIME = I
.. .. ._. ._. ._. 1- - _ ........
........ ,
,
,
,
.
I
.
.
......... , , , ,, , ,
.. .. ,. . . . ,. ,. . I
,
,
.. .. .. .. .. .. .. .. .. ......... 0.00
I: ..: :..:..: .. .. .. . _ . 0
,
I
I ... .. ..: :._._ .. . 1
s
If:
.
.
:::::::
I______.
..... .~,,,,,,
........ .. .. .. .. .. . . . . .. .. .. .. .. .. .. .. .. ......... .........
I . . . . . . . . I , , , . , ,,
, , , ,, ,, ,, , ,. ,.. 1
1
1 I , , , , . ,
I
I .
......... .. .. .. .. .. .. .. .. .. ... ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. I
0.75
0.50
0.25
4
.. .. .. .. .. .. ...... ......... .. .. .. .. .. .. .. .. ..
.on
TIME = 10
.. .. .. .. .. .. .. .. ..
.. .. .. .. .. .. .... ........ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ... ... ... ... ... ... ... ........
.. ,. ,. .. .. ,. ..,. ........ I 0.00
... ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ...... .........
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .........
.. .. .. .. .. .. .. .. .. -~
-
0.50 TIME = 7 5
........
........ .
,
,
,
,
.
.
,
,
#
.
.
,
,
,
,
,
,
.
I
,
#
,
,
,
I
*
,
,
,
,
,
I
I
.
I
I
.
,
I
,
I
I
,
,
,
.
.
I ,
d
, I
*
I
(
. I
I
I I
.
I
. .. .. .. .. .. .. .. .. ,, , ,,,,,, 1
.
,
,
,
,
I
I
.
I
I
I
I
,
,
,,,,,*.*
........ 0.75
.........
... ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. ......... .. .. .. .. .. .. .. .. .. ......... ......... ......... .........
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. I .on
= 100 CM/S
N=2600,650,20
FIG.4. : Current vectors at various levels from surface to bottom, s = 0 (0.25) 1 . 0 , at times t = 5 , 10, 7 5 hours. Origin of vector marked by small cross.
44
for five depth levels, corresponding to s = z/h = O.O(O.25)l.O. The time variation of the current pattern in a vertical NorthSouth cross section through the centre of the basin (the vertical component of current being scaled by a factor of 100) at nine equidistant points from sea surface to sea b e d , is given in Fig.5.
The vertical cross sections through the basin,
illustrate the initial high currents in the upper layer, accompanied by up-welling at the southern end of the basin, and down-welling at the northern end.
After 7 5 hours only the ver-
tical components of current adjacent to the basin walls have a n appreciable magnitude, the circulation pattern being characterised by a flow in the upper layer, at a slight angle to the WIND
1,I
\ \ \ \\\\\\
!
\\I:::::'/'/"' ; ;
1
\\\\\\-"'""
, \ \ \ \ \ \ \ I t \ \ \
............... i ' . . .. . . . . . . . . . . 1 ,
& $ * \ . . , , # , , , ,
l
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. I
=
TIHE
5
WINO
,-. \...---...,... ..\...----..,,\....-.,,,,,.
!
.......... ............... ............... I I
,
I
,
,
I
.
.
a
.
b
.
.
.
,
,
,
,
,
.
................. ( . . . . . . . . . . . . . . . I
=
TIHE
10
WIND
-,"""""""' _--------------. .. .. .. .. .. .. .. .. .. .. .. .. ...... ............... ......,........ I ,............--.................
I
.
.
.
-
.
.
.
.
.
.
.
I
.
.
.
I I
TIME Y
=
75
= 100 C W S
N=2600,650,20 FIG.5. : Time variation of current i n a vertical North-South section through the centre of the basin, the vertical component of current has been scaled by a factor of 100.
wind direction, with a return f l o w in the bottom half of the basin, falling rapidly to zero within a few metres of the sea bed. Expt. (vii) A further reduction i n N2 to 3 . 2 5
2 c m /sec has a pronounced
effect on the current profile close to the sea bed, producing a very high shear in the bottom two metres, Fig. 2f.
The time
variation of sea surface elevation, Fig.6, illustrates the very small damping produced with this low value of eddy viscosity at the sea bed. The series of numerical experiments clearly illustrates the importance of surface eddy viscosity in determining the transfer of wind energy from the upper surface layer into the interior of the water mass, and the magnitude of the surface wind induced current.
The value of eddy viscosity at the sea bed
clearly plays a n important part in determining the current profile and magnitude close to the sea bed, and
affects the dam-
ping of the motion w i t h i n the basin. Although the formulation presented here for the solution of the three-dimensional hydrodynamic equations with depth varying eddy viscosity has involved a basin of constant depth, the extension to a sea a r e a having a realistic bottom topography can be readily accomplished as described by Davies (1976a). CONCLUDING R E V A R K S The ability of the present method to represent eddy viscosity as a smooth continuous function of a quite arbitrary form permits a physically realistic variation of this parameter in both surface and bottom boundary layers to be modelled.
Since
the method produces a continuous vertical current profile i t can reproduce the bottom logarithmic change in current magnitude with depth, a major advantage over grid box models, and has considerable advantages over analytical models in that arbitrary depth variations in eddy viscosity and bottom topography c a n be included.
46
N = 2600,650,3-25 FIG.6. : T i m e v a r i a t i o n o f s u r f a c e e l e v a t i o n 6 a t t h e c o r n e r p o i n t B : N o = 2 6 0 0 , N , = 6 5 0 , N 2 = 3.25 c m Z / s e c .
47
The importance of surface and bottom eddy viscosity in determining the current profile has been clearly demonstrated, and comparisons between the damping of the system for various
values of bottom eddy viscosity and that obtained using a bottom slip boundary condition have been made, yielding comparisons
between bottom eddy viscosity and linear bottom friction
coefficient k
.
A n investigation into the use of a collocation method to
calculate the vertical structure of the horizontal components of current is presently in progress (Davies 1976b) which will provide a valuable alternative to the present method, A CKN 0W L E D G E'IE M T S The author is indebted to Dr. N,S. Heaps, f o r suggesting a number of particularly valuable numerical experiments, and for a number of long and enlightening discussions about the work. The assistance of M r . R . A . Smith in preparing the diagrams is much appreciated. REFERENCES Bowden, K.F.. Fainbairn, L . A , Astr. SOC., 1, 288-305.
and Hugues, P., 1975, Geophys. J.
R.
Bowden, K.F., and Hamilton,P.,1975, Estuarine and Coastal Marine Science, 2 , 281-301. Charnock, H., 1959. Geophys. J . R .
Astr. SOC.,
2,
215-221.
Sci. Lisge, S C r . 6,
8, to
Davies, A . M . , published.
1976a. MGm. SOC. R .
Davies, A . M . ,
1976b. I.O.S. Internal Report in preparation.
be
Hansen, N.O., 1975. Journal of the Hydraulics Division, ASCE, NO HY8, 1037-1052.
101,
Heaps, N.S., Johns, B.,
1971.
MQm.
SOC. R. Sci. LiSge, S 6 r . 6 ,
1966. Geophys. J . R . Astr. SOC.,
2,
143-180.
11, 103-110.
Kagan, B.A.,
1966. Izv. Atm. and Oceanic Phys..
Lesser, K . M . ,
1951. Trans. Amer. Geophys. Un.,
2, 32,
956-969. 207-211.
48
Liggett, J.A., 1970. Journal of the Hydraulics Division, ASCE, N'HY3, 725-743.
96,
Munk, 1 J . H . and Anderson, F.R., 276-295.
7,
1948. Journal of Marine Research
Neumann, 6 , and Pierson, W.J., 1966. "Principles of Physical Oceanography", Engle-wood Cliffs, N.J. XII, 545 p p . Nihoul, J.C.J., I 15-1 25. Thomas, J . H . , 142.
1973. M6m. SOC. R. S c i . LiSge, S6r. 6 , 1975. Journal of Physical Oceanography,
6,
5,
136-
TABLE I. Values of eddy viscosity used in the numerical experiments.
No
cm 2/ s e c
N2
2
cm / s e c
2
130
130
130
650
130
3600
130
2600
65
2600
20
2600
3.25
49
BOTTOM TURBULENCE IN STRATIFIED ENCLOSED SEAS I.D. LOZOVATSKY, R.V. OZMIDOV
P.P. Skyrshov Institute of Oceanology, V.S.S.R.
Acad. Sc.,
Moscow, U.S.S.R. Jacques C.J. NIHOUL Geophysical Fluid Dynamics, University of Lisge, Belgium. 1,
T'I?ID MIXIPG AND VERTICAL STRATIFICATION I n the North Sea,intensive long waves (tides and storm
surges),generally
travelling down f r o n the North, produce a n
important mixing of the water c o l u m n ,
Except for localized
areas and limited periods of time, the turbulence extends to the bottom and the existence of a n upper wind-mixed layer,separated from the water below by a thermocline, is not a n essential feature of the North Sea Hydrodynamics. I n semi-enclosed seas like the Baltic and the Mediterranean the absence of strong tidal currents and subsequent bottom friction,combined w i t h the existence of fresh water inflows and thermodynamic exchanges between the atmosphere and the sea which may play a more significant role, creates the conditions of vertical density gradients of considerable importance. Typically, in the absence of wind, the water is smoothly and regularly stratified from the surface to the bottom, IJhen the wind starts blowing, i t exerts a stress o n the water surface, momentum is transferred to the sea, a turbulent layer develops which extends downwards by entraining water from below.
Eventually this process slows down a n d , for constant
wind energy, the thickness of the upper mixed layer appears to tend to some maximum value presumably determined by the balanced competition of turbulent mixing and inhibition by the stratification in the presence of Coriolis effects (e.g. Nihoul, 1976).
60
The lower boundary of the turbulent mixed layer is then marked by a sharp density gradient usually referred to as the thermocline because temperature is, in most cases, the essential factor affecting density. Below the thermocline, a smooth stratification may prevail but. in general
-
in shallow areas, at least
-
temperature and
density profiles reveal significant vertical mixing in a bottom surface layer which may extend appreciably upwards. Fig. 1 shows for instance a typical temperature profile taken in the Mediterranean off the Island of Corsica where the Oceanographic Research Station of LiCge University is situated. In this case 10 m/sec
-
-
corresponding to a fairly constant wind of
, salinity effects are irrelevant and temperature
data may b e interpreted directly as density data. IS
20
FIG.1 : A typical temperature profile in the Mediterranean off Corsica according to long series of observations made by the Oceanographic Research Station of LiSge University at Calvi.
51 The diagram shows two well-mixed layers separated by a sharp thermocline. Mean currents in the two layers are very small and the question arises of the mechanisms which are responsible for the turbulence in the lower layer. 2.
VERTICAL STRUCTURE OF TURBULENCE IN TI!E
BALTIC
Measurements of small-scale velocity fluctuations were carried out in July 1975 southwards of the Barnholm island.
The
measurements have been effected via sounding by means of a hydroresistant sensor with a bandpass of I-250Hz and space resolution up to 2mm.
The device also incorporated a thermistor
with time constant of 0 . 1 sec. and noise level of 0.025OC. The s e a depth at the measurement area was 52m.
The soundings
were carried out every two minutes from an anchored vessel. Besides, recordings of the temperature fluctuations at fixed depths have been performed during several hours.
This allowed
the evaluation of the characteristics of internal waves in the investigated sea area. Fig. 2 shows the results of two pairs of soundings (a, b and c , d) carried out at
1
hour interval.
The vertical dis-
tribution of temperature was characterized by the presence of an upper mixed layer A (0-17m), a thermocline B (17-27m),
a
layer C with a developed fine structure and a lower temperature gradient (27-42m) and a bottom boundary layer D (below 42-48m) with a homogeneous temperature or even with a temperature inversion ; the density uniformity in this case being maintained by the increase of salinity with depth.
The thick-
ness of layer D and the structure of the temperature profile in i t , as shown in Fig. 2,changed considerably between two pairs of soundings.
The vertical profiles of velocity fluc-
tuations u' and their root-mean square values have the following features : In the surface boundary layer down to 5-7m, one observes an increased turbulence level which may be explained by the influence of surface waves.
62
FIG.2 (a,b) : V e r t i c a l p r o f i l e s o f m e a n t e m p e r a t u r e , v e l o c i t y f l u c t u a t i o n s u' and t h e i r r o o t - m e a n s q u a r e v a l u e s S u' *
53
d
(c,d) : Vertical profiles of mean temperature, velocity f l u c t u a t i o n s u' and t h e i r r o o t - m e a n s q u a r e v a l u e s S u'
PIG.2
-
54
Below this layer and d o w n to the upper limit of the thermocline, the intensity of small-scale velocity fluctuations, as a rule, is not great.
I n layer B,the velocity fluctuations
sharply increase; the turbulence here having a n intermittent character,
Below layer B, there usually occurs a decrease in
the intensity of turbulent fluctuations which again considerably increases in the bottom boundary layer D. high turbulence is divided into 5 - 7
This layer of
sublayers about 2 m thick
which have approximately constant levels of turbulence in them. It is worth noting
that a layer of 1-2m i n thickness with a n
increased turbulence is always found close to the bottom. Unlike the described typical vertical structure of the field of small-scale fluctuations, Fig. 2b shows a n example of a situation when a strong intermittent turbulence is observed throughout the whole water column.
It is interesting to note
that such reconstruction of the turbulent
structure occured
very quickly, say, within 2 minutes, i.e. the time between two adjacent soundings. Generation of turbulence w i t h i n a z o n e of maximum temperature gradient (layer B) is likely to happen due to a hydrodynamic instability of the short-period internal waves, the existence o f which i n this layer has been detected by measurements with a thermistor sensor. The energy density spectra (fig.3) decrease first as w or w
-2
and then fall-off very rapidly (as w - ~ or w
-7
)
-3
for fre-
quencies larger than the m e a n Brunt-VEisslE frequency 5
.
These results may be compared with observations reported by Voorhis ( 1 9 6 8 )
and Neshyba et al. ( 1 9 7 1 )
cal predictions of Nihoul ( 1 9 7 2 )
and the theoreti-
based o n the hypothesis of a n
erratic field of internal waves interacting with a m e a n shear flow. 3.
GENERATION OF BOTTOM TURBULENCE The existence of a bottom turbulent layer i s documented by
long series of observations in the Mediterranean and in the Baltic.
55
FIG.3 : S p e c t r a l d e n s i t i e s o f t e m p e r a t u r e f l u c t u a t i o n s i n t h e Baltic thermocline.
56 O n the basis of velocity measurements
in the ocean bounda-
ry layer, Munk ( 1 9 7 1 ) suggested the presence of a bottom Ekman layer of a thickness of the order of 10 meters. Observations of the diffusion of a d y e tracer i n the Black Sea bottom boundary layer at the depth of 25 m (Labeish and Burnashev 1 9 7 0 ) .
of the distribution of suspended particles
over the continental slope southwards of N e w York (Ichiye 1966)
and of the diffusion of nutrients below the thermocline
in the Mediterranean off the Corsican Coast (Nihoul 1 9 7 6 ) ,
all
converge towards a model of a turbulence intensity increasing first with increased distance from the bottom to decrease or flatten out to some constant value above some critical depth. These observations,
-
although they do not exclude the pos-
sibility of some momentum transfer through the thermocline by local instabilities, transient perturbations and internal waves breaking
- ,
seem to suggest the generation of turbulen-
ce near the bottom. T'ith
very weak m e a n flows, in general, below the thermocli-
ne, bottom friction cannot explain a l l the observations. Generation of turbulence i n the Baltic bottom layer (layer D) may result from flow instabilities according to the mecha-
nism proposed by l!imbush
(1971).
Instability of the internal waves existing in the sea bottom boundary layer may serve as a possible mechanism of turbulence production.
The instability of the waves may either be
produced by interaction with irregularities of the bottom surface (Munk, 1 9 6 6 ) or by the breaking of the waves spreading over shallow water with small bottom slopes. Munk and Wimbush ( 1 9 6 9 )
According to
such breaking of internal gravity
waves of frequency w takes place at sin Q = w/N
,
where Q is
the bottom slope and N the Brunt-Vsiszlz frequency. sin
Values of
i n the region of observations in the Baltic did not ex-
ceed 0.01 and this mechanism of formation of bottom turbulence in the Baltic is quite possible.
57
REFERENCES Ichiye T., 1966. TurbuIent diffusion of suspended particles near the ocean bottom. Deep Sea Res., 13: 679-685. Labeish V.G., Burnashev V.Kh., 1970. The observation of turbulence in the sea bottom boundary layer. Turbulent currents. "Nauka", M O S C O W , pp.233-235. Munk W.H., 1966. Abyssal recipes.
Deep Sea Res., 13: 707-730.
Munk ?.J.H,, 1971. The circulation near the sea floor. "The ocean world". Tokyo, p.230. Munk W.H., Wimbush M., 1969. A simple criterium of breaking the waves o n a shore slope. "Okeanologia", 9. Neshyba S . , Neal V.T., Denner W.W., 1971. Spectra of internal waves : in situ measurements in a multiple-layered structure. J. Phys. Ocean, 2: 91-95. Nihoul J.C.J., 1972. O n the energy spectra of a random field of internal waves, Tellus, 24, 2, 161-163. Nihoul J.C.J., 1976. The upper mixed layer and the vertical distribution of nutrients and primary production in t h e Bay of Calvi, in Dynamics of the Planetary Boundary Layer and the Ocean Thermocline, Euromech 78, Sept. 7-8 1976, Paris. 968. Measurements of vertical motion and partiVoorhis A.D., tion of energy in the New England slope water. Deep Sea Res. 1 5 . 599-668. Wimbush M., 19 I . Tokyo, p.230.
The abyssal boundary layer. "The ocean world"
This Page Intentionally Left Blank
59 TURBULENT STRUCTURE I N A BEN'I'HJC BOUNDARY LAYER
C.M.
GORDON a n d J . W I T T I N G
Ocean S c i e n c e s D i v i s i o n , N a v a l R e s e a r c h L a b o r a t o r y , W a s h i n g t o n , DC
20375
(U.S.A.)
ABSTRACT
D i r e c t m e a s i i r e m e n t s of t h e t u r b u l e n t s t r u c t u r e o f a h e n t h i c , t i d a l boundary l a y e r h a v e b e e n made by r e c o r d i n g s e v e r a l t i m e s e r i e s o f s i m u l taneoiis h o r i z o n t a l and v e r t i c a l v e l o c i t y f l u c t ~ l a t i o n sa t various d i s t a n c e s above t h e b o t t o m .
For tlie d u r a t i o n of t h e experi ment t h e l a y e r t h i c k n e s s
5 was a p p r o x i m a t e l y 5 m and t l i e mean c u r r e n t s p e e d r a n g e d b e t w e e n 5 0 and
120 cm s e c - l . hotirs.
V e l o c i t i e s were sampled e a c h second f o r approxi mat el y foilr
A n a l y s i s o f tlie e x p e r i m e n t a l d a t a i n d i c a t e s t h a t t h e v e l o c i t y
f l u c t u a t i o n s w e r e s t a t i s t i c a l l y s e p a r a b l e i n t o t w o main c o m p o n e n t s ,
(1) l a r g e - s c a l e , q u a s i - o r d e r e d s t r u c t u r e s w h i c h are p r i m a r i l y r e s p o n s i b l e f o r v e r t i c a l momentum t r a n s p o r t , a n d ( 2 ) a b a c k g r o u n d o f r e l a t i v e l y i s o t r o p i c turbulence of smaller s c a l e .
The f r e q u e n c y o f o c c u r r e n c e o f t h e
l a r g e , c o h e r e n t s t r u c t u r e s was f o u n d t o b e somewhat d e p e n d e n t on t i d a l phase.
I n g e n e r a l , t h e p e r i o d b e t w e e n s u c h e v e n t s s c a l e s on t h e o u t e r
f l o w v a r i a b l e s ,U
a n d 6, i n a manner a n a l o g o u s t o t h e b u r s t i n g phenomenon
observed i n l a b o r a t o r y experiments.
The i n t e r m i t t e n c y o f t h e v e r t i c a l
momentum t r a n s p o r t a s s o c i a t e d w i t h t h e s e l a r g e - s c a l e s t r u c t u r e s i s shown t o b e a l i m i t i n g f a c t o r i n t h e a c c u r a c y of d i r e c t m e a s u r e m e n t s o f R e y n o l d s s t r e s s i n such n a t u r a l , s h e a r flows.
INTRODUCTION
I n 1956 Bowden a n d F a i r b a i r n ( 1 9 5 6 ) p u b l i s h e d t h e f i r s t , d i r e c t , eddy c o r r e l a t i o n measurements of Reynolds stress i n a m a r i n e , b e n t h i c , boundary layer.
Later, Bowden ( 1 9 6 2 ) n o t e d t h a t t h e s t r e s s v a l u e s c a l c u l a t e d f r o m
such t w o - d i m e n s i o n a l , v e l o c i t y f l u c t u a t i o n m e a s u r e m e n t s e x h i b i t e d a n i n e x p l i c a b l y l a r g e v a r i a t i o n from o n e 1 0 - m i n u t e r e c o r d t o a n o t h e r .
Bottom
s t r e s s e s i n m a r i n e e n v i r o n m e n t s m e a s u r e d by S t e r n b e r g ( 1 9 6 8 ) u s i n g t h e c l a s s i c a l l o g a r i t h m i c p r o f i l e method h a v e a l s o b e e n f o u n d t o d i s p l a y much s c a t t e r , e v e n when mean f l o w c o n d i t i o n s h a d r e m a i n e d r e l a t i v e l y c o n s t a n t . Recent work by H e a t h e r s h a w ( 1 9 7 4 ) a n d Gordon ( 1 9 7 4 , 1 9 7 5 ) h a v e a d i r e c t b e a r i n g on t h e s e e a r l i e r r e s u l t s , i n d i c a t i n g t h a t t h e y may b e c h a r a c t e r i s t i c
60
o f high-Reynolds-number,
g e o p h y s i c a l boundary l a y e r s .
The r e l e v a n t f a c t o r
t h a t t h e s e r e c e n t measurements have i d e n t i f i e d i s t h a t t h e u n d e r l y i n g p r o c e s s o f momentum t r a n s p o r t i s a h i g h l y i n t e r m i t t e n t o n e .
Figure 1 is an i l l u s t r a -
t i o n of t h e d e g r e e of i n t e r m i t t e n c e p o s s i b l e i n such n a t u r a l fl ows.
It
shows c o r r e l a t e d h o r i z o n t a l u and v e r t i c a l w t u r b u l e n t v e l o c i t y f l u c t u a t i o n s measured i n a b e n t h i c boundary l a y e r a t a sampl i ng rate of one second.
When
averaged, t h e s e d a t a d e f i n e t h e Reynolds stress f o r t h e t i m e i n t e r v a l sampled.
The p o i n t t o n o t e is t h a t a b o u t 70% of t h e s t r e s s v a l u e i s c o n t r i b -
u t e d by t w o i n t e r m i t t e n t , m o m e n t u m - t r a n s p o r t i n g e v e n t s t h a t o c c u r d u r i n g
30 s e c o n d s o u t o f t h e e i g h t m i n u t e r e c o r d .
I t is i n t u i t i v e l y c l e a r t h a t
l a r g e v a r i a t i o n s i n m e a s u r e d R e y n o l d s s t r e s s v a l u e s , s u c h as Bowden ( 1 9 6 2 ) o b s e r v e d , w o u l d b e e x p e c t e d f r o m a r e c o r d o f t h i s k i n d , d e p e n d i n g upon t h e
t i m e i n t e r v a l d u r i n g w h i c h t h e m e a s u r e m e n t s were made, a n d c h a n c e .
-
-40%
-
10 SEC
The r e a s o n s f o r i n v e s t i g a t i n g i n t e r m i t t e n t momentum t r a n s p o r t a n d d i s c u s s i n g t h e t u r b u l e n t s t r u c t u r e of a b e n t h i c , b o u n d a r y l a y e r i n terms o f d i s c r e t e e v e n t s are b a s e d o n r e l a t i v e l y r e c e n t d e v e l o p m e n t s i n l a b o r a t o r y boundary l a y e r r e s e a r c h .
The r e l e v a n t new c o n c e p t s h a v e b e e n d e r i v e d
p r i m a r i l y from t h e a p p l i c a t i o n of f l o w v i s u a l i z a t i o n t e c h n i q u e s a n d c o n d i t i o n a l s a m p l i n g m e t h o d s t o t h e s t u d y of b o u n d a r y l a y e r t u r b u l e n c e .
The
most s i g n i f i c a n t r e s u l t o f t h e s e v a r i o u s a p proaches i s t h e d i s c o v e r y t h a t t h e p r o d u c t i o n o f t u r b u l e n c e a n d t h e t r a n s p o r t o f momentum i n b o u n d a r y l a y e r s i s d o m i n a t e d by t h e i n t e r m i t t e n t o c c u r r e n c e of a r e p e t i t i v e s e q u e n c e
of h y d r o d y n a m i c e v e n t s i n v o l v i n g l a r g e - s c a l e , o r g a n i z e d s t r u c t u r e s .
A t the
61 p r e s e n t t i m e , t h e s e p r o c e s s e s are u s u a l l y r e f e r r e d t o c o l l e c t i v e l y as t h e " b u r s t i n g " phenomenon.
A g e n e r a l overview of t h e p r o g r e s s i n t h i s l i n e of
contemporary, boundary l a y e r r e s e a r c h c a n b e gai ned from a r e v i e w of t h e work o f a few m a j o r c o n t r i b u t o r s , s u c h as S c h r a u b a n d K l i n e ( 1 9 6 5 ) ; K l i n e . Reynolds, S c h r a u b a n d R u n d s t a d l e r ( 1 9 6 7 ) ; C o r i n o a n d Brodkey ( 1 9 6 9 ) ; Rao Narasimha a n d B a d r i N a r a y a n a n ( 1 9 7 1 ) ; K i m , K l i n e a n d R e y n o l d s ( 1 9 7 1 ) ; G r a s s ( 1 9 7 1 ) ; W a l l a c e , Eckelmann a n d Brodkey ( 1 9 7 2 ) ; W i l l m a r t h a n d Lu ( 1 9 7 2 ) ; Lu and W i l l m a r t h ( 1 9 7 3 ) .
T h i s phenomenon may p r o v e t o b e p a r t i c u l a r l y
r e l e v a n t t o b o t t o m t u r b u l e n c e b e c a u s e M o l l o - C h r i s t e n s e n ( 1 9 7 3 ) , Grass (1971) and Lahey and K l i n e ( 1 9 7 1 ) h a v e g i v e n b o t h t h e o r e t i c a l a n d e x p e r i m e n t a l evidence t h a t i n t h e t r a n s i t i o n from l a b o r a t o r y s c a l e boundary l a y e r s t o t h e v e r y high-Reynolds-number,
boundary l a y e r s c h a r a c t e r i s t i c o f geophysi cal
f l o w s , b o t h t h e i n t e r m i t t e n c e and t h e d e g r e e of o r g a n t z a t i o n i n t h e fl ow pattern increase.
With t h i s b r i e f i n t r o d u c t i o n s e r v i n g as b o t h b a c k g r o u n d a n d m o t i v a t i o n ,
l e t u s now p r o c e e d t o d e s c r i b e some m e a s u r e m e n t s o f t u r b u l e n t v e l o c i t y f l u c t u a t i o n s made i n t h e b o u n d a r y l a y e r o f a n e s t u a r i n e , d i s c u s s t h e m e t h o d s u s e l t o look f o r i n t e r m i t t e n t .
t i d a l c h a n n e l and
large-scale, quasi-
ordered s t r u c t u r e s i n t h e t u r b u l e n t motions.
FIELD MEASUREMENTS
The m e a s u r e m e n t s were made w i t h a p i v o t e d - v a n e ,
c u r r e n t meter.
Details
o f t h e d e s i g n o f t h i s d e v i c e h a v e b e e n p u b l i s h e d p r e v i o u s l y by Gordon a n d Dohne ( 1 9 7 3 ) .
The p r e s e n t model o f t h e c u r r e n t meter w e i g h s a b o u t 22 pounds
i n w a t e r a n d i s d e s i g n e d t o c p e r a t e w h i l e s u s p e n d e d on a l i n e . i t is a t h r e e - d i m e n s i o n a l , u n d e r w a t e r w e a t h e r v a n e .
In essence,
The p r i m a r y sensors are
a n a x i a l d u c t e d i m p e l l e r t h a t m e a s u r e s t h e c u r r e n t speed and a n i n t e r n a l pendulum t h a t o r i e n t s t h e a n g l e o f t h e v a n e w i t h r e s p e c t t o t h e v e r t i c a l . Thus, knowing t h e a x i a l c u r r e n t s p e e d a n d i t s a n g l e w i t h r e s p e c t t o t h e h o r i z o n t a l , i t i s p o s s i b l e t o r e s o l v e a g i v e n c u r r e n t i n t o two c o m p o n e n t s ,
a h o r i z o n t a l component i n t h e streamwise d i r e c t i o n a n d a v e r t i c a l component. Each component i s decomposed i n t o two p a r t s :
a s t e a d y , sI.owly v a r y i n g
component c o r r e s p o n d i n g t o t h e mean f l o w , and a r a p i d l y f l u c t u a t i n g component due t o t u r b u l e n c e .
I n o u r a n a l y s i s we i d e n t i f y t h e mean f l o w as t h e t h i r d
d e g r e e p o l y n o m i a l f i t t e d t o t h e r a w d a t a f o r e a c h component.
The t u r b u l e n t
p a r t is t h e n t h e d i f f e r e n c e b e t w e e n t h e mean f l o w a n d e a c h s p e c i f i c d a t a point.
The p r o c e s s e d d a t a s e t i s t h e n t w o mean f l o w c o m p o n e n t s and two
62 t u r b u l e n t v e l o c i t y components f o r each s a m p l i n g i n t e r v a l , which i n m o s t o f t h e c a s e s d i s c u s s e d h e r e is o n e s e c o n d .
The t u r b u l e n c e m e a s u r e m e n t s were made i n a n e s t u a r i n e c h a n n e l , t h e T h i s experimental s i t e h a s been des-
Choptank R i v e r n e a r C h e s a p e a k e Bay.
c r i b e d i n e a r l i e r work by Gordon a n d Dohne ( 1 9 7 3 ) .
I n operation the current
meter w a s s u s p e n d e d a t v a r i o u s d e p t h s f r o m a b r i d g e a c r o s s t h e r i v e r .
The
o v e r a l l d e p t h o f t h e c h a n n e l a t t h e p o i n t o f measurement w a s 8 - 9 m , t h e b o u n d a r y l a y e r t h i c k n e s s ( 6 ) was a b o u t 5 m a n d t h e d a t a r e p o r t e d h e r e w e r e t a k e n f o r t h e m o s t p a r t w i t h t h e c u r r e n t meter a p p r o x i m a t e l y 2 m a b o v e t h e bottom.
C u r r e n t s ranged between
t i d a l c y c l e is n e a r 0 cm s e c
-1
.
_+
120 cm s e c
-1
; t h e mean c u r r e n t o v e r a
The v e l o c i t y " s i g n a t u r e s " w h i c h l a r g e - s c a l e ,
m o m e n t u m - t r a n s p o r t i n g s t r u c t u r e s i n d u c e i n t h e c u r r e n t s e n s o r as t h e y are a d v e c t e d p a s t a r e u s e d t o d e t e c t them, i d e n t i f y them, c h a r a c t e r i z e t h e n , a n d c l a r i f y t h e i r r o l e i n terms o f b o u n d a r y l a y e r d y n a m i c s .
EXPERIMENTAL RESULTS
I f l a r g e f l o w m o d u l e s e x i s t as c o h e r e n t s t r u c t u r e s i n t h e b o t t o m t u r b u l e n c e , t h e y must h a v e some k i n d o f m e a s u r e a b l e d i m e n s i o n s , f o r e x a m p l e , a n o m i n a l a v e r a g e d i a m e t e r or e d d y s i z e .
One d e f i n i t i o n o f t h e " a v e r a g e s i z e
o f e d d i e s ' ' i s t h a t s u g g e s t e d by t h e l a t e G . I .
Taylor (1935), t h e auto-
c o r r e l a t i o n method f o r d e t e r m i n i n g t h e m a c r o s c a l e o f t u r b u l e n c e . E u l e r i a n v e r s i o n o f t h i s t e c h n i q u e , which we u s e h e r e , LT
In the
is t h e a v e r a g e
eddy d u r a t i o n a t a p o i n t m
R ( 7 ) dT, w h e r e R ( 7 ) =
LT =
J',
T
q ( t ) q ( t + T ) d T / (4" )
.
Here, T i s t h e d e l a y t i m e i n se c o n d s and q d e n o t e s t u r b u l e n t v e l o c i t y fluctuations.
Figure 2 is a g r a p h i c a l r e p r e s e n t a t i o n of t h e a u t o c o r r e l a t i o n s
for a 2400 s e c o n d t i m e series o f h o r i z o n t a l ( u ) a n d v e r t i c a l (w) v e l o c i t y f l u c t u a t i o n s and t h e i r p r o d u c t .
The h i g h c o r r e l a t i o n f o r c l o s e s p a c i n g ,
f a l l i n g t o zero at t h e l i m i t s of t h e l a r g e eddies is e a s i l y seen i n t h e figure.
I n t e g r a t i n g under t h e c u r v e s and m u l t i p l y i n g by t h e a d v e c t i o n
v e l o c i t y , v a l u e s f o r a v e r a g e e d d y s i z e s are o b t a i n e d .
The d o t t e d c u r v e
p r o v i d e s a h o r i z o n t a l d i m e n s i o n f o r c o h e r e n t v e r t i c a l l y moving f l u i d .
The
s o l i d l i n e i n d i c a t e s t h e dimensions of t h e l a r g e e d d i e s r e s p o n s i b l e f o r R e y n o l d s s t r e s s , s i n c e t h e R e y n o l d s s t r e s s i s d e f i n e d as t h e t i m e a v e r a g e o f uw m u l t i p l i e d b y t h e f l u i d d e n s i t y .
The s i g n i f i c a n t r e s u l t t o n o t e i s
63 t h a t t h e s t r e s s is r e l a t e d t o l a r g e - s c a l e s t r u c t u r e with a macroscale
(4.5 m) t h e o r d e r of t h e b o u n d a r y l a y e r t h i c k n e s s .
I.2/
I
I
I
I
2 4 0 0 POINTS U- I10 cmlsec
I
I
1
I
1
EDDY SIZE 13.4M 4.2M
--- u ..... ... w
3
LAG TIME IN SECONDS F i g . 2. A u t o c o r r e l a t i o n of h o r i z o n t a l ( u ) a n d v e r t i c a l (w) v e l o c i t y f l u c t u a t i o n s a n d t h e i r p r o d u c t (uw).
T a y l o r ' s a u t o c o r r e l a t i o n t e c h n i q u e i s a p u r e l y s t a t i s t i c a l method f o r e x t r a c t i n g a g e n e r a l i z e d d i m e n s i o n from a n e n s e m b l e o f amorphous m o t i o n s i n which i t i s p o s t u l a t e d t h a t t h e r e e x i s t s some t r a n s i e n t s t r u c t u r e a b o u t which v e r y l i t t l e i s known.
I n h e r e n t i n t h i s approach is a n implied admission
t h a t t h e s t r u c t u r e i s t o o complex t o b e m e a s u r e d d i r e c t l y .
However, l e t u s
assume t h a t t h i s i s n o t a c t u a l l y t h e c a s e a n d c o n s i d e r t h e p o s s i b i l i t y o f l e a r n i n g s o m e t h i n g a b o u t t h e s t r u c t u r e of i n d i v i d u a l " e d d i e s " .
I t i s c l e a r f r o m t h e a u t o c o r r e l a t i o n o f uw p r o d u c t s t h a t t h e r e
l a r g e - s c a l e s t r u c t u r e c o n t r i b u t i n g t h e R e y n o l d s stress.
& 3
So a c o m p u t e r p r o -
gram h a s b e e n w r i t t e n t o s e a r c h for s p e c i f i c s t r u c t u r e s i n t h e t i m e s e r i e s
of uw c r o s s c o r r e l a t i o n a n d e s t a b l i s h whether t h e r e are c o r r e l a t e d m o t i o n s i n two d i m e n s i o n s t h a t c a n b e i d e n t i f i e d w i t h s t r e s s - p r o d u c i n g , l a r g e - s c a l e structure.
F i g u r e 3 i s a t y p i c a l t i m e series o f o n e - s e c o n d , uw p r o d u c t s
t a k e n from o u r J u n e 1975 d a t a .
I t is n o t n e c e s s a r y t o b e l a b o r t h e p o i n t
t h a t t h e r e i s a c o n s i d e r a b l e amount of r e c o g n i z a b l e s t r u c t u r e i n t h i s r e c o r d .
64 Appropriate segments of t h e s e r i e s have been shaded f o r c l a r i t y .
The " c r o s s -
h a t c h e d " s t r u c t u r e s r e p r e s e n t c o h e r e n t v o l u m e s o f s l o w e r f l u i d moving away from t h e bottom.
T h e s e are t e n t a t i v e l y r e f e r r e d t o as e j e c t i o n s .
The
" s t i p p l e d " areas a r e f a s t e r f l u i d moving t o w a r d t h e b o t t o m f r o m t h e o u t e r f l o w or i n r u s h e s .
F i g u r e 4 shows t h e t h i r d s t i p p l e d e v e n t l o c a t e d a t
a p p r o x i m a t e l y 140 s e c . on a n e x p a n d e d s c a l e .
The f l u c t u a t i o n s i n h o r i z o n t a l
and v e r t i c a l v e l o c i t i e s t h a t p r o d u c e d t h e l a r g e c r o s s c o r r e l a t i o n uw a r e a l s o included i n t h e f i g u r e . and a ( - ) w
I t i s e v i d e n t t h a t t h e c o m b i n a t i o n o f a (+) u
c a n b e i n t e r p r e t e d as a f a i r l y l a r g e , c o h e r e n t volume o f f l u i d
moving t o w a r d t h e b o t t o m a t a s p e e d f a s t e r t h a n t h e mean f l o w a t t h e p o i n t o f measurement.
Knowing t h e t i m e s c a l e a n d t h e mean f l o w v e l o c i t y , a l i n e a r
d i m e n s i o n f o r t h e eddy o r f l o w module c a n b e o b t a i n e d . c o r r e l a t e d motion i s about 3
-4
In t h i s case the
meters i n h o r i z o n t a l e x t e n t .
A second
c h a r a c t e r i s t i c t h a t c a n b e a t t r i b u t e d t o t h i s s p e c i f i c "eddy" o r e v e n t i s a n a m p l i t u d e , t h a t i s , t h e area u n d e r t h e uw c u r v e .
The d a t a - p r o c e s s i n g ,
c o m p u t e r p r o g r a m s e a r c h e s t h e uw t i m e s e r i e s , p i c k s o u t i n d i v i d u a l e d d i e s , c a t e g o r i z e s them as t o t y p e a n d m e a s u r e s t h e i r d u r a t i o n s , a m p l i t u d e s a n d o t h e r c h a r a c t e r i s t i c s i n much t h e s a m e w a y a s d e s c r i b e d a b o v e .
tl
TIME I N SECONDS
F i g . 3. A t y p i c a l t i m e series o f o n e - s e c o n d m e a s u r e m e n t s o f L u w s h o w i n g i n t e r m i t t e n t , momentum-transporting events.
65
-uw -1'
\ I
JUNE,1975
I30 -149 X561 - 620
3 TIME IN SECONDS Fig. 4 .
A momentum-transporting e v e n t on a n expanded t i m e s c a l e .
The c o m p u t e r s e a r c h o f t h e 10,000 d a t a p o i n t s i n t h e 3.6 h r i n t e r v a l examined d e t e c t e d a b o u t 350 c a s e s of l a r g e - s c a l e , structures o r eddies.
two-dimensional flow
F i g u r e 5 i s a h i s t o g r a m of t h e d u r a t i o n s of t h e s e
s t r u c t u r e s as t h e y w e r e a d v e c t e d p a s t t h e v e l o c i t y s e n s o r .
It should b e
p o i n t e d o u t t h a t t h e r e .is n o significant d i f f e r e n c e i n t h e t e m p o r a l c h a r a c t e r of t h e two k i n d s o f m o m e n t u m - t r a n s p o r t i n g " e v e n t s " , and e j e c t i o n s t a k e a b o u t t h e same t i m e .
t h a t i s , t h e inrushes
On t h e a v e r a g e , t h e i r d u r a t i o n i s
a b o u t n i n e s e c o n d s , w i t h s e v e n s e c o n d s t h e most f r e q u e n t l y e n c o u n t e r e d v a l u e . I n t e r m s o f l i n e a r d i m e n s i o n s t h i s i s e q u i v a l e n t t o t h e 4-7 meter r a n g e , a v a l u e i n r e a s o n a b l e a g r e e m e n t w i t h t h e g e n e r a l i z e d d i m e n s i o n o f 4.5 meters computed e a r l i e r u s i n g G.I. T a y l o r ' s p u r e l y s t a t i s t i c a l a p p r o a c h .
Figure 6
shows t h e c h a r a c t e r i s t i c o f t h e c o h e r e n t s t r u c t u r e s r e f e r r e d t o as t h e i r amplitude.
The f i g u r e i s a h i s t o g r a m o f t h e n o r m a l i z e d p e r c e n t c o n t r i b u t i o n
o f e a c h l a r g e - s c a l e s t r u c t u r e t o t h e t o t a l stress d u r i n g t h e 9 - m i n u t e t i m e i n t e r v a l i n which i t o c c u r s .
The r e a s o n for t h i s k i n d o f n o r m a l i z a t i o n i s
t o accommodate t h e r a n g e o f c u r r e n t s p e e d s e n c o u n t e r e d d u r i n g t h e 3.6 h r s , w h i l e t h e d a t a were accumulated.
This took p l a c e over a s u b s t a n t i a l p a r t of
a t i d a l c y c l e a n d t h e f l o w v a r i e d b e t w e e n 50 a n d 1 2 0 cm s e c
-1
.
As was t h e
case f o r d u r a t i o n s , t h e r e i s r e l a t i v e l y l i t t l e d i f f e r e n c e i n t h e amplitude d i s t r i b u t i o n s o f t h e two k i n d s o f momentum t r a n s p o r t i n g " e v e n t s " .
66
60 50
I
I
I
I
1
-
v)
E w
1
I JUNE 10,1975 +cEJECTIONS -A-dINRUSHES
AVERAGE DURATION
N
9 SEC
40-
-
k 30[L
g 203
z
10-
0
2
4
6
8 10 12 14 DURATION IN SECONDS
16
18
Fig. 5. Durations of 350, large-scale, momentum-transporting structures detected in the 3.6 hr interval examined.
--I A
30
-
JUNE 10, 1975 EJECTIONS 4-4- INRUSHES
AMPLITUDE IN PERCENT CONTRIBUTION TO A TIME INTERVAI Fig. 6. Amplitude distribution of 350, momentum-transporting events.
20
67 The most s i g n i f i c a n t r e s u l t o f t h e c o m p u t e r a n a l y s i s i s t h a t t h e s e 3 5 0 , l a r g e - s c a l e s t r u c t u r e s so i d e n t i f i e d , h a v i n g a s u m e d t o t a l d u r a t i o n o f a b o u t 55 m i n u t e s , c o n t r i b u t e a l l t h e v e r t i c a l momentum t r a n s p o r t f o r t h e whole 3.6 h r i n t e r v a l o f t h e m e a s u r i n g r u n .
F o r t h e r e m a i n i n g 75% o f t h e
t i m e e s s e n t i a l l y n o t h i n g o f any r e l e v a n c e t o t h e f l o w dynamics a p p e a r s t o occur a t t h e p o i n t i n s p a c e b e i n g sampled and w i t h i n t h e t i m e r e s o l u t i o n of T h i s f i n d i n g i s of c o n s i d e r a b l e i m p o r t a n c e f o r t w o r e a s o n s .
t h e sensor used.
F i r s t , i t r e c o n f i r m s t h a t t h e u n d e r l y i n g p r o c e s s o f t u r b u l e n t momentum t r a n s p o r t i n t h e g e o p h y s i c a l b o u n d a r y l a y e r i s a h i g h l y i n t e r m i t t e n t o n e , and
&
that t h i s process
associated with the large-scale s t r u c t u r e .
Second, i t
p r o v i d e s e v i d e n c e t h a t t h e t u r b u l e n t m o t i o n s i n t h e b o u n d a r y l a y e r may b e viewed as a d u a l p o p u l a t i o n ; i n f r e q u e n t , l a r g e - s c a l e ,
dynamically a c t i v e
e v e n t s and a p a s s i v e , b a c k g r o u n d t u r b u l e n c e w h i c h d o e s n o t c o n t r i b u t e s i g n i f i c a n t l y t o momentum t r a n s p o r t .
I n a l l t h e f o r e g o i n g q u a l i t a t i v e a n a l y s e s , a c o n s i d e r a b l e amount o f s u b j e c t i v e judgment w a s e x e r c i s e d i n d e t e r m i n i n g which c o h e r e n t motions were c a l l e d "events",
s p e c i f y i n g p r e c i s e l y when a n e v e n t s t a r t e d o r e n d e d and
estimating other derived parameters.
A s Van A t t a ( 1 9 7 4 ) a n d O f f e n and
K l i n e (1973) h a v e p o i n t e d o u t , t h e u n s o l v e d problem of r i g o r o u s l y d e f i n i n g
some u n i v e r s a l c r i t e r i a f o r s p e c i f y i n g t h e e x i s t e n c e o r n o n - e x i s t e n c e o f q u a s i - o r d e r e d s t r u c t u r e s i n t u r b u l e n t b o u n d a r y l a y e r s i s f a r from t r i v i a l .
I n o r d e r t o e s t a b l i s h some o f t h e c h a r a c t e r i s t i c s o f t h e t u r b u l e n t m o t i o n i n a more o b j e c t i v e way, a 1 1/2 h r n o r m a l i z e d uw t i m e s e r i e s w i t h
2 . 2 6 s e c s a m p l i n g i n t e r v a l s h a s b e e n examined m o r e m a t h e m a t i c a l l y .
The
o b j e c t i v e o f t h i s a n a l y s i s is t o d e t e r m i n e some o f t h e s t a t i s t i c a l p r o p e r t i e s
of t h e i n t e r m i t t e n t , momentum-transporting e v e n t s and a p p l y t h i s i n f o r m a t i o n t o t h e problem o f s t r e s s p r e d i c t i o n and t h e i n t e r p r e t a t i o n of t h e t u r b u l e n t m o t i o n as a w h o l e .
S i n c e many o f t h e p o w e r f u l t h e o r e m s from p r o b a b i l i t y
t h e o r y a r e d e r i v e d f o r s e r i e s i n w h i c h t h e members a r e s t a t i s t i c a l l y independent, t h e f i r s t s t e p i n t h i s approach i s t o g e n e r a t e from t h e e x p e r i m e n t a l d a t a set a t i m e s e r i e s composed o f s t a t i s t i c a l l y i n d e p e n d e n t members. For s t a t i s t i c a l r e l i a b i l i t y , possible.
t h e s e r i e s s h o u l d c o n t a i n as many members a s
From i n s p e c t i o n o f a t y p i c a l segment o f t h e r e c o r d , F i g u r e 3 f o r
example, i t i s s e e n t h a t t h e p o s i t i v e a n d n e g a t i v e numbers are u s u a l l y grouped i n s e t s o f two o r more.
For t h i s r e a s o n , any a n a l y s i s t h a t treats
s u c c e s s i v e d a t a p o i n t s as i n d e p e n d e n t i s a p r i o r i i n c o r r e c t .
One m i g h t a l s o
group d a t a p o i n t s b e t w e e n c o n s e c u t i v e z e r o c r o s s i n g s t o form members o f a
68 new t i m e s e r i e s , b u t t h e s e c a n n o t b e s t a t i s t i c a l l y i n d e p e n d e n t , f o r a d j a c e n t members o f t h i s new series m u s t h a v e o p p o s i t e s i g n .
It is c l e a r t h a t the
m i n i m a l d i v i s i o n t h a t c a n p o s s i b l y y i e l d i n d e p e n d e n t members, g r o u p s d a t a between t w o c o n s e c u t i v e z e r o c r o s s i n g s .
We d e f i n e t h e a m p l i t u d e o f e a c h
" e v e n t " i n a n " e v e n t s e r i e s " as t h e t o t a l momentum t r a n s p o r t e d b e t w e e n two zero crossings:
k +di i
k = k
i
is t h e v a l u e a s s o c i a t e d w i t h t h e i t h e v e n t , k . i d e n t i f i e s i t h e value i n t h e o r i g i n a l t i m e s e r i e s w h i c h m a r k s t h e a p p r o p r i a t e p o s i t i v e I n Eq.
(1) T
z e r o c r o s s i n g , a n d di is t h e number o f d a t a p o i n t s w h i c h e x i s t up t o t h e next p o s i t i v e zero crossing. The set
ri
t h u s d e f i n e s a n "event"
series r e c o r d .
i s g r e a t e r t h a n or e q u a l t o 2 . i series a n a l o g o u s t o t h e i n i t i a l t i m e
Obviously, d
The s p e c i f i c 1 1 / 2 h r r e c o r d u n d e r e x a m i n a t i o n c o n t a i n s a
s e q u e n c e o f 4 2 8 e v e n t s o f t h i s k i n d , w h i c h is a m e a n i n g f u l l y l a r g e s a m p l e .
We now e x a m i n e w h e t h e r t h e e v e n t s are s t a t i s t i c a l l y i n d e p e n d e n t b y p e r f o r m i n g tests. F i r s t , c o n s i d e r t h e a u t o c o r r e l a t i o n f u n c t i o n as shown i n T a b l e 1. l a g , k , indexes t h e event series ( 0 5 I k , 5 4 2 8 ) .
The
According t o J e n k i n s and
Watts (1968), i f t h e s e e v e n t s are s t a t i s t i c a l l y i n d e p e n d e n t ( w h i t e n o i s e ) , the resulting autocorrelation function f o r k u t e d w i t h a mean o f z e r o a n d
8
2
1 shoul d be normally d i s t r i b -
v a r i a n c e e q u a l t o 1 / N , where N = 4 2 8 - k .
The
q u e s t i o n i s w h e t h e r t h e s l i g h t L y p o s i t i v e mean a n d t h e v a r i a n c e o f t h e members o f t h e a u t o c o r r e l a t i o n f u n c t i o n a r e i n c o n s i s t e n t w i t h t h e h y p o t h e s i s t h a t t h e e n t r i e s i n T a b l e 1 are n o r m a l l y d i s t r i b u t e d w i t h t h e a p p r o p r i a t e variance.
The s l i g h t l y p o s i t i v e a v e r a g e v a l u e o f t h e a u t o c o r r e l a t i o n
c o e f f i c i e n t s w o u l d d e p a r t f r o m z e r o a b o u t 10% o f t h e t i m e , i f t h e e v e n t
series i s w h i t e n o i s e .
The v a r i a n c e of t h e a u t o c o r r e l a t i o n s i n Tabl e 1 i s
e v e n more c o n s i s t e n t w i t h t h e h y p o t h e s i s of w h i t e n o i s e .
Thus, t h e hypothesi:
t h a t t h e e v e n t s e r i e s i s composed o f s t a t i s t i c a l l y i n d e p e n d e n t e n t r i e s r e m a i n : v i a b l e a f t e r t h e a u t o c o r r e l a t i o n test.
P e r h a p s o f more p r a c t i c a l s i g n i f i c a n c c
i s t h e f a c t t h a t n o l a r g e d e p a r t u r e s from r a n d o m n e s s show up f o r s m a l l k, which would b e t h e c a s e i f t h e o c c u r r e n c e of
8
l a r g e momentum t r a n s p o r t i n g
e v e n t g r e a t l y e n h a n c e d or d i m i n i s h e d t h e p r o b a b i l i t y of o c c u r r e n c e o f a nearby l a r g e event.
69 Table 1 Autocorrelation of the 428 event series
Lag (K) 0
Autocorrelation coefficient
Number of standard deviations x [Coefficient/(428 - K)']
1.000
1
+O. 043
+O. 89
2
+0.061
+1.25
-0.031
-0.63
H.013
+O. 27
-0.050
-1.02
6
+O. 082
+l. 68
7
+0.027
+o. 55
8
+0.067
+1.37
-0.013
-0.26
10
+o.
000
+O.oo +0.6a
9
11
+O. 033
12
-0.014
-0.29
13
+0.065
+1.32
14
M.069
+1.41
15
+O. 015
+0.31
16
+O.044
+0.90
17
+O. 070
+1.42
18
+O. 024
+0.48
19
-0.054
-1.10
20
-0.054
-1.10
21
+O. 107
+2.15
22
-0.015
-0.30
23
-0.028
-0.57
24
-0.037
-0.74
25
-0.37
-0.74
26
m.007
+O.
27
-0.020
-0.39
28
+O. 062
+l. 24
29
+o. 001
+0.02
30
-0.033
-0.67
13
A p o s s i b l y more s e n s i t i v e t e s t o f t h e h y p o t h e s i s of s t a t i s t i c a l i n d e p e n d e n c e i s shown i n T a b l e 2 .
Out o f t h e 428 e v e n t s we h a v e c h o s e n t h e 1 1 9
which h a v e t h e l a r g e s t p o s i t i v e v a l u e .
T h i s s e t o f e v e n t s c o n t r i b u t e s more
t h a n 90% of t h e t o t a l momentum t r a n s p o r t d u r i n g t h e 1 1 / 2 h r s e r i e s .
If
t h e s e 119 e v e n t s are s t a t i s t i c a l l y i n d e p e n d e n t , t h e n t h e p r o b a b i l i t y of o c c u r r e n c e of o n e o f t h e s e e v e n t s s h o u l d n o t b e a f f e c t e d by t h e e x i s t e n c e o r non-existence of a n o t h e r l a r g e momentum-transporting event nearby. S p e c i f i c a l l y , t h e “ w a i t i n g t i m e s ” b e t w e e n t h e s e 1 1 9 members o f t h e e v e n t
s e r i e s s h o u l d b e d i s t r i b u t e d a c c o r d i n g t o a g e o m e t r i c d i s t r i b u t i o n (see, e . g . F e l l e r (1950). pp 2 1 8 ) .
I t is s e e n f r o m t h e t a b l e t h a t t h e d i s t r i b u t i o n o f
w a i t i n g t i m e s is p r e c i s e l y g e o m e t r i c , w i t h i n f a i r l y t i g h t s t a t i s t i c a l u n c e r tainties.
F u r t h e r m o r e , t h e a v e r a g e a m p l i t u d e of p a i r s o f e v e n t s w h i c h a r e
c l o s e t o g e t h e r is a p p r o x i m a t e l y t h e s a m e as t h e a v e r a g e v a l u e o f e v e n t s which are w i d e l y s e p a r a t e d .
The r e s u l t s of t h e a u t o c o r r e l a t i o n t e s t a n d
t h e w a i t i n g t i m e test g i v e no r e a s o n t o abandon t h e h y p o t h e s i s t h a t t h e
series f o r m s a s t a t i s t i c a l l y i n d e p e n d e n t d a t a s e t when t r e a t e d i n t h i s way. Table 2 D i s t r i b u t i o n of w a i t i n g t i m e s f o r h i g h e s t 119 e v e n t s Number of e v e n t p a i r s Waiting t i m e
Observed
Expected, i f s t a t i s t i c a l l y independent
1
35
36.1
2
28
25.1
3
20
17.4
4
6
12.1
5
9
8.4
6
5
5.8
7
5
4.0
8
1
2.8
9
6
1.9
10
0
1.3
11
1
0.9
12
0
0.6
13
1
0.5
14
1
0.3
> 14
0
0.7
-
-
118
117.9
71 The p r i m a r y r e a s o n f o r e s t a b l i s h i n g t h e s t a t i s t i c a l i n d e p e n d e n c e o f t h e e v e n t series is t o j u s t i f y t h e a p p l i c a t i o n o f t h e c e n t r a l l i m i t t h e o r e m o f p r o b a b i l i t y t o t h i s d a t a set.
T h i s w i l l make i t p o s s i b l e t o c a l c u l a t e
whe the r t h e l a r g e v a r i a t i o n s i n m easu r ed R e y n o l d s stress f r o m o n e time t o a n o t h e r , s u c h as w e r e o b s e r v e d by Bowden ( 1 9 6 2 ) an d i n t h e d a t a p r e s e n t e d h e r e , are p r e d i c t a b l e f r o m t h e s t a t i s t i c a l p r o p e r t i e s o f t h e event series.
This theorem s t a t e s t h a t a v e r a g e stresses o v e r f i n i t e s a m p l i n g i n t e r v a l s are n o r m a l l y d i s t r i b u t e d , w i t h a mean e q u a l t h e stress o b t a i n e d f r o m a n i n f i n i t e l y l o n g m easu r em en t an d a v a r i a n c e e q u a l t o 1 / N t i m e s t h e v a r i a n c e of t h e i n d i v i d u a l d a t a p o i n t s u s e d t o c a l c u l a t e t h e a v e r a g e stresses (see, e.g.
F e l l e r (1950) pp 1 9 2 ) .
N is t h e t o t a l number o f s u c h d a t a p o i n t s .
Applying t h e c e n t r a l l i m i t t h e o r e m t o t h e 1 1/2 h r s e r i e s , w h i ch h a s a n -2 a v e r a g e d stress 7 o f 5 . 6 4 d y n e s cm , a v a r i a n c e o f 148/N is o b t a i n e d , w h er e 148 is t h e v a r i a n c e o f t h e i n d i v i d u a l d a t a p o i n t s a n d N i s 428.
From s t a n d a r d
s t a t i s t i c a l t a b l e s f o r n o r m al d i s t r i b u t i o n s i t i s s e e n t h a t a t t h e 90% -2 c o n f i d e n c e level 7 = 5 . 6 4 f 0 .9 7 d y n e s cm For t h e s h o r t e r sampling i n t e r -
.
v a l s , t h e o r d e r o f 8 t o 10 m i n d t e s , t h e e r r o r i n stress would b e p r e d i c t e d t o be about f 3 dynes cm
-2
a t t h e 90% c o n f i d e n c e l e v e l ( i t s h o u l d b e p o i n t e d
o u t t h a t f o r s u c h s m a l l n cm b er s o f p o i n t s t h e c e n t r a l l i m i t t h e o r e m i s n o t p a r t i c u l a r l y p r e c i s e at t h e f l a n k s of t h e d i s t r i b u t i o n ) . n o t e is t h a t t h i s s e m i - q u a n t i t a t i v e ,
The k e y p o i n t t o
s t a t i s t i c a l a n a l y s i s p r e d i c t s stress
v a r i a t i o n s of t h e same o r d e r as t h o s e o b s e r v e d i n t h e e x p e r i m e n t a l measu r ements.
I n d e e d , t o o b t a i n errors as s m a l l as f 10% a t t h e 90% c o n f i d e n c e
level, a s a m p l i n g t i m e a t l e a s t as l o n g as 4 . 5 h r i s r e q u i r e d .
Therefore,
i t c a n b e c o n c l u d e d t h a t m e a s u r e m e n t s o f q u a n t i t i e s l i k e R e y n o l d s stress,
f o r which t h e u n d e r l y i n g p r o d u c t i o n mechanisms are i n t e r m i t t e n t , are i n h e r e n t l y imprecise.
As h a s b e e n p o i n t e d o u t by S t e w a r t ( 1 9 7 3 ) . t h i s is a
p a r t i c u l a r l y d i f f i c u l t p r o b l e m i n g e o p h y s i c a l e n v i r o n m e n t s s u c h as t i d a l flows i n which s t e a d y - s t a t e c o n d i t i o n s d o n o t e x i s t s u f f i c i e n t l y l o n g f o r a n a d e q u a t e s e r i e s o f m easu r em en ts t o b e made.
12
500
3 v, 0
3 200
0 POSITIVE EVENTS
A NEGATIVE EVENTS 0 POSITIVE -BACKGROUND
&
:1
100
I %ACCOUNTED OF TOTAL STRESS FOR
0
50 J
a
z
p
20
I
$
10
P 6 5 >
W
LL
0
a m W
2
0
4%
10
20
30
40 50 60 70 NORMALIZED puW
80
90
100
3
Fig. 7. The integral distribution of events with respect to their contribution to the Reynolds stress. Figure 7 illustrates some of the other statistical properties of the event series.
Shown here is the distribution function of stress amplitude.
Half the total stress is attributable to about 30 events having normalized uw greater than 25. Consider first the events with negative sign. The
t
largest nine appear to follow one exponential distribution while the rest of the data fall into an exponential distribution with a different slope. A minimum interpretation is that the large ones are somehow different from
the small ones.
As a tentative classification, the uppermost nine negative
points will be called "momentum-transporting" negative events and the rest of the data "background" negative events.
The events with positive sign
can be decomposed similarly into two groups.
The distribution is again
exponential for at least the highest 100- 150 events; these contribute between 90 and 95% of the stress, even when the net stress is reduced by the negative momentum-transporting events. These large events with positive sign will be referred to as "momentum-transporting" positive events.
For
purposes of this discussion the highest 119 events will rather arbitrarily be termed "momentum transporting", and the remainder, "background". us hypothesize that this background is isotropic.
Let
Under this assumption
there will be small amplitude events of both positive and negative sign
13
which should be regarded as part of the momentum transporting portion of the curve, and non-momentum transporting events which are assumed to be isotropic. By extrapolating the negative data to the origin it is estimated that there are some 19 negative momentum transporting events and approximately 105 negative background events.
If this number of background negative points
is subtracted from the positive data points, assuming that the background is isotropic, the data points designated by the diamonds are obtained.
Note
that by removing what might be considered an isotropic background, the distribution of positive momentum transporting events follows a straight line all the way to the origin.
By noting the intercepts, the sample can
be considered to contain 183 positive momentum-transporting events, 19 negative, momentum-transporting events, and about 110 background events of each sign.
This kind of procedure is potentially quite useful in that it
can provide a reasonably objective method for specifying which turbulent motions are "events" and which are background. There are two other, well-defined data sets that we have examined statistically. They are the two sets produced by grouping the uw points between successive zero crossings and placing alternate groupings in each set, i.e. all those with positive sign in one set and all those with negative sign in the other.
This method of subdividing t'le record approximates
the original subjective approach to finding momentum-transporting structures. For purposes of the following discussion, members of each set will be separated into two classifications, structures (those of large amplitude) and aggregates (those of small amplitude).
LDWEST a mslTIVE AQQREOATES 1773 DATA POINTS)
30
?
ME%&RIC
DISTTRIBUTIDN
-
AQQREQATES (013 DATA PDINTSI MEAN l.%
I
I
HICHEST 101 POSITIVE STRUCTURES 1730 DATA POINTS1 MEAN-7.23
I
11[1 m
c
II '!
,
HIQHESTONEQATIVE STRUCTURES
I I DATA POINTS) MEAN-422
ii
F i g . 8. The number o f d a t a p o i n t s b e t w e e n s u c c e s s i v e z e r o c r o s s i n g s i n t h e uw t i m e series.
F i g u r e 8 shows t h e d u r a t i o n s o f series members d e f i n e d b y s u c c e s s i v e zero crossings.
The d u r a t i o n s are e x p r e s s e d i n terms o f t h e number o f uw
d a t a p o i n t s i n each grouping, M ( r e c a l l t h a t t h e sampling i n t e r v a l h e r e
i s 2.26 s e c ) .
The f i g u r e p r e s e n t s h i s t o g r a m s o f f o u r p r e s e l e c t e d s a m p l e s
from t h e new t i m e series; n i n e u n a m b i g u o u s l y m o m e n t u m - t r a n s p o r t i n g ,
negative
s t r u c t u r e s , t h e l a r g e s t 101 p o s i t i v e , m o m e n t u m - t r a n s p o r t i n g s t r u c t u r e s , a n d t h e r e m a i n i n g p o s i t i v e and n e g a t i v e g r o u p s d e s i g n a t e d a g g r e g a t e s .
Because
t h e r e are o n l y a f e w n e g a t i v e s t r u c t u r e s , i t is a n t i c i p a t e d t h a t most of t h e w e a k e r n e g a t i v e p o i n t g r o u p s , t h e a g g r e g a t e s , are p a r t o f t h e b a c k g r o u n d turbulence.
The i m p o r t a n t p o i n t t o n o t e i s t h a t t h e d u r a t i o n s o f t h e
n e g a t i v e a g g r e g a t e s are d i s t r i b u t e d g e o m e t r i c a l l y ( w i t h i n s t a t i s t i c a l u n c e r t a i n t i e s ) , t h e numbers w i t h a d u r a t i o n N b e i n g o n e - h a l f t h e number w i t h d u r a t i o n N-1.
T h i s i s e x a c t l y t h e l a w t h a t would r e s u l t f r o m i n f r e q u e n t
s a m p l i n g o f a random p r o c e s s i n which t h e g r o u p i n g of d a t a p o i n t s i n t h e a g g r e g a t e s o c c u r r e d by c h a n c e .
I t can t h e r e f o r e be concluded t h a t t h e
n e g a t i v e a g g r e g a t e s are a t t r i b u t a b l e t o h i g h f r e q u e n c y o r h i g h wavenumber t u r b u l e n c e t h a t t h e s a m p l i n g i n t e r v a l is t o o c o a r s e t o r e s o l v e .
By a s i m i l a r
a r g u m e n t t h e p o s i t i v e a g g r e g a t e s are p r o b a b l y a c o m b i n a t i o n of t h e same, high-frequency,
t u r b u l e n t b a c k g r o u n d p l u s some s m a l l m o m e n t u m - t r a n s p o r t i n g
15
events. The series members plotted in the lower part of this figure have a very different distribution, one in which durations are well resolved.
It
should be pointed out that because of the way this series was defined, it is possible that the mean duration of the positive momentum-transporting structures may be longer than the durations of the ejections or inrushes mentioned earlier. The experimental data are not adequate to demonstrate whether there is any difference in the mean durations of positive and negative momentum-transporting structures. The geometric, statistical distribution of the aggregates and the well resolved durations of the structures provide additional support for the interpretation of the turbulence as a dual population, consisting of a highfrequency, isotropic background (the aggregates) and large-scale, momentumtransporting events (the structures). Although the foregoing analysis of this limited sample by no means establishes definitive criteria for unambiguously specifying che presence of these quasi-ordered, momentum-transporting structures, it does provide encouraging indicat'ons that an objective, statistical approach to this problem is possible. Thus far the linear dimensions, durations. amplitudes and some of the statistical properties of the organized flow modules responsible for momentum transport in this particular geophysical boundary flow have heen investigated. Since these large-scale structures recur in a more or less recognizable form, another parameter that can be assigned to them is an average rate of occurrence. The most obvious way to determine this value
is to take the 350 events in the 3 . 6 hr sample series and perform a simple division.
This gives an average period between events of about 3 7 seconds.
By the definition used to produce Figure 7 the equivalent average time between events is 27 sec.
If the structures of Figure 8 that contribute
90% of the Reynolds stress are considered the events, the average period becomes about 45 sec. Obtaining a number for this quasi-period does not in itself reveal very much, however, it should be kept in mind that this value along with the descriptive morphology are the only clues available for interpreting these large-scale structures in terms of laboratory results.
16
INTERPRETATION
I n t h e i n i t i a l s t u d i e s o f t h i s k i n d o f c o h e r e n t m o t i o n s in b o t t o m t u r b u l e n c e , b o t h H e a t h e r s h a w ( 1 9 7 4 ) a n d Gordon ( 1 9 7 4 , 1 9 7 5 ) t e n t a t i v e l y i n t e r p r e t e d t h e i n t e r m i t t e n t , m o m e n t u m - t r a n s p o r t i n g s t r u c t u r e s as l a r g e s c a l e a n a l o g s o f t h e " b u r s t i n g phenomenon".
The a s s o c i a t i o n w a s b a s e d
p r i m a r i l y o n a f e w o f t h e c h a r a c t e r i s t i c , m o r p h o l o g i c a l f e a t u r e s of b u r s t i n g
as p r e s e n t l y u n d e r s t o o d f r o m f l o w v i s u a l i z a t i o n m e a s u r e m e n t s .
In particular,
by f o l l o w i n g t h e t r a j e c t o r i e s o f s p e c i f i c f l u i d v o l u m e s , l a b o r a t o r y s t u d i e s have found t h a t t u r b u l e n c e i s n o t g e n e r a t e d c o n t i n u o u s l y i n t h e boundary region near the w a l l but r a t h e r t h i s inner l a y e r periodically experiences
a v i o l e n t d i s r u p t i o n or b u r s t of turbulence generation.
This event is
a s s o c i a t e d w i t h t h e e j e c t i o n o f slower moving f l u i d away f r o m t h e n e a r
w a l l r e g i o n i n t o t h e o u t e r p a r t of t h e boundary.
This kind of coherent
m o t i o n p r o d u c e s a n i n t e r m i t t e n t i n t e r v a l of h i g h momentum t r a n s p o r t .
It
i s a l s o b e l i e v e d t h a t t h e s e b u r s t s are e i t h e r t r i g g e r e d by o r f o l l o w e d by i n r u s h e s of h i g h s p e e d f l u i d f r o m t h e o u t e r b o u n d a r y l a y e r .
These e v e n t s
h a v e b e e n t e r m e d s w e e p s a n d t h e y a l s o g i v e r i s e t o p e r i o d s of h i g h momentum transport.
The m o r p h o l o g i c a l s i m i l a r i t y o f t h e s e i n t e r m i t t e n t f e a t u r e s o f
t h e b u r s t - s w e e p c y c l e t o t h e e j e c t i o n a n d i n r u s h e v e n t s f o u n d i n t h i s geop h y s i c a l boundary l a y e r i s immediately e v i d e n t .
The q u e s t i o n r e m a i n s
w h e t h e r t h i s q u a l i t a t i v e s i m i l a r i t y h o l d s up u n d e r more q u a n t i t a t i v e exami n a t ion.
Consider f i r s t t h e c o n t r i b u t i o n s of v a r i o u s k i n d s of e v e n t s t o t h e t o t a l R e y n o l d s stress, w h e r e "kind" r e f e r s t o t h e v a r i o u s c o m b i n a t i o n s o f u a n d w t h a t g o i n t o making u p a g i v e n uw p r o d u c t .
T a b l e 3 c o m p a r e s some
o f o u r measurements of g e o p h y s i c a l i n r u s h - e j e c t i o n e v e n t s w i t h c o r r e s p o n d i n g m e a s u r e m e n t s o f t h e b u r s t i n g phenomenon made by Wallace, Eckelmann a n d Brodkey ( 1 9 7 2 ) .
T h e i r m e a s u r e m e n t s were made w i t h a h o t f i l m f l o w meter
i n a n o i l c h a n n e l , w h e r e t h e b o u n d a r y l a y e r w a s a few c e n t i m e t e r s t h i c k . The a g r e e m e n t shown h e r e is c o n s i s t e n t w i t h a n a n a l o g o u s p r o c e s s .
Table 3 R e l a t i v e c o n t r i b u t i o n s of v a r i o u s k i n d s o f e v e n t s t o t h e R e y n o l d s stress (%)
Distance o f f bottom
2
2
6
- 1 5 2 4
- 1 5 2 4
7 3 2 8
6 5 2 8
- 1 7 2 6
- 2 0 2 8
- 68
- 60
- - 15
- - 15
l m
65
2.25 m W a l l a c e e t a1 ( 1 9 7 2 )
3
67
A s e c o n d q u a n t i t a t i v e i n d i c a t i o n t h a t t h e s e may b e a n a l o g o u s phenomena
c a n b e d e r i v e d from t h e q u a s i - p e r i n d o f t h e s t r u c t u r e s , t h a t is t h e 3 7 seconds between e v e n t s o b t a i n e d b e f o r e .
Even t h o u g h t h e " p e r i o d i c " a s p e c t
of momentum t r a n s p o r t i n t h i s g e o p h y s i c a l b o u n d a r y l a y e r h a s b e e n d i s c u s s e d p r e v i o u s l y bv Gordon ( 1 9 7 5 ) . a b r i e f r e v i e w o f t h e r e l e v a n t l a b o r a t o r y
results is a p p r o p r i a t e h e r e .
I n compilations of experimental d a t a on
b u r s t i n g Rao, N a r a s i m h a a n d B a d r i N a r a y a n a n ( 1 9 7 1 ) a n d L a u f e r and B a d r i Narayanan (1971) h a v e f o u n d t h a t i f t h e mean p e r i o d b e t w e e n b u r s t s T i s made n o n - d i m e n s i o n a l by s c a l i n g w i t h o u t e r f l o w v a r i a b l e s (U,,
stream f l o w r a t e , a n d 6 o r E 9 : ,
the free
measures o f t h e boundary l a y e r t h i c k n e s s )
t h e n o n - d i m e n s i o n a l p e r i o d is i n d e p e n d e n t o f R e y n o l d s number
~
These rela-
t i o n s h i p s are e x p r e s s e d as
T
a
32 (.,/Urn)
or
T
a
5 (&/Urn)
.
I n the marine boundary l a y e r , t h e a p p r o p r i a t e o u t e r flow v a r i a b l e s a r e a c u r r e n t s p e e d o f a b o u t 70 cm s e c
-1
,
a boundary l a y e r t h i c k n e s s (C)
of about 5 m e t e r s , and a d isp la c e m e n t t h i c k n e s s ( 6 , )
o f a b o u t 6 0 cm.
S u b s t i t u t i n g t h e s e v a l u e s i n t o t h e f o r m u l a s would p r e d i c t a p e r i o d b e t w e e n b u r s t i n g e v e n t s o f T = = 36 o r 27 s e c , r e s p e c t i v e l y .
As m e n t i o n e d p r e v i o u s l y ,
t h e r e r e m a i n s c o n s i d e r a b l e a m b i g u i t y r e g a r d i n g t h e d e f i n i t i o n o f what i s meant by t h e p e r i o d b e t w e e n b u r s t s , so t h e s e v a l u e s s h o u l d n o t b e t a k e n as precise.
The s i g n i f i c a n t p o i n t is t h a t w i t h i n a f a c t o r o f t w o or so, t h e
s c a l i n g b a s e d on l a b o r a t o r y e x p e r i m e n t s seems t o b e c o n s i s t e n t o v e r a t
least two o r d e r s o f m a g n i t u d e i n f l o w d i m e n s i o n s .
A final piece of evidence in support of the burst-sweep analogy concerns the possible influence of the longitudinal pressure gradient of the mean flow on the frequency of bursts.
Actually, laboratory experiments
have not as yet quantitatively established this relationship, but in a qualitative sense Schraub and Kline (1965) have shown that an adverse pressure gradient enhances bursting while a favorable gradient suppresses it.
Although pressure gradients were not measured directly i n the tidal
flow described here, there are accelerating and decelerating currents.
If
acceleration is considered equivalent to a favorable pressure gradient and deceleration considered equivalent to an adverse gradient, bursting rates and other turbulent parameters can be compared under the two conditions. Table 4 shows the results of such a comparison.
In qualitative agreement
with the burst-sweep analogy, there are more momentum-transporting structures and higher Reynolds stress on the decelerating tidal flow than on the accelerating phase.
It should be pointed out that Everdale (1976).
in a
recent study of the tidal phase dependence of turbulent velocity structure at a site in Long Island Sound, found no significant difference between accelerating and decelerating phases except at low currents (below 25 cm -1 sec ). Therefore, the universality of this effect in marine bottom turbulence is as yet not well documented. Table 4 The influence of pressure gradient on boundary layer turbulence Measurement
dP/dx, Favorable
Average current -1 U in cm sec Acceleration
-
dLJ/dt i n cm sec
Inrush-ejection events I cuw I > 15 dynes cm-2 Reynolds stress
-2 -ouw, in dynes cm
dP/dx, Adverse
51;5
+
51.7
-
9 10
1.0
6x 51
2.9
I9 I t is o u r o p i n i o n t h a t t h e o v e r a l l similarities i n morphology, q u a s i -
p e r i o d , and p r e s s u r e g r a d i e n t e f f e c t s p r o v i d e adequat e grounds f o r i n t e r p r e t i n g t h e l a r g e - s c a l e s t r u c t u r e s i n t h i s g e o p h y s i c a l b o u n d a r y l a y e r i n terms o f t h e b u r s t i n g phenomenon o b s e r v e d i n l a b o r a t o r y - s c a l e e x p e r i m e n t s .
It
i s c l e a r , h o w e v e r , t h a t much more e v i d e n c e i s n e e d e d t o p r o v e t h e c a s e o n e way or a n o t h e r .
The real v a l u e o f t h i s t e n t a t i v e i n t e r p r e t a t i o n i s t h a t
i t s e r v e s as a c o n c e p t u a l framework f o r d e s i g n i n g f u t u r e e x p e r i m e n t s i n
o t h e r b o u n d a r y l a y e r s of g e o p h y s i c a l i n t e r e s t .
SLIMMARY
A n a l y s i s o f several t i m e s e r i e s o f c o r r e l a t d , two-dim n s i o n a l , t u r b u l e n t v e l o c i t y f l u c t u a t i o n s measured i n t h i s e s t u a r i n e , t i d a l channel
allows t h e f o l l o w i n g c o n c l u s i o n s t o b e d r a w n . 1) T h e r e are c o h e r e n t , r e p e t i t i v e s t r u c t u r e s i n t h i s m a r i n e b e n t h i c boundaGy l a y e r . 2) They are l a r g e s c a l e , w i t h d i m e n s i o n s t h e o r d e r o f t h e b o u n d a r y
layer thickness.
3 ) They o c c u r h i g h l y i n t e r m i t t e n t l y a n d a p p a r e n t l y are i n d e p e n d e n t of e a c h o t h e r .
4) I n t h e p o r t i o n o f t h e boundary l a y e r where measurements have b e e n made, t h e y d o m i n a t e t h e v e r t i c a l t r a n s p o r t p r o c e s s e s .
5) I n terms o f p o r p h o l o g y , q u a s i - p e r i o d a n d p r e s s u r e g r a d i e n t r e s p o n s e , t h e y b e a r a r e m a r k a b l e r e s e m b l a n c e t o t h e s o - c a l l e d " b u r s t i n g phenomenon." 6 ) The o v e r a l l t u r b u l e n t m o t i o n s i n t h i s g e o p h y s i c a l b o u n d a r y l a y e r may b e t r e a t e d as a d u a l p o p u l a t i o n , c o n s i s t i n g o f l a r g e - s c a l e ,
dynamically
a c t i v e , c o h e r e n t s t r u c t u r e s , s u p e r i m p o s e d on a p a s s i v e b a c k g r o u n d o f r e l a t i v e l y s m a l l - s c a l e , i s o t r o p i c t u r b u l e n c e t h a t c o n t r i b u t e s l i t t l e or n o t h i n g t o t h e v e r t i c a l t r a n s p o r t o f momentum.
7) As a consequence of t h e i n t e r m i t t e n t n a t u r e of t h e u n d e r l y i n g momentum-transporting p r o c e s s ,
t h e r e is a n i n t r i n s i c l i m i t t o t h e p r e c i s i o n
o f R e y n o l d s s t r e s s m e a s u r e m e n t s i n s u c h u n s t e a d y f l o w s as m a r i n e , b e n t h i c boundary l a y e r s .
80 REFERENCES Bowden, K.F., 1962. M e a s u r e m e n t s o f t u r b u l e n c e n e a r t h e sea bed i n a t i d a l c u r r e n t . J . Geophys. Res., 67: 3181-3186. Bowden, K.F., a n d F a i r b a i r n , L . A . , 1 9 5 6 . M e a s u r e m e n t s o f t u r b u l e n t f l u c t u a t i o n s and Reynolds s t r e s s e s i n a t i d a l c u r r e n t . P r o c . Roy. SOC. London, S e r . A , 237: 422-438. C o r i n o , E . R . , and Brodkey, R.S., 1969. A v i s u a l i n v e s t i g a t i o n of t h e wal l 1-30. r e g i o n i n t u r b u l e n t f l o w . J . F l u i d Mech., 37: ,
E v e r d a l e , F.G., 1 9 7 6 . The n e a r b o t t o m t u r b u l e n t v e l o c i t y s t r u c t u r e ; v a r i a t i o n o v e r a t i d a l c y c l e a t a s i t e i n e a s t e r n Long I s l a n d Sound. M.S. T h e s i s , Univ. of C o n n e c t i c u t , 80 pp. F e l l e r , W . , 1950. An I n t r o d u c t i o n t o P r o b a b i l i t y T h e o r y a n d I t s A p p l i c a t i o n . V o l . I , J o h n W i l e y a n d S o n s , New York, 4 1 9 pp. Gordon, C . M . , 1 9 7 4 . I n t e r m i t t e n t momentum t r a n s p o r t i n a g o e p h y s i c a l 393-394. b o u n d a r y l a y e r . N a t u r e , 248: Gordon, C . M . , 1 9 7 5 . P e r i o d b e t w e e n b u r s t s a t h i g h R e y n o l d s number. Phys. F l u i d s , 1 8 : 1 4 1 - 1 4 3 . Gordon, C . M . , a n d Dohne, C . F . , 1 9 7 3 . Some o b s e r v a t i o n s o f t u r b u l e n t f l o w i n a t i d a l e s t u a r y . J. Geophys. R e s . , 78: 1971-1978. Gordon, C . M . , a n d Dohne, C . F . , 1 9 7 3 . Ocean ' 7 3 I E E E , New Y o r k , 46-49.
A p i v o t e d - v a n e c u r r e n t meter.
Grass, A . J . , 1 9 7 1 . S t r u c t u r a l f e a t u r e s o f t u r b u l e n t f l o w o v e r smooth a n d r o u g h b o u n d a r i e s . J . F l u i d Mech., 5 0 : 233-255.
Heathershaw, A.D., 394-395.
1974.
" B u r s t i n g " phenomena i n t h e sea.
N a t u r e , 248:
J e n k i n s , G . M . , a n d Watts, D . G . , 1 9 6 8 . S p e c t r a l A n a l y s i s a n d I t s A p p l i c a t i o n s . Holden-Day, S a n F r a n c i s c o , 525 pp. K i m , H . T . , K l i n e , S . J . , a n d R e y n o l d s , W . C . , 1 9 7 1 . The p r o d u c t i o n o f t u r b u l e n c e n e a r a smooth w a l l i n a t u r b u l e n t b o u n d a r y l a y e r . J. F l u i d Mech., 50: 133-160.
K l i n e , S . J . , R e y n o l d s , W . C . , S c h r a u b , F.A., a n d R u n d s t a d l e r , P.W., 1967 The s t r u c t u r e o f t u r b u l e n t b o u n d a r y l a y e r s . J . F l u i d Mech., 30: 741-773. L a h e y , R.T. J r . , and K l i n e , S . J . , 1 9 7 1 . A s t o c h a s t i c wave model i n t e r p r e t a t i o n of c o r r e l a t i o n f u n c t i o n s f o r t u r b u l e n t s h e a r fl ows. S t a n f o r d Univ. Rep. MD-26, 235 pp. L a u f e r , J . , a n d B a d r i N a r a y a n a n , M . A . , 1971. Mean p e r i o d o f t h e t u r b u l e n t p r o d u c t i o n mechanism i n a b o u n d a r y l a y e r . Phys. F l u i d s , 1 4 : 182-183. L u , S . S . , and W i l l m a r t h , W . W . , 1 9 7 3 . M e a s u r e m e n t s o f t h e s t r u c t u r e o f t h e R e y n o l d s s t r e s s i n a t u r b u l e n t b o u n d a r y l a y e r . J . F l u i d Mech., 6 0 : 481-511.
81 Mollo-Christensen, E., 1973. Intermittence in large-scale turbulent flows. Ann. Rev. Fluid Mech., 5: 101-118. Offen. G.R., and Kline, S.J., 1973. Experiments on the velocity characteristics of "bursts" and on the interactions between the inner and outer regions of a turbulent boundary layer. Stanford Univ. Rep., MD-31, 230 pp. Rao. K.M., Narasimha, R., and Badri Narayanan, M.A., 1971. The '%ursting" phenomenon in a turbulent boundary layer. J. Fluid Mech., 48: 339-352. Schraub, F.A., and Kline, S.J., 1965. A study of the structure of the turbulent boundary layer with and without longitudinal pressure gradients. Stanford Univ. Rep., MD-12, 157 pp. Sternberg, R.W., 1968. Friction factors in tidal channels with differing bed roughness. Mar. Geol., 6: 243-260. Stewart, R.W., 1973. The air-sea momentum exchange. Boundary Layer Meteorol., 6: 151-167. Taylor, G . I . , 1935. 151: 421-454.
Statistical theory of turbulence.
Proc. Roy. SOC., A
Van Atta, C.W., 1974. Sampling techniques in turbulence measurements. Ann. Rev, Fluid Mech., 6: 75-91. Wallace, J.M., Eckelmann, H., and Brodkey, R.S., 1972. The wal turbulent shear flow. J. Fluid Mech., 54: 39-48.
region in
1972. The structure of Reynolds stress Willmarth, W.W., and Lu, S.S., near the wall. J. Fluid Mech., 55: 65-92.
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83 AN ACOUSTIC SENSOR OF VELOCITY FOR BENTHIC BOUNDARY LAYER STUDIES*
Albert J. W i l l i a m s 3rd and John S . Tochko Woods Hole Oceanocraphic I n s t i t u t i o n
ABSTRACT
The t e c h n i q u e s of f l o w measurement which have been s u c c e s s f u l i n l a b o r a t o r y s t u d i e s of boundary l a y e r t u r b u l e n c e a r e d i f f i c u l t t o use in t h e ocean; and t h e c u r r e n t meters p e n e r a l l y used i n t h e ocean a r e n o t s u i t e d t o measurinp bottom boundary l a y e r flow.
A s u i t a b l e s e n s o r f o r bottom t u r -
bulence measurements s h o u l d measure v e c t o r components, respond l i n e a r l y t o these components, m a i n t a i n a n a c c u r a t e z e r o p o i n t , d i s t u r b t h e flow negl i g i b l y o r i n a w e l l p r e d i c t e d way, and sense a s m a l l enouph volume t o r e p r e s e n t t h e i m p o r t a n t s c a l e s of t h e flow.
W e have c o n s t r u c t e d a n a c o u s t i c
t r a v e l t i m e s e n s o r i n a c o n f i p u r a t i o n t h a t w i l l a l l o w v e c t o r components of t h e flow t o be measured w i t h s u f f i c i e n t a c c u r a c y t o compute Reynolds stress at
B
p o i n t 50 c m above t h e bottom.
This sensor responds l i n e a r l y t o hori-
z o n t a l and v e r t i c a l f l o w s i n flume t e s t s .
When t h e flow i s n e i t h e r h o r i -
z o n t a l n o r v e r t i c a l , t h e wake from one a c o u s t i c t r a n s d u c e r may i n t e r f e r e with t h e measurement a l o n p one s e n s i n p p a t h b u t t h e r e i s s u f f i c i e n t redundancy i n t h e d e t e r m i n a t i o n t o r e j e c t t h i s p a t h and s t i l l r e s o l v e t h e vector velocity.
An i n s t r u m e n t u s i n p f o u r of t h e s e s e n s o r s is b e i n g de-
s i p e d t o measure Reynolds stress i n t h e lower s i x meters of t h e ocean.
V e l o c i t y Sensor Requirements f o r Ocean B e n t h i c Boundary Layer (BBL) S t u d i e s
The g o a l of BBL measurements i s a n u n d e r s t a n d i n p of i n t e r a c t i o n s of t h e flow w i t h t h e b e n t h i c boundary; t h e flow b e i n g c h a r a c t e r i z e d by a t i m e
series of r e l e v a n t flow p a r a m e t e r s a t enouph d i s t a n c e s from t h e sea f l o o r to infer a profile.
The t i m e series must b e l o n p enouch t o sample t h e i m -
p o r t a n t p r o c e s s e s i n t h e flow-boundary i n t e r a c t i o n s , presumably " b u r s t i n ? " b e i n c one of t h e most i m p o r t a n t ones, and t h e samples must be f r e q u e n t enough t o p r e v e n t a l i a s i n g of h i g h frequency components o f t h e flow. Furthermore, i t i s d e s i r a b l e t h a t t h e series be long enough t h a t many independent samples o f t h e l o w e s t frequency of i n t e r e s t i n t h e flow mipht
*Woods Hole Oceanographic I n s t i t u t i o n C o n t r i b u t i o n Number 3843.
84 b e o b t a i n e d and t h e series mipht w e l l b e r e p e a t e d a t i n t e r v a l s o v e r s e v e r a l days t o d e t e c t t h e flow v a r i a b i l i t y w i t h d r i v i n p f o r c e s of l o n p e r p e r i o d s such as t i d e s , mid-ocean e d d i e s , and chanpes of i n t e r n a l s e a s t a t e . To p r e v e n t a l i a s i n g of h i p h frequency components of t h e flow.
the
v e l o c i t y must b e sampled a t t w i c e t h e maximum f r e q u e n c y t h a t c a n b e s e n s e d by t h e s e n s o r .
C o n v e r t i n g t h e maximum f r e q u e n c y t o a wavenumber
by d i v i d i n g by t h e e x p e c t e d a d v e c t i o n v e l o c i t y of t h e flow g i v e s t h e maximum wavenumber which c a n b e s e n s e d w i t h o u t a l i a s i n p .
I t is n e c e s s a r y
t o make a s p a t i a l a v e r a g e of t h e flow v e l o c i t y t o p r e v e n t t h e s e n s i n g of h i g h e r wavenumber components t h a n d e s i r e d , and t h e s i z e o f t h e a v e r a g i n g volume or a v e r a g i n g l e n g t h c a n n o t b e reduced w i t h o u t i n c r e a s i n g t h e s a m pling rate o r risking a l i a s i n g .
T h i s minimum volume or maximum wavenumber
sets t h e l i m i t on t h e f i n e s t s c a l e of t u r b u l e n c e which can b e probed.
As-
suming t h e scale of t h e most e n e r p e t i c eddy i n a boundary l a y e r f l o w i s t h e o r d e r of t h e d i s t a n c e from t h e w a l l , t h e s e n s o r s h o u l d b e p l a c e d s e v e r a l
times i t s a v e r a g i n p l e n p t h from t h e sea f l o o r t o sample t h e e n e r g e t i c port i o n of t h e turbulence spectrum.
Thus t h e f i r s t r e q u i r e m e n t of a v e l o c i t y
sensor i s t h a t i t b e small enouRh t o sample t h e scales of t u r b u l e n c e of i n t e r e s t and t h a t t h e s e n s o r be sampled o f t e n enough t o a v o i d a l i a s i n g c o n s i d e r i n g i t s a v e r a g i n g l e n g t h and t h e e x p e c t e d a d v e c t i o n v e l o c i t y . The two most i m p o r t a n t flow p a r a m e t e r s which s h o u l d b e measured a r e t h e mean v e l o c i t y and t h e Reynolds stress.
I f t h e Reynolds stress i s t o be
c a l c u l a t e d from i n s t a n t a n e o u s v e c t o r v e l o c i t i e s , t h e r e q u i r e m e n t s imposed
on t h e v e l o c i t y s e n s o r by t h i s measurement a r e more demanding t h a n t h o s e of measuring t h e mean v e l o c i t y .
T h i s w i l l b e assumed t h r o u g h o u t .
T i l t of t h e measurement c o o r d i n a t e s w i t h r e s p e c t t o t h e a c t u a l mean flow d i r e c t i o n c a u s e s a f a l s e c o n t r i b u t i o n t o t h e Reynolds stress from f l u c t u a t i o n s i n t h e component o f c u r r e n t i n t h e mean flow d i r e c t i o n .
A
d i g r e s s i o n i s n e c e s s a r y t o c l a r i f y t h e terms used f o r t h e c o o r d i n a t e s .
The
v e l o c i t y components u, v , and w a r e t h e h o r i z o n t a l downstream, h o r i z o n t a l c r o s s - s t r e a m and v e r t i c a l v e l o c i t y components r e s p e c t i v e l y i n a flow a l o n g
a h o r i z o n t a l boundary.
I f t h e boundary i s n o t h o r i z o n t a l , t h e t e r m s
" h o r i z o n t a l " and " v e r t i c a l " are m i s l e a d i n p b u t c o u l d b e r e p l a c e d by t h e
terms " p a r a l l e l t o t h e boundary" and "normal t o t h e boundary".
The l a t t e r
terms r e t a i n t h e n o t i o n t h a t t h e Reynolds stress i s a measure of momentum exchanpe between t h e flow p a r a l l e l t o t h e boundary and t h a t boundary. However, i f t h e boundary i s n o t even f l a t , t h e s e terms may b e i n a d e q u a t e and one must go back t o d e f i n i n p t h e downstream d i r e c t i o n , x, as t h e d i r e c t i o n t h a t produces z e r o mean i n t h e v e l o c i t y a l o n g t h e two o r t h o g o n a l
85
-
-
d i r e c t i o n s y and z , i . e . v = w = 0 .
This defines x but not y o r z .
If
t h e mean flow changes d i r e c t i o n , t h e o l d and new mean flow d i r e c t i o n s def i n e t h e xy p l a n e and t h e z d i r e c t i o n i s t h e normal t o t h i s p l a n e .
Op-
e r a t i o n a l l y , t h i s i s a c o n v e n i e n t d e f i n i t i o n of t h e measurement c o o r d i n a t e s .
To summarize:
-
z i s t h e d i r e c t i o n a l o n p which t h e a v e r a g e v e l o c i t y , w ,
is
z e r o f o r any d i r e c t i o n of mean flow; y i s t h e d i r e c t i o n p e r p e n d i c u l a r t o z
-
a l o n e which t h e a v e r a g e v e l o c i t y , v , i s z e r o ; and x i s t h e d i r e c t i o n per-
-
p e n d i c u l a r t o y and z a l o n p which t h e mean flow, u, i s measured.
Errors
i n t i l t a r e s u f f i c i e n t l y s e r i o u s t h a t one s h o u l d v e r i f y t h a t t h e mean v e l o c i t y i n t h e d i r e c t i o n assumed t o b e z i s z e r o .
I f i t is n o t , a n in-
s t r u m e n t a l o r c o o r d i n a t e r o t a t i o n s h o u l d b e performed t o a c h i e v e t h i s result
.*
Zero p o i n t u n c e r t a i n t y i n t h e z a x i s i s i n d i s t i n p u i s h a b l e from a t i l t e r r o r and t h u s p r e v e n t s v e r i f i c a t i o n of t h e a l i g n m e n t and i n t e r f e r e s w i t h r o t a t i n g t h e c o o r d i n a t e s t o remove t i l t .
The z e r o p o i n t e r r o r should n o t
exceed ttie allowed d e v i a t i o n from z e r o of t h e mean of t h e w component of velocity.
A z e r o o f f s e t i n w of 1%of t h e mean v e l o c i t y i s e q u i v a l e n t t o
a t i l t e r r o r of 1 / Z 0 and i s p r o b a b l y t h e l a r g e s t e r r o r a c c e p t a b l e . Leakage of u s i g n a l i n t o t h e w c h a n n e l c o n t r i b u t e s a n e r r o r d i r e c t l y t o t h e c a l c u l a t e d Reynolds stress. non-orthogonality
Three s o u r c e s of l e a k a g e are p o s s i b l e :
of t h e c h a n n e l s , flow d i s t u r b a n c e by t h e s e n s o r , and
e l e c t r o n i c c r o s s - t a l k i n t h e samplinF c i r c u i t r y .
Geometric c o n t r o l s may
n o t n e c e s s a r i l y e n s u r ? t h e o r t h o g o n a l i t y of t h e c h a n n e l s and t h i s s h o u l d be checked i n a tow t a n k o r flume 'by r o t a t i n g t h e s e n s o r 90' the n u l l i n the u channel.
and v e r i f y i n g
T h i s t e s t w i l l n o t d e t e c t flow d i s t u r b a n c e by
t h e s e n s o r as t h e flow i n u s e w i l l n o t b e i n t h e s e o r t h o g o n a l d i r e c t i o n s . The b e s t check on flow d i s t u r b a n c e p r o b a b l y remains c h e c k i n g of c o s i n e response f o r each v e c t o r c h a n n e l .
E l e c t r o n i c c r o s s - t a l k is evidenced as a
s i g n a l a p p e a r i n p i n a c h a n n e l w i t h a dummy s o u r c e p r e s e n t i n p l a c e of t h e
*The Reynolds s t r e s s is p u " .
be computed.
The q u a n t i t i e s p u "
and p v "
can a l s o
They would r e p r e s e n t momentum t r a n s p o r t t h a t c o u l d n o t b e
e x t r a c t e d from a mean flow t h a t w a s homopeneous i n t h e y d i r e c t i o n and
in t h e x d i r e c t i o n and t h u s they s h o u l d be s m a l l i n p r a c t i c e . s u g g e s t s a n a l t e r n a t e way t o d e f i n e t h e c o o r d i n a t e s .
This
The y a x i s i s
chosen so t h a t t h e c o v a r i a n c e of t h e flow a l o n p t h a t a x i s w i t h t h e flow along any a x i s p e r p e n d i c u l a r t o y is minimum. dicular to y so that
y and
Z.
w
=
0.
Then z i s chosen perpen-
F i n a l l y x i s chosen p e r p e n d i c u l a r t o
86
normal s e n s o r w h i l e a n o t h e r c h a n n e l is normally connected and measuring flow. Non-linearity
is u n a c c e p t a b l e in measurements used in c a l c u l a t i n g
Reynolds stress and for t h i s r e a s o n w e b e l i e v e o n l y i n h e r e n t l y l i n e a r dev i c e s are s u i t a b l e f o r deep sea Reynolds stress s e n s o r s .
S e n s i t i v i t y re-
quirements a r e h i g h f o r low v e l o c i t y flows, a n approximate l i m i t b e i n g 1% of t h e mean flow.
F a i l u r e of t h e w s e n s o r t o d e t e c t small flow components
w i l l s e r i o u s l y u n d e r e s t i m a t e t h e stress.
Classes o f Sensors
-
Advantages and L i m i t a t i o n s
Two c l a s s e s of s e n s o r s d e s e r v e c o n s i d e r a t i o n (and indeed are s u i t a b l e ) f o r BBL s t u d i e s by v i r t u e of t h e i r i n h e r e n t l y l i n e a r r e s p o n s e and minimized i n t e r f e r e n c e w i t h t h e measured flow.
The f i r s t class is s c a t t e r i n g s e n s o r s
such as a c o u s t i c d o p p l e r and laser d o p p l e r v e l o c i m e t e r s .
The second c l a s s
is volume a v e r a g i n p s e n s o r s such as e l e c t r o m a p e t i c and a c o u s t i c t r a v e l t i m e velocimeters. S c a t t e r i n g s e n s o r s a c t u a l l y measure t h e v e l o c i t y of p a r t i c l e s suspended
in t h e f l u i d .
Though t h e s e may n o t have t h e same v e l o c i t y a s t h e f l u i d , in
open ocean water t h e m a j o r i t y of t h e s c a t t e r e r s are v e r y small and t h e d i f f e r e n c e between t h e i r v e l o c i t y and t h e f l u i d ' s v e l o c i t y c a n u s u a l l y b e neglected.
The e x c e p t i o n is when t h e s e t t l i n g v e l o c i t y o f t h e p a r t i c l e s is
more t h a n a s m a l l f r a c t i o n , s a y 1%. of t h e measured v e r t i c a l component. The advantages of s c a t t e r i n g s e n s o r s are g r e a t , t h e p r i n c i p l e ones b e i n g a n a c c u r a t e z e r o p o i n t ; a s e n s e d volume t h a t is remote from s t r u c t u r e s ; reduced c r o s s - t a l k ;
and i n t h e c a s e of t h e laser d o p p l e r v e l o c i m e t e r ,
a very s m a l l sampling volume.
The f a c t t h a t t h e s i g n a l is i n t h e form o f a
frequency s h i f t is c o n v e n i e n t f o r d i p i t a l sampling w h i l e t h e dependence of s e n s i t i v i t y on geometry a l o n e makes flume c a l i b r a t i o n unnecessary.
The c h i e f d i s a d v a n t a p e of t h e s c a t t e r i n g s e n s o r s is signal dropout due t o t h e i n t e r m i t t a n t p r e s e n c e of p a r t i c l e s in t h e volume hence complexity of t h e signal p r o c e s s i n p n e c e s s a r y t o overcome t h e dropout problem.
A second
d i s a d v a n t a g e is a loss o f d i r e c t i o n s e n s e w i t h o u t a n added frequency s h i f t e r , a n e x p e n s i v e complexity in laser d o p p l e r v e l o c i m e t e r s and s o f a r unexplored o p t i o n in a c o u s t i c d o p p l e r v e l o c i m e t e r s .
A t h i r d disadvantage, t h a t the
s c a t t e r i n g is weak and r e q u i r e s high power l e v e l s , is probably n o t s o s e r i o u s f o r most BBL s t u d i e s which can have b r i e f deployments.
Because t h e scat-
t e r i n g volume i s v e r y small, p o s s i b l y much smaller t h a n t h e smallest scale of i n t e r e s t , t h e r a p i d sampling n e c e s s a r y t o a v o i d a l i a s i n g w i l l r e q u i r e e x c e s s i v e d a t a s t o r a g e c a p a c i t y u n l e s s o n - l i n e p r o c e s s i n g is performed.
81
This adds a d d i t i o n a l c o m p l e x i t y .
However, f o r t h e most demandinp ap-
p l i c a t i o n s , s u c h a s v e r y n e a r t h e bottom or v e r y slow f l o w s , laser d o p p l e r o r a c o u s t i c d o p p l e r v e l o c i m e t e r s are p r o b a b l y n e c e s s a r y .
Laser d o p p l e r
v e l o c i m e t e r s are a b o u t 100 times as s e n s i t i v e as a c o u s t i c d o p p l e r velocim-
eters and can form much smaller s c a t t e r i n p volumes.
Thus t h e s e w i l l
probably become t h e s t a n d a r d a g a i n s t which o t h e r sensors a r e compared. One of t h e a d v a n t a p e s of volume a v e r a p i n p s e n s o r s is t h e s i m p l e samp l i n g which is a l l o w e d when t h e sampling volume c o r r e s p o n d s t o t h e smallest l e n g t h s c a l e of i n t e r e s t in t h e flow.
T h i s volume in t h e EM s e n s o r is re-
l a t e d t o t h e volume o v e r which t h e f i e l d is s o l e n o i d a l o r a l e n g t h s c a l e approximately t h e d i a m e t e r of t h e f i e l d c o i l .
U n f o r t u n a t e l y , t h e non-
u n i f o r m i t i e s i n flow in r e g i o n s o u t s i d e t h e s o l e n o i d a l f i e l d c o n t r i b u t e t o t h e measurement b u t i n a c o m p l i c a t e d way.
Ducted EM s e n s o r s a v o i d t h e com-
p l i c a t e d volume a v e r a g e b u t s u f f e r a reduced r a n p e o v e r which a c o s i n e response a p p l i e s .
S t i l l , under many c o n d i t i o n s , t h e measurements of EM
s e n s o r s a p p e a r t o r e p r e s e n t t h e flow averaped o v e r t h e l e n p t h s c a l e o f t h e sensor diameter.
A c o u s t i c t r a v e l t i m e s e n s o r s a v e r a g e t h e flow o v e r t h e
a c o u s t i c p a t h between t h e t r a n s d u c e r s .
Except when t h e wake of one t r a n s -
ducer d i s t u r b s t h e p a t h , t h i s a v e r a g e is a s i m p l e one. An i m p o r t a n t a d v a n t a p e of t h e EM and a c o u s t i c s e n s o r s is t h e r e l a t i v e
a v a i l a b i l i t y of t h e technolopy.
S e v e r a l p e n e r a t i o n s o f e a c h have been made
and e x p e r i e n c e h a s been accumulated w i t h t h e t e c h n i q u e s . The volume a v e r a g i n p t e c h n i q u e s have a l i n e a r r e s p o n s e t o t h e v e l o c i t y component a l o n p a s i n g l e a x i s b u t they do not have p e r f e c t z e r o p o i n t s .
Their s e n s i t i v i t i e s may d i f f e r from t h o s e which are c a l c u l a t e d from physic a l dimensions and e l e c t r o n i c component v a l u e s s o t h e y must b e c a l i b r a t e d i n a tow t a n k .
However, d i r e c t i o n sense is n o t a problem as t h e s i p n a l
changes s i p when t h e f l o w r e v e r s e s . I f c a r e is t a k e n w i t h t h e d e s i p n , t h e flow t h a t is s e n s e d is l i t t l e d i s t u r b e d by t h e p h y s i c a l s t r u c t u r e of t h e s e n s o r s .
This helps reduce
c r o s s - t a l k as w e l l as minimizinp d i s t u r b a n c e t o t h e n a t u r a l flow.
Acoustic T r a v e l 'Time S e n s o r As w e have t h e g r e a t e s t e x p e r i e n c e w i t h t h e a c o u s t i c t r a v e l t i m e sen-
s o r , we are e x p l o i t i n p t h a t t e c h n i q u e f o r BBL s t u d i e s .
Our e x p e r i e n c e h a s
been w i t h a two-axis f r e e - f a l l v e l o c i t y s h e a r meter d e s i p n e d by Trygve Gytre (Gytre, 1975) a t C h r i s t i a n Michelson I n s t i t u t e , Bergen.
signals were low p a s s e d a t 0.2 Hz and sampled a t 5 Hz. between t r a n s d u c e r s w a s 15 cm.
The a n a l o g
The a c o u s t i c p a t h
Two h o r i z o n t a l p a t h s a t r i g h t a n g l e s were
88
used which o p e r a t e d i n u n d i s t u r b e d water due t o t h e v e r t i c a l s i n k i n g of the instrument.
F i g u r e 1 shows t h e i n s t r u m e n t , SCIMP (Williams, 1 9 7 4 ) ,
w i t h t h e a c o u s t i c s h e a r m e t e r mounted v e r t i c a l l y , i t s t r a n s d u c e r s a t t h e end of t h e t e t r a p o d p r o j e c t i n p below t h e s h o r t v e r t i c a l c y l i n d e r which houses t h e e l e c t r o n i c s .
SCIMP w a s equipped w i t h a r e c o r d i n p CTD a s w e l l
as t h e s h e a r m e t e r t o measure m i c r o s t r u c t u r e a s s o c i a t e d w i t h v e l o c i t y s h e a r .
FIGURE 1:
F r e e - f a l l i n s t r u m e n t , SCIMP, c o n t a i n i n p a c o u s t i c v e l o c i t y s e n s o r as a s h e a r m e t e r .
89 F i g u r e 2 shows one such c o r r e l a t i o n as a n example, p r i n c i p a l l y , of the performance of t h e a c o u s t i c t r a v e l t i m e s e n s o r .
A shear sheet located
a t the d e n s i t y i n t e r f a c e i s r e c o r d e d as a s h a r p i n c r e a s e of 2 cm/sec f o r t h e t i m e d u r i n g which t h e v e l o c i t y s e n s o r is in t h e lower l a y e r b u t t h e c e n t e r of d r a g of t h e i n s t r u m e n t s t i l l in t h e upper l a y e r .
The v e l o c i t y
r e t u r n s t o z e r o as t h e c e n t e r of dray! (50 cm above t h e v e l o c i t y s e n s o r ) Velocity s t r u c t u r e is a l s o apparent i n the l a y e r s
e n t e r s t h e lower l a y e r .
on e i t h e r s i d e of t h e i n t e r f a c e .
11.8 -
SCIMP VII 8 20
\
&$ 2 Y Y
\
s
L
‘0
L
L
-
$
-
11.4
11.2 -
-
0
A
27.1
DENSITY
\ 730 74 0
11.0
FIGURE 2:
3
a
5 k!
3
11.6 -
720 DEPTH lrneiersl
a 2 Y Y
2 \
27.0
P r o f i l e o f t e m p e r a t u r e , d e n s i t y , and h o r i z o n t a l v e l o c i t y d i f f e r e n c e o v e r 50 c m v e r t i c a l s e p a r a t i o n .
The a i r backed p i e z o e l e c t r i c c r y s t a l t r a n s d u c e r s used i n t h e s h e a r -
meter were s a t i s f a c t o r y t o 2000 M d e p t h b u t f o r deep ocean work w e p r e f e r p r e s s u r e compensated t r a n s d u c e r s .
F i p u r e 3 shows t h e epoxy e n c a p s u l a t e d
c r y s t a l we now use which performs w e l l a c o u s t i c a l l y and is n o t d e p t h limited. The geometry of t h e s e n s o r head used i n t h e s h e a r m e t e r would d i s t u r b t h e flow i n a s t a t i o n a r y mount so t h e peometry i l l u s t r a t e d in F i g u r e 4 w a s devised which i n c l u d e s f o u r a c o u s t i c p a t h s and s e n s e s an u n d i s t u r b e d flow
90 i f t h e flow i s a p p r o x i m a t e l y h o r i z o n t a l o r v e r t i c a l .
Four p a t h s p r o v i d e
s u f f i c i e n t redundancy f o r a v e c t o r v e l o c i t y measurement t h a t one can b e discarded i f necessary.
I f t h e flow i s n e i t h e r h o r i z o n t a l n o r v e r t i c a l ,
t h e wake of one t r a n s d u c e r may l i e a l o n p o r n e a r one a c o u s t i c p a t h . path w i l l then b e discarded i n determining the v e l o c i t y vector.
This
The wake
may c r o s s a n o t h e r p a t h b u t c a n n o t l i e a l o n p o r n e a r i t f o r any s i g n i f i c a n t d i s t a n c e s o w i l l n o t d i s t u r b e i t t o any g r e a t e x t e n t .
P V C Mount
\
FIGURE 3:
Piezoelectric
fi
Epoxy-insulated pressure-exposed transducer.
1.0 cm diameter 1.3 mm thick
piezoelectric acoustic
Tow t a n k tests have been made on t h e model o f F i g u r e 5 and have v e r i f i e d t h e e x p e c t e d c o s i n e r e s p o n s e f o r h o r i z o n t a l and v e r t i c a l f l o w s and f o r d i a g o n a l f l o w s up t o a b o u t 20'
from t h e a x i s o f t h e a c o u s t i c p a t h .
A d i s c u s s i o n of t h e e l e c t r o n i c arranpement s h o u l d s t a r t w i t h t h e pen-
era1 p r i n c i p l e of t h e a c o u s t i c t r a v e l t i m e s e n s o r .
As i l l u s t r a t e d i n
F i g u r e 6 , two p i e z o e l e c t r i c t r a n s d u c e r s s e p a r a t e d a d i s t a n c e d a r e e x c i t e d simultaneously.
The component of flow a l o n g t h e i n t e r t r a n s d u c e r a x i s de-
c r e a s e s t h e t r a v e l time o f t h e a c o u s t i c p u k e p r o p a g a t i n g w i t h t h e c u r r e n t and i n c r e a s e s t h e t r a v e l t i m e of t h e p u l s e p r o p a g a t i n g a g a i n s t t h e c u r r e n t . The d i f f e r e n c e i n t r a v e l t i m e i s A t = 2dv/c2 t o f i r s t o r d e r where c i s t h e speed o f sound i n t h e medium.
1500 m / s
R e f r a c t i o n due t o c u r r e n t s h e a r n e a r
t h e t r a n s d u c e r s does n o t e f f e c t t h i s r e s u l t d i r e c t l y because t h e p a t h , though b e n t , i s t h e same f o r e a c h p r o p a g a t i o n d i r e c t i o n and t h e t i m e d i f f e r e n c e is a l i n e i n t e p r a l between t h e t r a n s d u c e r s .
An i n d i r e c t e f f e c t may
-
o c c u r , however, throuph a m p l i t u d e r e d u c t i o n .
%, FIGURE 6:
V
F l u i d v e l o c i t y component, v , a l o n p p a t h between t r a n s d u c e r s A and B s e p a r a t e d by d i s t a n c e d r e t a r d s one p u l s e and advances t h e o t h e r .
91
I
,
\\
1
FIGURE 4 :
Acoustic sensor with four diaponal sensing paths.
FIGURE 5:
'ho-path
model (used i n tow t e s t s ) .
The p u l s e s a r e g e n e r a t e d by a p p l y i n g a h i g h v o l t a g e t r a n s i e n t t o t h e c r y s t a l s which changes t h e i r t h i c k n e s s and produces a c o m p r e s s i o n a l wave i n the f l u i d .
I t i s d i f f i c u l t t o d e l i v e r enough e n e r g y t o t h e c r y s t a l in-
s t a n t a n e o u s l y t o a c h i e v e a measurable a c o u s t i c p u l s e s o i n p r a c t i c e t h e c r y s t a l forms t h e c a p a c i t a n c e of a tuned c i r c u i t , t h e tuned c i r c u i t b u i l d i n k up a m p l i t u d e d u r i n g t h e f i r s t q u a r t e r c y c l e of i t s r e s o n a n t p e r i o d and
producing a much h i g h e r v o l t a g e t r a n s i e n t on t h e second q u a r t e r c y c l e than can be o b t a i n e d i n s t a n t a n e o u s l y .
T h i s is t h e p u l s e t h a t i s a c t u a l l y used.
I t is, however, d e l a y e d by one-half
c y c l e from t h e t r i g g e r p u l s e .
Simi-
l a r l y , t h e r e c e i v e d p u l s e e x c i t e s t h e tuned c i r c u i t s o t h a t t h e second q u a r t e r c y c l e h a s a g r e a t e r a m p l i t u d e than t h e f i r s t and t h e comparator is
s e t t o t r i g g e r on t h i s edRe.
Thus, t h e r e are two d e l a y s due t o t h e p e r i o d s
92 o f two tuned c i r c u i t s added t o t h e t r a v e l times.
The t r a n s m i t t i n g c i r c u i t
a d d s e l e m e n t s n o t p r e s e n t in t h e r e c e i v i n e c i r c u i t s o r e c i p r o c i t y does n o t q u i t e h o l d and t h e d i f f e r e n c e s i n t r a n s m i t t i n p p e r i o d s and i n r e c e i v i n p p e r i o d s produce a z e r o p o i n t e r r o r .
Careful tuning of the inductors t o
the c r y s t a l s can reduce t h i s e r r o r .
Temperature and p r e s s u r e terms i n t h e
r e s o n a n c e must be s i m i l a r l y matched.
The r e c e i v e d s i g n a l s are d e t e c t e d by a p a i r of c o m p a r a t o r s . make
P
These
TTL l e v e l t r a n s i t i o n a s h o r t b u t f i x e d t i m e a f t e r t h e v o l t a p e from
t h e c r y s t a l exceeds a t h r e s h o l d which i s set t o r e j e c t n o i s e and t h e weak precursor pulse.
The f i x e d d e l a y i s n o t t h e same f o r t h e two comparators
and t h i s i n t r o d u c e s a second p o t e n t i a l z e r o p o i n t e r r o r .
T h i s c a n b e re-
moved by i n t e r c h a n g i n p t h e c r y s t a l s between t h e two comparators i n t h e scheme i l l u s t r a t e d in F i g u r e 7. S E N
s
0 R
5 E
N S
0 R
FIGURE 7:
Block diagram of a c o u s t i c v e l o c i t y s e n s o r w i t h t r a n s d u c e r t r a n s position f o r zero d r i f t correction.
The d i f f e r e n c e i n d e t e c t i o n t i m e s i s t h e measure of v e l o c i t y .
The
t r a n s i t i o n of t h e lower comparator may o c c u r b e f o r e o r a f t e r t h e t r a n s i t i o n of t h e upper comparator so a one-shot
t i m e r is t r i g g e r e d by t h e upper com-
p a r a t o r t o add a d e l a y somewhat l o n g e r than t h e p r e a t e s t e x p e c t e d t i m e d i f ference.
C o n s t a n t c u r r e n t ramp i n t e p r a t o r s are s t a r t e d by e a c h comparator
and s t o p p e d by t h e one-shot timer.
V a r i a t i o n s in t h e i n t e r v a l of t h e one-
s h o t timer do n o t e f f e c t t h e measurement d i r e c t l y s i n c e they simply e x t e n d b o t h ramps by t h e same amount.
However, n o n - l i n e a r i t i e s i n t h e ramp o r
93 d i f f e r e n c e s i n ramp shape between t h e two i n t e p r a t o r s c a n produce a z e r o p o i n t e r r o r w i t h one-shot
timinp v a r i a t i o n s .
Apain t h i s can b e removed by
i n t e r c h a n g i n g t h e t r a n s d u c e r s between t h e c o m p a r a t o r s .
The o n l y e r r o r
which remains i s t h e second o r d e r e r r o r due t o j i t t e r i n t h e one-shot i n t e r v a l and t h i s w i l l a p p e a r as n o i s e .
Two c y c l e s o f t r a n s m i s s i o n are r e q u i r e d f o r t h e measurement.
The
f i r s t c y c l e i s performed w i t h t r a n s d u c e r A connected t o t h e upper comparat o r and t r a n s d u c e r B connected t o t h e lower c o m p a r a t o r .
The s e q u e n c e r
t r i g p e r s t h e t r a n s m i t p u l s e and t h e r e c e i v e d p u l s e s a r e d e t e c t e d by t h e comparators.
G a t i n g ( n o t shown) p r e v e n t s p r e m a t u r e d e t e c t i o n and sup-
p r e s s e s d e t e c t i o n of t h e echoes of t h e f i r s t p u l s e .
The i n t e g r a t o r s c h a r g e
and a r e d i f f e r e n c e d i n a n o p e r a t i o n a l a m p l i f i e r , t h e o u t p u t b e i n g s t o r e d i n
one sample and h o l d c i r c u i t .
Then t h e t r a n s d u c e r s a r e i n t e r c h a n g e d and B
connected t o t h e upper c o m p a r a t c r and A connected t o t h e lower comparator. The s e q u e n c e r t r i p p e r s a n o t h e r t r a n s m i t p u l s e and t h e measurement i s made a p a i n , t h e r e s u l t b e i n p s t o r e d i n t h e o t h e r sample and h o l d c i r c u i t .
The
two sample and h o l d c i r c u i t s are d i f f e r e n c e d and t h e o u t p u t of t h e d i f f e r e n c e a m p l i f i e r i s d i g i t i z e d and r e c o r d e d .
Any e r r o r s i n t r o d u c e d i n t h e
e l e c t r o n i c s between t h e comparators and t h e f i r s t d i f f e r e n c e a m p l i f i e r which have remained t h e same f o r t h e two c y c l e s a r e n u l l e d w h i l e t h e t i m e d i f f e r e n c e s i g n a l s from t h e t r a n s d u c e r s are doubled. In the four path sensor described before, the transducer p a i r s w i l l be s e r v i c e d s e q u e n t i a l l y , t r a n s d u c e r s 3 and 4 r e p l a c i n g 1 ar.d 2 as t h e A and B c h a n n e l s , e t c .
A c y c l e r e q u i r e s 2 m s f o r t h e echoes t o d i e and a
r e c o r d i n g of a sample t a k e s 15 ms, t h u s t h e r e is ample t i m e t o o b t a i n t h e v e l o c i t y a l o n g t h e f i r s t p a t h , s w i t c h t o t h e second p a i r of t r a n s d u c e r s , and s o on, r e c o r d i n g t h e f o u r v e l o c i t y components i n 60 ms.
In f a c t , f o u r
s e p a r a t e f o u r - p a t h s e n s o r s c a n be m u l t i p l e x e d t o a s i n g l e r e c e i v e r and r e corded i n 240 ms. E l e c t r o n i c c r o s s - t a l k between t h e r e c e i v e d sipnals i s a problem as t h e
risetimes a r e s h o r t and larEe c u r r e n t s a r e needed t o c h a r g e even small capacitances.
The c o m p a r a t o r s a r e v o l t a g e s e n s i n p d e v i c e s so induced v o l t -
ages o r common ground v o l t a p e s p r e s e n t on t h e s i g n a l l e a d s change t h e d e t e c tion time.
I f two p a t h s are sampled s i m u l t a n e o u s l y , v e l o c i t y components on
one p a t h w i l l change t h e a r r i v a l times of t h e s i p n a l s which t h e n c r o s s - t a l k i n t o t h e o t h e r p a t h s i g n a l s t o c a u s e an a p p a r e n t change i n a r r i v a l t i m e on t h a t channel.
S e q u e n t i a l l y c o n n e c t i n g o n l y one p a i r of t r a n s d u c e r s a t a
time i n a m u l t i p l e x e r removes t h i s e r r o r .
94 C r o s s - t a l k between c h a n n e l s A and B s t i l l o c c u r s and i s d i f f i c u l t t o d e t e c t , g e n e r a l l y c a u s i n g a lower s e n s i t i v i t y t h a n c a l c u l a t e d ; i n our s h e a r -
meter t h e d e c r e a s e - i n s e n s i t i v i t y amounted t o 15%. A c a r e f u l l y l a i d o u t p r o t o t y p e r e c e n t l y t e s t e d w a s much improved.
The concern w i t h c r o s s - t a l k
between c h a n n e l s A and B i s n o t so much t h a t t h i s i n t r o d u c e s a n e r r o r i n t h e Reynolds stress c a l c u l a t i o n p e r s e b u t t h a t t h e s e n s i t i v i t y mipht v a r y w i t h a c o u s t i c s i g n a l amplitude and t h u s b e a f f e c t e d by chanpes i n alignment, f o u l i n g , and b a t t e r y v o l t a g e .
Benthic Acoustic S t r e s s Sensor (BASS)
We p l a n t o c o n s t r u c t a n i n s t r u m e n t u s i n g f o u r of t h e a c o u s t i c v e l o c i t y s e n s o r s t o s t u d y b e n t h i c boundary l a y e r flows on t h e deep c o n t i n e n t a l s h e l f , c o n t i n e n t a l s l o p e , and c o n t i n e n t a l r i s e .
The f r e e s t r e a m v e l o c i t y i n t h e s e
a r e a s i s expected t o b e t h e o r d e r of 10 cm/sec. s u b l a y e r w i l l probably n o t e x i s t .
I n t h i s case, a viscous
The c o n s t a n t s t r e s s l a y e r w i l l be ap-
proximately 2 meters t h i c k and t h e l o g a r i t h m i c l a y e r about 10 meters t h i c k . The f o u r s e n s o r s w i l l b e spaced throuph t h e s e l a y e r s i n an a t t e m p t t o obtain a profile. 1.2 M,
The l o w e s t s e n s o r w i l l be 50 cm above t h e bottom, t h e n e x t
t h e n e x t 2.5 M, and t h e top s e n s o r 6 M above t h e bottom.
Figure 8
i l l u s t r a t e s t h i s instrument. A s t a f f rises from a weiphted t r i a n g u l a r frame, t h e s e n s o r s b e i n g secured t o t h e s t a f f .
The b a s e c o n t a i n s t h e buoyancy, e l e c t r o n i c s package,
and b a t t e r i e s , t h u s t h e flow w i l l be d i s t u r b e d by t h i s roughness element f o r about 1 meter above t h e bottom.
However,
t h e upstream d i s t u r b a n c e due
t o t h e b a s e should be minimal and t h e s e n s o r s are upstream f o r flow d i r e c t i o n s c o v e r i n g perhaps 240'.
Except f o r t h e lowest s e n s o r , t h e flow i s
only d i s t u r b e d behind t h e s t a f f .
There w i l l b e p e r i o d s of d i s t u r b e d flow
when t h e measurements cannot b e used; however, w i t h luck.
these w i l l be infrequent
We f e e l a r i g i d mounting i s w c e s s a r y f o r measurements a t t h i s
s c a l e and t h u s must s u f f e r t h e consequences of i n t e r f e r e n c e by t h e s t r u c ture.
The b u l k of t h e s t r u c t u r e h a s been p u t low f o r s t a b i l i t y .
s t r u m e n t w i l l b e lowered by c a b l e .
The i n -
T i l t and a s i n g l e v e l o c i t y component
w i l l be a c o u s t i c a l l y t e l e m e t e r e d t o t h e s u r f a c e so t h e s u i t a b i l i t y of a
s e l e c t e d s i t e can b e determined b e f o r e t h e c a b l e is r e l e a s e d . v e l o c i t y w i l l be noted w i t h t h e BASS n e a r t h e bottom.
First, the
I f the velocity is
r e a s o n a b l e , i t w i l l b e lowered t o t h e bottom and some s l a c k p a i d o u t .
If
t h e t i l t is r e a s o n a b l e and t h e v e l o c i t y remains r e a s o n a b l e , t h e c a b l e w i l l be r e l e a s e d .
Otherwise, BASS w i l l b e recovered and a new s i t e s e l e c t e d .
95
FIGURE 8:
B e n t h i c A c o u s t i c Stress S e n s o r . 6 M above t h e b a s e .
The t o p v e l o c i t y s e n s o r is
Recovery o f BASS by a c o u s t i c command w i l l e n t a i l d r o p p i n g t h e weighted base o f t h e frame.
The i n s t r u m e n t w i l l be t r a c k e d t o t h e s u r f a c e acous-
t i c a l l y where i t w i l l b e r e c o v e r e d w i t h t h e a i d of a f l a s h i n g l i p h t and r a d i o beacon. A round o f measurements w i l l b e made e a c h 750 m s t o a v o i d a l i a s i n g
c u r r e n t s up t o 10 cm/sec.
Each round g e n e r a t e s 192 b i t s of d a t a which,
in
96
w i t h housekeepinp b i t s and i n t e r r e c o r d gaps, a l l o w s something more t h a n 12 h o u r s of c o n t i n u o u s r e c o r d i n g w i t h a Sea Data d i p i t a l cassette r e c o r d e r . I n i t i a l l y a s i n g l e e o n t i n u o u s r u n w i l l be most u s e f u l as i t w i l l c o v e r a f u l l t i d a l period.
Subsequently, t h e sampling w i l l be programmed,to ob-
t a i n i n f o r m a t i o n o v e r 3 days t o n o t e v a r i a t i o n s w i t h change i n mesoscale a c t i v i t y o r i n t e r n a l sea s t a t e . The program f o r p r o c e s s i n g t h e d a t a i s as f o l l o w s :
f o r each sample
t h e v e l o c i t y w i l l b e r e s o l v e d i n t o u, v, and w components u s i n g t h r e e paths.
I f t h i s v e c t o r i s n e a r one o f t h e p a t h s used, i t w i l l be recomputed
s u b s t i t u t i n g a n o t h e r p a t h f o r t h e d i s t u r b e d one. checked f o r z e r o mean.
The w component w i l l b e
I f t h e r e is a sysrematic o f f s e t , the d a t a w i l l be
transformed by a c o o r d i n a t e r o t a t i o n t r a n s f o r m a t i o n .
I t may be n e c e s s a r y
t o do t h i s s e p a r a t e l y f o r p i e c e s of t h e d a t a where t h e flow is from t h e
same d i r e c t i o n . Then f o r each s e n s o r t h e u and v components w i l l be reduced t o a n amplitude and azimuth.
Averages of t h i s a m p l i t u d e w i l l b e t a k e n f o r s e c -
t i o n s of t h e d a t a between 10 m i n u t e s and 1 hour l o n g , and v a r i a t i o n s from t h i s mean w i l l b e r e c o r d e d alonpr w i t h t h e p r o d u c t of t h e u v a r i a t i o n and w component a t each sample.
d a t a f o r t h e experiment.
T h i s i n f o r m a t i o n w i l l form t h e t i m e series
I t w i l l c o n t a i n t h e u mapnitude and azimuth, t h e
w v e l o c i t y , t h e u speed v a r i a t i o n , and a p r o d u c t c o r r e s p o n d i n g t o a n i n -
s t a n t a n e o u s Reynolds stress. P i e c e s of t h e t i m e series w i l l t h e n b e s e l e c t e d which a p p e a r t o behave s i m i l a r l y , f o r example. a n a c c e l e r a t i n g t i d a l i n t e r v a l o r a d e c e l e r a t i n p t i d a l i n t e r v a l , and a s e t of frequency a n a l y s e s w i l l b e performed on t h e data:
t h e spectrum of u f o r each s e n s o r , t h e cospectrum of u and w f o r
each s e n s o r , and t h e coherency and phase of u v e l o c i t y and of w v e l o c i t y between p a i r s o f s e n s o r s . With t h i s i n s t r u m e n t we hope t o e x t e n d b e n t h i c boundary l a y e r v e l o c i t y o b s e r v a t i o n s d e e p e r i n t o t h e sea where mean v e l o c i t i e s and s h e a r stresses
are lower.
W e hope t o o b s e r v e mean p r o f i l e s , Reynolds stress l e v e l s , and
t r a n s i e n t phenomena i n t h e s e environments. Discussion D i s c u s s i o n of t h i s and a n o t h e r p a p e r opened t h e q u e s t i o n of how s m a l l a n a c o u s t i c t r a v e l time s e n s o r might r e a s o n a b l y be made.
A s t h e s i z e is
d e c r e a s e d , t h e t i m e d i f f e r e n c e f o r any piven v e l o c i t y d e c r e a s e s b u t t h e t i m i n g e r r o r s remain f i x e d .
This means t h e v e l o c i t y u n c e r t a i n t y i n c r e a s e s .
A t t h e same t i m e , t h e small scale e d d i e s a c c e s s i b l e w i t h t h e smaller s e n s o r
97 have c h a r a c t e r i s t i c v e l o c i t i e s t h a t a r e less t h a n t h o s e of t h e l a r g e r eddies.
A t some s c a l e , t h e v e l o c i t y u n c e r t a i n t y e q u a l s t h e t u r b u l e n t
v e l o c i t y f l u c t u a t i o n s one e x p e c t s t o see.
T h i s c r o s s o v e r p o i n t depends on
the d i s s i p a t i o n c o n s t a n t s one u s e s h u t a n estimate of t u r b u l e n t v e l o c i t y f l u c t u a t i o n s o f 5 mm/sec f o r a n eddy s c a l e of 3 c m seems a b o u t r i p h t and t h i s matches a p r o b a b l e v e l o c i t y u n c e r t a i n t y (1 Hz bandwidth) o f 5 mm/sec f o r a 3 cm p a t h l e n p t h .
Thus a s e n s o r smaller t h a n 3 cm w i t h o u r p r e s e n t
e l e c t r o n i c s w i l l be unable t o resolve t h e v e l o c i t i e s associated with 3 c m scale velocity fluctuations.
A s e n s o r w i t h an a c o u s t i c p a t h 5 cm l o n g
would h e p r a c t i c a l f o r o c e a n i c work.
REFERENCES
1. Gytre, Trygve (1975) " U l t r a s o n i c Measurements of Ocean C u r r e n t s Down t o 1 mm/sec," Conference P r o c e e d i n p 832 of t h e IERE Conference on I n s t r u m e n t a t i o n i n Oceanography, 23-25 September 1975, U n i v e r s i t y College N . Wales, Bangor, U . K . , pp 69-80. 2.
Williams, A . J . (1974) "Free-Sinkinp Temperature and S a l i n i t y P r o f i l e r f o r Ocean M i c r o s t r u c t u r e S t u d i e s , " Ocean '74 I E E E I n t e r n a t i o n a l Conf e r e n c e on E n g i n e e r i n g i n t h e Ocean Environment, V o l 11, I n s t i t u t e of E l e c t r i c a l and E l e c t r o n i c E n g i n e e r s , I n c . , 345 East 4 7 t h S t . , New York New York 10017, c a t a l o p 674, CH0873-0 OCC, pp 279-283.
ACKNOWLEDGEMENTS
W i l l i a m s r e c e i v e d s u p p o r t f o r t h i s work from ONR C o n t r a c t N00014-74C0262 NR 083-004.
Tochko w a s s u p p o r t e d by t h e Woods Hole Oceanographic
Institution/Massachusetts I n s t i t u t e of Technolopy J o i n t Program i n Ocean Engineerinp
.
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99
TURBULENT BOUNDARY LAYER BEARING SILT IN SUSPENSION
Jacques C.J.
NIHOUL
Institut de MathEmatique, UniversitC de LiSge, Avenue des Tilleuls, 1 5 , B-4000 LiSge, Belgium.
ABSTRACT
Essential characteristics of a turbulent boundary layer, bearine silt flocks in suspension, are described and interpreted with the help of a simple steady state model calibrated for the test region of the Math. Modelsea project off the Northern Belgian Coast in near critical conditions : no net f l u x of particles through the bottom boundary. The model emphasizes the existence between the viscous sublayer and the classical Prandtl-Karman logarithmic layer, of an "elastic sub-layer" where gravity acting on the suspended load provides the necessary restoring Iorce.
T h i s p a p e r was a l s o p r e s e n t e d a t t h e I U T A M Symposium o n S t r u c t u r e o f T u r b u l e n c e and Drag R e d u c t i o n 1 7 - 1 2 J u n e 197.6,
a n d w i l ! be p u b l i s h e d i n f u l l i n a s u p p l e m e n t of P h y s i c s of F l u i d s . Pashington D . Z . )
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101 THE BOTTOM MIXED LAYER ON THE CONTINENTAL SHELF
R.D.
PINGREE
I n s t i t u t e o f Oceanographic S c i e n c e s , Brook Road, Wormley, Godalming,
Surrey
D.K.
GRIFFITHS
Marine B i o l o q i c a l A::sociation o f I h e Uni t.ed Kingdom, The L a b o r a t o r y , C i t a d e l H i l l , Plymouth, PL1 2PB
ABSTRACT' An i n v e s t i g a t i o n i s mdde o f t h e q t r u c t u r e of t h P bottom mixed l a y e r o n t h e corilinent a 1 s h e l f . f i n e r e s o l u t i o n S.T.D.
Within t h e Lurhulent r e g i o n below t h e thermocline, measurement'. show homogeneous s a l i n i t y , and
temperature q r a d i e n t s near adiabatic.
A t k:everal pos:it ion.. i n t h e C e l t i c
Sea and EngliG-h Channel, r e c o r d i n g c u r r e n t meter moorinqc, were used t o
d e t e r m i n e t h e v e r t i c a l F t r u c t u r e of v e l o c i t y i n t h i = l a y e r .
The s t a b i l i s i n q buoyancy f l u x , c,hown t o be predominantly Chat o f h e a t ,
i s e s t i m a t e d from the-. r a t e of warminq of t h e l a y e r .
C u r r e n t meter
rneasuremeni-, of 1 he f r i c l i o n v e l o c i t y i n t h e t u r b u l e n t houndary l a y e r g i v e a v a l u e f o r t h e r a t - . e o f p r o d u c t i o n of t u r b u l e n t k i n e t i c eneryy.
In
t h e bottom l a y c r t h i : ; i:: much g r e a t e r t h a n t h e : ; l a b i l i s i n q buoyancy f l u x ,
t h e Richardson f l u x numhei- h e i n q 0.05,
consi-.+enI w i t h t h e e x t e n t and
p e r s i s t e n c e o f thi:: t u r b u l e n t houndary l a y e r .
Mearurementr. w e r e e x t e n d 4 t o t h e t o p of t h e bottom mixed l a y e r and
into t h e thermocline.
Here l a r q e v a l u e s o f i h e q r a d i e n t Richardson
number reflect t h e t e m p e r a t u r e q r a d i e n t b u t a t t i m e s l a r q e v e l o c i t y sliear
allow:: t h e valiie t o d r o p below t h e c r i i ic-a1 Richard-.on number of 0.75, t h u s e n a h l i n q heat
t o p e n e t r a t e down i n t o t h e bottom l a y e r .
102 A d e t a i l e d study of t h e i n s i t u temperature g r a d i e n t i n t h e bottom l a y e r
was made by averaging about f i f t y temperature p r o f i l e s a t two s e p a r a t e positions.
The mean temperature i n c r e a s e with depth i r ; r a t h e r les:: than
t h e a d i a b a t i c r a t e , h e a t t h e r e f o r e p e n e t r a t i n q t h e bottom l a y e r , allowing warminq while a thermocline exi::ts.
An e s t i m a t e of +.he mean p o t e n t i a l temperat-ure g r a d i e n t by equating eddy thermal d i f f u s i v i t y and eddy v i x o s i t y gives good agreement clone t o t h e bottom with t h e experimental value?.
It was not p o s s i b l e to corre1at.e
t.he variation:; i n measured temyrrrdlure gradient. with t h e s t a t e of t h e tide.
103
ON THE IMPORTANCE OF STABLE STRATIFICATION TO THE STRUCTURE OF THE BOTTOM BOUNDARY LAYER ON THE WESTERN FLORIDA SHELF Georges L. Weatherly Department of Oceanography, Florida State University Tallahassee, Florida 32306 U.S.A. and John C. Van Leer Rosensteil School of Marine and Atmospheric Chemistry University of Miami Miami, Florida 33149 U.S.A. ABSTRACT This is a preliminary report of bottom boundary layer
(BBL) observations made with a cyclosonde on a continental shelf in the summer when the water is relatively stably stratified. Vertical profiles of temperature, current direction and speed show persistent and large temperature changes (1.5Q-4.00C), large direction changes (3Oo-75O) during periods of long-isobath flow and a low speed "jet" in the lowest 6-11 m above the bottom. Thc BBL is interpreted to be a turbulent Ekman layer of depth ranging from 6-11 m in which stable stratification is very important in determining its depth, Ekman veering and speed profile. The Temperature in the BBL is seen to change with time at a faster rate than outside the BBL. This is explained by upwelling (downwelling) of colder (warmer) water in the BBL due to Ekman veering in the boundary layer. Such upwelling (downwelling) is expected when the isopynals are nonparallel to the bottom and the geostrophic current above the boundary layer is aligned predomj-nantly along isobaths with deep water to the right (left). INTRODUCTION The observational investigations of the oceanic bottom boundary layer when compared to comparable studies of the atmospheric boundary layer are limited both in number and in vertical resolution, particularly in the lower part of the boundary layer. This is especially true for the bottom boundary layer (BBL) on continental shelves. Although limited, the observations from continental shelves are not inconsistent with the BBL being a turbulent Ekman layer (e.g., Smith and Long, 1976, Kundu, 1976, Mercado and Van Leer, 1976) in that the layer thickness is approximately . 4 u~,/fand Ekman-like veering, order loo to ZOO, in the current directions is observed in the outer region of the boundary layer. These values of layer thickness and Ekman veering are approximately those f o r a neutrally stratified turbulent Ekman layer suggesting that stratification played only a minor role in determining the structure of these boundary layers. In this paper we report some observations which indicate that density stratification may at times markedly change the structure and thickness of the BBL from its neutrally strati-
104
fied analog on a continental shelf. The observations, made on the Western Florida Shelf (WFS) in the summer of 1975, show relatively large density gradients, large current direction changes, and a-low speed jet occurring within 6-11 m of the bottom. The BBL observations are interpreted in terms of this layer being a turbulent Ekman layer in which the stable stratification is very important in determining the current direction changes (Ekman veering) in the boundary layer and the depth of the boundary layer. The temperature and temperature gradient in the lowest 10 m is observed to change with time at a faster rate than at heights further removed from the bottom. When upwelling (downwelling) of colder (warmer) water in the BBL induced by Ekman veering may occur is discussed and it is suggested that this process is responsible for the observed changes or lack of changes of temperature and temperature gradient at various heights in the water column. It is also suggested that the resulting stratification changes in the BBL due to upwelling (downwelling) can appreciably modify the structure of the BBL. DATA The observations reported here were made between 3 and 8 July, 1976 on the Western Florida Shelf at 26°00'N, 83'49'W. in water of depth %101m. They were obtained during an experiment designed primarily to intercompare cyclosondes as well as to compare them with other instruments. The cyclosonde, an unattended vertical profiler for measuring the current and density fields in the upper ocean, is described in detail by Van Leer et a1 (1974) to which we refer the reader for more detail. A subsiderary objective of this cyclosonde intercomparison experiment was to study the BBL formed on the WFS in summer conditions. In this preliminary report we present data collected by only one instrument, a cyclosonde. Fig. 1 depicts the bathymetry of the WFS in the vicinity of the site of the experiment. The currents in the vicinity of 2 6 ' N between the 100 m and 200 m isobaths are generally oriented along isobaths, i.e. the currents generally flow northward or southward, Niiler (1976). Large trans-shelf motions are sometimes observed (ibid); however, during our experiment, which was located near the 100 m isobath at 26ON, the currents outside of the BBL were primarily oriented along isobaths (cf Fig. 5). Fig. 2 shows a hydrographic transect taken on 2 June, 1972 along 26ON on the WFS. During our experiment hydrographic surveys were made on a horizontal scale of about 25 km (comparable to the baroclinic radius of deformation) and we must rely on other surveys to consider summer time hydrographic conditions over larger scales. Several features are noteworthy. First, in summer time conditions the lower part of the water column at the site of the experiment - denoted by an X in Fig. 2 - is stably stratified. Second, the isopycnals intersect the bottom down slope of our site. Later we shall present observations which are consistent with isopycnals also intersecting the bottom upslope of the site during our experiment. Third, the bottom is sloped with the site of the experiment being approximately at a shelf break point where the bottom slope
105
Fig. 1. Map of the Western Florida Shelf taken from a United States Coast and Geodedic Survey map by E. Uchupi. The site of the experiment is indicated by an "X". Depths are in meters.
changes from 0.26 x upslope to 2.4 x downslope. Fourth, the isotherms are approximately parallel to the isopycnals. Hence the thermal stratification is also an index of the density stratification and we can use the term temperature stratification and density stratification interchangeably. With the objective of studying the BBL a cyclosonde was programmed to have a sampling rate of 30 seconds which when combined with a vertical speed of about 10 cm/s gave an average vertical resolution of about 2.5m. In addition the accoustic release used was positioned ciose enough to the anchor weight to permit the cyclosonde to descend to within about 2.5m of the bottom. For comparison, in a similar study with a cyclosonde, Mercado and Van Leer ( 1 9 7 6 ) , the average vertical resolution was about 5m and measurements were made to within 5m of the bottom. This cyclosonde made two profiles each hour with each profile covering the depth range 12m& z ,& 98.5m. At the beginning of each hour the cyclosonde began to ascend from its rest-
106
'Fig. 2. Hydrographic data along 2 6 ' N collected on June 2, 1972 (Courtesy of C.N.K. Mooers, Unlv. of Delaware, Newark, Longitude of site is denoted by arrow. Delaware, U S A ) . ing position of z = 98.5m. Upon reaching z = 12m it began its down profile. The total transit time f o r both profiles was % 25 minutes. During the up-profile no current directions were recorded due to an instrumental malfunction; only data obtained in the down profiles is presented here. The cyclosonde began profiling at about 0600 hours local time on 3 July, 1975 and continued until about 1100 hours local time on 8 July, 1975. Between about 1400 and 2000 hours local time it failed to profile below z = 50m. Since the region of interest in this paper is the lower part of the water column only data obtained for depths z > 50m is presented. In time series plots present-
107
ed here, formed by linearly interpolating values for fixed depths from individual profiles, linearly interpolated values were inserted in the time series in the 6 hour interval in which the cyclosonde failed to profile below z = 5Om. A total of 119 down profiles of horizontal speed, current direction, temperature, conductivity, and pressure were obtained with this instrument. In this study measurements were made to within ~ 2 . 5 mof the bottom. For comparison in the studies of the BBL on continental shelves of Smith and Long (19761, Kundu (19761, and Mercado and Van Leer (1976) no observations were made closer than 5m from the bottom. The impetus for our making measurements closer to the bottom was the observation of appreciable Ekman-like veering (1Oo-3O0) between 1 and 3m above the bottom in the Florida Current BBL (Weatherly 1972). Cur making measurements lower tnan 5m above the bottom was fortuitous because, as is shown in Section 5 , persistent and relatively large changes in the lower part of the water column of current direction as well as density stratification frequently occurred within 6m of the bottom. REVIEW In this section we review BBL processes which we think may have been active and responsible for certain features in our data. From the description of the previous section we inquire into the structure of a boundary layer formed above a sloping bottom in a stably stratified fluid in which the isotherms are non-parallel to the bottom. €or simplicity we restrict the discussion to a boundary layer formed under a steady geostrophic current flowing along isobaths over a uniformly sloping bottom. The density stratification in the geostrophic interior is taken to be constant and due to temperature alone. The situation considered is shown in Figures 3 b, c where the interior geostrophic flow is perpendicular to the plane of this figure. Before investigating the BBL for such a current we first consider whether in the absence of a geostrophic current does the interaction of the stratification and sloping bottom spontaneously generate flows. If the bottom is thermally conductive and all the heat flowing down the water column goes into the bottom then the situation depicted in Figure 3a with the isotherms non-normally intersecting the bottom is statically stable. Allowing the overlying water to heat the bottom in summer conditions may not be unrealistic for continental shelves. If, however, the bottom is non-conductive then the isotherms must intersect the bottom at right angles, and, as Wunsch (1970) has shown, this spontaneously induces flow both upslope and alongslope in a rotating system. Of the two types of bottoms we choose for simplicity the former, the conducting bottom, which permits a state of static equilibrium to exist. The purpose of this section is not to present a detailed review but to heuristically discuss what may be occurring. If one takes the bottom to be thermally non-conductive and tries to apply the results of Wunsch (1970) (henceforth Wunsch) to our site complications arise. First, Wunsch considers the case of kinematic viscosity and thermal diffusivity constant through out the water column, i.e. the values for the boundary layer are the same as for the interior region. If 3 Reynolds number
108
a
b
Fig 3. Idealization of transect shown in Fig. 2 . The bottom slope is a . Initially isotherms are everywhere horizontal as shown in (a), and stratification is stable and constant. The Ekman layer formed under a long-isobath current can induce downwelling of warmer water (b) or upwelling of colder water (c) in the bottom boundary layer. See text.
is formed for his boundary layer one obtains Re (sin2 a + F2 c0s2 a ) - % where K=thermal diffusivity, u=kinematic viscosity, a=bottom slope, F=f/PN, f=coriolis p rameter, and N=Brunt Vaisala frequency. For our site a Z l 0 - ' , f=.6 x 10-3sec-1 and N"1.6 x sec-l which gives Re z K l o 3 . u Thus if the Prandtl number U / K is 0(1), Re lo3. This implies that the boundary layer is turbulent, and hence that U / K for the boundary layer should be considerably larger than the corresponding values for the interior. Second, it is unclear from Wunsch what the response time is for the fluid column to readjust for an initial stratification as depicted in Figure 3a If one takes the response time to be the diffusive time td= 0 ( D 2 / t c ) , where D is the water depth, the? for our site (D=100m) with K-10cm2/s td%3 years. Taking ~ = 1 0cm2/s reduces the 100 days; however such a large thermal difresponse time to fusivity suggests meteorological forcing and consequently meteorological induced currents which are not included in the initial static formulation. Thus, due to the turbulent nature of the boundary layer it is somewhat difficult to determine the final state that would evolve for an initial static situation
26
-
-
109
as shown in Fig. 3a for the case of a non-conducting bottom. In addition since the estimated response times are large compared to periods when on WFS there are no imposed currents (order several days, cf. Niiler (1976) Fig. 8 1 , it is doubtful whether effects induced by the BBL of these currents can be neglected. It is these effects that are now considered. Consider an imposed, steady barotropic current flowing along isobaths. If the geostrophic current has magnitude representative for the WFS of severals time of cm/s then simple dimensional considerations show that the Ekman layer formed under this current should be turbulent. If the current is directed into the plane as shown in Fig. 3b, which for our site would correspond approximately to a northward current, then cross-isobaric flow in the turbulent Ekman layer would result in advection of warmer water down-slope in this boundary layer as indicated in Fig. 3b. Conversely if the exterior current were directed out from the plane as indicated in Fig. 3c, approximately a southward current at our site, cross-isobaric flow in the BBL would result in advection of colder water upslope. If the heat advected down- or up-slope is not diffused completely back into the interior or bottom the temperature in the BBL would either increase or decrease with time. The appropriate heat equation for the BBL in a coordinate system aligned with the coordinate system is (see Pedlosky, 1974) A
a0 = -
B - ~ S U
at
+
2 a;
C K?
ar;
where 0=temperature, S=temperature stratification outside the B B L , u-cross-isobaric flow induced in the turbulent Ekman layer, r; is the normal coordinate to the bottom, and K is the eddy thermal diffusivity. If there is a balance between terms B and C in (l), i.e. all the heat advected down- or up-slope is diffused back into the interior. Then A=O and the interior temperature should slowly change with time. If on the other hand there is a balance between terms A and B , then the temperature in the BBL should change in the BBL (increase for a northward flow, decrease for a southward flow) and remain fixed outside In this case (1) becomes the B B L . 30 at
-aSu
(1')
In the next section we present data which indicates that the latter situation, Equation (l'), applied approximately during our experiment. Thus in a case where (1')applies the temperature in the BBL should change with time and the temperature outside the layer should remain unchanged when the isotherms intersect the bottom and the geostrophic flow outside the BBL is oriented along isobaths. Since u is not constant within the BBL ae/at should vary with height in the layer and the thermal stratification thus change with time in the B B L . At the beginning of the experiment the BBL is stably stratified. It is appropriate then to review how the structure of a stably stratified turbulent Ekman layer varies with changing stratification. To do
110
this we must rely on studies of the stably stratified atmospheric boundary layer. The source of the stable stratification f o r the atmospheric boundary layer is quite different from our case. It is due t o radiative cooling of the ground. Hence the heat flux at the earth's surface, Ho, is an important parameter for the atmospheric case and results are often categorized in terms of parameters determineg from Ho. For example, the MoninObukhov length scale, L -ug (kg(HO/pcp))-l where ug = friction velocity, k = von Karman's constant =.4, g = gravitational acceleriation, p = density, and cp = specific heat at constant pressure, is often used (cf. Businger and Arya, 1974). At our site the stratification in the BBL depends primarily on horizontal advective processes rather than surface heating and it is doubtful whether Ho (and hence L) is the appropriate parameter to use to categorize our observations. Hence we use results of studies of the stably stratified atmospheric boundary layer to give some physical insight into how changing the stratification modifies the structure of a BBL rather than to make quantitative comparisons. Studies of the stably stratified atmospheric boundary layer indicate that a stably stratified turbulent Ekman layer changes as follows by increasing the stable stratification. The layer thickness decreases and the cross-isobaric flow in the boundary layer (the Ekman veering) increases (up to a limit discussed below). In addition a low level speed in the boundary layer becomes more pronounced with increasing stratification. The results displayed in Fig. 4, taken from Businger and Arya (19741, a theoretical study of a steady, stably stratified atmospheric boundary layer, demonstrate these features. Fig. 4 shows BBL speed and Ekman veering profiles for various values of a stratification parameter p .I ug /fL, where uQ is the value of the friction velocity for p=O or Reutral stratiFication. In Fig. 4b B = O o represents the direction of the wind outside the boundary layer. Note how appreciably the Ekman veering increases with increasing p and a "jet"-like structure to the Ekman veering profile for larger p . Together with these profiles and other results given in Businger and Arya (1974) it can be seen that the thickness of the boundary layer h < k u,/f for p > o , i.e. h decreases faster than ug with increasing stratification. For example for p 150, ug = .3u and h = (.17) k ~ , ~ / f . *O It is of interest to consider how much Ekman veering can be expected in a stably stratified turbulent Ekman layer and in what region of the boundary layer most of this veering should occur. Below by examining the u (cross-isobaric) momentum equation we attempt to show that if the boundary layer is sufficiently stratified, h < < k u*/f, the total Ekman veering approaches 90° and that most of the veering should occur where the speed shear is larger. For comparison in a neutrally stratified turbulent Ekman layer the total veering is less than 4 5 O and occurs mostly in the outer region of the layer where the speed shear is small (Deardorff, 1970). The u (cross-isobaric) momentum equation for the boundary layer is
111
la
I
lb
F i g . 4. P r e d i c t e d s p e e d ( a ) and wind d i r e c t i o n ( b ) p r o f i l e s f o r a s t e a d y , s t a b l y s t r a t i f i e d a t m o s p h e r i c boundary l a y e r t a k e n from r e s u l t s r e p o r t e d i n B u s i n g e r and Arya ( 1 9 7 4 ) . p is a s t r a t i f i c a t i o n parameter (p = 0 denotes n e u t r a l s t r a t i f i c a = 0, f = Coriolis t i o n ) , ug i s t h e f r i c t i o n v e l o c i t y f o r h e i g h t above t h e g r o u n d , and $ = 0 d e n o t e s t h e parameter: z d i r e c t i o n o f t h e g e o s t r o p h i c wind. See t e x t .
where v i s t h e v e l o c i t y component i n t h e d i r e c t i o n o f g e o s t r o p h i c f l o w o u t s i d e t h e boundary l a y e r , V g i s -magnitude of t h e g e o s t r o p h i c c u r r e n t ( V g = ( p f 1 - l a p / a x ) and U ' W I i s t h e a p p r o p r i a t e Reynolds s t r e s s . By c o n s i d e r i n g t h e n e u t r a l l y s t r a t i f i e d case f i r s t w e c a n see t h a t r e g i o n i n t h e boundary l a y e r where t h e second t e r m o f ( 2 ) i s e s s e n t i a l l y z e r o i s a l s o t h e r e g i o n where c u r r e n t d i r e c t i o n i s c o n s t a n t . We d i v i d e t h e boundary l a y e r i n t o a n i n n e r a n d o u t e r r e g i o n . For t h e i n n e r r e g i o n t h e a p p r o p r i a t e v e r t i c a l l e n g t h s c a l e i s t h e bottom roughness param eter. I n t r o d u c i n g a non-dimenEPonal l e n g t h 0 f o r t h e i n n e r r e g i o n by
and s c a l i n g t h e Reynolds s t r e s s by u g 2 as f o l l o w s TX
Z
-
u'w'/k
u * ~
(3)
E q u a t i o n (2) becomes
I
I
The maximum v a l u e o f v-V ] / u g i s IV / u which i s t y p i c a l l y ( i b i d ) f o r n e u t r a l l y s t r a f i f i e d p l a g e t b y boundary l a y e r s . r e p r e s e n t a t i v e v a l u e of z o f / k ug f o r o c e a n i c c o n d i t i o n s i s
.- 2 5 A
112
-
(Wimbush and Munk, 1 9 7 0 ) and W e a t h e r l y ( 1 9 7 2 ) . Hence t h e first t e r m i n ( 4 ' ) is order Thus a T X / a r l = O t o O(10-4). It i s i n t h e i n n e r r e g i o n where most o f t h e s p e e d s h e a r o c c u r s a n d i n t h i s r e g i o n t h e - c u r r e n t d i r e c t i o n c h a n g e s v e r y l i t t l e (Deardorff, 1970). In the outer region t h e appropriate length scale i s the boundary l a y e r t h i c k n e s s h . Introducing a non-dimensional h e i g h t 5 by
and a g a i n s c a l i n g t h e Reynolds s t r e s s as i n ( 3 ) E q u a t i o n ( 2 ) becomes
F o r a n e u t r a l l y s t r a t i f i e d l a y e r h 1 k u , / f a n d h e n c e t h e coe f f i c i e n t of t h e f i r s t term i n ( 2 " ) 1. A s n o t e d e a r l i e r t h e r e i s l i t t l e s p e e d s h e a r i n t h e o u t e r r e g i o n , i . e . v-Vg i s s m a l l and i n f a c t lv-Vgl /u, = 1. Hence i n t h e o u t e r r e g i o n where most o f t h e d i r e c t i o n c h a n g e s o c c u r i n a n e u t r a l l y s t r a t ified layer
F o r a v e r y s t a b l y s t r a t i f i e d t u r b u l e n t Ekman l a y e r h < < k u,/f. Hence t h e f i r s t t e r m i n ( 2 " ) h a s a c o e f f i c i e n t <<1 a n d t h i s term i s O(1) and h e n c e t h e s e c o n d t e r m aTX/35 = O(1) o n l y when I v - V g l / u n >>1. T h u s , i n c o n t r a s t t o t h e n e u t r a l l y s t r a t i f i e d c a s e w e do n o t e x p e c t much d i r e c t i o n change t o o c c u r i n b u t where << t h a t p a r t o f t h e o u t e r r e g i o n where Iv-V lVg.l Thus i n t h e o u t e r r e g i o n a p p r e c i a b f e = s E : a r i n v i s e x p e c t e d . I n t h e l o w e r p a r t o f t h e o u t e r r e g i o n where w e e x p e c t ( v 1 < < IVgl E q u a t i o n 2 c a n be a p p r o x i m a t e d a s
I
1.
T h i s i s t h e same e q u a t i o n f o r t u r b u l e n t f l o w i n a p i p e . Thus i n t h e l o w e r p a r t o f s u c h a boundary l a y e r t h e f l u i d m o t i o n i s a p p r o x i m a t e l y down t h e p r e s s u r e g r a d i e n t . Since t h e motion i s o u t s i d e t h e boundary l a y e r a t r i g h t a n g l e s t o t h e p r e s s u r e g r a d i e n t , i . e . g e o s t r o p h i c , t h i s i m p l i e s t h a t t h e t o t a l Ekman veering a c r o s s t h e o u t e r region of a s t a b l y s t r a t i f i e d turbul e n t Ekman l a y e r f o r which h < < k u R / f i n a b o u t 9 0 ° . OBSERV AT I ON S
F i g u r e 5 shows e s t i m a t e d p r o g r e s s i v e v e c t o r d i a g r a m s f o r To form t h e s e f i g u r e s t h e d e p t h s 50m, 7 0 m , 9 0 m , a n d 98.5m. v e l o c i t y v e c t o r s a t t h e s e r e f e r e n c e d e p t h s were e s t i m a t e d by l i n e a r i n t e r p o l a t i o n f o r each ascending p r o f i l e . The ' i n s t a n t a n e o u s ' h o u r l y v a l u e s were assumed t o be r e p r e s e n t a t i v e o f t h e h o u r l y a v e r a g e d v e l o c i t y v e c t o r t o form t h e s e f i g u r e s . We see t h a t t h e currents f o r depths 9 0 m were v e r y s i m i l a r . For t i m e s 0 5 h o u r .& 4 8 t h e c u r r e n t w a s f l o w i n g t o t h e n o r t h - n o r t h -
a
n
E
v
ln
m
m rd
c
a
E 0 m
113
114
w e s t approximately p a r a l l e l t o t h e i s o b a t h s . For t i m e s 4 8 & hour 7 2 t h e c u r r e n t w a s f l o w i n g westward a p p r o x i m a t e l y t r a n s v e r s e t o t h e i s o b a t h s , and f o r t i m e s 7 2 & hour & 125 t h e c u r r e n t flowed south-southwest a g a i n approximately p a r a l l e l t o t h e i s o baths. The c u r r e n t away f r o m t h e b o t t o m ( d e p t h s & 9 0 m ) w i l l be d e s i g n a t e d as n o r t h w a r d , westward or s o u t h w a r d i n t e r i o r or geostrophic current, respectively, for these intervals. For depths > 9 0 m t h e f l o w i s a p p r e c i a b l y d i f f e r e n t . The v a r i a b i l i t y i n c u r r e n t d i r e c t i o n and speed seen i n t h e i n t e r i o r flow, due e i t h e r t o i n e r t i a l m o t i o n s or d i u r n a l t i d e s , i s by c o m p a r i s o n nearly absent. The d i r e c t i o n o f t h e ' b o t t o m ' f l o w i s s i g n i f i c a n t l y d i f f e r e n t from t h a t o f t h e i n t e r i o r f l o w when t h e l a t t e r i s r e l a t i v e l y s t r o n g and s t e a d y , i . e . d u r i n g t h e p e r i o d s o f n o r t h w a r d and s o u t h w a r d i n t e r i o r f l o w . T a b l e 1 shows computed a v e r a g e d i r e c t i o n d i f f e r e n c e s or veering r e l a t i v e t o t h e c u r r e n t a t 90m depth f o r t h e t h r e e i n t e r i o r c u r r e n t regiemes. During t h e p e r i o d s of northward and southward g e o s t r o p h i c c u r r e n t s t h e v e e r i n g between 9 0 a n d 9 8 . 5 m d e p t h i s l a r g e , % 30° and % 7 S 0 , r e s p e c t i v e l y , a n d l a r g e r I n Table 1 t h a n a n y o f t h e v e e r i n g s between 9 0 a n d 50m d e p t h . n e g a t i v e v a l u e s d e n o t e v e e r i n g s i n t h e wrong s e n s e f o r Ekman veering. F o r n o r t h w a r d f l o w e s s e n t i a l l y a l l o f t h e "Ekman" v e e r i n g o c c u r s w i t h i n 6 m o f t h e b o t t o m ; f o r s o u t h w a r d f l o w most o f t h e v e e r i n g o c c u r s w i t h i n 6 m o f t h e b o t t o m ( a b o u t 55O) w h i l e o c c u r s between 6 a n d l l m a b o v e t h e t h e r e m a i n d e r ( a b o u t 20') b o t t o m . When t h e i n t e r i o r f l o w i s westward it i s c o m p a r a t i v e l y weak and v a r i a b l e t h e v e e r i n g between 9 0 a n d 98.5m i s s m a l l % 3.5', occurs i n t h e lowest 6m, and i n t h e c o r r e c t sense f o r Ekman v e e r i n g .
T a b l e 1. Average d i r e c t i o n d i f f e r e n c e s r e l a t i v e t o 9 0 m d e p t h computed a c c o r d i n g t o t h e methods o f Kundu ( 1 9 7 6 ) and W e a t h e r l y (1972). The f o r m e r v a l u e s a r e g i v e n w i t h c o r r e l a t i o n c o e f f i cient i n parentheses. See t e x t . Depth
(m) 50 60 70 80 95 98.5
Average D i r e c t i o n D i f f e r e n c e R e l a t i v e t o 90M Depth Northward Flow Westward Flow Southward Flow -16.6(.82) -19.6 3.0C.86) 0.9 4.0(.92) 3.2 2.4C.98) 2.5 2.8(.99) 0.0 31.5(.92) 30.3
-
-
-50.4(.69) -35.4(.62) -15.3(.70) 7.1C.87) 2.5(.88) 2.6(.79)
-
-70.7 -57.5 -41.9 -16.4 0.5 4.7
7.6(.87) 8.8(.89) 4.2(.90) -2.9C.89) 18.9(.92) 65.2(.76)
9.0 12.7 5.9 -0.5 25.1 84.8
A c c o r d i n g t o t h e d i s c u s s i o n l e a d i n g t o (1') t h e t e m p e r a t u r e i n t h e BBL s h o u l d i n c r e a s e ( d e c r e a s e ) w i t h t i m e f o r n o r t h w a r d (southward) i n t e r i o r flow while remaining c o n s t a n t i n t h e i n terior. F u r t h e r l a e / a t ) s h o u l d be g r e a t e r f o r s o u t h w a r d t h a n % a , t h e bottom f o r northward i n t e r i o r flow. From (1') a B / a t slope angle. From F i g . 2 i t i s a p p a r e n t t h a t t h e a p p r o p r i a t e a f o r northward f l o w i s t h e a n g l e u p s l o p e of t h e s h e l f b r e a k w h i l e f o r s o u t h w a r d f l o w it s h o u l d be t h e a n g l e down s l o p e o f
I
I
115 the shelf break. A s noted e a r l i e r t h e l a t t e r angle i s about t e n times l a r g e r . T e m p e r a t u r e t i m e s e r i e s a r e shown i n F i g . 6 f o r t h e d e p t h s 5 0 , 6 0 , 7 0 , 80, 9 0 , a n d 38.5111. T h e s e t e m p e r a t u r e s were d e t e r mined e x a c t l y a s t h e v e l o c i t y v a l u e s u s e d i n F i g . 5 . Although t h e r e i s much d e t a i l g e n e r a l t r e n d s a r e a p p a r e n t . Comparing t h e t i m e s e r i e s a t 5 0 a n d 98.5111d e p t h o n e s e e s : 1) f o r t i m e s O
7 2 a e / a t i s n e g a t i v e and a8/at i s much l a r g e r n e a r t h e b o t t o m . Although t h e c u r r e n t d i r e c t i o n a t 50m d e p t h s t a r t s c h a n g i n g f r o m w e s t ward t o s o u t h w a r d a t h o u r z 7 2 , t h e d i r e c t i o n n e a r t h e b o t t o m d o e s n o t b e g i n t o c h a n g e u n t i l hour-81, a n d t h e t e m p e r a t u r e a t 38.5m s t a r t s t o change a f e w h o u r s l a t e r . It i s important t o note t h a t while t h e temperature a t a l l l e v e l s i s variable t h e l a r g e s t t e m p e r a t u r e changes o c c u r n e a r . t h e bottom. From T a b l e 1 a n d F i g . 6 as w e l l as p r o f i l e p l o t s p r e s e n t e d below w e i n f e r t h a t t h e BBL w a s a b o u t 6 m t h i c k d u r i n g t h e p e r i o d s o f n o r t h w a r d a n d w e s t w a r d f l o w a n d a b o u t llm t h i c k when t h e i n t e r i o r f l o w was s o u t h w a r d . B e c a u s e o f t h e d i r e c t i o n v e e r i n g w e i n f e r t h a t t h i s b o u n d a r y l a y e r i s a t u r b u l e n t Ekman layer. According t o arguments p r e s e n t e d i n t h e l a s t s e c t i o n t h e t e m p e r a t u r e s t r a t i f i c a t i o n a c r o s s t h i s BBL s h o u l d d e c r e a s e ( i n c r e a s e ) w i t h t i m e when t h e i n t e r i o r f l o w i s n o r t h w a r d ( s o u t h ward). I n F i g . 6 , w h i l e n o t a s a p p a r e n t f o r n o r t h w a r d f l o w , it c a n b e s e e n t h a t t h e s t r a t i f i c a t i o n a c r o s s t h e BBL c h a n g e s i n a manner c o n s i s t e n t w i t h t h e s e a r g u m e n t s . I n F i g . 7 a r e shown v e r t i c a l p r o f i l e s o f t e m p e r a t u r e , c u r r e n t s p e e d and c u r r e n t d i r e c t i o n r e p r e s e n t a t i v e f o r t h e three i n t e r i o r flow regiemes. The r e a s o n f o r t h e t e m p e r a t u r e i n v e r s i o n which c o n s i s t e n t l y a p p e a r s a t d e p t h s o f a b o u t 80m i s n o t known. I n t h i s r e g i o n t h e water i s s t i l l s t a b l y s t r a t i f i e d because t h e d e s t a b i l i z i n g temperature gradient i s countered by a s t a b i l i z i n g s a l i n i t y g r a d i e n t . A g e n e r a l f e a t u r e o f t h e p r o f i l e s i s t h a t t h e l a r g e s t , p e r s i s t e n t t e m p e r a t u r e changes o c c u r i n t h e BBL. Associated with t h e l a r g e temperature gradi e n t s i n t h e BBL a r e l a r g e c h a n g e s i n t h e c u r r e n t d i r e c t i o n . The d i r e c t i o n c h a n g e i s l a r g e s t f o r s o u t h w a r d i n t e r i o r f l o w (Fig. 7d) t h a n f o r northward flow ( F i g . 7 a , b ) . The d i r e c t i o n c h a n g e s a r e s m a l l e r a n d more e r r a t i c d u r i n g w e s t w a r d f l o w ( F i g . Except 7 c ) when t h e i n t e r i o r f l o w i s w e a k e r a n d more v a r i a b l e . f o r t h e p e r i o d o f westward g e o s t r o p h i c f l o w , t h e d i r e c t i o n prof i l e s i n t h e l o w e s t 2 10m a r e r e m a r k a b l y s i m i l a r t o t h e t e m p e r I n t h e lowest a t u r e p r o f i l e s . A s t r i k i n g example i n F i g . 7d. 6m t h e w a t e r i s n e a r l y i s o t h e r m a l a n d l i t t l e v e e r i n g o c c u r s . Between a b o u t 6 a n d llm a b o v e t h e b o t t o m t h e t e m p e r a t u r e g r a d i e n t i s l a r g e ( U ° C / 5 m ) a n d it i s h e r e where a l a r g e d i r e c t i o n I n many o f t h e c u r r e n t d i r e c t i o n p r o f i l e s change ( 2 6 5 O ) o c c u r s . t h e r e i s a weak " j e t " a t t h e t o p of BBL s i m i l a r t o t h a t s e e n i n Fig. 4b. A c o n s i s t e n t f e a t u r e o f most o f t h e v e r t i c a l speed p r o f i l e s i n d e p e n d e n t of t h e d i r e c t i o n o f t h e i n t e r i o r f l o w i s a weak " j e t " i n t h e l o w e s t 9 10m. I t i s a t p r e s e n t d i f f i c u l t t o quant i t a t i v e l y d e s c r i b e t h i s " j e t " b e c a u s e some o f t h e v a r i a b i l i t y i n t h e speed p r o f i l e s i s due t o v a r i a b i l i t y i n t h e a s c e n t r a t e of t h e cyclosonde during any p a r t i c u l a r p r o f i l e . A j e t i n t h e speed p r o f i l e s f o r l a r g e s t r a t i f i c a t i o n v a l u e s i s a l s o i n d i -
I
I
116
18 ! 0
I
I
24
I
I
I
1
I
72 TIME (HOURS)
48
Fig. 6. T e m p e r a t u r e t i m e s e r i e s for d e p t h s 5 0 m , 9 0 m , and 98.5m.
c a t e d i n F i g . 4a.
1
1
I
96 60m, 70m,
I 20 80m,
The n e g a t i v e s p e e d s h e a r s e e n i n F i g . 7 a , b
for z<90m may be due t o a t h e r m a l wind e f f e c t a s s o c i a t e d w i t h t i l t i n g upwards of t h e i s o t h e r m s g o i n g e a s t .
1 1 \1 ,j, :;,I a
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: 22
-.
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TEYPERATURE ? C )
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SPEEC!(C+)
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TEMPERATURE (.c)
C
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2
0
SPEED@+)
W
N
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DIRECTION
d I
TEYPERATURE',t)
SPEED@+)
DIRECTION
TEMPERATURE(-)
SPEED@+)
Fig. 7. R e p r e s e n t a t i v e t e m p e r a t u r e , s p e e d and c u r r e n t d i r e c t i o n p r o f i l e s f o r p e r i o d s of ( a ) , ( b ) northward i n t e r i o r flow, ( c ) westward i n t e r i o r f l o w , and ( d ) southward f l o w . Number i n u p p e r l e f t of e a c h p r o f i l e s e t i s h o u r i n t o t h e e x p e r i m e n t t h e p r o f i l e was made.
DIRECTION
118
BOTTOM HOMOGENEOUS LAYERS Bottom homogeneous layers (BHL) in areas where repeated observations of them have been made are more often seen than not (Weatherly and Niiler (1974) and Armi and Millard (1976)). At our site, however, these layers are more the exception than the rule. Of the 119 cyclosonde profiles examined, 47 ( W O ) had a layer immediately above the bottom resembling a BHL. Of these 47 only 28 had features similar to the BHL in Fig. 7d in which the water was fairly isothermal (lae/azl & .05OC/5m. The remaining 19 show a near.bottom laver with lae/azl nearly constant and equal to &0.1°C/5m. Here a BHL is one in which lae/azl = constant ,$ O.l0C/5m. Of the 47 observed BHL 38 ( ' ~ 8 0 % ) had a thickness of 7m or less. Fig. 8 shows temperature profiles in the lowest 30m for every other hour in a period when BHL were common. These layers cannot be accounted for solely by turbulent mixing since the temperatures of these layers typically do not increase when they thicken. For example, Fig. 8j has a layer about twice as thick as that in Fig. 8a yet both layers have about the same temperature. Some layers cool as they thicken. Further, a sharp 'elbo' at the top of the layer (e.g. Fig. 8a) would be indicative of a forming or recent layer, and a curved 'elbo' at the top (e.g. Ffg. 8e) would be indicative of a decaying layer. Fig. 8ghi suggests this is not the case. A layer with a curved 'elbo' is seen to thicken with time. We believe horizontal advective processes similar to those described in Weatherly (1975) played a major role in the formation of the observed BHL. The appearance of such layers in over half the observed instances were associated with relatively weak and thus comparatively time-dependent interior flows. An essential factor seen by Weatherly (1975) for formation of homogeneous layers is strong time-dependence in the geostrophic current. CONCLUSIONS Data on the velocity and density field in the lower half, depths > 50m, of the water column obtained at one location on the Western Florida Shelf by a cyclosonde have been used to study the BBL. The data were obtained over a five-day period in summer conditions when the water column was stably stratified. Data from a total of 119 hourly profiles were used. T The currents. The currents away from the bottom, depths < 90m, flowed initially ( 0 I hour & 48) approximately along isobaths with deep water to the left facing downstream (northward interior flow). For times 48 5 hour ,$ 72 they flowed approximately westward transverse to the isobaths (westward interior flow). For times 72 ,$ hour k 125 they again flowed approximately along isobaths but with deep water to the right facing downstream (southward interior flow). During the westward interior flow interval the water at depths > 90m also tended to flow westward. During the northward and southward interior flow intervals the water at depths
119
d'c) d
m
e
n
f
0
Fig. 8. Temperature profiles in the lowest 30m for every other hour in a period when bottom homogeneous layers were common. Circled numbers denote hour into the experiment.
> 9 0 m flowed in a direction ranging from 30° to 90' counter clockwise looking down to the interior flow. Although the transition f r o m westward to southward flow in the interior occurred at time = 7 2 hours a comparable change below z = 90m
120
occurred approximately 1 0 hours l a t e r . p e r i o d t h e c u r r e n t s were weak.
During t h i s t r a n s i t i o n
Bottom boundary l a y e r t h i c k n e s s . We i n f e r t h a t t h e BBL t h i c k n e s s w a s a b o u t 6m f o r t h e p e r i o d s o f n o r t h w a r d a n d westward i n t e r i o r f l o w s and a b o u t l l m f o r t h e p e r i o d o f s o u t h w a r d i n t e r i o r f l o w . These t h i c k n e s s e s a r e s i g n i f i c a n t l y l e s s t h a n t h a t e x p e c t e d i f t h e BBL were a n e u t r a l l y or n e a r l y n e u t r a l l y s t r a t i f i e d t u r b u l e n t Ekman l a y e r . The l a t t e r t h i c k n e s s s h o u l d be a b o u t ku,/f = k (.03) V / f ( W e a t h e r l y ( 1 9 7 2 ) ) which w i t h Vg = 1 0 - 2 0 c m / s , r e p r e s e n t a f i v e v a l u e s f o r t h e g e o s t r o p h i c c u r r e n t a t o u r s i t e , i s about 20-40m. Ekman v e e r i n g . L a r g e , p e r s i s t e n t c u r r e n t d i r e c t i o n c h a n g e s , i n a s e n s e c o n s i s t e n t w i t h Ekman v e e r i n g , were o b s e r v e d d u r i n g t h e p e r i o d s of northward and southward i n t e r i o r flow i n t h e BBL. D u r i n g n o r t h w a r d f l o w t h e a v e r a g e Ekman v e e r i n g w a s % 30' a n d f o r s o u t h w a r d f l o w it was % 7 5 0 . The v e e r i n g o c c u r r e d i n t h a t p a r t o f t h e BBL which w a s t h e r m a l l y s t r a t i f i e d and t h e amount o f v e e r i n g w a s d i r e c t l y p r o p o r t i o n a l t o t h e temperature g r a d i e n t . D u r i n g t h e p e r i o d o f westward f l o w when t h e i n t e r i o r f l o w w a s weaker a n d more v a r i a b l e t h e a v e r a g e Ekman v e e r i n g was s m a l l , % 3.5'. For c o m p a r i s o n t h e e x p e c t e d Ekman v e e r i n g f o r a n e u t r a l l y s t r a t i f i e d t u r b u l e n t Ekman l a y e r w i t h V g = 10-20cm/s and a g e o s t r o p h i c d r a g c o e f f i c i e n t .03 i s about l o o ( M I . u*/V g Was t h e BBL a v e r y s t a b l y s t r a t i f i e d t u r b u l e n t Ekman l a y e r ? S e v e r a l f e a t u r e s common t o s u c h boundary l a y e r s s u g g e s t it was. I t s i n f e r r e d d e p t h was l e s s t h a n t h a t f o r a n e u t r a l l y s t r a t i f i e d t u r b u l e n t Ekman l a y e r . The Ekman v e e r i n g s f o r n o r t h w a r d and s o u t h w a r d i n t e r i o r f l o w s w a s a p p r e c i a b l y g r e a t e r t h a n t h a t f o r a n e u t r a l l y s t r a t i f i e d t u r b u l e n t Ekman l a y e r . The Ekman v e e r i n g i n c r e a s e d when t h e s t r a t i f i c a t i o n i n c r e a s e d , i . e . , t h e veering w a s g r e a t e r f o r southward flow t h a n f o r n o r t h w a r d f l o w . The t o t a l v e e r i n g i n p e r i o d s o f s t r o n g e s t s t r a t i f i c a t i o n , e n c o u n t e r e d when t h e i n t e r i o r f l o w was n o r t h ward, r a r e l y e x c e e d e d 90°, t h e l i m i t i n g v a l u e f o r a v e r y s t a b l y s t r a t i f i e d t u r b u l e n t Ekman l a y e r . A I ' j e t - l i k e ' ' s t r u c t u r e w a s o f t e n observed i n v e r t i c a l p r o f i l e s of h o r i z o n t a l s p e e d and d i r e c t i o n .
-
Upwelling ( d o w n w e l l i n g ) i n t h e BBL. We have p r e s e n t e d arguments t h a t i f t h e i n t e r i o r , geostx>ophicflow i s along i s o b a t h s w i t h d e e p water t o t h e r i g h t ( l e f t ) l o o k i n g downstream u p w e l l i n g ( d o w n w e l l i n g ) o f c o l d e r ( w a r m e r ) w a t e r may o c c u r i f t h e isotherms are n o n - p a r a l l e l t o a bottom with p o s i t i v e s l o p e . P r o v i d e d t h e r e i s n o t a c o m p l e t e b a l a n c e between h o r i z o n t a l a d v e c t i o n a n d v e r t i c a l d i f f u s i o n of h e a t , t h e BBL t e m p e r a t u r e s h o u l d d e c r e a s e ( i n c r e a s e ) w i t h t i m e . Our d a t a a p p e a r s c o n s i s t e n t with such a p r o c e s s occuring. F u r t h e r , w e have suggested t h a t s i n c e t h e bottom s l o p e a n g l e f o r southward flow i s about t e n t i m e s l a r g e r t h a n t h a t f o r northward flow l a 9 / 3 t l f o r t h e f o r m e r s h o u l d be a b o u t t e n s t i m e s l a r g e r t h a n f o r t h e l a t t e r . Our d a t a i s a l s o c o n s i s t e n t w i t h t h i s i d e a . I t i s i n t e r e s t i n g t o n o t e t h a t u p w e l l i n g i n t h e BBL c a n c a u s e t h e BBL t o become more s t a b l y s t r a t i f i e d t h a n t h e f l u i d a b o v e
121
the boundary layer. Bottom homogeneous layers. Such layers were the exception rather than the rule. They were seen in about 40% of the profiles and 80% of these layers had depths < 7m. Several features are inconsistent with their formation being due solely to turbulent mixing: (a) generally the BHL layer temperatures remained constant or cooled as they thickened and (b) whether these layers were capped hy sharp 'elbos' or smooth curves was not indicative of whether they were thickening or decaying. Horizontal advective processes seem essential to their formation. Finally this is a preliminary report. Possible complicating features (tiltle-dependence, baroclinicity, insulated bottom, internal gravity waves, implications of the site being at a shelf break) need to be considered further. It is encouraging that the relatively simple arguments presented here account for many of the observed features. ACKNOWLEDGEMENTS This research was sponsored by the Office of Naval Research under contract N000-14-75-C201 and by the National Sciance Foundation, Continental Shelf Dynamics Program, under grant GA-34009. REFERENCES Armi, L. and R.C. Millard, Jr., 1976. The bottom boundary layer of the deep ocean. J. Geophys. E . ,81, 49834990. Businger, J.A. and S.P.S. Arya, 1974. Height of the mixed layer in the stably stratified planetary boundary layer. Advances & Geophysics, H.E. Landsberg and J. Van Mieghem, ed., Academic Press, New York, pp. 73-92. Deardoff, J.W., 1970. A three-dimensional numerical investigation of the idealized planetary boundary layer. 1,377-410. Geophys. Fluid I)+., Kundu, P.K., 1976. E man veering - observed near the ocean bottom. J. Ph s. Oceanogr., 6, 238-242. Mercado, A. ana n*J Leer, 1976.- Near bottom velocity and temperature profiles observed by cyclosonde. Submitted to Geophys. Res. Letters. Niiler, P.P., 1976. Observations of low-frequency currents on the Western Florida coritinental shelf. Memoires de la Societe Royale des Sciences de Liege, Tome X, pp. 331-358. Pedlosky, J., 1974. Long shore currents, u m n g and bottom topography. J. Phys. Oceano r., 4, 217-226. Smith, J.D. and C.F. Long, h e zffect of turning in the bottom boundary layer on continental shelf sediment transport, Memoires de la Societe Royale des Sciences de Liege, Tome X , 369-396. Van Leer, J., W. Duing, R. Erath, E. Kennelly, and A. Speidel, 1974. The cyclosonde: an unattended vertical profile
122
for s c a l a r a n d v e c t o r q u a n t i t i e s i n t h e u p p e r o c e a n . Deep-sea R e s . , 2 1 ( 5 ) : 385-400. W e a t h e r l v . G . L T 1 9 7 r A s t u d v of t h e b o t t o m b o u n d a r y l a y e r of t h e F l o r i d a C u r r e n t . Phys. Oceano r . , 2 , i4-13. Weatherly, G . L . , 1975. A numei?icGtu&me-aependent t u r b u l e n t Ekman l a y e r s o v e r h o r i z o n t a l a n d s l o p i n g J. Oceano r . , 5 , 2 8 8 - 2 9 9 . bottoms. W e a t h e r l y , G.L.-and P . P . Niile:, 1 9 7 4 . Bottom homogeneous l a y e r s i n t h e F l o r i d a C u r r e n t . Geophys. %. L e t t e r s , 1, 316-319. Winbuzh, M . and W . Munk, 1 9 7 0 . The b e n t h i c b o u n d a r y l a y e r . The S e a , Vol. 4 , P a r t 1, New York, W i l e y , p p . 731-758. Wuns~h,C.,1970. On o c e a n i c boundary m i x i n g . Deep-sea Res., 293-301. -
2.
-.
c,
123 BOUNDARY LAYER ADJUSTMENTS TO BOTTOM TOPOGRAPHY AND SUSPENDED SEDIMENT
J. Dungan Smith and S . R. McLean Department of Oceanography, University of Washington Seattle, Washington
ABSTRACT An accurate knowledge of flow in the immediate vicinity of the sea bed is important in marine geological, benthic ecological, geochemical, and sediment transport studies. However in many cases, the velocity field is complicated by the presence of ripples and dunes on the sea bed and suspended sediment-induced stratification in the flow.
Recently techniques
for handling these factors were developed by the authors, but they were applied only to a situation where the sea bed was comprised of a single size and specific gravity class.
In this paper these techniques are
extended to the case wi-?re the suspended material is characterized by an ensemble of settling velocities and critical shear velocities.
The results
are applied first to a flat sea bed and then to spatially averaged flow over a wavy boundary such as might be produced by natural bed forms. These calculations indicate that the sediment transport process can have a significant effect on flow near the sea bed and that proper account must be taken of the settling velocity distribution comprising the suspended sediment concentration field. INTRODUCTION In marine and fluvial systems, near-bottom velocity fields capable of eroding and transporting sediment can be modified relative to those in non-sediment bearing flods in three jmportant ways.
First, near-bed
particle motions substantially increase the apparent roughness of the bottom.
Second, when such flows carry suspended sediment, the vertical
eddy diffusive transfer of mass and momentum is inhibited by the maintenance of a stable density field and third, such flows inevitably produce This is contribution 931 from the University of Washington. The work described herein was supported by NSF Grant GA-14178 and DES-75-15154.
bed forms causing the near-bottom velocity to vary as a function of downstream position and the stress on the bottom to be supported, in part, by form drag on the topographic features.
In order to permit accurate mean
velocity and sediment transport computations, each of these effects must be accounted for.
In a recent paper, Smith and McLean (in press) have
presented a method by which the spatially averaged velocity profile in a sediment-bearing flow over two-dimensional topographic features can be computed. A s part of this theory, means of determining the apparent bed roughness, the suspended sediment concentration profile and the effect that the suspended sediment has on the mean velocity field
are provided.
For computational simplicity this was done using a single size class deemed representative of the entire sample.
In the paper at hand the
problem is generalized so that the bed sediment sample can be divided into any number of size and specific gravity classes, thus avoiding the somewhat arbitrary choice of an effective sediment size and specific gravity.
In applying the generalized theory, size distributions obtained from the crest and trough of a large sand wave in the Columbia River are used. Both samples were procured in 1969 as part of a comprehensive examination of sand wave dynamics carried out over a field of 2 to 3 meter high, 70 to 100 meter long dunes during a period of maximum, but nearly steady river flow.
These particular samples have been chosen for use here because data
from five Columbia River experiments were employed by Smith and McLean to set several coefficients in their spatially averaged flow theory, and subsequently in this paper, velocity and sediment concentration profiles computed using the proper size distributions will be compared to those obtained previously by Smith and McLean.
Bed sediment analyses indicate that the
river bottom was comprised of material with the same size and specific gravity composition from year to year.
However, the trough sample is more
representative of the bed composition under conditions of zero or near zero sediment transport because, under the high boundary shear stress part of a non-uniform flow, an erodible bed sample comprised of a wide variety of sediment particles always appears to be deficient in the lighter and finer classes.
This effect can be seen by comparing the two size distributions
shown in Fig. 1.
125
30
-
20
-
% 10
-
01
1
0.5
1
1.0
1
1
1.5
1
1
1
2.0
1
I
2.5
3.0
J
Fig. 1. Sediment size distrihution from the trough (A) and crest (B) of a 2.7 m high, 74 m long sand wave in the Columbia River. Size data for ten specific classes along with the settling velocity and critical boundary shear stress for each size class are given in Table 1. TABLE 1 Sediment Parameters for Each Size Class of Two Bed Samples xS(cm/sec)
D 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
.595 -500 .420 .354 .297 .250 .210 .177 .149 .125
7.14 6.04 5.01 4.04 3.28 2.62 2.07 1.65 1.28 0.99
2
‘ I(dynes/cm ~
3.12 2.71 2.40 2.18 7.00 1.86 1.72 1.58 1.45 1.32
% )
%
(Sample A)
-
2.5 4.5 8.5 13.0 21.0 19.5 15.0 9.5 5.0 1.5
(Sample B) 5.0 8.5 19.0 24.0 24.0 11.0
5.0 2.5
1.0
--
Included are sediment size, settling velocity, critical boundary shear stress, and percent of material in ten categories for each distribution. Sample A is from the trough of a 2.7 m high, 74 m long sand wave in the Columbia River and Sample B is from its crest.
126 Apparent Bed Roughness
In flow over a geometrically smooth, fixed boundary, the apparent roughness of the bed (2,) can be computed using the work of Nikuradse (1933);
however, once the transport of bed material has been instigated,
the characteristic sand grain diameter and the viscous sublayer thickness no longer provide the relevant length scales.
The presence of a solid
phase in the lower part of the flow tends to equalize the momentum distribution in this region making the velocity profile, when plotted with the logarithm of the distance from the boundary denoted
on
the ordinate, appear
concave from the right hand side, thus increasing the apparent boundary roughness for the flow in thenon4ediment transporting region.
Physically,
this tendency to equalize the momentum distribution in the near-bottom layer arises because the sediment comprising the river or sea bed is ejected in a near-vertical direction with zero horizontal velocity and extracts momentum from the flow until
1) it lands o n the bottom again or
2) it reaches the horizontal velocity of the fluid.
In the former
situation (the bed load case) the maximum velocity difference between the sedimentary particles and the transporting fluid occurs in the neighborhood of the top of the trajectory causing the maximum momentum defect to lie just below this level.
The sediment grains put a stress on the flow
and cause the velocity profile in the multicomponent fluid to increase less rapidly with distance from the boundary than would be the case in an otherwise analogous non-sediment transporting flow.
The characteristic
length scale in this situation is the thickness of the bed load layer. In the suspended sediment transport case, the particles attain the horizontal velocity of the fluid well before reaching their maximum elevation and under these circumstances maximum momentum loss to the transporting fluid occurs in the region of the maximum velocity difference so the appropriate length scale is the height of this zone.
In an examination of the bed load transport by wind, Owen (1964) recognized that the effective boundary roughness z should be proportional to the thickness of the bed load layer and argued that the bed load layer should be of order ~ , ~ / 2 g . Owen's expression is obtained by balancing potential energy at the top of the trajectory with the kinetic energy of the particle just after the flight has been initiated and assuming that the particle's vertical velocity at this point is proportional to u,. Using aeolian velocity profile data for the bed load transport situations studied by Bagnold (1941), Chepil (1945a,b,c) and Zingg (1953), Owen determined the constant of proportionality between zo and ~ , ~ / 2 g to be
127 2.07 x
Even when t h e d e n s i t i e s that c a n c e l f o r t h e a e o l i a n s i t u a t i o n
are r e t u r n e d so t h a t z
-
= psu,'/2(p
p)g, a quick c a l c u l a t i o n using
t y p i c a l v a l u e s f o r t h e r e l e v a n t p a r a m e t e r s i n d i c a t e s t h a t Owen's e x p r e s s i o n
i s n o t v a l i d under f l u v i a l and m a r i n e c o n d i t i o n s .
However, i t i s l i k e l y
t h a t t h e i n i t i a l k i n e t i c e n e r g y p e r u n i t volume of t h e sediment p a r t i c l e (psuo2)
i s n o t p r o p o r t i o n a l t o psu,2
b u t r a t h e r t h a t i t depends upon t h e
work done p e r u n i t p a r t i c l e volume i n l i f t i n g t h e sediment g r a i n from t h e bed.
-
T~
2 A s t h e f o r c e on a sediment g r a i n i s of t h e o r d e r ( T -~ T )D where . c T~ i s t h e e x c e s s boundary s h e a r stress averaged o v e r a few t e n s of
g r a i n d i a m e t e r s and D i s t h e p a r t i c l e d i a m e t e r , and as t h e d i s p l a c e m e n t i s of 3 t h e o r d e r D , t h e work done i s s c a l e d by ( T -~ rc)D . T h e r e f o r e t h i s approach y i e l d s a n e x p r e s s i o n of t h e form
z
=
zN
for
'b
-
'Ic
+
T
b
< -
T
c
and
z
0
=
(Ps
zN
for
P)g
T
b
T
where zN i s t h e v a l u e o b t a i n e d from t h e e x p e r i m e n t s of Nikuradse. Good agreement between Owen's r e s u l t s and t h o s e from a s e t of f i e l d measurements made by Smith and McLean i n t h e Columbia R i v e r i s o b t a i n e d when Owen's method i s s o m o d i f i e d .
The c o e f f i c i e n t a
a v a l u e of 2 6 . 3 i n rhe Columbia R i v e r d a t a .
w a s found t o have
In c o n t r a s t , i f t h e c r i t i c a l
boundary s h e a r stress f o r t h e i n i t i a t i o n of sediment motion were t a k e n t o b e z e r o and p case, a
were t a k e n t o b e much g r e a t e r t h a n p a s i n t h e a e o l i a n
= 2 2 . 4 p u t s ( l b ) i n q u a n t i t a t i v e agreement w i t h t h e e x p r e s s i o n
Here T = P U , ~ is t h e s h e a r stress on t h e sediment b bed averaged o v e r a r e g i o n of o n l y a few t e n s of g r a i n d i a m e t e r s i n s c a l e
o b t a i n e i by Owen.
and denoted s u b s e q u e n t l y i n t h i s paper a s t h e l o c a l boundary s h e a r stress, whereas motion.
T
i s t h e v a l u e of t h i s p a r a m e t e r a t t h e i n i t i a t i o n of sediment I t should be n o t e d t h a t T
depends upon t h e s i z e and s p e c i f i c
g r a v i t y of t h e sediment p a r t i c l e s c o m p r i s i n g t h e bed and i s denoted by T
when r e f e r r i n g t o a p a r t i c u l a r component of t h e sediment sample. Smith and McLean ( i n p r e s s ) d i d n o t a d d r e s s t h e problem of what t o
do w i t h ( l b ) when t h e bed i s comprised of more t h a n a s i n g l e s i z e and specific gravity class.
However t h e model p r e s e n t e d above i s based on
s i n g l e p a r t i c l e mechanics and i t i s c l e a r t h a t t h e n e t momentum d e f e c t due to
a d i s t r i b u t i o n of sediment s i z e s i n t h e bed l o a d l a y e r must be made up
128 of the sum of the individual defects. Moreover, if the sediment distribution is monomodal and if the sample is reasonably well-sorted, the height of the maximum momentum defect cannot differ significantly from that given by multiplying the height to which the grains in each size class rise times the number of grains in that size class and averaging over all size classes, that is, taking the concentration weighted average of a
(‘cb
-
T~)”
For greater accuracy or for use with more complicated sediment (p, - p)g. size and specific gravity distributions, a complete model of the bed load transport process must be used to evaluate the momentum defect; however, no such model is yet available and in most practical situations, the
procedure that has just been outlined here is sufficient.
In the suspended sediment case the situation is somewhat more complicated because the grains no longer return directly to the river or sea bed, and the region over which momentum exchange occurs between the sediment and fluid components is more broad.
Nevertheless, order of magnitude cal-
culations using the coupled particle-fluid equations of motion indicate that the finer bed load grains would attain the horizontal velocity of the fluid by the end of one flight and that particles destined to be transported as suspended load also attain the horizontal velocity of the fluid at a height of the order of magnitude given by (lb). ments permit the constant of proportionality between
Although these argu(ib
-
iC)/(pS
- p)g
and z o to vary between the bed load and suspended load cases and probably between situations characterized by well and poorly sorted sediment distributions, no evidence of a systematic variation is available to date and it is suggested that the value a
=
2 6 . 3 determined for a well-sorted
bed material being transported as bed load can be employed in all problems. Two factors assist in permitting more general use of (lb) with the abovementioned coefficient. First, in most suspended sediment transport cases T
is small relative to
ib
and does not vary widely, making (lb) relatively
insensitive to particle size, thus to the nature of the sediment distribution.
Second, the velocity field varies logarithmically with zo so
small errors in the latter parameter are not of great importance.
Perhaps
experimental and theoretical work to be carried out in the future will provide a more accurate means of finding z but for the present the use of (lb) with a coefficient of 26.3 should suffice in most practical sediment transport problems.
129 The E f f e c t s o f Suspended Sediment-Induced S t r a t i f i c a t i o n on Near Boundary Flow
Theore t i c a 1 Cons i d e r a t i o n s Smith and McLean ( i n p r e s s ) do n o t p r o v i d e a means of d e t e r m i n i n g what happens i n a suspended sediment problem when t h e bed sediment is comprised of a d i s t r i b u t i o n of p a r t i c l e s i z e s and s p e c i f i c g r a v i t i e s , n o r do t h e y account f o r t h e d i f f e r i n g d i f f u s i v i t i e s between sediment and momentum. I n t h i s s e c t i o n t h e s e d e f i c i e n c i e s a r e r e c t i f i e d t h u s p r o v i d i n g a complete t h e o r y f o r d e t e r m i n i n g v e l o c i t y and sediment c o n c e n t r a t i o n d i s t r i b u t i o n s i n h o r i z o n t a l l y uniform flow. Following Hunt (1969) o r Smith (1976, p. 5 6 0 ) , t h e e q u a t i o n s f o r cons e r v a t i o n of sediment and f l u i d mass i n a h o r i z o n t a l l y
uniform, m u l t i -
component f l o w can be w r i t t e n as
and
Here, €
is
e
i s t h e average. sediment c o n c e n t r a t i o n of component c l a s s n , and
tie
total
c o n c e n t r a t i o n of suspended m a t e r i a l , w
v e l o c i t y of component n , K nent n, K
W
is the s e t t l i n g
i s t h e mass d i f f u s i o n c o e f f i c i e n t f o r compo-
i s t h e mass d i f f u s i o n c o e f f i c i e n t f o r t h e water, and w
W
is the
For s t e a d y f l o w t h e s e two e x p r e s s i o n s can
v e r t i c a l v e l o c i t y of t h e water. be combined t o g i v e
A s t h e K ' s and K
W
a r e a l l d i f f u s i o n c o e f f i c i e n t s f o r mass, t h e y must b e
s e t e q u a l f o r r e a s o n s g i v e n by Smith and Hopkins (1972):
however, t h e y
need n o t b e e q u i v a l e n t t o t h e d i f f u s i o n c o e f f i c i e n t f o r momentum, Km. R e s u l t s from suspended sediment t r a n s p o r t s t u d i e s a s w e l l as from i n v e s t i g a t i o n s of s t a b l e a t m o s p h e r i c boundary l a y e r s i n d i c a t e t h a t t h e y may d i f f e r
.
Both s u g g e s t t h a t K = K = a K Suspended sediment s t u d i e s (Hunt, 1969) n w m s u g g e s t t h a t a i s a c o n s t a n t and i n d i c a t e t h a t i t l i e s between 1 . 2 and 1 . 5 .
130 The atmospheric investigations are concerned with the diffusion of heat rather than sediment particles, but heat is carried by the fluid and should diffuse with the mass diffusion coefficient thereof, so it is likely that the effects are the same.
In the latter case, a is found to depend upon
the degree of stratification parameterized by the distance from the boundary divided by the Obukhov length.
a =
Thus
1
+
cl
+ B5
55
where
and 3 L =
pu*
Here, p is the fluid density, u* is shear velocity, g is the acceleration due to gravity, k is von Karman’s constant, and flux into the fluid from the boundary.
(PX’)~ is the buoyancy
Businger, et a1 (1971) show that
5 is related to the gradient Richardson number at the boundary in the
following manner,
In general the gradient Richardson humber depends upon the density given in the sediment transport situation by the concentration field through
In sediment transporting problems, the values of a obtained from (4a) vary from 1.2 to 1.35.
The agreement between results from suspended sediment
studies and calculations based on meterological formulae appears not to be fortuitous and suggests that the use of (4a) and (5), which are derived from comprehensive atmospheric boundary layer experiments, is likely to be fruitful in sediment transport problems.
In addition to the concentration dependence of the coefficient a, the momentum diffusion coefficient K also varies with flow stratification, m
131 t h u s depends on € ( z ) .
Smith and McLean show t h a t n e a r t h e boundary
where
t h e n a r g u e t h a t ( 7 ) p r o b a b l y i s v a l i d t h r o u g h o u t t h e e n t i r e boundary l a y e r . Here
5
=
z / h i s t h e n o n d i m e n s i o n a l d i s t a n c e from t h e boundary and f 2 ( C ) i s
t h e n o r m a l i z e d d i s t r i b u t i o n o f eddy v i s c o s i t y w i t h h e i g h t under n e u t r a l conNear t h e boundary f (5) 2 5 i n agreement w i t h s i m i l a r i t y t h e o r y . 2 Above 0 . 1 h t h e eddy v i s c o s i t y i n c r e a s e s a t a d e c r e a s i n g r a t e and e v e n t u a l l y
ditions.
f a l l s o f f w i t h h e i g h t , a p p r o a c h i n g z e r o a t t h e t o p of t h e boundary l a y e r .
I n a r i d e r t h e boundary l a y e r t h i c k n e s s i s e q u i v a l e n t t o t h e f l u i d d e p t h ( h ) whereas on t h e c o n t i n e n t a l s h e l f i t i s t h e Ekman d e p t h g i v e n a p p r o x i m a t e l y by u,/2f
that l i m i t s it.
*
The form of f 2 g i v e n i n (7b) w a s o b t a i n e d by f i t t i n g a two-part polynomial t o t h e d a t a o f Klebanoff
(1954) and Townsend (1951), as shown by
Although t h e s e d a t a are f o r a growing boundary l a y e r
Hinze (1959, p. 493).
on a f l a t p l a t e r a t h e r t h a n f o r a f l o w o f f i n i t e d e p t h , t h e s h a p e o f (7b) i s i n r e a s o n a b l e agr,eement w i t h less a c c u r a t e d a t a from c h a n n e l f l o w s , f o r
example, see Vanoni (1946).
U s e of ( h a ) ,
( 5 ) . and ( 7 a ) f o r R
i
defined
w i t h i n t h e f l o w r a t h e r t h a n a t t h e boundary rests on a n unconfirmed postulate a t t h i s point.
However, t h e s e e x p r e s s i o n s must b e s a t i s f i e d asymptot-
i c a l l y and c a n n o t b e v e r y f a r i n e r r o r i f a t a l l .
With t h e a s s i s t a n c e o f
t h i s a s s u m p t i o n , a g e n e r a l i z e d v e r s i o n o f 5 c a n b e d e f i n e d and u s i n g ( 5 ) , (6),
( 7 ) and a u / a z = r / p K
m
i t becomes
112
c=--"
20
(8)
T h i s new p a r a m e t e r i s no l o n g e r d e f i n e d by (4b) and i s no l o n g e r d i r e c t l y r e l a t e d t o L as d e f i n e d i n ( 4 c ) . c a n b e r e p l a c e d by r b ( l
-
For a steady, uniform channel flow
~(5)
5).
*See Smith and Long (1976) f o r a d i s c u s s i o n o f t h e bottom boundary l a y e r on c o n t i n e n t a l s h e l v e s and i t s e f f e c t on s e d i m e n t t r a n s p o r t .
132 In order to solve ( 3 ) , a boundary condition must be applied at the top of the momentum defect layer.
To a first approximation this ought to be
proportional to the level of maximum momentum defect which in turn ought to be proportional to the value of z
taken above.
Moreover, an apparent value
of sediment concentration at level zo can be used just as w e l l as the actual value at level mzo, where m is a constant of proportionality, as long as the expression for the suspended sediment concentration profile remains fixed in Indeed, this is the case in all'problems
form for a given value of zo.
except those involving extremely small scale boundary topography. The use of a fictitious rather than a real reference concentration has the advantage of permitting the boundary conditions on the flow and sediment transport problems to be applied at the same level, a result that is most useful in complicated situations. The solution to ( 3 ) using (4a),
(7) and Kn
=
K w
=
aK is m
where w n Pn = ku*
and solving T
= T
b
(1
- 5)
= pK ( a u / a z ) where Km is given by (7) yields
m
When z<
and u =
u* In 0
Furthermore, when B<<
If an approximate analytical expression
133 for the uncorrected case is desirable, a constant momentum diffusion coefficient can be used above 0.2 h and the resulting expressions can be matched to (lla) and (llb) as described by Smith (in press, p. 561 ff). The sediment concentration at the reference level z is obtained by using the empirical expression
where
s =
b'
- n' T
n
generalized from the one component form suggested by Smith and McLean (in -3 however, the constant y has been changed from 1.95 x 10 to make
press);
the mean concentration 10 cm from the sea bed equal to that measured at this heig'ht in the Columbia River. Here € is the concentration of sediment b in the bed and i is the fraction of the bed sample in size class n. A s n Smith and McLean assumed that the sediment concentration field could be represented satisfactorily by a single size class, their concentration field was in error in the region near the bed where the coarsest fractions produce a noticeable effect.
Therefore, the concentration that they used at the
reference level, hence the value of y
that they used in (12) was based
upon an incorrect extrapolation of,the data from 10 cm to z Yo = 2.4 x
.
The value of
employed in this paper presumably is of genzral validity
as a reasonable number of size classes (ten) has been used to construct the distribution.
The form of (12) was obtained by noting that for low
to moderate concentrations, the empirical information used in bed load and total load formulae suggest a linear increase in reference concentration with respect to normalized excess boundary shear stress Sn and that the sediment concentration is bounded by the value for a fixed permeable bed. Other forms of the relationship between en(zo) and Sn certainly are possible, but the evidence available to date suggests that the one used here is quite good over the range of interest in most natural sediment transport problems.
In order to obtain complete solutions to (9) and (10). it is necessary to sum (9) over all N size classes, then solve for f? get
and differentiate to
134
r
When (12) and (13) are combined and substituted into (8), the latter can be solved iteratively to get 5
=
C(5).
Once known this function can be
inserted into (9) and (10) and the integrals can be evaluated to get the sediment concentration and mean velocity profiles. Hunt (1969) has derived an expression similar to the one presented in (lla) and has pointed out some of its characteristics. For example, he notes that €
decreases with height;
therefore, 1
-
€
increases with dis-
tance from the boundary making it possible for very fine particles in the presence of much coarser ones to decrease in concentration in the immediate vicinity of the bed.
This effect is potentially present in the numerical
suspended sediment concentration profiles of the more general problem; however, in most cases, the size distribution for the bed material is such that the concentrations never actually decrease. Discussion of results
In order to display some of the characteristics of these equations, computations have been carried out with the size distributions of Fig. 1. First concentration profiles for each of the ten components, along with the profile of e
S’
are displayed in Fig. 2 for both distributions.
In this case
a shear velocity of 4.52 cm/sec and a stratification parameter from (4a) of u
=
0.74 are used.
The effect of the stratification correction as well
as that of differing mass and momentum diffusion coefficients is examined in Fig. 3 .
Although comparison of results obtained using the size distri-
bution for the sand wave trough to those obtained using the crest sample (thus containing less material in the finer size fractions) is possible, the salient differences are better displayed in Fig. 4 where the velocity and suspended sediment concentration profiles are plotted relative to the same axes.
In Fig. 5 one of the most important aspects of this study is
displayed.
Here for the same shear velocity used in previous figures, the
135 dependence of the suspended sediment concentration profiles on number of classes is examined for the two cases. Finally in Fig. 6 the importance of flow stratification and differing mass and momentum diffusion coefficients is examined as a function of shear velocity as the latter is increased in three steps from 2.25 cm/sec to 10 cm/sec.
4
I
I
I
I
I
I
I
Fig. 2. Suspended sediment concentration profiles for u = 0.74 at a shear velocity of 4 . 5 2 crn/sec for each size class of the sand wave trough ( A ) and sand wave crest (B) size distributions of Fig. 1 and Table 1. The total concentration of suspended sediment, denoted 8 , is shown in each part of the figure. Note that € is comprised of mediEm and large particles near the bed and smaller partfcles further toward the surface.
Profiles in Figs. 2a and 2b show that the concentration of material in the coarser size classes falls off rapidly with distance from the boundary; whereas those for the-finer components are much less steep as expected. Moreover it is clear from these figures that the overall sediment concentration profile is affected significantly by the presence of the dominant coarse material near the bed, but is comprised primarily of the dominant finer fractions at greater distance from the boundary. that makes the constant Y validity.
It is this result
found previously by Smith and McLean lack general
The effect of using an eddy viscosity that goes to zero at the
river or sea surface can be seen in the drastic reduction of sediment concentration near the tops of the profiles.
The main difference between
results from the two distributions is a lack of finer material, thus a faster fall-off in overall sediment concentration, for the crest sample (B) relative to that for the trough (A). In order to emphasize the importance of suspended sediment-induced flow stratification as well as to indicate the specific effect of differing mass
and momentum diffusion coefficients on the velocity and suspended sed-
iment concentration profiles, curves for each of the two size distributions are shown in Fig. 3. mass
u
Those labeled u
=
1.0 represent cases in which the
and momentum diffusion coefficients are equal; whereas, those labeled
= 0.74 represent cases in which the best available information on ther-
mally
stably stratified boundary layers has been employed.
Use of a con-
stant value in the range of 1.2 to 1.4 instead of the variable value given by (4a) has only a small effect on the results.
The profiles labeled
R. = 0 are uncorrected for flow stratification effects. noted from (4a) and (6) that a
=
1.35 when u
=
Here it should be
0.74 and Ri = 5
= 0.
Again
the two sediment distributions yield similar results and the salient features shown in Figs. 3a and 3b are about the same.
Ri
= 0 case is the same whether
The velocity field for the
u = 1.0 or 3.74.
Examination of the velocity
profiles indicates that the flow speed at a given level is increased for a fixed boundary shear stress relative to what would occur at that level in an unstratified situation. Karman's constant.
This results in an apparent decrease in von
The sediment concentration profiles clearly show that
the effect of flow stratification and that of having unequal mass and momentum diffusion coefficients are opposite, the former acting to inhibit upward diffusion of suspended sediment and the latter acting to enhance it.
137 Uicm/secl 50
0
4
100
I50
I
1
I
1
I
I
10
200
250
I
I
I
0
c
IdZ
lo3
4
10
t
RI
.o.
J
I
I
Fig. 3 . Comparison of corrected and uncorrected velocity and suspended sediment concentration profiles for the sand wave trough (A) and crest (B) sediment distributions. The curves labelled R. = 0 have an uncorrected eddy diffusivity used in them and the curves labeldd u = 1.0 are for the case in which mass and momentum have the same diffusion coefficient. A better comparison of the velocity and suspended sediment concentra-
tion profiles for the two different size distributions is provided by Fig.
4.
The additional fine material in size distribution A has the effect of
increasing the suspended sediment concentration and flow speed at each level, showing the sensitivity of the correction to the specific character of the size distribution. This being the case, it is important to find out exactly how many size classes a typical sediment sample must be divided into in
138 order to obtain reasonably accurate results. U(cm/sec) 50
0
150
I00
200
250
I
I0
c
Id2
-4
10
1
I
I
I
I
I
I
Fig. 4 . Comparison of corrected velocity and suspended sediment concentration profiles for u = 0 . 7 4 and u* = 4.52 cm/sec between the trough '(A) and the crest (B) sediment samples. Note that the absence of fine material in the crest sample substantially increases the slope of the concentration profile and that it reduces the slope of the velocity profile. Fig. 5 constructed for a shear velocity of 4 . 5 2 cm/sec and a stratification parameter of 0 . 7 4 suggests that a single size class characterized by the median grain diameter is a very poor choice, but that the use of only three size classes is quite sufficient in most problems.
I n effect, if a sediment
sample is comprised of material having the same specific gravity, then dividing it into three size classes permits a reasonable treatment of the effects produced by the coarser fractions near the bed and those due to the finer fractions well into the interior of the flow.
Use of additional
classes does not increase the accuracy very much because the coarsest and finest classes, which otherwise would he expected to have the greatest effect, do not have much material in them.
When the sediment samples are
comprised of components with substantially different specific gravities, then the best procedure is to use a two-dimensional distribution function based on settling velocity and critical shear velocity.
Three settling
139 v e l o c i t y c l a s s e s and t h r e e c r i t i c a l s h e a r v e l o c i t y c l a s s e s a l m o s t always a r e s u f f i c i e n t and i n many c a s e s a mean o r median c r i t i c a l boundary s h e a r
stress f o r e a c h s e t t l i n g v e l o c i t y c l a s s i s s a t i s f a c t o r y .
0
I
,o’
1
,oh
1
lo5
lo4
I
1
10’
lo2
10’
I
€5
Fig. 5. F u l l y c o r r e c t e d sediment c o n c e n t r a t i o n p r o f i l e s f o r u = 0 . 7 4 and u* = 4 . 5 2 cm/sec i n t h e cases where t h e sediment s i z e d i s t r i b u t i o n h a s been d i v i d e d i n t o 1, 3 and 9 o r 1 0 s p e c i f i c s i z e classes. A s t h e case o f N = 1 does n o t p e r m i t t h e e f f e c t s o f f i n e m a t e r i a l d i f f u s i n g t o t h e i n t e r i o r of t h e water column t o b e t r e a t e d , i t d i f f e r s s u b s t a n t i a l l y from t h e o t h e r two. R e s u l t s s u c h as t h i s are o b t a i n e d o v e r a wide r a n g e o f p a r a m e t e r s and i n d i c a t e t h a t under most c i r c u m s t a n c e s d i v i s l o n of t h e sediment s i z e d i s t r i b u t i o n i n t o three c l a s s e s is s u f f i c i e n t . Up t o t h i s p o i n t t h e s h e a r v e l o c i t y h a s been h e l d f i x e d a t 4 . 5 2 cm/sec, but i t i s of i n t e r e s t t o know how t h e e f f e c t s of suspended sediment-induced momentum d i f f u s i o n c o e f f i c i e n t r e d u c t i o n and t h o s e of sediment d i f f u s i o n c o e f f i c i e n t enhancement v a r y w i t h boundary s h e a r stress.
This question is
a d d r e s s e d i n F i g . 6 through t h e p r e s e n t a t i o n of v e l o c i t y and suspended
140 sediment concentration profiles at four separate shear velocities.
In the
discussion of previous figures, the differences that arise between calcullations using each of the two size distributions have been considered in some detail; therefore, Fig. 6 presents curves only for the sand wave trough sample.
u Icmlsec)
u ( cmlsec) 0
50
100
150
200
250
0
IW
50
u..
U. . 7 0 cm/,.(.
U.
150
2W
250
300
4 52cm/im
.10ocm/s.c
Fig. 6. Variation of velocity and suqpended sediment concentration profiles with boundary shear stress with u = 0 . 1 4 . Note that the effect o f unequal mass and momentum diffusion coefficients dominates at low shear velocities and that the stratification correction is much more important at high shear velocities. Fig. 6a is constructed using a shear velocity so low (2.25 cm/sec) that the suspended sediment concentration field does not affect the velocity profile. Furthermore, the suspended sediment concentration profiles are not affected substantially either and these differ primarily depending upon whether the suspended sediment diffusion coefficient is assumed to be the same as that for momentum or whether it is enhanced as characterized by (4a).
At a shear
velocity of 4.52 cm/sec as shown in Fig. 6a, distinct stratification effects
141 are evident.
The d e t a i l s of t h i s p a r t i c u l a r f i g u r e have been d e s c r i b e d
p r e v i o u s l y a s t h e c u r v e s i n i t a r e t h e same a s t h o s e p r e s e n t e d i n F i g . 3a. By t h e t i m e t h e s h e a r v e l o c i t y h a s reached 7.0 cm/sec, t h e s t r a t i f i c a t i o n c o r r e c t i o n s on b o t h t h e v e l o c i t y and suspended sediment c o n c e n t r a t i o n f i e l d s a r e l a r g e r t h a n t h o s e due t o unequal mass and momentum d i f f u s i o n c o e f f i c i e n t s . These e f f e c t s a r e even more pronounced a t a s h e a r v e l o c i t y of 1 0 cm/sec. P r o f i l e s f o r t h e l a t t e r two c a s e s a r e shown i n F i g . 6c and 6d r e s p e c t i v e l y . It i s o b v i o u s from t h e sequence of p r o f i l e s i n F i g . 6 t h a t t h e suspended
sediment s t r a t i f i c a t i o n c o r r e c t i o n h a s o n l y a minor e f f e c t a t r e l a t i v e l y low s h e a r v e l o c i t i e s , b u t t h a t i t i s of major importance a t h i g h boundary s h e a r stress v a l u e s .
Moreover i t i s c l e a r t h a t c a r e f u l e x p e r i m e n t s t o t e s t t h e
a p p l i c a b i l i t y of ( 4 a ) should b e c a r r i e d o u t a t r e l a t i v e l y low c o n c e n t r a t i o n s and t h a t e x p e r i m e n t s t o t e s t q u a n t i t a t i v e p r e d i c t i o n s of t h e t h e o r y p r e s e n t e d i n t h i s paper s h o u l d be c a r r i e d o u t a t h i g h r e l a t i v e boundary s h e a r stresses. In problems such a s t h e one a t hand t h e boundary s h e a r stress can be nond i m e n s i o n a l i z e d u s i n g t h e mean s e t t l i n g v e l o c i t y of t h e sediment sample, t h u s p r o v i d i n g a q u a n t i t a t i v e means of d e f i n i n g r e l a t i v e boundary s h e a r stress.
S p a t i a l l y Averaged Flow o v e r Two-Dimensional Topographic F e a t u r e s
I n o r d e r t o g e t a h a n d l e on t h e t h i r d d i f f i c u l t y t h a t a r i s e s i n s e d i ment t r a n s p o r t i n g f l o w s , namely t h e d i s t u r b a n c e caused by t h e i n e v i t a b l e bed forms, Smith and McLean ( i n p r e s s ) n o t e t h a t t h e primary e f f e c t of t h e s e f e a t u r e s i s on t h e downstream averaged v e l o c i t y and suspended sediment concentration fields.
Using d a t a from t h e comprehensive s e t of Columbia R i v e r
sand wave e x p e r i m e n t s , t h e y show t h a t t h e s p a t i a l l y averaged v e l o c i t y p r o f i l e s can be thought of a s b e i n g comprised of s e m i - l o g a r i t h m i c segments, each c h a r a c t e r i z e d by ( l l b ) b u t w i t h d i s t i n c t v a l u e s of u* and z most l a y e r t h e a p p a r e n t bed roughness z
.
I n t h e inner-
i s g i v e n by (1) and t h e s h e a r v e l o c -
i t y i s t h a t a s s o c i a t e d w i t h t h e boundary s h e a r
stress averaged o v e r a n a r e a
of a few t e n s of g r a i n d i a m e t e r s i n s c a l e , t h a t i s , w i t h t h e s h e a r s t r e s s comprised of s k i n f r i c t i o n and form d r a g on t h e i n d i v i d u a l g r a i n s .
In the
n e x t l a y e r t h e a p p a r e n t roughness of t h e bed i s t h a t due t o t h e s m a l l e s t bed forms and t h e s h e a r v e l o c i t y i s based upon t h e boundary s h e a r stress t h a t i n c l u d e s form d r a g on t h e s e f e a t u r e s .
The n e x t l a y e r h a s a n a p p a r e n t rough-
n e s s l e n g t h r e l a t e d t o t h e n e x t l a r g e s t s c a l e of bed forms, i f any, and a s h e a r v e l o c i t y r e l a t e d t o t h e boundary s h e a r s t r e s s t h a t i n c l u d e s t h e form d r a g of t h e s e l a r g e r f e a t u r e s .
The l a y e r s c o n t i n u e outward u n t i l a l l s c a l e s
of d i s c r e t e bed forms have been i n c l u d e d .
For example, i n t h e c a s e of a
142 rippled bed only two layers exist; whereas in the case of a bed comprised of ripples, dunes, and sand waves, four separate layers are required.
The
Columbia River experiments upon which Smith and McLean's theory is based were carried out over large sand waves that had smaller dunes on their backs and three distinct layers were identified. The velocity profiles must match at the boundaries between the layers; therefore, procedures to compute the height of each matching level and the values of the apparent roughness parameter for each of the outer layers are required.
The former was accomplished by noting that the top of each layer
had to be associated with the average height of a growing internal boundary layer and that theories such as those of Elliot (1958) or Smith (1969) all yielded about the same results, even though specific problems to which they were addressed were substantially different. layer j and layer j
+
layer j is given by
(z ).
If the matching level between
1 is denoted ( z * ) ~ , ~ +the ~ , apparent bed roughness of and the wavelength of bed forms is denoted by A ,
O J
the approach taken by Elliot (1958) yields
The coefficient al ranges from 0.3 to 0.5 according to Elliot's theory; however, the problem at hand is somewhat different from the one with which he was concerned, and in particular the conditions at the origin of the internal boundary layer are somewhat less well-defined;
therefore, Smith
and McLean used ( 1 4 ) but left al to be set by non-uniform flow data. At the matching height the velocity must be continuous
so
versions of
(10) for each level can be equated and rearranged to yield
Now in order to find the apparent roughness parameter ( z ) . for one of the O
J
outer layers, form drag on the topographic features giving rise to this parameter can be expressed in terms of a drag coefficient, yielding an equation of the form
143 o r rearranging
i s t h e h e i g h t of t h e bed forms c a u s i n g
Here CD i s t h e d r a g c o e f f i c i e n t , Hj+l l a y e r j + l , and Xj+l
i s t h e wavelength of t h e bed forms c a u s i n g l a y e r j + l .
A least squares f i t of equations (14),
(15) and (16b) t o t h e s p a t i a l l y
averaged Columbia R i v e r v e l o c i t y and Reynolds s h e a r stress p r o f i l e s g i v e s al = 0.0995, k = 0.38, C
D
s e p a r a t e d flow.
=
0.840 f o r u n s e p a r a t e d f l o w , and CD = 0.212 f o r
As used i n (16a) t h e d r a g c o e f f i c i e n t i s d e f i n e d i n terms
of t h e v e l o c i t y a t t h e matching l e v e l and i s a f a c t o r of f o u r o r s o h i g h e r than would be t h e c a s e were i t d e f i n e d i n terms of t h e s u r f a c e v e l o c i t y (the nearest equivalent t o a f r e e stream velocity).
The r e a s o n f o r t h e
l a r g e d i f f e r e n c e between t h e d r a g c o e f f i c i e n t s i n t h e s e p a r a t e d and unsepa r a t e d c a s e s h a s n o t been e x p l a i n e d s a t i s f a c t o r i l y , b u t i s c l e a r l y e v i d e n t i n t h e measured v e l o c i t y p r o f i l e s shown i n F i g . 7 .
1
UNSEPARATEG A 0
0
I971 W I 1971 W 2 1972 W I
-
Fig. 7. Normalized v e l o c i t y p r o f i l e s f o r s e p a r a t e d and u n s e p a r a t e d f l o w 1 0 0 meter l o n g dunes i n t h e Columbia o v e r a set o f 1.5 - 3 meter h i g h , 70 River. The s o l i d l i n e s r e p r e s e n t t h e f i t o f Smith and McLean ( i n p r e s s ) t o t h e s e p o i n t s , o b t a i n e d by a v e r a g i n g t h e r a w d a t a i n t i m e and t h e n o v e r one wavelength a t c o n s t a n t d i s t a n c e from t h e boundary.
-
144 The c u r v e s p a s s i n g through t h e d a t a p o i n t s i n t h i s f i g u r e a r e t h o s e g i v e n by the theory described i n t h i s section.
Measured and p r e d i c t e d s p a t i a l l y
averaged s h e a r s t r e s s p r o f i l e s a r e p r e s e n t e d i n F i g . 8 .
The i n i t i a l i n c r e a s e
i n Reynolds s h e a r stress w i t h d i s t a n c e from t h e boundary i s due t o a t o p o g r a p h i c a l l y induced z-dependent p r e s s u r e g r a d i e n t and i s a n i n e v i t a b l e consequence of t h e bed forms. I .c
..C
.€
.1
.6
-.
r
rlJ
.5
.4
.3
.2
.I
F i g . 8. S p a t i a l l y averaged Reynolds s h e a r stress p r o f i l e s o b t a i n e d d u r i n g t h e 1972 Columbia R i v e r s a n d wave experiment. The s o l i d l i n e i s t h e p r o f i l e p r e d i c t e d by t h e approach of Smith and McLean ( i n p r e s s ) .
145
The velocity field given by (10) can be generalized for layer j by changing the limits to yield
5
Using (17) for 5
=
0, (15) and (16) can be generalized to
and ./2
D' 1+ 2k2
"j+l
Aj+l
(5,)
is the actual matching level discussed in a subsequent paragraph. j,j+l Equating (18a) and (18b) yields an expression for (So)j+l in terms of known parameters.
The' local stratification parameter 5 has been deleted
from these equations because the drag coefficient must be defined in terms of an equivalent, unstratified flow velocity profile.
If the actual velocity
at some reference level were to be used, then as the stratification became stronger, the reference velocity could become higher implying a higher form drag; however, the force on the boundary does not change under these conditions.
This being the case, it is equivalent unstratified flow velocity
at the reference level to which drag coefficient must be related. The generalized expression for aes/ag to use in (8) can be written in the form
146
where (5*)-l,o
S i m i l a r l y t h e i n d i v i d u a l suspended sediment con-
=
c e n t r a t i o n p r o f i l e s a r e g i v e n by
A = en (5*).-1 . '' 1 - €
exp
l-es ( C*) j -1, j
I
-pn
5
j%
(20)
dc]
(C*)j-I,j
In o r d e r t o o b t a i n s o l u t i o n s f o r t h e v e l o c i t y and sediment c o n c e n t r a t i o n f i e l d s i n a non-uniform, be s o l v e d t o g e t h e r .
s t r a t i f i e d flow (8),
( 1 4 ) , ( 1 8 ) , (19) and (20) must
The s o l u t i o n b e g i n s w i t h t h e v a l u e f o r (u,)
.
If it is
t h e v e l o c i t y a t some l e v e l i n t h e i n t e r i o r of t h e w a t e r column t h a t i s known, t h e n t h i s p a r a m e t e r must be e s t i m a t e d , t h e v e l o c i t y p r e d i c t e d a t t h e
(u,)
r e q u i r e d l e v e l and a new e s t i m a t e of
t h e c a l c u l a t e d and measured v e l o c i t i e s . and converges f a i r l y r a p i d l y . Once a v a l u e of
d e r i v e d from a comparison between The p r o c e d u r e i s s t r a i g h t f o r w a r d
T h i s i s t h e outermost l e v e l of i t e r a t i o n .
(u,)~ i s s u p p l i e d o r e s t i m a t e d ,
( 1 ) and t h e v a l u e s of ( u )
* j
can b e c a l c u l a t e d from
and ( z ) . can be computed f o r a l l l e v e l s u s i n g O
J
( 1 4 ) , (18a) and ( 1 8 b ) . Using (u*)~, aes/ag
can b e fuund f o r t h e lowest l a y e r i n terms of c,
from (19) and t h e l a t t e r can b e used i t e r a t i v e l y i n c o n j u n c t i o n w i t h (8) t o f i n d ~ ( 5 ) . A s soon a s ~ ( 5 )i s known t h e v e l o c i t y and c o n c e n t r a t i o n p r o f i l e s can be computed u s i n g (17) and ( 2 0 ) . pute
Next (14) i s used t o com-
t h e matching l e v e l a t t h e t o p of t h e j = 0 l a y e r .
t h e procedure i s r e p e a t e d f o r t h e j = 1 l a y e r .
Once t h i s i s known,
Here t h e c o n c e n t r a t i o n a t
t h e t o p of t h e lower l a y e r f o r e a c h s i z e c l a s s i s used a s t h e boundary condition for z
.
Equation (14) g i v e s t h e matching l e v e l f o r an u n s t r a t i f i e d
s i t u a t i o n and when t h e v e l o c i t y and sediment c o n c e n t r a t i o n f i e l d s a r e c o r r e c t e d , t h e i r a c t u a l matching p o i n t i s somewhat d i f f e r e n t .
T h i s means
147
that the position at which € (5*)o,1
has been evaluated, thus its value at
the proper matching level, is not quite correct.
As it is used as a bound-
ary condition on (2O),the error must be corrected iteratively. After the velocity and sediment concentration profiles have been joined smoothly at the actual matching level, these fields are known to the top of the j layer.
=
1
The next unstratified flow matching level is computed from (14),
2C / a 5 is found for the next layer frorr (19), 5(5) is found from (19) and
(8), u and 8 are found from (17) and (20),and the matching level is determined iteratively. The procedure proceeds until all levels have been included. At this point if the velocity in the interior is the known parameter, a new value of (u,)
is chosen and the entire procedure is
repeated. As mentioned above Smith and McLean (in press) used a single size class, chosen to be representative of sediment sample A.
This was done
because they had to solve the problem posed here backwards to the coefficients al, k, and C
D
and to obtain a subsidiary specification of the sedi-
ment transport parameters
ci
and y
.
The numerical accuracy of the method
that they describe rests upon the validity of this guess and their approach needs to be checked using the equations presented in this paper.
This is
done in Figs. 9 and 10. The former compares the original result (denoted
N = 1) with the more general result for the case u = 1 and the latter compares the original calculation to what is presumed to be the most accurate estimate, namely thaF for which u = 0.74.
In both cases the concentration
has been fixed at the measured value 10 cm above the sea bed. Comparison of the curves in Fig. 9 shows that the concentration fields -4 are in reasonable agreement in the range of 5 = 5 x 10 to 5 = lo-* and that the velocity fields are in good agreement above 5 = 5 x 10-2 ; however, some discrepancy occurs in the lower part of the velocity profile.
148
U (cmlsec 1 0
50
100
I50
2 00
250
I
-1
10
E
-2 10
10
-4
10
Fig. 9. Comparison of velocity and suspended sediment fields calculated with u = 1.0 (and ten sediment size classes included) to those calculated by Smith and McLean (in press) for Columbia River sand wave data. The former used ten sediment size classes (N = 10) whereas the latter used only one (N = 1). The concentration measured 10 cm from the bed was matched in both cases so the values of yo differed. In Fig. 10 the velocity fields match almost perfectly; however, the suspended sediment concentration field is underestimated in the upper part of the flow. The fact that the velocity field obtained using the techniques of this paper is in good agreement with that based on Smith and McLean's one class estimate, which in turn is based on a least squares fit to data,means that the coefficients set by that least squares) fit, and previously listed in this paper, accurately represent the spatially averaged flow investigated in the Columbia River experiments.
149
U (crnkec) 0
50
I00
150
200
250
I
lo3
-4
10
Fig. 10. Comparison of velocity and suspended sediment fields calculated with u = 0.74 to those calculated by Smith and McLean (in press). The former is based on ten sediment size classes (N = 10) whereas the latter uses only one (N = 1) and equates the mass and momentum eddy diffusion coefficients. Note that the N = 1 velocity profile is in good agreement with the fully corrected version obtained using the techniques described herein. Conclusions Procedures for treating three important complications that arise in boundary layer flow have been outlined in this paper.
Each is in need of
quantitative verification under diverse natural flow conditions. Nevertheless, even as the methods now stand, they permit a reasonable treatment of effects that hitherto have been ignored in regard to both boundary layer and sediment transport mechanics. It is hoped that these techniques and the ideas on which they are based will serve as an aid in the design and interpretation of future boundary layer experiments. If they prove accurate, the phenomena that they explain can be quantified and those that remain unexplained can be isolated, then examined more thoroughly.
150 Bibliography Bagnold, R.A., 1941. The Physics of Blown Sand and Desert Dunes. London, 265 pp.
Methuen,
Businger, J.A., Wyngaard, J.C., Izumi, Y., Bradley, E.F., 1971. Fluxprofile relationships in the atmospheric surface layer. Journal of Atmospheric Science, 28:181-189. Chepil, W.S., 1945a. Dynamics of wind erosion: of soil by wind. Soil Science, 60:305.
I. nature of movement
Chepil, W.S., 1945b. Dynamics of wind erosion: movement. Soil Science, 60:397.
11.
Chepil, W.S., 1945c. Dynamics of wind erosion: capacity of the wind. Soil Science, 60:475.
111.
initiation of soil the transport
Elliot, W.P., 1958. The growth of the atmospheric internal boundary layer. Transactions of American Geophysical Union, 38:1048. Hinze, J.O., 1959. Turbulence.
McGraw-Hill, New York, 586 pp.
Hunt, J.N., 1969. On the turbulent transport of a heterogeneous sediment. The &uarterly J o u m Z of Mechanics and Applied kthematics, 22: 235-246. Klebanoff, P.S., 1954. Characteristics of turbulence in boundary layer with zero pressure gradient. National Advisory C o d s s i o n Aeronautical Teclrnical Notes, No. 3178. Nikuradse, J., 1933. Stromungsgesetze in rauhen rohren. VDI-Forschungsheft 361. Beilage zu Forschng auf dem Gebiete des Ingenieuruesens, Ausgabe B, Band 4. Owen, P.R., 1964. Saltation of uniform grains in air. Mechanics, 20:225. Smith, J.D., 1969.
Jourmal of Fluid
Studies of non-uniform boundary layer flows.
In:
Investigations of TurbuZent Boundary Layer and Sediment Transport Phenomena as Related t o Shallow Marine Environments, Part 2. U.S.A.E.C. Contract AT(45-1)-1752. Ref: A69-7, Department of Oceanography, University of Washington. Smith, J.D., 1976. Modeling of sediment transport on continental shelves. In: E. D. Goldberg (Editor), The Sea, John Wiley, New York (in press). Smith, J.D., and Hopkins, T . S . , 1972. Sediment transport on the continental shelf off of Washington and Oregon in light of recent current measurements. In: D. J. P. Swift, D. B. Duane, and 0. H. Pilkey (Editors), Shelf Sediment Transport, Dowden, Hutchinson & Ross, Inc., Stroudsburg, Pennsylvania. Smith, J.D., and Long, C.E., 1976. The effect of turning in $he bottom boundary layer on continental shelf sediment transport. Memoires Socie'te' Royale des Science de Lisge, 6e sgrie, 10:369-396. Smith, J.D., and McLean, S.R., in press. Spatially averaged flow over a wavy surface. J o u m Z of Geophysical Research.
161 Townsend, A.A., 1951. The structure cf the turbulent layer. of Cambridge Philosophical Society, 47: 375. Vanoni, V.A.,
1946.
Proceedings
Transportation of suspended sediment by water.
TMnsactwns of American Society of Civil k g i n e e r s , 111:67-133. Zingg, A.W.,
1953. Wind tunnel studies of movement of sedimentary material.
University of Iowa Studies i n Engineering Bulletin, 34:111.
This Page Intentionally Left Blank
153
THE DYNAMICS OF THE BOTTOM BOUNDARY LAYER OF THE DEEP OCEAN* LAURENCE ARM1 Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 0 2 5 4 3 , U.S.A. ABSTRACT Profiles of salinity and temperature from the center of the Hatteras Abyssal Plain have a signature that is characteristic of mixing up a uniformly stratified region:
a well-mixed layer above the bottom, bounded
by an interface. The penetration height of the mixed-layer varies from about 10 m to 100 m and has been correlated by Armi and Millard (1976) with the one day mean velocity, inferred from current meters located above the bottom boundary layer. Here the dynamics of such layers is discussed.
A model of entrainment
and mixing for a flat bottom boundary layer is outlined; this model is however incomplete because we find too little known of the structure of turbulence above an Ekman layer. An alternate model is suggested by the estimate, from the correlation of penetration height with velocity of the internal Froude number of the mixed layer, F
1.7.
This value indicates
that the large penetration height may be due to the instability of the well mixed layer to the formation of roll waves. INTRODUCTION In a recent study of the bottom boundary layer of the deep ocean Armi and Millard (1976) have described aspects of this layer as observed with a CTD profiler.
An example of a salinity and potential temperature profile
taken over the smooth Hatteras Abyssal Plain is shown in figure 1. Here the well-mixed region extended to about 55 meters above the bottom and was bounded by a sharp interface across which the salinity, potential temperature and potential density changed.
Above the interface there was a
nearly uniformly stratified region.
*
Contribution No. 3857 from the Woods Hole Oceanographic Institution, Woods Hole MA 0 2 5 4 3 .
154
Salinity,
Figure 1. A salinity, potential temperature profile (from Armi acd Millard, 1976) in the middle of the Hatteras Abyssal Plain. Dotted line indicates structure could have formed by mixing up the stratified region above. The traces from both the lowering and raising of the profiler are shown.
The signature of the bottom boundary layer, with its well-mixed region bounded by an interface and a stratified region above, i s distinctive.
A
signature like that of F i g . 1 i s also seen in laboratory experiments in which a uniformly stratified fluid in a tank i s mixed up by stirring with a grid as in the experiments of Cromwell (1960) and Linden (1975), or a surface stress as in the experiments of Kato and Phillips (1969).
The
signature i s typical of a mixing process, the penetration of which i s bounded by the interface which forms as a result of mixing the uniformly stratified fluid.
155 The thickness of the characteristic well-mixed region of the bottom boundary layer on the smooth abyssal plain was correlated by Armi and Millard (1976) with the one-day mean velocity measured just above the layer to show that the layer is of dynamic origin. The region does not form a pool or have a distinctive water mass characteristic. The thickness of the layer is large compared with estimates of the turbulent Ekman layer height. Some quantitative relationships. Quantitative data characterizing the observed bottom mixed layers and the adjacent stratified fluid are as follows. A correlation of penetration height and velocity f o r the smooth Hatteras Abyssal Plain gives h/U typical values of h
and
U
%
1.2
X
10’ sec;
(1)
being 50 m and 4 cm/sec. The celerity of
long interfacial waves can be calculated knowing the Brunt-Vaisgla frequency in the stratified region above the mixed layer. The Brunt-Vaisalz frequency, N is . 4 c.p.h. (7
X
=
m
sec-’) and is nearly constant for all the profiles
taken over the center of the Hatteras Abyssal Plain.
By continuity of mass
the reduced gravitational acceleration for the mixed layer is given by
Neglecting the stratification above the interface, the celerity of a long wave on the interface is c =
m,
about 2.5 cm/sec for the typical penetration height of 50 meters.
(4)
An
internal Froude number, F, can be defined for the bottom-mixed layer, c.f. Turner (1973, p. 12):
F
(5)
= U/c
Using (3) and the ratio of penetration height to velocity we find
F
%
1.7.
156 Turbulent Ekman layer and stratified Ekman layer models.
Theories for
the bottom boundary layer have considered the effects of rotation alone and in combination with-the stabilizing effect of the sharp interface formed by entrainment of the stratified fluid above. We will now review these theories for the bottom boundary layer, limiting the review primarily to those theories for a flat bottom. Can the bottom mixed layer be treated as a classical turbulent Ekman layer? Such a treatment has been suggested by Wimbush and Munk (1971) and Weatherly (1972, 1975).
The experiments of Caldwell, Van Atta, and Helland
(1972) and Howroyd and Slawson (1975) give the height of the turbulent Ekman layer, he, as h
=
u* .4-. f
(For the experiments of Caldwell st. he :6 9 9 , the height at which the velocity is 99% of the geostrophic velocity; for the experiments of Howroyd and Slawson he
is defined as the height at which the velocity is
parallel to the geostrophic velocity.)
Biscaye and Eittreim (1974) report
that photographs on the Hatteras Abyssal Plain display a monotonous, flat, mud bottom showing only some "lebenspuren" with relief of about 1 cm. With the Coriolis parameter f = 7
X
sec-'
and the friction velocity
uj,
for a smooth bottom given by uj, = (1/30)U, (c.f. Csanady, 1967)
(7)
the Ekman layer height to velocity ratio is he - % 2 x 10' U
sec.
Using the empirical result of (1). we see the penetration height of the mixed layer is about six times the turbulent Ekman layer height. The Ekman layer can thus constitute only the lower sixth of the well-mixed region of the bottom boundary layer.
For the typical velocity of 4 cm sec-'
the Ekman height is only 8 meters. The effects of unsteadiness are discussed by Wimbush and Munk (1971) who note the time scale for the entire Ekman layer is 2i~/f;therefore those features of the boundary layer, in particular the logarithmic layer, with time scales very much less than 2n/f can be approximated by steady-state theory.
As suggested by Munk, Snodgrass and Wimbush (1970), the simplest
157 procedure is to reinterpret the turbulent Ekman height (6) with u* dependent
OIL
the local mean current. Even if' u*
were dependent on the
maximum value of the velocity, the Ekman layer height given by ( 8 ) would be larger by at most 6 m since the most energetic inertial or tidal velocities are
%
3 cmlsec.
We must also be cautious about using an Ekman layer to model even the lower portion of the bottom boundary layer on a slope of only one in a hundred. Then an advective term, say v
aU -, aY
in the mean value equations of
motion, will scale in the Ekman layer like aU2/he, where a
is the slope.
The Rossby number in the turbulent Ekman layer can then be found using (6) and (7).
It is of order
102a. The Rossby number is therefore of order
lo-';
unity for a slope of only
inertial effects then must be included in
a model of this laver. Nonetheless, the vertical length scale defined bv (7) is probablv still important. It is the length scale for which the Rossby number of the most energetic bottom generated turbulence is of order unity.
At this length scale the effects of rotation will be felt by the
largest turbulent "eddies". A model, of the bottom boundary layer, combining the effects of stratification and rotatioi. has been proposed by Thompson (1973).
This is a slab
model of the homogeneous well-mixed region, with the penetration height given by a bulk Richardson number closure assumption based on the height of the layer and jumps across the sharp interface of density and velocity. The velocity difference is that between the geostrophic velocity above and the mean velocity in the layer. This bulk Richardson number closure is also used by Pollard, Rhines and Thompson (1973) to whom Thompson refers for a detailed explanation. The penetration height of 8 meters, for u
%
5 cm sec-',
as suggested by Thompson is much too small.
Csanady (1974) finds the parameterization of interfacial stability, used by Pollard g&. (1973) and Thompson (1973), unattractive because the control of the mixed layer depth may be independent of the mechanism that maintains the stability of the interface. Csanady suggests that a limit to entrainment is set by tne turbulent length and velocity scales in the mixed layer and the density difference across the interface. This asymptotic limit for the entrainment is deduced from the experiments of Kato and Phillips (1969).
The limit used by Csanady is given by = 500 u*2
where g> is the reduced gravitational acceleration at the interface,
(9)
158 h
is the friction velocity or turbulent
u*
is the layer thickness, and
velocity scale. We note however that with u**
=
lo-' U2, used by Csanady, F = 1.4, not very
equation 9 is equivalent to a Froude number closure of
different from the value chosen by Pollard et al. (1973) of unity or the
F
empirical result found here of
%
1.7.
The existence of an asymptotic
entrainment limit, such as expressed by (9) above and used by Csanady, is certainly not established and must be questioned in light of the experiments of Turner (1973, p. 291) in which no tendency towards a limit is actually observed. Effects of convection. The effects of convection due to geothermal heat flux on the ocean bottom have been estimated using the Monin-Obukov length,
41,
by Wimbush and Munk (1971).
H = 1.5
x
cal om-' sec-'
With a geothermal heat flux
and u* = .1 cm sec-l,
$
=
-10' m.
This
scaling must however be approached with some caution since the dimensional argument used to derive the Monin-Obukov length contains neither the effects of the smaller length scale, he
1
KU*/f,
due to rotation, nor the exis-
tence of a well-mixed layer bounded by an interface. We note also that the time required to raise the temperature of a wellmixed layer 50 m in height by 1 m°C due to geothermal heating alone is about 40 days.
Temperature variability due to the mesoscale variation on
a time scale of 40 days can be as large as 40 m°C. One might argue that convection due to the variability of 25 m°C of the background temperature gradient moving across a constant temperature bottom might create the 50 m thick layers by penetrative convection into the ambient .5 m°C m-l
gradient.
However such convection due to a constant
sediment temperature and varying temperature due to advection of the mixed layer by mesoscale motions will also fail to explain the unusually large penetrative heights of the mixed layers because of the short diffusive length scale and hence small heat content within the sediments, over the time scale of the eddy.
The available maximum temperature contrast due to
variability of the background temperature field will only penetrate the sediments to a depth scale L
%
or about 50 cm in 20 days.
small depth compared with a typical layer depth of 50 m.
Indeed a
The signatures of
the profiles also do not support such an argument. Differential advection. We can estimate the slope of the interface that would be associated with any differential advection within the mixed layer and above the bottom Ekman layer, assuming such differential advection is a gostrophic flow and no interfacial stress exists. A change in height of the interface, on the order of the mixed-layer depth, h, will then occur over
159 a distance,
x,
given by
I f t h e assumed d i f f e r e n t i a l v e l o c i t y i s o n l y one t e n t h t h e g e o s t r o p h i c
U,
velocity,
U
=
we can u s e t y p i c a l v a l u e s
4 c m sec-'
and f i n d
x
Q,
2 km
and
c = h/x
=
Q,
1/50.
2.5 cm s e c - l ,
h = 50 m,
V a r i a t i o n s over t h i s
s h o r t a l e n g t h s c a l e w e r e n o t observed, and we conclude t h a t any g e o s t r o p h i c d i f f e r e n t i a l v e l o c i t y above t h e Ekman l a y e r must be slower t h a n about
.4 cm sec-'
.
Some p r e l i m i n a r y t h o u g h t s on modeling t h e bottom boundary l a y e r .
The
t r e a t m e n t , of t h e entrainment by t u r b u l e n c e a t an i n t e r f a c e i n terms of e s t i m a t e d t u r b u l e n c e v e l o c i t y and l e n g t h s c a l e s a t t h e i n t e r f a c e ( c . f . Turn e r , 1973, chap. 9) i s a t t r a c t i v e .
The i d e a i s p a r t i c u l a r l y a t t r a c t i v e i f
a s t r o n g feedback mechanism e x i s t s between t h e entrainment mechanism and t h e gener'ation mechanism f o r t h e t u r b u l e n c e .
It i s u s e f u l t h e r e f o r e t o ex-
p l o r e t h e p o s s i b i l i t y of c h a r a c t e r i z i n g t h e t u r b u l e n c e i n t h e well-mixed r e g i o n of t h e bottom boundary l a y e r and t e s t i n g t h e f e a s i b i l i t y of e s t a b l i s h i n g e i t h e r an entrainment v e l o c i t y t h a t i s s m a l l but f i n i t e , o r even a hard entrainment l i m i t based on t h e t u r b u l e n c e reaching t h e i n t e r f a c e . The well-mixed
r e g i o n of t h e bottom boundary l a y e r h a s been shown t o con-
s i s t of a lower p a r t , about one s i x t h t h e t o t a l h e i g h t , which can be modeled
a s a t u r b u l e n t Ekman l p y e r .
The t u r b u l e n c e l e v e l i n a t u r b u l e n t Ekman
l a y e r has been measured i n t h e experiments of Howroyd and Slawson (1975) and C a l d w e l l s g .
(1972).
But what of t h e t u r b u l e n c e above t h e h e i g h t of
t h e Ekman l a y e r and below t h e i n t e r f a c e ?
Here t h e experiments of Howroyd
and Slawson i n d i c a t e d t h a t t h e t u r b u l e n c e v e l o c i t y f l u c t u a t i o n s approach a n e a r l y c o n s t a n t v a l u e of
u'
.025 U; T a t r o and Mollo-Christensen (1967)
and more r e c e n t l y Ingram (1971) have r e p o r t e d s i m i l a r r e s u l t s .
Some con-
t r o v e r s y r e g a r d i n g t h e s e experiments h a s been p o i n t e d o u t by C e r a s o l i (1975).
T a t r o and Mollo-Christensen a t t r i b u t e t h e dominant frequency of
t h e f l u c t u a t i o n s , which i s j u s t less t h a n t h e i n e r t i a l frequency, i n e r t i a l o s c i l l a t i o n s i n t h e i n t e r i o r region.
f,
to
They s u g g e s t t h a t a n i n -
s t a b i l i t y i n t h e Ekman boundary l a y e r , which always h a s a frequency h i g h e r t h a n t h e i n e r t i a l frequency, can e x c i t e a f i r s t subharmonic w i t h less t h a n t h e i n e r t i a l frequency, and t h e r e s u l t i n g i n e r t i a l wave i s found t o propagate throughout t h e r e g i o n above t h e Ekman l a y e r .
It should be noted
t h a t i n a t u r b u l e n t Ekman l a y e r t h e l a r g e s t t u r b u l e n t f l u c t u a t i o n s may a l s o have t h e c o r r e c t v e l o c i t y and l e n g t h s c a l e s t o e x c i t e r a d i a t i n g i n e r t i a l
160
With a turbulence frequency, w
waves. he = 0
(2),
=
);
,
O[
the tuGbulence frequency is given by
and the length scale, w = O(f);
the tur-
bulence could thus excite inertial waves and it will become difficult to distinguish between such waves and true turbulence. With, as yet, so little known about the fluctuations which may occur in the mixed region above the turbulent Ekman height, it is perhaps premature to attempt to characterize an entrainment limit. Perhaps it would be more appropriate to characterize the stability of the radiated inertial waves. Nonetheless, if the fluctuations (either turbulent or radiated) scale with and their vertical length scales with h the Ekman height, then the e’ value of the turbulent Richardson number, Ri’, (c.f. Turner, 1973, p. 291) u*,
that corresponds to the observed empirical bulk Richardson number of Ri
%
.4, is Ri> % 70. Turner (1973, fig. 9.3) finds that for this tur-
bulent Richardson number the entrainment velocity has decreased to about 5
x
lo-’ times the unstratified entrainment velocity. Perhaps entrainment
just matches the rate at which the mixed layer interface i s eroded. We believe the erosion mechanism is likely to be internal waves breaking at the interface, a region of higher relative Brunt-VEisgli frequency. The instability of the bottom well-mixed region to the formation of roll waves; a possible mechanism controlling the penetration height of the wellmixed layer.
The immediate purpose of the arguments to be presented below
is to suggest a possible mechanism by which a limiting vertical penetration height may be established. The appeal of the mechanism is that it does not depend on a detailed understanding of the entrainment and mixing mechanisms within the well-mixed layer; it was shown in the previous section that the details of this mixing are somewhat elusive. We are strongly attracted by the empirical correlation which yielded the result that the internal Froude number of the bottom boundary layer, F
%
1.7.
What kind of mechan-
ism has an internal Froude number with a limiting critical value given by F
%
1.7?
We suggest that the mechanism may be the instability of the mean flow, in the well-mixed layer, to the formation of intermittent surges or roll waves. Such roll waves have been known for many years to occur in open channel flows on supercritical slopes; a classic picture of them can be found in the book of Cornish (1934).
[See also Dressler (1949) and Stoker (1957,
161 p. 465) f o r t h e same photograph.]
R o l l waves o r i n t e r m i t t e n t s u r g e s form
when t h e Froude number of t h e flow is s u f f i c i e n t l y s u p e r c r i t i c a l t h a t t h e b a l a n c e between s l o p e , o r p r e s s u r e g r a d i e n t , and f r i c t i o n i s no l o n g e r s t a b l e ; t h e c r i t i c a l Froude number f o r f o r m a t i o n i s
F = 2
when t h e Chezy
i f t h e Manning formula i s used ( c . f .
r e s i s t a n c e law is assumed and
F = 1.5
L i g h t h i l l and Whitham, 1955).
Analyses of r o l l waves can a l s o b e found i n
many t e x t s , f o r example S t o k e r (1957, p. 466) and Whitham (1974, p. 85). A l l of t h e a n a l y s e s c l o s e l y f o l l o w t h e o r i g i n a l of D r e s s l e r (1949, 1952). W e t e n t a t i v e l y p i c t u r e t h e growth of t h e bottom well-mixed
ceeding as f o l l o w s :
r e g i o n pro-
The l a y e r f i r s t grows t o t h e Ekman l a y e r h e i g h t by
normal t u r b u l e n t mixing g e n e r a t e d a t t h e bottom. b e r of t h e well-mixed
The i n t e r n a l Froude num-
l a y e r i s g i v e n , u s i n g ( 3 ) , (4) and (5) by F =
f i g N
h '
A t t h e p e n e t r a t i o n h e i g h t of t h e Ekman l a y e r ,
i s s m a l l , approximately
h
one s i x t h of t h e f i n a l p e n e t r a t i o n h e i g h t observed; y e t above t h e Ekman l a y e r h e i g h t t h e v e l o c i t y must always approach t h e g e o s t r o p h i c v e l o c i t y . T h e r e f o r e , t h e i n t e r n a l Froude number i s l a r g e ,
F
10.
A s l o n g as t h e
i n t e r n a l Froude number of t h e l a y e r i s l a r g e r t h a n t h e c r i t i c a l v a l u e f o r t h e f o r m a t i o n of r o l l waves, t h e s e i n t e r m i t t e n t s u r g e s o r b o r e s form.
The
b o r e s have mixing a s s o c i a t e d w i t h them which c o n t i n u e s u n t i l t h e mixed l a y e r i s deepened s u f f i c i e n t l y t h a t t h e i n t e r n a l Froude number i s j u s t less t h a n t h e c r i t i c a l v a l u e f o r t h e formation of t h e r o l l waves. Although t h e i n s t a b i l i t y of t h e mean f l o w t o t h e f o r m a t i o n of r o l l waves
is a c a n d i d a t e f o r t h e mechanism which c o n t r o l s t h e u n u s u a l l y l a r g e penet r a t i o n h e i g h t of t h e bottom boundary l a y e r , t h e i n s t a b i l i t y does n o t prov i d e an e x p l a n a t i o n f o r how a once-formed l a y e r may d e c r e a s e i n h e i g h t when t h e geostrophic v e l o c i t y gradually decreases.
The mechanism of r o l l wave
f o r m a t i o n o n l y p r o v i d e s an e x p l a n a t i o n as t h e v e l o c i t y i n c r e a s e s o r remains c o n s t a n t .
W e n o t e however t h a t any i n t e r f a c e formed a t t h e bottom
of t h e ocean w i l l b e a r e g i o n of r e l a t i v e l y h i g h Brunt-Vaisalg I t i s a t such r e g i o n s of h i g h r e l a t i v e Brunt-Vaisala
frequency.
frequency t h a t i n t e r n a l
waves w i l l break; c . f . Turner (1973, p. 120) f o r a d i s c u s s i o n of s h e a r i n s t a b i l i t y produced by i n t e r n a l waves a t i n t e r f a c e s .
The b r e a k i n g of
i n t e r n a l waves a t t h e i n t e r f a c e i s an e r o d i n g mechanism which may always be present.
162 CONCLUSION We have reviewed-anumber of existing models for the bottom boundary layer and find that none predict the large penetration height of the wellmixed region. This region extends about six times the height of what is considered to be a typical turbulent Ekman layer height.
Because of the
large penetration height, differential advection between the mixed layer and the water immediately above must be small; the layer is, we believe, advected over the flat Hatteras Abyssal Plain with the mesoscale motions. We have outlined how one might model mixing and entrainment in the homogeneous layer; unfortunately we find that too little is known about the structure of turbulence above a turbulent Ekman layer for us to complete such a model.
The correlation of penetration height with velocity has
allowed us to estimate, knowing the Brunt-Vgisalg frequency of the stratification which was mixed to form the bottom layer, the value of the internal Froude number of this layer: F
2 1.7.
This value suggests that the pene-
tration height may be controlled by the instability of the mean flow, in the bottom mixed layer, to the formation of roll waves or intermittent surges. ACKNOWLEDGMENTS
I am very grateful for financial support as a postdoctoral scholar from the Woods Hole Oceanographic Institution, and for the assistance of the W.H.O.I.
Buoy Group with the collection and presentation of these results.
The data used was collected with support from the Office of Naval Research under contract NO00 14-66-CO241 NR 083-004 and from the International Decade of Ocean Exploration Office of the National Science Foundation
(GX 29054).
My sincere thanks to the University of Lisge for making available to me a travel grant to participate in the Eighth International Lisge Colloquium on Ocean Hydrodynamics. REFERENCES Armi, L. and R. C. Millard, Jr., 1976. The bottom boundary layer of the deep ocean. J. of Geophysical Res., 81,27, 4983-4990. Biscaye, P. E. and S. L. Eittreim, 1974. Variations in benthic boundary layer phenomena: Nepheloid layer in the North Atlantic Basin, In: Suspended solids in water, R. J. Gibbs, Ed., Plenum Publ. Co., N.Y., 227-260.
163 Caldwell, D. R., L. W. Van Atta and K. N. Helland, 1972. A laboratory study of the turbulent Ekman layer, Geophys. Fluid Dyn., 3, 125-160. Corasoli, C. P., 1975. Free shear layer instability due to probes in rotating source-sink flows. J. Fluid Mech., 72, 559-586. Cornish, V., 1934. Ocean Waves and Kindred Geophysical Phenomena, Cambridge University Press. Cromwell, T., 1960. Pycnoclines created by mixing in an aquarium tank. J. Mar. Res., 18, 73-82. Csanady, G. T., 1967. On the "Resistance Law" of a turbulent Ekman layer. J. Atm. Sciences, 24, 467-471. Csanady, G. T., 1974. Equilibrium theory of the planetary boundary layer with an inversion lid. Boundary-Layer Met., 6, 63-79. Dressler, R. F., 1949. Mathematical solution of the problem of roll-waves in inclined open channels, Corn. on Pure and Applied Math., 2, 149-194. Dressler, R. F., 1952. Stability of uniform flow and roll-wave formation. U. S. Nat. Bur. of Standards, NBS Circular 521, Gravity Waves, 237-241. Howroyd, G. C. and P. R. Slawson, 1975. The characteristics of a laboratory produced turbulent Ekman layer. Boundary-Layer Met., 8, 201-219. Ingran, R. G., 1971. Experiments in a rotating source-sink annulus, Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution Rep. 71-001N (unpublished manuscript). Kato, H. and 0. M. Phillips, 1969. On the penetration of a turbulent layer into a stratified fluid. J. Fluid Mech., 37, 643-55. Lighthill, M. J. and G. B. Whitham, 1955. On kinematic waves I. Flood movement in long rivers. Proc. Roy. SOC. of London, A229, 281-316. Linden, P. F., 1975. The deepening of a mixed layer in a stratified fluid. J. Fluid Mech., 71, 385-405. Munk, W., F. Snodgrass and M. Wimbush, 1970. Tides off-shore: Transition from California coastal to deep-sea waters. Geophys. Fluid Dyn., 1, 161-235. Pollard, R. T., P. B. Rhines and R. Thompson, 1973. The deepening of the wind-mixed layer. Geophys. Fluid Dyn., 3, 381-404. Stoker, J. J., 1957. Water Waves, Interscience Publishers, Inc., New York. Tatro, P. R. and E. L. Mollo-Christensen, 1967. instability. J. Fluid Mech., 28, 531-543. Thompson, R., 1973.
e., 5, 201-210. Turner, J.
S.,
1973.
Experiments on Ekman layer
Stratified Ekman boundary layer models.
Geophys Fluid
Buoyancy effects in fluids, Cambridge University Press.
164
Weatherly, G. L., 1972. A study of the bottom boundary layer of the Florida Current. J. Phys. Oceanogr., 2, 54-72. Weatherly, G. L., 1375. A numerical study of time-dependent turbulent Ekman layers over horizontal and sloping bottoms. J. Phys. Ocean., 5, 288-299. Whitham, G. B., 1974. Linear and Nonlinear Waves, Wiley-Interscience, New York. Wimbush, M. and W. Munk, 1971. The benthic boundary layer,.The Sea, Vol. 4, Part 1, 731-758, Wiley, New York.
165 I4ASS U R I V E l l FLUCTUATIOIJS WITHIN THE EKIAN BOUiJOARY LAL'LX"
WILLIAM 0. C R I I I I N A L E , J R .
D e p a r t m e n t of O c e a n o g r a p h y , G e o p h y s i c s P r o g r a m , and A p p l i e d H a t h e m a t i c s Group, U n i v e r s i t y o f W a s h i n g t o n , S e a t t l e , WA
93195
A t h e o r e t i c a l m o d e l t o d e s c r i b e t h e f l u c t u a t i o n s w i t h i n t h e s u r f a c e Ek-
man b o u n d a r y l a y e r b e l o w t h e A r c t i c i c e c o v e r h a s b e e n f o r m u l a t e d w i t h t h e n o v e l f e a t u r e b e i n g a s o u r c e o f mass a t t h e i c e boundary.
The f u l l p h y s i c s
of t h i s p r o b l e m i s d e v e l o p e d t o g e t h e r w i t h n u m e r i c a l i n t e g r a t i o n of t h e governing e q u a t i o n s t h a t are used i n o r d e r t o e x p l o r e s a l i e n t c o r r e l a t i o n s and t h e s t r u c t u r e of t h e s u r f a c e l a y e r .
The s u r f a c e l a y e r p e r se i s con-
s i d e r e d w e l l mixed a n d bounded b e l o w by a s t r o n g p y c n o c l i n e .
Using l i n e a r
t h e o r y , i t i s shown t h a t f o r c e d u n s t e a d y m o t i o n s c a n b e s i g n i f i c a n t f o r modest v a l u e s o f t h e mass f l u x a n d p r e s e n t d i f f e r e n t c h a r a c t e r i s t i c s t h a n would h a v e b e e n p o s s i b l e i f t h e b o u n d a r y h a d b e e n s i m p l y p a s s i v e .
The con-
c l u s i o n i s t h a t t h e c o m p l e t e p e r t u r b a t i o n f i e l d i n s u c h a r e g i o n (and i n d e e d a n a l o g o u s l a y e r s w h e r e a c t i v e b o u n d a r y c o n d i t i o n s c a n b e i n v o k e d ) must b e d u e t o a c o m b i n a t i o n t h a t i s made up of b o t h t h e f o r c e d and t h e more convextional
self-excited
oscillations.
INTRODUCTION
In t h e i c e - c o v e r e d
area o f t h e A r c t i c Ocean t h e i n s i t u m e a s u r e m e n t s
made by S m i t h d u r i n g t h e 1 9 7 0 , 1 9 7 1 , a n d 1 9 7 2 A I D J E X p i l o t s t u d i e s ( S m i t h , 1974) a r e r e m a r k a b l e i n o c e a n i c r e s e a r c h and have t h e l u x u r y o f h a v i n g been done from a s t a b l e p l a t f o r m .
Data h a v e b e e n o b t a i n e d t h a t p r o v i d e a knowl-
a d g e o f b o t h t h e f l u c t u a t i n g a n d mean f i e l d s t h r o u g h o u t t h e s u r f a c e bounda r y l a y e r (and somewhat b e y o n d ) t h a t i s b e n e a t h t h e i c e c o v e r d u r i n g t h e s p r i n g t i m e of t h e y e a r .
The q u a l i t y o f t h e e x p e r i m e n t s i s e x t r e m e l y h i g h
and sets a s t a n d a r d f o r a n y a t t e m p t t o a s s e s s t h e p h y s i c s o f s u c h m o t i o n a s it occurs i n nature.
O v e r a l l , i n terns o f t h e mean v e l o c i t y , S m i t h h a s c o n f i r m e d t h e p r e s ence o f a moderately well-developed
Ekman s p i r a l , a t l e a s t d u r i n g c e r t a i n
" C o n t r i b u t i o n No. 935 from t h e D e p a r t m e n t o f O c e a n o g r a p h y
periods.
For a s p i r a l t o e x i s t , i t w a s n e c e s s a r y f o r t h e wind above t h e
i c e t o be a c t i v e enough i n o r d e r f o r s u f f i c i e n t s h e a r stress t o b e t r a n s mitted t o t h e water'below.
U n l i k e t h e well-known Ekman r e s u l t , however,
t h e s p i r a l t h a t was found was d i s t o r t e d and spanned o n l y t h e r e g i o n between t h e 4 m and 54 m l e v e l s , r e s p e c t i v e l y , w i t h t h e f l o w d e c i d e d l y t u r b u l e n t . The mean d e n s i t y w a s found t o be made up of two p r i n c i p a l p o r t i o n s , namely a mixed l a y e r t h a t extended from t h e i c e cover down t o a d e p t h o f 4 0 m,
r o u g h l y s p e a k i n g , and, below t h i s l e v e l , a well-developed p y c n o c l i n e . The o t h e r s a l i e n t f e a t u r e o f t h e measurements w a s t h e f i n d i n g s f o r t h e d i s t r i b u t i o n of t h e f l u c t u a t i o n f i e l d w i t h i n t h e s u r f a c e l a y e r .
In the
n e a r i c e r e g i o n , t h a t p o r t i o n of t h e boundary l a y e r from t h e i c e down t o a p p r o x i m a t e l y 4 m, t h e t u r b u l e n c e i n t e n s i t y was of t h e o r d e r o f 10% of t h e mean c u r r e n t .
I4oving f u r t h e r downwards, t h e i n t e n s i t y dropped t o a v a l u e
of 6% i n t h e upper 1 5 t o 2 0 m of t h e mixed l a y e r .
Ultimately, t h e value
d e c r e a s e d t o 3X i n t h e lower p a r t of t h e mixed l a y e r and t h e n remained cons t a n t throughout t h e pycnocline. Because of t h e methods employed by Smith i n t h e c o l l e c t i o n of t h e d a t a , a l l components of t h e f l u c t u a t i n g stress t e n s o r were d e t e r m i n e d .
complete structed.
a
Hence, as
p i c t u r e as t h a t found i n some l a b o r a t o r y c o n d i t i o n s can b e con-
The s t r u c t u r e found, though, i s d i f f e r e n t from t h a t o f any o t h e r
t u r b u l e n t measurements and cannot b e simply e x p l a i n e d by t h e l o g i c t h a t i s a t t r i b u t e d t o b a s e s t h a t a r e consequences o f t h e s t a t e of t h e knowledge. For example, b e s i d e s t h e r a t h e r weak f a l l o f f n o t e d f o r t h e i n t e n s i t y ,
stress components were found t o change from p o s i t i v e t o n e g a t i v e i n s i g n a t o t h e r l o c a t i o n s t h a n t h o s e where z e r o e s of t h e mean v e l o c i t y g r a d i e n t
o c c u r , making t h e common eddy t r a n s f e r h y p o t h e s i s a l t o g e t h e r unuseable. The s t a t e of t h e a r t of u n d e r t s a n d i n g t u r b u l e n t f l o w i s r e s t r i c t e d t o a m o d e s t u n r a v e l l i n g of t h e s i m p l e s t kind of f l o w t h a t c a n b e r e s e a r c h e d , namely i n c o m p r e s s i b l e shear-flow
turbulence.
Almost u n e q u i v o c a l l y t h e ma-
j o r p o r t i o n of t h e f a c t s r e v o l v e s around l a b o r a t o r y - a n a l y t i c a l g e n e r a t i o n u s i n g s u c h a flow a s a b a s i s t o g e t h e r w i t h t h e more s p e c i a l s u b problem of i s o t r o p i c turbulence.
Beyond t h i s p o i n t , i t becomes s p e c u l a t i o n .
The
kind of s i t u a t i o n s t h a t e x i s t i n n a t u r e i n c l u d e body f o r c e s and even t h e e x t e n s i o n of t h e few means t h a t are a v a i l a b l e are n o t r e a l l y v a l i d , s t r i c t l y s p e a k i n g , f o r t h i s kind of environment.
It is not s u r p r i s i n g , then,
t h a t ariy a t t e m p t t o f o r c e t h e problem i n t o an a l r e a d y f i x e d framework cann o t be b e n e f i c i a l and, i n s t e a d , n o v e l m o d e l l i n g t h a t i s c o m p a t i b l e w i t h t h e p a r t i c u l a r s i t u a t i o n must b e sought. Turbulence r e s e a r c h h a s u t i l i z e d t h r e e main modes of i n v e s t i g a t i o n .
In e s s e n c e t h e means a r e complementary and i d e a l l y s h o u l d be combined f o r meaningful r e s e a r c h i n t h i s a r e a .
F i r s t , t h e f u l l problem i s s t r i p p e d of
a l l b u t t h e v e r y e s s e n t i a l p h y s i c s and t h e n s o l u t i o n i s s o u g h t .
Homogene-
o u s i s o t r o p i c t u r b u l e n c e i n a n i n c o m p r e s s i b l e f l u i d a t h i g h Reynolds num-
b e r s w i t h no e x t e r n a l i n p u t i s t h e p r o t o t y p e of t h i s approach b u t h a s n o t r e s u l t e d i n any major advancement. r e c t l y w i t h t h e Navier-Stokes
Second, t h e computer i s used e i t h e r d i -
e q u a t i o n s o r i n d i r e c t l y a s a t o o l f o r making
t h e many and cumbersome c a l c u l a t i o n s t h a t are needed i n a n a l y s i s and e x p e r i -
In a d d i t i o n , t h e machine p l a y s a s i g n i f i c a n t r o l e as a n u m e r i c a l s i m -
ment.
u l a t o r of t u r b u l e n t - l i k e problems.
In p r i n c i p l e , more of t h e n a t u r a l com-
p l i c a t i o n s a r e p e r m i t t e d i n t h i s manner b u t t h e s u c c e s s h a s been e q u a l l y s h o r t due t o i t s own l i m i t a t i o n s . The t h i r d means of e x p l o r a t i o n d e a l s w i t h t h e g r o s s f e a t u r e s of many k i n d s o f p h y s i c a l c o m p l i c a t i o n s by t h e u s e of f i r s t - o r d e r p e r t u r b a t i o n theory.
B a s i c a l l y , i t i s assumed t h a t a r e l a t i v e body of knowledge e x i s t s f o r
t h e s i m p l e s t problem ( r e s t r i c t e d , o p t i m i s t i c a l l y s p e a k i n g , t o s h e a r flow t u r b u l e n c e in a n i n c o m p r e s s i b i e medium w i t h no e x t e r n a l d r i v i n g a t b e s t ) and t h e n a s k s two s a l i e n t q u e s t i o n s of t h e a n a l y s i s : ( 1 ) In what way w i l l well-established
f e a t u r e s b e modified o r t h e known s o l u t i o n s , i f t h e r e a r e
any, b e a l t e r e d by c e r t a i n c o m p l e x i t i e s ?
( 2 ) What e n t i r e l y new o r a d d i -
t i o n a l phenomena w i l l b e p r e s e n t t h a t are due t o t h e a d d i t i o n a l p r o p e r t i e s of t h e f l o w b e i n g c o n s i d e r e d ?
Although n o t t r u l y t u r b u l e n t by d e f i n i t i o n
t h i s s t r a t e g y does c o v e r more examples and h a s t h e c a p a b i l i t y of l e a d i n g t o physical insight.
More i m p o r t a n t l y , o t h e r i n t e r e s t i n g phenomena do n o t
tend t o b e r u l e d o u t because a l t e r n a t i v e s have n o t been e v a l u a t e d i n a cons i s t e n t manner b e f o r e an a p r i o r i c o n j e c t u r e of t u r b u l e n c e dominates. A c r i t i c a l p a r a m e t e r i n t h e s t u d y of t h e s t a b i l i t y of a s t r a t i f i e d
s h e a r f l o w ( o r f o r t h e maintenance of t u r b u l e n c e i n a s t r a t i f i e d f l u i d ) i s t h e Richardson number.
T h i s number i s a measure of t h e r e l a t i v e s t r e n g t h s
of t h e buoyancy e f f e c t s d u e t o t h e s t r a t i f i c a t i o n t h a t i n h i b i t s d e s t a b i l i z a t i o n and t h e s h e a r e f f e c t s t h a t promote i n s t a b i l i t y .
M i l e s (1961) and
Howard (1961) have shown t h a t s u f f i c i e n t c o n d i t i o n f o r s t a b i l i t y i n a s t r a t i f i e d s h e a r f l o w i s t h a t t h e l o c a l Richardson number be everywhere g r e a t e r than 1/4.
Even though t h i s c r i t e r i o n was d e t e r m i n e d by c o n s i d e r i n g a l i n -
e a r system, i t h a s been both a d e q u a t e and u s e f u l when d e a l i n g w i t h t h e f u l l problem.
An i n s p e c t i o n of t h e p r o f i l e s from S m i t h ' s work shows t h a t , with-
i n t h e lower p a r t of t h e s u r f a c e Ekman l a y e r , t h e Richardson number is less then 1 / 4 i n d i c a t i n g t h a t t h e t u r b u l e n c e i s p r o b a b l y s h e a r induced w i t h mod-
168 e r a t i o n by p r e s s u r e g e n e r a t e d by i c e topography i n t h e v e r y n e a r i c e p a r t of t h e flow.
Indeed, t h e mean p r o f i l e i s l o g a r i t h m i c and t h e r e i s no s p i -
r a l i n g ; t h e v a l u e s - o f t h e t u r b u l e n c e found a r e comparable t o s t a n d a r d f l a t p l a t e rlieasurements.
I n t h e d e e p e r p a r t of t h e niixed l a y e r t h e Richardson
number i s much g r e a t e r t h a n t h e c r i t i c a l v a l u e w h i l e t h e t u r b u l e n t s t r e s s e s a r e s t i l l q u i t e r e a s o n a b l e ; 6% as compared t o
lo%, w i t h
no r a p i d d e c r e a s e .
Such v a l u e s a r e n o t c o m p a t i b l e w i t h t h e s i m p l e i n s t a b i l i t y c r i t e r i o n and r e q u i r e a n o t h e r e x p l a n a t i o n f o r t h e s o u r c e of t h e f l u c t u a t i o n s . A c l o s e r i n s p e c t i o n of t h e Smith d a t a h e l p s t o c o n s t r u c t an h y p o t h e s i s
f o r t u r b u l e n c e g e n e r a t i o n under t h e c o n d i t i o n s t h a t a r e d i c t a t e d by t h e situation.
The c o n j e c t u r e i s t h a t t h e g e n e r a t i o n i s due t o b r i n e - d r i v e n
c o n v e c t i o n , an a c t i o n t h a t i s q u i t e a c t i v e i n t h e s p r i n g a t t h e t i m e t h e measurements w e r e made. w e l l and i s based on
Such a p r o p o s a l was p u t f o r t h by Smith (1974) a s
( a ) t h e b e h a v i o r of t h e mean d e n s i t y p r o f i l e s where
d u r i n g m e t e o r o l o g i c a l l y q u i e t p e r i o d s t h e o c c u r r e n c e of v a r i a t i o n s penet r a t e t o g r e a t e r d e p t h s than when t h e r e i s i n c r e a s e d a t m o s p h e r i c a c t i v i t y above t h e i c e ; and
(b) t h e j e t - l i k e
s t r u c t u r e of t h e mean v e l o c i t y t h a t
c o r r e s p o n d s t o a r e p l e n i s h m e n t of f r e s h e r water from t h e s i d e s a t c e r t a i n d e p t h s a f t e r t h e b r i n e h a s f a l l e n v e r t i c a l l y t o a d i f f e r e n t l e v e l below. I n t e r m s of t h e f l u c t u a t i o n s , t h e c o n v e c t i v e a c t i v i t y c r e a t e s a downwards f l u x of h o r i z o n t a l momentum t h a t is d r i v e n by b r i n e plumes.
The plumes a r e
g e n e r a t e d a t random l o c a t i o n s and t i m e a t t h e i c e c o v e r by t h e f r e e z i n g of sea water.
T h i s c o n c e p t i s d i f f e r e n t from t h a t of t h e c l a s s i c a l s t r a t i f i e d
( o r n o n - s t r a t i f i e d ) s h e a r f l o w where t h e b o u n d a r i e s a r e o n l y p a s s i v e . The g e n e r a l problem h a s two major n o n l i n e a r i t i e s : e q u a t i o n s and cluded.
(a) t h e governing
(b) t h e boundary c o n d i t i o n s , i f t h e topography i s t o be in-
S o l u t i o n of t h e combination i s beyond t h e p o s s i b i l i t y of c u r r e n t
t e c h n i q u e s and i s n o t n e c e s s a r i l y t h e mnst d e s i r a b l e r o u t e t o f o l l o w even i f t h i s were n o t t h e c a s e .
I n s t e a d , a model i s s y n t h e s i z e d t h a t is t r a c t -
a b l e b u t s t i l l r e t a i n s t h e f e a t u r e s t h a t are b e l i e v e d t o be of prime importance.
For t h i s p u r p o s e , t h e s p e c u l a t i o n t h a t c r i t i c a l p h y s i c s a r e con-
n e c t e d w i t h t h e b r i n e s o u r c e a t t h e i c e cover i s c e n t r a l t o t h e i s s u e and the construction. The immediate and c o n v e n i e n t consequences of t h e h y p o t h e s i s i s t h e f a c i l i t y t h a t t h e i c e topography c a n b e ignored.*
I n f a c t t h a t t h e s i s can
* I t s h o u l d he n o t e d t h a t Smith h a s r e p o r t e d t h a t , i n c o n t r a s t t o t h e mean flow, t h e t x b u l e n c e f i e l d i s l e s s s e n s i t i v e t o t o p o g r a p h i c d i s t u r b a n c e s .
169 be i n i t i a l l y t e s t e d u s i n g t h e l a m i n a r Ekman l a y e r as t h e mean d i s t r i b u t i o n for the velocity.
S i n c e t h e f i e l d d a t a have a l s o shown t h a t t h e mean zone
o f s h e a r i s w e l l w i t h i n t h e mixed l a y e r where t h e f l u i d i s o n l y weakly s t r a t i f i e d , i. e .
n e u t r a l , t h e v e l o c i t y can b e t h e e x a c t s o l u t i o n coupled be-
low by c o n s t a n t f l o w i n t h e r e g i o n o f t h e p y c n o c l i n e . The second n o n l i n e a r i t y is more fundamental and i s i m p o s s i b l e t o circumv e n t u n l e s s t h e system is l i n e a r i z e d .
T h i s s t e p i s made by p e r t u r b i n g t h e
mean f i e l d and t h e n l i n e a r i z i n g i n terms of t h e f l u c t u a t i o n s .
An a s s e s s m e n t
of b r i n e - d r i v e n c o n v e c t i o n is c l e a r l y w i t h i n t h e realm of t h e model. C l a s s i c a l l i n e a r s t a b i l i t y t h e o r y i s concerned w i t h i n s t a b i l i t i e s i n l a m i n a r f l o w s ( c f . Betchov and C r i m i n a l e , 1967, f o r a n in-depth r e v i e w of t h e s u b j e c t ) and i s i m p o r t a n t i n t h e u n d e r s t a n d i n g of t h e p h y s i c s of why a l a m i n a r f l o w c a n n o t be m a i n t a i n e d as a f u n c t i o n of c e r t a i n p a r a m e t e r s .
It
a l s o h e l p s i n t r a c i n g t h e o r i g i n s of t u r b u l e n c e i n c e r t a i n k i n d s of f l o w s , f o r t h e l i n e a r and t h e n o n l i n e a r problems have much i n common when t h e y are examined i n terms of energy arguments.
Extending t h e s t r a t e g y t o f o r c e d
flows is not a t a l l unnatural, p a r t i c u l a r l y i f t h e physics warrants t h i s treatuient.
I t i s i m p o r t a n t t o now n o t e t h a t s t a b i l i t y i s no l o n g e r an is-
sue when o s c i l l a t i o n s a r e f o r c e d b e c a u s e t h e b o u n d a r i e s can now p r o v i d e a n i n f i n i t e s o u r c e of energy.
And, even though t h e r e i s no exchange of energy
between components i n l i n e a r i z e d e q u a t i o n s , i t is p o s s i b l e through a c t i v e boundary c o n d i t i o n s . L i l l y (1966) h a s made t h e n e c e s s a r y c a l c u l a t i o n s f o r t h e s t a b i l i t y of t h e n o n s t r a t i f i e d l a m i n a r Ekman l a y e r and found, c o n t r a r y t o a B l a s i u s l a y -
e r , t h e r e a r e t h r e e p o s s i b l e modes of i n s t a b i l i t y .
Brown (1970) has extend-
ed t h e a n a l y s i s t o t h e t u r b u l e n t Ekman problem* ( w i t h a l a t e r c o n s i d e r a t i o n of t h e r a m i f i c a t i o n s of n o n l i n e a r i t y as w e l l ) under t h e same c o n d i t i o n s . The p r e s e n t a n a l y s i s is similar t o t h e s e works o n l y from t h e b a s i s of l i n earity.
Because t h e r e must b e a s o u r c e o f b r i n e a t t h e i c e c o v e r , t h e prob-
l e m is n o 4 o n g e r one o f t h e e i g e n v a l u e v a r i e t y .
With inhomogeneous boundary
c o n d i t i o n s , a n i m p o r t a n t p a r t i c u l a r s o l u t i o n i s i n t r o d u c e d and i s d r i v e n by a s o u r c e t h a t i s p r e s c r i b e d g p r i o r i w i t h t h e s p e c i f i c a t i o n of t h e mass f l u x
a s a known f u n c t i o n of t h e v a r i a b l e s i n t h e p l a n e of t h e i c e cover and t i m e . F i n a l l y , t h e manner i n which t h e g o v e r n i n g e q u a t i o n s are d e r i v e d r e t a i n s t h e complete t h r e e - d i m e n s i o n a l i t y .
B e s i d e s n o n l i n e a r i t y , three-dimension-
*Although t h e a c t u a l r q r e s e n t a t i o n of t u r b u l e n c e h e r e w a s o n l y token i n t h a t t h e l a m i n a r s o l u t i o , . w a s combined w i t h a n eddy v i s c o s i t y . The r e s u l t s of s u c h a combination have q u e s t i o n a b l e v a l u e .
170 a l i t y h a s always been one of t h e major c h a r a c t e r i s t i c s of t u r b u l e n c e t h a t i s e s s e n t i a l t o t h e mechanics o f t h e f l o w , e s p e c i a l l y i f c o r r e l a t i o n s are
t o be evaluated. The s t r a t i f i e d Ekman boundary l a y e r has been c o n s i d e r e d f o r i n s t a b i l i t y , w i t h t h e p a p e r s of Brown (1972) and Kaylor and F a l l e r ( 1 9 7 2 ) b e i n g r e p r e sentative.
For t h e s u r f a c e boundary l a y e r under t h e i c e c o v e r , however, i t
i s f o r t u n a t e t h a t s t r a t i f i c a t i o n i s n o t s t r o n g o v e r t h e e n t i r e d e p t h and
c o n s e q u e n t l y u s e of t h i s work i s o n l y m a r g i n a l l y r e l e v a n t .
On t h e o t h e r
hand, t h e r e is a n i n t e r f a c e i n t h e mean d e n s i t y f i e l d where t h e mixed l a y e r The p h y s i c s
j o i n s onto t h e pycnocline, causing a l a r g e density gradient.* connected w i t h t h e a d d i t i o n of t h e g r a v i t y f o r c e and s u b s e q u e n t wave g e n e r a t i o n c a n b e e x p e c t e d .
internal
The Smith measurements of t h e f l u c t u a t i n g
stresses e x t e n d w e l l beyond t h e d e p t h f o r t h e change i n t h e mean d e n s i t y and no doubt a l r e a d y b e a r some i m p r i n t r e l a t i n g t o t h i s t r a n s i t i o n .
In
terms of t h e l a m i n a r model, t h e r e l a t i v e d e p t h of t h e p y c n o c l i n e l o c a t i o n can be c r i t i c a l t o t h e p h y s i c s . S o l u t i o n s f o r t h e l i n e a r e q u a t i o n s t h a t govern t h e o u t l i n e d s y s t e m can be o b t a i n e d f o r a l l t h e dependent v a r i a b l e s ( v e l o c i t y , d e n s i t y , p r e s s u r e ) by F o u r i e r decomposing i n terms of t h e s p a c e v a r i a b l e s of t h e p l a n e of t h e i c e and timeand t h e n i n t e g r a t i n g t h e s e t o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s t h a t d e t e r m i n e t h e a m p l i t u d e s as a f u n c t i o n of t h e d e p t h .
Correlations
(energy, stress, buoyancy, f l u x ) o f t h e s e v a r i o u s q u a n t i t i e s c a n be c a l c u l a t e d by a v e r a g i n g t h e v a r i o u s p r o d u c t s .
These o p e r a t i o n s d e t e r m i n e t h e
s t r u c t u r e a s a f u n c t i o n of t h e p a r a m e t e r s t h a t o c c u r n a t u r a l l y ( o r a s i n p u t s ) w i t h i n t h e s u r f a c e boundary l a y e r a s a f u n c t i o n of t h e v e r t i c a l coordinate.
Among t h e l i s t of p o s s i b i l i t i e s , t h e p r i n c i p a l p a r a m e t e r s d e a l t
w i t h are t h e Reynolds and Richardson n u n b e r s t o g e t h e r w i t h an a n g l e t h a t d e s c r i b e s t h e t h r e e - d i m e n s i o n a l i t y a l o n g w i i h f r e q u e n c y and s c a l e of t h e oscillations. Thermodynamics are n o t e x p l i c i t l y i n c l u d e d i n t h e model.
It i s through
s u c h a c t i o n a t t h e i c e cover t h a t e x p l a i n s t h e s o u r c e of t h e b r i n e , b u t t h e d e t a i l s a r e n o t r e l e v a n t and a mass f l u x i s t a k e n as known. t h e problems uncouple.
In o t h e r words,
In a d d i t i o n , t h e f l u i d i s assumed i n c o m p r e s s i b l e i n
t h e Boussi:iesq s e n s e , removing any f u r t h e r need f o r c o n s i d e r a t i o n of temper-
*The l o c a t i o n of t h i s v a r i a t i o n i s n o m i n a l l y below t h e r e g i o n of s h e a r of t h e t u r b u l e n t boundary l a y e r , i.e. where t h e mean v e l o c i t y i s a p p r o x i m a t e l y c o n s t a n t e x c e p t f o r p o s s i b l y some remnant of t h e s p i r a l ; t h e Elanan d e p t h i s more o r l e s s t h e same a s t h a t of t h e mixed l a y e r .
171 ature.
D i f f u s i o n o f t h e d e n s i t y by e i t h e r m o l e c u l a r o r t u r b u l e n t (eddy d i f -
f u s i v i t y ) means, i s n o t e s s e n t i a l and can be n e g l e c t e d s i n c e i t i s t h e adv e c t i v e t r a n s p o r t o f t h e mass t h a t l e a d s t o a downwards d i f f u s i o n of t h e T h i s s t e p i s c o n s i s t e n t w i t h t h e h y p o t h e s i s of t h e model.
momentum.
It
w i l l a l s o become c l e a r t h a t t h i s o p e r a t i o n h a s a f u r t h e r a d v a n t a g e i n t h a t i t s i m p l i f i e s t h e study.
I t i s t r u e t h a t t h e f i e l d measurements have l i m i t a t i o n s .
For example,
i t w a s impossible t o obtain absolutely t r u e steady-state values f o r t h e
mean v e l o c i t y and d e n s i t y b e c a u s e , even w i t h t h e s e v e r a l weeks spent i n t h e f i e l d , u n s t e a d i n e s s was s t i l l p r e v a l e n t .
Large and moderate-scale
convec-
t i v e a c t i v i t y i s p r e s e n t i n t h i s p a r t of t h e world and is a c a u s e of some of t h e undue v a r i a t i o n s t h a t are s e e n i n t h e d e n s i t y p r o f i l e . r e n c e of l e a d s caused h o r i z o n t a l motion.
Then, occur-
On t h e o t h e r hand, t h e measure-
ments were made d u r i n g a p p r o x i m a t e l y t h e same t i m e of y e a r f o r t h r e e succ e s s i v e y e a r s and s u b s t a n t i a l l y t h e same r e s u l t s were o b t a i n e d and t h e t i m e s c a l e s f o r t h e dynamic c h a r a c t e r i s t i c s of t h e f l u c t u a t i o n s are s h o r t e r t h a n t h o s e f o r t h e e v o l u t i o n of t h e mean f i e l d , a l l o w i n g them t o b e i n v e s t i g a t e d separately.
I n t h i s s p i r i t t h e i n v e s t i g a t i o n i s an e x p e r i m e n t a l analogue
for theory.
PERTURBATION EQUATIONS
D e f i n e a set of a x e s such t h a t t h e z i s p o s i t i v e upwards and p a r a l l e l t o t h e f o r c e of g r a v i t y ; x and y t h e n d e f i n e t h e p l a n e of t h e i c e t h a t i s perpendicular t o z . functions
The mean v e l o c i t y and d e n s i t y a r e assumed g i v e n by t h e
U ( z ) J + T C z j J , and BIZ! r e s p e c t i v e l y .
The g o v e r n i n g e q u a t i o n s
f o r t h e f l u c t u a t i o n s are o b t a i n e d by superimposing s m a l l p e r t u r b a t i o n s o n t o t h e mean f i e l d , l i n e a r i z i n g , and t h e n s u b t r a c t i n g t h e mean f i e l d from t h e instantaneous equations. L e t u s adopt t h e following notation f o r t h e
ponents: -
p
+ p'.
(z+ u ' ,
-
V
+ v',
w'J,
( a ) v e l o c i t y v e c t o r com-
-
(b) p r e s s u r e :
A l l primed q u a n t i t i e s are f u n c t i o n s of
P
+p',
and
(c) density:
(x,y,z,t). The d e r i v a t i o n
of t h e e q u a t i o n s i s now s t r a i g h t f o r w a r d , and, i n t h e s e terms w i l l r e a d
172
where
-
0' = p'/po,
with
0 = F/po
p0
defined as a reference density and
f the rotation factor that includes the factor of 2 . proximation is explicit and
V*
The Boussinesq ap-
is the three-dimensional Laplace operator.
The set of equations (1) to (5) can be transformed into a set of ordinary differential equations as a result of the linearity and the fact that the coefficients depend only upon z . in the variables
.c,:j,
Accordingly, we can Fourier decompose
and t by assuming an expansion o f the form (or al-
ternatively, Fourier transform a l l quantities)
etc. for
I > ' w', , p',
8'.
Making the substitution, we now have
where the pressure amplitude,
=
p'
/pO.
with respect to z and A = d2/dz2- (a'
+
The primes denote derivatives y'),
the transformed Laplace opera-
tor. Solving the above equations in the full form has not been done.
In-
stead, additional simplifications have been made and then solutions are sought to the reduced system.
For example, for the nonstratified problem,
Lilly (1966) performed a rotation of the coordinates and then neglected the dependence of the dependent variables on one of the new variables.
Under
the current scheme, this is done by changing the system to an alternate problem which, except for the stratification, is no more complex than the Lilly computations. Define the transformations
173
and ( 6 ) to (10) can now be rewritten as
Elimination of the pressure between (13) and (15) and invoking (12) produces the set of three equations for 3,
;and 6 .
The transformations, as given by (ll), can be interpreted as a projection of both the mean and fluctuation velocity both perpendicular and parallel to the oblique Fourier wave front.
A s a result, equation (12) for the
incompressibility condition is aualogous to a two-dimensional problem and it is by this means that derivation of equations (17) to (19) is eased considerably.
The component i)
is more analogous to a vorticity and hence
the reason that the pressure is absent in equation (14). 2
and
5
The defining of
i s tantamount to assuming that the velocity in the
x-y plane is
decomposed into the sum of an irrotational and a solenoidal vector.
Once
these solutions for the decomposed portions are known, the velocity proper can be determined.
Introducing the additional definitions. lil
=
ac
,
$I
=
tan-l(v/a)
G
=
w
,
g
=
8 ,
(20) w
=
w
174 a f u l l y e q u i v a l e n t two-dimensional system becomes
T h i s s e t of e q u a t i o n s i s now e x p r e s s e d s o l e l y i n terms of (C,$),
the polar
wave c o o r d i n a t e s and t h e t r a n s f o r m e d dependent v a r i a b l e s . A comparison of t h e L i l l y e q u a t i o n s w i l l show a f o r m a l c o r r e s p o n d e n c e
f o r ( 2 1 ) , ( 2 2 ) w i t h s t r a t i f i c a t i o n removed.
To f i n d answers n u m e r i c a l l y ,
L i l l y chose t h e s c a l e s o t h a t t h e r a t i o of t h e Reynolds t o t h e Rossby numbers was unity, selected values for the rotation angle t i o n s is r e l a t e d t o $ a s computed
c
required
($)
+
$ = E
a s a f u n c t i o n of
b.
n/2)
i n t h e L i l l y no-
and t h e Reynolds number and t h e n
I n t h e p r e s e n t c a s e , an a n g l e i s s t i l l
a s w e l l as a Reynolds number ( t o g e t h e r w i t h a Richardson
number when t h e s t r a t i f i e d problem i s t r e a t e d ) t o f i n d 5.
(E
c
a s a f u n c t i o n of
Such computations a r e o n l y n e c e s s a r y , however, when t h e e i g e n v a l u e prob-
l e m i s t o be s o l v e d .
scales
Otherwise, j u d i c i o u s c h o i c e s of f r e q u e n c y
(a)
and
(5) must b e s e l e c t e d f o r p h y s i c a l l y meaningful r e s u l t s f o r t h e non-
e i g e n v a l u e problem.
T h i s remark i s t o be t a k e n w i t h c a u t i o n s i n c e t h e i n -
homogeneous s y s t e m c a n be s o l v e d f o r any f r e q u e n c y o r s c a l e .
The eigen-
v a l u e problem, by c o n t r a s t (and d e f i n i t i o n ) , can o n l y admit p a r t i c u l a r v a l u e s and s t i l l s a t i s f y t h e boundary c o n d i t i o n s . There are s e v e r a l ways t o s o l v e (21) t o ( 2 3 ) f o r 2 , 3, and 8 . t h e s y s t e m of t h r e e
Either
e q u a t i o n s is programmed f o r n u m e r i c a l i n t e g r a t i o n o r
a s i n g l e e q u a t i o n f o r one unknown
i n t e g r a t i o n i s performed.
--
preierably
w -- is o b t a i n e d and
then
The b e s t c h o i c e i s n o t always a p p a r e n t and de-
pends t o some d e g r e e on t h e a d r o i t n e s s of t h e programmer and t h e v a l u e s of t h e parameters t o be considered. t h e t a s k done and
For t h e moment, however, l e t u s c o n s i d e r
3 , 3 , and 6 are known.
An i n v e r s i o n of t h e t r a n s f o r m a -
t i o n s (11) and (20) p r o v i d e s t h e o r i g i n a l complex v a r i a b l e a m p l i t u d e s . Having done t h i s w e have, a f t e r e l i m i n a t i n g
u
by u s e of (12),
175 and
Li
=
D,
6 = ;
still follows. A nondimensional form of t h e e q u a t i o n s can be e s t a b l i s h e d by c h o o s i n g a
2, and a t i m e s c a l e based on t h e v e l o c i t y 40 = [E(O,’ + L e t z = I n b e t h e new c o o r d i n a t e i n t h e v e r t i c a l , K = b l the
l e n g t h scale,
7(0)2]4
.
new wave number r a d i u s , and
$ = qow,
t h e v e l o c i t y components r e s p e c t i v e l y ; dimensional.
= qoV,
(?
= qoC,
and
8 =
0 =
5
=
qoU,
P
=
qov
b a r e a l r e a d y non-
The p a r a m e t e r s can b e d e f i n e d as
Reynolds number
Richardson number
: R
e
:
=
q l 0 J
io=
3I
Rossby number
Substitution gives
F A + KEe(U BA
KRe
(u -
.Re (U - c)e Now, i f t h e r a t i o
=
i R o w .
Re//%
(29)
e n
i s s e t e q u a l t o u n i t y i n (27) and (28) we w i l l
have
Thus,
2
= (v/f)’
i n a c c o r d a n c e w i t h t h e d e f i n i t i o n of t h e Ekman d e p t h .
The i m p o r t a n t t o p i c of boundary c o n d i t i o n s must b e d i s c u s s e d .
An in-
s p e c t i o n of t h e governing e q u a t i o n s r e v e a l s t h a t t h e system i s s i x t h - o r d e r i n terms of t h e d e r i v a t i v e s w i t h r e s p e c t t o z .
Hence, we are f r e e t o d e s i g -
n a t e s i x c o n d i t i o n s i n t e r m s of t h e dependent v a r i a b l e s .
I t s h o u l d be re-
marked i n p a s s i n g t h a t t h e system would have been e i g h t h - o r d e r
i f the dif-
f u s i o n of mass had n o t been n e g l e c t e d , r e s u l t i n g i n t h e a d d i t i o n of t h e
176 Schmidt number a s a n a d d i t i o n a l parameter.
Normally, f o r a boundary l a y e r
w i t h homogeneous boundary c o n d i t i o n s , t h e r e q u i r e m e n t s a r e a t t h e s o l i d boundary ( z = 0 here],
u ' = u'
U' = 0.
p'
The p r e s s u r e
=
w'
=
0.
Far away, i . e .
and t h e d e n s i t y
z =
u ' = 0' =
-m,
are a u t o m a t i c a l l y determined
p'
through t h e v e l o c i t y c o n s t r a i n t s . P u t t i n g t h e homogenous boundary c o n d i t i o n s i n t h e t r a n s f o r m e d v a r i a b l e s of e q u a t i o n s (21) t o (23) the fact that
e
=
U
( o r 27-29)
leads t o
5
=
must v a n i s h a t b o t h l o c a t i o n s i n z.
0 w i l l a l s o come a b o u t from t h e
5
= 0 and
2'
=
0
by
I t can be s e e n t h a t
condition.
= 0
The inhomogeneous problem i s a r r i v e d a t by r e c o g n i z i n g t h a t t h e r e i s a mass f l u x a t t h e i c e c o v e r .
Taking t h e f l u x t o be
F , a known f u n c t i o n o f
( x , y , t ) , w e can w r i t e
at
z = 0.
I n F o u r i e r s p a c e t h e e q u i v a l e n t of ( 2 8 ) i s
I t i s assumed t h a t a form
can be d e t e r m i n e d , a s any o t h e r
S = S,(G,$,w)
c a s e p r e v e n t s t h e f u l l d e t e r m i n a t i o n of t h e problem i n terms of t h e p o l a r coordinates
d , $.
Using e q u a t i o n (23) i t can be s e e n t h a t (29) can b e
i n t e r p r e t e d a s a new c o n d i t i o n on must s t i l l v a n i s h a t t h e i c e c o v e r .
i)
at
z
=
0
since
3 and
ii, i . e . , 5'
O p e r a t i n g on (23) e n a b l e s u s t o w r i t e
o r , i n a s l i g h t l y d i f f e r e n t form, (31) i s
The i n f i n i t y c o n d i t i o n s a r e u n a l t e r e d , d e m o n s t r a t i n g t h a t a l l p e r t u r b a t i o n dependent v a r i a b l e s must b e p r o p o r t i o n a l t o S O .
This, then, defines t h e
p a r t i c u l a r s o l u t i o n w i t h So assuming t h e r o l e of t h e d r i v e r .
SOLUTIONS (28),
The s e t o f e q u a t i o n s (27),
and ( 2 9 ) w a s combined
LO
form one s i x t h
o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r t h e complex v e l o c i t y component
#.
T h i s e q u a t i o n w a s n u m e r i c a l l y i n t e r g r a t e d s u b j e c t t o t h e p r e s c r i b e d boundary c o n d i t i o n s t o g e t h e r w i t h t h e f o l l o w i n g p a r a m e t e r s a n d c o e f f i c i e n t s t h a t appear:
( 1 ) l e n g t h s c a l e was f i x e d i n a c c o r d a n c e w i t h t h e Ekman d e p t h d e f i n i -
t i o n , ( 2 ) t h e mean v e l o c i t y w a s t a k e n f r o m t h e e x a c t s o l u t i o n f o r a b o u n d a r y a t r e s t , ( 3 ) t h e mean d e n s i t y was a p p r o x i m a t e d by a n a l y t i c a l ‘ e x p r e s s i o n s t h a t p r o v i d e f o r a w e l l - m i x e d l a y e r f r o m t h e bou.idary down t o t h e end o f t h e s h e a r z o n e o f j u s t b e l o w a n d t h e r e a f t e r f o l l o w e d by a p y c n o c l i n e , and
( 4 ) t h e R e y n o l d s number,
Re,
2 , t h e wave number,
R i c h a r d s o n number,
K,
and
t h e a n g l e o f o b l i q u i t y ( o r s p i r a l i n g ) , 0. w e r e f i x e d and t h e n t h e complex p h a s e v e l o c i t y , C, (b) pre-selected
was ( a ) d e t e r m i n e d i f t h e e i g e n v a l u e was c o n s i d e r e d o r
f o r t h e f o r c e d problem.
Once t h i s t a s k i s c o m p l e t e , t h e
real agd i m a g i n a r y p a r t s o f t h e e i g e n f u n c t i o n
d e t e r m i n e d as a f u n c t i o n o f of e q u a t i o n (28) t o f i n d
v;
#
m d i t s d e r i v a t i v e can be
I n t u r n , t h e s e s o l u t i o n s permit i n t e g r a t i o n
1..
0
i s f o u n d d i r e c t l y fror.. ( 2 9 ) .
Finally,
t h e s e t h r e e f u n c t i o n s are s u b s t i t u t e d i n t o ( 2 4 ) t o ( 2 6 ) i n o r d e r t o e v a l u a t e t h e complex a m p l i t u d e s o f a l l t h e f l u c t u a t i n g q u a n t i t i e s w i t h r e s p e c t t o t h e o r i g i n a l f i x e d set o f c o o r d i n a t e s . Although t h e i n d i v i d u a l behavior of each F o u r i e r amplitude is i n t e r e s t i n g i t i s easier, f o r t h e p u r p o s e s o f t h i s s t u d y , t o i n t e r p r e t t h e o u t p u t i n terms o f c o r r e l a t i o n s .
More’ s p e c i f i c a l l y , t h e k i n e t i c e n e r g y i n t e n s i t y
o f e a c h component a n d t h o s e c r o s s - c o r r e l a t i o n s
t h a t lead t o energy t r a n s f e r
f r o m t h e mean m o t i o n t o t h e f l u c t u a t i n g f i e l d a r e c e n t r a l t o t h e i s s u e o f p o s s i b l e t u r b u l e n c e g e n e r a t i o n w i t h i n such a flow.
I t i s i n t h i s manner
t h a t t h e l a m i n a r f l o w c a n breakdown and u l t i m a t e l y become t u r b u l e n t o r , a t t h e o t h e r e x t r e m e , t h a t i s when t h e f l o w i s t a k e n a s f u l l y t u r b u l e n t , i t is t h e same mechanism t h a t c a n m a i n t a i n t h e m o t i o n .
The c o r r e l a t i o n s are
d e f i n e d by a v e r a g e s t h a t a r e o b t a i n e d by i n t e g r a t i n g v a r i o u s p r o d u c t s o v e r t h e x-y p l a n e a n d t i m e .
For t h e s o l u t i o n s of t h e l i n e a r e q u a t i o n s , t h e s e
c o m p u t a t i o n s a r e t h e same as a v e r a g i n g t h e F o u r i e r p r o d u c t s o v e r o n e wave l e n g t h i n t h e c. a n d y d i r e c t i o n s a n d o n e ! e r i o d i n t i m e . h a v e ( a s t e r i c k s d e n o t i n g complex c o n j u g a t e s )
Thus, w e w i l l
178
f o r t h e component i n t e n s i t i e s and
-u" = -Real[&;*] -uIw'= -Real [;d* 1 -u" = -Real [%* ] f o r t h e stress components.
402 qi
(34)
qg
The buoyancy f l u x is
For t h e most p a r t , ( 3 3 ) . ( 3 4 ) , and ( 3 5 ) p r o v i d e s u f f i c i e n t i n f o r m a t i o n by which a n a s s e s s m e n t of t h e dynamics can b e made. A s p o i n t e d o u t b e f o r e , L i l l y h a s c a l c u l a t e d t h e s t a b i l i t y of t h e l a m i n a r
non-stratified
Ekman boundary l a y e r and i t i s w i s e t o summarize h i s f i n d -
ings a t t h i s p o i n t as r e f e r e n c e bases.
The r e s u l t s f o r t h i s system show
t h a t t h e r e a r e t h r e e b a s i c ways t h a t t h e l a y e r can b e u n s t a b l e :
(1) Dynamic i n s t a b i l i t y .
The o r i g i n of t h i s t y p e of i n s t a b i l i t y i s w e l l
known from t h e Rayleigh theorem t h a t r e q u i r e s an i n f l e x i o n i n t h e mean velocity profile t o exist.
Within t h i s framework, i n f l e x i o n s a r e due t o t h e
s p i r a l i n g and can b e q u a n t i t a t i v e l y d e s c r i b e d by i n v i s c i d e q u a t i o n s .
Ig-
n o r i n g t h e s t r a t i f i c a t i o n and t a k i n g t h e l i m i t f o r i n f i n i t e Reynolds numb e r , e q u a t i o n s ( 2 7 ) , ( 2 8 ) , and ( 2 9 ) r e d u c e t o t h e uncoupled p a i r of equations,
(U and
-
C)AW
(U-C)U
- UQnt) =
i$
=
(36)
0
.
(37)
Consequently, t h e s t a b i l i t y i s c o m p l e t e l y d e t e r m i n e d by s o l v i n g ( 3 6 1 , o r t h e Rayleigh e q u a t i o n from which t h e o r i g i n a l theorem on t h e i n f l e x i o n p o i n t i s determined.
(2)
Resistive instability.
T h i s mode of o s c i l l a t i o n i s u s u a l l y re-
f e r r e d t o as T o l l m i e n - S c h l i c h t i n g waves and r e q u i r e s v i s c o s i t y .
A good
a p p r o x i m a t i o n f o r f i n d i n g t h e s e e i g e n v a l u e s ( t e s t e d by L i l l y ) n e g l e c t s t h e e f f e c t s of r o t a t i o n and u s e s t h e f o u r t h - o r d e r e q u a t i o n t h a t r e s u l t s .
For-
mally, t h i s l i m i t i s p r o v i d e d by h a v i n g f i n i t e b u t l a r g e enough Reynolds number.
T h i s means t h a t ( 2 7 ) becomes
179 o r an Orr-Sommerfeld t y p e e q u a t i o n .
Hence, j u s t as i n t h e i n v i s c i d c a s e ,
(38) depends o n l y upon t h e component o f t h e mean v e l o c i t y t h a t is perpend i c u l a r t o t h e o b l i q u e wave f r o n t , a r e s u l t known from o t h e r r e a s o n s by t h e S q u i r e theorem. (3)
Parallel instability.
(1) and ( Z ) ,
T h i s t y p e of i n s t a b i l i t y i s n o v e l f o r a l -
L i l l y termed i t p a r a l l e l b e c a u s e , u n l i k e c a s e s
most a l l p a r a l l e l flows.
t h e e i g e n v a l u e s r e l y on t h e component of t h e mean v e l o c i t y
t h a t i s p a r a l l e l t o t h e o b l i q u e wave f r o n t .
In a d d i t i o n , t h e i n s t a b i l i t y
i s v i s c o u s , r e q u i r e s r o t a t i o n , and must b e computed from t h e f u l l s i x t h -
U
A simplified treatment? allows
o r d e r system.
r e p l a c e d by a c o n s t a n t .
t o b e set t o z e r o and
vrl
The r e s u l t i s a system t h a t is s t a t i c a l l y u n s t a b l e
f o r c e r t a i n v a l u e s of t h e wave number and Reynolds number w i t h t h e c r i t i c a l v a l u e o f t h e Reynolds number b e i n g l e s s t h a n t h a t of t h e Tollmien-Schlichting oscillations.
A t t h e o u t s e t , t h e n , t h e r e is t h e p o s s i b i l i t y of a com-
p l e t e l y d i f f e r e n t mechanism f o r c a u s i n g e n e r g y t o b e t r a n s f e r r e d t o t h e f l u c t u a t i o n s t h a t h a s h e r e t o f o r e been a p p r e c i a t e d .
The e x a c t r o l e t h a t
such t r a n s f e r might p l a y i n t h e t u r b u l e n t problem i s completely unknown. Adding t h e p y c n o c l i n e a t t h e edge of t h e s h e a r zone and s t i l l r e q u i r i n g t h a t t h e e i g e n v a l u e problem t o b e computed is n o t a n unduecomplication. S i n c e t h e s t r a t i f i c a t i o n i s s t a b l e , i t w a s found t h a t t h e e i g e n v a l u e s t h a t d e n o t e i n s t a b i l i t y f o r t h e Ekman l a y e r are reduced.
T h i s e f f e c t i s ex-
p e c t e d on p h y s i c a l grounds w i t h t h e r e d u c t i o n in t h e a m p l i f i c a t i o n r a t e d i r e c t l y due t o t h e c o n v e r s i o n t o p o t e n t i a l e n e r g y by t h e buoyancy f l u x of t h e f l u c t u a t i n g motiok.
A s a p h y s i c a l a n a l o g u e , t h e p y c n o c l i n e becomes es-
s e n t i a l l y a second boundary, making i t d i f f i c u l t f o r energy t o p e n e t r a t e below t h i s d e p t h .
Using a n o b l i q u e a n g l e t h a t i s r e l e v a n t f o r t h e non-stra-
t i f i e d Elcman l a y e r , namely wave number,
=
K
pared t o t h i s is for
2,
=
49074
0.35,
$ = 110".
t h e e i g e n v a l u e s are
C = -.05716
+
i.01802
C
=
-.05338
R
= 350, and a
+ i.04784.
Com-
when t h e p y c n o c l i n e is i n c l u d e d
o r , i n terms of t h e b u l k Richardson number,
t h e v a l u e i s 1 5 i n t h e p y c n o c l i n e and t h e boundary.
a Reynolds number,
.005
R.2
= 0
n 5o '
between t h e p y c n o c l i n e and
Although n o t checked i n d e t a i l f o r e v e r y e i g e n v a l u e of t h i s
problem, t h e t h r e e d i s t i n c t modes o f i n s t a b i l i t y s t i l l e x i s t w i t h t h e m o d i f i c a t i o n s i n t h e a m p l i f i c a t i o n r a t e s and, t o a lesser e x t e n t , an a l t e r a t i o n i n t h e phase v e l o c i t y caused by t h e buoyancy f l u x .
tcf.
Greenspan (1968)
The e i g e n f u n c t i o n s
180 must, iiorrcver, a d j u s t t o t h e p r e s e n c e of t h e p y c n o c l i n e and t h e r e b y f a l l o f f much more r a p i d l y t h a n when t h e mean d e n s i t y i s c o n s t a n t . F i g u r e 1 d e p i c t s t h e c r o s s c o r r e l a t i o n s f o r t h e c a s e when t h e p y c n o c l i n e is l o c a t e d a t
rl =
,
4.5
a p o i n t !.hat r o u g h l y c o r r e s p o n d s t o where t h e mean
v e l o c i t y has decreased t o l / e .
The s c a l e s a r e r e l a t i v e w i t h t h e normaliza-
t i o n s e t by t h e l a r g e s t of t h e t h r e e p o s s i b i l i t i e s and t h e n o t i n g of t h e inflexion is for reference.
In o r d e r t o p u t t h e s e d i s t r i b u t i o n s i n prospec-
t i v e w i t h r e s p e c t t o energy g e n e r a t i o n , v a l u e s f o r t h e mean g r a d i e n t s must
-I
be included. -
I"
for
> 0
graphs.
For a l l c a s e s , L' > 0
0 5 rl
<
1.15
i n t h e "-9
and n e g a t i v e beyond;
and t h e n n e g a t i v e i n t h e r a n g e shown on t h e
Thus, f o r t h e t o t a l e n e r g y ,
The p r o d u c t
0 5 q < 3.35
for
-
-uw
i s more p r o d u c t i v e t h a n
-
-VW.
-uv d o e s n o t a p p e a r when t h e c o n s e r v a t i o n of t h e t o t a l e n e r g y
place is taken separately.
T h i s means t h a t energy i s exchanged
between t h e s e components through t h e c o r r e l a t i o n m u l t i p l i e d by t h e r o t a t i o n ; when t a k e n t o g e t h e r , t h e r e i s c o n s e r v a t i o n s i n c e t h e C o r i o l i s e f f e c t c a n n o t c r e a t e o r d e s t r o y energy.
Looking a t t h e v e r t i c a l d i r e c t i o n r e q u i r e s t h e
Re=
350
K = .35
=
110'
Cr =-.05716417 CI = .01801811
J = 49074
point
--
-1.0 -.8
-.6 - A
-2
.2
.4
.6
.8
1.0
F i g u r e 1. Reynolds stresses f o r s e l f - e x c i t e d system. c a t e d a t n = 4.5
Pycnocline is lo-
c o r r e l a t i o n s of t h e d e n s i t y f l u c t u a t i o n s w i t h t h e v e l o c i t y i n t h i s d i r e c t i o n and i s shown i n F i g u r e 2.
C l e a r l y , t h e r e i s an energy d r a i n a s t h e
181 work a g a i n s t t h e f o r c e of g r a v i t y d i c t a t e s in a s t a b l y s t r a t i f i e d f l u i d .
Pycnocl ine
\ 4.-
-Re = 350 K = .35 = 110"
4
Cr -.05716417 Ci = .01801811 J
= 49074
't
1 1 1 1 1 ( 1 1 1 1 1 -1.0 -.8-6 74 :2 0 .2 4 .6 .8 1.0 F i g u r e 2.
Buoyancy f l u x i n s e l f - e x c i t e d system.
Looking a t t h e k i n e t i c energy by componcnts g i v e s an i n d i c a t i o n of t h e exchange t h a t is t a k i n g p l a c e i n t h i s s y s t e n .
T h i s is provided in F i g u r e
3 where a g a i n , t h e n o r m a l i z a t i o n is by t h e l a r g e s t of t h e v a l u e s . seen t h a t t h e and, t h e
-
-uu
I t .is
c o n t e n t is a l n o s t n e g l i g i b l e in comparison t o t h e o t h e r s exchange is from t h e u t o t h e u component of t h e f l u c t u a t i n g
v e l o c i t y , a t l e a s t f o r t:ie s c a l e s and f r e q u e n c y used i n t h e s e computations. C l e a r l y , t h e n e t p r o d u c t i o n f o r t h e t o t a l energy i s g r e a t e r t h a n t h e l o s s e s since
Ci
>
0, i n s u r i n g t h a t t h e f l u c t u a t i o n s are a m p l i f i e d .
An e q u i v a l e n t set of d a t a f o r a f o r c e d problem are shown in F i g u r e s 4 ,
5, and 6.
The v a l u e s o f t h e p a r a m e t e r s remain t h e same as f o r t h e p a s s i v e
boundary case s o t h a t a r e a s o n a b l e comparison can b e made.
The e x a c t form
of t h e f o r c i n g a t t h e boundary was chosen t o b e Gaussian i n t h e p l a n a r varia b l e s z and y as w e l l as i n t i m e .
As a r e s u l t ,
come new p a r a m e t e r s t h a t must be s p e c i f i e d .
t h e s t a n d a r d d e v i a t i o n s be-
F o r t h e n u m e r i c a l example de-
p i c t e d , t h e c h o i c e s c o r r e s p o n d to a l e n g t h scale of 8.5 cms w i t h t h e z and
182
-R e s 350
-
K = .35 110.
4-.
I
Cr .-.05716417
ci =
L
I
-1.0 -.B
1
I
1
-.6 -.4 - 2
0
.01801811
= 49074
J
I
1
1
I
2
.4
.6
.B
I
1.0
F i g u r e 3. K i n e t i c energy by components f o r s e l f - e x c i t e d system. is l o c a t e d a t n = 4.5.
Pycnocline
y v a r i a t i o l i s e q u a l and a t i m e s c a l e c o r r e s p o n d i n g t o 20 seconds.
Field data
(Martin, p r i v a t e communication) i n d i c a t e t h a t t h e s p a c e s c a l e i s t y p i c a l of what i s o b s e r v e d whereas t h e t i m e s c a l e i s n o t a s w e l l knorm and depends on a g r e a t many f a c t o r s .
Adjustment of t h e t i m e s c a l e can b e made i f t h e an-
p l i t u d e of t h e d r i v i n g d u e t o t h e mass f l u x is allowed t o vary.
For exam-
w
p l e , by l e t t i n g t h e f o r c e d and t h e f r e e o s c i l l a t i o n a m p l i t u d e s f o r t h e
component of t h e v e l o c i t y b e e q u a l ,
ae'/az
= .30
x
second p e r i o d ; i n c r e a s i n g t h e p e r i o d t o 2 h o u r s r e q u i r e s cn-l under t h e same c i r c u m s t a n c e s . able.
cm-l f o r t h e 20
a B ' / a z = .30
x
The l a t t e r i s p r o b a b l y more reason-
I f i t i s r e c a l l e d t h a t o n l y t h e l a m i n a r problem i s b e i n g i n v c s t i -
g a t e d , t h e n t h e s e a m p l i t u d e s are small indeed. Three major f e a t u r e s are r e v e a l e d when t h e s t r u c t u r e of t h e f o r c e d prob-
l e m i s examined.
F i r s t , t h e Reynolds stress c o r r e l a t i o n s of F i g u r e 4 have
s i g n i f i c a n t v a l u e s closer t o t h e s o l i d boundary t h a n t h o s e of t h e f r e e problem.
Second, t h e buoyancy f l u x ( F i g u r e 5) now c a u s e s a t r a n s f e r of energy
t o the fluctuations. f e r i o r t o its
-
-z)w
Third, t h e counterpart.
c; orrelation -
( F i g u r e 4 ) i s now in-
The r e l a t i v e v a l u e s of k i n e t i c e n e r g y of
t h e components ( F i g u r e 6 ) , on tne o t h e r hand, f o l l o w t h e p a t t e r n of t h e self-excited oscillations.
Some u n d e r s t a n d i n g of t h e r e s u l t s can b e found
by n o t i n g t h a t t h e r e i s a profound d i f f e r e n c e i n t h e two probleiis.
The
183
- - - -1nflpxion point
-1.0 -.B
- 6 -.4
Figure 4.
-.2
0
I
1
1
1
1
.2
.4
.6
8
1.0
Reynolds stresses f o r f o r c e d o s c i l l a t i o n s .
c
Re = 350 K = .35 4 = lloo Cr =-.05716417 Ci 0.0
I
I
I
I
I
I
1
1
'
1
1
-1.0 -.8-.6-.4 -.2 0 .2 4 .6 .8 1.0
F i g u r e 5.
Buoyancy f l u x f o r f o r c e d o s c i l l a t i o n s .
184
I
R e = 350 K * .35 110.
4
-
Cr =-.05716417
ci
I
l
l
1
1
-1.0 -.8 -.6 -.4 -.2
F i g u r e 6.
= 0.0
0
.2
.4
.6
.8
1.0
Forced o s c i l l a t i o n k i n e t i c energy.
homogeneous problem i s one t h a t i s u n s t a b l e and t h e r e f o r e t h e energy produced by t h e working t h e t h e Reynolds s t r e s s e s on t h e mean g r a d i e n t s must exceed t h e d r a i n by t h e buoyancy f l u x and t h e v i s c o u s d i s s i p a t i o n .
The
f o r c e d problem i s one t h a t i s n e u t r a l , t h a t i s , t h e r e i s a s o u r c e of cnergy a t t h c boundary s o l o n g a s t h e i c e d i s c h a r g e s b r i n e , and t h i s i n p u t must be e x a c t l y b a l a n c e d by t h e l o s s - g a i n
from t h e same s o u r c e s - s i n k s t h a t a r e
common t o t h e a c c o u n t i n g i n t h e p a s s i v e boundary s i t u a t i o n . r e a s o n t h a t t h e phase i n t h e buoyancy f l u x i s r e a d j u s t e d .
It is for t h i s
Secondly, t h e r e
i s one more p r o d u c t term i n t h e t o t a l energy c o n s e r v a t i o n e q u a t i o n f o r t h e f o r c e d problem t h a t c a n n o t a p p e a r when cbe boundary c o n d i t i o n s a r c iiono-
__
geneous.
Specifically, t h e pressure-velocity
correlation
v a n i s h a t t h e boundary b e c a u s e t h e r e i s a f i n i t e negative;
w
p'w'
cannot
(either positive or
i t is p o s i t i v e f o r t h e case i l l u s t r a t e d ) a t t h e i c c cover.
This
a c t i o n a l s o c o n t r i b u t e s t o t h e phase a l t e r a t i o n .
Although o n l y two examples arc q u a n t i t a t i v e l y p r e s e n t e d , i t s h o u l d be n o t e d t h a t t h e s t r u c t u r e can b e a l t e r e d even more i f one i s w i l l i n g t o al-
low n complete v a r i a t i o n of a l l t h e v a r i a b l e s , even though t h e b a s i s f o r t h e c a l c u l a t i o n s i s l i n e a r mathematics.
The i l l u s t r a t i o n s p r o v i d e d s e r v e
t o emphasize t h a t any s t r u c t u r e i n t h e s e c i r c u m s t a n c e s can be v e r y compli-
185 c a t e d a n d a r e s u l t a n t must i n some way b e a c o m b i n a t i o n o f a l l t h e i n g r e d i ents.
Besides t h e physical requirements, t h i s i s p r a c t i c a l l y guaranteed
f o r , a t t h e v e r y minimum, t h e r e a l s p a c e e q u i v a l e n t o f a n y c o r r e l a t i o n must b e a sum ( o r a n i n t e g r a t i o n ) o v e r a l l wave numbers a n d f r e q u e n c i e s .
The
d a t a o f F i c u r e s 1 t o 6 a r e f o r b u t o n e F o u r i e r component f r o m t i i t ! p o s s i b l e band of u n s t a b l e o s c i l l a t i o n s t h a t c a n o c c u r b e t w e e n
O.d35 2
K
5 0.707.
I n a d d i t i o n L h e r e a r e numerous n e u t r a l and damped s o l u t i o n s t h a t must b e t a k e n i n t o a c c o u n t i n a c o m p l e t e summation. One l a s t comment i s i n o r d e r .
The t u r b u l e n t g e o p h y s i c a l b o u n d a r y l a y e r
t h a t h a s b e e n o b s e r v e d u n d e r t h e i c e c o v e r o f t h e A r c t i c p r e s e n t s o n e more c r i t i c a l l e n g t h s c a l e beyond t h a t o f t h e r e l a t i v e l o c a t i o n o f t h e mixed l a y e r w i t h r e s p e c t t o t h e s o l i d boundary and t h e r e g i o n of s h e a r .
This
length is t h e depth of t h e non-spiraling
o r l o g a r i t h m i c p o r t i o n of t h e mean
v e l o c i t y t h a t is c l o s e t o t h e boundary.
I f t h i s r e g i o n e x t e n d s f a r enough,
t h e dynamics o f t h e t u r b u l e n t b a l a n c e can be compl et el y a l t e r e d , j u s t a s happens i n a c o n v e n t i o n a l f l a t p l a t e boundary l a y e r t h a t i s f u l l y t u r b u l e n t .
I n o t h e r w o r d s , as i s a l r e a d v known f o r t h e f l a t p l a t e b o u n d a r y l a y e r ( B e t c h o v and C r i m i n a l e , 1 9 6 4 ) , i t i s c o n c e i v a b l e t h a t a s t a b i l i t y a n a l y s i s
of t h e t u r b u l e n t p r o b l e m w i l l r e v e a l t h a t a l l o s c i l l a t i o n s w i l l b e c o m p l e t e l y stabilized.
i f t h i s is t r u e , t h e s y s t e m assumes a n e q u a l l o o t i n g w i t h
nore conventional turbulence,
t h a t i s , t h e r e is a s t a b l e l i m i t c y c l e .
I'he
f o r c i n g d u e t o t h e mass f l u x b o u n d a r y c o n d i t i o n s i s a mechanism t h a t must b e i n c l u d e d o v e r a n d beyond t h o s e t e r m s t h a t n o r m a l l y l e a d t o a m a i n t e n a n c e of t h e t u r b u l e n c e . be d u e t o
The l a m i n a r c a l c u l a t i o n s r e v e a l t h a t new s t r u c t u r e can
lie phenomenon.
A c o m p l e t e t r e a t m e n t i s t h e t o p i c o f work t h a t
is i n progress. ACKIlOWLEDGEi IEiJTS I s h o u l d l i k e t o t h a n k t h e d i l i g e n t a s s i s t a n c e i n t h e n u m e r i c a l conipu-
t a t i o n s by G.
S p o o n e r and t h e s u p p o r t o f t h e A e r o m e c h a n i c s D i v i s i o n o f t h e
A i r F o r c e O f f i c e o f S c i e n t i f i c R e s e a r c h f o r t h e i r s u p p o r t u n d e r AFOSR G r a n t
74-2579.
REFERENCES B e t c h o v , R.
a n d W.0. C r i m i n a l e ,
B e t c h o v , R. a n d W.O. P r e s s (1967).
P h y s i c s o f F l u i d s , S ( 1 9 6 4 ) 920.
C r i m i n a l e , S t a b i l i t y of P a r a l l e l Flows
Academic
186 Brown, R.A.,
J. A t m s .
Sci.,
27(1970)742.
Brown, R.A.,
J. A t m s .
Sci.,
29(1972)850.
G r e e n s p a n , H.P., (1968). Howard, L.N.,
T h e - T h e o r y of R o t a t i n g F l u i d s , Cambridge U n i v e r s i t y P r e s s ,
J. F l u i d Mech.,
10(1961)509.
K a y l o r , R. a n d A.J. F a l l e r , J. A t m s . L a n d a h l , M.T.,
J. F l u i d Mech.,
Sci.,
L i l l y , D.K.,
J. A t m s .
Miles, J.W.,
J. F l u i d Mech.,
S m i t h , J.D.,
Rapp. P. -v.
Sci.,
29(1972)497.
29(1967)441.
23(1966)481. 10(1961)496.
R6um. Cons. i n t . E x p l o r . Mer., 1 6 7 ( 1 9 7 4 ) 5 3 .
187 ON THE IETPORTANCE OF FRICTION IN T\JO TYPICAL COP!TTNENTAL WATERS : OFF OREGON AND SPANISH SAHARA PIJI!SH
K. KUNDU
School of Oceanography, Oregon State University, Corvallis, Oregon 97231 ABSTRACT The current meter data at various depths near the coasts of Oregon (water depth 100 m) and northwest Africa (water
-
depth < 6 7 m) have been analyzed. raged over about seven day periods,
The results have been aveso
that stationary Ekman
layer-like characteristics could be detected.
It has been
concluded that the entire water column off Africa is frictional, whereas the Oregon coastal dynamics are not s o .
This is due to
the lower Coriolis parameter, larger friction velocity u* the much lower stratification off Africa.
,
and
The horizontal den-
sity gradients can explain the observed vertical velocity shears off Oregon, but not off Africa.
A typical
U~
near
Africa is about 0 . 8 cm/s, whereas that near Oregon is about
0.3 cm/s.
The thickness of the bottom Ekman layer is estimated
to be about 60 m off Africa and 1 2 m off Oregon, whereas the thickness of the logarithmic layer is estimated to be about 9 m off Africa and 2 m o f f Oregon. been observed off Africa.
Ekman turnings of 2 5 ' - 4 0 "
have
The upper surface layer data near
Oregon display hodographs resembling the classical Ekman spiral, rather than the "slab" type mixed layer. INTRODUCTION The objective of the present note is to examine the current meter data from two typical coastal upwelling regions off Oregon and Spanish Sahara, in order to ascertain whether the entire water column is frictionally dominated for time scales long compared to the inertial period, of the order a week or more.
An idealized coastal upwelling problem in the northern
188
hemisphere is the following (Fig.1)
: A southward wind stress
acts parallel to the coast, driving an offshore Ekman flux at the ocean surface-, which therefore needs a compensating onshore return flow.
The question that we want to answer is : Is the
return flow through a bottom Ekman layer ?
Or is it through a
frictionless geostrophic interior generated by a northsouth pressure gradient ?
Nonlinear forces will be neglected in this
study, since the Rossby number for the Oregon region has been found to be less than 0 . 1 5 (Kundu et al., 1 9 7 5 ) and our calculation shows that it is also small off Africa. Some of the various possible situations are listed in Fig. 1 by means of profiles in the u-z, v-z and u-v planes, where u,
v,
w are the velocity components in the
northward),
upward) directions respectively.
eastward), The geostro-
phic interior in Case 1 can be of negligible thickness, so that the top and bottom Ekman layers may be almost adjacent to each other, in which case the profiles in Case I
(ii), say, may look
very similar to those in Case 2 (iii) or 2 (iv).
It is, there-
fore, not a trivial task to determine whether the return flow is frictional or geostrophic. The analytical model of Garvine ( 1 9 7 1 )
for upwelling re-
gions assumed that the bottom Ekman layer is dynamically unimportant, and that the return flow is accomplished geostrophically by means a northsouth pressure gradient.
Because of the
absence of such a pressure gradient, on the other hand, the theory of Allen ( 1 9 7 3 )
predicted that for large times the re-
turn flow is through a bottom Ekman layer, assumed thin compared to the water depth. (1976).
The recent vork of Smith and Long
however, suggested that the entire water depth on the
Oregon-Washington shelf is frictionally dominated, that is, the flow field is simply two Ekman layers one on top of the other, the bottom layer balancing the offshore mass flux of the top layer. Some
hodograph profiles calculated by Smith and Long are
reproducedlin Fig. 2 .
The pressure gradient was taken to be
'Smith and Long had a northward wind stress. Their solutions have been replotted here with signs changed, so as to correspond to our southward wind stress.
189
r, = o 7y=
I. Onshore Return Through Bottom Ekman Layer ( i ) No thermal wind:
( i i l Thermal wind:
p, = - a
p, = -0 -bz p =o
Py=0
Y
U
V
I
V
U
2. Onshore Return Through Geostrophic Interior ( i i ) Arbitrary Vg added:
( i ) No V.
p, = - a py = - A
P, = 0 py 5 - A
V
U
(tii) Vg having thermal wind: p, = - 0 -bz
FIG.1.:
V
U
(iv) Both Ug ,Vg having thermal wind p, = -a-bz
V a r i o u s p o s s i i i l e f l o w c o n f i g u c a t i o n s due t o a southward wind s t r e s s a c t i n g p a r a l l e l t o t h e c o a s t . The p r o f i l e s i n t h e u - z , u - z , and u-v p l a n e s a r e s k e t c h e d .
190
BAROTROPIC
$'
L4
U
20
(i
a r 1; ' -V
U
(;;;
-V FIG.2.:
WATER DEPTH h=80m
8Om
-V
/ 70m
M( IRE THE W
THERMAL WIND
I-V
U
/70m
V
U
h= l70m
50
f
/mm
'00
V
Hodograph p r o f i l e s c a l c u l a t e d b y S m i t h and Long ( 1 9 7 6 ) .
191 px
=
-
a o r -a-bz
(a,b>O),
py = 0 (using subscripts for deri-
vatives) and was determined by requiring a mass balance perpendicular to the coast.
The eddy viscosity was assumed to be l i -
nearly increasing near the sea bed, and constant above that. The solutions suggested that for the shallow water case (water depth h
=
80 m) the two Ekman layers are very nearly abutting
each other, whereas for the deep water c a s e (h = 1 7 0 m) there is a thick geostrophic interior from heights z = 50 m to
z = 100 m .
Fig. 3 shows the experimental data of Smith and
L o n g , taken off the coast of Washington during winter months w h e n the wind stress w a s northward.
I t appears that there is
some qualitative agreement of these data with the "two Ekman layer" picture, but the agreement is n o t very good. OBSERVATIONS The current meter data examined here were taken off the coasts of Oregon (inertial frequency f Spanish Sahara ( f 4).
=
=
1.03
x 1 0 - 4 s - 1 ) , and
0.536 x 10-4s-') i n northwest Africa (Fig.
The Oregon data were collected during the summer of 1973
and are part of the CUE-2
program.
The Africa data were collec-
ted during the winter of 1 9 7 4 , and are part of the JOINT-I experiment.
All of the Africa data and part of the Oregon data
were measured by Drs. R o b e r t ' L . Smith and Dale Pillsbury of the Oregon State University, whereas the remainder o f the Oregon data were measured by Dr. David Halpern of the Pacific Marine Environmental Laboratory a t Seattle.
The heights o f current
meters above sea bed were 10 m , 2 5 m at station R 4 5 (h = 4 5 m);
7 m , 2 7 m at U67 (h = 67 m) ; 6 m , 1 2 m , 2 7 m at 1.167 (h = 6 7 m); and 5 m , 20 m, 40 m , 60 m , 80 m, 82 m , 8 4 m , 90 m , 92 m , 97 m at ClOO (11 = 100 m)2.
A l l frequencies higher than the inertial,
including the tides, were removed by means of a low pass filter having a half power point of 40 h.
This was done s o that one
c a n determine the Ckman layer-like characteristics ;
the
fluctuations slower than inertial should behave like in a L
The station names used i n the present paper differ from those used in the data reports and earlier publications. The original names were, C l O O = Carnation, R45 = Rhododendron, W67 = Weed, and U 6 7 = Urbina.
192
v A
v
c
i
0 h = l64m FIG.3.:
h =77m
E x p e r i m e n t a l d a t a of S m i t h and L o n g (1976) in the c o n t i n e n t a l shelf off W a s h i n g t o n in the west coast T h e wind s t r e s s w a s n o r t h w a r d . of N o r t h America.
20
193
c
9)
.A
U
m m
c U m m
P 0 .r(
m
0 0
m
U
.rl
m
U
U 9)
E
01
U
m
c
U
m
c
0 .A
u
m
U
0 cl
194 quasi-steady turbulent Ekman layer, whose time scale of adjustment is of order of the inertial period (Tennekes and Lumley 1972,
p.12). Fig. 5 shows typical hydrographic sections (reproduced
from Huyer 1 9 7 6 ; see also Mittelstaedt et al. 1 9 7 5 , and Barton et al. 1 9 7 6 ) ,
where it is evident that the Oregon water is
about 10 times more stratified than the water off Africa.
Many
other points of contrast have been noted in Huyer's'work, although the question of friction has not been dealt with. AFRICA RESULTS Figs. 6 and 7 show the average velocity profiles at the three African stations during two time intervals, each about a week long, during which the currents were least variable. cept for the station R45 in Fig. 7 ,
Ex-
the velocity profiles are
consistent with a picture of a bottom Ekman layer.
It may be
argued that the frictional influence is confined to only a few meters from the sea bed, and the shear above it is simply thermal wind.
To check this point, an upper bound for the density
gradient near the sea bed is estimated from Fig.4 to be about
1,
lo-"
gm/cm4
,
which would produce a thermal wind of This is much smaller than the obvz = gp,/fp 2 2 x 10-3s-1. served shear at 7 m of 10-2s-1. In the absence of data of denpx
sity sections parallel to the coast, i t is not possible to determine if the shear in u can be a thermal wind effect, but the small stratification off Africa makes this possibility less likely than off Oregon. The veering is 25'-4O0,
which is larger than those pre-
viously reported near the ocean bottom (e.p. Weatherly 1 9 7 2 , Kundu 1 9 7 6 ) .
However, Weatherly and van Leer ( 1 9 7 6 ) have re-
cently reported observed veerings as high as 4 O o - 6 O 0 off the coast of Florida. An estimate of the thickness of the neutral turbulent Ekman layer is given by 6 2 0 . 4 u,/f, where u,+ is the friction velocity. We shall first take ,u to be about 4 X of the geostrophic velocity V above the boundary layer (Weatherly 1 9 7 2 , g
260
26 5
. .
. NW
Africo
.. . -27 March . . * .. . . . . . . .
270Q:.
. . .+. .
272 A
0
8
8
0
1
1974
268
Oregon SIGMA-T
2 70
13 July 73
? 50
DIS T A M E FROM SHORE /&mJ
FIG.5.:
Typical hydrographic s e c t i o n s perpendicular t o the c o a s t s o f A f r i c a and O r e g o n , r e p r o d u c e d f r o m Huyer ( 1 9 7 6 ) .
196
AFRICA DATA MARCH 19-25
T
w+1 . 2 dyne /cm2
-0
-- +20 E
I
a I- -40 W
0
._ 60 - 80
’qy
-20
0
t
20
U (cm/s)
- 20 *k
55
-40 -40
S T A . U67
FIG.6.:
- 20
V (cm/s)
STA. W 6 7
STA. R45
A v e r a g e p r o f i l e s in t h e u-z, v-z a n d u-v p l a n e s a t t h e t h r e e A f r i c a n s t a t i o n s d u r i n g M a r c h 19-25.
197
AFRICA DATA MARCH 3 0 - A P R I L 5
IN
r = 2.0 dyne/cm' -40
-20
0
10 -40
-20
-0
--.
-IE I-
a
-40
h = 45m
W
0
--
-60
h =67m
I,,
U (crn/s)
- 20
60
-40
61
-40
'55
- 40
V (cm/s)
STA. U 6 7
FIG.7.:
STA. W 6 7
STA. R 4 5
A v e r a g e p r o f i l e s i n t h e u-z, v - C a n d u-v p l a n e a t t h e t h r e e A f r i c a n s t a t i o n s d u r i n g March 3 0 - A p r i l 5 .
198
1975), although a more accurate determination of u) will be attempted in a later Section. Taking Vg to be about 25 cm/s from Figs. 6 a n d - 7 , this gives a bottom layer thickness of 6 2 80 m. The stable stratification might decrease the thickness below this value, but not by much because of the very low stratification off Africa. The bottom Ekman layer is therefore estimated to be of the order of the water depth, and it appears that off Africa the two Ekman layers are superimposed on each other. OREGON RESULTS Fig. 8 shows the average profiles at C l O O during two time periods. The July 10 -16 period was chosen because this was a typical upwelling "event" (Halpern 1976), caused by a fairIy large wind stress. The August 15-24 period was chosen because of the minimum variability of currents observed during this time.
In the u-z profiles in Fig. 8, the maximum u does not
occur where it is expected of a bottom Ekman layer, but much above. It seems that the bottom Ekman layer may be about 10 m thick, and the shear in the velocity above it may be geostrophic. The hodographs also have a kink around 10 m , suggesting that it might be a point of transition between two kinds of dynamics. A typical u) during the upwelling event is estimated to be about 0.3 cm/s (see next Section), giving an estimate of the boundary layer thickness of 6 2 0.4 ux/f 2 1 2 m. The stratification, moreover, might decrease the boundary layer thickness even further. If it were assumed that all the onshore flow was frictional, as was suggested off Africa, then one might have taken V 35 cm/s (at 6 0 m). Taking u , ~0.04 Vg, this would result g in u* 2 1.3 cm/s , which is larger than the value of u u - 2 0 . 3 Q
cm/s found by a direct calculation of uy in the next section. The boundary layer therefore seems thinner than 60 m. The strong horizontal density gradients off Oregon can explain the velocity shears above 10 m. From Fig.5, the middepth density gradient at the current meter station is about p x 2 6 x 1 0 ' ' gm/cm4 which gives a thermal wind of ~
199
OREGON DATA AUGUST 15-24
OREGON DATA JULY 10-16
w$k.2 w-$?l.l
-40
-60
-20
0
dyne/cm2 10 cm/s
0
dyne /cm2 10cm/s
a
'p W
0
h=lOOm
h =lOOrn
U (cm/s)
-20
0
- 60 v FIG
a
(cmls)
: A v e r a g e p r o f i l e s i n t h e u - z , v-z Oregon s t a t i o n d u r i n g J u l y 1 0 - 1 6
40 14
t
-60 V(cm/s) and u-v p l a n e a t t h e and A u g u s t 1 5 - 2 5 .
200 -3 - I vz % 6 x 10 s shear in Fig. 8 .
-
in excellent agreement with the observed Such validity of thermal wind balance had al-
ready been supported by Smith ( 1 9 7 4 ) from 1972 data off Oregon. A
verification of the thermal wind balance for u is more
difficult.
A density section parallel to the coast taken during
July 13-14 was prepared from the hydrographic data of Huyer and Gilbert ( 1 9 7 4 ) , and is shown in Fig. 9.
During this time the
observed shear at 4 0 m height (Fig. 8 ) was u z - 10-3s-1 , which would require that the ut surface at 4 0 m go down southwards only about 7 m in a horizontal distance of 10 km.
Such a small
gradient, however, is impossible to detect within experimental accuracies of plots like Fig. 9 . There is some hint in Fig. 9 that the upward-toward-south slope of the isopycnals around the 8 0 m
-
8 5 m height may be
responsible for the larger observed shear of uz that height.
%
lo-'
s
at
If this is true, then the upper frictional layer
extends to about 1 8 m from the ocean surface ( 8 2 m from sea bed) and the shear below it is geostrophic.
The hodographs in Fig.
8 suggest that this might be true.
The curvature of the hodograph in the upper 15 m ( 8 5 m < z < 100 m) of the ocean surface (see also Fig. 14 of Halpern
1 9 7 6 ) is very interesting, suggesting a behavior like in a classical Ekman layer rather than in the slab model of Pollard, Rhines and Thompson ( 1 9 7 3 ) . All observations thus far lead to the conclusion that the bottom boundary layer thickness is about 1 0 - 1 5 m . the veering in this Ekman layer ' i
What is then
Fig. 8 shows that the angle
between the velocity vectors at 5 m and 20 m during July 10-16 was about 8 '
in the "Ekman" sense.
During August 15-24 the bot-
tom boundary layer thickness was apparently less than 5 m, and therefore no computation of the veering is possible ; moreover, the bottom velocities were so small during this time that a calculation of the veering would be meaningless.
(Just for the
sake of record, the angle between 5 m and 2 0 m during August 1524 turned out to be 24" in the non-Ekman s e n s e ) .
A weighted
average of the veering between the 5m-20m pair during the entire length of the record had been estimated to b e about 2" in
w 45
CURRENT METER STATION
"
10km NoRrH
4
Y 450IO'N
;=24./ 24.6
--
26.0
V E R T I C A L SECTION OF FIG.9.:
40
PARALLEL TO COAST
S i g m a - t c o n t o u r s d u r i n g J u l y 13-14 i n a p l a n e p a r a l l e l to the O r e g o n Coast and passing through s t a t i o n CIOO.
N
s
202
the Ekman sense by Kundu (1976).
The lower computed values of
veerings off Oregon than off Africa may be due to the fact that undetected veerings may have occurred below 5 m, since the log layer was only about 2 m thick (see next Section). FRICTION VELOCITY Many of the previous estimates were based on a rough guess A more accurate estimate of u,will now be of u y % 0.04 V g' attempted by fitting the log law u/u, = (1/K) log(z/zo) through the data, where K = 0.4 roughness parameter.
,
u is the speed at z, and z o is the
The calculations were done assuming both
a smooth bottom (for which
zo
= 0.1
v/u,
,
v
being the kinema-
tic viscosity of sea water), and a rough bottom (for which z o = d/-30, where d is the average height of the roughness elements, taken here as 1 cm following Weatherly 1972). Fig. 10 shows the regression plots of u,
obtained by fit-
ting the logarithmic law at two heights, as was done by Wimbush and Munk ( 970) and Weatherly (1972).
If both heights
are within the log layer, then the points fall on a 45" line through the origin
It is apparent that the behavior at sta-
tion W67 is fairly reasonable, suggesting that both 6 m and 12 m are within the log layer.
The behavior at ClOO is somewhat
worse, and the huge spread at U67 suggests that at least the 2 7 m was far beyond the log layer.
The spread for the 10 m-25 m
pair at R45 was also very high, and is not shown.
It is seen in Fig. 10 that the regression plots are not very sensitive to the smooth and rough wall assumptions made. The average difference between the uy calculated by the two assumptions was about 25 X .
This is not surprising. since it is easy to show that a 100 X error in z o causes only about a 10 X
error in .,u Table 1 shows the values of u,calculated
by fitting the
log law to the lowest current at each station, averaged over the time periods referred to in Figs. 6, 7 and 8.
Although some
of these currents may actually be above the log layer, the values in Table 1 can certainly be used as a guide.
Using typical
203
moom 08
W67 (AFRICA)
c
0
=* 0 4 02
02
0
04
06
08
10
U, ot 6 m
U, ot 6m
08
U 67
E b
(u
06
0
( AFRICA)
02
0.4 0.6 U, ot 7m
08
1.0
0
02
04 06 U,, ot 7m
08
10
08
10
E 06
IOREGON)
c
0
02
FIG.10.:
04
06
U,ot
5m
08
10
0
02
04 06 U, 01 5m
R e g r e s s i o n p l o t s o f u) d e t e r m i n e d b y f i t t i n g t h e l o g a rithmic l a w through current m e t e r data at the two heights nearest the sea bed. Calculations were perf o r m e d a s s u m i n g h o t h a s m o o t h and a roiigh w a l l .
204
values of uy and V estimates of the thickness of the bottom g ’ 2 Ekman layer 6 % 0.4 ux/f , the log layer 6 % 2 u ,/f Vg, and log the viscous sublayer 6vis % 12v/u, are listed below3 :
ux
V
g
6
‘vis
Oregon
Africa
0.3 cm/s
0.8 cm/s
10
cm/s
25
cm/s
12
m
60
m
2
m
9
m
0.6
cm
0 . 2 cm
The boundary layer thickness, of course, would be smaller than that listed above for untypically small values of the wind stress, for example during August 1 5 - 2 4
in Fig. 8.
SUMMARY AND CONCLUSIONS The current meter data at various depths near the coasts of Oregon (water depth 100 m) and Spanish Sahara in N.W. Africa (water depth ( 6 7 m )
have been analyzed.
These are upwelling re-
gions characterized by a strong southward wind stress causing an offshore flow in the upper layers of the ocean surface.
The
objective was to study the characteristics of the low frequency components (time scales of the order of a week or more), and determine whether the onshore return flow is through a frictional Ekman layer, or whether it is geostrophic. Off Africa, the extremely small stratification has been found to be incapable of explaining the velocity shears, which would therefore need friction for an explanation. velocity has been estimated to be about 0.8 cm/s
The friction
,
and thick-
nesses of the bottom Ekman layer and logarithmic layer have been estimated to be 6 0 m and 9 m respectively. ’The 9 m thickness of the log layer off Africa, although larger than those quoted in Wimbush and Munk ( 1 9 7 0 ) . Weatherly ( 1 9 7 2 ) and Kundu ( 1 9 7 6 ) . does not appear to be too high. Pingree ( 1 9 7 4 ) has verified that the tidal currents of about 50 cm/s satisfies the log law in the lowest 33 m of a water column of 1 8 0 m on the continental shelf near England.
205
Off Oregon
the strong stratification c a n explain the shears
upwards of 1 0 - 1 5 m from the sea bed by thermal wind balance.
A
typical fr ction velocity has b e e n estimated to be about 0.3 cm/s
,
and the thicknesses of the bottom Ekman and log layers
have been estimated to be 1 2 m and 2 m respectively.
The cur-
rents within the top 15 m near the ocean surface fall on a nice hodograph resembling a classical Ekman layer rather than a "slab" mixed layer. O n the basis of the above facts, it has been concluded that
off Oregon the onshore return flow is mostly geostrophic, whereas off Spanish Sahara the return flow is through a bottom Ekman layer.
In fact, for the latter case the entire water column is
frictionally dominated, the flow is simply two Ekman layers superposed o n each other.
The three factors that seem to cause
this difference i n the dynamics of these two coastal waters are : ( I ) drag or u)
a much smaller stratification, ( 2 ) a larger bottom and (3) a smaller f off Africa.
It is therefore suggested that the influence of bottom friction can be given due considerations i n the analytical models of coastal dynamics. Table I
: u*
calculated by fitting the l o g law through the
lowest current at various stations. Station
Height (m)
Date
u*
(smooth) cml s
uy (rough) cml s
ClOO
5
July 10-16
0.31
0.38
ClOO
5
Aug 1 5 - 2 4
0.14
0.16
W6 7
6
Mar 1 9 - 2 5
0.68
0.87
W6 7
6
Mar 30-Apr 5
0.79
I .02
U67
7
Mar 1 9 - 2 5
0.71
0.90
U67
7
Mar 30-Apr 5
0.81
1.04
R4 5
10
Mar 1 9 - 2 5
0.72
0.91
R4 5
10
Mar 30-Apr 5
0.65
0.82
206
ACKNOWLEDGEMENTS
I am grateful-to Dr. John S . Allen for suggesting the problem and helpful discussions, and to Drs. Robert L. Smith, Dale Pillsbury and David Halpern for allowing me to use their current meter data,
This research was supported by the CUEA Program of
the IDOE Office of the National Science Foundation under Grant OCE76-00596.
REFERENCES Allen, J.S., 1 9 7 3 . Upwelling and coastal jets in a continuously stratified ocean. J . P h y s . O c e a n o g r . , 3 , 2 4 5 - 2 5 7 . Barton, E.D., A. Huyer and R.L. Smith, 1 9 7 6 . Temporal variation observed in the hydrographic regime near Cab0 Corveiro in the NW African upwelling region, February-April. Deep-sea R e s . (in press). Garvine, R., 1 9 7 1 . A simple model of coastal upwelling dynamics. Oceanogr., I , 169-179.
J . Phys.
Halpern, D.. 1 9 7 6 . Structure of a coastal upwelling even observed off Oregon during July 1 9 7 3 . Deep Sea R e s . ( t o be published). Huyer, A., 1 9 7 6 . A comparison of upwelling events in two locations : Oregon and Northwest Africa. Submitted to J. Mar. R e s . Huyer, A., and W.E. Gilbert, 1 9 7 4 . Coastal upwelling hydrographic data report. Rep. 7 4 - 8 , School of Oceanography, Oregon State University. Kundu, P.K.. 1 9 7 6 . Ekman veering observed near the ocean bottom. 3 . P h y s . Oceanogr.. 6 . 238-242. Kundu, P.K., J.S. Allen, and R.L. Smith, 1 9 7 5 . Modal decomposition of the velocity field near the Oregon coast, J . P h y s . Oceanogr,, 5 , 683-704. Mittelstaedt, M., D. Pillsbury and R.L. Smith, 1 9 7 5 . Flow patterns in the Northwest African upwelling area. D e u t s c h e Xydrographische Z e i t s c h r i f t , 2 8 , 145-167. Pingree, R.. 1 9 7 4 . The turbulent boundary layer on the continental shelf. N a t u r e . 2 5 0 , 7 2 0 - 7 2 2 . Pollard, R.T., P.B. Rhines and R.O.R.Y. Thompson, 1 9 7 2 . The deepening of the wind-mixed layer. Geophys. Fluid Dyn., 4 , 3 8 1 404.
Smith, J.D. and C.E. Long, 1 9 7 6 . The effect of turning in the bottom boundary layer on continental shelf sediment transport. Mkmoires d e l a S o c i S t S R o y a l e d e s S c i e n c e s de L i B g e , 6 e slrie. 10, 3 6 9 - 3 9 6 .
Smith, R.L., 1 9 7 4 . A description of current, wind and sea level variations during coastal upwelling off the Oregon coast, JulyAugust 1 9 7 2 . J . Geophys. R e s . , 7 9 , 4 3 5 - 4 4 3 .
207
Tennekes, H. and J.L. Lumley, 1972. A F i r s t C o u r s e on T u r h l e n c e , the MIT Press, 300 pp. Weatherly, G.L., 1972. A study of the bottom boundary layer of the Florida Current. J . P h y s . O c e a n o g r . , 2, 54-72. Weatherly, G.L., 1975. A numerical study of time-dependent turbulent Ekman layers over horizontal and sloping bottoms. J . Phys. O c e a n o g r . 5 , 288-299. Weatherly. G.L. and J.C. van Leer, 1976. On the importance of density stratification to the bottom boundary layer on the West Florida continental shelf, Mdrnoires S o c i d t d R o y u l e d e s S c i e n c e s d e L i B g e , 6e s6rie, I 1 (to be published). Wimbush, M., and W. Munk, 1970. The benthic boundary layer. The S e a , Vol. 4. Part 1 , Wiley, Chap. 1 9 .
This Page Intentionally Left Blank
209 MASS TRANSFER PROPERTIES IN SEDIMENTS NEAR THE BENTHIC BOUNDARY LAYER.
J.P. VANDERBORGHT and R. WQLLAST Universitg Libre de Bruxelles (Belgium).
INTRODUCTION The turbulent nature of momentum transport in the bottom boundary layer has been pointed out by several authors and recently reviewed by Bowden et a1.(1976). A theoretical approach of this mechanism is presented by J.C.J.Nihou1 in this volume. It is now admitted that the viscous sublayer is periodically disrupted by a sequence of motions referred to as "bursting". Experimental evidences of the intermittent resuspension were obtained by Kline et a1.(1967), (1973).
Seitz
Heathershaw (1974) and Gordon (1974). who observed periodical
ejection of solid particles and related it to peaks of high shear stresses. This phenomenon must also affect the entrapped interstitial water and modify deeply the mass transfer properties of dissolved substances through the benthic boundary layer. It is well known that chemical and biochemical reactions between interstitial water and sediments have a major effect on the composition of the pore water. Production or consumption of various compounds during these reactions are rfsponsible for vertical concentration gradients in the interstitial water with consecutive upwards or downwards fluxes of the dissolved species. When the rates of reaction are known, the modelling of vertical concentration profiles provides an indirect estimation of the mass transfer coefficients in the sedimentary column and of the exchange of dissolved substances through the benthic boundary layer. We intend to show in this paper that the vertical concentration profiles in cores taken in the shallow area of the southern bight of the North Sea, reflect the mixing of the upper sedimentary layer provoked by momentum transfer in that layer.
THE DIAGENETIC EQUATION Most of the freshly deposited material is out of equilibrium with respect to the sedimentary environment. It is thus submitted to various phy-
210 sical, chemical or biochemical reactions which affect both the nature of the solid phases and the composition of the interstitial water. The modelling of the vertical concentration profiles by means of diagenetic equation was first applied by Berner ( 1 9 7 1 ) . In the interstitial water of a sedimentary layer where mass transfer occurs according to Fick's laws and where a reaction proceeds at a rate r, the concentration profiles c(z,t) of the various chemical species can be calculated from the equation :
where D(z)
is the mass transfer coefficient in the sediment (z axis
oriented downwards from the sediment
- water
interface).
This formulation implies some assumptions, mainly that the problem can be treated in terms of one dimensional depth model , which is justified by the fact that vertical gradients in pore water composition are generally much greater than lateral gradients. On the other hand, when the deposition rate of fresh sediments is high, the burial of the sedimentary layers has to be considered. This can be done by introducing a term of the form -w 6c/6z which account for the vertical translation of the depth axis with respect to a fixed point. Furthermore, most of the diagenetic models applied in the past were based on the assumptions of steady - state and constant mass transfer coefficients, which means that the general diagenetic equation can be written :
Most of the models presented in the literature assumed that the mass transfer was directly related to the molecular diffusion in the porous medium. The mass transfer coefficient D was thus related to the molecular diffusion coefficient in sea water Dm by the expression : D = -Dm
'
e2 where Q and 0 respectively stand for the porosity and the tortuosity of the sediment. Typical values of D for most of the compounds fall in the vicinity of 10-6cm2.s-'. It must be pointed out that the diagenetic models were generally used in order to verify qualitatively a kinetic mechanism or to evaluate the rate of reaction r. assuming a constant molecular diffusion coefficient
211 all over the sedimentary column. Our approach consisted in determining independently the values of the rate of reaction r,
so
that the diagenetic model could be used in order
to evaluate the mass transfer coefficient.
EXPERIMENTAL RESULTS Vertical profiles of nutrients concentration were obtained for sandy and muddy sediments of the southern bight of the North Sea. These cores were carefully taken by divers, deep-frozen and sliced in very close vertical spacings, so that the structure of the sediment was preserved to the most possible extend. In fact, with the classical gravity corer. and to a lower extend with the b?x corer, the turbulences caused by the corer itself is able to resuspend the top sediments. On the other hand, the removal of the supernatant water may carry away another fraction of the surface layer,
so
that
a careful coring technique and treatment of the cores are necessary to avoid any loss of information. This is particularly true for unconsolidated muddy sediments. The vertical profiles exhibit two patterns corresponding respectively to sandy and muddy sediments, which will thus be discussed separately. Sandy sediments The diagenetic reactions involving nitrogen species were intensively studied by Billen ( 1 9 7 6 ) . who determined the net rate of ammonium production and of nitrification as a function of depth in the considered cores. The experimental vertical distribution of ammonium and nitrate were then compared with the diagenetic model. Figure 1 shows the experimental and calculated NH
4
and NO
3
curves for a set of D values. For both species,
the best agreement is found for a value of the mass transfer coefficient equal to 0.8 10-4~m2.s-’. This value is approximately 50 times greater than the coefficient estimated for q molecular diffusion process. Muddy sediments The case of muddy sediments seems much more complicated. In the North Sea cores, the porosity profiles show a rapid decrease from a value close to 100 % at the interface to a nearly constant value comprised between
212
PM N H ~ 50
100
a
b
c
d
CJM N O i 50
0
100
r
10’
e
f
9
h
Fig.1. Experimental and calculated NH; and NO; vertical profiles in the interstitial water of a sandy sediment 0114, 11.12.1975), for a set of D values : (a) 1.5 (b) (c) 0.8 (d) 0.6 2 -1 (e) 2 4 0 (f) lom4; (9) 0.8 (h) 0.5 an .s
.
213 40 and 65 % at a depth below a few centimeters (Fig.2). This can be compared to the vertical profile of silica concentration (Fig.31, where a sharp gradient discontinuity is observed at a depth of about 4 cm. with a very low gradient in the upper part of the sediment. This profile differs from those usually described by other experimenters, who obtained high gradients of dissolved silica at the interface. Other profiles display the same pattern, for example sulfate (Fig.3), Nitrates are an exception (Fig.5) : they exhibit a maxi-
ammonium (Fig.4).
mum value at a depth of 2' cm. This maximum can be related to a rapid production of nitrates by nitrifying bacteria in the upper layer (Vanderborght and Billen, 1976). Its existence proves that the low gradient observed in the upper layer for the other elements is not the result of a washingout of these elements during the coring and subsequent manipulations. The concentration profiles suggest that the mass transfer coefficient is strongly dependent on the depth within the sediment. The existence of a sharp discontinuity for most of the profiles has let us to consider a two layer model, where each layer is characterized by a constant but different mass transfer. It is evident that the division of the system into two layers is still a simplistic approximation, mainly because it does not take into account the dependence of the mass transfer coefficient with depth suggested by the porosity profile in the upper layer. For the lower layer, the model is probably more realistic as it appears from the low and constant porosity. Separate measurements or estimations of the various reaction rates enable then to determine the value of both mass transfer coefficients.
In the case of silica, for example, the reaction rate can be expressed by
r
k (cm
=
where cm constant
.
-
c)
is the equilibrium concentration for
t =
m,
and k the kinetic
The diagenetic equations :
for the upper layer ( z
zn), and
for the lower layer (z
z ) , can be solved by writing the continuity of
214 mM SOA
porosity (%)
40
5
Fig.2. Porosity profile in a muddy sediment of the North Sea (M1149, 12.03.1974) mM N H i 0
1
2
3
4
5
15’
Fig.3. Vertical profiles of sulfate and silica concentration in a muddy sediment of the North Sea (Y1149)
5-
10
-
f
Y
5
15-
8
0
20-
25-
30-
Fig.4. Vertical profile of annnonium concentration in a muddy sediment of the North Sea (M1149, 12.03.1974)
1
15
Fig.5. Vertical profile of nitrate concentration in a muddy sediment of the North Sea 011 149, 1 2.03.1974)
216
concentration and of flux at z = zn, i.e. : C
-
Iz-= clz+
dc
Dl
xz
z
-
=
-
D2
dc z
Iz
n
+ n
and by imposing the boundary conditions : c = seawat er at c =
‘m
at
z = O z =
m.
The solution of this system can be compared to the experimental profiles for various values of D l (Fig.6) and D2. The best fit is obtained when the mass transfer in the lower part corresponds to molecular diffusion : if porosity and tortuosity are tnken into account, the apparent diffusion coefficient in the lower layer should be approximately equal to 10-6cm2/s. The value of the mass transfer coefficient D 1 in the upper layer must then be 100 times greater to fit the observed profiles. The same mathematical approach can be applied to the other chemical species, leading to the same conclusions (Fig.7).
DISCUSSION It should be first noted that the concentration profiles observed could possibly result from successive erosion and redeposition during storm surges, with entrapment of surface sea water at constant composition. Persistence of a constant composition could be observed only if the time to reach a new stationnary state was great compared to the time interval between two storms. However, it can be demonstrated that, for the values of the kinetic parameters considered in the models. a new stationnary state would be attained after approximately 10 days (Vanderborght and Billen, 1976). which is much less than the time interval between two storms. This is also confirmed by the existence of a sharp maximum of nitrate concentration in the upper layer.
In addition to the effects of bottom turbulence, the biological activity can also be suggested to explain the large increase of the mass transfer coefficient in the upper layer of the sediment in shallow waters. Bioturbation has been evoked by numerous authors in order to explain
216
C
2
4
6 h
E 2
%a Q)
‘0
10
1;
Fig.6. Experimental and computed concentration profiles of dissolved silica in the pore water of a muddy sediment (M1149,12.03.1974) for 2 -1 a set of D1 values : (a) (b) (c) lo-’; (d) cm .s 2 -7 -1 The values of D2 and k are respectively cm .s-’ and 5.0 10 s ,
.
217
0
I
0 I
10 mM SOiI
4
mM NHi I
100
pM
20
30
I
1
1 NOi
I-
200
6
A .
A
Fig.7. Experimental and computed concentration profiles of sulfate, ammonium and nitrate in the pore water of a muddy sediment of the North Sea (M1149,12.03.1974). The values of D1 and D2 are respecticm2.s-'. vely lf4and
218 intensive reworking of the solid phase marine sediments. This activity must also have an influence on the distribution of dissolved substances, but this effect is quantitatively poorly known. Some complementary experiments were thus performed in order to estimate the relative importance of the physical and the biological factors on the increase of the mass transfer. Some freshly collected sediments mixed with a solution of chemical tracer (rhodamine)were placed in an open container and replaced in their previous environment, situated at the Sluice Dock at Ostend ( = 2 m depth). The rate of tracer release was measured for different meteorological conditions. Global mass transfer coefficients for the upper layer calculated from these experiments, ranged from 2.9
cm2.s-1 in the absence of wind to 6 . 2 lo-'
weather, with a mean value of 1 . 1
cm'.~-~ for stormy
crn'.~-~ ( 1 4 experiments)(Billen,
1 9 7 6 ) . Although this experimental procedure is not entirely comparable
with the natural conditions, it shows undoubtedly the preponderant effect of the overlying water hydrodynamics upon the upper layer of the sediment.
CONCLUSION The present work suggests that careful analysis of the pore water composition can provide an interesting approach to the study of the hydrodynamical properties of the benthic boundary layer in shallow waters. There is an urgent need for more comprehensive hydrodynamic studies in this field, because of the importance of the reactions occuring in the upper sedimentary layer, and of the subsequent fluxes between the sediments and the overlying water. It becomes now apparent that the fluxes of dissolved species accross the benthic boiindary layer, have until now been largely underestimated in shallow waters, where their impact on the water composition and on the productivity may be particularly important. Fluxes classicaly computed by considering only molecular diffusion are 50 to 100 times lower than those computed with our model for the region considered.
219 REFERENCES 1.
R.A. Berner, 1971. Principles of chemical sedimentology. Mc Graw
- Hill,
New York, 240 pp. 2 . G. Billen, 1976. Etude Gcologique des transformations de l'azote dans
les s6diments marins. Universite Libre de Bruxelles, Ph.D. Thesis. 3 . K.F. Bowden and R.C. Seitz, 1976. Velocity variations,turbulence and
stability. Working Group Report, 231-260.
I" I.N.
McCave (ed.), The
benthic boundary layer, Plenum Press, New York, 323 pp. 4 . C.M. Gordon, 1974. Intermittent momentum transport in a geophys cal
boundary layer. Nature, 248 : 392-394. 5 . A.D. Heathershaw, 1974. "Bursting" phenomena in the sea. Nature
242 :
394-395. 6 . S .J.
Kline, W.C. Reynolds, F.A. Schaub and P.W. Runstadler, 1967.
The structure of turbulent boundary layers. J.Fluid Mechanics, 30 : 74 1-773. 7 . R.C. Seitz, 1973. Observations of intermediate and small-scale turbulent
water motion in a stratified estuary, Parts I and 11. Chesepeake Bay Institute, John Hopkins University, Technical Report 7 9 . 8 . J.P. Vanderborght and G. Billen, 1976. Vertical distribution of nitrate
concentration in interstitial water of marine sediments with nitrification and denitriftcation. Limnology and Oceanography, 20 : 953-961.
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221 SEDIMENT TRANSPORT PHENOMENA IN THE ZAIRE RIVER.
J.J. PETERS Laboratoire de Recherches Hydrauliques, Borgerhout(Be1gium).
SUMMARY Sediment transport measurements have been made in the estuary of the Za'ire river since 1968. The results show that bed and channel morpholop,y is a predominant factor of the sediment distribution and transport, Vertical distribution of sediment transport near the bed reveals high rates in a layer of a thickness of a few decimeters, moving close to the bed, and a very erratic transport on the bed. At constant velocity, the influence of varying bottom shear stresses is shown in time series of bed load and suspended load. These observations could be related to the burst-sweep cycle. The successful application of Bagnold's approach for computation of sediment transport at single stations tends to prove the validity of power-related theories for large alluvial streams.
INTRODUCTION Most of the experimental laws governing sediment transport are based on the results of flume tests. There is a lack of reliable data of field
measurements especially for large rivers, partly due to the imperfection and inefficiency of devices and instruments used in sediment transport observaLions, partly to the complexity and diversity of the transport phenomena, and finally partly to the almost non stationnary conditions of natural flows. In the upstream reach of the Za'ire estuary, the flow regime may be considered as stationnary with very high discharges and because of the small tidal effect. Channels are wide and deep, and provide an exceptionnal site for investigations on sediment transport. The Zaire (or Congo) river is the second largest river of the world regarding the annual fresh water inflow to the oceans. River discharges 3 3 range from 23000 m / s to 80000 m / s and floods evolve very slowly. The
partition of the watershed at each side of the equator explains the very
222
small ratio between the extreme river discharges. The mean sediment concentration is one of the lowest of the large rivers. Since 1967, a research project of the Belgian State Hydraulic Laboratory involves the study of the improvement of the navigation in the braided and meandering area near the mouth of the Zaire estuary. Physical model studies and field investigations provided informations about the sediment transport mechanisms and allowed prediction of the future evolutions of the meanders. In this way, dredging operations could be reduced, becoming more efficient. The area under investigation is located in the estuarine reach which is 150 km long (Fig.1 and Fig.2).
The main part of the watershed of the
Zaire constitutes what is called the "central basin" (cuvette centrale) separated from the Atlantic Ocean by the "Cristal Mountains". The mean river slopes are generally very small, and only fine sands reach Kinshasa and pass through the rapids and waterfalls of the cristal Mountains to Matadi at the head of the estuary. This configuration regulates the river sediment input of the estuary. The braided area, 60 km long, begins at the harbour of Boma, 60 km downstream the harbour of Matadi. and ends at the head of a large submarine ca6on.
CHARACTERISTICS OF THE BRAIDED AREA Flow. -
3
During average floods, discharges vary from 30000 m / s in August to 3 60000 m /s in December (Fig.3). The tidal amplitude at the mouth amounts to I m, decreasing rapidly upstreams under the influence of the freshwater flow. River sediments fill up progressively the very deep submarine ca6on in which the salt wedge is located. -4
The mean surface slope drops from 10
near Boma to less than
near Malela at the downstream limit of the braided area. Mean velocities range from I to 2 m/s but surface velocities locally reach 3 m/s and more at high discharges, even in the deepest regions (30 m). Near Boma, tidal variations are small and flow conditions may be considered as almost stationnary. Near Malela, tidal variations of velocities are high, often of the same order of magnitude as the mean velocity. Shear velocities vary from 0.01 to 0.15 m/s, but are generally less than 0.10 m/s. The large bedforms, and the intricated channel morphology cause very im-
P
P
0
m
N h)
W
N N rp
Fig.2. The Zaire estuary.
mauo imV.1
80 000 Im3/51
60.W
70 000 60 000
5oOOo
50 WO LOMY)
10000
30 000 30 COO
20 wo
20 000
KlW 0
10 000
1 1 1 1 1 1 1 1 1 1 1 J
1 1 1 1 1 1 1 1 1 [ 1
1 1 1 1 i I I I I 1 I
F M A M J J A S O N 0 J F M A M J J A S O N D J F M A M J J
1967
1968
1 1 1 1 1 1 1 1 1 1 1
A S O N D J F M A M J J A S O N 1
1969
Fig.3. Fresh water inflow of the Zaire estuary.
1970
0
225
portant secondary currents and a strong turbulent flow. Sed iment s
.
Sediments entering the braided area contain pebbles, gravels, sands, and silts. The inflow rate of the sand fraction at Boma vary between 3 3 3 10000 m /day to 500000 m /day and amounts to 50000 m /day at mean river
discharge. Selective sedimentation occurs along the braided area and sediment sizes drop regularly from I mm to 0.3 nun over a distance of 40 km (Fig.4). Only the smallest particles reach the 0cean.Large differences of sediment sizes in and between cross-sections are due to the influence of secondary currents. Bed and channel morphology. Bedforms are generally large scale dunes; their wavelength and -amplitudes average respectively 100 m and 2 m. They move at a velocity ranging from 2 m to 10 m a day. At hip,h river discharges and for sediment sizes smaller than 0.5 mm, these dunes are flattened and other bedforms appear similar to small scale dunes. Their wavelengths and -amplitudes average then respectively 20 m and 1 m. The characteristics and the behaviour of these bedforms are poorly known, however correspondinp, sediment transport is always high. These bedforms exist on shoals as well as in deeper channels. During a high flood in 1968, these small scale bedforms were developping in the Northern channel, where the sediment size was smaller than 0.5 nun, while the large scale bedforms remain in the Southern channel where the sediment size was larger than 0.5 nun (Fig.5). By plotting the mean power of the flow per unit area versus sediment size, it can be seen that the small scale bedforms develop in the Southern channel at conditions of upper flow regime (Fig.6). The Froude number of the flow average 0.1,
and the accepted classification of bed-
forms would suggest a plane bed. Meanders move sometimes very quickly : concave banks in bends may erode at a rate of 100 m per year, or even more. Many rocks and rocky bars influence strongly the meandering of the different channels of the braided area, and therefore analysis of the meander characteristics are not very meaningful. Length, meander belt width, meander radius and channel width, average respectively 12 km, 3 km, 3 km, and 1.5 km. The interconnections between the different branches of the braided area, where sediments have different sizes and move at different speeds, complicate
226
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Selective sedimentation in the braided area.
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Fig.5. Bedforms evolving during the flood of 1968-1969 in two main channels of the braided area.
227
228
D Sirnons a n d F R i c h a r d s o n (1966)
__--
TU
J R Allen (1968). H Guy,D Sirnons and E R i c h a r d s o n (1966)
Puissance du fleuvc par unit4 dc surface du lit
d Diametre moyen du sediment
-
-
S t r e a m powcr
Median fall diameter
F i g . 6 . C l a s s i f i c a t i o n o f bedforms observed during the f l o o d o f 1968-1969 i n two 'main channels o f the braided area ( s e e fig.5).
229
the analysis and the prediction of the evolution of the meanders.
SEDIMENT TRANSPORT MEASUREMENTS Instrumentation and techniques. As the goal of the investigations was the improvement of .the navigation with dredging and eventually hydraulic structures, measurements were chiefly conducted in relation to the transformations of the bed morphology, i.e. near the bed. Sediments moving close to the bed are almost fine sands, containing a very small percentage of clay, and sediment transport rates are very low. For these reasons, continuous samplers were prefered to instantaneous ones.
Two*types of instruments were used : the Delft bottle (D.F.) and the Bed load Transport Meter Arnhem (R.T.M.A.),
both developped in the Nether-
lands. The Delft bottle was used in two versions. For measurements close to the bed, it was mounted on a sleigh, and the inlet, having a diameter of 0.015 m or 0.022 m depending on the velocity, may be positioned at 0.05 m, 0.15 m, 0.25 m
or 0.35 m from the bed. For the rest of the water
column, i.e. 0.40 m from the bottom to the water surface, an integrated sample is taken w i t h a suspended Delft bottle, at 4 to 8 levels depending on the total depth. At each level, the sampling time was fixed to 5 minutes. At each station four time
-
integrated samples were taken
with the Delft bottle mounted on a sleigh (D.F.21, and one time depth
-
-
and
integrated sample with the suspended Delft bottle (D.F.1).
Sample volume varies from a fraction of a cubic centimeter to a few hundreds cubic centimeters. The B.T.M.A.
sampler, sometimes called the Dutch sampler, is compo-
sed of a rigid rectangular entrance, 0.085 m wide and 0.05 m high, connected
by a diverging rubber neck to a basket of 0.2 mm mesh. The 8.T.-
M.A. samples a layer of a thickness of 0.05 m on the bottom, usually during 2 minutes.
Only the sand fraction of the sediments is withhold in these instruments. The sizes of the samples are determined with the visual accumulation tube. The Delft bottle on a sleigh and the B.T.M.A. are lowered on the
230
bottom very carefully without disturbing the bed. However, it is not possible to know if the entrance of the B.T.M.A. do not scoop the bottom
se-
diment. Therefore, four samples are taken at each station. Special attention was paid to the location of D.F.2 and B.T.M.A. on the bedforms using echosoundings. Indeed, measurements in the dead zone of the brough of a dune are meaningless.
At each station, the velocity profile was measured and the bottom shear velocity computed.
RESULTS Three types of results will be discussed : variation in bed load and suspended load discharges at constant mean velocity; distribution of transport rates near the bottom; distribution of sediment transport in a cross-section. Variation of sediment transport at constant mean velocity (Fip,.7). During periods of high river discharges, the influence of tides on the mean current is negligible at the head of the braided area. During the flood of 1973, variations of sediment transport were measured during a 7 hours period. The tidal amplitude at the sampling station was 0.03 m, and the mean velocity, 1.3 m/s, did not vary significantIy. Two Del t bottles on a sleigh mounted with the inlet at 0.05 m from the bottom, were used simultaneously in order to increase the number of observations. The distance between these instruments was about 50 m in a 3000 m wide channel with depths of about 9 m.Hourly averages of the se-
diment transport were estimated for the two Delft bottles D.F.2a and D.F.2b. and f o r the bed load sampler B.T.M.A. Using Bagnold's approach
I I I , bed
load and suspended load were calcu-
lated. The method was modified in order to provide results in an isolated station, using the power of the flow as the product of the mean velocity and bottom shear stress deduced from the vertical velocity distribution. Sediment sizes used in the calculations were hourly weighted averaged values. Bed load measured with sampler B.T.M.A. evolves erratically. Mean values are higher during the 4 hours period preceding low water, than during the 3 hours period following low water. Sediment transport rates at 0.05 m from the bottom measured with samplers D.F.2 are less irregular,
01
I
IEZ
but mean values have the same order of magnitude as bed load data. Suspended load measured with sampler D.F.l varies regularly, higher values being observed before low water. Measured sediment transport rates and computed sediment transport capacities of bed load b' evolve similarly. The ratios between these transport capacities computed with the modified Bagnold's approach and sediment transport rates measured with the samplers, are just given here a s an indication, these data havinE not the same meaning. The influence of bottom shear stress seems to be clearly demonstrated. Unfortunately, direct measurements of Reynold stress and turbulent kinetic energy could not be achieved in the boundary layer. Nevertheless the observed variations could be related to a burst
-
sweep cycle, and
further research will be oriented in this direction. Distribution of transport rates near the bed (Fig.8). Generally, sediment transport is intense in a layer of a thickness of a few centimeters to a few decimeters, moving close to the bed. It is difficult to make a distinction between bed load and suspended load, but probably most of the solid particles in this layer moving by saltation rather than suspended, contribute to the displacement of the bedforms, the large variations of bed load sampled with the B.T.M.A.
sampler, and
the less erratic transport measured with the D.F.2 sampler in this layer strongly suggests a possible contribution of the burst
-
sweep cycle to
these phenomena. Distribution of sediment transport in a cross-section. Several water and sediment discharp,e measurements were performed in various channels in order to relate their spatial distribution and their variability during floods. The sediment input of the main channel of the braided area is controlled in the cross-seceion of Ntua-Nkulu, where the river discharge represents 86 2 of the total fresh water inflow of the estuary. An example of field data and calculations is given in figure 9. The plume of suspended particles observed near the left bank is due to the presence of a rock, inducing intense secondary currents. The remarkable distribution of sediment sizes in the 1800 m wide cross-section is also due to morphological factors and an analysis of sediment transport distribution should take these factors into account. The agreement between sediment transport rates measured with samplers D.F.2 and D.F.1,
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234
% ’,\
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Fig.9. Distribution of sediment transport in the control section of Ntua-Nkulu. Gauging of November 1973.
235
and calculated sediment transport capaclties is satisfaztory, with exception for stations 1 1 , 7 and 6 , for the reasons mentioned above. Using Bagnold's approach with mean characteristics of flow and sediments, calculated total transport capacity amounts to 245000 m of 145000 m
3
3
per day instead
per day using data of each station, while sediment transport
measured with D.F. samplers amounts to 82000 m
3
per day.
CONCLUSIONS Sediment distribution and
-
transport in the Zaire estuary seem to
be controlled by morphological factors. Measurements performed with Dutch samplers B.T.M.A.
and Delft bottles, provide useful informations
about the spatial and temporal distribution of transport.
The burs-
ting phenomenon is able to explain the very erratic transport observed on and near the bed, the existence of a layer close to the bed where
sediment transport is intense, the difficulty to make distinction between bed load, suspended load, and sediment participating to the bed transformations. In further i;.vestigations, an attempt will be made to relate sediment transport to the Reynold stress and turbulent kinetic energy. In large alluvial streams, computations of sediment transport rates with mean flow and sediment data are not satisfactory. Even if Ragnold's approach is somewhere subject to criticism, the application of his formula to the Zaire estuary is almost successful, and tends to confirm the validity of power-related theories.
REFERENCES
II1
R.A. Bagnold, 1966. A n Approach to the Sediment Transport Problem from general Physics. United States Geological Survey Professional Paper 422-1, Washington, 37 pp.
ACKNOWLEDGEMENTS The author wishes to acknowledge Mr.A.Sterling, Director of the Belgian Hydraulic State Laboratory for his continuous stimulation during this work. He is also indebted to Dr.R.1Jollast for helpful discussions. This work is supported by the Belgian Foreign Ministry. Field measurements were performed with the help of the RdRie des Voies Maritimes of the Rdpublique du Zaire.
236 SYMBOLS USED 1N.FIGURES 7 AND 9 b
Transport rate of solids (total load) calculated with modified Bagnold's approach.
b'
Transport rate of solids (bed load) calculated with modified Bagnold's approach.
b"
Transport rate of solids (suspended load) calculated with modified Bagnold's approach.
B.T.M.A.
Sediment transport rate measured with samplers B.T.M.A., D.F.2 and D . F . I .
D.F.2 D.F. I
d50
Size of sediments sampled with B.T.M.A.,
(d50)
Size of bed load used in calculations with modified Bagnold's approach.
h
Water depths.
M.H.
Hourly average data.
9
Water discharge per unit width = u m Solid discharge per unit width.
% Rts
Rc
D.F.2
or D . F . I .
. h.
Ratio between sediment transport rate of solids calculated with Bagnold's approach and sediment transport rate measured with Dutch samplers. b' b Rc a Rt D.F.2 + D.F.l D.F.2
-
T.L.
Tidal level.
U
Mean velocity.
m
Uf
Velocity measured at 0 . 4 m from the bottom.
U
Velocity measured at 0.3 m from the water surface.
V.H.
Hourly data.
T
Bottom shear stress. T=PU:.
237
BOTTOM BOUNDARY LAYER OBSERVATIONS IN THE FLORIDA CURRENT Georges L. Weatherly Dept. of Oceanography, Florida State University Tallahassee, Fla. 32306 U.S.A. ABSTRACT This is a report of some bottom boundary layer (BBL) observations made in the Florida Current over a 7 day period at one area in the Florida Straits. The location is about 82 km away from the area where Weatherly (1972) made similar BBL observations and in comparison is in a region of more uniform topography and closer to the axis of the Florida Current. During the first part of the experiment the bottom 5 and more variable currents were weaker (mean speed ~ 1 cm/s> than in the latter part of the experiment (mean speed ~ 4 0 cm/s). For the first part uR typi al=.45 cm/s and for the latter part uR tyRic 1z.80 cm/s. As in Weatherly (1972) persistent veering tge correct sense for Ekman veering was observed in the logarithmic layer which for the first part of the experiment had mean value of ~7~ and f o r the latter part ~ 2 7 ~ Since . the mean current direction at the top of the logarithmic layer was aligned approximately in direction of isobaths it is inferred that, as in Weatherly (19721, most of the Ekman veering occurred in the lower part of the boundary layer. INTRODUCTION In Weatherly (1972) (hereafter referred to as W) are reported some observations of the bottom boundary layer (BBL) of the Florida Current at one location in the Florida Straits. Not unexpectedly this study indicated that the BBL had many features of a stationary turbulent Ekman layer. Specifically, these observations indicated a BBL of thickness h = .4u / f = 25m, where uR = the friction velocity and f = the Coriofis parameter; a total average Ekman veering a o = sin-l (Cu /Vg) = loo, where C = an emperical constant = 4.5 and V = *he geostrophic velocity outside the BBL; and a geostr%phic drag u,/V = 04. The surface Rossby number Ro = coefficient cf V /fz where z o = th!? bottom roughness parameter, was x and the above values of .a and cf are not inconsistent with the BBL being a turbulent Ekman layer characterized by such a value of Ro (Deardorff (1970)). Certain features, however, of the study of W were inconsistent with the BBL being a quasi-stationary turbulent Ekman layer. The observed Ekman veering was strongly and directly proportional to Ro rather than being a weak function of R0-l. In addition all the mean veering occurred in the lower part of the BBL rather than in the upper part. Fig. 1 shows the location where W made his observations (hereafter referred to as Site A). In order to test whether the above, unexpected features about the observed Ekman veering could be due to the location of the observations a similar Q!
?66
238
experiment w a s r e p e a t e d a t a n o t h e r s i t e a l s o i n d i c a t e d i n F i g . 1. A t t h e s e c o n d s i t e ( h e r e a f t e r r e f e r r e d t o as S i t e B) t h e b o t t o m t o p o g r a p h y , as i n d i c a t e d i n F i g . 1, b o t h u p s t r e a m ( s o u t h ) and downstream ( n o r t h ) as w e l l a s c r o s s - s t r e a m i s
F i g . 1. S i t e of e x p e r i m e n t o f W e a t h e r l y ( 1 9 7 2 ) ( A ) and o f e x p e r i m e n t r e p o r t e d h e r e (B) i n t h e S t r a i t s of F l o r i d a . Depths a r e i n f a t h o m s .
239
more regular. Site A is in the Miami-Bimini transect of the Florida Straits, a region characterized by a horizontal constriction in the Straits, a shoaling up of the bottom in the middle of the Straits, and a deep trench in the western region of the Straits. The purpose of this paper is to report the BBL observations made at Site B and to compare them with those made at Site A by W. REVIEW In the preceeding section some of the results. of W were discussed as background for the present study. In this section the results of this study are summarized in more detail. Bottom currents. The mean (over a 64 day period) current 14m a b o v e h e bottom had a magnitude of ~ 1 cm/s 0 and direction ~ 3 4 3 ~ This . direction is aligned approximately along the direction of isobaths which in vicinity of Site A, looking . of the low frequency, large downstream, is ~ 3 4 0 ~ Much amplitude fluctuations in the speeds were due to the K1 and O1 constituents of the tidal currents as predicted using the amplitudes and phases given by Smith et a1 1969. The highest 0 speeds were ~ 3 cm/s. Friction velocit A typical value of u* was ~ 0 . 4 cm/s; this v a l d z i h t to be representative for the Florida Current at this site. Peak values around 1.0 cm/s were observed. The above u* values were computed using a value of 0 . 4 0 for von Karman's constant, as were the values for Site B given later. If one takes von Karman's constant 20.35 (cf. Businger et a1 1971) then these u* values should be scaled by the faczra.875. The bottom roughness parameter Rou hness ammeter z z w a k o h d ? variable. However, for periods o? comparatively strong, steady flow a value zo = .03 cm was found. Such a zb value suggests that the bottom roughness elements had sizes Q 1 cm (cf. W Eq. (6)) which was consistent with bottom photographs. Geostro hic drag coefficient 9. With u*=.4cm/s and Vg=lO&average current speed at 14 cm above the bottom, cf I u /Vg = .04. This value is consistent with a steady, neutraf ly stratified planetary boundary layer characterized by a surface Rossby number Ro = V /fzo = 5x106 (Dearg dorff (1970)). Lo arithmic la er thickness. A logarithmic layer of thickw h s e r v e d . This is consistent with the relation 61 given in W for a neutrally srratified turbulent Eknan layer. However, as pointed out by M. Wimbush (personal communication) the theoretical justification for the expression (cf. Monin and Yaglom 1970 Eg. (6.61)) is incorrect. Emperically the logarithmic layer thickness is 10%-15% the thickness of a neutrally strati,ied planetary boundary layer (cf. Businger and Arya (1974) p. 79). The observed 6in is also consistent with this relation (see BBL thickness below). BBL thickness. Speed profile data indicated that a representative thickness of the BBL was 25m. This is consistent with the relation for the thickness of a neutrally stratified planetary boundary layer h = .4u*/f.
.
Q
240
The mean v e e r i n g f o r t h e e x p e r i m e n t w a s Ekman v e e r i n g . 100 i n t h e c o r r e c t s e n s e f o r Ekman v e e r i n g ( c o u n t e r c l o c k w i s e l o o k i n g down). T h i s v a l u e i s c o n s i s t e n t w i t h t h a t p r e d i c t e d by s i m i l a r i t x t h e o r y , u s i n g t h e c o n s t a n t of D e a r d o r f f (19701, f o r Roz5xlO However, i n c o n t r a s t t o s i m i l a r i t y t h e o r y , t h e v e e r i n g o c c u r r e d i n t h e l o w e s t p a r t of t h e BBL ( t h e l o g a r i t h m i c l a y e r ) r a t h e r t h a n above i t , and w a s d i r e c t l y p r o p o r t i o n a l t o Ro r a t h e r t h a n i n v e r s e l y p r o p o r t i o n a l .
.
EXPERIMENT The o b s e r v a t i o n s r e p o r t e d h e r e were made i n a r e g i o n a b o u t 26O30'N, 79O4O'W where t h e bottom t o p o g r a p h y i s r e g u l a r b o t h i n t h e l o n g - i s o b a t h and c r o s s - i s o b a t h d i r e c t i o n ( c f . F i g . 1). I n t h e c r o s s - i s o b a t h d i r e c t i o n s t h e bottom s l o p e s downward toward t h e east w i t h a s l o p e % - 1 . 5 ~ 1 0 ' ~ . The n e a r b y u n i f o r m l y spaced and s t r a i g h t i s o b a t h s a r e a l i g n e d %go e a s t of n o r t h which i s assumed t o be t h e a p p r o x i m a t e d i r e c t i o n of f l o w o f t h e mean bottom c u r r e n t s j u s t above t h e BBL. According t o H o l l i s t e r (1973) t h e bottom i n t h i s r e g i o n i s a s i l t y sand o f c o m p o s i t i o n 60%-80% c a r b o n a t e sand o f d i m e n s i o n s .062mm 2.0mm and t h e r e m a i n d e r s i l t of d i m e n s i o n s .004mm .062mm. The w a t e r d e p t h a t S i t e B i s a b o u t 640 m . The two moorings d e s c r i b e d i n W were s e t a t S i t e B. They a r e b r i e f l y d e s c r i b e d h e r e , f o r more d e t a i l see W. One mooring, d e s i g n a t e d C I I , c o n s i s t e d o f n i n e Savonius r o t o r s l o c a t e d a t t h e h e i g h t s above bottom g i v e n i n T a b l e 1. The o t h e r mooring, d e s i g n a t e d CM, c o n s i s t e d of four Geodyne f i l m c u r r e n t meters; t h e h e i g h t above bottom o f t h e Savonius r o t o r and vane of e a c h c u r r e n t m e t e r i s g i v e n i n T a b l e 1. The f u n c t i o n of t h e C I I mooring w a s t o make d e t a i l e d s p e e d p r o f i l e neasurements from which t h e h e i g h t o f t h e l o g a r i t h m i c l a y e r (and hence u* and zo from Eq. (1)) and t h e h e i g h t o f t h e BBL c o u l d be i n f e r r e d . The CM mooring w a s deployed t o i n f e r Ekman veering.
-
-
T a b l e 1. H e i g h t s above t h e bottom o f t h e mid-point o f t h e c u r r e n t s e n s o r s on e a c h mooring. The u n c e r t a i n t i e s of t h e s e h e i g h t s i n c l u d e t h e a c c u r a c y w i t h which t h e y were measured and e s t i m a t e d s e t t l i n g of e a c h mooring - i n t o t h e bottom. CII Savonious Rotor
1 2 3 4 5 6 7 8 9
Mooring
CM Mooring
Height (m)
1.07 1.43 1.79 2.21 3.24 4.28 6.31 10.38 20.40
f .03 f .03 f .03 f .01, fi . 0 4 f .04 f
.04
f .05 f .06
Current Meter CM1 CM3 CM18 CM34
Rotor Height
(m> 1.12 2.93 16.49 32.78
f .03
-+ .03 f .10 f .15
Vane H e i g h t (m)
1.34 3.16 17.81, 34.18
f f f f
.03 .03 .10 .15
241
The speeds measured by the CII rotors were recorded every 178 seconds ( % 3 minutes). The current meters were set on the continuous mode which in this case resulted in the current direction being recorded every 5 seconds and the rotor revolutions being recorded continuously. The film record from the CII mooring was read by eye. The current meter film records were read by an automated film reader by the manufacturer1. From this data one-minute averaged current speeds and directions were formed, and it is these one-minute averaged current meter values that were used as input data in this study. In order to obtain information on the temporal variability of temperature in the BBL two thermistors were placed on the CM mooring, at 5 m and 19 m above the bottom, and aqthird thermistor was placed on the CII mooring at 20 m above the bottom. The thermistor packages, made by C. Wilkins at Nova University, Dania, Florida, were self-contained units in their own pressure housings and recorded continuously on a Rustrack recorder with an absolute accuracy of s.l°C and a The sensors were Yellow Spring relative accuracy of s.02'C. model 44-00-33 thermistors. The three thermistor instruments returned with full temperature records; their temperature data was digitized every 15 minutes. A freefall STD (see W p. 58) was to have been used to provide data on the density distribution in the BBL as well as to calibrate the drift in the thermistor instruments. However, the STD developed major problems at the beginning of the experiment and no STD profiles were obtained. Five bottom color pictures were taken with a freely dropped camera near Site B after the experiment. A l l were taken within several hours of each other and indicated a silty sand bottom somewhat smoother than that seen at Site A by W. Fig. 2 is a black and white reproduction of one of the photographs. The moorins were set on 20 July 1972 and recovered on 27 July 1972. The CM moor'ing was set 4.2 km from the CII mooring in a direction 22O west of north. Approximately 164 hours (6.8 days) of speed data was obtained from the CII mooring. No data was obtained from the lowest rotor because a wire was damaged during launch. The second rotor from the bottom failed to give data between hours 42 and 85 of the experiment. It is not known whether this was due to something obstructing the rotor or to an intermittent, unknown electrical problem. Current meters CM1, CM3, CM18, and CM34 (at heights above the bottom, respectively, 'L 1, 3, 18, and 34m) returned, respectively, 162, 158, 49, and 0 hours of current speed and direction data.
Geodyne Division, EGEG International, Waltham, Massachusetts.
242
P i g . 2. R e p r o d u c t i o n o f c o l o r p h o t o g r a p h made 1. 1 . 8 m above Note c l o u d o f s i l t a b o u t t h e b a l l a s t t h e b o t t o m a t S i t e B. weight.
RESULTS AND C O N C L U S I O N S
Bottom c u r r e n t s . S i m i l a r t o t h e observations a t S i t e A by W , t h e o t t o m c u r r e n t s a t S i t e B d u r i n g t h e p e r i o d o f t h e e x p e r i m e n t c a n be c h a r a c t e r i z e d by a mean n o r t h w a r d f l o w o r i e n t e d a p p r o x i m a t e l y a l o n g i s o b a t h s . However, u n l i k e t h e s t u d y o f W , t h e mean f l o w , which i s a s s o c i a t e d w i t h t h e F l o r i d a C u r r e n t , changed d u r i n g t h e e x p e r i m e n t . The a v e r a g e s p e e d a t 20 m above t h e bottom f o r t h e f i r s t 72 h o u r s o f t h e e x p e r i m e n t w a s 1 5 . 4 c m / s , and f o r t h e remaind e r of t h e e x p e r i m e n t t h e a v e r a g e s p e e d a t t h i s h e i g h t w a s F i g . 3 shows 4-hour a v e r a g e d s p e e d s a t z = 2 0 m p l o t 3 8 . 5 cm/s. t e d as a f u n c t i o n o f t i m e . A l s o shown a r e t h e p r e d i c t e d t i d a l c u r r e n t s u s i n g t h e a m p l i t u d e s and p h a s e s g i v e n i n Smith et a 1 (1969) p l u s , r e s p e c t i v e l y , a mean s p e e d o f 1 5 . 4 cm/s and 3 8 3 cm/s. A s c a n be s e e n from F i g . 3 f o r t h e f i r s t 1. 72 h o u r s t h e bottom c u r r e n t s c a n be c h a r a c t e r i z e d by a mean c u r r e n t o f s p e e d 1.15 c m / s modulated by t h e p r e d i c t e d , p r i m a r i l y d i u r n a l , t i d a l c u r r e n t o f a m p l i L u d e ~ 1 c2m / s ; f o r t h e r e m a i n d e r o f t h e e x p e r iment t h e y a r e c h a r a c t e r i z e d by a mean c u r r e n t w i t h s p e e d 1.39 c m / s modulated by t h e same t i d a l c u r r e n t . The f i r s t 72 hours of t h e experiment, s t a r t i n g a t 1155 2 5 minutes l o c a l s t a n d a r d t i m e on 20 J u l y 1972, h e r e a f t e r i s d e s i g n a t e d as P e r i o d 1; t h e r e m a i n i n g 1.92 h o u r s o f t h e e x p e r i m e n t h e r e a f t e r i s d e s i g n a t e d a s P e r i o d 2. I n F i g . 4 a r e shown p r o g r e s s i v e v e c t o r d i a g r a m s f o r t h e c u r r e n t meters a t 3 and 1 8 m above t h e b o t t o m , CM3 and CM18. For CM18 o n l y 1.49 h o u r s o f d a t a was o b t a i n e d and t h u s o n l y t h e
243
TIME (hours) Pig. 3 . Time series of 4-hour averaged speeds at 20m above the bottom solid curve. The dotted and dashed curves are of a mean current of speed 15.4 cm/s and 39.5 cm/s, respectively, modulated by the predicted tidal current. Time 0 is 1 1 5 5 EST 20 July 1972.
_first 4 9 hours of data from CM3 is shown in this figure. A l though there is much variability, the flow at both levels is similar in that the direction of mean flow is about the same, slightly east of north, and oriented approximately along isobaths (cf. Fig. 1). The mean current directions at these two levels for this period is given in Table 2.
KM NORTH
00
20 I
1
M 3
a.
-0
L o
..
.Ao
0
0
0 . 00.
10 L Fig. 4 . Progressive vector diagram for current meters moored 3 and 18m above the bottom f o r the period when the current meter at 18m functioned. Symbols ( O f o r z = 3m, A for z = 18m) are given after each 24 hour period.
244
I n F i g . 5 a r e shown p r o g r e s s i v e v e c t o r d i a g r a m s f o r t h e f u l l r e c o r d s from t h e c u r r e n t meters a t 1 and 3 m , C M 1 and C M 3 . A s can be s e e n from t h i s f i g u r e i n t h e l a t t e r p a r t o f t h e e x p e r i m e n t , i n P e r i o d 2 , n o t o n l y i s t h e c u r r e n t s t r o n g e r it i s l e s s v a r i a b l e i n d i r e c t i o n as w e l l . A r e p r e s e n t a t i v e BBL t h i c k n e s s f o r P e r i o d 1, d u r i n g which CM18 f u n c t i o n e d f o r ~ 6 7 % of t h e t i m e , was % 2 0 m (see s u b s e q u e n t BBL T h i c k n e s s d i s c u s s i o n ) . S i n c e CM18 w a s a t 18m, it f u n c G n e d t o r a p p r o x i m a t e l y an i n t e g r a l number of d i u r n a l t i d a l p e r i o d s , and i t s a v e r a g e d i r e c t i o n of f l o w was a p p r o x i m a t e l y p a r a l l e l t o t h e i s o b a t h s , it i s i n f e r r e d t h a t t h e mean b o t t o m c u r r e n t d i r e c t i o n j u s t above t h e BBL f o r P e r i o d 1 w a s %15O, t h e a v e r a g e d i r e c t i o n v a l u e g i v e n f o r CM18 i n T a b l e 2 . Although no c u r r e n t d i r e c t i o n d a t a n e a r t h e t o p o f t h e BBL was o b t a i n e d d u r i n g P e r i o d 2 it seems r e a s o n a b l e t o e x p e c t t h a t t h e mean bottom c u r r e n t d i r e c t i o n f o r t h i s period w a s a l s o p a r a l l e l t o t h e d i r e c t i o n o f t h e i s o b a t h s (%,9O e a s t of n o r t h l o o k i n g downstream) t o + 6 O . m.
1 I
__.... e __.....,
u 0
5 I
m.
(..._........' __.. I
0.'
-..ax.''
'".. .? 1
'. . .
......
'0.
.......-
Fig. 5. P r o g r e s s i v e v e c t o r d i a g r a m from c u r r e n t meters moored 1 and 3 m above t h e bottom. Open c i r c l e s a r e drawn a f t e r each 24 hour i n t e r v a l .
That t h e bottom c u r r e n t s c a n b e c l a s s e d i n t o two f l o w regiemes i s a l s o i n d i c a t e d i n t h e t e m p e r a t u r e r e c o r d s . In F i g . 6 a r e shown t h e t e m p e r a t u r e t i m e s e r i e s o b t a i n e d from t h e t h e r m i s t o r s on t h e CM mooring. During P e r i o d 1 t h e tempe r a t u r e was w a r m e r ( b y % 0 . E o C ) and more v a r i a b l e t h a n i n Period 2 . The peaks i n P e r i o d 1 a r e a s s o c i a t e d w i t h p e r i o d s of weaker, e a s t w a r d f l o w . The l o g a r i t h m i c l a y e r a t S i t e B was Logarithmic J a y e r . e x p e c t e d t o be t h i c k e r t h a n a t S i t e A b e c a u s e t h e b o t t o m c u r r e n t s a t t h e f o r m e r s i t e were e x p e c t e d t o b e l a r g e r s i n c e it i s c l o s e r t o t h e a x i s of t h e F l o r i d a C u r r e n t . The o b s e r v a t i o n s showed t h a t a t S i t e B t h e bottom c u r r e n t s w e r e i n d e e d s t r o n g e r . The l o g a r i t h m i c l a y e r was a n t i c i p a t e d t o be t h i c k e r by t h e f o l l o w i n g l o g i c . S i n c e u,/V i s very nearly a c o n s t a n t f o r a l a r g e r a n g e o f R o , t h e t h i g k n e s s of t h e BBL
245
I
W OZ
3
s [L
w
5
1
1
I
I
I
I
1
+
a I W
I-
5 0
40
80
I20
I60
TIME (hours) Fig. 6. T i m e series p l o t s of t h e t e m p e r a t u r e r e c o r d s o b t a i n e d a t 1 9 m ( a ) and 5 m ( b ) above t h e b o t t o m . The c o a r s e a p p e a r a n c e i s due t o t h e l a r g e t i m e i n c r e m e n t ( 1 5 m i n u t e s ) u s e d when d i g i t i z i n g t h e cont,inuous t e m p e r a t u r e r e c o r d s .
w a s e x p e c t e d t o be h = . 4 u , / f , t h e t h i c k n e s s of t h e l o g a r i t h m i c l a y e r 61n = ( . 1 0 - . 1 5 ) h , and V was e x p e c t e d t o be a p p r e c i a b l y l a r g e r , t h e t y p i c a l 6 l n fog S i t e B should be l a r g e r t h a n f o r S i t e A. Using t h e r e l a t i o n g i v e n i n W , 6 l n = 2u,*/fV , which a l t h o u g h b a s e d on q u e s t i o n a b l e grounds g i v e s r e a s o n a g l e a n s w e r s , and t a k i n g u,/Vg = . 0 4 , a r e p r e s e n t a t i v e v a l u e f o r S i t e A ( s e e W F i g . 16) V = 1 5 - 4 0 c m f s , ( s e e p r e c e e d i n g sec i o n ) , and f =. 6 3 ~ 1 0 - ~ ; - ~ ~ p r e d i tc ht as t f o r S i t e B 4-llm $ The o b s e r v e d l o g a r i t h m i c l a y e r t h i c k n e s s , 5 m i n P e r i o d 1 and -3.2m i n P e r i o d 2 , w a s t h i n n e r t h a n e x p e c t e d . The t h i c k n e s s o f t h e o b s e r v e d l o g a r i t h m i c l a y e r as e x p e c t e d i n c r e a s e d w i t h i n c r e a s i n g c u r r e n t s p e e d . A t S i t e A b i n w a s s e e n t o be a s l a r g e a s 8m i n p e r i o d s o f s t r o n g f l o w .
-
.
The r e l a t i o n 6 1 = 2 ~ , ~ / f V i s e s s e n t i a l l y t h e same a s t h e frequently e x p r e s s i o n 6 1 n = ? . 1 0 - . 1 5 ) h For v a l u e s o f cf=u,/V e n c o u n t e r e d i n t h e a t m o s p h e r i c and t h e o c e a n i c boftom boundary layers.
246
. .
. .
*
. .
.
NUMBERS DENOTE TIME IN uouas
.Fig. 7. Four consecutive profiles of hourly averaged speeds on In z scale for every other hour.
In contrast, at Site B the largest value of 61n observed was -4m. Examples of logarithmic layers are given in Fig. 7. Friction velocit and rou hness arameter 5 . Because the l o g a r i t h m i d t % c E s & b e c t e d to be s5m or greater six rotors were placed within 5m of the bottom. Even with no data from one rotor this would give five points through which to fit the equation u(z) = u*/k
In z / z o ,
(1)
where u(z)=observed speed at height z above the bottom and k = von Karman's constant, which for comparison with W is taken to be . 4 0 , to determine u* and z o . However, as noted previously
247
61n was observed to be ~ 2 . 5 mto 3.2m, the rotor which failed was the lowest one, and the second lowest rotor did not always function properly. Thus typically usually three and sometimes four speed measurements were made in the logarithmic layer. For comparison W typically had five speed measurements in the logarithmic layer. As a result the u* and z0 values presented here are not considered as good as those determined by W at Site A. In Fig. 8 a,c are shown histograms of u* values determined from Eq(1) using hourly averaged speeds from Rotors 2 and 3 for periods when Rotor 2 functioned and the speeds in the logarithmic layer >4cm/s, the approximate thresh-hold speed for a Savonius rotor. For comparison histograms of uitvalues for the same period computed using for input data hourly averaged Rotor 3 and 4 speeds are shown in Fig. 8 b,d. To ~ 0 . 1cm/s the peaks in Fig. 8 a,b and Fig. 8 c,d occur at the same value of u*. For reasons given below the u* values determined for Rotors 2 and 3, the lowest rotors, when available, are thought to be better than those determined from Rotors 3 and 4. However, Rotors 3 and 4 functioned through-out the experiment while Rotor 2 did not. In Fig. 8a, are shown histograms of u* values determined from Rotors 3 and 4 hourly averaged speeds for, respectively, Periods 1 and 2. For Period 1 the peak is at u* = .45cm/s and for Period 2 it is at u* = .80cm/s. These values, thought to be representative for the Florida Current, may be underestimated by O.lcm/s since they were determined from Rotor 3 and 4 speeds (compare Fig. 8 a,b and Fig. 8 c,d). With u* = .45cm/s and Vg = 15cm/s, = .8Ocm/s and V 39cm/s, 6ln 3 2 ~ , ~ / f V = 4.3m, 5.2ma;odruperiods 1 and 2 , rgspectively. That the obgerved 6ln was less than predicted may be due to density stratification being significant in the BBL (cf. Monin and Yaglom (1970) Fig. 52). Without concurrent STD profiles or detailed bottom photographs it is not possible to infer if the BBL was stratified o r to infer the source of the stratification (temperature and salinity or suspended sediments). If stratification were important in the lower part of the BBL then Eq.(l) should include a linear correction term (E Eq. 7.33) Q
u(z)
u,/k[ln(z/zo)
+ Az)
(1')
where A is a measure of the stratification. The linear correction term in (1') is smaller for smaller z, thus the ub values determined from Eq.(l) using Rotors 2 and 3 speeds may be more accurate than those determined using Rotors 3 and 4. The geostrophic drag coefficient cf u+!Vg for periods 1 and 2, using the u* and V values used previously is .030 and .021, respectively. Sifice c is a slowly varying function of Ro = V /fzo (cf. W Fig. 16) tf;e rather large difference in the cf va'iues for these periods cannot be due to changes in Vg alone. Such a change in cf implies a change in Ro at least several orders of magnitude with Ro being considerably larger for Feriod 2. This suggests that z for Period 2 was appreciably smaller than for Period 1. fn Fig. 9 is shown zo as a function of time for those periods when Rotor 2 functioned
248
= 0
0
0
1
2
0
1
2
1
2
3
0
1
0
2
1
0
2
3
4
1
2
3
u, ( c m / s ) Fig. 8. Histogram of u* values determined from E q (1) and hourly averaged (a) Rotors 2 and 3 speeds for intervals in Period 1 ( 0 < hour d 7 2 ) when Rotors 2 worked and Rotor 2 speeds > 4cm/s, (b) Rotors 3 and 4 speeds for the same intervals in (a), (c) Rotors 2 and 3 speeds for intervals when Rotor 2 worked in Period 2 ( 7 2 < hours < 1 6 4 ) (d) Rotors 3 and 4 speeds for the same intervals in (c), (el Rotors 3 and 4 speeds in Period 1, and (f) Rotors 3 and 4 speeds in Period 2 .
properly and the speeds in the logarithmic layer >4 cm/s. Hourly averaged speeds were plotted as a function of In z and the zero intercept ( z o ) determined from straight lines fit by eye. This subjective method of estimating z,may yield values off by as much as an order of magnitude. Nonetheless some patterns are discernable in Fig. 9. The z o values for Period 1 are generally at least one order of magnitude larger than for Period 2 . Peaks ii: the z values often are associated with maximums in the currents ?cf. Figures 3 and 8, lines ~ 2 4 , 100, 1 2 5 , and 1 5 0 hours). After hour 1 0 5 in Period 2 there is a tendency for zo to decrease with time. Thus while the bottom at Site A appears to be characterized by a zo z.03 cm no comparable single value appears to be charcterize the bottom at Site B. The bottom at Site A is quite different than that at Site B. At Site A the bottom is a basement rock on which there is a thin film of sediment which does not always cover the hard bottom (personal communication C. Neuman, D. Cacchione and W. Gardner). At Site €3 the bottom consists of a silty sand of sufficient thickness such that no underlying hard surface is exposed (personal communication D. Cacchione and W. Gardner). In the vicinity of this site alternating strips of rippled and non-rippled Q
249
_Fig. 9. Time series plot of z o values read from plots of hourly averaged speed profiles on In z scale (cf. Fig. 6 ) for periods wIIenRotor 2 worked and Rotor 2 speeds 7 4cm/s. sediment, oriented downstream with widths order tens to hundreds of meters, are sometimes observed (ibid.). M. Wimbush Tpersonal communication) has observed small ripples of heights ?r 4cm forming, migrating and eroding away at a deep site in the Florida Straits, 26O6'N, 79031'W, where the bottom is also a silty sand. In addition he observed sediment going into suspension during periods of strong current. The reason for the large zo values in Fig. 9 for Period 1 is not known. zo values ranging from 1-10 cm may imply large bed forms with amplitudes .3 to 3m if the emperical relation zo =dl30 for rough surfaces ussd in W is assumed. Large sand ripples with amplitudes of several meters have been observed south of Site B in the Miami Trough (personal communication D. Cacchione and W. Gardner). However, no such features have been observed in the vicinity of Site B. It has been assumed in this study that the anchor weight on the CII mooring did not sink appreciably into the bottom. Assuming that it did sink as much as 40 cm only reduces the z o values in Period 1 by about a half. During times 8 4 hour < 130 zo 2, .lcm to within an order of magnitude. This value in not inconsistent with the bottom
250
being roughened by small ripples of amplitude % 4cm. F o r times 130 < hour < 164 zo % .001cm to within about an order of magnitude. This value is not inconsistent with the bottom being smooth and un-rippled. That there is a trend for zo to decrease after hour % 105 may be indicative of eroding small-scale ripples. Smith (1976) gives the following relation for zo when there is bedload transport 20
= 26.4 (Tb-Tc)/(Ps-P)g
+Zn,
(2)
where T b = the bottom stress (PU,~), = the critical bottom stress for initiation of bedload transport, p s = the sediment density, p = the water density, g = gravitational acceleration, and zn = the Nikuradse roughness parameter. The coefficient was determined for the quartz sand bottom of the Columbia River and is not expected to apply for the silty-sand bottom at Site B. However, Eq.(2) states that z o should vary directly In Fig. 10 u* values given in Fig. 8 with Tb and hence u*. a,c are plotted as a function of time. A comparison of this figure with Figures 3 and 9 shows that at times of strong current (e.g. hours % 24, 100, 1 2 5 , 150) zo is directly proportional to u* suggesting that during these periods bedload transport occurred.
4.0 -
3.0h
m
\
! 2.0i
V
v
S
3
I.0-
TIME ( h o u r s )
Fig. 10. Time series plot of uB values displayed in Fig. 7 a,c. Error bars are due to uncertainty in rotor heights above the bottom-and resolution of the speed values.
261
Thickness of the BBL. Using the u* values previously given as r e p r e s e n t x v e f o r Periods 1 and 2 in the expression h=.4u,/f gives a thickness of 29m and 51m for Periods 1 and 2 respectively. The speed profiles for Period 1 indicate h=20m which is slightly less than predicted. The highest z at which speed data was obtained was 20m. Consistent with the above predicted value, the speed profiles for Period 2 indicate a BBL thicker than 20m. It should be noted that the above BBL thicknesses are for mean conditions. During periods of strong flow when the tidal current reinforces the Floriaa Current h can be appreciably greater.
.
The predicted representative total Ekman Ekman veerin v e e r i G d = s i n - l ( C c f ) for Periods 1 and 2 respectively, is '8 and S C . Because of the limited data return from the current meters above z=3m (49 hours of data from the CM18 and 0 hours of data from CM34) only inferences about a. can be made from the data. Before discussing direction differences it is appropriate to discuss the accuracy of the current meter directions. The instantaneous direction values are resolved to f 2.8O. Twelve of these values were averaged to form a one minute averaged direction. Hence one might expect the one minute averaged directions to be resolved to f2.8°/(11)4 or 2, f0.84O and henze direction differences to be resolved to 2, f1.7O. Examination of the one-minute averaged direction histograms of CM1 and CM3 suggests that the directions are resolved to about a degree. The directim and veering values summarized in Table 2 are thought to be resolved to 2, f1.5O and 5 +3O, respectively. Table 2. Average currents and direction differences. The number in the current meter label is the nominal height above bottom in meters of the current meter vane. See text for explanation of time intervals and accuracy of direction and direction difference values. The average direction differences were computed by the method of Kundu (1976) and Weatherly (1972); the former values are given with correlation coefficients in parenthesis. Positive direction differences are consistent with Ekman veering. Time Interval Hours
0-49 0-49 0-49 0-72 0-72 72-158 72-158 0-158 0-158
Average Currents Current Speed Meter cm/s CM18 CM 3 CM 1 CM 3 CM 1 CM 3 CM1 CM 3 CM 1
14.2 9.2 6.4 9.2 6.6 24.2 15.8 17.3 11.4
Average Direction Differences
Direction Current Oo=North Meter Pair 14.7 26.6 17.4 18.2 8.2 12.0 345.3 13.6 351.3
Degrees
CM18,CM3-12.6(.963)-11.9 CM3,CMl 7.4(.997) 7.4
CM3,CMl
8.6C.991) 10.1
CM3,CMI
27.7(.972)
26.7
CM3,CMl
24.1(.966)
22.1
262
For the first 49 hours of the experiment, when CM18 functioned, the average veering between 3 and 18 m was ~ - 1 2 ~ where the negative sign indicates veering in the wrong sense for Ekman veering-. For the same period the average veering between 1 and 3 m was s7O. This value is nearly the same as the value for these levels for Period 1. Thus for Period 1 the average veering between 3 and 18m was probably close It is to the value for the period 0-49 hours, 1.e. -12O. interesting that the ovserved veering for Period 1 between 1 and 3m is the same, to within experimental accuracy, as the predicted .a for that period. W also found the observed and predicted average .a for his experiment to agree and to occur in the logarithmic layer, However, during Period 1 the currents were quite variable. The fluctuations in speed were comparable to the mean, the current flowed alternately eastward and northward, and the time scale of the fluctuations, 24 hours is comparable to the time scale for a planetary boundary layer 2n/f. Thus the agreement with stationary theory may be coincidental. From Fig. 5 and Table 2 it is apparent that the veering between 1 and 3m during Period 2 was substantial, ~ 2 7 ~ This . is over five times larger than the predicted .a for this period. That the observed veering was larger than that expected for a neutrally stratified BBL and that it occurred within the logarithmic layer may be indicative that the logarithmic layer was appreciably stably stratified (see the Weatherly and Van Leer paper in this volume). The veering between these levels is plotted as a function of time in Fig. 11. In particular for Period 2 and as noted also by W the veering in the logarithmic layer is directly proportional to the current speed. That the mean direction at 3m f o r Period 2 was along the direction of isobaths to within experimental accuracy suggests that little veering occurred in the BBL above z=3m during this period. Suitabiiit of the Savonius rotor as a BBL sensor. The current SDeeds : n T h n B L of the m d a C i k r e n t a r e u fficiently aiove the thresh-hold speed of =2-4cm/s to spin a Savonius rotor over 90% of the time. This is not always the case in the oceanic BBL (cf. Armi and Millard (1976)). This rotor has been calibrated for uniform flows; however, I am unaware of its being calibrated for boundary layer shear flows. Further, how much of the speed signal is due to rectification of the shear induced turbulent motions needs to be studied. For lack of information these effects have been assumed negligible. Intercomparison studies are in order to see if indeed this is the case. Q
I
253
S
‘ W
E
E
S
S
100
120
140
I 160
F i g . 11. T i m e s e r i e s p l o t o f h o u r l y a v e r d g e d c u r r e n t d i r e c t i o n ( a ) and s p e e d ( c ) d a t a from t h e c u r r e n t m e t e r a t 3m a b o v e t h e b o t t o m , a n d ( b ) h o u r l y a v e r a g e d d i r e c t i o n I n (b) p o s i t i v e values are d i f f e r e n c e between 3 a n d l m . c o n s i s t e n t w i t h Ekman v e e r i n g .
ACKNOWLEDGZMENTS The o b s e r v a t i o n s were made w i t h s u p p o r t f r o m t h e Office o f Naval R e s e a r c h u n d e r C o n t r a c t M 0 0 0 1 4 - 6 7 - A - 0 3 8 6 - 0 0 0 1 and t h e a n a l y s i s w i t h s u p p o r t from t h e N a t i o n a l S c i e n c e Foundat i o n u n d e r G r a n t GA-36458X and from t h e O f f i c e o f Naval Res e a r c h u n d e r C o n t r a c t N000-14-75-C201. P a r t of t h e a n a l y s i s was done w h i l e I w a s a v i s i t i n g s c i e n t i s t a t t h e I n s t i t u t e o f Oceanology o f t h e USSR Academy o f S c i e n c e s i n a program s p o n s o r e d by t h e N a t i o n a l Academy o f S c i e n c e s . The m o r a l and p r a c t i c a l s u p p o r t o f D r s . W.S. R i c h a r d s o n a n d P. N i i l e r d u r i n g t h e c o u r s e o f t h i s work i s g r a t e f u l l y a c k n o w l e d g e d . W . Campbell, S . F u r g a n g , D . Hunley a n d E . T a n k a r d , J r . a r e t h a n k e d for t h e i r I warmly t h a n k D r . W. a s s i s t a n c e i n making t h e o b s e r v a t i o n s . Powers, J. DeSzoeke a n d J . W e a t h e r l y for t h e i r a s s i s t a n c e i n the data analysis.
264
REFERENCES Arm?, L. and R.C. Millard, Jr. 1976. The bottom boundary layer of the-deep ocean. 2. Geophys. E . , 49834990.
e,
Businger, J.A., Wyngaard, J.C., Izumi, Y., and Bradley, E.F. 1971. Flux-profile relationships in the atmospheric surface layer. J . Atmos. 30, 788-794.
s.,
Businger, J.A. and S.P.S. Arya. 1974. Height of the mixed , layer in the stably stratified planetary boundary layer. Advances in Geo h sics, H.E. Landsberg and J. Van Mieghem, York, pp. 73-92. ed., Acad=i*ew Deardorff, J.W. 1970. A three-dimensional numeric investigation of the idealized planetary boundary layer. J. Geophys. Fluid Mech., 1,377-410. Hollister, C.D. 1973. Atlantic continental shelf and slope of the United States - texture of surface sediments from New Jersey to Southern Florida, Geological Survey Professional Paper 529-M, U.S. Government Printing Office, Washington, 23 pp. Kundu, P.K. 1976. bottom. 2.
-.
Ekman veering observed near the ocean Oceanogr., 6 , 238-242.
Monin, A.S. and A.M. Yaglom. 1970. Statistical Fluid Mechanics : Mechanics of Turbulence, MIT Press, Cambridge, Massachusetts, 769 PP
-
Smith, J.A., B . D . Zetter and S. Broiaa. 1969. Tidal modulation of the Florida Current flow. Marille Tech. J., 3 , 41-46. -
*.
Smith, J.C. 1976. Modeling of sediment transport on continental shelves, The Sea, Vol. 6, in press. Weatherly, G.L. 1972. A study of the bottom boundary layer of the Florida Current. 2. Oceanogr., 2, 54-72.
w.
266 COASTAL J E T S ,
M.
A N D PHYTOPLANKTON PATCHINESS
FRONTS,
J . BOWMAN a n d W .
E.
ESAIAS
Marine S c i e n c e s Research C e n t e r , S t a t e U n i v e r s i t y o f New York, Stony Brook,
11794
N e w York
ABSTRACT
A f r o n t a l s y s t e m h a s b e e n d i s c o v e r e d i n Long I s l a n d S o u n d , forming t h e inshore boundary o f a s t r o n g t i d a l l y induced c o a s t a l jet.
Regenerated each ebb t i d e , t h e f r o n t e x t e n d s f o r s e v e r a l around a l o c a l promontory,
kilometers
and a d j a c e n t t o a h i g h l y
p r o d u c t i v e s h a l l o w embayment. Chlorophyll-a
c o n c e n t r a t i o n s measured i n A p r i l w i t h i n t h e
j e t were t y p i c a l l y t w i c e b a c k g r o u n d ,
and suggest t h a t t h e system
may b e a n e f f e c t i v e m e c h a n i s m f o r t h e p e r i o d i c i n j e c t i o n a t t i d a l frequencies
o f h i g h c o n c e n t r a t i o n p h y t o p l a n k t o n p a t c h e s from t h e
i n s h o r e embayment i n t o t h e i n t e r i o r o f t h e S o u n d .
INTRODUCTION
Long I s l a n d S o u n d i s a m a j o r e s t u a r y some on t h e U . York,
S.
1 6 5 km i n l e n g t h ,
e a s t e r n s e a b o a r d , l y i n g b e t w e e n Long I s l a n d , N e w
and C o n n e c t i c u t .
The main c o n n e c t i o n t o t h e A t l a n t i c
Ocean i s a t t h e e a s t e r n mouth w h e r e t h e t i d e s ,
principally
d i u r n a l , are t r a n s m i t t e d t o t h e i n t e r i o r o f t h e Sound. t h e Sound, transport
strait
the tides
semi-
Within
approximate a q u a r t e r s t a n d i n g wave, w i t h
d e c r e a s i n g westward towards t h e East R i v e r ,
a tidal
(Bowman, 1 9 7 6 1 , c o n n e c t i n g t h e S ound a n d N e w York H a r b o r .
A horizontal salinity
R i v e r and an i n f l o w of t h r o u g h t h e East R i v e r ,
gradient,
derived from t h e Connecticut
low s a l i n i t y Hudson R i v e r e s t u a r y w a t e r drives a classical estuarine circulation.
L o n g i t u d i n a l h y d r o g r a p h i c s e c t i o n s ( W i l s o n , 1 9 7 6 ) show a p a r t i a l s t r a t i f i c a t i o n i n t h e c e n t r a l r e g i o n o f t h e Sound d u r i n g s p r i n g and summer; t r a n s v e r s e p r o f i l e s
across t h i s
central basin also
i l l u s t r a t e t h e s t r o n g summer p y c n o c l i n e s e p a r a t i n g s u r f a c e a n d bottom water (Cordon and Pilbeam,
1975).
Near s h o r e w a t e r s a r e
u s u a l l y w e l l m i x e d or p o s s e s s a w e a k l i n e a r v e r t i c a l g r a d i e n t .
256 These mixing zones are r e g i o n s where bottom g e n e r a t e d t u r b u l e n t t i d a l k i n e t i c e n e r g y i s s u f f i c i e n t t o mix downward t h e s u r f a c e buoyancy a r i s i n g from v e r t i c a l s a l i n i t y g r a d i e n t s Hunter,
(Simpson and
1974; Fearnhead, 1975).
The z o n e s
contract
(Gordon and Pilbeam, over the 9 m isobath.
and expand between neaus and s p r i n g s
1975) w i t h t h e boundary u s u a l l y c e n t e r e d T i d a l l y i n d u c e d f r o n t s a r e commonly
a t t h e s e mixing zone b o u n d a r i e s ;
found
t h i s paper d e s c r i b e s one such
r e g i o n w h i c h p o s s e s s e s s o m e i n t e r e s t i n g f e a t u r e s a n d w h i c h may b e i m p o r t a n t i n e j e c t i n g medium s c a l e patches
(2
-
1 0 km) p h y t o p l a n k t o n
i n t o t h e i n t e r i o r o f t h e Sound from a nearby h i g h l y
p r o d u c t i v e s h a l l o w embayment.
THE STUDY R E G I O N
The a r e a shown i n F i g u r e s 1 sistent tidal
f r o n t which
Old F i e l d P o i n t .
740
-
3 w a s chosen t o study a per-
f o r m s e a c h e b b a r o u n d C r a n e Neck
and
P r e v i o u s a e r i a l a n d s h i p s u r v e y s h a d shown
7 30
72O
v
10
MONTAWK POINT
41'
41'
ATLANTIC O C E A N
I 0 0 km
40 .- O
740
7 30
72O
F i g , 1. L o c a t o r map f o r L o x g I s l a n d S o u n d . s h e area shown i n F i g u r e 2 .
40' 71 O The i n s e t r e p r e s e n t s
257
Fig. 2. Long I s l a n d S o u n d c e n t r a l b a s i n . The i n s e t r e p r e s e n t s The s o l i d b a t h y m e t r i c l i n e i s t h e s t u d y a r e a shown i n F i g u r e 3 . 2 0 f e e t ; t h e d a s h e d l i n e i s 60 f e e t . Arrows d e l i n e a t e t i d a l c u r r e n t d i r e c t i o n s ; numerals speeds i n knots. Note t h e 1 . 4 k n o t t i d a l J e t a r o u n d Crane Neck. Local t i m e is "slack; ( f r o m NOS, 1 9 7 3 ) . e b b b e g i n s a t The Race sharp discontinuities c o n t r a s t s and
'L
i n chlorophyll-a
(easily visible color
3 : l c h l - a c o n t r a s t s ) a c r o s s t h e f r o n t a l zone
d u r i n g t h e s p r i n g d i n o f l a g e l l a t e bloom. The r e g i o n i s one of s h a r p l y s l o p i n g topography ( F i g . where d e p t h s d r o p s h a r p l y t o of
%
1 km.
35 m o v e r a h o r i z o n t a l d i s t a n c e
C r a n e Neck i s t h e e a s t e r n l i m i t o f S m i t h t o w n B a y ,
s h a l l o , ~( % 2 0 m )
embayment b o u n d e d by t h e E a t o n ' s
1 8 km t o t h e w e s t . t h e Bay
%
31, a
Neck p r o m o n t o r y
T i d a l c u r r e n t s a r e i n v a r i a b l y weak i n s i d e
(NOS, 1 9 7 3 ) .
Two u n g a u g e d s o u r c e s o f f r e s h w a t e r ,
Nissequogue R i v e r and Stony Brook Harbor,
although of small d i s -
charge represent. impartant n u t r i e n t sources.
(Long I s l a n d i s a
h i g h l y p o p u l a t e d g l a c i a l t e r m i n a l m o r a i n e ; most waste d i s c h a r g e i n the study region seepage.)
,
principally
r e s i d e n t i a l , is v i a c e s s p o o l
P r e l i m i n a r y o b s e r v a t i o n s h a v e s h o w n t h a t t h e Bay
268
F i g . 3. L o c a l s t u d y a r e a , s h o w i n g t h e s a m p l i n g t r a n s e c t , mean s u r f a c e convergence p o s i t i o n r e l a t i v e t o t h e bottom contours.
s u s t a i n s r e l a t i v e l y h i g h primary p r o d u c t i o n compared t o t h e i n t e r i o r o f t h e Sound. Tidal current charts
(NOS, 1 9 7 3 ) s h o w a l o c a l i z e d e a s t w a r d
t i d a l j e t a r o u n d C r a n e N e c k , t h a t commences some 2 1 1 2 h o u r s b e f o r e t h e t i d e i n t h e c e n t r a l b a s i n b e g i n s t o ebb ( F i g .
2).
Phase d i f f e r e n c e s due t o i n e r t i a l e f f e c t s i n t h e b a r o t r o p i c t i d e between t h e d e e p e r o f f s h o r e water and t h e s h a l l o w n e a r s h o r e
water c o n s t r a i n t h e j e t t o a narrow band e x t e n t , w i t h maximum c u r r e n t s where i n t h e c e n t r a l b a s i n .
%
Q ,
1 m sec-l,
1 . 5 km i n l a t e r a l t h e s t r o n g e s t any-
On e b b t i d e , S m i t h t o w n Bay w a t e r i s
f u n n e l l e d i n t o t h e j e t , c a r r i e d around Crane Neck,
and d i s p e r s e d
o f f s h o r e and downstream. Our h y p o t h e s i s i s t h a t t h i s t i d a l j e t i s a h i g h l y e f f e c t i v e mechanism f o r p e r i o d i c e j e c t i o n , a t t i d a l f r e q u e n c i e s , o f h i g h concentration phytoplankton patches i n t o t h e i n t e r i o r of t h e Sound.
R e s u l t s r e p o r t e d i n t h i s p a p e r do n o t d e f i n i t i v e l y
259 s u p p o r t t h i s c o n c l u s i o n ; however, w e have g a t h e r e d enough d a t a t o enJoy
some s p e c u l a t i o n on t h i s mechanism a n d t o d e s i g n f u t u r e
experiments f o r next spring.
DETAILS OF THE EXPERIMENT
(Fig. 3 )
A t r a n s e c t c o n s i s t i n g of f i v e anchored s t a t i o n s
was c o n d u c t e d o n A p r i l 1 4 , 1 9 7 6 , c e n t e r e d a r o u n d l o c a l s l a c k
water a f t e r f l o o d i n t h e c e n t r a l b a s i n . s a l i n i t y , chl-a,
and n u t r i e n t s
(NO2,
In s i t u temperature,
NO3,
P O b ) were d e t e r m i n e d
from s a m p l e s drawn w i t h a s e l f - c o n t a i n e d pumping s y s t e m (Hulse, 1975). T e m p e r a t u r e and s a l i n i t y were m e a s u r e d w i t h a P l e s s e y 6600 E s t i m a t e s o f p h y t o p l a n k t o n b i o m a s s were
T Thermosalinograph.
made w i t h a T u r n e r D e s i g n m o d e l 1 0 - 0 0 5 R
meter, .regularly
flow through fluoro-
calibrated via filtered extracts.
Frozen water
NO3 a n d
s a m p l e s were l a t e r a n a l y z e d i n t h e l a b o r a t o r y f o r N O 2 , PO4 u s i n g a T e c h n i c o n A u t o a n a l y z e r ,
C u r r e n t v e l o c i t i e s were
d e t e r m i n e d w i t h a d e c k r e a d o u t Endeco model 110 c u r r e n t meter lowered from t h e s h i p . navigation
A low f l y i n g a i r c r a f t p r o v i d e d r e a l t i m e
i n f o r m a t i o n t o t h e s h i p a n d was u s e d f o r a e r i a l
photography o f a computer c a r d seeding experiment. The e n t i r e e x p e r i m e n t l a s t e d t h r e e h o u r s .
A strong front
w a s c l e a r l y v i s i b l e b o t h from t h e a i r c r a f t and t h e s h i p d u r i n g t h e duration of the experiment.
Although
c o n s i d e r a b l e meander-
i n g was e x h i b i t e d , i t s mean p o s i t i o n , a s d e t e r m i n e d b y s u r f a c e c o n v e r g e n c e o f f l o a t a b l e s , i s s k e t c h e d i n F i g u r e 3. The component o f t i d a l v e l o c i t y t a n g e n t i a l t o t h e f r o n t
at station 5
(60' T ;
Fig.
4)
consisted of a remarkable j e t ,
limited i n horizontal extent t o c i t y c o r e ( z 7 0 cm s e c - l ) ,
z
1 . 5 km, a n d w i t h a h i g h v e l o -
3 m below s u r f a c e .
Both i n s h o r e
a n d o f f s h o r e c u r r e n t s were n e a r z e r o . Surface velocities perpendicular c o n s i s t e d o f a s h a l l o w (?.
t o the front
(Fig.
2 m), v i g o r o u s l y m i x e d ( R i
s u r f a c e l a y e r w i t h s t r o n g l a t e r a l and v e r t i c a l s h e a r . convergent
currents
?.
5 0 cm s e c - l
%
5) 0.02)
Surface
outside t h e front agree w e l l h a p 112
w i t h t h e t h e o r e t i c a l i n t e r f a c i a l p r o p a g a t i o n velocity(*)
260
T i d a l c u r r e n t v e l o c i t y (cm s e c - l ) F i g . 4. s u r f a c e f r o n t at s t a t i o n 5 (60° T ) .
tangential t o the
T i d a l c u r r e n t v e l o c i t y (cm s e c - l ) p e r p e n d i c u l a r t o t h e Fig. 5. The s l o p e of t h e f r o n t a l i n t e r f a c e s u r f a c e f r o n t at s t a t i o n 5. i n s h o r e of s t a t i o n 6 i s i n f e r r e d from t h e d e n s i t y f i e l d . The arrows i n d i c a t e streamlines.
261 I,
( G a r v i n e , 1 9 7 4 ) w h e r e hp i s t h e d e n s i t y c o n t r a s t
30 c m s e c - 1
a c r o s s t h e f r o n t , and h i s t h e s c a l e t h i c k n e s s of t h e l i g h t pool
of water i n s h o r e ( a shallow rocky bottom inshore of s t a t i o n 6 made i t i m p o s s i b l e t o s a m p l e t h e r e ; t h e d e p t h a n d s l o p e aD/ ax
of t h e f r o n t a l i n t e r f a c e have been i n f e r r e d from t h e d e n s i t y structure;
Fig.
8).
The c o n v e r g i n g s u r f a c e c u r r e n t u t h e n s a n k
beneath t h e front (with a t h e o r e t i c a l velocity sec-l;
3~ uaD
0.5
%
cm
G a r v i n e , 1 9 7 4 ) , a n d r e t u r n e d s e a w a r d as a d i f f u s e l a y e r
a t d e p t h w i t h maximum v e l o c i t i e s
'L
2 0 cm s e c - l .
Another f e a t u r e
o f t h e n e a r s u r f a c e c i r c u l a t i o n was u p w a r d e n t r a i n m e n t b e t w e e n s t a t i o n s 2 and
4.
S u r f a c e c u r r e n t s p e e d s t o w a r d t h e foam l i n e were c o n f i r m e d by a n a l y s i s o f a e r i a l p h o t o g r a p h s
of a seeding experiment.
L a r g e numbers o f c o m p u t e r c a r d s were d r o p p e d by s h i p , p a r a l l e l and p e r p e n d i c u l a r t o t h e f r o n t a l l i n e , gence determined. almost
zero;
and t h e r a t e o f conver-
S u r f a c e c o n v e r g e n c e i n s i d e t h e f r o n t was w e a k ,
v i g o r o u s d o w n w a r d a d v e c t i o n o f t h e c a r d s was n o t e d
a t t h e s u r f a c e f r o n t by t h e o b s e r v e r s a b o a r d s h i p . H y d r o g r a p h i c s e c t i o n s a r e shown i n F i g u r e s
T'C
2
0
3
4
--
--
6
5
0-
-
..
Fig.
6.
10
--_6.50
Vertical temperature (OC) section.
.
8.
-
Early
6
7.0
1
262
Fig. 7. Vertical salinity (O/oo) section. c o n t o u r i n t e r v a l between s t a t i o n s 5 and 6.
Note t h e b r e a k i n
0
5
10
15
20
25
30
F i g . 8. V e r t i c a l d e n s i t y (sigma-T) s e c t i o n . contour i n t e r v a l between s t a t i o n s 5 and 6 .
Note t h e b r e a k i n
263 development of t h e s e a s o n a l thermocline i s e v i d e n t both o f f s h o r e and i n s h o r e o f t h e j e t stream ( F i g . 6 ) .
Waters a r e r e l a t i v e l y
w e l l mixed w i t h i n t h e j e t w i t h t e m p e r a t u r e s
The s a l i n i t y s e c t i o n ( F i g .
%
6.5
-
6 . 7 5 O C.
7 ) i l l u s t r a t e s a weak h a l o c l i n e o f f -
s h o r e , b u t s t r o n g s t r a t i f i c a t i o n a t t h e f r o n t a l i n t e r f a c e , which
i s ‘ a t t r i b u t e d t o a plume o f low s a l i n i t y Stony Brook Harbor water e n t r a i n e d a r o u n d C r a n e Neck o v e r l y i n g u p w e l l e d Sound b o t t o m water ( s a l i n i t y
%
26.0).
T h e c h l - a maximum B t 1 2 m ( F i g . 9 ) l o c a t e d a t t h e 0 . 0 1 % l i g h t l e v e l , w e l l below t h e p h o t i c zone ( 0
-
6 m ) i s f o u n d so m e-
what i n s h o r e o f t h e j e t c o r e i n an area o f s t r o n g n u t r i e n t gradients ( F i g s . 10 20.25
at
%
-
20.30.
-
12), a n d i n w a t e r o f d e n s i t y ( s i g m a - T )
%
Outside t h e j e t , water of t h i s d e n s i t y i s found
5 m ; t h u s t h e o b s e r v e d d i s t r i b u t i o n s s u g g e s t s t r o n g down-
ward and eastward advection o f phytoplankton i n t o t h e j e t . Downstrean one might e x p e c t s h e d d i n g o f e d d i e s w i t h t h e c h l o r o p h y l l c o r e r e t u r n i n g back i n t o t h e p h o t i c zone.
0
5
10
15
20
25
30
Fig.
9.
V e r t i c a l c h l o r o p h y l l a (mg m - 3 )
section.
0
5
10
15
20
. . . .
.
25
30
Fig.
10.
Vertical inorganic n i t r i t e
( N O 2 ; u g m a t 1-I) s e c t i o n .
Fig.
11.
Vertical inorganic nitrate
“03;
Llgm a t 1-l) s e c t i o n .
0
2
4
3
5
6
5
10
15
20
25
30
Fig.
12.
V e r t i c a l l n o r g a n i c p h o s p h a t e (Pol,; ugm a t + - ‘ ) s e c t i o n .
Although w e y e t have no d i r e c t e v i d e n c e o f v o r t e x s h e d d i n g , f o u r evenly spaced (by one t i d a l e x c u r s i o n ) cusps i n t h e sandy s h o u l d e r on t h e n o r t h s h o r e o f t h e I s l a n d ( d e n o t e d A ,
B,
C,
D
i n F i g . 2 ) s u g g e s t t h e p r e s e n c e o f s c o u r i n g by p e r s i s t e n t e d d i e s of s i g n i f i c a n t dimension.
Lekan a n d W i l s o n ( i n p r e s s ) i n v e s t i -
g a t e d t h e s p a t i a l s t r u c t u r e of s u r f a c e (1 m) c h l - a ,
temperature,
a n d s a l i n i t y a l o n g a n e a s t w a r d t r a n s e c t i m m e d i a t e l y e a s t of t h e study zone. (contrasts
Several (scale ?J
i n t h i s area.
?J
8 km) p h y t o p l a n k t o n p a t c h e s
2 : l a b o v e b a c k g r o u n d ) were o b s e r v e d i n A u g u s t , 1 9 7 5 T h e s e p a t c h e s were f o u n d t o b e p o o r l y c o r r e l a t e d
with temperature and s a l i n i t y , suggesting p h y s i c a l mixing of
waters w i t h similar h y d r o g r a p h i c p r o p e r t i e s b u t d i f f e r i n g s t a n d ing stocks. DISCUSS I O N It i s i n t e r e s t i n g t o compare t h e s i m i l a r i t i e s of o u r s t u d y w i t h o t h e r e x a m p l e s o f c o a s t a l e n t r a p m e n t o f d i f f u s i n g f i e l d s by
266 shallow water fronts (e.g.
Csanady,
1971) and v o r t e x s h e d d i n g
i n t h e v i c i n i t y of l o c a l promontories ( e . g . Helseth,
1975).
Ebbesmeyer and
M e d e l s o f p h y t o p l a n k t o n h e t e r o g e n e i t y show t h a t
a minimum c r i t i c a l p a t c h s i z e e x i s t s b e l o w w h i c h e x p o n e n t i a l c e l l growth i n e x c e s s o f g r a z i n g by zooplankton i s u n a b l e t o k e e p p a c e w i t h a t t r i t i o n by d i f f u s i o n
( K i e r s t e a d and S l o b o d k i n ,
1 9 5 3 ; Okubc, 1 9 7 2 ; P l a t t , 1 9 7 5 ; P l a t t a n d Denman, 1 9 7 5 ; D u b o i s , 1975a, b ; Wroblewski and O ' B r i e n , 1975, 1 9 7 6 ) . However, none of these theories
e x p l a i n how a n o b s e r v e d p h y t o p l a n k t o n p a t c h g r e w
t o critical size in the first place, but "spontaneous
a s s u m e some f o r m o f
creation".
P a t c h g e n e r a t i o n mechanisms s u c h as s u d d e n n u t r i e n t e n r i c h ment r e s u l t i n g f r o m u p w e l l i n g e v e n t s o r b r e a k i n g o f s h e l f waves are w e l l documented ( e . g . ,
et al.,
in press).
internal
Beers e t a l . , 1 9 7 1 ; Walsh
We s u b m i t t h a t e j e c t i o n
o f waters w i t h h i g h
p h y t o p l a n k t o n b i o m a s s i n t o w a t e r s o f l o w b i o m a s s , v i a t h e mechanism o f t i d a l j e t c u r r e n t s ,
can r e p r e s e n t another important
mechanism f o r t h e g e n e r a t i o n
o f medium s c a l e p a t c h i n e s s i n e s t u -
a r i n e and c o a s t a l environments.
ACKNOWLEDGEMENTS
We t h a n k A k i r a O k u b o a n d P e t e r K . commenting on t h e p a p e r . pilot
M.
Gwinner,
Captain H.
Weyl f o r r e a d i n g a n d
Stuebe of t h e R / V
and p e r s o n n e l f r o m t h e Marine S c i e n c e s
Research Center are thanked f o r t h e i r r e s p e c t i v e r o l e s experiment.
Onrust,
in the
T h e p r o j e c t was p a r t i a l l y s u p p o r t e d b y t h e J o i n t
Awards C o u n c i l / U n i v e r s i t y Awards Committee o f t h e S t a t e U n i v e r s i t y
o f N e w Y o r k (SUNY) a n d t h e R e s e a r c h F o u n d a t i o n o f SUNY. C o n t r i b u t i o n 1 7 5 o f t h e M a r i n e S c i e n c e s R e s e a r c h C e n t e r (MSRC) o f t h e S t a t e U n i v e r s i t y o f New York a t S t o n y B r o o k .
267 REFERENCES
Beers, J. R . , S t e v e n s o n , M. R . , Eppley, R . W. and Grooks, E . R . , 1971. P l a n k t o n p o p u l a t i o n s and u p w e l l i n g o f f t h e c o a s t o f P e r u , June 1969. Fishery B u l l e t i n , 69:859-876.
M . J . , 1 9 7 6 . T h e t i d e s o f t h e E a s t R i v e r , New Y o r k . J o u r n a l of G e o p h y s i c a l R e s e a r c h , 8 1 : 1 6 0 9 - 1 6 1 5 .
Bowman,
G. T . , 1971. C o a s t a l e n t r a p m e n t i n Lake Huron. In: P r o c e e d i n g s of t h e F i f t h I n t e r n a t i o n a Z W a t e r P o Z l u t i o n Research Conference, JuZy-August 1970. III:11/1-11/7,
Csanady,
Pergamon P r e s s L t d . Dubois, D. M., 1975. A model o f p a t c h i n e s s f o r p r e y - p r e d a t o r plankton populations. EcoZogicaZ ModeZZing, 1 : 6 7 - 8 0 . Dubois, D. M . , 1975. Simulation of t h e s p a t i a l structuration of a patch of prey-predator plankton populations i n t h e southern b i g h t of t h e North Sea. Mkmoires S o c i d t d R o y a l e d e s S c i e n c e s d e L i e ' g e , V I I :7 5 - 8 2 . C . C . , and H e l s e t h , J . M . , 1975. A S t u d y of C u r r e n t P r o p e r t i e s and M i x i n g U s i n g Drogue Movements O b s e r v e d Durlng Summer and W i n t e r i n C e n t r a Z P u g e t S o u n d , W a s h i n g t o n . E v a n s - H a m i l t o n , I n c , S e a t t l e , 81 p p .
Ebbesmeyer,
.
On t h e f o r m a t i o n o f f r o n t s by t i d a l Fearnhead; P. G . , 1975. mixing around t h e B r i t i s h Isles. Deep-sea Research, 2 2 : 311-321. R. W., 1974. Dynamics o f s m a l l - s c a l e o c e a n i c f r o n t s . JournaZ o f PhysicaZ Oceanography, 4 : 5 5 7 - 5 6 9 .
Garvine,
Gordon, R . B . , and Pilbeam, C . C . , 1975. Circulation in central Long I s l a n d S o u n d . J o u r n a Z of G e o p h y s i c a Z R e s e a r c h , 8 0 : 414-422. Hulse, G. L . , 1975. The P l u n k e t : a shipboard water q u a l i t y monitoring system. Marine S c i e n c e s Research C e n t e r Technical Report, #22, 1 2 4 p p . The s i z e o f w a t e r K i e r s t e a d , H . , and Slobodkin, L. B . , 1953. masses c o n t a i n i n g p l a n k t o n b l o o m s . J o u r n a l of M a r i n e Research, 12:141-147. Spatial variability L e k a n , J. F., a n d v ! i l s o n , R . Z . , i n p r e s s . o f p h y t o p l a n k t o n b i o m a s s i n t h e s u r f a c e w a t e r s o f Long Island.
1 9 7 3 . T i d a Z C u r r e n t C h a r t s : Long I s l a n d Sound and B l o c k I s Z a n d S o u n d . National Oceanic a n d A t m o s p h e r i c A d m i n i s t r a t i o n , R o c k v i l l e , M a r y l a n d , 14 p p .
N a t i o n a l Ocean S u r v e y ,
Okubo, A . , 1 9 7 2 . A n o t e on s m a l l o r g a n i s m d i f f u s i o n a r o u n d an a t t r a c t i v e c e n t e r ; a m a t h e m a t i c a l model. J o u r n a l of t h e
Oceanographic S o c i e t y o f Japan, 2 8 : l - 7 .
268 P l a t t , T . , 1975. The p h y s i c a l e n v i r o n m e n t a n d s p a t i a l s t r u c t u r e of p h y t o p l a n k t o n p o p u l a t i o n s . MBmoires S o c i d t g RoyaZe
d e s S c i e n c e s de LiBge, V I I : 9 - 1 7 . P l a t t , T . , a n d Denman, K . L . , 1 9 7 5 . A general equation for the mesocale d i s t r i b u t i o n of phytoplankton i n t h e sea. Me m oir e s S o c i e t e R o y a l e des S c i e n c e s de L i e g e , V I I : 3 1 - 4 2 . Simpson, J . H . , and H u n t e r , J . R . , Sea. Nature, 250:404-406.
1974.
Fronts i n the I r i s h
W a l s h , J . J . , W h i t l e d g e , T . E . , C o d i s p o t i , L . A . , Howe, S . O . , W i r i c k , C . D . , and C a s t i g l i o n e , L. J . , i n p r e s s . The b i o l o g i c a l response t o t r a n s i e n t f o r c i n g s of t h e s p r i n g bloom w i t h i n t h e New York B i g h t . Wilson, R. E., 1976. G r a v i t a t i o n a l c i r c u l a t i o n i n Long I s l a n d Sound. E s t u a r i n e and C o a s t a l M a r i n e S c i e n c e s , 4 : 4 4 3 - 4 5 3 . A s p a t i a l model Wroblewski, J . S . , and O ' B r i e n , J . J . , 1976. of phytoplankton patchiness. Marine B i o l o g y , 35:161-175.
Wroblewski, J. S . , O ' B r i e n , J . J . , and P l a t t , T . , 1975. On t h e p h y s i c a l and b i o l o g i c a l s c a l e s o f phytoplankton patchiness i n t h e ocean. MBmoires S o c i & t 6 R o y a l e d e s S c i e n c e s de L i B g e , V I I I : 4 3 - 5 7 .
269
INTERNAL WAVES I N THE
NW
AFRICA UPWELLING
J. SALAT and J. FONT I n s t i t u t o de I n v e s t i g a c i o n e s Pesqueras, B a r c e l o n a ( S p a i n )
SUMMARY
Temperature p r o f i l e s t a k e n i n t h e t h e p r e s e n c e o f i n t e r n a l waves.
NW
A f r i c a u p w e l l i n g r e g i o n show
Some p r e l i m i n a r y arguments and hypo-
t h e s i s a r e drawn t r y i n g t o e x p l a i n t h e g e n e r a t i o n o f t h e s e waves and t h e i r r e l a t i o n s h i p w i t h coastal upwelling. INTRODUCTION I n t e r n a l waves a r e o b s e r v e d o f f C.Bojador
(26O 10' N, 14' 30' W )
d u r i n g an e x p e r i m e n t c o n c e r n i n g w a t e r mass c i r c u l a t i o n i n a s t r i p o f s t r o n g upwelling,
10 NM wide, a d j a c e n t t o t h e shore. I n t h i s r e g i o n , but i t
t h e c o n t i n e n t a l s h e l f has a g e n t l e s l o p e a n d i s v e r y narrow, w i d e n s p r o g r e s s i v e l y southwards,
(25O 10' N).
being
60
NM w i d e o f f C.Pen'a Grande
D u r i n g t h e time o f t h e experiment,
l y weak ( 4 m/s)
t h e wind was extreme-
o r i g i n a t i n g an e x t r e m e l y q u i e t sea s u r f a c e .
Such an e x p e r i m e n t c o n s i s t e d i n f o l l o w i n g a p a r c e l o f w a t e r t a g g e d by a d r i f t i n g f l o a t w i t h a l a r g e vane l o c a t e d a t 5 m below t h e s u r f a r e l e a s e d 6 NM o f f s h o r e ,
ce,
t h e l o c a l isobaths, d i f f e r e n t points,
i t moved southwards (190°),
parallel t o
w i t h an a v e r a g e v e l o c i t y o f 30 cm/s (Fig.1).
d u r i n g t h e f i v e h o u r s o f t h e experiment,
At 8
temperatu-
r e p r o f i l e s were r e c o r d e d i n t h e down a n d up e x c u r s i o n s o f a M a r t e k EBT sensor,
t h r e e o f them were r e p e a t e d i m m e d i a t e l y a f t e r r e a c h i n g
t h e surface. OBSERVATIONS
The t e m p e r a t u r e p r o f i l e s e r i e s r e v e a l s o s c i l l a t i o n s t h a t show a n e t upwards p r o p a g a t i o n . Such waves seem t o be g e n e r o t e d a t t h e b o t tom,
breaking a t t h e surface,
p r o b a b l y due t o a b s o l u t e l a c k o f s t r a -
tification.
By s t u d y i n g s e v e r a l o f such p r o f i l e s ( F i g . 2,3) t h e wave p e r i o d and a m p l i t u d e a s w e l l as t h e n e t upwards t r a n s p o r t ,
can be e s t i m a t e d
270
26.
r
16'
1s 1C
13'
Fig.1.
Map showing t h e p a t h o f t h e f l o a t d u r i n g t h e experiment i n d i -
c a t i n g t h e p o i n t s where t h e p r o f i l e s were t a k e n and t h e time, nutes,
i n mi-
a f t e r t h e b e g i n n i n g o f t h e experience.
"gross0 modo" by assuming t h a t t h e mouvement has an e q u a t i o n such as: x = xo
+
+
v t
A s i n (at
+ 'Q
),
where x i s t h e p o s i t i o n a t t h e i n s t a n t t, x,is
p u l s a t i o n and
'0
the i n i t i a l position,
A the amplitude o f the o s c i l l a t i o n ,
v the v e r t i c a l velocity,
its
i t s phase angle.
Since t h e p r o f i l e s a r e repeated i n m e d i a t e l y ,
we can t a k e 4 p o i n t s
o f e q u a l temperature a t 4 d i f f e r e n t i n s t a n t s . Then,
we have t h e f o -
l l o w i n g s e t o f equations: x. = x 1
tl Reducing,
0
+
< t2 <
v ti
t3
+A
s i n ( w ti
+ 'Q ),
i = 1,2,3,4,
and
< t4'
by s u b t r a c t i o n ,
t h e f i r s t two equations,
we o b t a i n t h i s
expression : x1
- x2 =
v(tl
-
t2) + A ( s i n (
tl
+ '4 )
- s i n ( L3 t2+ 'p
)).
271
F i g . 2 and 3. Two temperature p r o f i l e s showing t h e i r four excursions.
212
S i n c e t h e s t r o n g t e m p e r a t u r e g r a d i e n t i s near t h e l o w e r end o f t h e profile,
t,- t2 i s small, v(tl-
a l l o w i n g u s t o s i m p l i f y by assuming t h a t :
t2)(
-
sin(wt2+u))
and thus:
- arcsin(x2/A)
arcsin(xl/A)
tl
-
t2
The v a l u e o f A can be e s t i m a t e d f r o m t h e r e c o r d s a s b e i n g 10 m,
resul-
t i n g then:
a % m / 6 4 , and t h e p e r i o d T z 1 3 0 s. On t h e o t h e r hand: xl-
x3= v(tl-
t3) + A(sin(wtl+P) A
A
but i n t h i s caseat,
i s very close t o o t j ,
v(tl
-
xl-
x3~v(tl-
t3)>>A( sin(W,+'?)
-
-
sin(wt,+'f'))
then: sin(cJt3+Y))
consequently: t
3 ), and a l s o s i m i l a r t o x2- x4, g i v i n g
v ~ 0 . 8cm/s. Thus,
l o o k i n g a t such v a l u e s even o n l y a s an a p p r o x i m a t i o n , we c a n
e l l i m i n a t e t h e p o s s i b i l i t y o f such o s c i l l a t i o n s b e i n g a consequence o f s h i p a r sensor motion or,
because o f t h e i r p e r i o d ,
r e l a t e d w i t h ti-
d a l motions. The v a l u e s o b t a i n e d a r e v e r y d i f f e r e n t f r o m t h e ones o b s e r v e d by Johnson e t a l .
(1972) a t t h e same l o c a t i o n i n A u g u s t 72. The p e r i o d
o f t h e i r o b s e r v a t i o n s was t h a t o f t h e t i d a l s e m i d i u r n a l o s c i l l a t i o n . On t h e o t h e r hand,
t h e upwards n e t p r o p a g a t i o n o f t h e o b s e r v e d wa-
ve can be r e l a t e d t o t h e phenomenon o f c o a s t a l u p w e l l i n g q u o t e d by
S m i t h (1968) and a t t r i b u t e d t o some k i n d o f K e l v i n wave p r o p a g a t i n g p o l e w a r d s a n d h a v i n g a p r e c i a b l e a m p l i t u d e when t h e r e i s a resonance between t h e wave a n d t h e f o r c i n g d i s t u r b a n c e , f e s t w i t h o u t a p p a r e n t wind, Finally,
and c o u l d perhaps mani-
as o c c u r s i n o u r case.
we can a l s o say t h a t Mc N i d e r a n d O ' B r i e n
(1973) found,
i n t h e n u m e r i c a l s o l u t i o n o f t h e i r t h e o r e t i c a l model o f c o a s t a l upwe-
lling,
waves i n t h e l o n g s h o r e v e l o c i t y f i e l d whose upwards v e l o c i t y
o s c i l l a t e s between 1 and 2 cm/s,
which i s a l i t t l e h i g h e r than o u r es-
273
t i m a t e d value. CONCLUS IONS A l t h o u g h such o c c a s i o n a l o b s e r v a t i o n s can o n l y be considered as a p r e l i m i n a r y approach t o more s p e c i f i c studies,
t h e y a l l o w us t o draw
t h e f o l l o w i n g h y p o t h e s i s t o e x p l a i n t h i s phenomenon:
A c o n s t a n t h o r i z o n t a l c u r r e n t o v e r t h e bottom i n a sea. o f decreas i n g depth g i v e s p l a c e t o an o n d u l a t i o n t h a t propagates a g a i n s t such a c u r r e n t . T h i s phenomenon c o u l d be c o n s i d e r e d as i n t e r m e d i a t e between pure o s c i l l a t i o n and bottom t u r b u l e n c e , Then a c c o r d i n g w i t h Cox
(1963) we a r e i n presence o f a t u r b u l e n t motion t h a t i n v o l v e s v e r t i c a l oscillation. On t h e o t h e r hand,
t h i s o s c i l l a t i o n c o u l d a l s o be produced i n t h e
edge o f t h e c o n t i n e n t a l s h e l f p r o p a g a t i n g northwards f o l l o w i n g t h i s edge. T h i s f a c t i s a l s o quoted by Cruzado (1976) and i t a l s o agrees w i t h t h e o b s e r v a t i o n o f waves o f s i m i l a r c h a r a c t e r i s t i c s i n t h e acoust i c s c a t t e r i n g l a y e r s near t h e s h e l f edge i n t h i s r e g i o n . We a r e p r e s e n t l y d e s i g n i n g experiments t o be performed i n t h i s ar e a i n o r d e r t o o b t a i n a complete s e t o f d a t a a l l o w i n g a b e t t e r e x p l a n a t i o n o f t h i s phenomenon, p r o v i d i n g t h e c r i t i c a l p o l i t i c a l s i t u a t i o n o f t h e Sahara a l l o w s us t o c o n t i n u e o u r r e s e a r c h on c o a s t a l u p w e l l i n g processes o f f FW A f r i c a . REFERENCES
1963. I n t e r n a l waves. I n : M.N.Hil1
Cox, C.S.,
(Editor),
The Sea,
1:
752-763. Cruzado, A.,
1976. A f l o r a m i e n t o c o s t e r o en e l A t l C I n t i c o N o r o r i e n t a l . U n i v e r s i d a d de Barcelona, 97 pp.
Tesis Doctoral, Johnson,
D.R.,
Barton,
E.D.,
Hughes, P. and Mooers,
C.N.K.,
1975. C i r -
c u l a t i o n i n t h e Canary C u r r e n t u p w e l l i n g r e g i o n o f f Cab0 B o j a d o r i n August 1972. Deep Sea Res. 22(8): Mc Nider,
R.T.
and O'Brien,
o f c o a s t a l u p w e l l i n g . J.Phis. Smith,
R.L.,
J.J.,
547-558. 1973. A m u l t i - l a y e r t r a n s i e n t model
Oceanogr. 3(3):
258-273.
1968. Upwelling. 0ceanogr.Mar.Biol.Ann.Rev.
6: 11-46.
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276
A R E P O R T ON E N V I R O N M E N T A L S T U D I E S O F D R E D G E S P O I L D I S P O S A L S I T E S P A R T I : AN I N V E S T I G A T I O N OF A D R E D G E S P O I L D I S P O S A L S I T E P A R T 1 1 : D E V E L O P M E N T A N D USE O F A B O T T O M B O U N D A R Y L A Y E R P R O B E G.S.
COOK,
R.W.
and A . T .
MORTON,
MASSEY
Naval Underwater Systems Center, R h o d e I s l a n d 02840,
Newport Laboratory,
Newport,
USA
The c h a n n e l d r e d g i n g of m a t e r i a l from t h e Thames R i v e r i n New London, C o n n e c t i c u t , USA r e q u i r e d d r c d g c s p o i l d i s p o s a l b a r g c dumping a t a p r i m a r y s i t e 5 kin s o u t h of t h e r i v e r e n t r a n c e and p o s s i b l y a t an a l t c r n a t c s i t e
15 km s o u t h c a s t o f N e w London on t h e c o n t i n e n t a l s h e l f . S u r v e y s w c r c madc a t t h e p r i m a r y and a l t e r n a t e d i s p o s a l s i t e s i n o r d e r t o a s s e s s ambient e r o s i v c and t r a n s p o r t c o n d i t i o n s . The s u r v c y r e s u l t s showed t h a t a f t e r a n i n i t i a l s t a b i l i z a t i o n p e r i o d f o l l o w i n g s p o i l s dumping t h e r c was no m a j o r c h a n g e i n t h e s h a p c or a r e a o f the spoil pile.
T h i s was c o n f i r n i c d b y u n d e r w a t e r t e l e v i s i o n s u r v e y s t h a t
r e v e a l e d n o major c r o s i o n a l p r o c c s s c s o c c u r r i n g o n t h e s u r f a c c of t h c s p o i l pile.
E v i d e n c e w a s n o t c d , I i o v c v c r , o f l o c a l s o r t i n g and e r o s i o n a r o u n d
s p o i l c l u m p s d u r i n g pc.aIc
t i d a l flow.
i n d i c a t e d s i m i l a r c u r r c n t rcgimcs.
C u r r e n t m c a s u r c m e n t s a t t h c two s i t c s
Peak v c l o c i t i e s w c r c a b o u t 30-40 c m / s c r
a t b o t h sites b u t o c c u r i n d i f f e r e n t d i r e c t i o n s producing n e t c a s t c r l y d r i f t a t t h e New I.ondon D i s p o s a l S i t e and w c s t c r l y d r i f t a t t h e E a s t h o l e a l t e r nate s i t e . I n o r d e r t o d c t e r m i n e i f t h e s c l o c a t i o n s would a c t a s s p o i l c o n t a i n m c n t s i t e s i t was r e q u i r e d t o m c a z u r e t h o d y n a n i c p r o p e r t i c s o f t h e c u r r e n t s i n t h e bottom boundary l a y c r .
An i n s t r u n c n t was d c s i s n e d a n d c o n z t r u c t c d t o
m e a s u r e b o t h t h e v e r L i c a l c u r r e n t s h e a r and t t l r b u l e n t v e l o c i t y s t r u c t u r e w i t h i n one meter of t h e s e a f l o o r . Froin t h c b o t t o m c u r r e n t d a t a ::cynulds
were d c t e r n i i n c d by two m c t h o d s : 1)
s t r c s n c s i n t h c boundary l a y c r
f r o m p a r a m e L r i c f i t or t h e o b s e r v e d mean
v e l o c i t i c s t o a l o g a r i t h c i i c v c l o c i t y p r o f i l e a n d , 2 ) from c s t i m a t c s o f t h c r a t e o f k i n c t i c c n c r g y d i s s i p a t i o n u s i n g t h c Kolmogoroff H y p o t h e s i s f o r t h e i n e r t i a l s u b r a n g c and by a s s u u i n g a b a l a n c e bctwccn p r o d u c t i o n and d i s s i p a t i o n o f e n e r g y w i t h i n t h e boundary l a y e r .
276 R e y n o l d s s t r e s s e s o f 4-6 mum t i d a l c u r r e n t .
d y n c s / c m 2 wcrc e s t i m a t e d d u r i n g timcs of maxi-
I t was d c t c r m i n c d f r o m f l u m c t a n k s t u d i e s t h a t R e y n o l d s
s t r e s s e s o f 1 6 d y n c s / k m 2 o r g r c a t c r would b e r e q u i r e d t o e r o d e t h c p a r t i c u l a r s p o i l material.
I t is t h u s concluded t h a t for t h e s e sites under normal
c u r r e n t c o n d i t i o n s n o major e r o s i o n and t r a n s p o r t w i l l o c c u r .
INTRODUCTION Thc T h a n e s R i v e r i n N e w London, C o n n e c t i c u t , USA i s b e i n g d r c d g c d o f a b o u t o n e m i l l i o n c u b i c mctcrs o f m a t e r i a l t o i n c r c a s c c h a n n e l d e p t h .
This
p r o j e c t h a s c r e a t e d a r c q u i r c m e n t f o r a l o r n 1 d i s p o s a l s i t c t o accommodate t h e dredged s p o i l s .
The Mer~ London D i s p o s a l S i t c ( h e n c c f o r t h r c f c r r c d t o
a s NLDS) l o c a t e d a p p r o x i m a t c l y 5 km S o u t h o f t h c Tnames R i v e r c n t r a n c c ( F i g . 1 ) h a s b e e n d c s i g n a t c d a s a p r i m a r y s i t c a n d h a s r e c c i v c d a l l of t h e d r e d g e s p o i l s removed d u r i n g t h e f i r s t p h a s c s o f t h c p r o j c c t ( a b o u t one half million
i d ) .
Tnc E a s t H o l e D i s p o s a l S i t e ( h c n c c f o r t h r c f c r r c d t o
AS
EHUS) i n B l o c k I s l a n d Sound ( F i g . 1 ) has b c c n d c s i g n a t c d as a n a l t e r n a t e
site f o r possibly receiving f u t u r c drcdgc s p o i l s . The N a v a l U n d c r w a t c r S y s t c m s C c n t c r (NUSC) a t N c w p o r t , Ichodc I s l a n d h a s conducted cnvironmcntal s t u d i e s a t b o t h s i t c s s i n c e August, 1574.
A t the
NLDS f o u r b a t h y m e t r i c s u r v e y s wcrc made o v c r a two y c a r p c r i o d t o d e f i n c t h e b o u n d a r i e s of t h c d c p o s i t c d material and t o m o n i t o r c h a n g e s i n i t s volumc a n d g c o g r a p h i c d i s t r i b u t i o n .
A s u r v c y a t t h e OHDS w a s a l s o made t o
d e t e r m i n c a b a s e l i n e t o p o g r a p h y f o r u s e i n e s t i m a t i n g volumes o f s p o i l s i n t h c c v c n t t h a t t h c EHDS d c p r e s s i o n i s u s e d a s a f u t u r c d i s p o s a l s i t e . B c c a u s c o f t h c p o s s i b l y I i i g h c o n c e n t r a t i o n of p e t r o l e u m a n d h c a v y mct-
a l s i n t h e Thames R i v e r s c d i r n e n t s i t w a s a g r c e d t h a t t h e d i s p o s a l a r e a s h o u l d b e a c o n t a i n m c n t s i t c w h c r c tiic s e d i m e n t w i l l r c m a i n i n t h e a r e a o f dumping.
The c l i a r a c t c r i s t i c s o f t i d a l r u r r e n t s wcrc s t u d i e d a t t h e NLDS
a n d EHDS t o d e t e r m i n c t h e d c g r c c t o w h i c h t h c c u r r c n t r e g i m e i s c a p a b l e o f e r o d i n g and t r a n s p o r t i n g t h e d r e d g e s p o i l s . v i s i o n p i c t u r e s of
I n a d d i t i o n , u n d e r w a t e r tclc-
t h e s u r f a c c o f t h e NLDS s p o i l p i l e werc made t o o b s c r v c
e r o s i o n a l f e a t u r e s t h a t m i g h t f u r t h e r i n d i c a t e r e m o v a l o f s p o i l s by c u r rents.
C u r r e n t o b s e r v a t i o n s were made a t t h e s i t e s b y two m e t h o d s :
first,
c o n v e n t i o n a l t a u t - w i r c m o o r s w i t h t i m e a v e r a g i n g c u r r c n t mctcrs were i n s t a l l e d t o e v a l u a t e t h e l o n f tcrm c u r r c n t r e g i m e ; a n d s e c o n d , a s p e c i a l b o t t o m b o u n d a r y l a y e r c u r r c i i ~mctcr was d c v c l o p c d t o m c a s u r c s h o r t term t u r b u l e n t c u r r e n t f l u c t u a t i o n s w i t h i n o n c mctcr of t h e b o t t o m .
277 P a r t one p r e s e n t s
3
b r i e f r c v i c w of t h c b o t t o m c h a r a c t c r i s t i c s and
g e n e r a l c i r c u l a t i o n i n t h e d i s p o s a l areas.
P a r t I1 p r e s e n t s a d e t a i l e d
d i s c u s s i o n of t h e b o t t o m boundary l a y c r s t u d i e s and i n s t r u m e n t a t i o n .
72.05'
72.00'
71.55'
71.50'
41. 20'
41.15'
Fig.1.
L o c a t i o n o f t h e New L o n d o n , D i s p o s a l S i t c (NLDS) a n d t h e E a s t H o l e
D i s p o s a l S i t e (EHDS)
PART I:
CONVENTIONAL ENVIRONMENTAL STUDIES OF THE DISPOSAL SITES
BATHYMETRIC SURVEY F o u r b a t h y m e t r i c s u r v e y s were made a t t h e NLUS b e t w e e n November, 1 9 7 4 a n d A u g u s t 1 9 7 5 , a n d a b a s e l i n e s u r v e y of t h e EHDS w a s made i n A u g u s t , 1 9 7 6 The n a v i g a t i o n u s e d i n a l l s u r v e y s w a s a Decca D c l N o r t e Model 202A T r i s p o n d e r s y s t e m c a p a b l e of m e a s u r i n g d i s t a n c e s w i t h a n a c c u r a c y o f
meters o v e r a maximum r a n g e o f 40 km.
5
3
The e c h o s o u n d e r s y s t e m c o n s i s t s o f
a n ED0 Western Nodel 4034A u n i t p r o v i d i n g d i g i t a l d e p t h o u t p u t .
The n a v i -
g a t i o n s y s t e m , e c h o s o u n d e r a n d a d i g i t a l c l o c k were i n t e r f a c e d so t h a t a l l d a t a w a s r e c o r d e d on d i g i t a l t a p e f o r s u b s e q u e n t computer p r o c e s s i n g .
The
b a t h y m c t r i c d a t a were p l o t t e d a n d h a n d c o n t o u r e d a f t e r c o r r e c t i o n s were app l i e d f o r t i d a l l e v e l a n d s p e e d of s o u n d .
The e r r o r o f a b s o l u t e d e p t h i s
278
(t 1 f t . ) .
30 cm
estimated t o be
The f i r s t NLDS sur-vey i n November, 1 9 7 4 was made a f t e r t h e dumping o f t h e Thames R i v e r d r e d g e s p o i l s h a d b e g u n , a n d s i n c e t h e NLDS h a s b e e n a dumping g r o u n d f o r many y e a r s n o b a s e l i n e d a t a w a s a v a i l a b l e t o d e t e r m i n e t h e o r i g i n a l b o t t o m t o p o g r a p h y o f t h e area.
This survey revealed t h c pres-
e n c e o f two t o p o g r a p h i c h i g h s ( F i g . 2 ) i n t h e a r e a of t h e dumpinR g r o u n d : a r e l i c p i l e t o t h e N o r t h e a s t w i t h a minimum d e p t h of 1 1 . 0 m a n d o n e i n t h e c e n t e r of t h e c h a r t w h i c h i s t h e s p o i l p i l e f r o m t h e r e c e n t d r e d g i n g w i t h a minimum d e p t h o f 1 5 . 5 m .
72' -.
OS'Oor T--
1
04'30' I
I
I
41' 16'10'
41' 16'00'
20
Fig.2.
I
I
B a t h y m e t r y of 1 6 November 1 9 7 4 .
NLDS
279 The s p o i l s from c u r r e n t d r e d g i n g o p e r a t i o n s s t a n d o u t a s a d i s t i n c t c i r c u l a r mound w i t h s t e c p s l o p e s , p a r t i c u l a r l y toward t h e S o u t h e a s t and a comp a r a t i v c l y f l a t t o p bctwecn 15.5 and 16.5 m .
The d i a m e t e r of t h c mound
US-
i n g t h e 2 1 m c o n t o u r as a g u i d e i s on t h e o r d c r of 300 m . Thc second s u r v e y of F e b r u a r y , 1975 ( P i g . 3 ) shows d i f f e r e n c e s from t h e f i r s t s u r v e y : t h c minimum d e p t h of t h e s p o i l s p i l c dcsccnded from 1 5 . 5 m t o 17.5
ni
and t h e mound had widcncd t o a d i a m e t e r o f 400-450 m , tlie t o p s u r f a c e
became e x t r e m e l y f l a t w i t h more g r a d u a l s l o p e s . I t s h o u l d be n o t e d t h a t i n t h e November 1974 s u r v e y t h e e n t i r c s p o i l p i l c
was Wcst of 72O 4 ' 50" \I w h i l e i n t h i s s u r v c y t h c r e i s a s i g n i f i c a n t amount of m a t c r i a l e a s t of t h a t l i n e .
Another i m p o r t a n t f e a t u r c i s t h e 21 m con-
t o u r NE of tlie p i l e had s h i f t e d t o SW i n d i c a t i n g some s h o a l i n g toward t h e East.
Betwecn F e b r u a r y 1975 and August 1975 s u r v c y s t h e p o s i t i o n of t h e r e f e r e n c e dumping buoy w a s moved 100 m southward.
Tne August 1975 topography
shows t h a t d r e d g e s p o i l s dumped w i t h t h e new r e f e r e n c e d r a s t i c a l l y a l t e r e d t h e s h a p e of tlic o r i g i n a l s p o i l p i l c c o n f i g u r a t i o n g i v i n g i t an e l l i p t i c a l s h a p e t r c n d i n g t o t h e SE ( F i g . 4 ) . The August 1975 s u r v e y shows t h a t t h e o r i g i n a l mound a p p e a r s t o have sett l e d 0.5
-
1 . 0 m , however, t h c r c a p p e a r s t o be no s i g n i f i c a n t i n c r e a s e i n
t h e d i a m c t c r of t h a t p o r t i o n of t h e p i l e .
Again, t h e r e scems t o be s p o i l
t r a n s f e r t o t h e c a s t a s s o c i a t e d w i t h t h e s h i f t of t h e 20 m contour a t t h c b a s e of t h e s p o i l mound.
I n F e b r u a r y 1975 t h e maximum e a s t w a r d e x t e n t of
t h i s l i n e w a s a p p r o x i m a t e l y 7 2 O 4 ' 48" w h i l e i n August 1975 t h e l i n e moved a b o u t 100 m e a s 2 w a r d . The a d d i t i o n a l s p o i l s dumped t o t h c S o u t h e a s t of t h e o r i g i n a l mound have formed a n o t h e r t o p o g r a p h i c h i g h t h a t i s c o n t i n u o u s w i t h t h e f i r s t r e a c h i n g a minimum d e p t h of 1 6 . 5 m w i t h s t c c p e r s l o p e s t o t h e S o u t h e a s t .
Thc f i n a l s u r v e y of Scptcmbcr 1975 i s c s s c n t i a l l y t h c same a s t h e August survey (Pig. 5 . ) month.
Very l i t t l e dumping had t a k e n p l a c c d u r i n g t h e i n t e r v e n i n g
111 t h e n o r t h p o r t i o n of t h c p i l e t h e d e p t h i n c r e a s e d a b o u t 60 cm.
However, tlir o v e r a l l d i m e n s i o n s of t h e p i l e a r e a p p r o x i m a t e l y t h e same a n d \ t h e s l o p c s a t t h e edge of t h c p i l c a r c somcwhat l e s s .
The bottom topography
100 m from tlie p i l e edge rcmaincd unchanged. From t h e e v i d e n c e shown by tlic bathymctry t h e d r c d g c s p o i l s were dumpcd o v e r a r e l a t i v e l y s m a l l a r e a and have g c n c r a l l y m a i n t a i n e d t h e i r o r i g i n a l
280 72.
l------
05'00'
04'30'
41' 6'30'
41. 8'00.
I
*L1L O W * U U O Y L YTL
@ '
-
CWTOU INTLRVAL im DATUM Y W
mom
RCFfRENCE BUOY
41. 1'30.
Fig.3.
Bathymetry of 4 February 1975
-
NLDS
72' O'OQ
OC30'
I
, ,"I
NEW L W D W DISPOSAL SITE CONTOU
INTERVAL im
D A T W YLW
,
,
0 REfEREWCC 8uOI
Fig.4.
41. 15'30'
Bathymetry of 7 August 1975
-
NLDS
281
lm
C O N T O M INTERVAL DATUM YLW
"
0 REFERENCE BUOY
.- -Fig.5.
Bathymetry of September 1975
configuration.
-
NLDS
T h e r e a p p e a r s t o bc a p c r i o d of s e t t l i n g o r r o m p a c t i o n f o l -
l o w i n g d i s p o s a l t h a t r c o u l t s i n a f l a t t c n i n y , and d e c p c n i n g o f t h e t o p o f the p i l c .
A s s o c i a t e d w i t h t h i s , t h e p i l c sccms t o s p r c a d s l i g h t l y and t h e
b o r d e r i n g s l o p e s bccomc l c s s s t c c p .
Thcsc e f f e c t s a r e probably a s s o c i a t e d
w i t h c o m p a c t i o n of t h e s e d i m e n t s . UNDERWATER TELEVISION OBSERVATIONS The b a t h y m e t r i c s u r v e y s o f tlic d r c d g c s p o i l s madc a t s c v e r o l months o r y e a r l y i n t e r v a l s i n d i c a t e t i l e l o n g term s t a b i l i t y o f t h e s p o i l p i l c , howe v e r , t h c s e s u r v e y s c a n n o t r c v c a l s m a l l s c a l e p r o c c s s e s t h a t may b e o c c u r r i n g on t h e s p o i l s u r f a c e .
C o n s e q u e n t l y , a s e r i e s o f s c u b a d i v e s w e r e made
w i t h a hand h c l d u n d c r w a t e r t c l c v i s i o n s y s t e m t o d i r e c t l y o b s c r v e tile s t a t e of t h e s p o i l s .
The d i v i n g c x p l o r a t i o n was l i m i t e d b e c a u s e o f t h e p o o r v i s -
i b i l i t y w h i c h r a n g e d f r o m o n e t o t h r e e mctcrs d e p e n d i n g upon t h e s t a t e o f t h e t i d e and s e a c o n d i t i o n s . The t e l e v i s i o n u s c d was m a n u f a c t u r e d by t h e R e a l 8 C o r p o r a t i o n , w i t h a w i d e a n g l e R i b i c o f f l e n s w i t h a f o c a l l e n g t h ( i n w a t e r ) o f 60 cnl.
Attached
t o t h e camera was a n u n d e r w a t e r l i g h t t o p r o v i d e s u f f i c i e n t i l l u m i n a t i o n f o r operations i n the turbid water.
The c a m e r a was c o n n e c t e d by 100 q of
n e u t r a l l y b u o y a n t c a b l c t o a Sony V i c d o t a p c R c c o r d c r a n d a Sony T c l c v i s i o n . The v i d c o p h o t o s r c v c a l c d t h a t two d i s t i n c t t y p c s o f s u r f a c c s wcrc g c n c r a l l y f o u n d o n t h e t o p - o f t h e s p o i l s p i l c ; a t h i n l a y c r of f i n c . s i l t t h a t i m m c d i a t c l y w e n t i n t o s u s p c i i s i o n wlicn d i s t u r b c d ; arid
c o n s i s t i n g of s m a l l g r a v c l s t o n c s and s h c l l i r n p c n t s .
Fcaturcless d
surface
Tlic b o u n d a r y be-
t w e e n t h c s e two t y p e s o f s u r f a c e s w a s o f t c n c x t r c m c l y s h a r p a n d c x t c n d c d f o r t e n s o f meters i n a s t r a i g h t l i n e . The q u c s t i o n a s t o w h e t h e r t h e c o a r s c m a t c r i a l i s a l a g d c p o s i t l e f t from winnowing of f i n e m a t e r i a l o r s i m p l y a d e p o s i t o f c o a r s e s p o i l s i s d i f f i c u l t t o answcr.
C c r t a i n l y winnowing o c c u r s , howcvcr,
t h c s c d i m c n t bc-
low t h c f i n e s u r f a c c was s i m i l a r t o t h e s u r f a c e m a t c r i a l , a l t h o u g h more coh e s i v e a n d c o n t a i n e d no s t o n c s o r s h c l l m a t c r i a l .
Similarly, the coarsc
m a t e r i a l e x t e n d s a t l e a s t t o 5-10 c m d c c p a l t h o u g h t h e r e i s a g e n e r a l i n c r e a s e i n s i l t y material w i t h d e p t n . A n o t h e r common f e a t u r c of t h e s p o i l s a r e a i s t h c p r c s c n r e o f l a r g e c l u m p s o f f i n e c o h c s i v c m a t e r i a l t h a t r a n g c i n d i a m e t c r f r o m a b o u t 10-300 cm.
S m a l l c h a n n e l s 10-20 cm d c c p wcrc f o u n d a r o u n d t h e c l u m p s p r o b a b l y
c a u s e d by e r o s i o n a l t u r b u l c n c c c r c a t c d by i n t c r a c t i o n o f t h c mcati f l o w w i t h clumps.
I t w a s f o u n d t h a t d u r i n g slaclc water a n a r c a o f f i n c , s i l t y s a n d
d e p o s i t c d on t h e d o w n s t r c a m s i d c o f t h e c l u m p s .
I t was a n area of f i n c ,
s i l t y s a n d d e p o s i t c d o n t h c d o w n s t r c a m s i d c of t h e c l u m p s ( i . c . ,
following
an ebb o r f l o o d t i d c ) f i l l i n g t h e e r o d e d c h a n n e l and c x t c n d i n g outward t o a p p r o x i m a t e l y o n c h a l f t h c d i a m c t c r of t h c c l u m p .
The p e r m a n e n c e o f t h i s
d e p o s i t i s q u e s t i o n a b l e a s i t w a s o n l y o b s c r v c d d u r i n g p e r i o d s o f low c u r r e n t s ( s l a c k t i d e ) ; t h e s e d i m e n t may b c r c s u s p c n d c d d u r i n g p e r i o d s o f strong current. These o b s e r v a t i o n s i n d i c a t c t h a t t h c s u r f a c c o f t h c s p o i l p i l e h a s somc
l i m i t e d small s c a l e e r o s i o n and d c p o s i t i o n proccsscs o c c u r r i n g , but t h a t i n g e n e r a l t h e c o h e s i v e n a t u r e of t h c d r c d g c s p o i l s t h c m s c l v c s p r e v e n t s any m a j o r e r o s i o n a n d t r a n s p o r t a t i o n of m a t e r i a l u n d e r n o r m a l c u r r c n t c o n d i tions. CONVLNTIONAI, CURRENT MEASUREMENTS Thc b a t h y m e t r i c s u r v c y s a n d u n d c r w a t e r t e l e v i s i o n p i c t u r e s p r o v i d c d a n e s t i m a t e o f t h e s t a b i l i t y of d r c d g c s p o i l s i n t h c m a r i n e e n v i r o n m e n t i n a q u a l i t a t i v e manner.
S i n c e t h c c r o s i o n and t r a n s p o r t of t h c s p o i l s is solc-
l y due t o t h e ambient c u r r e n t s .
I n f o r m a t i o n i s r e q u i r c d o f t h c mean r u r -
r e n t a n d t h e t i d a l f l o w t h r o u g h o u t t h e water c o l u m n .
F o r t h e l o n g term
283 m e a s u r e m e n t s c o n v e n t i o n a l t a u t w i r e moors w i t h t i m e a v e r a g i n g c u r r e n t meters ! u e r e i n s t a l l e d a t t h e d e s i r e d l o c a t i o ~ ~; ~si - p ~ r i u l i sUI 14-35 d a y s .
I l c a s u r e m e n t s were made a t b o t h t h e NLDS a n d EllDS d u r i n g t h e p e r i o d s 10 ,December 1974
-
22 J a n u a r y 1975 and 6 August 1 3 7 5 t o 2 Scptcmber 1975.
T h r e e c u r r e n t meter n o o r i n p were d c p l o y c d a r o u n d t l i c p i l e d u r i n g e a c h of t h e measurement p e r i o d s ( F i g . 6 ) .
Each m o o r i n g c o n t a i n e d t h r e e c u r r e n t
meters; o n e n e a r s u r f a c e ( 3 m) o n e l o c a t e d a t t h e a p p r o x i m a t e d e p t h of t h e t o p o f t h e P i l e ( 1 5 m); a n d o n e 1 . 5 a b o v e t h c b o t t o m .
72O04' SURFACE BUOY
I
I
72°04'
06'
Fig.6.
C o n f i g u r a t i o n o f C u r r e n t Mcters a r o u n d t h e NLDS area
The ENDECO Type 1 0 5 C u r r e n t I k t e r s were u s e d w h i c h a r e a x i a l - f l o w
ducted
i m p e l l e r s y s t e m s d e s i g n e d f o r s h e l f and e s t u a r i n e e n v i r o n m e n t a l s t u d i e s . The c u r r e n t s p e e d and d i r e c t i o n i s r e c o r d e d o n c a r t r i d g e l o a d c d 16 mm f i l m
at
4 hour
intervals.
Data r e d u c t i o n i s d o n c b y t h e m a n u f a c t u r e r , a n d d a t a
a n a l y s i s w a s performed a t NUSC. A summary of t h e mean b o t t o m c u r r e n t v e l o c i t y , mean maximum f l o o d a n d
e b b v e l o c i t i e s a n d the h o r i z o n t a l k i n e t i c c n e r g y of t h e mean f l o w i s shown i n Table I.
284 TAGLL I
MEAN BOTTOM CURRENT VELOCITY (cm/sec)
-
NLDS
NLDS
Dec. 1974 7.1 cm/sec lO8'T
EHDS
Aug. 1975 7.5 c d s c c 0 8 8 O
June-Aug. 1975 8.6 rmleec 240° 1'
'r
Mean/Maximum Ebb & Flood C u r r e n t Spceds (cm/sec) EBB
43.2 cm/sec
E
41.4 cm/sec FLOOD
E
28.6 cm/sec
E
33.3 d s e c
F
37.1 cm/sec
F
39.8 cmlsec
F
R e p r e s e n t a t i v e time s e r i e s p l o t s of t h e mean s u r f a c e ( 3 m) and bottom
(20 m ) c u r r e n t s a t NLDS d u r i n g tile 10 December 1974 - 2 2 J a n u a r y 1975 ( F i g . 7) p e r i o d s show a s t r o n g , s e m i - d i u r n a l t i d a l component a t b o t h t h e s u r f a c e and bottom
T h i s i s p r o b a b l y c h a r a c t e r i s t i c of t h e mean f l o w i n
t h i s area throughout t h e y e a r .
As can be s e e n , q u i t e d i f f e r e n t c h a r a r t e r -
i s t i c s o c c u r a t t h e s u r f a c e and t h e bottom; a s e x p e c t e d , t h e c u r r e n t s p c e d s peak much h i g h e r a t t h e s u r f a c e .
A progressive vector plot for the current
meters NU & NL i s shown a t 3 m and a t t h e bottom i s shown i n F i g . 8.
The
n e t t r a n s p o r t i s much h i g h e r a t t h e s u r f a c e a s i n d i c a t e d by t h c p r o g r e s s i v e
-
v e c t o r diagrams. 0
tE
s
90
so 30
W
n m
n
v
0
I
0' 0
I
2
1
3
1
-.
I
2
3
4
I
4
so
I
so
I
2
1
I
I
2
3
1
3
4
1
4
5
1
5
F i g . 7 . R e p r e s e n t a t i v e t i m e series o b s e r v a t i o n s of c u r r e n t speed and d i r e c t i o n a t NLDS: a . c u r r e n t v e l o c i t y a t 3 m ; b . c u r r e n t v e l o c i t y a t 20 m North array.
285
200 W 0
-
100
2
100-
0
0
i
:\
z
1
1
i
1
1
1
9
0
1
1
1
1
,
1
1
1
DISTANCE (KM)
DISTANCE (KM) Fig.8.
1
P r o g r e s s i v e v e c t o r p l o t (PVP) of c u r r e n t v e l o c i t i e s shown in F i g . 7
Left, PVP a t 3 m ; R i g ' l t , PVP a t 20 m . PART 11: HEASUREMENTS OF THE BOTTOM BOUNDARY LAYER INSTRUMENTATION The i n s t r u m e n t d e s i g n e d and c o n s t r u c t e d t o measure t h e c h a r a c t e r of t h e near-bottom c u r r e n t s i n t h e d i s p o s a l s i t e areas i s shown i n F i g . 9 .
The
s y s t e m c a l l e d t h e "Boundary Layer Thing" (BLT) c o n s i s t s of 3 d u c t e d impel-
l e r c u r r e n t meters (DICEl's) mounted on h o r i z o n t a l s h a f t s which p i v o t v i a b a l l b e a r i n g mounts a b o u t t h e v e r t i c a l s t a i n l e s s s t e e l s u p p o r t s h a f t .
Each
h o r i z o n t a l s h a f t a l s o supports an instrument cylinder containing recording electronics.
T h i s u n i t i s f a i r e d w i t h h o r i z o n t a l p l a t e s and s e r v e s a l s o a s
a t r a i l i n g vane from which t h e d r a g of t h e c u r r e n t d i r e c t s t h e DICM d i r e c t l y i n t o t h e mean f l o w .
The e n t i r e s y s t e m i s envcloped in a shrouded cage f o r
p r o t e c t i o n and f o r ease i n s h i p b o a r d lrandling.
Tlie 1 . 5 m d i a m e t e r s t e e l
b a s e p l a t e (weight a b o u t 1 7 3 k g ) s e r v e s t o anchor and s t a b i l i z e t h e system even i n s t r o n g c u r r e n t s . The vane h o u s i n g s were trimmed s o t h e i r s l i g h t n e g a t i v e buoyancy o f f s e t t h e ( i n w a t e r ) weight of t h e DICM's,
t h u s minimizing b e a r i n g f r i c t i o n
against the vertical shaft. The D I C M u n i t s developed by S h o n t i n g (1968) f o r wave o r b i t a l
286
k’ig.9. The a s s e m b l e d BLT s y s t e m showing t h e c a g e d h o u s i n g e n c l o s i n g the t h r e e DICMs
281 m o t i o n s c o n s i s t o f a s i x b l a d e d m i c a r t a i m p c l l c r s c o n t a i n i n g m i n i a t u r e (b.2 0 . 5 gm) A l n i c o m a g n e t s .
A s t h e i m p e l l e r r o t a t e s t h e magnet f i c l d g e n e r a t e s
a v o l t a g e p u l s c as t h e y c u t t h r o u g h a m i n i a t u r e p i c k u p c o i l p l o t t e d i n a s m a l l p i l l b o x mounted on t h c s i d e o f t h e c y l i n d e r .
The s i g n a l s a r e l e a d
t h r o u g h e x t e r i o r w i r e s t o t h e c l e c t r o n i c s i n t:ic v a n e h o u s i n g . Each DICM was c a l i b r a t e d i n a tow t a n k o v e r s p e e d s o f 3-100 c m / s e c and e x h i b i t a very l i n e a r responsc with pulse frcqucnce d i r e c t l y proportioncd t o flow speed.
The r e s p o n s e of t h c D I C M t o o f f - a x i s f l o w v a r i e s a s t h e co-
s i n e of t h e o f f - a n g l e
f r o m 0 t o a b o u t 80 d e g r e e s .
Thc d i s t a n c e c o n s t a n t o b t a i n e d from t h e c a l i b r a t i o n s ( i d e n t i c a l f o r a l l s e n s o r s ) was 3 c m / s e c p e r c y c l c / s e c o r 18 cm p c r i m p e l l e r r o t a t i o n .
Since
t h i s i s a b o u t e q u a l t o t h e g e o m e t r i c p i t c h o f t h c i m p e l l c r t h e h i g h l y res p o n s i v e c h a r a c t e r o f t h e d u c t e d mctcrs is e v i d e n t . Thc r c s p o n s e o f t h e D I C M t o f l u i d a c c c l e r a t i o n s i s s p c c i f i c d by t h c r e s p o n s e d i s t a n c e ; t h e a x i a l l e n g t h of w a t c r p a r t i c l e s t r a v e r s e f o r tlie DICM o u t p u t t o r e g i s t e r 63% of t h e c h a n g c t o a s t e p i n c r c a s c i n s p e e d .
The
a c t u a l t r a v e r s e d i s t a n c e f o r d e t e c t i o n of t h e c h a n g e o f v c l o c i t y a l o n g t h e a x i a l f l o w must b c n o w c v c r , a t l e a s t t w o p u l s e s e p a r a t i o n s , o r 6 cm.
More-
o v e r , t h c p h y s i c a l d i m e n s i o n s of t h e DICM r e a l i s t i c a l l y l i m i t i t s ' a b i l i t y t o r c g i s t c r small velocity fluctuations.
Thus, t h e c y l i n d e r d i m e n s i o n s of
10 cm d i a m e t e r and 1 5 cm l e n g t h p r o h i b i t r e g i s t c r i n g t u r b u l e n t s c a l e s much smaller t h a n 20-30
C?,
along the flow a x i s .
DATA LOGGING AND PROCESSING
The p u l s e s i g n a l s o u t p u t from t h c DICM a r c t r a n s m i t t e d i n t o t h e v a n c h o u s i n g t o a c i r c u i t t o r e g i s t e r c a c h s i x t h p u l s e and t h e n t o a wave p e r i o d processor.
T h i s p r o d u c c s a s c r i e s of d i g i t a l v a l u e s p r o p o r t i o n a t c t o t h e
t i m e s p a c i n g b c t w c c n e a c h p u l s e p a i r w h i c h i s r c c o r d c d on a Mcmodyne Model
201 d i g i t a l c a s s e t t e r e c o r d c r .
Tile e n t i r e e l e c t r o n i c s , i n c l u d i n g a t i m e
s e q u e n c e s w i t c h and DC b a t t e r y s u p p l y i s c o n t a i n e d w i t h i n e a c h v a n e h o u s i n g Each s y s t e m b e i n g i n d e p e n d e n t .
Upon r e t r i e v a l o f t h e BLT e a c h c a s s e t t e
t a p e i s r e a d and c o n v e r t e d t o 7 t r a c k m a g n e t i c t a p e t o b e a n a l y z e d on a CDC
3300 Computer. The s a m p l i n g r a t e of t h e BLT c a n r c c o r d t h e D I C N p u l s c s c o n t i n u o u s l y o r b y usc o f t l i e s w i t c h i n : ; c i r c u i t s a m p l e a t g i v e n i n t e r v a l s a t p r e s e t spacinl;.
The amount o f d a t a ( i . c . ,
p u l s c s ) rccordcd is l i m i t e d by t h e t a p c
c a p a c i t y ; rougiily 1 6 , 0 0 0 d i z i t a l villucs o f time i n t e r v a l s p c r t a p c .
Note
t h e f a s t c r t m mean c u r r e n t t h e s h o r t e r t l i c r e a l t i m e r e c o r d , e.g.,
fq'r a
288 c u r r e n t speed of 90 cm/sec t h e r c c o r d i n g t i m c i s a b o u t 6 h o u r s ; f o r 10 cm/sec, 54 h o u r s .
For o u r a p p l i c a t i o n w e chose a r e c o r d i n g c y c l e of 17.5
This i n t e r v a l p r o v i d e d a b o u t 7 days re=
minutes on and 52.5 m i n u t e s o f f .
c o r d i n g f o r t h e normal t i d a l c u r r e n t s . DATA REDUCTION AND ANALYSIS
Since d a t a p o i n t s ( i . e . ,
instantancous velocity values) a r e ootained at
t h e r a t e of one f o r c v c r y 1 8 cm advance of w a t e r through t h e D I C M , t h e res u l t a n t sequence i s a p p r o p r i a t c f o r s p a t i a l / w a v c n u m b c r s p e c t r a l a n a l y s i s . The e n t i r e r e c o r d i s d i v i d e d i n t o non-overlapping scgments o f 120 p o i n t s e a c h ( z e r o f i l l i n g t h e l a s t scgment i f n e c e s s a r y ) .
Thc c n c r g y d e n s i t y
spectrum is found f o r e a c h segment and t h e r e s u l t i n g s p e c t r a a r e ensemble averaged t o o b t a i n a f i n a l s i n g l c e n c r g y d c n s i t y spectrum f o r t h e r e c o r d . Before s p e c t r a l a n a l y s i s f o r e a c h segment, w i l d d a t a p o i n t s a r e r e p l a c e d w i t h t h e a r i t h a t i c a v e r a g e of t h e good p o i n t s f o r t h a t segment and t h e a r i t h m e t i c a v e r a g e i s t h e n removed from t h e segment.
A 10%: t a p e r c o s i n e
d a t a window i s a p p l i e d and t h e e n e r g y d e n s i t y s p e c t r u m is c a l c u l a t c d u s i n g t h e FFT a l g o r i t h m .
The s p e c t r a a r e normalized s u c h t h a t t h e a r e a under t h e
curve i s equal t o t h e variance
0
of t h e o r i g i n a l series.
A p l o t of t h e e n c r g y d e n s i t y s p c c t r u m i n l o g / l o g form i s g e n e r a t e d .
The
o b j e c t i v e i n computing e n e r g y d e n s i t y s p e c t r a from t h e BLT d a t a is t o e s t i -
mate t h e r a t e of d i s s i p a t i o n of t u r b u l e n t k i n e t i c e n e r g y by v i s c o s i t y u s i n g t h e Kolmogoroff i n e r t i a l s u b r a n g e h y p o t h e s i s .
In t h e i n e r t i a l s u b r a n g e t h e
energy d c n s i t y spectrum f o r t h e l o n g i t u d i n a l v e l o c i t y component i s e x p e c t e d t o b e of t h e form @(Xi
where €
= 0.137 f 2/3 ~ 5 / 3
i s t h e d i s s i p a t i o n r a t e and k i s t h e wavenumber ( c y c l e / c m ) .
Among
t h e r e q u i r e m e n t s of t h e t u r b u l e n t f i e l d f o r Eq. ( 1 ) t o b e a p p l i c a b l e i s t h a t Reynold's number f o r t h e mean motion be l a r g e and t h a t t h e t u r b u l e n c e b e homogeneous and i s o t r o p i c t h r o u g h o u t t h e s m a l l s c a l e r a n g e , s p e c i f i c a l l y , t h e n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e of a n i n e r t i a l s u b r a n g e i s p r e c i s e l y ( B a t c h e l o r , 1960)
(y)
3/8 > > l
289 where U i s t h e R1.E v a l u e of tile t u r b u l e n t v e l o c i t y .
is t h e l e n g t h cor-
r e s p o n d i n g t o t h e wavenumber a t which t h e maximum i n t h e e n e r g y d e n s i t y spectrum o c c u r s and
v
=
1.3 x lo-'
cm2/sec is t h e k i n e m a t i c v i s c o s i t y .
Using a v a l u e of 4 . 0 cm/sec f o r U ( r e p r e s e n t a t i v e of t h e s t a n d a r d d c v i a t i o n s f o r t h e v e l o c i t y o b t a i n e d from t h e BLT measurements) and 40 cm f o r
4
( t h e energy containing s c a l e s should be approximately equal t o the
p r o d u c t o f t h e d i s t a n c e from t h e bottom and Von Karman's c o n s t a n t , 0 . 4 ) .
(gy'8P 34
(3)
s o t h a t t h e p r e c e d i n g c o n d i t i o n i s somewhat s a t i s f i e d and a l i m i t e d i n e r t i a l s u b r a n g e can b e e x p e c t e d . Thc observed s p e c t r a a r e , i n g e n e r a l , c o n s i s t e n t w i t h a -5/3 power law f o r a range of s c a l e s around 1 meter, a l t h o u g h t h i s v a l u e is l a r g e r t h a n would be e x p e c t e d f o r a l e g i t i m a t e i n e r t i a l s u b r a n g e a c c o r d i n g t o t h e prec e d i n g argliment.
The more r a p i d d e c r e a s e i n energy d e n s i t y w i t h i n c r e a s -
i n g wavenumber a t wavelengths g r e a t c r t h a n 0.01 r y c l c s / c m i s a t t r i b u t e d t o t h e DICM r e s p o n s e c h a r a c t e r i s t i c s .
The s p e c t r a i n d i c a t e no s i g n i f i c a n t i n -
p u t of e n e r g y a t t h e h i g h e r wavenumbers and i f t h e e n e r g y d e n s i t y spectrum
i s a monotonic, d e c r e a s i n g f u n c t i o n o f wavenumber w i t h a power law nowhere g r e a t e r t h a n -5/3 t h e n t h e a c t u a l d i s s i p a t i o n r a t e s h o u l d n o t exceed t h e v a l u e s d e t e r m i n e d from t h e s p e c t r a , t h u s p r o v i d i n g an upper l i m i t f o r t h e Reynold's stress and f r i c t i o n v e l o c i t y . The e n e r g y d i s s i p a t i o n r a t e f o r t h e r e c o r d i s e s t i m a t e d from t h e ensemb l e average energy density spectrum using
e v a l u a t e d f o r K = 0.01.
Every r e c o r d t h u s y i e l d s v a l u e s f o r t h e a v e r a g e
s p e e d , v a r i a n c e (and s t a n d a r d d e v i a t i o n ) and d i s s i p a t i o n r a t e . The d a t a o b t a i n e d from t h e BLT a r e t h e n r c l a t c d t o t h e mean c u r r e n t f i e l d and i t s ' e f f e c t on t h e bottom through c a l c u l a t i o n of t h e f r i c t i o n v e l o c i t y (U,). FRICTION VELOCITY Various techniques a r e a v a i l a b l e f o r estimating the f r i c t i o n veloFity
290
u* 3 where
7
1.
i s t h e K e y n o l d s ' s t r c s s and
(5)
p
is t h e water d e n s i t y .
L o g a r i t h m i c V e l p-c i-~ t y P r o f-~ ilc. I f t h e mean v e l o c i t y p r o f i l e i s
l o g a r i t h m i c t h e n , a c c o r d i n g t o P r a n d t l ' s mixin;:
length thcory (Schlicting,
1960).
for w h i c h
u2 - 8, u*l
2. tlie
Quadrcitir -
-
S-___ t r e s s Lciw.
5 . 7 5 log (z2/z,)
T h e r o u z h n c s s , h e i g h t , Zo, is d e f i n e d a s
d i s t a n c e above t h e b o t t o m a t w h i c h t h e m e a n v e l o r i t y i s z e r o .
6 then
,
u*2log(;)
U(Z) =
5.75
u*2
5 . 7 5 log (z/zo)
E -
and
'
=
'*2
w h e r e t h e d r a g coefficient
2
=
(7)
[
5 . 7 5 log (Z/ZO)
From E q .
291 Measurements by ( S t e r n b c r z ,
1969, 1 9 7 2 ) i n d i c a t e t h a t a t a h e i g h t o f 100 cm
a b o v e t h e b o t t o m CD i s a p p r o x i m a t e l y 3.1 x
T = p(3.1 x
therefore
qoo
The r o u g h n e s s h c i g l i t c o r r e s p o n d i n g t o t h i s v a l u e o f CD i s 7 . 5 5 x 10The v a l u e s o f CD d c t c r r , d n c d f r o m (10) u s i n g Zo a n d a t 2 5 cm a r e 3 . 5 5 x
3.
Dissipation katc.
and 4.76 x
-
7.58 x
2
cm.
cm a t 6 2 . 5 cm
respcrtively.
I f a b a l a n c e is a s s ume d t o e x i s t b e t w e c n p r o -
d u c t i o n a n d d i s s i p a t i o n of t u r b u l c i i t k i n e t i c e n e r g y i n t h e b o u n d a r y l a y e r (Hinze, 1959) t h e n
from which
I f P r a n d t l ' s mixing l e n g t h h y p o t h e s i s is also v a l i d then
w h e r e K = 0 . 4 is Von Karman's
constant.
Sincc T/p
=
U,2
,
Eq.(14)
yields
I f tile d i s s i p a t i o n rates d e t e r m i n e d f r o m t h c s p e c t r a a r e a c c u r a t e t h e n Eq.
(13) g i v c s p o t e n t i a l l y tlie b e s t mct hod f o r e s t i m a t i n g tlic f r i c t i o n v e l o c i t y o r b o t t o m stress b e c a u s e : d c p c n d e n c c on a l o g a r i t h m i c v e l o c i t y p r o r i l e i s n o t n e c e s s a r y a n d ; a n e s s e n t i a l l y t u r b u l e n t q u a n t i t y is f o u u d f r o m t u r b u -
292 l e n c e measurements i n s t e h d of mean flow mcasurements o n l y . Because of b e a r i n g p r o b l e m s no more t h a n two D I C M ' s o p e r a t e p r o p e r l y a t t h e same t i m e s o i t was n o t p o s s i b l c t o d c t c r m i n e i f t h e v e l o c i t y p r o f i l e s were a c t u a l l y l o g a r i t h m i c .
A t Lest, t h e mean v e l o c i t i e s could o n l y be
p l o t t c d on s e m i l o g g r a p h papcr and t h e s t r a i g h t l i n e s e x t r a p o l a t e d t o z e r o v e l o c i t y t o f i n d t h e roughness h e i g h t s , which v a r i e d from 3 t o 5 cm, somcwhat l a r g e r t h a n b u t n o t i n c o n s i s t e n t w i t h v a l u e s o b t a i n e d by ( S t e r n b e r g , 1972). For a l l c a s e s where r e l i a b l e d a t a was o b t a i n e d s i m u l t a n e o u s l y from two D I C M ' s t h e f r i c t i o n v e l o c i t y was c a l c u l a t e d from b o t h Eq.
( 5 ) and ( 1 3 ) .
For t h e s e c a s e s t h e f r i c t i o n v e l o c i t y was a l s o c a l c u l a t e d i n d e p e n d e n t l y f o r b o t h DICll's u s i n g Eq.
(5) and Eq. (11) w i t h t h e a p p r o p r i a t e d r a g c o e f f i -
i e n t , depending on t h e h e i g h t of t h e DICM.
For t h e remaining c a s e s where
o n l y one DIW gave r e l i a b l e d a t a t h e f r i c t i o n v e l o c i t y was found u s i n g Eqp. (15) and ( 1 1 ) . RESULTS The ULT was p l a c e d a t t h e NLDS f o r a p e r i o d of one week b e g i n n i n g 22 September 1975 and a t t h e EHDS f o r 2!5 days b e g i n n i n g on 4 August 1975.
In
b o t h c a s e s one of t h e D I C M ' s was n o t o p e r a t i n g due t o e l e c t r o n i c malfunct i o n s t h a t o c c u r r e d w h i l e t h e i n s t r u m e n t was on t h e bottom.
A t t h e NLDS
t h e d a t a w a s r e c o v e r e d from t h c meters a t 100 cm and 25 cm above tlie b o t tom, w h i l e a t t h e EHDS t h e meters a t 62.5 and 25 cm p r o v i d e d good d a t a . R e p r e s e n t a t i v e t i m e s e r i e s p l o t s of s p e e d a t t h e 25 cm and 100 cm l e v e l s
are shown i n Fig:. 10.
The e n e r g y s p e c t r a ( F i g . 11) computed f o r b o t h rec-
o r d s are v e r y s i m i l a r w i t h s i m i l a r t o t a l e n e r g y d e n s i t y l e v e l s and t h e de-
crease of e n e r g y w i t h i n c r e a s i n g f r e q u e n c y i s t h e same.
The c u r r e n t d a t a o b t a i n e d from t h e BLT i s most e a s i l y r e l a t e d t o t h e e f f e c t of t h e c u r r e n t s on t h e bottom through t h e f r i r t i o n v e l o c i t y o r Reynolds stress (Cq. 5 ) . I f t h e boundary s h e a r stress
To
r e q u i r e d t o e r o d e t h e sediment i s
known t h e n tlie t h r e s h h o l d f r i c t i o n v e l o c i t y the
U,
u,,
can b e c a l c u l a t e d .
If
v a l u e s computed froin t h e c u r r e n t d a t a a l o n e are g r e a t e r t h a n t h i s
threshhold f r i c t i o n v e l o c i t y , then e r o s i o n w i l l occur, i f n o t , the sediment can be c o n s i d e r e d s t a b l e .
293
2
40
z
9
30
t
l3
z 0
2o 10
' 0
20
+o
6.0
8.0
100
I20
U Q I60
10.0
TIME (MINUTES)
Fig.10.
BLT record at 25 cm (upper) and 100 cm (lower)
I
I -I
-2 LOG WAVEWM9ER
Fig.11.
(CY-')
Wavenumber spectrum for BLT current speed 100 cm above bottom
294
Measurements in a flume tank at the Massachusetts Maritime Academy have shown that for dredge spoils taken from the Thames River a mean velocity of approimately 52.5 cm/sec at a height of 15.25 cm above the bottom (half the height of the flume tank) was sufficient to cause significant erosion and material transport., By applying the Quadratic Stress Law to this data it is possible to calculate the threshhold friction velocity.
From Equation
'10, the drag coefficient Co at 15.24 cm is 5.69 x
T~ = p C D
d = 15. 68'dy/crn2
16 dy/cm
2
.
and from Equation ( 5 )
U,o
=
3.96 cm/sec
.
The major objective of the BLT, therefore, is an accurate assessment of the friction velocity or stress values for the NLDS and the EHDS to determine whether or not the currents in either location are large enough to produce a friction velocity greater than 4 cm/sec or a stress greater than
16 dy/cm2. A summary of the BLT measurements at the NLDS are presented in Table 2 and those from the EHDS in Table 3 .
Wherever possible, the friction velo-
city for each record was calculated by the different methods discussed above. The variability of the friction velocity among these estimates for any record and the variability between records is not unexpected.
The reasons
for this variability are:
(1) The assumption of a logarithmic velocity profile is probably not Previous work by
valid in approximately 15% of the cases given here.
others has indicated that up to 40% of the profiles measured were not logarithmic. (2)
The drag coefficient used in the quadratic stress equation is known
to vary from less than 2 x
to more than 4 x 10-3 depending on the bed
configuration with corresponding variation in the roughness height. (3)
Von Karman's constant used in methods 5 and 6 is unknown for fluids
containing suspended sediment. Furthermore, it should be noted that the quadratic stress law is derived
296 from t h e l o g a r i t h m i c v e l o c i t y p r o f i l e e q u a t i o n and t h e r e f o r e i s n o t a t r u l y i n d e p e n d e n t c a l c u l a t i o n of f r i c t i o n v e l o c i t y f o r t h e same roughness The d i f f e r e n c e i n t h e methods i s i n t h e v a l u e s of t h e roughness
height.
heights used.
The d r a g c o e f f i c i e n t (CD) i n t h e q u a d r a t i c stress e q u a t i o n
i s a n e m p i r i c a l v a l u e of 3.1 x
s p e c i f i e d f o r a h e i g h t of 100 cm which
c o r r e s p o n d s t o a roughness h e i g h t o f .0758 c m .
However, p l o t s of mean
v e l o c i t y p r o f i l e i n d i c a t e roughness h e i g h t s an o r d e r of magnitude g r e a t e r than t h i s value.
C o n s e q u e n t l y , f r i c t i o n v e l o c i t i e s c a l c u l a t e d from t h e
q u a d r a t i c s t r e s s e q u a t i o n a r e , i n most c a s e s , s m a l l e r t h a n t h o s e c a l c u l a t e d from t h e l o g a r i t h m i c v e l o c i t y p r o f i l e e q u a t i o n . Because of t i m e l i m i t a t i o n s o n l y one t h i r d of t h e e n e r g y d e n s i t y s p e c t r a
were examined t h r o u g h t h i s method t o o b t a i n k i n e t i c e n e r g y d i s s i p a t i o n rates.
F r i c t i o n v e l o c i t i e s c a l c u l a t e d t h r o u g h t h i s method t e n d t o a g r e e
more c l o s e l y w i t h v a l u e s c a l c u l a t e d from t h e q u a d r a t i c stress e q u a t i o n t h a n t h o s e o b t a i n e d from t h e l o g a r i t h m i c p r o f i l e method. The f r i c t i o n v e l o c i t i e s c a l c u a l t e d from t h e d i s s i p a t i o n r a t e s do n o t depend on v a l u e s of e i t h e r Von Karman's c o n s t a n t o r t h e d r a g c o e f f i c i e n t (CD) i n t h e q u a d r a t i c stress e q u a t i o n o r on a l o g a r i t h m i c v e l o c i t y p r o f i l e . T h e r e f o r e , t h e s e v a l u r s a r e a c o m p l e t e l y independent measure of f r i c t i o n v e l o c i t y and a r e c o n s i d e r e d t o be t h e most r e l i a b l e e s t i m a t e s . A t b o t h t h e EHDS and NLDS t h e e n e r g y d e n s i t y s p e c t r a and d i s s i p a t i o n
rates f o r t h e lower D I C M ' s are g e n e r a l l y g r e a t e r t h a n t h o s e f o r t h e upper m e t e r , which f o r a c o n s t a n t ( w i t h h e i g h t ) f r i c t i o n v e l o c i t y i s c o n s i s t e n t with theory. If 9
u,
=
( x z q 1/3
where K i s Von Karman's c o n s t a n t e q u a l t o 0 . 4 , Z t h e h e i g h t above t h e b o t tom and
C
is the dissipation rate.
Then f o r a c o n s t a n t
U,
and
which i s g r e a t e r t h a n rates ( C
C2
since
Z2
is g r e a t e r than Z 1 .
the dissipation
) f o r t h e EHDS f o l l o w t h i s r e l a t i o n s h i p q u i t e c l o s e l y ; t h o s e
a t t h e NLUS n o t a s w e l l .
296 The v a l u e s f o r f r i c t i o n v e l o c i t y o b t a i n e d f r o m e q u a t i o n (13 ) c a l c u l a t e d from t h e d i s s i p a t i o n r a t e method a l o n e , b u t s i m i l a r t o t h o s e o b t a i n e d from t h e l o g a r i t h m i c v e l o c i t y p r o f i l e s .
This suggests t h a t t h e value of
0 . 4 f o r Von Karman's c o n s t a n t i s h i g h s i n c e a l o w e r v a l u e would r e d u c e t h e
u,
from t h e l o g a r i t h m i c v e l o c i t y p r o f i l e e q u a t i o n t o v a l u e s t h a t would
b e more i n a g r e e m e n t w i t h t h e
u,
from t h e o t h e r m e t h o d s .
F u t u r e work
a l o n g t h e s e l i n e s m i g h t g i v e a n e s t i m a t e of Von Karman's c o n s t a n t f o r w a t e r s c o n t a i n i n g suspended m a t e r i a l . An e x a m i n a t i o n o f T a b l e s 2 and 3 i n d i c a t e s t h a t most o f t h e f r i c t i o n v e l o c i t i e s m e a s u r e d a t b o t h t h e NLDS and t h e EHDS a r e less t h a n 2 c m / s e c and t h a t a p r a c t i c a l upper l i m i t might be set a t approxi mat el y 2.5 cm/sec. T h i s v a l u e i s o b v i o u s l y much less t h a n t h e 4 c m / s e c m e a s u r e d a s t h e t h r e s h hold f r i c t i o n v e l o c i t y i n t h e flume t a n k , hence i t r a n be concluded t h a t t h e s p o i l s d e p o s i t e d a t t h e NLDS a r e c o m p a r a t i v e l y s t a b l e u n d e r n o r m a l cond i t i o n s and v e r y l i t t l e e r o s i o n , i f a n y , s h o u l d o c c u r d u e t o r u r r e n t f l o w .
CONCLUSIONS
The r e s u l t s o f t h i s s t u d y d e m o n s t r a t e t h a t t h e e x i s t i n g BLT i s a u s e f u l d e v i c e w i t h s e v e r a l l i m i t a t i o n s , f o r s t u d y i n g t h e b o t t o m b o u n d a r y l a y e r and w i t h f u r t h e r development h a s t h e p o t e n t i a l f o r b e i n g a n e x c e l l e n t boundary layer instrument.
S p e c i f i c a l l y , measurements a r e o b t a i n e d of o n l y t h e
l o n g i t u d i n a l t u r b u l e n t v e l o c i t y component a n d , a l t h o u g h t h i s i s a n o r d e r o f m a g n i t u d e improvement o v e r t e c h n i q u e s m e a s u r i n g o n l y t h e mean f l o w , t h e Reynolds stress is s t i l l d e te r m in e d i n d i r e c t l y .
The n e x t v e r s i o n o f t h e
BLT w i l l a l l o w measurement o f b o t h t h e v e r t i c a l and h o r i z o n t a l v e l o c i t y components s o t h a t t h e R e y n o l d s s t r e s s c a n b e d e t e r m i n e d d i r e c t l y .
Sensors
o t h e r t h a n t h e DICM a r e b e i n g i n v e s t i g s t e d , f o r e x a m p l e , t h e e l e c t r o m a g n e t i c c u r r e n t meter, i n o r d e r t o a c h i e v e improved s m a l l s c a l e r e s o l u t i o n and a l o w e r v e l o c i t y t h r e s h h o l d ; t h e p r e s e n t BLT r e q u i r e s a mean c u r r e n t of 1 5 cm/sec o r g r e a t e r t o y i e l d r e l i a b l e measurements whereas t h e n e x t v e r s i o n of t h e B1.T w i l l b e d e s i g n e d t o o p e r a t e a t a much l o w e r v a l u e ; c o n s e q u e n t l y i t s h o u l d a l s o b e p o s s i b l e t o e x a m i n e t h e e f f e r t s o f wave a c t i o n on t h e b o t t o m w i t h o r w i t h o u t t h e p r e s e n c e o f a mean c u r r e n t , w h i c h ( e s p e c i a l l y d u r i n g s t o r m c o n d i t i o n s ) may c o n t r i b u t e h e a v i l y t o s e d i m e n t t r a n s p o r t .
TABLE I1
%oo*Q 22
Sep
’75
1341
cmlsec
37.453.5
UZ5*U
Q’UlOO
- NEW u’u25
cmlsec
30.5%4.1
.u9
LONDON DISPOSAL S I T E
Aii
U*l
u*2
u*4
cmlsec
cm/sec
cm/scc
cm/scc
u*5
E
cm/sec cmlsec
.13
6.9
2.0
2.1
2 .o
2.2
.27
3.1
2.2
1.5
2.0
.21
1451
37.455 .O
28.554.6
.13
.16
10.6
1601
27.952.6
16.052.6
.09
.16
6.6
1.9
1.2
1.1
1.1
.04
.21
6.2
1.8
1.2
1.1
1.4
.07
1711
21 .9+2 .8
15.723.3
.13
1821
6.754.9
.73
0.4
1931
22.352.5
.11
1.2
2041
36.7+4.4
.12
2 .o
2151
22.622.8
.12
1.3
2301
10.2k1.3
.13
0.6
0011
S.7+6.2
.71
0.5
0121
14.652.2
.15
0.8
0231
31.124.3
27.154.9
.14
.18
4 .O
1.2
1.7
2.3
2.2
.26
0341
26.623 .5
19.753.1
.13
.16
0451
25.153.3
.13
.17
5.9 6.6
1.7 1.9
1.4 1.4
1.3 1.5
1.8
18.753.1
2.0
.14 .21
0601
8.253.9
6.552.7
.48
.42
1.7
0.5
0.5
0711
14.251.9
.26
8.2
2.4
0.8 1.4
2 .o
2.4
.33
23 Sep ’75
0821
24.354.6
0931
24.6T3.6
.13 16.154.2
.19 .15
1.4
3
’I
Aug. ‘ 7 5
TJ25fa
T725fu
cm/scc
cm/scc
1837
27.123.2
21.753.1
1947
11.652.1
2.353.0
TABLLIII
u/Uloo
-
EAST HOLE DISPOSAL SITES
u / U ~ ~A
0
U*1 cm/scc
.ll
.18
u*2 cm/sec
cmtscc
.14
5.4
2.4
1.6
1.o
1.28
3.3
1.4
0.7
0.7
cm/scc
c m2/SCC 3
1.4
.07
2057 2207
26.e3.0
16.723.0
.ll
.17
9.3
4.1
1.6
2317
33.e3.7
26.953.6
.ll
.13
6.1
2.7
2 .o
1.2
1.5
-08
0027
17.852.3
14.2+2 - .9
.15
.20
3.6
1.6
1.1
0.9
1.o
.02
0137
14. e 2 . 2
10.952.3
.15
.20
3.1
1.4
0.8
1517
21.424.2
15.224 .2
.19
.27
6.2
2.7
1.3
1.5
1617
32 .M_3.3
26 .2&5.4
.10
.20
5.8
2.5
1.9
1.2
1.4
.07
1757
34 .0+3 .5
28.123.7
.10
.12
5.9
2.6
2 .o
1.1
1.4
.07
1907
34.855.2
27.554.8
.14
.17
7.3
3.2
2.1
1.2
1.9
.16
2017
27.e5.4
19.754.0
.19
.20
8.1
3.5
1.7
0.8
1.2
.03
2127
11.153.5
2237
13.823.2
12.422.7
.23
.21
1.4
0.6
0.8
2347
26.e3.3
4.5+3.8
.12
-17
4.5
2 .o
1.6
0.9
1.2
.03
5 Aug.
‘75
.31
299 ACKNOWLEDGEMENTS
We are i n d e b t e d t o Mr. John Roklan f o r d e s i g n of t h e BLT e l e c t r o n i c s and t o D r . David S h o n t i n g f o r h i s h e l p i n p r e p a r i n g t h e m a n u s c r i p t . T h i s work was s u p p o r t e d by U. S . Army Corps of E n g i n e e r s , Waltham, M a s s a c h u s e t t s and t h e U. S. Naval F a c i l i t i e s E n g i n e e r i n g Command, Philadelphia, Pennsylvania. REFERENCES Batchelor, G. K . ,
1960.
The Theory o f Homogeneous T u r b u l e n c e
Cambridge U n i v e r s i t y P r e s s , London H i n z e , J. O . ,
1959.
Turbulence
McGraw-Hill Book Company, N e w York Schlichting, H.,
1968.
Boundary l a y e r t h e o r y
McGraw-Hill Book Company, New York S h o n t i n g , D. C . ,
1968. A u t o s p e c t r a of Observed P a r t i c l e Motions i n Wind
Waves,
J. Mar. R e s . Vol 26(1):43-65 S t e r n b e r g , R. W . ,
1968.
F r i c t i o n f a c t o r s i n t i d a l channels with d i f f e r i n g
bed r o u g h n e s s . Marine Geology 6:243-260 S t e r n b e r g , R. W . ,
1972. P r e d i c t i n g i n i t i a l m o t i o n and b e d l o a d t r a n s p o r t o f
sediment p a r t i c l e s i n t h e shallow marine environment.
In:
S h e l f Sediment T r a n s p o r t , S w i f t , Duane and P i l k e y , Eds.
Dowden, H u t c h i n s o n and Ross, I n c . , S t r a n d s b u r g , PA 61-82
This Page Intentionally Left Blank
301 SUBJECT INDEX 62,
Autocorrelation,
63,
68,
approach, 2 2 1 ,
70.
230,
232,
235.
load Transport Meter, 2 2 9 ,
230,
232,
Bagnold's
123,
Bearing flows,
124.
Bed
- form, 2 2 5 , 2 3 0 , 2 3 2 .
-
load, 2 3 0 ,
232,
235. 235.
Bottom
-
boundary layer, 2 7 , 99, 209,
-
103-105, 237.
107,
239-241,
coefficient, 2 , current, 2 3 7 , friction, 4 ,
29,
109, 244,
24,
43,
240,
49,
40,
115,
245,
16, 40,
239,
37,
30, 110,
42,
247,
45,
120,
118,
51,
54,
153-157,
252,
275-277,
275,
284.
285,
56,
83,
159-162, 296.
47.
242,
244.
205.
118, 2 2 1 .
homogereous layer,
- m i x e d l a y e r , 101, 1 0 2 .
-
topography, 2 3 8 ,
240.
Boundary condition, 2 , 146,
147,
165,
4-6, 169,
168,
Boundary layer, 5 9 - 7 9 , 136,
-
142,
149,
atmospheric,
103,
benthic, 8 3 ,
86,
bottom,
83,
166,
176,
94,
184,
107-112,
129,
110,
18, 2 7 - 3 2 ,
175-177, 84,
170,
87,
16,
13,
185,
36,
40,
47,
132,
185, 215.
115, 194,
123,
118,
198, 232,
131,
239,
256.
130.
96,
209,
218.
see bottom boundary layer.
surface, 2 7 - 2 9 ,
37,
turbulent, 9 9 ,
101.
42,
Boussinescq approximation, Brunt-Vaisala
45,
51,
170,
frequency, 5 4 ,
165,
170.
172.
56,
108, 1 5 5 ,
160-lh2.
B-spline,
1-5,
14-18,
Buoyancy,
167,
170,
177,
179,
182,
184, 256.
76-79,
83,
209,
221,
232,
Bursting, 6 1 ,
24,
Chebyshev polynomials, Chlorophyll-a
27,
14,
36. 235.
15.
concentration,
Coriolis parameter, 4 9 ,
30-32,
108,
255,
257,
259,
263,
265.
156,
172,
187,
191,
237.
Current
- measurements , 61, 6 2 , 83, 8 6 , 9 0 ,
92-95,
275,
276,
282.
302
-
meter, 83, 244,
-
86,
251,
profile, 103,
104,
143,
145,
Cyclosonde,
87,
259,
I,
276,
15,
153,
101,
187,
283-285,
16,
18,
191,
198,
204,
240-242,
296.
24,
27,
28,
36,
38,
42,
44,
45,
47,
107,
115,
118,
120,
124,
126,
132,
134,
136,
147,
194,
198,
230,
240,
251,
272,
275,
290-292.
247,
290-92.
103-107,
115,
118.
104,
Density distribution,
140-
194,
198,
241,
261.
Diffusion coefficient, s e e eddy diffusivity. 120,
Drag coefficient, Eddy, 62-65,
-
141,
-
84,
-
157,
107-109,
145,
237,
239,
158,
166,
263,
265,
266.
129,
130,
133,
135,
136,
139-
statistical independence, 70. 1-4,
viscosity, 47,
-
97,
143.
171.
102,
131,
6,
bottom, 56,
158,
16,
18,
22-25,
27-31,
40,
42,
43,
45,
191.
157,
159,
boundary,
7,
136,
153,
Ekman layer,
-
96, 102,
diffusivity,
142,
202.
187,
165,
188,
170,
194,
198,
204,
205.
178.
frictional, 204. geostrophic, 204. laminar,
169,
179.
surface,
165,
167,
103,
turbulent, 161,
162,
194,
Ekman veering, 200,
Estuary,
202,
104, 237,
103,
237,
221,
188,
109,
112,
110,
115,
120,
155-157,
159,
239.
104,
240,
222,
191.
107,
251,
232,
112,
110,
114,
120,
187,
194,
252.
235,
255, 9,
Finite difference scheme, 1 ,
266.
I I ,
13,
24,
35,
36.
Fluorometer, 259. 170,
Fourier decomposition, Frictional flow, 188, 237,
156,
110,
239,
240,
246,
247,
250,
Fronts, 255,
256,
259,
260,
266.
Froude number,
153,
177.
198.
101,
Friction velocity,
172,
155,
158,
158, 251,
160-162,
187, 289,
194, 291,
202, 292,
204,
225.
Geostrophic
-
current, 103,
107,
109,
- drag coefficient, 3,
30,
I l l , 120.
114,
115,
118,
120,
205,
294-296.
205.
303
-
interior, 1 8 8 ,
191.
shear, 2 0 0 . velocity, 1 5 6 - 1 5 9 ,
161,
194,
198,
237.
Geothermal heat flux, 1 5 8 . Halocline, 2 6 3 . Ice boundary, 1 6 5 . Interfacial propagation velocity, 2 5 9 . Intermittence, 5 4 ,
-
60,
61,
67,
71,
76,
79,
86,
241.
quasi-period, 7 5 - 7 7 .
Internal waves, see waves. Isobaths, 2 3 7 ,
239,
Isopycnals, 1 0 3 ,
240,
105,
242-244,
252.
200.
Jet
-
-
coastal, 2 5 5 ,
259.
stream, 2 6 3 . tidaf, 2 5 8 .
Kinematic viscosity, 1 0 7 ,
108,
Laplace transform, I ,
172.
Lilly equations, 1 7 4 ,
179.
Logarithmic layer, 9 9 ,
187,
202,
202,
289.
204,
237,
239,
240,
244-248,
252.
Mass transfer coefficient, 2 1 0 ,
211,
213,
177,
180.
215,
218.
Mean
-
density, 1 6 6 ,
168,
170,
-
field, 1 6 5 ,
-
surface slope, 2 2 2 .
-
velocity, 8 4 ,
169,
171,
85,
96,
171,
282-285,
124,
296.
134,
168,
171,
173,
177-180,
247,
269,
275,
282,
290,
292,
Mixed-layer, 1 5 6 ,
157,
166,
168-170,
-
well, 1 5 3 ,
-
wind, 4 9 - 5 1 .
slab, 1 8 7 ,
185,
205.
surface, 2 5 9 . 162,
177.
Monin-Obukov length, 1 3 0 , Nutrient, 2 5 7 ,
259,
263,
158. 266.
Orr-Sommerfeld equation, 1 7 9 .
153,
222,
155,
157,
230,
237,
185,
256.
294.
165,
239,
166,
242,
243,
304
255,
Phytoplancton patchiness, Pressure gradient, 7 8 , 165,
166,
Reynolds stress, 5 9 ,
157,
flux,
101.
177-179,
288.
112,
161,
144,
170,
177,
180, 2 5 5 .
170,
174,
175,
167, 60,
112,
I l l ,
bulk,
266.
79,
62,
143,
160,
63,
144,
167,
Richardson n u m b e r , 101,
-
265,
188.
205.
107,
Reynolds number, 96,
263,
169,
Return flow, 188, 2 0 4 ,
94,
258,
108.
Prandtl number,
Pycnocline,
256,
71,
182,
168,
72,
75,
76,
184, 232,
170,
174,
78,
235.
175,
79,
276,
83-86, 289-292
177.
179.
130.
gradient,
- turbulent, 1 6 0 . Roll waves, see waves 157,
Rossby number,
174,
250,
290-292,
294,
-
rate, 6 0 ,
230,
67,
142,
239,
202,
240,
237,
244,
239,
247.
240,
246-
247,
255,
256,
259,
265.
232.
84,
86,
87,
105, 2 8 7 .
time, 2 2 9 . 176.
Schmidt number,
Secondary currents, 2 2 5 , Sediment, 1 2 4 , 209-212,
-
188, 2 3 7 ,
141,
296.
Salinity profile, 2 4 1 , Sampling, 2 2 9 ,
175,
126,
Roughness parameter,
232.
126-130,
2 1 8 , 221,
132-134,
222,
225,
136-139, 229,
concentration profile, 2 0 9 - 2 1 1 , 123,
suspended,
124,
128-130,
230,
213, 132,
141,
146,
147,
232,
235,
248-250.
149,
215.
134-137,
139-141,
146,
148, 247. Sensor, 51, 5 4 , 269,
-
61,
62,
65,
67,
acoustic travel time, 8 7 - 9 0 .
- benthic acoustic stress, 9 4 ,
-
95.
scattering, 86. thermistor, 5 4 . volume averaging, 8 6 ,
Settling velocity,
129,
Shear, 2 9 ,
45,
-
83-87,
272.
42,
flow, 5 4 ,
43,
59,
87.
138, 1 3 9 , 252,
166-168,
259.
252.
141.
89,
90,
93-97,
241,
252,
306
-
meter, 8 7 - 8 9 , stress, 3 ,
. . -
94.
15,
bottom, 2 ,
27,
221,
boundary, 1 2 4 ,
velocity, 1 2 3 , 205.
222,
130,
30,
96,
232,
125,
134,
127,
135,
166,
209.
291,
292.
250. 133,
136,
138-141,
139,
187,
141.
194,
198,
200,
204,
120,
123,
230.
Stratification, 4 9 , 129,
29, 230,
132,
50,
103-105,
134-136,
162,
167,
168-170,
205,
247,
255,
174,
263,
107-110,
138,
140-141,
178,
179,
115, 145,
181,
118, 146,
187,
153-157,
194,
198,
204,
269.
Stress
- fluctuating stress tensor, 1 6 6 ,
170.
- shear stress, see shear.
-
wind stress, 4 , 198,
12,
13,
16,
28,
36,
30,
37,
42,
188,
191,
204.
Sub 1aye r
-
elastic, 9 9 . viscous, 9 9 ,
126,
204,
209.
Surface
-
elevation, I ,
-
waves, 5 1 .
current, 2 6 1 . layer, 1 6 5 ,
.
13,
15,
17,
29,
36,
40,
42,
43,
45.
187.
boundary layer, see boundary layer.
Suspension, 9 9 ,
-
3,
166,
249,
282.
suspended load, 2 3 0 ,
232,
235.
suspended particles, 2 3 2 . suspended sediment, see sediment.
Temperature profile, 5 0 , 247.
259,
265,
51,
102-104,
107,
115,
118,
153,
241,
272.
Thermal
-
diffusivity, see eddy diffusivity.
-
wind balance, 1 9 4 ,
Thermocline, 4 9 - 5 1 ,
198,
54,
56,
200,
205.
101,
102,
263.
Thickne s s
-
boundary layer, 5 1 ,
62,
126,
131,
198,
251,
261.
155,
158,
77,
103,
200,
110,
221,
112,
229,
118,
232,
120,
237,
121,
239,
244,
306
-
Ekman layer, 5 6 ,
103,
115,
110,
logarithmic layer, 2 3 9 ,
187,
188,>194, 204,
205.
245,
246.
78,
239,
242,
251,
257-259,
83, 96,
106,
107,
115, 221,
Tidal
-
-
amplitude, 2 2 2 , current, 2 7 , front, 256,
230.
28,
49,
71,
jet, see jet. kinetic energy, 256. motion, 272. period, 2 4 4 ,
255,
Time series, 6 0 , 252,
284,
Upwelling,
64,
258, 67,
272. 68,
292.
269,
Velocimeter,
272,
273.
see current meter.
Velocity profile, see current profile. Vortex, s e e eddy. Waves
-
-
276.
257.
internal, 5 1 ,
54,
56,
160,
161,
Kelvin, 272. roll,
153,
160-162.
surface, 5 1 . Tollmien-Schlichting,
Wind stress, see stress. Zooplankton, 266.
178,179.
170,
266,
269.
244,
