Ocean Hydrodynamics: Hydrodynamics of Estuaries and Fjords 9th: International Colloquium Proceedings (Elsevier oceanography series ; 23)

Hydrodynamics of Estuaries and Fjords

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 O F OCEANS AND ESTUARIES

4 G.NEUMANN OCEAN CURRENTS

5 N.G. JERLOV OPTICAL OCEANOGRAPHY

6 V.VACQUIER GEOMAGNETISM IN MARINE GEOLOGY

7 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

1 3 E. STEEMANN NIELSEN MARINE PHOTOSYNTHESIS

14 N.G. JERLOV MARINE OPTICS

15 G.P. GLASBY MARINE MANGANESE DEPOSITS

16 V.M. KAMENKOVICH FUNDAMENTALS O F OCEAN DYNAMICS

1 7 R.A. GEYER SUBMERSIBLES AND THEIR USE IN OCEANOGRAPHY AND OCEAN ENGINEERING

18 J.W. CARUTHERS FUNDAMENTALS O F MARINE ACOUSTICS

19 J.C.J. NIHOUL BOTTOMTURBULENCE

20 P.H. LEBLOND and L.A. MYSAK WAVES IN THE OCEAN

2 1 C.C. VON DER BORCH (Editor) SYNTHESIS O F DEEP-SEA DRILLING RESULTS IN THE INDIAN OCEAN

2 2 P. DEHLINGER MARINE GRAVITY

Elsevier Oceanography Series, 23

Hydrodynamics of Estuaries and Fjords PROCEEDINGS OF THE 9th INTERNATIONAL LIEGE COLLOQUIUM ON OCEAN HYDRODYNAMICS

Edited by JACQUES C.J. NIHOUL Professor of Ocean Hydrodynamics, University o f Liege, LiBge, Belgium

ELSEVTER SCIENTIFIC PUBLISHING COMPANY Amsterdam - Oxford - New York 1978

ELSEVIER SCIENTIFIC PUBLISHING COh5PANY 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

l . i l i r a r , o l ('iingrv\\

(

a i a l o g ~ n gin I'uhlirat,,,,,

~

)

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Colloquium on Ocean Hydrodynamics, 4th, 1977 FIydrodynamics of e s t u a r i e s and f j o r d s .

Li&e

( i : l s e v i e r o c e a n o g r a p h y s e r i e s ; 23) B i b l i o g r a p h y : p. Includes index. 1. C s t u a r i ne o c e a n o g r a p h y - - C o n g r e s s e s . ?. rj ords - - C o n g r e s s e s . 3. Hydrodynamics-Conqresses. I. N i h o u l , J a c q u e s C. J. 11. T i t l e . G C ~ G . ' ~ . L > 1377 ~ 551.h1609 78-1405

ISBN 0-444-4168: -X

ISBN 0-444-41682-x (Vol. 23) ISBN 0-444-41623-4 (Series)

o Elsevier Scientific Publishing Company, 1978 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 Printed in The Netherlands

V

Foreword

The I n t e r n a t ional L i P g e Colloquia o n Ocean Hydrodynamics are o r ganized annually.

T h e i r topics d i f f e r from one y e a r t o another and

try to a d d r e s s , a s much a s possible, r e c e n t problems and incentive new s u b j e c t s in physical oceanography. Assembling a group o f active and eminent scientists from d i f f e rent c o u n t r i e s and often different disciplines, they provide a forum for d i s c u s s i o n and foster a mutually beneficial exchange o f informa-

tion o p e n i n g o n to a survey of major recent discoveries, essential mechanisms, i m p elling question-marks and valuable suggestions for future r e s e a r c h . Es t u a r i e s and F j o r d s have been extens ively studied in the past and t h e c h o i c e o f t h i s subject for t h e 1977 Colloquium may appear a little o u t o f line with the tradition. Es t u a r i e s a n d F j o r d s however play an essential r o l e in m a n ' s activities.

C o n n ecting the o c e a n s and the inland r i v e r s , they are

natural t r a n s p o rtation channels.

They provide rational sites for

harbors and i n d ustrial developments and simultaneously natural n u r sing g r o u n d s , r i c h in n u t r i e n t s , for mar ine plants and animals. Es t u a r i e s and F j o r d s , o n the o t h e r hand, have become increasingly vulnerable, receiving the impact o f modern expansion

:

continu-

ously growing p o pulation, production and u s e o f p o w e r , manufacture o f n e w and m o r e diversified materials, intensification of transpor-

tation and f i s h ing effort. Di v e r s i o n s o f r i v e r s , land reclamatio n, excessive siltation, dredging, d u m p i n g of chemical and biolog ical w a s t e s , while c r e a ting severe t h r e ats o n the estuarine env ironment

,

have produced

continuous, o f t e n d r a s t i c , modifications calling for further, more extensive, m o r e elaborate and more interdisciplinary research. The p e r f e c t i n g of n e w equipment, t h e constitution o f more exhaustive d a t a b a n k s , coinciding w i t h the dev elopment o f mathematical modelling t e c h n i ques and the intensive u s e o f modern computers, have p r o v i d e d t h e m e a n s o f a better unde rstanding o f estuarine' d y namics.

V I The Scientific Organizing C o m m i t t e e of the N i n t h I n t e r n a t i o n a l L i G g e Colloquium on O c e a n H y d r o d y n a m i c s s a w the d e s i r a b i l i t y o f

bringing together, on the i m p o r t a n t and p r e s s i n g s u b j e c t of E s t u a r i e s and F j o r d s , specialists f r o m d i f f e r e n t f i e l d s , e x p e r i m e n t a l i s t s and m o d e l l e r s , h y d r o d y n a m i c i s t s , c h e m i s t s and biologists. The present book w h i c h m a y be r e g a r d e d a s the o u t c o m e of t h e colloquium c o m p r i s e s the p r o c e e d i n g s of t h e meeting and s p e c i a l l y commissioned c o n t r i b u t i o n s o n o b s e r v a t i o n s , p a r a m e t e r i z a t i o n and modelling o f e s t u a r i n e d y n a m i c s .

J a c q u e s C.J.

NIHOUL.

V I 1

T h e Scientific Organizing Committee

of the

LiPge Colloquiu m

Ninth on

International

Ocean Hydrodynamics

and all the participants wish to express their gratitude to the

Belgian Minister

of E d u c a t i o n , t he National Science Foundation

of

LiS?ge and

Belgium,

the University

the Office of

of

Naval Research

for their most valuable support.

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I X

LIST O F P A R T I C I P A N T S

Ir.

Y. A D A M , S e c r e t a r i a t d ' E t a t d l ' E n v i r o n n e m e n t , Belgium.

Dr.

G. A L L E N , C e n t r e O c e a n o l o g i q u e d e B r e t a g n e , B r e s t , F r a n c e .

M.

A. H A H , U n i v e r s i t e d e L i e g e , Belgium.

Dr.

W. B A Y E N S , S e c r e t a r i a t d ' E t a t d l ' E n v i r o n n e m e n t , Belgium.

Ir.

G. B E L H O M M E , U n i v e r s i t e d e L i e g e , Belgium.

M.

A. B E R Q U I N , Mission d ' A m e n a g e m e n t Basse N o r m a n d i e , C a e n , France.

M.

G. B I L L E N , V r i j e U n i v e r s i t e i t B r u s s e l , Belgium.

Dr.

R. B O N N E F I L L E , E l e c t r i c i t e d e F r a n c e , C h a t o u , France.

Pr0f.M.J. Dr.

B O W M A N , S t a t e U n i v e r s i t y of N e w Y o r k a t S t o n y B r o o k , U.S.A.

G.A. C A N N O N , P M E L / N O A A , S e a t t l e , U.S.A.

Pr0f.G. C H A B E R T D ' H I E R E S , I n s t i t u t d e M e c a n i q u e , G r e n o b l e , France. Dr.

P.C.

C H A T W I N , U n i v e r s i t y o f L i v e r p o o l , U.K.

Dr.

P . B . C R E A N , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , Canada.

M.

D.K.

D E M P S T E R , W a t e r R e s e a r c h C e n t e r , H e r t s , U.K.

Pr0f.A. D I S T E C H E , U n i v e r s i t e d e L i e g e , Belgium. Dr.

J. D R O N K E R S , R i j k s w a t e r s t a a t , R i j s w i j k , T h e Netherlands.

Dr.

K.R. D Y E R , I n s t i t u t e of O c e a n o g r a p h i c S c i e n c e s , T a u n t o n , U.K.

Dr.

A.J.

Dr.

D . M . F A R M E R , I n s t i t u t e of O c e a n S c i e n c e s , V i c t o r i a , Canada.

E L L I O T T , N A T O A S W R e s e a r c h C e n t r e , L a S p e z i a , Italy.

Pr0f.H.G. G A D E , G e o p h y s i c a l I n s t i t u t e , B e r g e n , Norway. M.

Y. G A L L A R D O , Ant. O R S T O M C e n t r e O c e a n o l o g i q u e d e B r e t a g n e , B r e s t , France.

Dr.

R.F. G R A M E N D E , E L S E V I E R S c i e n t i f i c P u b l i s h i n g C o m p a n y , Amsterdam, The Netherlands.

M.

P. H E C Q , u n i v e r s i t e d e L i e g e , Belgium.

M.

H.B.

H E L L E , U n i v e r s i t y of B e r g e n , N o r w a y .

Dr.

D.O.

H O D G I N S , R i v e r and H a r b o u r Lab., T r o n d h e i m , Norway.

X

Dr.

J.L. H Y A C I N T H E , C N E X O , P a r i s , France.

Pr0f.R.G.

I N G R A M , Mc G i l l U n i v e r s i t y , M o n t r e a l , C a n a d a .

Ir.

B. J A M A R T , University of W a s h i n g t o n , S e a t t l e , U.S.A.

Dr.

M. K A R E L S E , D e l f t H y d r a u l i c s L a b o r a t o r y , T h e Netherlands.

Ir.

A. L A N G E R A K , D e l f t H y d r a u l i c s L a b o r a t o r y , T h e N e t h e r l a n d s

Dr.

G. L E B O N , U n i v e r s i t e d e L i e g e , Belgium.

Dr.

J.J. L E E N D E R T S E , R A N D Corp., S a n t a M o n i c a , U.S.A.

Dr.

C. L E P R O V O S T , I n s t i t u t d e M P c a n i q u e , G r e n o b l e , F r a n c e .

Dr.

D. LIU, R A N D Corp.,

IT.

A. L O F F E T , U n i v e r s i t e d e L i e g e , Belgium.

M.

J.P. M A T H I S E N , River and H a r b o u r L a b . , T r o n d h e i m , Norway.

M.

D. M I C H E L , U n i v e r s i t e L i b r e d e B r u x e l l e s , Belgium.

M.

L.R. M U I R , O c e a n

Pr0f.J.C.J. Dr.

&

S a n t a M o n i c a , U.S.A.

Aquatic S c i e n c e s , B u r l i n g t o n , Canada.

N I H O U L , U n i v e r s i t e d e L i e g e , Belgium.

J.P. O ' K A N E , U n i v e r s i t y C o l l e g e , D u b l i n , Ireland.

Pr0f.F.B.

P E D E R S E N , T e c h n i c a l U n i v e r s i t y of D e n m a r k , L y n g b y -

C o p e n h a g e n , D.K. Dr.

P.A. P E R R E L S , D e l f t H y d r a u l i c s Lab., ?he Netherlands.

Dr.

J.J. P E T E R S , W a t e r b o u w k u n d i g L a b o r a t o r i u m , B o r g e r h o u t , Belgium.

Ir.

G. P I C H O T , S e c r e t a r i a t d ' E t a t 21 l ' E n v i r o n n e m e n t , Belgium.

or.

D. P R A N D L E , I n s t i t u t e o f O c e a n o g r a p h i c S c i e n c e s , B i r k e n h e a d , U.K.

Dr.

H.G. R A M M I N G , U n i v e r s i t y of H a m b u r g , Germany.

M.

R.P. R E I C H A R D , U n i v e r s i t y of N e w H a m p s h i r e , D u r h a m , U.S.A.

Dr.

F.C. R O N D A Y , U n i v e r s i t e d e L i e g e , Belgium.

M.

Y . RUNFOLA, Universite d e Liege, Belgium.

Dr.

J.C.

Dr.

H.M. van S C H I E V E E N , R i j k s w a t e r s t a a t , R i j s w i j k , T h e Netherlands.

S A L O M O N , Lab. d ' O c 6 a n o g r a p h i e P h y s i q u e , B r e s t , F r a n c e .

P r 0 f . J . D . S M I T H , U n i v e r s i t y of W a s h i n g t o n , S e a t t l e , U.S.A. P r 0 f . N . P . S M I T H , U n i v e r s i t y o f T e x a s , P o r t A r a n s a s , U.S.A. Dr.

R. S M I T H , U n i v e r s i t y of C a m b r i d g e , D A M P T , U.K.

Ir.

J. S N I T Z , U n i v e r s i t e d e L i S g e , Belgium.

X I M.

J. S T R O N A C H , U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , Canada.

Dr.

P.J.

M.

H. S V E N D S E N , U n i v e r s i t y o f B e r g e n , Norway.

M.

M.J. T U C K E R , I n s t i t u t e of O c e a n o g r a p h i c S c i e n c e s , T a u n t o n , U.K.

Pr0f.R.E. Dr.

S U L L I V A N , U n i v e r s i t y o f W e s t e r n O n t a r i o , Canada.

U L A N O W I C Z , U n i v e r s i t y o f M a r y l a n d , S o l o m o n s , U.S.A.

R.J. U N C L E S , Inst. f o r Marine E n v i r o n m e n t a l R e s e a r c h , P l y m o u t h , U.K.

M.

J . V O O G T , R i j k s w a t e r s t a a t , R i j s w i j k , T h e Netherlands.

Pr0f.D.F. Dr.

W I N T E R , 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 , U.S.A.

J.T.F. Z I M M E R M A N , N I O Z , T e x e l , T h e Netherlands.

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XI11

CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . LIST OF P A R T I C I PANTS . . . . . . . . . . . . . . . . . . . . . FOREWORD

R.E. U L A N O W I C K Z and D.A. p l a i n e s t u ary J.C.J.

FLEMER

RONDAY, J.J.

P E T E R S and A. STERLING

H y d r o d y n a m i cs o f the Scheldt Estuary G. BILLEN and J. SMITZ

:

G.B. G A R D N E R and J . D .

. . . . . . . . . . . . . . .

SMITH

. . . . . . . . . . . . . . . . . . :

P.A.J. P E R R E L S and M. KARELSE

:

E L L I O T T a n d Dong-Ping WANG

:

f o r c i n g o n the Chesapeake Bay

D. PR A N D L E and J. WOLF S o u t h e r n N o r t h Sea R. BO N N E F I L L E

:

:

. . . . . . . . . .

:

T h e coupling between an

. . . . .

:

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

127

147

161

Residual phenomena in es tuaries, application

. . . . . . . . . . . . . . . . . .

187

A s ymmetry and anomalies o f circulation and

v e r t i c a l m i x ing in the branching of a lagoon-estuary ZIMMERMAN

:

. . .

197

Dispersion by tide-induced residual

c u r r e n t v o r t ices :

107

Surge-tide inte raction in the

to t h e G i r o nde Estuary

R. SM I T H

I9

The ef fect o f meteorological

re s p o n s e t o meteorological forcing

J.T.F.

63

L o ng-period, estuarine-shel f exchanges in

:

GA L L A R D O

55

A two-dimensional numerical

e s t u a r i n e system and i t s adjacent co astal waters

Y.

27

Turbulent mixing in a salt

model f o r s alt intrusion i n estuaries

SMITH

:

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

wedge e s t u a ry

N.P.

1

Modelling o f biological and chemical processes

:

in t h e S c h e ldt Estuary

A.J.

IX

Mathematical model of water quality

in a h i g h l y polluted estuary R. WO L L A S T

VII

A synoptic v i e w o f a coastal

:

. . . . . . . . . . . . . . . . . . . . . .

N I H O U L , F.C.

V

. . . . . . . . . . . . . . . . . . . . .

207

C o r i o l i s , curvature and buoyancy effects upon

d i s p e r s i o n in a n a r r o w channel

. . . . . . . . . . . . . .

217

XIV P.C. CHATWIN and P.J. SULLIVAN

:

H o w some n e w fundamental

results on relative turbulent diffusion c a n be relevant in estuaries and other natural flows L.R. MUIR

233

A o ne-dimensional t i d a l mode l for estuarine

:

. . . . . . . . . . . . . . . . . . . . . . . .

networks

B.M. J A M A R T and D.F.

WINTER

243

A n e w approach to the

:

computation o f tidal m o t i o n s in estu aries P.B. CREAN

. . . . . .

. . . . . . . .

261

A numerical model o f baratropic mixed tides

:

between V a n couver Island and the Mainland and i t s r e l a t i o n t o studies o f the estuarine circulation H.G. RAMMING

:

. . . .

283

Numerical investigations o f the influence of

c o a s t a l s t r uctures upon the dynamic off-shore process by a p p l i c a t i o n o f a nested t i d a l model R.P. REICHARD and B. CELIKKOL

:

. . . . . . . . . . .

315

Application o f a finite

element hydrodynamic model to the Gr eat Bay estuary s y s t e m , N e w Hampshire, U.S.A. M.J. BOWMAN

:

. . . . . . . . . . . . . .

349

S preading and mixing of the Hudson River

e f f l u e n t i n to the N e w York Bight J.J. L E E N D E R T S E and S.K.

LIU

:

. . . . . . . . . . . .

373

A three-d imensional turbulent

energy m o d e l for nonhomogeneous estuaries and coastal sea s y s t e m s

. . . . . . . . . . . . . . . . . . . . . . .

387

F1. Bo. P E D E R S E N : A brief review o f pres ent theories o f

. . . . . . . . . . . . . . . . . . . . .

Fjord d y n a m ics

H.G. G A D E and E. SVENDSEN

P r o p e r t i e s of the Robert R. Long

:

m o d e l o f estuarine circulation i n fj ords H. SVENDSEN and R.O.R.Y. in a fjord H.B. H E L L E

THOMPSON

:

423

Wind-driven circulation 439

S u mmer replacement o f d e e p water in Byfjord,

:

:

Mass exchange a c r o s s the sill induced

. . . . . . . . . . . . . . . . . .

by c o a s t a l upwelling D. FA R M E R and J.D.

SMITH

:

:

LAIRD

:

465

Two-layer analysis o f steady

c i r c u l a t i o n in stratified fjords C A N N O N a n d N.P.

441

Nonlinear int ernal waves in a

. . . . . . . . . . . . . . . . . . . . . . . . .

C.E. P E A R S O N and D.F. WINTER

G.A.

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

Western N o r w ay

fjord.

407

. . . . . . . . . . . .

495

Variability o f currents and

water p r o p e r ties from year-long observations i n a fjord estuary

. . . . . . . . . . . . . . . . . . . . . . . . .

Subje c t I n d e x

........................

515 537

1

A SYNOPTIC VIEW OF A COASTAL PLAIN ESTUARY* ROBERT E. ULANOWICZ AND DAVID A.

FLEMERS

University of Maryland, Center for Environmental and Estuarine Studies, Chesapeake Biological Laboratory, Solomons, Maryland 20688 James T. Allison James P. Brown Michael A. Champ Robert Cory

Donald R. Heinle John Klepper Donald W. Lear Charles E. Lewis

Curtis D. Mobley Kent Mountford John W. Pierce James L. Raper Susan K. Smith

ABSTRACT During October, 1972 the Patuxent River Estuary was monitored intensively and synoptically over two tidal cycles to determine the spatial and temporal patterns of various hydrodynamic, chemical and biological features. Forty-one depths at eleven stations along nine transects were sampled simultaneously at hourly intervals for salinity, temperature, dissolved oxygen, chlorohyll 5 , particulate nitrogen, nitrate, nitrite, total kjeldahl nitrogen, ammonia, particulate carbohydrate, dissolved organic carbon, total hydrolizablc phosphorous, dissolved inorganic phosphorous, suspended sediment, particle size distribution, and zooplankton. Tidal velocity was continuously monitored at each depth by recording current meters. Riverine input and meteorological conditions were relatively stable for two weeks preceeding the deployment. This communication describes the calculation of the intrinsic rates of change of the observed variables from their measured distributions in the Estuary. The steady-state, one-dimensional equation of species continuity is employed to separate the advection and tidal dispersion of a hydrodynamically passive substance from its intrinsic rate of change at point. A new spatial transform is introduced for the purpose of interpolation and extrapolation of data *Contribution No. 766 , University of Maryland, Center for Environmental and Estuarine Studies. $Present Address : Division of Biological Services, U. S . Fish and Wildlife Service, Washington, DC 20240.

2

The intrinsic rate of change profiles reveal a region of heavy bloom activity in the upper estuary and a secondary bloom near the point in the River that most of the suspended material settles out. The changes in Ammonia and nitrates are highly correlated to the productivity patterns. productivity.

Phosphorous rates are less closely correlated to

The perturbations that the Chalk Point steam electric

power plant have on the heat and oxygen balances are easily discernible. INTRODUCTION Practically every ecologist who has planned a field study has had to grapple with the limitations finite manpower and equipment impose upon his ability to adequately sample his system over its spatial and temporal domains. Marine and estuarine ecologists are particularly limited by the size and accessibility of their study areas from viewing the manifold physical, chemical and biological processes synoptically. While the developing technology of remote sensing is beginning to alleviate this difficulty, there is still no substitute for in situ sampling through the water column and over its areal extent. In the study described below the investigators have amassed a set of data on key physical, chemical and biological variables taken simultaneously over a net of stations along the Patuxent River Estuary, a tributary estuary of the Chesapeake Bay. The objectives behind such a data acquisition are threefold: 1. To serve as a data set for the purpose of calibrating a combined physical - chemical - biological model yet to be developed. 2. To enable the authors to estimate the magnitude of the rates

of various processes as they occur along the Estuary. 3.

To provide a reference set of data that investigators without

recourse to synoptic data collection may use to evaluate their own hypothesis about estuarine ecosystem dynamics. An opportunity to embark upon such an ambitious task occurred in the fall of 1972 during the acquisition of prototype data for the The Chesapeake

U.S. Army Corps of Engineer's Chesapeake Bay Study.

Biological Laboratory and the Chesapeake Bay Institute of the Johns Hopkins University were under contract to the Corps to deploy current meters and research vessels to measure tidal velocities and salinities in the mid-portion of the Bay.

3

TO monitor the stations prescribed by the Corps in the Bay stem and major tributaries usually required several deployments of the

available manpower.

The Patuxent Estuary, however, was small enough

to cover in a single deployment, yet large enough t'o serve as a replica of most estuarine processes. The study called for the deployment of thirty-four meters at eight stations on six "transects" along the axis of theEstuary. The current meters (Braincon #1301 Histogram Recording Current Meters) recorded tidal speed and direction automatically every ten minutes. The salinity beside each meter was to be measured with an induction salinometer lowered from a shipboard at hourly intervals for thirteen hours of three consecutive daylight periods. With all of the vessels and men deployed for this study it appeared to the authors that, for relatively little extra effort, a host of

chemical and biological variables could be measured simultaneously with the currents and salinities. The result would be a "snapshot" of the Estuary giving detailed information about a complex of phenomena for a short period of time. As extra manpower and equipment would be needed for such a survey beyond that of the two participating organizations, assistance was solicited from neighboring research groups in the Bay. The response was overwhelming.

Nine research institutions volunteered boats,

equipment and manpower to the effort. With the consent of the Corps the program was expanded to cover forty depths of eleven stations on nine transects along the Estuary. Some of the details concerning the study area, sampling location, variables measured and data reduction are given in the following

sections.

Later, the authors present the analysis of the process

rate profiles and attempt to relate these magnitudes to mechanisms occurring at various reaches of the Estuary. STUDY AREA

The Patuxent River is a significant tributary of the Chesapeake Bay some 160 km in length and draining some 2494 km 2 , all within the State of Maryland. The River rises some 48 km west: of the city of Baltimore and flows southeast and south through the Piedmont Plateau to the fall line 90 km above the mouth.

While the upper

32 km of this river is protected as a source of drinking water for

the Washington Metropolitan Area, approximately 200 million liters per day of treated sewage enter the next 56 km.

The region from

4

90 km to 48 km above the mouth is tidal, fresh-brackish water and is characterized by a narrow channel meandering through broad, marshy flats.

The lower 40 km of the Estuary is a drowned river

valley characterized by partially-mixed, two-layer flow, except near the mouth where occasional three-layer phenomena have been reported. The study area is confined to the lower 72 km of the Estuary ending at a point where the Western Branch sub-tributary joins with the main stream. Eleven stations were established at nine distances along the river as shown on Fig. I.

The coordinates of each station

are listed in Table I. along with the depth in meters at which each current meter was suspended. The vertical spacing between the sensors was nominally 3 meters. The lower four stations were sampled only for tidal current, salinity and temperature, whereas stations P-03-01 through P-07-01 were sampled for the full set of physical, chemical and biological data as described below. The study period was from 0600 on 17 October through 0700 on 18 October, 1972 with samples taken at hourly intervals over the two tidal cycles and one diurnal period.

Jug Boy

Nottingham

Fig. I.

Stations on the patuxent River Estuary

5

TABLE 1

Station location and depths

I Station Designation1

;yo

P-01-01 P-01-02

0.0

i I

I Longitude 76O 2 5 '

17"

76

17

25

Depths (M) 38O 1 8 ' 4 3 " 38

18

55

0.6, 3.7, 12.2

6.7,

9.8,

0.6,

6.7,

9.8,

12.8,

3.7, 15.9

P-02-01

10.0

76

29

33

38

20

50

0.6,

3.7,

6.7

P-02-02

10.0

76

29

08

38

21

00

0.6,

3.7,

6.7,

12.8,

15.9,

P-03-01

22.6

76

35

07

38

24

42

1.2,

3.7,

6.7

P-04-01

33.4

76

39

55

38

29

38

1.2,

3.7,

6.7,

P-04-02

39.3

76

40

32

38

32

30

1.2,

3.7

P-05-01

43.6

76

40

44

38

34

46

1.2, 3 . 7

P-05-02

53.3

76

41

03

38

39

23

1.2, 3 . 7 ,

6.7 6.7

P-06-01

61.3

76

42

02

38

42

33

1.2,

P-07-01

71.8

76

42

53

38

46

40

1.2

3.7,

9.8,

18.9,

21.5

9.8

SAMPLING PROCEDURE

At each hour beginning on the hour the following sequence of sampling procedures was carried out at each of the seven stations for every depth at which a current meter was moored: 1. Conductivity and temperature were measured in situ by lowering an induction coil and thermocouple apparatus (Inter-Ocean 503A CST or Beckman RS-5 salinometer). 2.

Dissolved oxygen was measured in situ at three stations equipped with the Inter-Ocean CST-DO units and from the remaining stations by immersing a YSI dissolved oxygen cell into freshly pumped water from the proper depth. 3. Approximately eight liters of water was pumped from the required depth and immediately processed as described below. 4. Zooplankton were filtered from 3 0 liters of water pumped from the prescribed depth through a 7 2 micron plankton net. 5. During the daylight hours Secchi disc extinction depths were

read. Aliquots of the water collected in step 3 above were immediately filtered and processed as follows: Chlorophyll - Mg C o g was added t o a 1 0 0 - 2 0 0 m l (exact amount recorded) aliquot and filtered through a GF/C glass fiber filter. The filtered material was immediately frozen for subsequent analysis in the laboratory.

6

Particulate Nitrogen - 100-200 ml of water was filtered on a different Millipore system and the GF/C filter and material were dehydrated for later analysis. Particulate Carbohydrate - 100-200 ml was filtered through a GF/C filter which had been fired to remove any carbon. The residual material was frozen for later analysis. Dissolved Nitrogen and Phosphorous - Two Whirl-Pax bags were filled with 75-100 ml of filtrate from the two preceeding filtrations and frozen for later chemical analysis. Total Phosphorous and Organic Carbon ~

-

Unfiltered samples of

50-75 ml volume were frozen to be analyzed later. Suspended Sediment - About 50-100 ml was filtered on a "tared" 47 mm Millipore filter to be dehydrated and weighed in the laboratory. ANCILLARY COOPERATIVE STUDIES In addition to the baseline measurements outlined above the schedules of three other Patuxent studies being conducted by cooperating institutions were altered to be cotemporaneous with the Patuxent synoptic survey. Heinle and Flemer (1976) were directing monthly observations of mass transfer between a section of marsh and the Patuxent River channel. The subject marsh was within the synoptic survey area and the sampling protocol was very similar to that described above. Therefore, the 24-hour marsh study took place simultaneous with the synoptic survey. The Philadelphia Academy of Natural Sciences, likewise, was conducting monthly cruises to measure the gross and net photosynthesis along the River by the oxygen-bottle method. Relative numbers of phytoplankton and bacterial taxa were also determined alongside the various stations of the synoptic survey while the study was underway (Mountford et. al. 1972). The National Aeronautics and Space Administration facilities at Wallops Island and Lanqley, Virginia realized an opportunity to acquire ground-truth data from the synoptic operations and arranged to fly two C-147 and one C-130 missions to take black and white, color IR photographs and multi-spectral scans of the Patuxent during the daylight hours of the deployment (Ohlhorst, 1976). The U.S. Coast and Geodetic Survey was also maintaining four automatic recording (six-minute interval) tidal height gauges along the Patuxent as part of the Corps' Chesapeake Bay Study.

7

CHEMICAL ANALYSIS OF SAMPLES Immediately upon the termination of the deployment the samples were sorted and sent to the laboratories of five of the cooperating institutions.

The Chesapeake Biological Laboratory performed the

analysis for chlorophyll a, particulate carbohydrate and particulate nitrogen; the Department of Biology of the American-University analyzed the samples for particulate and dissolved carbon; the Maryland State Water Resources Administration determined the values of total and dissolved phosphorous; and the Annapolis Field Office of the U.S. Environmental Protection Agency effected the measurement of ammonia, kjeldahl nitrogen, nitrate and nitrite. The sedimentology division of the Smithsonian Institution's Museum of Natural History weighed the sediment samples. Active chlorophyll g was determined fluorometrically with a Turner Model 111 fluorometer (Yentsch and Menzel, 1963 and Holm-Hansen et. al., 1965). A specific absorption coefficient of 12.8 was used in the primary spectrophotometric calibration. The Dumas method of high temperature oxidation was used to determine particulate nitrogen. Analysis were carried out on a Coleman Model 29A Nitrogen Analyzer equipped with a Model 29 combustion tube and syringe. Particulate carbohydrate was determined by the anthrone reaction as described in Strickland and Parsons (1972). Particulate and dissolved fractions of organic carbon were measured according to the methods described by Menzel and Vacarro (1964). The remaining fractions of phosphorous and nitrogen were measured on Technicon Auto Analyzers according to Methods for Chemical Analysis of Water and Wastes published by the U.S. Environmental Protection Agency (1974). The single reagent ascorbic acid reduction method (pp. 249-255) was used to obtain dissolved orthophosphorous, while the total hydrolyzable phosphorous values were the results of the colorometric ascorbic acid reduction method (pp. 256-263). Total kjeldahl nitrogen Values resulted from the automated phenate method (pp. 182-186); ammonia from the automated colorometric phenate method (pp. 168-172); and both nitrite and nitrate from the automated cadmium reduction method (pp. 207-212). In summary, the tidal speed and direction were recorded at each depth at ten-minute intervals. Other variables measured each hour at the forty depths include salinity, temperature, dissolved oxygen,

8

suspended sediment, chlorophyll

a,particulate

nitrogen, particulate

carbohydrate, nitrate, nitrite, ammonia, total kjeldahl nitrogen, total hydrolyzable phosphorous, dissolved orthophosphorous, particulate organic carbon, dissolved organic carbon and zooplankton density. Other variables observed on an opportunistic basis include gross and net photosynthesis, phytoplankton taxa and relative numbers, insolation, coliform counts, river flow and tidal height. Meteorological data from the Patuxent River Naval Air Station near the mouth of the Estuary are probably available, but have not been assembled to date. All processed data is available to the public through the National Oceanographic Data Center*. ESTIMATION OF PROCESS RATES The primary objective of the Patuxent Synoptic Study as cited in the introduction is to enable the development of a combined physical chemical - biological model of a coastal plain estuary. Ideally, if one is to set about modeling a system of such complexity, it is useful to develop a preliminary model based on fragmentary empirical data and other a_ priori _ _ estimates. Such an initial model is often a substantial aid in prescribing a data acquisition scheme. Unfortunately, the opportunistic and @ hot nature of this study did not allow for such preliminaries, and the authors must begin the modeling process after the data collection. The model structure (especially the chemical and biological sub-models) will thus be guided by the results of the initial data manipulations. The entire modeling procedure will then take on much of the nature of a posteriori modeling as described elsewhere (Ulanowicz et. al., 1975 and 1978). Under this approach the structure of the reaction kinetics results from comparing the rates at which species (inorganic, organic, and living) are appearing and disappearing with the amounts present. The data acquisition scheme described above will result in information on the stocks of the species. The rates at which they are intrinsically changing, however, is confounded by the association advection and dispersion in the Estuary. The remainder of this presentation will be devoted to the estimation of the process rates and the qualitative behavior evinced by the results.

*NOAA Master Reel #9008, Environmental Data Services, National Oceanographic Data Center, Washington, D.C. 20235, U.S.A.

9

The separation of the intrinsic rates from the hydrological transport requires a statement of mass balance. Since data was acquired from a string of single stations along the estuary, it is natural to begin with a one-dimensional mass balance, i.e., all variables are averaged over a cross-section of the estuary. Since concentrations and velocities are available at frequent intervals, it is possible to state the equation of species conservation at various times during the tidal cycle.

To do so,

however, would yield results with little statistical significance. Therefore, a one-dimensional, tidally-averaged equation of species continuity is chosen to begin with: A

(CQ) -

=

at where

ax

a (KA -)ac

+

R

ax

ax

C is the tidally-averaged concentration Q is the cumulative freshwater input up to point x A is the local cross-sectional area K is the longitudinal dispersion coefficient

R is the rate of appearance or disappearance of C

x is the t is the Mow the middle of the synoptic study,

distance upstream time October, 1972 was a propitious time to perform since the U . S . Geological Survey records indicate

that riverine input to the lower estuary was virtually constant for the two weeks preceeding the observations. Hence, the River was, most likely, as close to tidally-averaged steady-state conditions as one could hope to achieve.

During the measurement period a

meteorological high pressure front did pass through the area causing a net loss of water from the Estuary, but the effect of this short-term phenomenon upon the steady-state gradients was probably small. Henceforth, the Estuary will be assumed at steady-state, and equation (1) can thus be solved for the "reactions term" as: d dC R = K d x (A=)

dC d K + A -dx dx

-

3 dx

dC

Qdx

Each term on the right-hand-side of e q u a t i m (2) can be reasonably estimated - the concentration profiles are known from the measurements, the freshwater input profile can be evaluated with minor assumptions from USGS data, the areas are available from bathymetric charts, and the dispersion coefficient profile can be calculated from the observed salinities.

There are, however, a number of numerical

details associated with these estimates which should be discussed.

10

To begin with the values for the concentrations at a station are averaged over the station depths. In this averaging each point reading is weighted according to the fraction of the cross-sectional area associated with the particular depth. The resultant station values are subsequently averaged over the two tidal cycles (and one diurnal period) of the study. Each variable then has one "steadystate" value associated with each station at which measurements were taken. The calculated steady-state values are listed in Table I1 The longitudinal distances between stations are greater than is desirable, with distances of over 10 km separating the biological stations.

Furthermore, the lower 22 kilometers were not covered by

the chemical and biological sampling program. A rational method of interpolation and extrapolation of the variables and their derivatives is therefore, in order. Reasoning heuristically that longitudinal mixing becomes greater (in the absolute sense) as the Estuary crosssection increases, it would follow that longitudinal gradients are dampened as the Estuary widens. The cross-sectional area thus becomes a weighting factor for the existing gradients, and it is convenient to define a new independent variable, A , characterizing longitudinal distance as: dh

=-

dx A (x)

(3)

or equivalently:

This transformation of the independent variable has the advantage that the transformed descriptions of advection and dispersion become independent of estuary qeometry, i.e., equation (2) becomes:

Straightforward linear extrapolation of C ( A ) and its derivatives into the downstream range gave more plausible results than similar efforts using several different non-linear regression schemes an C (x). The areas used in this transform are graphed in Figure 2. Encouraged by the utility of this transform, the author proceeded to estimate C ( X ) and its two derivatives by the simplest means possible. Concentrations at any longitudinal distance were

TABLE I1

Averaged c o n c e n t r a t i o n s Species (Units)

33.4

39.3

11810

10430

10.25

7.464

0.081

0.166

K j e l d a h l N i t r o g e n (MGA/L)

0.567

D i s s o l v e d O r t h o P h o s p h a t e (MG/L)

0.026

T o t a l P h o s p h o r o u s (MG/L)

22.6

43.6

53.3

61.3

71.8

7640

6340

1410

300

203

16.85

22.46

40.20

68.50

8.715

0.130

0.100

0.153

0.114

0.075

0.759

0.174

0.101

0.118

0.224

0.717

2.150

0.502

0.629

0.507

0.416

0.609

1.387

0.052

0.055

0.019

0.058

0.071

0.767

0.052

0.134

0.114

0.100

0.070

0.262

1.187

D i s s o l v e d O r g a n i c Carbon (MG/L)

3.192

3.483

3.196

3.190

4.278

4.962

3.562

P a r t i c u l a t e Carbon (MG/L)

2.210

2.088

1.881

2.438

3.527

4.015

2.290

Suspended M a t e r i a l

S a l i n i t y (MG/L) C h l o r o p h y l l -A

( G/L)

Ammonia Nitrate-Nitrite

(MGA/L)

(KM)

18.651

-----

35.507

68.486

52.380

44.034

30.00

D i s s o l v e d Oxygen (MG/L)

9.42

9.34

8.98

8.62

10.80

12.11

7.13

H e a t C o n t e n t (KCAL/LITER)

16.83

16.38

15.69

16.02

15.12

14.62

13.59

(MG/L)

12

approximated by linear interpolation of the two nearest stations. The derivatives at the mid-point between two stations were estimated by the difference quotient of the concentration change and the interval of A . Derivatives at other points were acquired by linear inter polation and extrapolation. Second derivatives were calculated by a repeat application of the derivative scheme.

0.025 Potuxent Estuary

Cross -Sectional Areas

0.020

0.015

-0 0 ._ c

j

0.010-

n v)

?

U

0.005 10

OO

20

,

L

30 1

1 L ---

40

50

60

- L L - - . L

70

-

River Kilometer

Fig. 2.

Patuxent estuary cross-sectional areas.

Over 40% of the area of the Patuxent watershed lies adjacent to the study area, making it impractical to consider that all of the freshwater input occurs at the head of the model. By pro-rating the input according to area, one estimates that Q increases from about 4.75 M3/sec at the head of the Estuary to around 8.10 M3/sec at the mouth. Now observation of the watershed reveals that most of the feeder streams run perpendicular to the longitudinal axis of the Estuary and their mouths are evenly dispersed along both banks. An appropriate assumption, therefore, is that the rate of accumulation of freshwater input, dQ/dx, is nearly c3ntinuous along the main River axis and proportional to the width of the watershed at that point. Figure 3a shows the schematic representation of the watershed adjacent to the study area. Figure 3b below it illustrates the cumulative riverine input at any point in the Estuary. Under the above assumptions Q varies almost linearly along the region of interest.

13

1

Drainage Basin Schematic

, 0

20

1

,

I

40

60

I

River Kilometer

Fig. 3 . (a) Drainage basin schematic showing width as a function of distance upstream (b) Cumulative freshwater input as calculated along the estuary The only remaining terms from equation (4) to be estimated are K and dK/dh.

They may be calculated from the observed salinity profile. Salt, being a conservative substance, should have a zero intrinsic rate of change. There is a source term for salt, however, associated which arises from the input of residual salinity (C,) with the freshwater input. Equation (4) is then written as:

%(K

dC dX -

d or -[K dX

dQ CQ) = -C r dh

dC dh - (C-Cr ) Q ] = 0

14 Assuming that advection balances dispersion at steady-state allows one to estimate K as:

and subsequently calculate:

TO avoid the possibility of a negative value of K resulting from noise

in the derivatives of the salinity, the salinity was approximated by the implicit function: (S

-

.17)1.04776

[A

+

814.04(S-.17)

-

107941 = 3255.7

where S is the salt content in parts per thousand and in reciprocal kilometers. mated salinities.

A

is measured

Figure 4 shows the measured and approxi-

151

lo-

\

0

8 > 4

(S-0.171"'"[A

+ Sl4.04(S-O.I7)-lO794.0] ;3255.7

A

OO

10

20

30

Fig. 4 . Salinity as a function of reduced longitudinal coordinate (river month = 0.0).

15

The longitudinal variation of the dispersion coefficient is depicted in Figure 5. Qualitatively, the variation is similar to that obtained from the Escaut Estuary by Wollast (1973) and discussed by Ronday (1975). The dispersion coefficient declines upstream to a minimum near the point at which theEstuary narrows and rises thereafter to values higher than those found in the lower estuary

.

'20i

7-

COEFFICIENT OF L O N G I T U D I N A L DISPERSION

I

10

20

30 RIVER

Fig. 5.

40

50

60

70

KILOMETER

Calculated coefficient of longitudinal dispersion.

Now Q , K and their derivatives have been estimated independently of the outlined interpolation scheme. A useful test of how compatible the interpolation estimates are with the assumptions used on Q and K would be to calculate the reaction rates of the salt as if it were

a reactive substance.

Performing such a balance yields a total

gain of 0.41 metric tons of salt per day for the entire Estuary. This is an inconsequential fraction of the 5.7 million metric tons of salt present in the Estuary.

16

DISCUSSION OF THE INTRINSIC RATES OF CHANGE The calculated profiles for the rates of change are depicted in Figure 6. A positive value for the rate of change indicates a source of the given material and a negative value denotes a sink. The reader will notice that the term "reaction" has been avoided where possible so as not to infer .a priori the mechanism contributing to a given source or sink. Other mechanisms besides chemical or ___.

biological reactions which might contribute to the intrinsic rates of change include inputs associated with freshwater inflows and adsorption onto sedimenting material. Chlorophyll2 is often used as an indicator of primary productivity of an aquatic ecosystem. The appearance of chlorophyll 2 is then, indicative of an algal "bloom". A very significant bloom is observed in the upper estuary (60-72 km), and a secondary bloom is observed along the range from 39 to 45 km (Figure 6a). A sewage treatment plant introduces nutrients into the Western Branch which enters the mainstream of the River about two kilometers above the study area. It is reasonable to assume that the observed bloom is in response to this nutrient addition. The secondary bloom is coincident with the initial disappearance of suspended material and is possibly the result of light no longer being limiting to productivity.

Chlorophyll

a

is lost in the remainder of the Estuary presumably due to herbivorous uptake. On balance the Estuary as a whole is a source of approximately 0.04 metric tons of chlorophyll per day. It is of primary interest to follow the behavior of the nutrient species to see how they relate to the observed patterns of phytoplankton growth and death. The most striking correlation to the productivity is exhibited by ammonia, Figure 6b. Its rate of change is practically inversely proportional to that of chlorophyll a. With the exception of a small reach of the Estuary (53-57 km), the appearance of one microgram of chlorophyll g is accompanied by the disappearance of approximately ten microgram atoms of ammonia (and vice-versa)

.

Nitrate and nitrite also exhibit close correlation to the primary production patterns (Figure 6c). The loss of these species is slightly heavier than that of ammonia in the upper estuary (>45 km) and the inverse correlation with primary production breaks down more drastically in the stretch from 47-57 km. In the lower estuary nitrates appear on almost a mole-for-mole basis with ammonia.

17

I Chlorophyl

E

I

1

.-U c n ._

-c

t

I

I

c

I

- 0.02 0

10

20

30

40

50

,

__--

60

70

River Kilometer

0)

:

. E

Y

>.

1 U

-d

o-2i

Ammonia

I

0.11

I-

a

F

L

V c

0

a

L

0

DI

.-U C n ._ c L

-C _.

0

10

20

.

30

40

50

60

70

River Kilometer

Fig. 6. Daily rates of appearance ( + ) or disappearance ( - ) of vdrious substances for kilometer sequents of the Patuxent River Estuary.

0.2 -

Nitrates

6

Nitrita

0.1 -

0

-0.1

-0.2I

0

10

20

30

40

50

60

70

River Kilometer

0.l6

Total Kjeldohl Nitrogen (dissolved)

0.08

I

0

10

20

I

30

40

River Kilometer

50

60

70

19

0.2

Dissolved O r t h o p k q h t e

0.1

0

-0.1

-0.21 10

0

20

30

40

50

60

70

River Kilometer

Total Hydrolyzable Phosphorus

-0.08

-0.161

, L

0

,

10

'

,

20

'

'

-.A__ '

30

40

River Kilometer

50

60

70

Dissolved Organic Carbon

-0.4 40 River Kilometer

0.4

-0.2 -

-0A-

Porticubte C

h

50

60

70

21

0

10

20

30-

50

40

River Kilometer

1

.

0

6

7

-10

20

1

30

40

River Kilometer

1

,

50

,

-L

60

70

60

70

23

Total (kjeldahl) nitrogen is lost throughout the entire length of the Estuary (see Figure 6d) with the exception of the reach from 38-45 km.

The gain in total nitrogen coincides with the secondary bloom of phytoplankton just below the sediment trap. There is loss of all species of nitrogen from the Estuary as a whole. Cumulative loss of total nitrogen amounts to about 4.7 metric ton atoms per day with 1.9 metric ton atoms of nitrate-nitrite and about 0.5 metric ton atoms of ammonia disappearing from the study area each day. Phosphorous appears to be less correlated to productivity than was the case with the nitrogen species. Dissolved orthophosphate (see Figure 6e) was lost from the upper Estuary ('45 km) with heavy disappearance above G O km. The lower Estuary hosted a small gain in the same species. Apparently, the dissolved phosphorous does not remain long in the water column after its addition from the Western Branch. Total phosphorous (Figure 6f) behaves similarly, except that there is significantly more phosphorous gained in the lower Estuary (presumably in the particulate form). Total phosphorous is almost conserved over the whole range with a l o s s of only 0.4 metric ton atoms occurring per day. Dissolved phosphorous, in apposition, is lost at the rate of 1.7 metric ton atoms per day. There are several hypotheses which might explain the observed patterns of phosphorous behavior.

The phosphorous lost in the

upper ~ s t u a r yis likely due to adsorption to the suspended sediments. There does not appear to be any uptake of dissolved phosphorous in the region of the secondary bloom. The source of phosphorous in the lower Estuary is in question. It could originate in the main stem of the Bay, or it could conceivably be regenerated from the sediments. Dissolved and particulate carbon (see Figures Gg, h) follow similar patterns. Both are accreted in the upper regions ( > 5 5 km) and the lower Estuary ( < 3 5 km), but the forms are lost in the transitional region. The bloom and detrital contributions from the marsh are likely sources of carbon in the upper reach. Metabolic products could possibly explain the source of carbon in the lower Estuary. The disappearance of chlorophyll 5 in the lower Estuary does not imply the absence of carbon fixation in these regions. It simply states that losses (e.g. consumption by grazers) exceeds production by growth.

The productivity of the lower Estuary is

24

revealed by the carbon figures. Over 20 metric ton atoms of carbon are produced each day by the study area with 13.5 ton atoms appearing in the dissolved phase and 6 - 7 ton atoms in the particulate form. The calculations reveal (Figure 6i) that 150 metric tons of suspended material are lost to the system each day with most of that figure probably going to the sediments. The upper region where suspended material is accreted is well demarcated from the lower region (<43 km) where sedimentation dominates. The bloom appears to be a net producer of oxygen, even when averaged over the diurnal period. The marked contribution of oxygen from the bloom dominates Figure 6j. Immediately downstream of the bloom area is a region (41-56 km) of oxygen depletion. The region of depletion is probably a combination of the Streeter - Phelps oxygen sag and the lowered oxygen solubility associated with the warmer water in the vicinity of the power plant out fall (km 46). Estuary exhibitsa gradual recovery from the sag.

The lower

The assumption of steady-state is probably weakest when made upon the thermal structure of the Estuary. During the Autumn the upper River is dominated by the cooler runoff water, whereas the lower Estuary is kept warmer by exchange with the heat reservoir of the Bay. Traveling with a parcel of river water, one would observe a warming trend as one proceeded downriver. Figure 6k shows that this is indeed what is observed throughout most of the cE-s$"a~~ with the notable exception of a region downstream of the power plant effluent (37-43 km) where the excess heat from the power plant is being dissipated. FUTURE WORK

The logical next step in the modeling process is the correlation of the calculated rates of change with the observed concentrations (or prescribed functions thereof) to culminate in a kinetic scheme for the chemical and biological species. This would result in a complete steady-state, tidally averaged model of the Patuxent kinetics. Using the kinetic scheme thus derived it would be interesting to release the assumption of steady-state and see how the derived model would respond to changes in riverine flow and nutrient input. With any degree of success in this endeavor, the next progression would be to a one-dimensional dynamic model which could be calibrated to the hourly data.

The parameters derived from the

tidally-averaged model should provide a convenient starting point for the calibration of the dynamic model.

25

No matter how complete the Patuxent model may become, its applicability will always be limited to the fall season unless it can be calibrated against data from other seasons.

Hence, there is a

desire on the part of the authors to repeat such synoptic surveys during other seasons of the year. ACKNOWLEDGEMENTS Besides those contributors listed on the title page, the authors would like to thank a host data collection assistants at the following institutions: American University (Dept. of Biology) Johns Hopkins University (Chesapeake Bay Institute) The Chesapeake Biological Laboratory Environmental Protection Agency (Region 111, Annapolis Field Off ice) National Aeronautics and Space Administration (Langley and Wallops, Virginia) Philadelphia Academy of Sciences (Benedict Field Station) Smithsonian Institute (Museum of Natural History and Chesapeake Bay Center) U.S. Geological Survey Maryland Water Resources Administration The U.S. Army Corps of Engineers, Baltimore District, was most cooperative in allowing us to coordinate our activities with those prescribed under contract DACW31-70-C-0077. Finally, Professor Nihoul and the University of Liege were most generous in assisting one of the authors (Ulanowicz) with travel expenses incurred in attending the Ninth Colloquium on Ocean Hydrodynamics. REFERENCES Environmental Protection Agency.

1974.

Methods for Chemical

Analysis of Water and Wastes. EPA-625-16-74-003 Office of Technology Transfer, Washington, D.C. Heinle, D. R.

and D. A. Flemer.

1976.

Flows of Materials

between Poorly Flooded Tidal Marshes and an Estuary. Mar. Biol. 35:359-373. Holm-Hansen, O., C. J. Lorenzen, R. W. Holmes, and J. D. H. Strickland 1965. Fluorometric determination of chlorophyll. Int. Explor. Mer. 30:3-15.

J. Cons. Perm.

26

Menzel, D. W. and R. F. Vacarro.

1964.

The measurement of dissolved

organic and particulate carbon in seawater. 9:138-142.

Limnol. Oceanogr.

Mountford, K., R. S. Mullen, and R. S. Shippen.

1972.

Patuxent

River Primary Productivity Profiles and Surface Phytoplankton Data. Benedict Estuarine Laboratory, Benedict, Maryland. Unpublished Manuscript. Ohlhorst, C. W. 1976. Analysis of Six Broad Band Optical Filters for Measurement of Chlorophyll-a and Suspended Solids. NASATMX-339 National Aeronautics and Space Administration, Langley, Virginia. Ronday, F. C. 1975. Etude de l’envasement et de la variation -1ongitudinale du coefficient de dispersion dans des estuaires partiellement stratifies. Annales des Travaux Publics de Belgique, No. 4. Strickland, J. D. H. and T. R. Parsons. 1972. A Practical Handbook of Seawater Analysis. Fish. Res. Bd. Can., Ottawa. Ulanowicz, R. E., D. A. Flemer, D. R. Heinle, and C. D. Mobley. 1975. The A-Posteriori Aspects of Estuarine Modeling.

pp. 602-616 in Vol. 1, L. E. Cronin (ed.), Estuarine Research, Academic Press, New York. Ulanowicz, R. E., D. A. Flemer, D. R. Heinle and R. T. Huff. 1978. The Empirical Modeling of an Ecosystem. press). Wollast, R.

1973.

Ecol. Modelling, 4 (in

Origine et mecanismes de 1 envasement de

estuaire Insitut de Chimie Industrielle. Universite Libre de Bruxelles et Laboratorie de Recherches Hydrauliques - Bogerhout. Ministere des Travaux Publics

de 1 Escaut.

de Belgique.

Service d Oceanographie chimique.

27

OF T H E SCHELDT ESTUARY

HYDRODYNAMICS

Jacques C . J .

N I H O U L and FranGois

UniversitE! de Liege,

Jean J.

PETERS

Ministere

C.

RONDAY

Belgium

a n d A n d r e STERLING

des Travaux P u b l i c s ,

I.

H Y D R O D Y N A M I C CHARACTERISTICS

I.

1. Introduction

Belgium

O F THE

SCHELDT ESTUARY

t h e R h i n e - Meuse

The S c h e l d t E s t u a r y i s t h e S o u t h e r n b r a n c h o f

- Scheldt delta. a large extent, of

The n a t u r a l e v o l u t i o n of t h e d e l t a h a s b e e n , i n f l u e n c e d by m a n ' s

arms and c r e a t i o n Since the

Scheldt, Eastern

of

artificial 1867 of

closing in

activities f r e s h water

the

:

embanking,

closing

lakes.

c o n n e c t i o n w i t h t h e Western

t h e Scheur and t h e O o s t g a t a r e t h e o n l y o p e n i n g s of Scheldt t o the

sea.

to

the

The m o u t h of t h e R h i n e i s s i t u a t e d

80 k m t o t h e N o r t h .

t h e S c h e l d t and i t s t r i b u t a r i e s

The d r a i n a g e b a s i n o f some 2 1 . 5 8 0

km2 i n t h e N o r t h - W e s t

and t h e S o u t h - W e s t The motions of

flow of

of

t h e West o f

covers Belgium

the Netherlands.

t h e Scheldt River

in the estuary are large

i s generally small while t i d a l p r o d u c i n g a f a i r l y good mixing

f r e s h and s e a w a t e r s . A map o f

Walsoorden, "flood"

the Scheldt Estuary

curs during

Middelgat

fig.

1)

shows,

downstream o f

a c o m p l i c a t e d s y s t e m o f c h a n n e l s o f t e n r e f e r r e d t o as

and "ebb"

fication is,

c h a n n e l s a c c o r d i n g a s t h e main w a t e r m o t i o n oc-

flood-tide

o r ebb-tide,

respectively.

i n many c a s e s o v e r s i m p l i f i e d ,

Upstream o f r i z e d by a m a i n

channel between

.

Walsoorden, channel,

(Such a c l a s s i -

especially for the

c h a n n e l a n d t h e G a t Van O s s e n i s s e

Terneuzen and Hansweert)

of

France,

of

up t o G e n t b r u g g e ,

well defined,

secondary channels upstream.

with,

the river

is characte-

o c c a s i o n a l l y embryos

28

~ i y .1. Map o f

I.

2.

the Scheldt Estuary.

H y d r a u l i c a n d g e o m e t r i c a l p a r a m e t e r s of

IJ s i n g

t he

"

h y d r a u 1i c p a r ame t e r

i n t r o d u c e d by P r i t c h a r d r e n t zones

in

I'

and t h e

the Scheldt Estuary

" g e ome t r i c a 1 p a r ame t e r

( P r i t c h a r d 1967) one can d i s t i n g u i s h d i f f e -

the Scheldt Estuary,

c h a r a c t e r i z e d by d i f f e r e n t water

and s a l t c i r c u l a t i o n s . The h y d r a u l i c p a r a m e t e r

is the r a t i o of

flood

tide

lume o f

(mentioned a s

f r e s h water

" f l o o d volume"

water

t h e volume of

flowing upstream t h e e s t u a r y through a given

section during t h e

in the

fiyures)

t o t h e vo-

flowing i n t o t h e e s t u a r y upstream t h e

during a complete t i d a l

cycle

(mentioned a s

section

" f r e s h water volume"

in the figures). Schelle,

A t

immediately

g e n e r a l l y t h e upstream fresh water

downstream

l i m i t of

of

t h e mouth o f

t h e Rupel and

t h e b r a c k i s h water zone,

f l o w i s e s t i m a t e d a t some

1 0 0 m3/s,

t h e mean

corresponding

rough-

evacuated over a t i d a l period. Monthly averaged f r e s h water f l o w s l e s s t h a n 40 m3/ s o r h i g h e r t h a n l y t o a volume o f

3

4,5.106m3

3 5 0 m /s a t S c h e l - l e may b e

r e g a r d e d as e x c e p t i o n a l

(Coen 1 9 7 4 ) .

29

The t i d e i n t h e

S c h e l d t e s t u a r y i s e s s e n t - i a l l y t h e l u n a r semi-

d i u r n a l t i d e M2 w i t h a p e r i o d o f

12 hours,

25 minutes.

The t i d a l

a m p l i t u d e i n c r e a s e s from 4 m i n t h e open

s e a t o 5 m a t Hermiksen,

upstream of

t o v a l u e s of

Antwerpen and d e c r e a s e s t h e n

2 m a t Gentbrugge

ZONE 3

( T h e u n s a n d Coen

1973).

ZONE 2

Fig.

ZONE1

2.

t h e o r d e r of

30 The h y d r a u l i c p a r a m e t e r v a r i e s c o n s i d e r a b l y a s a f u n c t i o n o f f r e s h water flow and p o s i t i o n along t h e e s t u a r y f o r t h e same f r e s h w a t e r

given section,

the t i d a l amplitude.

F o r mean v a l u e s o f

( c l o s e t o W a l s o o r d e n ) , 10 n e a r

100 n e a r Hansweert

The g e o m e t r i c a l p a r a m e t e r

i s more d i f f i c u l t t o d e f i n e .

rectangular channel the r a t i o of s e c t i o n of

In an i r r e g u l a r

sand-banks

cross-sectional

width t o depth is meaningfull.

t h e r a t i o of

cross-sections

were

sometimes v e r y a b r u p t l y .

t h i s r a t i o o s c i l l a t e s between

100 and

downstream o f

and

a t t r i b u t e d t o t h e e x i s t e n c e o f nume-

can be

"dead w a t e r " .

U p s t r e a m of

Walsoorden,

Rupel t r i b u t a r y ,

In the

the width t o depth r a t i o

f o r m a t i o n of

region

of

the order of

upstream of

l o s e s much of

is p r a c t i c a l l y fresh w a t e r .

t h e d e p t h becomes v e r y

According

to Pritchard's

Scheldt e s t u a r y can

large

classification

(Pritchard 1967), the

t h u s be d i v i d e d i n t h r e e zones. the

Rupel

characterized

l o w v a l u e s of b o t h g e o m e t r i c a l a n d h y d r a u l i c p a r a m e t e r s .

(ii)zone 2,

from t h e mouth of

by i n t e r m e d i a t e v a l u e s o f mixing of (iiilzone 3 , of

is

small.

(i) z o n e 1 , f r o m G e n t b r u g g e t o t h e m o u t h o f by

the

i t s meaning.

The c r o s s s e c t i o n

v e r y s m a l l a t l o w t i d e a n d t h e r a t i o may b e c o m e a r t i f i c i a l l y where

ra-

t h e width-depth

and r e m a i n s r e g u l a r l y s m a l l ,

t h e mean d o w n s t r e a m s v a l u e s .

The w a t e r t h e r e

Walsoorden,

shallow areas with a f a i r l y complicated flow

t i o decreases r a p i d l y of

2).

1000 w i t h a n a p p r o x i m a t e m e a n

p a t t e r n around and over t h e banks and t h e p o s s i b l e

%

(fig.

The i r r e g u l a r o s c i l l a t i o n s o b s e r v e d i n t h e f i r s t

5 0 km f r o m V l i s s i n g e n

rous s a n d - b a n k s

the width t o

extreme water l e v e l s

r e g i o n of m u l t i p l e c h a n n e l s ,

value about 450.

Therefore the

s i m p l i f i e d and t h e v a r i a t i o n of

mean r a t i o w a s c o n s i d e r e d b e t w e e n

10

width t o average

d e p t h v a r i e s w i t h p o s i t i o n a l o n g t h e e s t u a r y and levels,

z o n e s of

In a

a s h a l l o w a n d w i d e e s t u a r y w i t h many

and t i d a l f l a t s ,

with the t i d a l

In the

t h e f r e s h water flow and

t h e Rupel and 1 or less a t Gentbrugge.

t h e mouth o f

channels,

a

A t

t h e h y d r a u l i c r a t i o v a r i e s from 200 a t

the t i d a l amplitude, Vlissingen t o

2).

(fig.

it v a r i e s w e a k l y w i t h

flow,

the

the

Rupel t o Walsoorden,

the parameters

characterized

indicating

a partial

the water.

from Walsoorden

to the

sea,

c h a r a c t e r i z e d by h i g h v a l u e s

t w o p a r a m e t e r s i n d i c a t i n g a good m i x i n g o f

(except perhaps during w i n t e r when

higher

the w a t e r

exceptional river flood periods

stratification occurs).

in the

31 The t h r e e z o n e s a r e i n d i c a t e d i n of

t h e wet

s e c t i o n A(x) of

fig.

3 w h i c h shows t h e v a r i a t i o n

the Scheldt a t half-tide

a s a function of

t h e d i s t a n c e x from V l i s s i n g e n .

Distribution

3.

I.

Schematically, low v a l u e s , values, The

of

salinity

the

s a l i n i t y of

c h a r a c t e r i s t i c of

f r e s h water

flow which l i m i t s

for extremely may b e

the

appreciably smaller than mixing

flows, the

the

but it a l s o

a t flood-tide

small f r e s h water

the

salinity

flows,

t h e Rupel;

a t Vlissingen

sea water value,

fig.

4

:

the

indicating

that

shaded r e g i o n s i n d i -

the expected displacements of

over half

the

longitudinal

a t i d a l period a t respectively ( r i g h t ) flow of

a typical-

fresh water.

The maximum s a l i n i t y d i f f e r e n c e s b e t w e e n t h e t w o b a n k s

r i v e r a r e o b s e r v e d between

Dutch-Belgian

border

to 4

found.

o/,,o

salt,

the

s a l i n i t y g r a d i e n t s a r e g e n e r a l l y n e g l i g i b l e upstream of

Walsoorden. of

f u n c t i o n of

s t i l l t a k e s p l a c e beyond V l i s s i n g e n ) .

( l e f t ) and a t y p i c a l l y small

Lateral

course a

i n t r u s i o n of

This s i t u a t i o n i s i l l u s t r a t e d i n cate the envelopes of

ly large

i s of

(For extremely

fresh water

a c o n s i d e r a b l e amount o f

from

near Gentbrugge t o l a r g e

s a l t may e x t e n d f a r t h e r t h a n t h e m o u t h o f

large

salinity profile

Scheldt Estuary varies

being s h i f t e d upstream

and downstream a t e b b - t i d e . i n t r u s i o n of

the

fresh water,

salinity profile

varies over a t i d a l period,

the

Scheldt Estuary.

sea water,near Vlissingen.

c h a r a c t e r i s t i c of longitudinal

i n the

are

(

%

Walsoorden

(

km 40)

Q

km 6 0 ) w h e r e o c c a s i o n a l l y ,

and t h e

d i f f e r e n c e s up

I n t h e mean, h o w e v e r , o v e r a t i d a l p e r i o d ,

t h e d i f f e r e n c e s remain v e r y s m a l l . Fig.

5 shows i n i l l u s t r a t i o n t h e d i s t r i b u t i o n o f

s e r v e d on M a r c h 3 1 s t

1971 a t slack water

in a cross

Scheldt E s t u a r y s i t u a t e d n e a r t h e Dutch-Belgian

c h l o r i n i t y obs e c t i o n of

border.

the

One c a n

see t h a t there i s a small l a t e r a l gradient. Downstream o f W a l s o o r d e n , s e p a r a t e d by l a r g e banks pattern lar,

may p r o d u c e

occasional

Vertical nity

t h e e x i s t e n c e of

and t h e

associated complicated c i r c u l a t i o n

l a t e r a l v a r i a t i o n s of

salinity profiles

cycle

a n d mean b o t t o m

s a l i n i t y and,

in particu-

s a l i n i t y d i f f e r e n c e s from one c h a n n e l t o t h e o t h e r . show,

in general,

from t h e s u r f a c e t o t h e bottom w i t h

over a t i d a l

f l o o d and ebb c h a n n e l s

(fig. 6).

an i n c r e a s i n g s a l i -

strong irregular variations

T h e d i f f e r e n c e b e t w e e n mean

s a l i n i t y i s generally l e s s than

1

surface with

32 maximum

v a l u e s up

and P e t e r s

to 2

near

the

Dutch-Belgian

border

( D e Pauw

1973).

100.000 (m*)

10.000

1.000

100

I1 -40 Fi.g.

Fig.

I

I

I

3.

4.

1

0

-20 :

:

1

1

20

I

I

I

40

Longitudinal p r o f i l e Estuary.

Envelopes of Estuary.

I

60

of

I

I

I

80 the

wet

I

100

I

I

I

120

s e c t i o n of

longitudinal salinity profiles

I

I

X I 10

140 (km) 160

the

in

Scheldt

the

Scheldt

33 20

15

10

5

0 VLI SSI N G EN

Fig. 5.

:

D i s t r i b u t i o n of c h l o r i n i t y o b s e r v e d on M a r c h 31St 1971 a t s l a c k w a t e r in a c r o s s s e c t i o n of t h e S c h e l d t E s t u a r y n e a r the D u t c h - B e l g i a n Border.

Surface

Bottom I

Fig. 6.

:

I

I

I

15

26

21

1 21 X.S

1 2B

V e r t i c a l s a l i n i t y p r o f i l e s in a section of t h e S c h e l d t E s t u a r y s i t u a t e d in t h e M i d d e l g a t C h a n n e l ( k m 30).

34 The o b s e r v a t i o n s o f Estuary

suggested i n ( i ) zone

the distribution

of

seem t o c o n f i r m t h e e x i s t e n c e o f section 2.

(ii)zone 2,

between

t h e Rupel and Walsoorden,

stratified estuary.

The

fresh

is representative

border

of

l o n g i t u d i n a l and v e r t i c a l

a r e maximum i n t h a t r e g i o n

salinity gradients

l a r l y n e a r t h e Dutch-Belgian

(

a n d more p a r t i c u -

km 6 0 )

2,

where

t h e maxi-

s a l i n i t y over a t i d a l p e r i o d are a l s o observed.

mum v a r i a t i o n s o f

tions of

is typically a region of

t h e Rupel,

t h e S c h e l d t and i t s t r i b u t a r i e s .

a partially

(iii)zone 3,

as

:

upstream of

1,

water of

salinity in the Scheldt

t h r e e d i s t i n c t zones

downstream o f

Walsoorden,

i s f a i r l y well-mixed.

s a l i n i t y are s m a l l except occasionally i n

Varia-

secondary

channels and around sand banks.

I.

4.

T i d a l and r e s i d u a l c u r r e n t s

The w a t e r tides.

circulation in

i n the Scheldt Estuary

the Scheldt Estuary

T i d a l c u r r e n t s can reach values of

while the order of t h e mean

magnitude of

a t t h e mouth o f

gion

of

tant

(Peters 1974).

t h e mouth of

t h e Rupel.

or more

the residual currents

c u r r e n t s over one o r s e v e r a l

l y from 1 c m / s e c

i s dominated by

1 m/sec

tidal periods)

the estuary t o Vertical

residue of

linity gradients.

They w i l l be

The d y n a m i c s o f

inflow upstream,

and t h e p r e s e n c e

sa-

of

TO T H E SCHELDT ESTUARY

and mixing,

of

the Scheldt Estuary strong tidal

i s d e t e r m i n e d by

currents with

and by t h e e x i s t e n c e on t h e o t h e r hand of

mentioned

least in

zone 2 ,

in part

I,

the estuary

t o begin with,

hydrodynamic

well-mixed

a t

equations des-

more c o m p l i c a t e d t h a t

e q u a t i o n s on w h i c h m a t h e m a t i c a l m o d e l s o f seas are b a s e d .

s a l i n i t y gra-

effects.

i s o n l y p a r t i a l l y mixed,

and t h e t r i - d i m e n s i o n a l

cribing the estuary are,

the presence,

subsequent turbulence

d i e n t s and s u b s e q u e n t s t r a t i f i c a t i o n and buoyancy

nental

impor-

Introduction

on t h e o n e h a n d ,

A s

t h e re-

in

d i s c u s s e d i n more d e t a i l s i n p a r t 111.

11. H Y D R O D Y N A M I C MODELS APPLICABLE

11.1.

( d e f i n e d as

v a r i a t i o n s may b e

tidal oscillations

7)

varies typical-

10 cm/sec

The r e s i d u a l c u r r e n t s a r e d u e t o t h e f r e s h w a t e r the non-linear

the

(fig.

the

shallow conti-

35 Depth

curves of equal velocity ebb velocity

100

150

flood ,velocity

1 50

0

50

f1004

ebb v e l o c i ty-

I--

velocrty

3 (cmh)

I! i0

100

su r f a c c

bottom

Fig.

7.

: Current

velocity profiles t h e Dutch-Belgian border,

However,

t h e morphology of

the

d e p t h and e x t e n d i n g e s s e n t i a l l y and it

plifications

and o n e - d i m e n s i o n a l depth. ther

;

is often models

S u c h m o d e l s may b e particular

In

the

integrated

zone

one

and zone

1 is less typical

nics).

3

the

and

and

suggests different with

sim-

two-dimensional

t h e width o r

(and) the

from one e s t u a r y

t o ano-

r i v e r determining the

impor-

can be n e g l e c t e d .

d i s c u s s e s t h e hydrodynamic of

station near

(limited i n width

over

different

c h a r a c t e r i s t i c s of

following, 2

estuary

streamwise)

j u s t i f i e d t o work

very

t a n t e f f e c t s and t h o s e which

b l e t o zone

measured a t a f i x e d O c t . 15th, 1970.

the Scheldt estuary can be

t r e a t e d by

S i m p l i f i c a t i o n s w i l l b e made

;

models

applica-

( t h e problem of

classical

r i v e r mecha-

b a s e d on t h e e x a m i n a t i o n o f

36 t h e o r d e r s of

m a g n i t u d e p r o v i d e d by t h e e x i s t i n g d a t a b a n k

Scheldt

Jager

(e.g.

Clearly, bank

they a r e amenable t o permanent r e v i s i o n

Boussinesq

of

reveal other priorities.

equations t h e S c h e l d t E s t u a r y i s d e t e r m i n e d by t h e

The H y d r o d y n a m i c s o f coexistence

as the data

and a s s i m u l a t i o n e x e r c i s e s , c o n f r o n -

t i n g models w i t h o b s e r v a t i o n s ,

11.2.

1975).

1 9 7 3 , Ronday

i s c o m p l e t e d and r e f i n e d

for the

f r e s h water

nerating turbulent

flow,

s t r o n g t i d a l and wind c u r r e n t s ge-

f l u c t u a t i o n s and mixing,

gnificant salinity gradients,sources

of

and s m a l l b u t

still si-

density stratification

and

gravity effects.

- and n o t even d e s i r e d - t o d e s c r i b e

i s obviously not possible

It

t h e dynamics of

the estuary i n every d e t a i l reproducing micro-turbulen

f l u c t u a t i o n s , t h e t i m e and l e n g t h s c a l e s o f r e d t o t h e main hydrodynamic p r o c e s s e s of

the order of

1 minute

c h a r a c t e r i s t i c time h o u r o r more

magnitude

smaller than

and l e n g t h s c a l e s s m a l l e r t h a n

s c a l e s of

t i d a l motions a r e of

and c h a r a c t e r i s t i c t i m e

r a l o r d e r s of

w h i c h a r e s m a l l a s compa-

(say,

larger).

s c a l e s of

the Scheldt)

1

r e s i d u a l c u r r e n t s seve-

from a t h e o r e t i c a l p o i n t o f

may b e d e f i n e d a s s t o c h a s t i c a v e r a g e d b u t w h i c h , averaged over a p e r i o d of

1 m e t e r while

the order of

The m a t h e m a t i c a l m o d e l i s b a s i c a l l y

c o n c e r n e d w i t h mean v a l u e s w h i c h ,

time

time s c a l e s

time

view,

practically,

( a couple of minutes,

are

say,

for

s u f f i c i e n t l y large t o eliminate the micro-turbulent

f l u c t u a t i o n s and s u f f i c i e n t l y s m a l l t o l e a v e t h e main p r o c e s s e s u n a f fected

(Nihoul 1 9 7 5 ) .

The m i c r o - t u r b u l e n t minated

fluctuations are not,

from t h e e q u a t i o n s by t h e

however,

averaging.

completely e l i -

They s u b s i s t i n t h e non-

l i n e a r t e r m s and t h e i r e f f e c t - which i s e s s e n t i a l l y an e n h a n c e d d i s persion, cient

-

d i f f u s i o n b u t many t i m e s m o r e e f f i -

s i m i l a r t o the molecular

m u s t be p a r a m e t e r i z e d w i t h t h e h e l p o f

di f f usiv i t ie s

so-called

"eddy

.

Although t h e water d e n s i t y of variations,

the

the Scheldt i s not

constant,

its

associated e s s e n t i a l l y with the s a l i n i t y gradients,

are

small and B o u s s i n e s q a p p r o x i m a t i o n c a n b e a p p l i e d . Thus i f

i s t h e mean v e l o c i t y ,

p t h e mean p r e s s u r e a n d a t h e mean

(buoyancy b e i n g d e f i n e d a s t h e d e v i a t i o n constant reference value,

of

w a t e r d e n s i t y from a

d i v i d e d by t h a t c o n s t a n t v a l u e

and m u l t i p l i e d

37 by t h e a c c e l e r a t i o n written,

of

gravity g)

t h e Boussinesq

t h e v e r t i c a l a x i s e3 p o i n t i n g upwards

e q u a t i o n s c a n be

(e.g.

.Nihoul and

Ronday 1 9 7 6 )

v.2

0

=

(1) (2)

X i s t h e eddy d i f f u s i v i t y , $ t h e e a r t h r o t a t i o n v e c t o r ,

where

eddy v i s c o s i t y and where, re

q

=

a n d p,

p

and t h e c o n s t a n t r e f e r e n c e

P

-+

denoting respectively

v

the

the pressu-

density,

gx,

P,

In eqs.

( 2 ) and

(3) horizontal

turbulent

d i s p e r s i o n h a s b e e n ne-

glected because c h a r a c t e r i s t i c length scales oE horizontal variations

a r e a l w a y s much

than

larger

vertical

length

s c a l e s and t h i s

e f f e c t i s n e g l i g i b l e a s compared t o v e r t i c a l t u r b u l e n t (One e m p h a s i z e s h e r e t h a t ,

with

an a v e r a g i n g t i m e o f

dispersion.

t h e order of

d i s p e r s i o n d i s c u s s e d h e r e i s t h a t of

a minute,

the turbulent

turbulent

f a i r l y homogeneous and i s o t r o p i c e d d i e s and t h a t

micro-

the cor-

eddy d i f f u s i v i t i e s are comparable i n t h e h o r i z o n t a l

responding

the vertical directions. t a l dispersion this effect

associated with in eqs.

is,

is, of

There

variable

and

an important horizon-

and i r r e g u l a r c u r r e n t s b u t

(3) included i n the advection

and

(2)

course,

terms

and w i l l a p p e a r e x p l i c i t l y l a t e r ) .

11.3.

Three-dimensional

Eqs. the

rectangular

In

ul,

au,

?a !

a

at

+

au, _- -t

at

~

--

ax,

appropriate

t o a r e c t i l i n e a r channel,

:

(4)

ax,

( u , a)

ax,

a

f i v e scalar equations f o r

q a n d a.

u3,

au,

- + - + - = O ax, ax, .-

form a system of

u2,

coordinates,

they can be w r i t t e n au,

(3)

( l ) , (2) a n d

f i v e unknowns

hydrodynamic models.

+

a ~

(u, a)

a

+

(u,

11,)

+

+

a ~

ax,

( u , u*)

2 (0, u, -

n,

(u, a)

~

=

ax3

ax2

+

u2)

~

a h ax, , : a

(

a

- ( u , u,)

=

3x3

-

as ~

ax,

+ -ax,a

(

1

I

v-

(5)

38

For t h e S c h e l d t E s t u a r y , b e n d s i s a l w a y s much of

where

larger than

t h e r a d i u s of

c u r v a t u r e R of

the

the characterlstic length scale 1 2 shown ( N i h o u l e t Ronday 1 9 7 6 )

transverse v a r i a t i o n s , it i s r e a d i l y

that

t h e xI - a x i s

t h e s e e q u a t i o n s remain v a l i d i f

central

s t r e a m l i n e of

the river,

i s

taken

along the

t h e x 2 - a x i s p e r p e n d l c u l a r t o ~t i n

t h e t r a n s v e r s a l d i r e c t i o n and t h e x - a x i s v e r t i c a l ( o r i e n t e d upwards 3 w i t h t h e o r i g i n a t t h e mean r i v e r l e v e l ) p r o v l d e d t h e v e r t i c a l component of

the earth rotation vector

e f f e c t of

curvature.

With a good a p p r o x i m a t i o n , equations,

where v

it i s s u f f i c i e n t t o r e p l a c e ,

is the circumferential velocity

some 10 k m , order

of

Im/sec

(4)

to

sections,

R

side of

c u r v a t u r e of

( 8 ) a r e of

the

i s i n f i n i t e and t h e c o r r e c t i o n v a n i s h e s . magnitude of

( 8 ) f o r d i f f e r e n t values of

observations i n the Scheldt.

a r e a l w a y s much T h i s can be Let,

and a r a d i u s o f

t h e two t e r m s i n t h e r i g h t - h a n d -1 and t h u s

One c a n e s t i m a t e t h e o r d e r s o f

range of

( N i h o u l and Ronday 1 9 7 6 ) .

sec

In rectilinear

eqs.

i n the

2 a 3 by

For a t y p i c a l v e l o c i t y v

of

is corrected t o include the

smaller than shown

i n MKS u n i t s ,

t h e d i f f e r e n t terms

the variables within the

One f i n d s t h a t

several terms

t h e o t h e r s and can be n e g l e c t e d .

s i m p l y by c o n s i d e r i n g a t y p i c a l c a s e . ( J a g e r 1973)

:

39 c h a r a c t e r i s t i c values of

be

and

I,

-

-

3 . 104; i 2

103;

1,

t h e v e l o c i t y components and t h e buoyancy

- 10: t, - 104 - f - ‘

c h a r a c t e r i s t i c l e n g t h s c a l e s and t i m e s c a l e s . The

R i c h a r d s o n number

stratification)

(measuring the

importance

of

the vertical

i s found t o be

aa ax,

The

s e p a r a t i o n b e t w e e n t h e mean v a l u e s u

t u r b u l e n t f l u c t u a t i o n s i s e f f e c t e d by time T of

t h e o r d e r of

According i n a non

where

s t r a t i f i e d medium,

i s the

E

eddies

h =

loL).

i s g i v e n by

the

eddy v i s c o s i t y v and t h e eddy d i f f u s i One c a n w r i t e

(Munk a n d

1948)

-vo

(I

+

(I

+ 3,33 Ri)”*

10 Ri)”’ vo

E

as a

21

( N i h o u l e t Ronday

-

?,

t u r b u l e n t e n e r g y t r a n s f e r from t h e l a r g e r

s t r a t i f i e d waters,

Taking

1

(T

t h e c o r r e s p o n d i n g eddy v i s c o s i t y ,

X depend on t h e R i c h a r d s o n number.

Anderson

=

rate of

a and t h e micro-

t o t h e s m a l l e r ones where d i s s i p a t i o n t a k e s p l a c e .

In vity

v

o n e o r two m i n u t e s

t o Kolmogorov t h e o r y ,

,

i n t e g r a t i o n over a period of

t y p i c a l value €or the Scheldt Estuary

19761, one

finds,

using

(10)

3.10-zm2/sec

IO-*

2

m /sec

These v a l u e s a r e i n good a g r e e m e n t w i t h o b s e r v a t i o n s 1965, F i s h e r

Comparing o r d e r s o f m a g n i t u d e , one f i n d s

1976) :

(e.g.

1972). ( N i h o u l a n d Ronday

Bowden

40 ( i ) t h a t a l l t e r m s in the left-hand s i d e o f eq. ( 8 ) and the v e r t i c a l t u r b u l e n t d i f f u s i o n in the r i g h t - h a n d s i d e are c o m p l e t e l y n e g l i gible a s c o m p a r e d t o b u o y a n c y w h i c h can o n l y be b a l a n c e d by the vertical g r a d i e n t o f q

,

i.e.

( i i ) t h a t the t r a n s v e r s e g r a d i e n t of q i s essentially b a l a n c e d b y C o r i o l i s and e v e n t u a l c u r v a t u r e e f f e c t s a n d , t o a s m a l l e r e x t e n t , by t u r b u l e n t d i f f u s i o n and t i m e v a r i a t i o n s of t h e t r a n s v e r s e velocity, yielding

(iii)that, in e q . ( 6 ) , t o t h e c o n t r a r y , e a r t h r o t a t i o n and c u r v a t u r e e f f e c t s can be n e g l e c t e d a s c o m p a r e d t o t u r b u l e n t diffusion. To e s t i m a t e t h e l o n g i t u d i n a l g r a d i e n t of q , i t i s c o n v e n i e n t to write

where q 1 i s the c r o s s - s e c t i o n a v e r a g e o f q and d e p e n d s on x 1 and t only t

;

q v i s the w i d t h - a v e r a g e o f q - q 1 and d e p e n d s o n x 1

q t i s equal t o q

;

-

q1

,

x 3 and

- q v and d e p e n d s on the f o u r v a r i a b l e s x

1'

x 2 , x 3 and t. S u b s t i t u t i n g in

(14) and

( 1 5 1 , one g e t s

An e s t i m a t e of q 1 c a n be based on the v a l d e o f g a t t h e f r e e s u r face

w h e r e r; i s the s u r f a c e elevation.

41 F o r e l e v a t i o ns o f the order o f 1 m ,

T o sum u p , t he equations of a three- dimensional hydrodynamic model of the S c h e l d t Estuary can b e written

a%

a4

-+fu, at

= - -

ax,

Eqs.

+-a

ax,

( ::jI v-

(18) to (22) f o r m a complete system o f five scalar equations

for the f i v e u n knowns u I , u2, u 3 ,

a, q .

11.4. T w o - d i m e n sional m o d e l s assuming transversal homogeneity

Ea r l y m o d e l s o f estuaries traditionally assumed a sufficiently good t r a n s v e r s a l homogeneity to n e g l e c t , i n the equations, a l l t e r m s

containing d e r i vatives with respect to x 2 These t e r m s , i n eqs.

(18),

(20) and

(e.g. Pritchard 1967).

(21) also contain the trans-

verse v e l o c i t y u2 and t h e hypothesis amo unts to considering that, in se c t i o n II.3., either the velocity u2 h a s been overestimated o r the c h a r a c t e r i s tic length scale l2 h a s been underestimated. If t h i s i s the c a s e , eq. all,

au,

3x1

ax,

(18) r e d u c e s to

-+-=o

and one c a n d e f ine a stream function UI =

IJJ

s uch that

a+ ~

8x3

The h y d r o d y n a mic equations can then be reduced to 3 , i.e.

42

aa + - a+ aa a+ -----=at ax, ax, ax, az + -__ a+ a2 41 atax, ax, ax, ax,

__ -+

aa a hax, ax, ),:a a+ a2 41 - - _ aq + -ax, a ax, ax,2 ax,

(

f o r t h e t h r e e v a r i a b l e s $,

(

v-

l a t e s t being a l s o regarded as

q ; the

a,

::,:1

a f u n c t i o n o f t , x1 and x3 o n l y . (22)

Eq.

i s i g n o r e d i n t h i s t y p e of model a s it p r o v i d e s a u s e -

l e s s r e l a t i o n between

v a r i a b l e s u 2 and q t .

uninteresting

a c l o s e r examination of

that

equation

shows t h a t

However,

one must t a k e

such

m o d e l s w i t h some r e s e r v a t i o n . Indeed

ful

reaches t y p i c a l values of order

l a n c e d by a n o t h e r

term a t l e a s t .

i s larger

(say, l o 4 ) , e i t h e r q t larger.

In the

f i r s t case,

q in equations l i k e de pe nd on x 2 . the decrease u2

a ax2 cannot

l2

(9,

Q

1 i n s t e a d of

lo-')

o r u2 is

i s a non n e g l i g i b l e c o n t r i b u t i o n t o

( 2 1 ) , a n d o n e may n o t a s s u m e t h a t q d o e s n o t

In the

of

qt

l o T 4 and must be ba-

l2 i s l a r g e r t h a n e s t i m a t e d

If

,

second c a s e ,

an i n c r e a s e o f

it is likely that

u2 compensating

terms containing

9 or 2x2

be n e g l e c t e d .

The m o d e l w o u l d r e m a i n v a l i d i f

u 2 was s m a l l e r t h a n e s t i m a t e d

(u2 << 3

but observations indicate that,

t h i s condition

i s f a r f r o m b e i n g r e a l i z e d i n many p l a c e s ,

11.5. If

Width-integrated

models

one i s n o t p r i m a r i l y

advantageous

interested i n transverse variations,it

a n d v e r t i c a l v a r i a t i o n s of

i n t e g r a t e d v a l u e s over t h e width of

and

x2 be

= -

is

t o i n t e g r a t e t h e e v o l u t i o n e q u a t i o n s from one bank t o

t h e o t h e r and t o s t u d y t h e l o n g i t u d i n a l mean o r

in the Scheldt Estuary,

A (x, x3)

t h e e q u a t i o n s of The b a n k s m u s t b e

t h e banks. stream surfaces.

Thus

the estuary.

43

Let

u, =

v,

=

L

V,

=

L uj

dx,

(33)

u3 dx2

(34)

a dx,

(35)

11,

= /:A

A

= L a

= /:A

~

-

where y

i s t h e width-averaged

y ( y = ul,

of

u2, a )

and where

L = A + D

(36) -

setting

u, = U) =

a

UI

+

=

y

- y

(y = ul,

u 2 , a ) , one w r i t e s

ii,

-u, +- u3

=a +a

Integrating

eqs.

(18) and

(20)

from - A

t o D,

o n e obtains

44 underlined t e r m s c a n c e l l i n g a s a r e s u l t o f the last f o u r t e r m s o f

a ~

ax3

(A

aA -)

ax3

.

Indeed

( 3 1 ) and

(32) and

(41) being n e g l i g i b l e as c o m p a r e d t o

:

W h i l e , for i n s t a n c e

assuming a m a x i m u m o f 10 m e t e r s of w i d t h v a r i a t i o n o v e r d e p t h . S i m i l a r l y , i n t e g r a t i n g eq.

( 2 1 ) , one o b t a i n s

q v being n e g l i g i b l e , t h e a p p r o x i m a t i o n q simplification r e s u l t s f r o m eq.

%

q h a s been made.

This

(22) w h i c h p l a y s n o o t h e r r o l e in

the integrated m o d e l . I n t e g r a t i n g eq.

( 1 9 ) , one f i n d s , w i t h t h e s a m e a p p r o x i m a t i o n

O n e can see t h a t in e q s ( 4 1 ) and d e v i a t i o n s a r o u n d t h e mean appear.

(42), i n t e g r a l s of p r o d u c t s o f O b s e r v a t i o n s r e v e a l t h a t these

t e r m s ( t h e s t r u c t u r e of w h i c h i s r e m i n i s c e n t o f t h e R e y n o l d s s t r e s s e s ) a r e r e s p o n s l b l e o f l o n g i t u d i n a l and v e r t i c a l d i s p e r s i o n s i m i l a r to turbulent d i s p e r s i o n b u t c o n s i d e r a b l y m o r e efficient. On the m o d e l of t u r b u l e n t d i f f u s i o n , one s e t s

where N 1 , N 3 , A l

, A 3 are n e w d i f f u s i v i t y coefficients.

45

According

N, N,

(e.g.

t o observations,

10’

i lo3

m’/s; A ,

- 10’

lo-’

;i

rn2/s; A,

-

1975)’

:

m’/s

l o - ’ m’/s

(40) suggests the dffinition

Eq.

Ronday

of

a stream f u n c t i o n

such

that

v,

=

a$

ax3

Eq.

(40) i s then

identically verified.

Eqs.

(41),

(42) and

(43)

can b e w r i t t e n

They form a complete

11.6.

Depth-integrated

Depth-integrated

systemforthe three variables

I$

,

in

shallow

models

models have been e x t e n s i v e l y used (e.g.

a partially

l i k e the Scheldt Estuary

stratified estuary

Nihoul

a t least), t h e i r u t i l i t y i s less evident. t e n d t o mask t h e

effects of

o f t e n p l a y an

t h e y a r e o f t e n more For

instance,

a result of

v e r t i c a l s t r a t i f i c a t i o n which, important

role.

2,

( i n zone

On t h e o n e h a n d ,

they in the

On t h e o t h e r h a n d ,

d i f f i c u l t to handle than width-integrated

models.

although both t y p e s o f models are two-dimensional

the

i n t e g r a t i o n over one space c o o r d i n a t e ,

i n t e g r a t e d models,

as

i n depth-

one c a n n o t i n t r o d u c e a stream f u n c t i o n and sub-

s e q u e n t l y r e d u c e t h e number steady

1975).

I n t h e case o f

c o n t i n e n t a l seas hydrodynamics

gravity field,

and

A

of

variables

i f

the

system i s n o t

in a

state.

A 1 a n d A3 a r e i n g e n e r a l n o t c o n s t a n t a l o n g t h e e s t u a r y . They must be r e g a r d e d a s c o n t r o l p a r a m e t e r s t o be determined by t h e o r e t i c a l r e f l e x i o n a n d s i m u l a t i o n e x e r c i s e s b a s e d on t h e d a t a bank.

46 Indeed,

if

x3 = 5

and t h e bottom,

and x3

=

respectively,

- h

a r e t h e e q u a t i o n s of

the surface

one mast have

Defining

(18) from - h t o 5

and i n t e g r a t i n g

using

( 5 3 ) and

I n wide, l i k e zone

e s t u a r i e s w i t h f l o o d - and ebb

the Scheldt,

the width-integrated

model,

t i o n a l i n f o r m a t i o n on t h e

i t may b e

vations.

a depth-integrated

to

depth-integration

i s only advanta-

t h e v e r t i c a l d i s t r i b u t i o n o f b u o y a n c y i s known f r o m o b s e r Otherwise,

t o any r e a l

the

integration over depth, transforming par-

s i m p l i f i c a t i o n , even i f

x-

, does not lead

i t a p p e a r s t o r e d u c e t h e number

variables.

11.7.

Cross-se-cJion

In a

o n e may u s e a c r o s s - s e c t i o n

The o p e r a t i o n o f mathematically, w i l l not

i n t e g r a t e d models

f i r s t a p p r o a c h , when o n e i s m a i n l y i n t e r e s t e d i n l o n g i t u d i -

nal variations,

*

channels,

model t o have a d d i -

t i a l differential equations into integral equations

of

-

i n t e r e s t i n g t o combine,

l a t e r a l d i s t r i b u t i o n of velocity.

I n p a r t i a l l y mixed e s t u a r i e s , g e o u s i.f

one o b t a i n s

(54).

well-mixed

3 of

,

be

i n t e g r a t e d model.

integration over the cross-section

t o the

is similar,

i n t e g r a t i o n over t h e width and t h e d e t a i l s

reproduced here.

47 The

f i n a l one-dimensional

Estuary are given i n

111. RESIDUAL

111.

CIRCULATION

Flocculation

1.

equations appropriate t o the Scheldt

( N i h o u l and Ronday

AND

1976).

S I L T DEPOSITION IN T H E SCHELDT ESTUARY

and v a r i a t i o n s of

t h e s u s ~ e n d e dl o a d i n t h e

Scheldt Estuary C r i t i c a l

s a l i n i t y values of

the central reach of

1 t o 5

the estuary

o / o o

(fig.

salinity,

intense flocculation

suspended

load decreases abruptly.

Variations of ment of

t u r b i d i t y a n d COD up-

f l o w r a t e Q and t h e subsequent d i s p l a c e -

c r i t i c a l s a l i n i t y r e f l e c t on t h e s e d i m e n t a t i o n s u s p e n d e d m a t t e r o v e r m o s t of

8 ) .

i s found however

It

the

(There i s about one o r d e r of

and t e n d t o s p r e a d t h e d e p o s i t i o n of (fig.

such values of

the c r i t i c a l region).

the river

t h e region of

zone 2

A t

and sedimentation t a k e p l a c e and t h e

magnitude d i f f e r e n c e between t h e v a l u e s of

stream and downstream o f

are t y p i c a l l y found i n

4).

a narrower region

t h a t bottom

than

sediments t e n d t o accumulate

one might expect

f o l l o w i n g w i t h t h e h e l p of

and t h i s

a two-dimension61

is explained

width-integrated

in

in the

resi-

d u a l c i r c u l a t i o n model.

Model of

111. 2 .

residual circulation

The t i m e - d e p e n d e n t been d e r i v e d i n ly the

width-integrated

section 11.5.

i n t e g r a l s of

ul,

av, + - av, =o ax, ax3 aA at

av,

-+ at

a

-(L ax,

a

(L-l

I

ax,

I

+

V, A)

a

( L - ' V, A)

~

ax3

a

v,

V,)

=-I-+-

+

as ax,

a

(L-'

~

ax3

a ax,

V 1 , V2

and A r e p r e s e n t respective-

u2 and a o v e r t h e w i d t h ,

can be w r i t t e n

-+

If

hydrodynamic e q u a t i o n s have

v,

V,)

these equations

48 Q

5

50 m 3 ~ a s c

2----I

+ZONE

bottom km

U

I

8

I

lation

Q

.*

2 5 0 n’lsec

i I

lFLOCULATION

Fig.

8.

:

Flocculation flows.

i n the

Scheldt Estuary for

different

river

49 The r e s i d u a l c i r c u l a t i o n over a p e r i o d of

time

c a n b e d e f i n e d a s t h e mean

circulation

sufficiently large t o eliminate t i d a l oscilla-

t i o n s and t r a n s i t o r y wind i n d u c e d c u r r e n t s . The r e s i d u a l c i r c u l a t i o n i s e s s e n t i a l l y r e l a t e d t o t h e r i v e r r a t e and t o g r a v i t y c u r r e n t s a s s o c i a t e d w i t h t h e w i d t h - i n t e g r a t e d model

is thus

appropriate.

i n t e g r a t e d e q u a t i o n s a r e d e r i v e d from e q s . averaging over time.

In t h i s process,

flow

stratification.

A

The r e s i d u a l w i d t h (58) t o

( 6 1 ) by f u r t h e r

time d e r i v a t i v e s a r e replaced

by f i n i t e d i f f e r e n c e s w h i c h c a n b e made a s s m a l l a s d e s i r e d b y i n creasing the period of

t i m e o v e r which t h e a v e r a g e s a r e t a k e n .

Thus

if

t h e r e s i d u a l c u r r e n t s a r e d e f i n e d over a s u f f i c i e n t l y long p e r i o d

of

time, If

<

t h e y can be d e s c r i b e d by s t e a d y s t a t e e q u a t i o n s . > i n d i c a t e s a t i m e a v e r a g e and i f

one g e t s ,

i n t e g r a t i n g eq.

(58) t o

One c a n s e e t h a t t h e a v e r a g e o f two c o n t r i b u t i o n s

; the

f i r s t of

(61) over time

:

t h e quadratic terms gives again

which c o n t a i n i n g t h e p r o d u c t of

m e a n s a n d t h e s e c o n d t h e mean p r o d u c t o f

the

fluctuations.

the

50

The latter contributes to the dispersion and combines to turbulent mixing and shear effect.

In a first approach,-one can parame-

terize the general effect of the dispersion processes, on the model of turbulent dispersion, introducing new dispersion coefficients K

1‘

K 3 , M 1 , M 3 such that

with, according to observations ( M K S ) (e.g.Ronday 1975)

K3

K1,

,

MI

:

and M3 must be regarded as control parameters to be

determined by inspection of the data bank.

In general, they are

functions of x 1 even if, in first approximation, they can be assumed independent of x

3‘

If l 1 and l3 are characteristic length scales o f respectively horizontal and vertical variations o f the velocity field, eq.

(67)

gives

In eq. L

I

(69), the terms in the left-hand side are o f the order

u, u, The second term of the right-hand side is o f the order

while the last term of the right hand side is of the order M3

u, Taking

I, - to;

I,

-

(Ronday 1 9 7 5 ) , 104: L - I

u,

-

10-2

51 one can s e e t h a t v e r t i c a l d i s p e r s i o n i s s e v e r a l o r d e r s of l a r g e r than advection a71

- L __

be b a l a n c e d by

.

s i m i l a r s i m p l i f i c a t i o n is n o t ,

A

which d o e s n o t

magnitude

d i s p e r s i o n and m u s t a c c o r d i n g l y

and h o r i z o n t a l

in general,

possible for eq.(68)

c o n t a i n a t e r m a n a l o g o u s t o t h e wres-sure g r a d i e n t

and where v e r t i c a l d i s p e r s i o n i s c o m p a r a t i v e l y l e s s i m p o r t a n t tical

s a l i n i t y gradients being

ver-

;

smaller than vertical velocity gra-

d i e n t s i n many c a s e s . Eq.

a0 u,

(67) s u g g e s t s t h e

such t h a t

introduction of

a r e s i d u a l stream function

:

a4Jo

-

=

(75)

8x3

u , = - - 890

(76)

8x1 Eliminating U

Eqs. Q0

and

111.

(77) and

and B ,

U3

one o b t a i n s

( 7 8 ) form a complete

s y s t e m f o r t h e two v a r i a b l e s

71.

3.

A p p l i c a t i o n t o t h e d e p o s i t i o n of s i l t i n t h e S c h e l d t E s t u a r y r e s i d u a l c i r c u l a t i o n model h a s been a p p l i e d t o a s e c t i o n o f

The

t h e S c h e l d t E s t u a r y from t h e Rupel t o t h e s e a king i n t o account

t h e geometry of

the basin

t i o n L(x1 , x 3 ) ) and t h e a p p r o p r i a t e boundary To d e t e r m i n e t h e d i s p e r s i o n Si02

*

d i s t r i b u t i o n s were u s e d .

coefficients,

( z o n e s 2 and 3 ) , t a -

(for instance,

o b s e r v e d s a l i n i t y and

The m o d e l was r u n

in preliminary

h i n d c a s t i n g e x e r c i s e s u n t i l by s u c c e s s i v e improvement o f M3

K1,

K3

and

a s a t i s f a c t o r y a g r e e m e n t was f o u n d b e t w e e n p r e d i c t i o n s a n d o b s e r -

vations

*

t h e func-

conditions.

.

The b u o y a n c y e q u a t i o n ( 6 8 ) i s e a s i l y a d a p t e d t o d e s c r i b e t h e d i s I n t h e r i g h t - h a n d s i d e two a d d i t i o n a l t e r m s t r i b u t i o n of S i 0 2 a p p e a r ; t h e f i r s t one r e p r e s e n t i n g t h e consumption by d i a t o m s , t h e second one t h e l a t e r a l i n p u t s .

.

52 Fig.

(9) r e p r e s e n t s t h e l i n e s o f e q u a l w i d t h - a v e r a g e d

h o r i z o n t a l v e l o c i t y c a l c u l a t e d by t h e model, t e d form,

for a

n o t e d n=4 i n d i c a t e s t h e

(-.-.-.)

i s r e a d i l y s e e n on t h e

gram

The c u r v e i n d o t a n d d a s h e s

l i n e of

zero residual current.

s a l i n i t y s c a l e p r i n t e d on t h e t o p o f

(and corresponding t o t h e

current goes t o zero,

calibra-

i t s l o w water value.

flow r a t e equal t o four times

C u r r e n t s a r e e x p r e s s e d i n rn/sec.

residual

i n its final,

same f l o w r a t e )

a t t h e bottom,

that

the residual

a t t h e v e r y p l a c e where

c a l s a l i n i t y v a l u e s and f l o c c u l a t i o n

criti-

a r e observed.

Mud s e d i m e n t a t i o n a n d a c c u m u l a t i o n t h u s o c c u r s i n a r e g i o n t h e average sediment For r e f e r e n c e , respectively,

linity scale is,

two c u r v e s i n d a s h e s

of

where

t r a n s p o r t on a n d n e a r t h e b o t t o m i s n e g l i g i b l e .

indicate the

r a t e s equal t o the

It

the dia-

l i n e s of

(-

n o t e d n=l a n d n = 1 0

- -)

zero residual current

f o r flow

low w a t e r v a l u e and 10 t i m e s t h a t v a l u e . course,

The

sa-

s h i f t e d a c c o r d i n g l y and t h e zone o f p r e -

c i p i t a t i o n a p p e a r s t o c o i n c i d e i n most c a s e s w i t h t h e r e g i o n

of

zero

bottom c u r r e n t .

5

-

2,5

0,25

r,

15

. ._ ... .....

....

....... .. -.__ ’. ... . . __ -__ .._ ..... . . _ _ .... ._._ .... .......... .............. 2 .

..

.-

n-l

* ./-: d

.-.

0.

/.

:,

R u m

:

L i n e s of equal width-averaged c i t y c a l c u l a t e d by t h e m o d e l .

........... M.5 .... _ ...... . :./ ... ... , . . . . _.-. ... Ha...’ ........ . ...::c/ . . . . .. .

_ _ c -‘

/.

9.

i

t.

0

Fig.

Salinity

-...

residual

....--:

.

. Sea

-

horizontal velo-

53 Hu w e v e r , when the f l o w r a t e increases o r decreases and comes back to typical v a l u es ( n

Q

4), a n e t motion near the bottom appears in

the r e g i o n s w h e re it w a s zero for lower and higher flow rates and direc t i o n s are such t h a t , as indicated by the arrows in fig. 9 , they tend t o c a r r y back the freshly deposited sediments t o the median zone.

IV. AC K N O W L E D G E MENT

T h i s work w a s conducted in the scope o f the Belgian National Envir o n m e n t P r o g r a m , sponsored by the Science P o l i c y Administration, Offic e o f the P r i m e Minister.

V. REF E R E N C E S

Bowden K . F . (1965), J. Fluid Mech. 2 1 , 83. Coen I. ( 1 9 7 4 ) , D e h i e t e n v a n h e t S c h e l d e b e k k e n , Antwerpse Zeediensten, Mi n i s t e r i e v a n Openbare W e r k e n , Belgium. D e P a u w N and P e t e r s J.J. (1973), C o n t r i b u t i o n t o t h e S t u d y o f t h e S a Z i n i t l i D i s t r i b u t i o n and C i r c u l a t i o n i n t h e W e s t e r n S c h e l d t E,’StUary, B e l gian National Environment P r o g r a m , Sea P r o j e c t , Mi n i s t r y f o r Science P o l i c y , Belgium. Fisher H.B. ( 1 9 7 2 ) , J . Fluid Mech., 5 3 , 6 7 1 . J a g e r Ph. ( 1 9 7 3 1 , D , u n a m i c a l p a r a m e t e r s o f t h e S c h e l d t E s t u a r u , Belgian N a t i onal Environment P r o g r a m , Sea P r o j e c t (compiled from “St o r m v l o e d e n of de S c h e l d e ” , 1966, Ministerie van Openbare We r k e n , vol. 11, I V , V), Ministry for Science P o l i c y , Belgium. Munk W.H. and A nderson E.R. ( 1 9 4 8 ) , J. Mar. Res. 7 , 276. Peter s J.J. (19?4), Mode2 v o o r d e s t u d i e v a n d e v e r o n t r e i n i g i n g v a n h e t S c h e Z d e e s t u n r i u r n , Symposium T h e Golden D e l t a , 2 , Pudoc. Nihoul J.C.;I. ( 1 9 7 5 ) , M o d e l l i n g o f Marine S ! y s t e m s , Elsevier Publ., Amsterdam. Nihou l J.C.J. and Ronday F.C. ( 1 9 7 6 ) , M o d B l e s d ‘ u n e s t u a i r e p a r t i e l l e m e n t s t r a t i f i d , in Recherche et Tec hnique a u Service de l ’ E n v i r o n n e m e n t , edited by L e Conseil Scientifique d e 1 ’ E n v i r o n n e m e n t , L i e g e , 315-338. Pritc h a r d D.W. ( 1 9 6 7 1 , E s t u a r i e s , AAAS P u b l . , USA. Ronday F.C. (1975), Annales d e s Travaux P u b l i c s d e Belgique, 4 , 1. T h e u n s J. and C o e n I. ( 1 9 7 3 ) , Annales d e s Travaux P u b l i c s de Be l g i q u e , 3 , 139.

This page intentionally left blank This Page Intentionally Left Blank

55

M A T H E M A T I C A L MODEL

G.

O F WATER Q U A L I T Y

H I G H L Y POLLUTED ESTUARY

A

BILLEN

Laboratorium voor E k o l o g i e en Brussel,

J.

I N

Systematiek, Vrije Universiteit

Belgium.

SMITZ

MPcanique

d e s F l u i d e s Geophysiques.

Mathematique,

Universiti. de Liege,

Environnement.

Institut de

Belgium.

I N T ROD U C T I O N

water

T h i s p a p e r d e s c r i b e s a g e n e r a l model o f polluted

rivers

and e s t u a r i e s ,

oxido-reduction ses.

balance,

Systematical

Antwerp

(km 8 0 ) b y

r a t i o n of

(120 km

load,

but

other oxidants tabolisms,

N e a r

is not

the

under

(Mn02

-

,

(km 1 2 0 - km kn 60,

:

oxygen

reduced

is rapidly

. . .)

forms

increasing

salinity,

t h e p r e s e n c e of

entirely depleted

++ ,

++ ,

NHq

Fe

++ , . . . )

i n c r e a s e of

flocculation

occur

;

are

and

(km 3 0 - km 6 0 ) , a c c e l e r a t i o n

sedi-

(Wollast 1973).

s a l i n i t y and d i s p a r i t i o n c f and a phase of by mixing w i t h

organic

recuperasea-water,

The d i f f e r e n t o x i d a n t s a r e s u c c e s s i v e l y r e g e -

saturated

i n oxygen.

nerated

the last step is the reapparition

;

of

by t h i s

70).

t h e b a c t e r i a l a c t i v i t y f a l l s down,

tion b e g i n s

The d e t e r i o -

the water

a r e u s e d b y a n a e r o b i c me(Mn

suspended o r g a n i c matter

the conjugated

proces-

estuary.

intense heterotrophic activity

NOj , F e ( O H ) ,

owing t o

the

organic m a t t e r .

properties of

the

the

i s h e a v i l y p o l l u t e d above

t h e d i r e c t consequence

r e s u l t of

and t h e r e l a t e d

mentation of

matter,

long)

i m p o r t a n t amounts o f

which d e g r a d e s t h i s c h a r g e

produced

i s made t o t h e E s c a u t r i v e r

c h e m i c a l and b i o l o g i c a l

domestic pollution organic

b a c t e r i a l a c t i v i t y and hydrodynamic

application

The S c h e l d t e s t u a r y

quality i n high

showing t h e r e l a t i o n s between

of

oxygen.

Typical

oxido-reduction

f o r m s a r e shown

i n Fig.

t h e c l a s s i c a l models of

river pollution

cannot be

longitudinal profiles

of

1.a.b. In t h i s case, used

:

56 Fe*'

u 0;

Mn+' am

Ir' 2c

200

. _ , ..

100

li

Sea

Fig. 1.a. Fe+

Mn' Pn

JUL Y

02

q0j

wm

im

? 50

200

20

150

100

10

50

_ . - . ..

,

. .

,

.., - ,

..

..

. - ..

.,

,-'

Fig. 1.b.

57

i) the o r g a n i c load i s very important, and i s n o t a limiting factor of t h e h e t e r otrophic activity

;

iilother o x i d a n ts than oxygen are u s e d , and m u s t be considered a s state variables. It i s h o w e v e r possible to describe c orrectly the evolution of

the c h e m i c a l c o mposition o f the water by a complete oxido-reduction bala n c e , using a few assumption about internal thermodynamic equilibrium.

O F THE MODEL

PRINCIPLE

The b a c t e r i a l a c t i v i t y , which contro ls the oxido-reduction process e s , can be considered d s a n electron flux H(s,t) imposed to the syst e m , a c c o r d i ng to the equation

:

T h e e l e c t r o n flux induces o n e o r several of the following r e a c tion s a) 4e

b) 8 e

c) 2 e

d) le

:

-

+ o2 + 4 ~ +

f

+ 2H20

-

-

-

+

N O 3 + 10H __

f -f

NH4 + 3H 0 2

+

Mn02 + 4H+ __

t

Mn + +

+

Fe(OH)3

Fe++

d')

e) 1 4 e -

+

-

x1

+

+

2S0-4 __

+

+

$ Fe

3H'

4

+

HCO-

3

+

Fe++

+

+

16H+

++

+

2H 0 2

+

3H20

F ~ C O 3

+

H+

$ FeS2 + 8H20

x2

x3

x4 X' 4

x5

So m e o f t h e s e r e a c t i o n s are biologically mediated, and the p r e sence of t h e r e sponsible o r g a n i s m s h a s been demonstrated in the Schel d t E s t u a r y tion,.

..)

(oxygen consumption, nitrification and denitrifica-

.

In s t a n t a n e o u s bacterial activity has been measured experimentally (usi n g a t e c h n i que of dark I4C bicarbonate incorporation). In the model , i t i s t h us possible to introduce t h i s activity, translated in electron € l u x , a s a c o n t r o l parameter.

58

T h e n , the t o t a l e l e c t r o n f l u x H i m p o s e d t o the system m u s t be divided between the d i f f e r e n t c h e m i c a l and b i o l o g i c a l p a t h w a y s , i.e. between r e a c t i o n s a ) to e). In a f i r s t s t e p , i t i s a s s u m e d t h a t an i n t e r n a l t h e r m o d y n a m i c equilibrium i s r e a c h e d f o r the r e a c t i o n s a ) to e ) in e v e r y section o f the e s t u a r y , so t h a t the following N e r n s t r e l a t i o n s hold Eh

=

EE

+

:

X.

b log

(2)

Y.

where E h i s the l o c a l r e d o x p o t e n t i a l , X . and Y . the c o n c e n t r a t i o n s of oxidized and reduced forms.

This assumption is equivalent to

consider that the b a c t e r i a o n l y u s e the t h e r m o d y n a m i c a l l y f a v o r a b l e half r e a c t i o n in their e n e r g y y i e l d i n g metabolism. t h i s a s s u m p t i o n has been d i s c u s s e d by Stumm ( 1 9 7 0 ) and verified by Billen

The validity of

(1966), Thorstenson

( 1 9 7 4 ) f o r t h e special c a s e o f n i t r i -

f i c a t i o n in the S c h e l d t Estuary. In a second s t e p , t o t a k e i n t o a c c o u n t the p o p u l a t i o n d y n a m i c s of the nitrifying b a c t e r i a , and t o o b t a i n a better c o r r e l a t i o n b e tween calculated and measured c o n c e n t r a t i o n s , i t is p o s s i b l e t o i n troduce a k i n e t i c l i m i t a t i o n o f the n i t r a t e p r o d u c t i o n term couple NH;

-

NO;

(the

being o u t s i d e the t h e r m o d y n a m i c equilibrium).

MATHEMATICAL F O R M U L A T I O N

I f X ( s , t ) i s a m e a n c r o s s - s e c t i o n c o n c e n t r a t i o n , we can w r i t e t h e conservation equation as

a a i a , t x + u - ax X = - - (A A as K - X )

:

aax

+

p

-

C

(3)

where t

i s the t i m e ,

s

i s the l o n g i t u d i n a l c o o r d i n a t e a l o n g e s t u a r y a x i s ,

u

is the v e l o c i t y ,

A

i s the s e c t i o n ,

K

i s the d i f f u s i o n c o e f f i c i e n t ,

P

and

C

are r e s p e c t i v e l y the p r o d u c t i o n and c o n s u m p t i o n terms.

T h e values o f u ( s , t ) , A ( s , t ) , K(s,t) are d e r i v e d f r o m hydrodynamic m o d e l s of the e s t u a r y ( W o l l a s t , 1973, R o n d a y 1 9 7 5 , N i h o u l and Ronday 1976).

59 S t a r t i n g from a f e w b o u n d a r y c o n d i t i o n s ( c o n c e n t r a t i o n s u p s t r e a m -

k m 120 - and d o w n s t r e a m - k m 0 - ) , . a n d f r o m the v a l u e s of t h e b a c -

terial a c t i v i t y , the m o d e l d e s c r i b e s the e v o l u t i o n of the w a t e r c o m position a l o n g a l o n g i t u d i n a l profile.

I t is not important a t this

stage to c a l c u l a t e the o r g a n i c load p r o f i l e , because

:

i) the o r g a n i c load i s s o h i g h that i t d o e s n o t c o n s t i t u t e t h e l i m i ting f a c t o r of b a c t e r i a l a c t i v i t y

;

ii)the f a t e of o r g a n i c m a t t e r i s d e t e r m i n e d m a i n l y b y f l o c c u l a t i o n and s e d i m e n t a t i o n p r o c e s s e s , and n o t by bacterial degradation. If

on

r e p r e s e n t s the " h y d r o d y n a m i c operator''

w e c a n write for e v e r y o x i d a n t s X . and e v e r y reduced form Y i ..

where the P . ' s ( s , t ) ( p r o d u c t i o n term) and the C i ' s ( s , t ) ( c o n s u m p tion term) are u n k n o w n functions. T h e p r o b l e m c a n be solved by the i n t r o d u c t i o n of a s s o c i a t e d v a r i a bles Z i and F i) 2 . AS

2

X.

+

:

(i = 2,5)

Y.

the o p e r a t o r V A

tive"

:

on

=

Zi

(6)

i s l i n e a r , the v a r i a b l e s Z . ( s , t ) are " C o n s e r v a -

( i = 2,5)

0

(7)

T h e s o l u t i o n s of t h e f o u r e q u a t i o n s (7) p r o v i d e d i r e c t l y the l o c a l m a x i m u m v a l u e f o r t h e X . ' s and t h e Y . ' s , a n d t h e v a l u e of t h e sum 1

x . is,t) +

Yi(S,t)

where the v . a r e s t o e c h i o m e t r i c c o e f f i c i e n t s f o r t h e e l e c t r o n fluxes. 1

As V A

is linear

V A F(s,t)

= Z

Vi

:

Pi -

V. C. 1

1

(9)

FEBRUClRV

, -... 100

50

sea

Fig. 2.a.

JUL Y

.loo

61 In ( 9 ) the sum of the c o n s u m p t i o n t e r m s C

>J

tron flux H ( s , t ) d u e t o t h e b a c t e r i a l a c t i v i t y

C . i s the t o t a l e l e c i i ; the s u m o f the p r o -

d u c t i o n t e r m s C v , P i r e d u c e s t o t h e r e a e r a t i o n , c a l c u l a t e d by a r e lation

where K i s a k n o w n c o n s t a n t . The e q u a t i o n ( 9 ) budget

-

VA F ( s , t )

-

which represents a global oxido-reduction

can be w r i t t e n =

Vl

sat K(X1 - XI)

-

T h e set of e q u a t i o n s ( 2 ) -

H(s,t)

(11)

( l l ) , with the corresponding boundary

c o n d i t i o n s , a l l o w s a c o m p l e t e d e t e r m i n a t i o n o f t h e X . ' s ( s , t ) and 1

t h e Y i ' s (s,t). T h e n u m e r i c a l d i s c r e t i z a t i o n of the e q u a t i o n s p r o v i d e s t r i d i a g o nal m a t r i c i a l e q u a t i o n s , w h i c h a r e c a l c u l a t e d b y r e c u r r e n t a l g o r i t h m s (Adam and Runfola 1 9 7 1 , Adam 1975).

A s the e q u a t i o n ( 1 1 ) c o n t a i n s

the o x y g e n c o n c e n t r a t i o n X I e x p l i c i t l y , the term

i s c a l c u l a t e d w i t h the v a l u e o f XI a t t h e p r e c e e d i n g t i m e s t e p ( i f X1

v a r i e s s l o w l y ) , o r w i t h a n i t e r a t i v e c o m p u t a t i o n scheme. During t h i s n u m e r i c a l i t e r a t i v e p r o c e s s , a k i n e t i c l i m i t a t i o n o f

the local n i t r a t e p r o d u c t i o n term is i n t r o d u c e d , to r e n r o d u c e t h e a c t i v i t y r a t e of n i t r i f y i n g b a c t e r i a in the d o w n s t r e a m p a r t of the estuary. T h e r e s u l t s of the c a l c u l a t i o n s are shown a t f i g . 2.a.b. situation

:

february

I

summer situation

:

(winter

july).

ACKNOWLEDGEMENT T h i s work w a s c o n d u c t e d in the s c o p e o f

the Belgian National

E n v i r o n m e n t P r o g r a m , s p o n s o r e d by the S c i e n c e P o l i l c y a d m i n i s t r a t i o n , Office

of the P r i m e Minister.

REF 3 ' REN C E S

A d a m , Y. and Runfola Y., '1971. N u m e r l c a l R e s o l u t l o n o f d i f f u s i o n e q u a t i o n , R a p p o r t N.9, P r o g r . Nat. E n v i r o n n e m e n t p h y s i q u e e t B i o l o q i q u e , P r o l e t Mer.

A d a m , Y., 1975. A H e r m i t l a n f i n i t e d l f f e r e n c e m e t h o d f o r the s o l u tion of p a r a b o l ~ c e q u a t i o n s , to be p u b l i s h e d .

62 B i l l e n , G., 1975. N i t r i f i c a t i o n in the S c h e l d t E s t u a r y ( B e l g i u m and the Netherlands). E s t u a r i n e a n d Coastal M a r i n e S c i e n c e , 3, 79-89. A m a t h e m a t i c a l m o d e l of oxido-reduction Billen G . and S m i t z J., 1975. p r o c e s s e s in the S c h e l d t E s t u a r y , M a t h M o d e l s e a - I C E S Hydrography Committee C.M. 1 9 7 5 , C : 2 1 .

B i l l e n , G., S m i t z , J., S o m v i l l e , M. and W o l l a s t , R., 1976. Degradation d e la m a t i e r e o r g a n i q u e e t p r o c e s s u s d ' o x i d o - r e d u c t i o n d a n s 1 ' E s t u a i r e d e 1'Escaut. P r o g . Nat. R-D E n v i r o n n e m e n t - P r o j e t Mer - R a p p o r t F i n a l - v o l X , p. 102 - 152. Redox p o t e n t i a l s , in T h e S e a , G o l d b e r g ed., vol B r e c k , W.G., 1974. 5 , W i l e y , N e w York. N i h o u l , J.C.J., Amsterdam.

1975. Modelling o f M a r i n e S y s t e m s , E l s e v i e r P u b l . ,

N i h o u l , J.C.,T., R o n d a y , F.C., S m i t z , J. and B i l l e n , G., 1977. H y d r o d y n a m i c and water q u a l i t y m o d e l o f t h e S c h e l d t E s t u a r y . Marsh-Estuarine S i m u l a t i o n Symposia. Georgetown, South Carolina, J a n v i e r 6-8, 1 9 7 7 , in press. O v e r b e c k , J. and D a l e y , R.J., 1973. S o m e p r e c a u t i o n a r y c o m m e n t s o n the Romanenko techniqu'e for estimating h e t e r o t r o p h i c b a c t e r i a l B u l l . E c o l . R e s . C o m m . ( S t o c k h o l m ) , 1 7 , 342-344. production. 1964. Heterotrophic CO a s s i m i l a t i o n by bacterial Romanenko, V . I . , f l o r a o f water. M i k r o b i o l . , 3 3 , 7 7 9 - 6 6 3 . R o n d a y , F.C., 1975. Ecude de l'envasement e t de la variation longitudinale d u coefficient d e dispersion dans les estuaires partiell e m e n t stratifies. A n n a l e s d e s T r a v a u n : Publics, 4 , 1975. T h o r s t e n s o n , D.C., 1970. E q u i l i b r i u m d i s t r i b u t i o n o f s m a l l o r g a n i c Geochim. Cosmochim. A c t a , 34, m o l e c u l e s in n a t u r a l waters. 745-770. Wollast, R . , 1973. C i r c u l a t i o n , a c c u m u l a t i o n e t bilan d e m a s s e d a n s l'estuaire d e l ' E s c a u t , R a p p o r t d e s y n t h e s e 1 9 7 2 , C o m m i s s i o n Interministerielle de la Politique Scientifique (Belgium).

63

MODELLING OF BIOLOGICAL AND CHEMICAL PROCESSES IN THE SCHELDT ESTUARY.

R. WOLLAST Laboratory of Oceanography, University of Brussels (Belgium). INTRODUCTION Research chemists and biologists involved in a dynamical description of the behaviour of chemical species or living organisms in estuarine systems are faced with the difficult problem of evaluating the in situ rates of transformation of the species considered, and distinguishing concentration changes due to the mixing of water masses from changes related to biological or chemical processes. Generally, they do not have the opportunity to use elaborated hydrodynamica1 models, for two reasons, either because these models are inexistant, or too complicated in order to include in a tractable manner, the kinetic terms describing the evolution of the chemical and biological parameters. Hereafter, we intend to show by means of a few examples, that the use of simplified unidimensional and stationnary models of the estuarine system constitute a first approach allowing a better understanding of the chemical and the biological processes occuring in the system, when considering long term evolutions of the parameters. The basic principle of these models is to use the distribution of salinity, which i s a conservative parameter, in order to evaluate the mixing processes of fresh water and sea-water. The longitudinal distribution is then simply reduced to : d (u-S) dx

=

Id - (A A dx

dS dx

K -)

The longitudinal mixing coefficient K deduced from the salinity profile then includes effects related to the complicated hydrodynamical circulation of estuarine system and, to a certain extent, non-stationnary effects due to changes of the fresh water discharge and the tidal amplitude. For a non-conservative substance, the longitudinal profile will be described by :

! !-(uC) dx

=

Id dC - (A K -)dx A dx

+

P

-

C

if P and C are respectively the production and the consumption terms affecting the considered constituant.

64

The principal aim of the model is however for the biologist or the chemist to evaluate their importance and their dependence on other environmental parameters. The utilisation of the same longitudinal mixing coefficient in order to describe the hydrodynamical behaviour of such substances, and particularly those introduced in the estuary by the fresh water flow, must be considered with great care.

It is for example easy to understand that such a simplifica-

tion is out of the question when the estuary is vertically stratified or when the residence

time of the fresh water is short in relation to the fluctuations

of the water discharge.

Thus, the model must be tested with other conservative

parameters, ideally characteristic of the fresh water flow. It will be shown in the following paragraphs that the simplified unidimensional model may be applied in the case of the vertically well mixed Scheldt estuary, and that it resulted in the identification of several important mechanisms and rate constants about the behaviour of various chemical species in the estuarine system. THE BEHAVIOUR OF DISSOLVED SILICA The estuaries constitute an important source of dissolved silica for the marine environment since river water has a mean content of 15 mgr SiO /l 2 compared to that of 6 mgr Si02/l for the mean ocean and values as low as 0 , l mgr SiO /l for surface sea-water. It is an essential nutrient for diatoms, 2 which are the dominant phytoplancton species in many areas. The behaviour of dissolved silica in the estuarine systems is at the present time a subject of controversy. The consumption of dissolved silica which varies from l O , 2 0 % in many estuaries (Burton and Liss, 1973) to 80, 90 i

in estuaries like the Scheldt (Wollast and De Broeu, 1971) and the Rhine (Van Bennekom, 1974) is explained either by chemical reactions with the suspended matter or by an intense activity of brackish diatoms. The conservative or non conservative character of chemical species is well illustrated by plotting the evolution of its concentration in the estuary as a function of the salinity (Fig. I ) .

In the case of dissolved silica, for the

month of February, one obtains a fairly good linear relation, corresponding to simple mixing law and a strong deviation from the mixing line for September indicating an intense consumption of dissolved silica in the estuarine zone. We will use the particular behaviour of this compound for testing the possibility of using the simplified unidimensional model to describe the longitudinal profile of a conservative substance, such as dissolved Si during the winter, as well as to evaluate the rate and consumption mechanism of the same compound during the period from spring to fall.

a

mg S i 0 2 aq

c

65

mg S i 0 2

l5

February 1973 10

5

September 1973

\ -

I 10

5

1s g c1-/1

)

g c1-/1

Fig. 1 . Evolution of dissolved silica as a function of salinity during February 1973 and September 1973 compared to the mixing line. The points A correspond to the composition of the coastal sea-water. Distribution of silica for conservative conditions The evolution of the longitudinal mixing coefficient K along the estuary calculated from the salinity profile during the month of February is given in figure 2.

It should be noted that the longitudinal concentration profiles are

always measured by following the low water slag starting from the mouth of the river. The calculated distribution of dissolved silica, considered as conservative and submitted to the same mixing processes as salinity (same K) is compared to the measured distribution in figure 3. The agreement is very satisfactory and justifies the use of the simplified model in the case of the Scheldt. Distribution of silica for non-conservative conditions The montly evolution of the longitudinal profile of dissolved silica over one year shows that, if this compound behaves as a fairly conservative substance during the winter, it undergoes an uptake in the estuary beginning in May, increasing during the summer and decreasing rapidly after September. This already suggests that the removal is related to biological processes. In order to estimate more quantitatively the rate of removal and to localize the area of silica uptake, the net consumption term (P

-

C)

of equation 2

was estimated €or successive 5 km long sections along sections along the estuary, using again the salinity profile as a tracer of the mixing properties of the water masses.

66

2

n Isec

K

200

100

1

I

50

100

bn

Fig. 2. Longitudinal mixing coefficient calculated from the salinity profile for the river Scheldt in February 1973.

1s

-

5 -

I

1 1 10 10

1 1

1

20 20

30 30

I 40 40

I I

II

w

m

I I

m

I I 00

I 8

I

im

b

Fig. 3. Calculated and measured profile of dissolved silica for the river Scheldt in February 1973 assuming a conservative behaviour for silica.

67

r

*In0

Production

PS'O2

0

50

Consumption

-IW *1W

-

75

March 1973

kT

1no

Production

gSi02

L

0

-100

-zoo

- 3oo +loo

+

0

1no

Sea

-loo

July 1973 -Jon

Consumption

Fig. 4 . Calculated production and consumption terms for dissolved silica as a function of the distance to the sea. The results of these computations for three typical months are represented in figure 4 .

These profiles indicate that the uptake of silica is restricted in

well defined zones of the lower part of the estuary.

This conclusion corres-

ponds to well known facts about the biological activity in the Scheldt estuary. The high turbidity (Fig. 5.) of the upper part of the estuary (above km 60) strongly inhibits the activity of the phytoplancton and explains the conservative behaviour of dissolved silica in that region.

On the other hand, the activity

of the diatoms is characterized by successive blooms of restricted expance usual-

ly starting early in the year near the mouth and progressing upwards during summer and fall. This is well demontrated in figure 6 which shows the evolution of the number of diatom cells recorded by De Pauw (1975) at two stations of the Scheldt during three successive years. Another direct proof of the role of the phytoplancton in the silica uptake observed in the Scheldt, as well as the validity of the consumption rate evaluated with the help of the model, was obtained by comparing the calculated uptake 14

of silica to the primary productivity measured in the zone of uptake by H incorporation.

68

Fig. 5. Longitudinal turbidity profile in the Scheldt estuary

I (10.

ono

I(1

.oon

I

.nnc

.

/--

196:

I

1968

1969

Fig. 6. Seasonnal evolution of the diatoms at Vlissingen (km 5) and Bath (km 4 0 ) (after De Pauw, 1975)

69

During the month of September 1973 the zone of highest primary producti2 vity was situated between k m 35 and km 65 from the mouth and reached 50 mg C/m .day. The mean rate of uptake of silica deduced from the model for the same region was 2 estimated at 138 mg S i O 2/m .day. The weight ratio of silica to C uptake is thus 2.76 in agreement with a weight ratio of Si02/C equal to 2.3 based on the mean composition of marine diatoms (Lizitsine, 1972). Taking these facts into account, we have used the unidimensional model in order to predict the longitudinal distribution of silica in the estuary, where the biological uptake of Si02 is simply proportionnal to the concentration of diatom cells (C),

as measured by De Pauw (1975).

The equation is then simply :

-

d2C + 1 dC K - (-K dX2

A dX

d dX

V)

dV + -1 (C - C') +

k C

(3)

A dX

where V is the fresh water discharge and C' the silica concentration in the various small tributaries. The results of the calculation are summarized in figure 7.

The upper curve

corresponds to a very low activity of the diatoms achieved during the winter and the lower curve to a maximum of the activity with a pronounced bloom of diatoms between km 50 and km 40 reached during the summer. As we can see, this model is very useful in order to evaluate the activity of the diatoms and allows one to relate this activity with various environmental

factors. The model was also used in order to evaluate the amount of dissolved silica discharge by the estuary into the North sea (figure 8). In fact, the amount of dissolved silica delivered to the North sea is considerably reduced from May until September. The same situation occurs in the river Rhine (Van Rennekom, 1974).

These unusual situations must b e related to

the eutrophication of these rivers due to their high concentration levels of dissolved nitrogen and phosphorus. This low input of dissolved silica has a considerable effect on the phytoplancton composition of the North sea where silica may become limitant

(Van Rennekom et al., 1975). NITRIFICATION IN THE SCHELDT ESTUARY Nitrification means the oxydation of ammonia into nitrite and nitrate caused by the activity of autotrophic bacteria.

This process is very important because

it modifies the speciation of the inorganic nitrogen and effects its assimilation

rate by the phytoplancton. Nitrification also consumes large quantities of dissolved oxygen and may affect the quality of the estuarine water.

Sea

150

100

50

)on

Fig. 7. Computed longitudinal profils of dissolved silica for winter (upper curve), summer (lower curve) and a medium situation. The vertical dashes correspond to the observed evolution of dissolved silica over one year (from De Pauw, 1975) g SiOz /sec

900

800

?on p.bny 1973 600

May 1973

son

400

300

200

1 no

SU

10

M

3 0 4 0 o s o

6

0

x

1

n

w

wa-

Fig. 8. Evolution o f the flux of dissolved silica in the Scheldt.

71 It is generally admitted that the nitrification process occurs only when

the heterotrophic degradation of organic matter is completed, but there is no physiological argument to support this hypothesis. We have thus tried to develop a model of the nitrification in the Scheldt estuary which occurs intensively in the lower part of the estuary where the organic load has severely decreased and oxydative conditions are restored. However, this model takes into account known physiological properties of bacteria and is based furthermore, on several in-situ or laboratory experiments (Billen, 1975, Somville, 1975). First of all it is important to underline that the oxydation of ammonia into nitrite and nitrate constitutes the sole source of energy in the metabolism of the nitrifying bacteria.

Thus nitrification can only occur in the area of

the estuary where this oxydation process is exoenergetic.

This condition may

be expressed thermodynamically by introducing the value of the oxydo-reduction potential Eh, above which nitrate and nitrite become more stable than ammonia. In the case of the Scheldt the zone where this condition is fulfilled is restricted to the lower part of the estuary and its extent depends upon the fresh water discharge, the pollution load, the temperature, etc... Even when the thermodynamical conditions are favourable in the estuary, the nitrification takes place rather slowly and nitrification is rarely complete. Some in-situ observations clarify this particular behaviour. Comparative counts of nitrifying bacteria grown on fresh water or sea-water mediums show that the nitrifying bacterial populations of the Scheldt estuary are essentially of continental origin.

Even in the coastal zone near the estua-

ry no halophile population is developing. The nitrifying organisms show no activity in the upper zone of the estuary because of the unfavourable redox conditions and nitrate appears only when the critical value of Eh is reached (figure 9).

On the other hand, the growth of

the fresh water populations is rapidly inhibed with the increase of salinity and the bacterial populations are more rapidely diluted by sea-water than they can expand by reproduction (figure 10a). The in-situ measured activity of the nitrifying bacteria (figure l o b ) reflects both the inhibition due to unfavourable redox condition in the upper part of the estuary and the rapid dilution of the fresh water population by the sea-water in the lower part of the estuary. Modelling of the nitrification A bacteriological model of nitrification must necessarily first of all

define the evolution of the biomass B of the nitrifying organisms, which may be decomposed into a growth term and a mortality term.

Fig. 9. Relation between nitrification and the oxydoreduction conditions in the Scheldt nitrification n n o nitrification

A A A

1 .o

0.5

Fig. 10a. Activity index of nitrifying bacteria as a function of salinity. Curves a, b,c correspond to the activity of cultures adapted progressively to higher salinities.

73 1.0

L wles/l.h

0.5

-

Fig. lob. In-situ measurements of the nitrification activity during October 1975.

If BA represents the "hydrodynamic" operator : BA

=

a + -

a - -l a (A ax A ax

u

at

a

K -)

(4)

ax

the evolution of the bacterial biomass may be written : VA B

=

KB - MB

where K and M are respectively the growth and mortality coeffecients. The growth coefficient is a function of temperature, salinity, concentration of ammonia and Eh. If k is defined as the optimal growth coefficient, the influence of these various parameters may be conveniently described by the following relation : K

= k

. fl

(T)

. f2

(S)

. f3

(NH4)

. ( Eh)

(5)

f l , f2 and f3 being equal to 1 for the optimal values of temperature, salinity and ammonia concentration. The function (Eh) expresses simply that nitrification is only possible above a given redox potential where the oxydation of ammonia becomes exoenergetic :

(Eh)

0

Eh <

1

Eh

EN

=

EN

and f were based on data from literature and 3

The value of k and f l , f

2

laboratory experiments.

-5

According to Painter ( I 970), k varies between 2 10

and 5

sec-l

.

Carlucci and Strickland (1968) have determined the influence of the temperature which may be described by

T El

=

10

-

28

15

(0

< T < 28" C)

f was adjusted from our laboratory experiments and approximated at : 2 f2

=

exp (- {C1-)/8)

ECl-1 being expressed in gr/liter.

74

The dependence of the activity with respect to the concentration of ammonia was described by the classical MichaElis-Xenten relation

Painter (1970) reports values for Km of between 70 to 700 iJmole/l with a reasonable mean situated around 250 iJmole/l. The mortality of the fresh water populations of nitrifying bacteria is also a function of salinity. This influence is again described empirically by an exponential equation based on experimental results :

M = m

(C1-) - 8 3

The nitrification activity is then simply related to the rate of growth of the bacterial population where the increase of nitrate + BKE is exactly compensated by a decrease of ammonia - BKB. -7

of the bacteria of between 10

B

and 5

is an efficiency factor characteristic pmoles/bact. according to Carlucci

and StrickIand (1968) and Watson (1965). This model was applied to the description of the profile of nitrate and ammonia in the Scheldt during the winter period

*.

The comparisons between the calculated and measured concentrations are shown in figure 1 1 .

The relation is very satisfactory.

The however model

shows to an interesting contradiction. If we use the admitted values for B -7 of between 10 and 5 umoles/bact., the number of the nitrifying bacteria predicted by the model is three orders of magnitude higher than the number observed in the water column.

Inversely, the adjustement in the model of the

number of nitrifying bacteria consistent with observed values also lead to a factor of

three order of magnitude higher, which in our opinion, is

unrealistic. Any attempt to improve the bacterial counts in the water column failed and this lead us to the suggestion that the benthic activity of the nitrifying bacteria is the predominant contribution in the oxydation of ammonia into nitrate.

The importance of this process has been suggested previously in the

case of the English rivers (Curtis et al., 1975) and in the coastal region of the North Sea (Billen

,

in press).

In fact, the concentration of the nitrogen species in the estuary, besides the winter, is further complicated by the uptake of these nutrients by the phytoplancton.

75 ebruary 1975

.

\ r(

4

a u

3 300

300

A

zoo

200

1no

1 no

A

L

n

so

0

Fig.

11.

n

lol

A A

sn

Calculated curves and measured points of the longitudinal

profiles of nitrate and ammonia.

The continuous lines correspond to

a computation based on the observed number of bacteria and an unusual 6 = 2 and 3 10-3 iJmoles/l). The dotted curves

high efficiency value (

correspond to a reasonable value of

(a

=

5

6 taken from the litterature

iJmoles/l) but a number of bacteria 1.000 times the

measured values.

76 CONCLUSIONS The chemical and biological processes occuring in natural environments are known to be very complicated according to the intricated interactions of

numerous factors.

In the case of the estuarine systems, the mixing of fresh

water and sea water induces furthermore large changes of ionic strenght, chemical composition, speciation and distribution of organisms along the estuary, which renders the situation even more complicated than in most more homogeneous environments. These processes are thus described generally by rather approximate semiempirical relations as shown for example in this paper by the behaviour of dissolved silica and nitrate in the Scheldt estuary. These approximations do not justify in many cases the use of elaborated hydrodynamical models and to our opinion a stationnary one dimensional model constitutes a sufficient first approach. Management decisions are often requiered urgently in order to prevent the increasing pollution of many estuaries.

In this respect, the use of

these simple models constitutes an useful1 tool realising a compromise between fiability, cost and time requirements. AKNOWLEDGEMENTS This research was supported by the Belgian National Program of Research and Development on the Physical and Biological Environment, spondered and conducted by the Department of Science Policy, Prime Minister's Office

(1970-1976).

REFERENCES Billen, G., 1975. Nitrification in the Scheldt estuary (Belgium and the Netherlands). Estuarine and Coastal Marine Science, 3,279-289. Billen, G., in press. A budjet of nitrogen recycling in the North sea sediments off Belgian coast. Submitted to Estuarine and Coastal Marine Science.

Burton, J.D. and Liss, P.S., 1973. Processes of supply and removal of dissolved silicon in the oceans. Geochim. Cosmochim. Acta, 37, 1761-1773. Carlucci, A.F. and Strickland, J.D.H., 1968. The isolation, purification and some kinetic studies of marine nitrifying bacteria. Journal of experimental marine Biology and Ecology, 2, 156-166. Curtis, E.J.C., Durrant, K. and Harman, H.M.I., 1975. Nitrification in rivers in the Trent basin. Water Res., 9, 255-268. De Pauw, C., 1975. Bijdrage tot de kennis van milieu en plancton in het Westerschelde estuarium, ThSse de doctorat, Rijksuniversiteit Gent. Lisitzin, A . P . ,

1972. Sedimentation in the world ocean, SOC. of Econ. Paleon.

and Mineralog., Special Public., 17. Painter, H . A . ,

1970. A review o f literature on inorganic nitrogen metabolism

in micro-organisme, Water Res., 4, 393-450. Somville, M., 1975. Nitrification dans l’estuaire de l’Escaut, M6moire de licence en sciences chimiques, UniversitG de Bruxelles. Van Bennekom, A . J . , Van Voorst, H . F . J . ,

Krijgsman-Van Hartingsveld,E., Van der Veer, G.C.M. and 1974. The seasonal cycles of reactive silicate and

suspended diatoms in the Dutch Wadden Sea, Neth. J. of Sea Res., 8 (2-3), 174-207. Van Bennekom, A . J . ,

Gieskes, W.W.C. and Thijssen, S.B., 1975. Eutrophication

of Dutch coastal waters, Proc. R. S o c . London, B 189, 359-374.

Watson, S.W., 1963. Autotrophic nitrification in the ocean, in : Symposium on Marine Microbiology, Oppenheimer, C.H. (Editor), CC. Thomas publ., Springfield. Wollast, R. and De Broeu, F., 1971. Study o f the behaviour of dissolved silica in the estuary of the Scheldt, Geochim. and Cosmochim. Acta, 35, 613-620.

This page intentionally left blank This Page Intentionally Left Blank

I9

TURBULENT MIXING IN A SALT WEDGE ESTUARY

GEORGE B. GARDNER AND J. DUNGAN SMITH Department of Oceanography, University of Washington

ABSTRACT

Experiments carried out by Partch and Smith (1977) at anchor stations in the Duwamish River in Seattle, Washington showed a period of intense vertical salt flux during the ebb tide.

In order to examine the longitudinal structure

of these mixing events, as well as to procure data relevant to the overall dyna-

mics of the estuary, a system was developed to permit measurement of the velocity and density fields from a moving vessel. experiment in March 1977.

The new system was used in an

Data obtained during this experiment indicate that

the intense mixing events may be triggered by a hydraulic jump that occurs at a sharp change in river depth.

The relatively high salinity water formed

at the jump advects downstream, but vertical mixing continues with a salt flux 2 of at least 0.5 gm/m 'sec. It is postulated that this continued mixing is related to an internal hydraulic instability.

As it occurs during periods

of minimum shear, shear instability is ruled out as a source of energy.

In

addition to the intense mixing events, an internal hydraulic jump that forms at the 16th Avenue bridge around the time of maximum ebb is described.

While

this apparently is not related to the intense mixing events, it is of interest in its own right, and indicates the importance of internal hydraulics to the dynamics of salt wedge estuaries. INTRODUCTION In order to determine the predominant mixing mechanisms in a typical salt wedge estuary, a project was instigated several years ago to investigate turbulent processes in the Duwamish River.

This estuary enters Puget Sound

in the southern part of Seattle, Washington.

Although heavily industrialized

along its banks, it also has a sizable salmon fishery and has been studied previously from several environmental points of view.

The background hydraulics

and oceanography are described by Dawson and Tilley (1972), Santos and Stoner (1972), and Stoner (1972).

This is contribution 1003 from the University of Washington. described herein was supported by NSF Grant DES-75-15154.

The work

80 3

For river discharges in excess of 30 m / s the estuary is of the salt wedge type throughout its length; whereas for lower river flows, such as normally occur in summer, the lower 5.6 km portion grades into a type 2B of the Hanscn and Rattray (1966) classification.

The salt wedge nature of the estuary

is due to high runoff velocities and occurs in spite of a large tidal range (up to 3.5 m).

These circumstances could produce qualitative differences

between the mixing processes found in the Duwamish and those in certain other salt wedge estuaries, such as the Mississippi, which have a much smaller tidal range.

Nevertheless, the former is, at very least, typical of an important

subclass of salt wedge estuaries. The lower 10 km of the Duwamish River have been dredged and straightened by the U.S. Army Corps of Engineers, producing an estuary comprised of several straight reaches connected by short curves, and an estuary of relatively uniform depth.

Figure 1 is a map of the lower portion of the Duwamish River.

The

segment of primary concern in this paper lies between the two lateral lines upstream and downstream of the 16th Avenue South bridge.

Figure 2 is a sketch

showing the depth profile and the general longitudinal salinity field under the flow conditions that existed during the experiment described in the present paper.

The relatively simple topography and its proximity to the University of

Washington make the Duwamish River an ideal laboratory for turbulent mixing studies. The first experiments conducted during the turbulent mixing project focused on elucidating the temporal variation of temperature, salinity, mean velocity, turbulent kinetic energy, Reynolds stress and turbulent salt flux profiles procured from anchor stations maintained for periods of two to four days.

Some results from this part of the project have been described by Partch

and Smith (1977). Their data were obtained at the 8.5 km point marked in Figure 1.

Partch and Smith found that 50% or more of the flux of salt across

the interface occurred during intense mixing events lasting only a few hours. These events were made evident through increases in surface salinity and surface mixed layer thickness as well as through direct salt flux calculations, the computed value of

s " varying

1.24 gm/m2.s during the events.

from 0.25 gm/m2-s during quiet periods to

The intense mixing events also were reflected

in greatly increased turbulent kinetic energies.

2

These ranged from 4 cm / s

during quiet periods to 20 cm2/s2 during intense mixing events.

2

The mixing

events occurred during periods of minimum shear so Kelvin-Helmholtz instabilities could not have been the main source of turbulent energy.

Further, the increas-

ing surface layer depth and surface salinity eliminated entrainment by bottominduced turbulence as the primary source.

However, Partch and Smith showed

that the mixing events corresponded to periods of critical or near critical

23 16

?2

2,

,?2~2@

I9

-a 36

fPUGET SOUNDl

- 35

!5 -

34 -

DISTANCE FROM MOUTH OF RIVER IN KILOMETERS 0 0

1

2

3

4

5

6

7

8 9 1 0 1 1 1 2 1 3

4i70

~

3

32 -

W

0

8-

3v-

I 23'

I 22

I 21'

51

I 122020

1

,9'

18

F i g u r e 1. Map of t h e Duwamish Waterway. The dashed, l a t e r a l l i n e s mark t h e l i m i t s of t h e s t u d y a r e a . S a l i n i t y d a t a from t h e numbered p o s i t i o n s are d e s c r i b e d i n t h e t e x t of t h e p a p e r . K i l o m e t e r 8 . 5 i s t h e l o c a t i o n a t which t h e measurements r e p o r t e d by P a r t c h and Smith (1977) were made.

LOWER LOW WATER 16

l

l

l

-

l

l

Figure 2. Schematic l o n g i t u d i n a l s e c t i o n s of t h e s a l i n i t y f i e l d a t h i g h and low w a t e r . T h i s s k e t c h i s based on t h e r e s u l t s o f Dawson and T i l l e y (1972), b u t h a s a d j u s t e d f o r t h e f r e s h water runoff conditions t h a t existed during the March 1 9 7 7 e x p e r i m e n t .

2

82

internal Froude number and suggested that they were the result of breaking internal waves, an internal bore or an internal hydraulic jump. If such intense mixing events occur over a significant segment of the estuary, and are common in other salt wedge estuaries, then they are of obvious importance in the understanding of circulation in such systems.

Unfor-

tunately procurement of data from one or several stations i s not sufficient to characterize these features satisfactorily. Therefore, to investigate the longitudinal structure of the intense mixing events, and also to examine the more general aspects of estuarine circulation, a system of instrument deployment that permits measurements to be made from a moving ship was developed.

With

this system the 3 km reach marked on Figure 1 could be covered with a round trip time of one hour. This underway measurement system was used for an experiment in the Duwamish between 11 and 14 March 1977. During this period the fresh water 3 3 3 3 inflow varied from 33 m / s to 35 m /s, compared to 13.3 m /s to 14.0 m / s during the period studied by Partch and Smith.

Under low runoff conditions

such as examined by Partch and Smith, the density profile is of type A, shown on Figure 3, at the start of each ebb.

The intense mixing event changes this

to a type B, and sometimes a second event results in a type C profile.

During

the experiment described in this paper, the type A profile rarely, if ever occurred, the runoff being sufficient to maintain a surface mixed layer at all times.

To confirm this statement, Figure 4 shows a sequence of salinity pro-

files covering one tidal day at intervals of approximately 2 hours.

The intense

mixing event is apparent as the increasing surface salinity between 1112 and 1444, 12 March 1977. EXPERIMENTAL TECHNIQUE

Instrumentation Two instrument systems were deployed during the March experiments. Both were designed to be used from a ship at speeds up to 4 kts, relative to the water.

To provide accurate measurements of velocity and salinity at

several, fixed, depths, a mast was suspended from an I-beam extending well to starboard from the bow of the research vessel.

This mast supported 9 triplets

of mechanical current meters as well as two pairs of temperature and conductivity cells.

In contrast, vertical profiles of salinity were obtained from a CTD

deployed from the stern of the research vessel.

A two-axis electromagnetic

current meter, oriented to measure longitudinal and vertical velocity components, was attached to the CTD.

This current meter obviously cannot yield

accurate measurements by itself due to unresolved motion of the instrument relative to the ship, but does provide a means of interpolating between the fixed depth velocity measurements from the mast.

In order to allow the

83

TYPE A

TYPE

B

s+

S-+-

m TYPE C

s -

Figure 3. Schematic diagram of the three types of salinity profiles found in the Duwamish River by partch and Smith.

I t 8Q

w

0

0

5 10 15 20 25 30

SALINITY

IN 700

Figure 4. Sequence of alinity profiles from position 2 obtainez between 0907, 12 March 1977 and 1147, 13 March 1977. The vertical lines at the top of the plot are origins for the curves, each of which is identified by the time at which it was taken.

84 profiling system to be lowered to within one meter of the bottom from the moving vessel, a depth sounder head was attached to the CTD, with the readout visible to the winch operator. During the past year the ship-mounted mast system was deployed on three major cruises and on each it proved to be a very useful tool for measuring the spatial structure of near-surface velocity and salinity fields. consists of a 13 m length of 2 1/2" OD thick walled steel tubing.

The mast itself This piece

is strengthened and fared with a 3" x 1/4" steel plate welded to the afterside. The 1/4" plate also provides attachment points for the current meter junction boxes and for the cables connecting the underwater sensors with the ship's electronics laboratory. The mast is attached to an I-beam by a clamp that holds the mast torsionally rigid yet allows it to pivot about an axis parallel to the supporting 1-beam.

Lateral support for the mast is provided by stays that attach to

the I-beam about 1 m from the clamp.

In operation, approximately 230 kgm of

lead in the form of 23 kgm donut-shaped weights is attached to the bottom of the unit.

The mast then is pulled back so that it makes an angle of 30'

with the vertical using a wire attached near its lower end.

This provides a

stable attitude for the current meters at relative speeds up to the point where drag on the mast causes it to swing back at an angle greater than 30'. Depending on how far the mast is extended into the water, this maximum speed varies from three to four kts. Duwamish estuary.

Figure 5 shows the mast in operation in the

Note the typical lack of surface waves in the background.

The current meters attached to the mast are triplets of velocity component sensing mechanical devices, Smith (1974), oriented so that all meters sense a portion of the mean flow.

The velocity is calculated using calibration

data, then is rotated into a normal u,v,w, coordinate system.

The technioues

for accomplishing these tasks are described by Smith (1974). Operation in shallow water with large tidally related or topographically induced variations in depth requires the depth to which the mast extends to be readily adjustable.

This is accomplished by loosening the clamp, and raising

or lowering the mast with the aid of a hand winch attached to the I-beam. latter is within reach of the research vessel.

The

Eyes are attached to the stays

at several points so that their length can be changed rapidly. With the mast pulled up to its shallowest setting (i.e. where the top triplet is just below the clamp) it can be rotated into the horizontal plane then swung inboard so that the sensors can be serviced.

This relatively easy

access is an important feature when using rather delicate current meters in an urban estuary at high speeds for periods of several days.

It is possible to

85

Figure 5.

Photograph of the ship mounted mast in operation in the Duwamish River

bring the mast over the deck, change one or two triplets and clean the rest, then resume operation within 30 minutes.

Figure 6 shows the system with the

mast raised for servicing. The I-beam supporting the mast is attached across the rails of the ship about 1.2 meters aft of the bow.

The mast enters the water 1.5 meters aft

of the point where the bow intersects still water, and is 2.3 meters, or about

1.7 times the vessel width, seaward of the hull at this point.

Interaction

of the bow wave produced by the mast with small surface waves makes measurement in the top 50 to 100 cm impractical. Below the surface, the ship hull appears even farther away due to its flair.

The bow wave from the ship is

considerably closer to the ship than to the mast, thus does not interfere with the current measurements.

Tests in still water have confirmed that errors

produced by the proximity of the ship's hull are small (less than 5%). In order to make accurate velocity measurements from a moving ship, the ship motion must be accurately determined and subtracted from the measured relative velocity.

This is accomplished with a microwave navigation system

which measures distances from shore based transponders.

This data like the

86

Figure 6. Photograph of the ship mounted mast raised for servicing the temperature, conductivity, and velocity sensors. current, temperature and salinity information is recorded on magnetic tape through a NOVA 1200 based system.

The same minicomputer is used for analysis

of data. Procedure The system described above provides the ability to sample velocity and density fields accurately from a moving research vessel.

In design-

ing an experiment to use this capability, one must consider the trade-off between the length of estuary to be covered and the time available to cover it. In a tidal estuary such as the Duwamish, it is necessary to sample each position often enough to follow tidal fluctuations. While a 12 hour tidal cycle could be sampled reasonably at a 3 hour interval, the short-lived intense mixing events described by Smith and Partch (1977) required an interval of one hour or less.

It was found that a 3 km reach of the estuary could be covered with

this round trip time.

In the March 1977 experiment, during which most of the

data described in this paper were obtained, this 3 km reach was sampled continuously for 66 hours, with no gaps in excess of 2 hours.

This unique set of

data,which comprises the basis of the rest of the paper, greatly increases our understanding of the dynamics and mixing of the Duwamish estuary, and, by extension, similar salt wedge estuaries elsewhere. LONGITUDINAL STRUCTURE OF INTENSE MIXING EVENTS The March 1977 experiment obtained salinity, temperature and velocity measurements over the 3 km reach shown on Figure 1; however, to date, only the salinity data have been analyzed in detail.

Not only are the intense mixing

events clearly evident in the measured salinity fields, but also the longitudinal extent of this data set provides insight into the nature of these important features.

Partch and Smith (1977) show that the intense mixing events are

charactertized by an increase in both surface salinity and thickness of the surface mixed layer.

In order to study the time and space relationships of

these mixing events, the surface salinity s

the salinity at the bottom of the

0'

surface mixed layer sl, and the thickness of the surface mixed layer h determined from the salinity profiles.

II II

I

I

5

10

1 were Figure 7 indicates the technique for

I 15

SALINITY Figure 7.

I

IN

20

I 25

I 30

O/oo

Sample salinity profile, showing the method of picking sl and h 1'

and h . Each profile was approximated by three straight line segments 1 1 representing the surface mixed layer, the halocline and the bottom mixed layer,

picking s

then s and h were found from the intersection of the upper two lines. This 1 1 technique provides consistent, objective estimates of the surface layer salinity and thickness.

The procedure was applied to profiles obtained each time the

research vessel passed one of the five locations marked on Figure 1.

This

88

provided about 80 samples of s , sl, and hl at each of the stations over the 66 hour experiment. The samples obtained in this manner were at unequal time intervals, and each position was sampled at a different time; however, it was desirable for the intended analysis to have the data equally spaced in time with common intervals between stations, so it was necessary to interpolate between samples.

The scheme

used for this interpolation, described by Akima (1970), minimizes spurious fluctuations between sample points.

The s , sl, and hl records were inter-

polated to six minute intervals, beginning at 1000, 11 March 1977.

To reduce

noise in the records as much as possible, they were each separated into several time series beginning at the high water before the lower low water and compounded; that is the records were split at the high water before the stronger ebb of each tidal day.

, s1 and h1 values then were plotted These are shown in Figure 8. Also marked on

Curves of average s

against time after high water.

this figure are the average times and heights predicted for the two low waters and other high water.

The experiment lasted slightly less than three tidal days,

so the curves represent averages of two or three samples at each time.

An

estimate of the variability was obtained by calculating the standard deviation for each mean, and averaging these for each smoothed curve.

The results are

shown by the error bars adjacent to the curves. The intense mixing events are evident in Figure 8 as the large peak in surface salinity during the strong ebb and the somewhat smaller but broader rise in salinity during the weak ebb.

The salinity does not drop off to a low value

after the weak ebb because the tidal excursion is insufficient to advect fresh water past the station after the intense mixing event ends.

The secondary peaks

at positions 4 and 5 during the strong flood probably are the result of a different process.

A feature of the s

curves, that may be related to this

secondary peak, is a phase lag in the intense mixing event with downstream distance.

The salinity peak produced by the intense mixing event will advect

downstream until the surface current reverses; then it will advect back upstream. Thus, the high salinity patch can be carried back to the downstream stations. Calculations for the flow conditions on 11 March indicate that this could have been the case.

As the maximum shear occurs on the flood, it also is possible

that the secondary peaks are the result of shear instabilities.

However, the

results of Partch and Smith (1977) suggest that significant mixing by shear instabilities is unlikely.

This question may be answered when the velocity data

are analyzed and the tidal excursion for the surface laver can be calculated more accurately.

In addition a gradient Richardson number field can be calculated and

regions of subcritical Richardson number will be evident. Figure 8 shows the same general features for sl, although this time

89

t

2 .o 1.5 I.o

0.5 0

f

Figure 8. Surface s a l i n i t y so, s a l i n i t y a t t h e bottom of t h e s u r f a c e mixed l a y e r s , and s u r f a c e mixed l a y e r depth h as f u n c t i o n s of t i m e a f t e r high water. 'These curves a r e compounded from h a t a over s l i g h t l y less than t h r e e t i d a l days. E r r o r b a r s i n d i c a t e t h e average of t h e s t a n d a r d d e v i a t i o n s f o r each curve.

90 series is noisier.

The additional noise is partly related to uncertainty in

defining the bottom of the surface layer and partly related to the effects of additional processes active at the base of this layer.

The cause of the

sharp spike during the strong flood at position 5 has been examined using the raw data and the spike appears to be spurious. There are two main differences between the hl curves and those for s and s . First, there is a greater similarity between the weak ebb and strong 1 ebb portions, and second, the phase shift noted for s and s1 is not apparent; indeed, the peak seems to occur sooner downstream than upstream.

It should be

noted that while a peak in salinity during the ebb is opposite to what advection would produce, advection of the salt wedge should produce a peak in h

1

at low

water, which is approximately the case in Figure 8. The phase shift in the s

and sl curves is particularly significant in 0

understanding the nature of the intense mixing events.

To get a somewhat

more quantitative measure of this phase shift, the cross-correlation was calculated between position l and the other positions, using the interpolated, not the compounded data.

Figure 9 shows the cross-correlation coefficients for lags

between -25 and +25 hours.

The difference between hl and s

more apparent in the correlations. diurnal peak, while h

and sl, is much

The two salinities show almost no semi-

has a relatively strong peak at this period.

1

This

difference, and the difference in the phase shifts, also apparent in the correlations, indicates that the hl curves may be dominated by advective effects, and changes in the slope of the pycnocline with the tidal phase, while the salinity curves are dominated by the intense mixing events. The lags at maximum correlation in the s

curves were used to calculate

an effective downstream speed for the intense mixing events. spatially uniform value of 35 cm/sec.

The result is a

Based on velocity data from previous

studies, this is consistent with an advective model for the mixing events. However, the maximum salinity increases downstream, so the data require the mixing to continue as the region of increased salinity advects down the estuary. The intense mixing event is well defined at the upstream-most position that was sampled.

Therefore it must have had its origin upstream of the experimental

section.

The rise in s

begins about 1 hour after high water, which indicates

0

a maximum upstream position 1.26 km from position 1 if the intense mixing does not begin before high water and does not move downstream faster than 35 cm/sec. There is a sharp change in the estuary from a shallow, undredged river to a dredged channel at 1.1 km upstream of position 1.

This suggests that an

internal hydraulic jump forming at the transition from shallower to deeper water initiates the intense mixing.

Such a jump would produce considerable turbulence

and a large turbulent salt flux.

Moreover, as the turbulent, higher salinity

91

0.5

0 -0.5 0.5

0 -0.5 0.5

0 -0.5 0.5

0 - 0.5

0.5 0 -0.5 L A G IN HOURS

and hl at positions 1 to 5 Fiqure 9. Cross correlation coefficients for s0, s1' and h1 at position 1. relative to s , s 0

1'

water is advected downstream from this point it would produce a peak in surface salinity. It is reasonable to assume that the low salinity water, which appears at position 1 about 1/2 hour before low water, indicates that the jump had disappeared by the time that water passed the step in the bottom topography.

If the salt

wedge extended beyond the step for all stages of the tide then the jump should continue until low water as long as the wedge extends beyond the step.

How-

ever, reference to Figure 2 indicates that it probably does not extend upstream of the step during the latter stages of the strong ebb.

In addition the

predicted tide curve indicates that there is less than 2m of water over the step during the last hour or two of this ebb and examination of hl in Figure 8 confirms that the salt wedge would not extend upstream of the step at this river surface elevation.

The hour between high water and the first rise in

salinity at position 1 also provides a reasonable estimate of the lag between the end of the jump and the arrival of minimum salinity water at this position. Therefore, the minimum salinitv water should be seen about one hour before low water, which is consistent with Fiuure 8.

On the weak nhh. the peak in s

occurs 0

at position 1 about 0.5 hours before low water, indicatinq that the peak intensity of the jump occurs about 1.5 hours before low water. peak salinity would move 1.89 km in 1.5 hours.

At 35 cm/sec, the

This is an upper limit for the

actual distance traveled because 35 cm/sec probably is too high for the advective velocity at the end of the weak ebb, but it does suggest that the peak in s should reach below position 1, and probably to the region of position 2.

This

also is consistent with Figure 8. This simple model provides a good explanation for several major features of the salinity data displayed in Figure 8. However, it still is necessary to account for the significant increase in the peak surface salinity as the patch of higher salinity water moves downstream from the jump.

Turbulence generated

at the jump would decay in much less than the available four hours so that an additional source of turbulent energy is required for the continued mixing. The near critical internal Froude number profile for the upper estuary, described by Partch and Smith (1977) suggests that an internal hydraulic instability is the most likely source.

The salinity increase appears to begin at times

consistent with the arrival of the patch of water generated by the jump suggesting that the instability must be triggered by that patch.

One possibility is

that the increased salinity in the surface layer reduces the internal wave speed sufficiently to change a near critical internal Froude number to a super critical one.

For a two layer system without shear and with small density

difference, the internal wave speed is given by:

93

where h and h' are the thicknesses of the two layers, Ap is the change in density across the interface and of salinity, h

p

is the mean density.

Using the measured values

and total depth for position 1, both before the salinity

1 increase begins and at the time of peak salinity, yields c = 51 cm/sec and

c = 53 cm/sec respectively.

The change in internal wave speed is too small to

be significant, moreover, the change in hl dominates over the change in A p , so the adjustment in wave speed is in the wrong direction to trigger an instability These values for phase speed will be modified slightly when shear is included in the calculation, however the times o f interest are those of minimum velocity difference between levels so it seems unlikely that an increased surface salinity, alone, can account for the instability that results in continued mixing as the turbulent patch advects downstream.

This, in turn, suggests that extraction of

turbulent energy from the near critical flow depends in some way on the already existing turbulence field. The magnitude of the vertical salt flux downstream of the jump can be estimated from the increase in salt content in the surface layer. the compounded time series of s

,

To this end,

sl, and hl were combined into a time series

of hl (so + s1)/2; that is they were combined into a time series of the product

of mean surface layer salinity and surface layer depth.

Graphs of this

estimate of upper layer salt content per unit horizontal surface area are shown in Figure 10.

The units in Figure 10 are noted as cm-gm/kqm; i.e., the product

of surface layer thickness measured in cms, and salinity expressed as grams of salt per killogram of water. However, the fluid density is approximately 3 3 kgm/m so to the accuracy of the analysis they also can be thought of as

10

salt contents measured in tens of grams of salt per square meter of river surface area.

Referring to Figure 10, we find that between position 1 and position

5, the peak surface layer salt content increases from 17770 gm/m2 to 20530 gm/m in 1.5 hours.

2

The width of the river is approximately the same at these two

positions, so the vertical salt flux is 0.51 gm/rn2.sec if the salinity anomaly curves are assumed to be the same width, similar in shape and scaled by the peak values.

However, the assumptions are negated to some degree by longitudinal

diffusion, so the estimate is a lower limit. A proper calculation usinq the 2 time averaged value yields 0 . 6 3 qm/m 'sec. In view of the difference in flow conditions, this estimate is in reasonable aqreement with the value of 1.24 gm/mL-sec measured by Partch and Smith (1977).

If, as postulated here, the mixing event is initiated by an internal hydraulic jump at the step in bottom topography, 1.1 km upstream of position 1,

94

TIME AFTER HIGH WATER IN HOURS Figure 10. Salt content of surface mixed layer per unit horizontal area as function of time after high water.

an estimate can be made of the amount of salt added to the upper layer in the immediate vicinity of the jump.

Figure 10 indicates that the increase in 2 2 . Using the value of 0.63 gm/m 'sec,

salt content at position 1 is about 14000 gm/m

calculated in the previous paragraph, as the salt flux downstream of the jump, one finds the amount of salt transported into the surface layer

in the approxi-

mately 1 hour required to advect the turbulent patch from the jump to position 1 2 to be 2268 gm/m . This indicates that something like 11732 gm/m2 of salt must be transported into the surface layer in the immediate vicinity of the step. Using 35 cm/sec as the surface layer velocity, the salt flux in the vicinity of 2 the jump must be 4100/L in gm/m -sec where L is the length of the mixing region at the jump measured in meters.

As the extent of the jump is likely to be

between 10 and 100 m it is clear that the intensity of mixing in the immediate vicinity of the step is several orders of magnitude greater than that further downstream. This advective model for the intense mixing events is consistent with all observations obtained to date, and provides additional insight into the nature of these events.

Analysis of velocity data from the ship-mounted mast system

no doubt will provide refinements, and hopefully it will elucidate the mechanism by which turbulence production continues downstream of the step.

With this

increased understanding it should be possible to deternine the general conditions necessary for the formation of intense mixing events, and to investigate the possibility of similar processes occurring in other salt wedge estuaries. E F F E C T S OF

A BRIDGE

In addition to turbulent exchange caused in the upper estuary by the intense mixing events, constrictions associated with bridge piers were found to be important in triggering local mixing.

The mechanism by which the latter is

accomplished was not known prior to the March cruise and it is of interest to examine transects taken in the neighborhood of the 16th Avenue bridge for information about this process.

By reference to the salinity profiles in

Figure 4, it can be seen that the upper conductivity cell, mounted 1.33 m below the river surface, was in the surface layer throughout the tidal cycle; whereas the lower conductivity cell, mounted 2.20 m below the surface, was in the pycnocline throughout the high flow part of the ebb.

Table 1 shows that

the intense mixing events peaked at the bridge site about 1.5 hours after maximum flow suggesting that they might be distinguishable from any hydraulic effects produced under the bridge at high flow.

96

TABLE 1 Times of intense mixing events at stations near the bridge

Date -

Position

11 March 1977

12 March 1977 13 March 1977

2 3 2 3 2 3

Time of Start _ 1130 1148 1212 1218 1300 1306

mixing events Peak End _ ~ 1300 1430 1324 1448 1418 1512 1448 1606 1548 1706 1554 1830

Time of - maximum ebb 1152 1152 1255 1255 1403 1403

River geometry in the neighborhood of the 16th Avenue bridge is shown in Figure 11.

In addition, two structures that protect the bridge piers from

-

0

100 200 300

METERS

Figure 11. bridge.

Plan view of the Duwamish River in the vicinity of the 16th Avenue

S.

barge collisions are indicated as solid lines extending out from the upstream and downstream sides of the bridge toward the north and south banks, respectively. These bumpers are comprised of a tight network of pilings, thus serve to restrict the flow as it passes under the bridge.

This effect is especially pronounced

in the upper layers due to the protrusion of these structures into shallow water.

It should be noted that the contours given on Figure 11 are for mean

lower low water; thus, the actual water depth at the 1 meter contour during the maximum ebb is 2.5 meters.

A tracing of an echo sounding profile made

down the axis of the channel through this region is shown in Figure 12. The inzreased depth and rough topography upstream of the 16th Avenue bridge occurs because the area was dredged fairly recently with a dragline.

97

16th AVE. S. BRIDGE 01

I

I

I

I

I

I

I

I

I

I

I

I

I

1

I

I I

E 2 I

I I I I

4-

l-

a

I

W D I

I

-600

I

I

-400

I

-200

I

I

0

/

I

I

I

200

I

I

400

600

DISTANCE, m (+ DOWNSTEAM) Figure 12. Longitudinal section showing bottom topography in the vicinity of the 16th Avenue S. bridge across the Duwamish River. Good quality records of the salinity and velocity fields in the neighborhood of the 16th Avenue bridge are available for 11, 12 and 13 March 1977.

These

indicate that additional mixing occurred only during the high velocity ebbs, that is only in the early afternoon on these three days.

A typical sequence of

salinity profiles for the mid pycnocline near the bridge on 13 March is given in Figure 13.

The bridge is taken as the origin of the distance axis and the

times at which it was passed are given on the right hand side of the figure. At the left a reference salinity is noted to which the scale at the far right of the figure can be applied to obtain actual salinity values at any time and location.

Profiles of the 2.2 m salinity, for the part of the tidal cycle not

shown, are similar to those for 1558 and 1032 and vary significantly only in their spatially mean values.

These follow the trend of so in Figure 8. Although

there appears to be an increase in low frequency internal wave activity by 1124 and higher frequency internal wave activity by 1231, the main period of interest is between 1230 and 1430.

During this interval, which also corresponds with

the earliest part of the intense mixing event, there is a distinct rise in salinity upstream of the bridge, a rapid drop at the bridge, and a subsequent rise downstream of the structure.

The peak of this event, around 1340, is

shortly before the predicted time of maximum current under the bridge (1403). At this time the salinity 500 meters upstream of the bridge is about 18 O/ao, whereas that 500 meters downstream is only about 10 O/OO. Furthemore, there is substantial, moderate amplitude internal wave activity shown in the

98

(TIMES INDICATE WHEN PASSED 16th AVE S BRIDGE)

BRIDGE I -

20%. 20%0

-fi 1032

A

--*

y

, 4

,I

i

1124

1231

20%0

1302

J

20%0

I

l -000

l

I

-600

l

l

-400

1

I

-200

1

I

0

I

l l 200

I

400

l

l

600

1

I

000

I

I

1000

I

I

I

1200

DISTANCE, m ( t DOWN STREAM 1

Figure 13. Salinity profiles in the neighborhood of the 16th Avenue S. bridge centered around the time of maximum ebb on 13 March 1977. records for the downstream section at 1340.

Here it should be noted that the

pycnocline is very steep and that a depression of only 30 cm results in a 10 O/oo change in salinity.

Also note that the ship-mounted mast moves with a speed

about 5 times that of the longest internal waves, thus, the spatial structure of these low frequency internal waves is not displayed accurately in the traces. Figure 14 is an analagous record showing the temporal structure of the salinity field between 1142 and 1343 on 12 March. Figure 15, are similar.

The data for 11 March, in

It is clear that the salient features of all three sets

of data are associated with the presence of the bridge, and that the spatial structure displayed by the 1259 section on 12 March is to be expected near the peak of each large ebb at least under runoff conditions such as those encountered in mid March 1977.

The general features of the 1259 trace are those to be expected

from a sensor in the pycnocline of a region in which there is a time dependent

99

( T I M E S INDICATE WHEN PASSED 16th AVE S BRIDGE)

I142

10%. %---

+

“

i n

1232

10%.

I

-600

I

I

-400

I

I

-200

I

1259

p

*

/

I

l

l

0

l

200

l

400

i

i

l

600

l

l

800

l

1000

l

l

1200

l

I

1400

I

I

I

1600

DISTANCE, rn ( + DOWN STREAM)

Fiqure 14. Salinity profiles in the neighborhood of the 16th Avenue S. bridge cenkered around the time of maximum ebb on 12 March 1977.

--

BRIDGE

“Vj-q&”

DISTANCE, m (

( T I M E S INDICATE WHEN PASSED 16th AVE S BRIDGE) n - A-

.A

n..

0941

+ DOWN STREAM)

Figure 15. Salinity profiles in the neighborhood of the 16th Avenue S. bridge centered around the time of maximum ebb on 11 March 1977.

100 internal hydraulic jump.

As a rise in salinity corresponds to a rise in

interface level, the hump centered around -200 m can be seen to represent the upstream propagating long wave necessitated by conservation of mass and momentum in the unsteady case.

Further, on the salinity traces for 1339 on 13 March

and 1050 on 11 March the interface at the top of this hump breaks down into very high frequency, high-mode internal waves or more likely into turbulence. This is shown by the disconnected salinity values on Figures 13 and 15 respectively.

The available data indicate a smooth progression from the rather

low frequency oscillations, shown by the 0941 trace for 11 March or by the 1302 trace for 13 March, to the relatively high frequency internal motions shown in the vicinity of the bridge at 1023 on 11 March and at 1132 on 12 March, to turbulence as indicated by the disconnected points at 1050 on 11 March or

1339 on 13 March.

Reference to Table 1 shows that in each case the climax

event occurred somewhat before the maximum ebb and several hours before the intense mixing peak. A series of downstream velocity component traces for various levels

beneath the river surface is presented for 1259 on 12 March in Figure 16. In addition, two salinity records procured with the conductivity and temperature sensors mounted on the mast are included.

The first of these provides a measure

of the salinity in the mixed layer and clearly demonstrates a 2 O/oo jump between

the upstream and downstream sides of the bridge.

The second duplicates the 1259

trace on Figure 15 and is included to show the phase of the velocity structure with respect to interface morphology. All velocity records have been corrected for ship motion.

To do this the

position data were block averaged for one record (0.4096 sec), edited to remove unreasonably large or small values, then averaged over a 30.72 sec interval. Ship speed was calculated by differentiating the 30.72 sec average position information.

Finally these were smoothed using a fifth order polynomial.

Although not our normal means of processing navigation data, this procedure allowed results to be procured rapidly and the quiet oceanographic conditions encountered in the Duwamish Waterway permitted it to yield an accurate ship velocity time series. The downstream velocity component profiles of Figure 16 can be grouped into three categories: those for z = 43, 87, 130 and 173 cm represent the surface mixed layer; those for z = 217, 260 and 303 represent the pycnocline and those for 346 and 389 cm represent the salt wedge.

From these traces it

is clear that the constriction due to the bridge piers and their protective bumpers causes a substantial increase in flow speed at all depths, although this effect is strongest in the surface layer and in the upper pycnocline agreement with what would be expected from the geometry of the river channel

101

BRIDGE

DEPTH

133 cm I

00

$20

-

v;

O 40

1

40

43 cm

I I

0.

-

87cm

130 cm 0 -

40 0 w

173 cm

-

0-

v, \

4

0

-

0

E u

I

0-

389 cm 4

0

0

1 1 -800

' I -600

1

I I -400

'

I -200

1

1

1

0

'

200 I

I

4'0 0

DISTANCE, m (+ DOWNSTREAM) Figure 16. Doynstream velocity component traces for 1259 on 12 March plus surface layer and mid pycnocline salinity profiles.

102 and the shape of the obstruction. Away from the bridge there is a distinct thinning of the surface layer and this is associated with a substantially increased downstream flow velocity, an effect that is especially clear over the large upstream hump centered near -200 m.

However, these enhanced flow velocities

are restricted to the pycnocline and surface mixed layer and primarily to the latter.

For example,in the middle of the surface layer the flow speed is

increased by nearly a factor of two over that in the mid pycnocline or below. This enhanced surface flow speed results in a drop in upper Pycnocline Pichardson number relative to what otherwise would have been the case at this location. However, the decrease is small and the Richardson number is still of order 1 in the middle of the layer. The Richardson number data of Table 2 indicate fully turbulent upper and TABLE 2 Richardson numbers for various depths and horizontal positions in the neighborhood of the 16th Avenue bridge

Downstream distance from bridge in meters

-900

-600

-130

0.27 0.33 4.70 1.81 0.36 11.43 1.34 0.31

3.42 0.38 0.28 1.13 3.89 35.49 0.50

0

40

0.51 0.55 16.84 38.01 0.86 22.29 0.77 0.23

0.32 0.16 0.26 0.98 6.66 6.94 1.54 0.31

Depth in meters

-

.65 1.09 1.52 1.95 2.39 2.82 3.25 3.68

0.42 2.98 4.66 3.29 61.13 1.92 0.25

0.13

lower layers separated by a region of limited mixing.

Near critical Richardson

numbers associated with the upper layer penetrate into the upper pycnocline at two locations; the first is over the crest of the upstream hump and the second is in the rapidly decelerating region of the hydraulic jump.

In both cases there

is evidence from other measurements that turbulence is produced at least intermittently in these regions, and it is suggested that the additional shears are provided by high frequency internal waves that propagate through the sites. Unfortunately the analysis scheme used to reduce the March data makes it difficult to resolve such features. Figure 17 shows all three velocity components at the 173 cm level, that is near the base of the surface layer.

There appears to be a small positive

vertical component of velocity associated with the upstream hump in the pycnocline as well as a flow towards the northeast bank in this region of the river.

A

103

16TH AVE. S. BRIDGE

- -

25

W,cm/sec

0

_.

-25 25 -v,cm/sec 0 =A-25 .-

- - - A

I

-

--.-!‘u----

U,cm/sec 0 ’

I

20

s Yo0 10 I

0

-800

-600

-400

-200

0

200

400

DISTANCE, m (+ DOWNSTREAM)

Figure 17. Velocity corponent profiles for a position near the base of the surface mixed layer centered around 1259 on 12 March. The horizontal velocity component u is positive in the downstream direction, the cross-stream velocity component v is positive toward the northeast bank and the vertical velocity component w is positive upward. complicated internal wave field, associated with the upstream hump, is evident in both the vertical and cross-stream velocity components.

Substantial vertical

velocity components also are found in the region downstream of the jump.

HOW-

ever, here they are associated with longer wavelength features. The cross-stream velocity component field can be best understood by reference to the series of profiles presented in Figure 18 and to the channel geometry displayed in Figure 11. The former shows that between -200 and -600 m the flow is toward the northeast bank in the surface layer and toward the southwest bank in the pycnocline and bottom layers.

However, it is in this region that the channel curves necessi-

tating a secondary flow.

In the case at hand the maximum return flow in the

secondary circulation occurs not in the bottom region as it does in an unstratified river but in the lower to central part of the pycnocline where the downstream velocity, hence the centrifugal force, is reduced.

The maximum flow

toward the outer part of the bend is in the mid to lower mixed layer with a near zero cross-stream component at the river surface. Superimposed on this general secondary circulation, is a substantial flow toward the northeast bank centered just under the upstream hump.

The

amplitude of this disturbance is maximum in the center of the pycnocline and falls off rapidly with distance into the surface layer.

Below the pycnocline

104

-

h

-

-

25

---

-

1 I -

-25 -

I I

43cm

-25

I

25 -2525 -

-

-25 -

I

25 \ m

E -25

' 0

25

-25 25 ~~

-

7 -25 25

h

-- -

/

-25-

7

303 cm

I I

346 cm

I

25 -25 -

h

-

---

h

"

fi

v

- 1

-

.,

389 cm

I I

-800

I -600

I -400

1 -200

I 0

1 200

I 400

DISTANCE, rn (t DOWNSTREAM)

Figure 18. Cross-stream velocity components for the period centered around 1259 on 12 March. Positive values correspond to flow in the northeast direction, that is toward the outside of the bend located between -200 and -600 m. Note the distinct secondary circulation in this region.

the cross-stream velocity component at this location falls off more slowly. The cause of this lack of two-dimensionality in the upstream hump is not known and cannot be investigated in much detail as only transects down the center of the channel are available.

Nevertheless, resoltuion of the velocity field is

sufficient to guarantee that these non zero horizontal and vertical velocity conponents are associated with thinning of the mixed layer and that they are not

105 spurious features.

From Figure 11 it can be seen that the reach upstream of

-600 m is straight and that in this region the spatially averaged vertical and cross-stream velocity components are zero as shown in Figures 17 and 18. As in the case of the intense mixing events, the data on flow near the 16th Avenue bridge reported in this paper put additional constraints on the nature of the processes causing a salt flux from the lower to the upper layer, but do not permit it to be characterized fully.

In the case of the disturbance near

the bridge, we have shown (1) that the pycnocline is distorted just before maximum ebb, but only on the strongest ebb each day, in a manner not unlike an internal hydraulic jump, (2) that the surface layer salinity increases by 2 O/oo over a region several 10's of meters in horizontal extent just downstream of the bridge during this period, (3) that acceleration of the surface layer due to the upstream hump in the pycnocline causes increased shear and reduces the Richardson number field in this region, and (4) that shear associated with the downstream end of the hydraulic jump likewise results in reduced Richardson numbers. In addition, two records with chaotic salinity traces in the pycnocline region are suggestive of turbulent flow if not conclusive demonstrations thereof.

These are

associated with the upstream region of low Richardson number just like the obvious 2 O/oo jump in surface layer salinity i s associated with the downstream end of the apparent hydraulic jump in which low Richardson numbers also are present.

The 2 O/oo increase in salinity is equivalent to a horizontal salt

flux of 2mLgm/kgm-sec which yields a vertical salt flux of 2000/L gm/mL.sec or 2 100 gm/m -sec when distributed over L = 20m. The 2000/L gm/m2-sec value is about half of the analagous salt flux at the upstream jump.

Although the down-

stream mixing process also acts over a shorter period of time it provides a measurable contribution to the overall salt balance of the estuary. REFERENCES Akima, H., 1970. A new method of interpolation and smooth curve fitting based on local procedures. Journal of the Association of Computing Machinery, 12: 589-602 Dawson, W.A. and Tilley, L . J . , 1972. Measurement of salt wedge excursion distance in the Duwamish River Estuary, Seattle, Washington, by means of the dissolved-oxygen gradient. U . S . Geological Survey Water Supply Paper, 1873-D, 27 pp. Hansen, D.V. and Rattray, M., Jr., 1966, New dimensions in estuary classification. Limnology and Oceanography 11: 319-326. Partch, E.N. and Smith, J . D . , 1977. Time dependent mixing in a salt wedge estuary. Estuarine and Coastal Marine Science, 5: Santos, J.F. and Stoner, J.D., 1972. Physical, chemical and biological aspects of the Duwamish River Estuary, King County, Washington, 1963-67. U . S . Geological Survey Water Supply Paper. 1873-C, 74 pp.

106

Smith, J.D., 1974. covered ocean.

Turbulent structure of the surface boundary layer in an iceRapp. P. -v. Reun, cons. int. Explor. Mer, 1967: 53-65.

Stoner, J.D., 1972. Determination of mass balance and entrainment in the stratified Duwamish River Estuary, King County, Washington. U.S. Geological Survey Water Supply Paper. 1873-F, 17 pp.

107

A TWO-DIMENSIONAL NUMERICAL MODEL FOR SALT INTRUSION IN ESTUARIES P.A.J. PERRELS and M. KARELSE Research Engineers, Delft Hydraulics Laboratory, The Netherlands

ABSTRACT A two-dimensional laterally integrated numerical model has been developed to represent the vertical velocity and salinity distribution along an estuary. The governing equations which express the conservation of mass, momentum and salt content, are solved by a finite difference method in combination with a splitting technique. The model has been applied to the Delft tidal salinity flume, which may be considered as a two-dimensional tidal flow characteristic for estuaries such as the Rotterdam Waterway. By this application several assumptions about the effect of stratification on the vertical diffusion were tested. Preliminary results of the comparison

of

computed and measured data will be shown in this

Paper. This Paper is the result of a study which is incorporated in a basic research programma T.O.W. (working group "Stromen en transportverschijnselen") executed by Rijkswaterstaat (Public Works and Water Control Department), the Delft Hydraulics Laboratory and other research institutes. 1

INTRODUCTION Estuaries are regions of water, which are connected to the sea or ocean

at one end and fed by sources of fresh water (rivers) at the landward boundaries. In these regions saline sea-water and fresh river-water meet each other. Because of mixing of salt and fresh water the distribution of salinity in an estuary is a gradually varying function of space and time. The major factors determining the salinity distribution are:

- the tidal motion within the estuary (as governed by the varying tidal elevation at the sea-entrance and the estuary geometry)

- the fresh river-water discharge - the density difference between fresh water and saline sea-water - the estuary geometry (tributaries, groins, harbours) -

wind influence

- Coriolis effects

106 Depending on the magnitude of these quantities varying degrees of stratification are possible. Most of the mathematical models, which have been developed to calculate the salinity distribution in estuaries, are descriptive rather than predictive. Mathematical models having predictive capability need physical information on the spatial and temporal functions for turbulent eddy and mass diffusivities and physical relevant boundary conditions which are general applicable. The present knowledge on for instance the effect of stratification on the vertical eddy and mass diffusivities is limited (Delft Hydraulics Laboratory, 1974 and Fischer, 1976), consequently models are descriptive rather than predictive. A general representation of salinity distribution in an estuary would require an unsteady, three-dimensional approach. For many situations however a simplified model may give satisfactory information in a much more economical way. For a lateral uniform situation, for instance, a vertical two-dimensional model is an obvious schematization. Many mathematical models used for salinity intrusion problems in estuaries are one-dimensional (Harleman et. al., 1974), in that they use cross-sectional integrated forms of the equations of mass, momentum and salt content. These models require as input data information on the dispersion, i.e. the integrated effect of variation of velocity and concentration over the cross-section. This in itself,

limits their predictive capability (Abraham et. al., 1975). For broad estuaries two-dimensional models with two horizontal dimensions have been developed (Leendertse, 1970). All these depth-integrated models are primarily applicable for mixed estuaries, in which the density differences between bottom and surface are small. For stratified estuaries, in which there are two fluids of different densities separated by a distinct interface, two-layer models are available (Vreugdenhil, 1970): models without mixing, in which the upper-layer consists of fresh water and the under-layer of salt sea-water and models with vertical

mixing, in which the vertical exchange of volume and mass between the two-layers results in a salinity that varies in longitudinal direction. For partly mixed estuaries, which are characterized by gradually varying density in both horizontal and vertical directions, two-dimensional, laterally integrated models have been developed. Of these models known from literature (Hinwood et. al., 1975), that of Hamilton (1973 and 1975) is the most detailed. This model however still has some drawbacks, such as the treatment of the free surface and the bottom configuration, where extrapolation is needed because for the numerical model a fixed grid is used. For the bottom stress the quadratic friction law is applied in which, however, a constant distance to the bottom i s used. At the downstream boundary a salt distribution is needed as function of place and time. At the free surface the first derivative of c is set equal to

109 zero instead of using the kinematic condition. Furthermore, a space-staggered grid is used, which makes the implementation of boundary conditions containing first or higher derivatives less straight forward. The more recent model of Boericke et. al. (1977) pays much attention to the exchange coefficients and also uses co-ordinate transformation. The computation of the velocities is rather simplified by neglecting the convective terms, although this yields inaccurate velocity profiles in tidal srreams (Abbott, 1959). Boundary conditions for the velocity and the concentration at the bottom and at the free surface are not mentioned. During the past two years a new two-dimensional laterally integrated model \as been developed at the Delft Hydraulics Laboratory for predicting the unsteady velocity and salinity distribution in a partly mixed estuary, based on the shallow water approximation and a mixing length approach for the closure problem. Within those limits the model is kept as general as possible with an exact representation of the bottom configuration and the free surface and the possibility to improve several boundary conditions at the upstream and the downstream boun-

daries. The free surface and the bottom configuration are reproduced exactly by means of a transformation. For the bottom stress a roughness coefficient is used, depending on the local circumstances. At the downstream boundary a function is needed, describing the transition of the salt concentration from an ebb to a flood situation. The objectives of the design of the two-dimensional laterally integrated model, presented in this paper, are

- To judge the application of this model with respect to a partly mixed estuary, in which the vertical transport of momentum and salt is dependent on the Richardson-number (Delft Hydraulics Laboratory, 1974). A necessary first step is to find generally applicable relations for the Richardson-dependency of the vertical turbulent exchange of momentum and mass and for the transition function in the sea-ward boundary condition of the salt balance.

-

To judge the application of this model with respect to a mixed estuary, in which the influence of the density differences on the vertical diffusion can be neglected; in this case the vertical exchange is known from literature (Delft Hydraulics Laboratory, 1973 and Fischer, 1973).

- To study the influence of density effects on the longitudinal dispersion coefficient of an one-dimensional model. Application of this knowledge will make the use of depth-averaged models for wide estuaries and of cross-sectional averaged models more justified.

- To get information for three-dimensional models about the ability to represent the vertical flow structure and salt distribution.

110 This Paper presents a description of the mathematical model and contains a comparison of the computed velocity and salinity distributions with experimental data from the Delft tidal salinity flume (Appendix I). This tidal flume may be considered as a nearly two-dimensional estuary in which it is possible to represent the characteristics (of the tidal motion and the salinity distribution) of estuaries like the Rotterdam Waterway. Tests in the tidal flume with bottom roughness have been used for the verification of the numerical model. Preliminary results of the comparison of computed and measured data will be shown in this Paper. The verification of the numerical model with flume data is still going on and will be published later when completed.

2 MATHEMATICAL MODEL Formulation After integration over the width and if the shallow water approximation is made, the equations for vertical two-dimensional density currents are: (Delft Hydraulics Laboratory, 1973) The horizontal momentum equation:

aU + -I a ( b u 2) + a (uw) - la (be at b ax a Z b ax x

aU a -aU) = - L a p a ~ )- a~ ( E Z aZ P ax

The hydrostatically distributed pressure:

The continuity equation:

The salt balance:

The equation of state:

p = po

+

=

po +

(3c in which B is of the order 0.75

(5)

Integration of (3) over the height, and substitution of the kinematic boundary condition, yields:

111

The width b is supposed to be a function of x only, because strong dependency on

z would give large errors by integration over the width (Vreugdenhil, 1974). TO compleie the model expressions are necessary for the turbulent viscosity and

diffusivity coefficients. For the present model a mixing length approach has been chosen which applies well to circumstances where the shallow water approximation can be made.

t

=

2 lm

Dz

=

2 lm

%( f (Ri) aZ

(7)

in which the mixing length 1 is defined by: m

1

m

= K

(z+z )

if 0 6 z 6 0.25 H

= K

(0.25 H+z,)

if 0.25 I3 s z 6 H

and the Richardson-number Ri by:

z

in which k

S

=

1/30 k s

is the roughness length.

For the damping functions f (Ri) en g (Ri) many formulas are available (Delft Hydraulics Laboratory, 1974). A definite choice is one of the subjects of investigation. The horizontal viscosity and diffusion terms:

‘b axa (bE

aU

-)

az

and

- (bDx zac)

l a

-

b ax

are commonly neglected because their influence is small; however, for long periods the horizontal turbulent diffusion may give a significant contribution. In the model both terms are included for numerical reasons (Delft Hydraulics Laboratory, 1975). Boundary conditions The mathematical model requires boundary conditions and initial conditions

112 for the tidal motion as well as for the salinity. The question of whether the model is descriptive or predictive ciepends on the formulation of the vertical momentum and mass exchange and on the way in which the boundary conditions are treated. If observed data are used which cannot be predicted, like the vertical tidal motion at sea can be predicted, the model is only descriptive. For the sea-boundary (x for the water-elevation:

5

=

0) the conditions are:

=

5,

(t)

: u = uo (t,z)

for the velocity

aLu or: 2

ax

: c = c max

for the salinity

go

(t,z) if u > 0 (flood tide)

2

and: The form of g

2 ax

0

=

if u

< 0 (ebb tide)

(t,z) is one of the subjects of investigation, as a first approach

a piece-wise linearized function of t, independent of At the upstream boundary of the estuary x

= L'

z

was used.

the following conditions

are valid: for the velocities

: u

for the salinity

: c = o

= UL,

(t,z)

Knowing the river-discharge and assuming a logarithmic velocity profile in the fresh water-region of the estuary, the function u L l has been determined (Appendix 11). The only restrictive aspect of this boundary condition is that L ' ought to be chosen large enough to be outside the zone of salinity intrusion At the bottom z

=

zb the conditions are: u

w

= o = o

At the water-surface the conditions are:

-a~- an

0 (no wind)

WC-D

z

- a2

- 0 (no salt through the water-surface)

At the water-surface also the kinematic boundary conditions holds:

a condition which is already used by the derivation of (6).

113

Transformation In general the longitudinal area of interest will not be a rectangle, because o f the variations in bottom and free surface. For a good representation of the flow a good description of the form of the free surface and the bottom is necessary. For a numerical approach with finite differences however a rectangular grid, which coincidences with the boundaries will be preferable. Therefore the area of interest is transformed by a simple transformation into a rectangle (Delft Hydraulics Laboratory, 1976 and Jamet, 1 9 7 0 ) . If the position of the free surface is described by:

and the position of the bottom by:

Zb = Zb

(4

then the transformation reads:

2' =

fb (x,t)

(2-2,)

where fb is given by:

This transformation adds some terms to the equations, but simplifies the finite difference approach at the boundaries considerable. Numerical approach The mathematical model is solved with a finite difference method, so a splitting method can be applied (Richtmeyer, 1967 and Roache, 1 9 7 2 ) . The momentum equation and the continuity equation for the salt are split into the spatial directions (Delft Hydraulics Laboratory, 1975). The part in the x-direction of the momentum equation is solved explicitly, while the part in the z-direction is solved implicitly. The implicit technique in the z-direction removes the most severe stability condition. The stability condition in the x-direction depends on the value of E

X'

which means that a suitable choice of

E

X

permits a larger time step

1970 and Delft Hydraulics Laboratory, 1 9 7 5 ) .

2E

X

Ax

T

(Jamet,

114 This mixed approach was found to be most economical due to the fact that the ratio of the horizontal and the vertical dimensions of the problem area is usually large. If this ratio is closer to one, the same approach in the x- as in the z-direction would be more likely. For the continuity equation of the salt, the same difference technique is used. Special attention is paid to the treatment of the continuity equation and of the boundary conditions, particularly the boundary conditions involving a derivative. By the use of this mixed approach the difficulty of finding boundary conditions at the intermediate level is circumvented. For the continuity equation a fourth order scheme in Ax and second order i n AZ is used,

and for the boundary conditions second order schemes are used.

In this way inconvenient perturbations are avoided and a good accuracy is

achieved. A special treatment is also applied near the bottom, where the gradients in the velocity profile are very large. So a very small step should be used, which in an uniform grid is not very attractive from an economical point of view. Therefore locally a special treatment is used.

Near the bottom the convective terms and the horizontal turbulent viscosity are neglected, so the momentum equation simplifies to:

aU

aU aZ

--+-(E

at

-aU) = - L a p

aZ

P

ax

which can be written as:

Now the right-hand side of ( 1 6 ) i s supposed to be independent of z, therefore

(16) can be solved by integrating twice. The constants that arise by the integration can be found by substituting the boundary conditions: u and: u

=

0 at z = 0

= u

(Az) at z

=

AZ

all

In this way an approximation for -near the bottom can be achieved, which will 22 then be substituted in the difference equation. The difference equations are given in Appendix 11. 3

VERIFICATION A s a first test for the numerical model. a comparison with flume data of

the Delft tidal salinity flume has been chosen mainly for the following reasons:

- In the tidal flume the tidal flow is nearly two-dimensional, in that there is no variation of density but some variation of velocity in transverse direction.

115

- The flow conditions are well adjustable. - The system has been measured systematically. For a description of the flume and the flume data see Appendix I. The verification is split into two phases, in order to have a clear distinction of the different mechanisms that influence the tidal motion and concentration distribution. In the first phase, computations for homogeneous circumstances are performed and compared with the measurements from the homogeneous test T22. These computations were meant to calibrate the water-level variations and the velocity profiles. After some adaptions of the bottom roughness, a reasonable agreement was achieved for the water-level variation (Figure I ) and for the velocity profiles (Figure 2). The deviation in the vertical tidal motion is mainly caused by the difference in the adjustment of the river-boundary between the numerical model (boundary at x tidal flume (boundary at x

=

=

L

i s

Q ) and the L

L' is Q (L',t), see Figure 3 ) . The velocity profiles are

shown at two characteristic times at two distances from the downstream boundary. The standard deviation in the measurement is of the order of

0

z 0.015 m/s.

The data for the computation were:

- Boundary conditions and flow parameters according to Table A2 -

z

=

-

E

=

-

E~ =

0.004 m D~

=

D

according Equations (7), (8) and (9) with f (Ri)

0.37 m2/s

- Ax

=

3.66 m ( = L/49)

-

Az

=

HI12

-

T

=

2.79375 s ( = 1/200 T)

=

800 (= 4T)

- Nt

=

g (Ri) = 1

- CPU-time = 160 s (at a CDC 6600) In the second phase of verifications the inhomogeneous test TI80 was used.

In this case the influence of the density differences on the flow ought to be represented. The major difficulty of these computations lies in correct representation of the Richardson-dependency of the coefficient of vertical, turbulent viscosity and diffusivity (Equations (7) and (8)). For the present computations the demping functions given by Van Rees (1975) turned out to give the best results. In Figure 4 the density distribution is shown at four characteristic times. The data for this computation were the same as in the homogeneous computation except: f (Ri)

=

exp (- 4Ri)

g (Ri) = exp (- 15Ri)

and CPU-time

=

according Van Rees (1975), see also Delft Hydraulics Laboratory (1974)

180 s

Conclusions Preliminary results are given of computed tidal motion and density distributior which show reasonable agreement with the experiments. Much work is still to be done primarily for determination of the physical coefficients. However, it may be concludec

116

h

lrnl

t

X=3.66m

h CmI

t X = 47.58m

FIG. 1

COMPARISON COMPUTED AND MEASURED VERTICAL TIDAL MOTIONS TEST T 22

-

---

MEA SURED COMPUTED

T : TIDAL PERIOD

117

I

X = 47.58m

0.20-

M. F. V.

0

D

0.15

-

0.10 -

-

0.10

-0

0.20

U Cm/sl

COMPARISON MEASURED AND COMPUTED VELOCITY PROFILES

FIG. 2

TEST T 22

M.E.V: :MAX. EBB VELOCITY

M.E V: :MAX. FLOOD VELOCITY

-

MEASUREMENT COMPUTATION

t

SNOllVlON H l l M 3Nnld A l I N I l V S 1Vall d0 d n l i ' S

-

6'9ld

t

(3l€lVlMVA) H19N37 3 W f l l d 3All3ld = 1

4

H19N31 NOISnYl Nl 11FS

119

M.E V

6-

-

4-

-

-

2-

-

0

I

l

2

l

1

4

6

10

8

12

l4

X/DX

M IV

I

l

l

I

1

12 Z/DZ

L.W.S.

I

8

DENSITY DISTRIBUTION IN THE VERTICAL PLAIN

FIG. 4

1

-X/DX

l4

I

I

10

I

I

12

I

I

I

1

l4

--+X/DX

-

MEASURED

---- COMPUTED D X Z 3.66m DZ= 1.66cm

120 on these preliminary results that the present numerical model is able to represent the

partly mixed tidal flow, regarding the tidal motion and salt distribution, in a descriptive way. More extensive comparison with the available flume tests is still going on and ought to prove the predictive character of this numerical model.

APPENDIX I: DATA FROM THE DELFT TIDAL FLUME Description of the €lume The lucid flume for the experiments has a rectangular cross-section, 0.672 m wide and 0.50 m high. Two straight sections and the bend between them have a total length of 100 m (Figure 3). Downstreams the flume ends in a sea-basin, 8 m long, 6 m wide with a bottom, I , ] m below the bottom of the flume. By means of a control valve any.periodic tidal movement of the water-level can be generated. The density of the sea-water is kept constant by means of a circulation system which pumps salt water into the basin through perforated tubes on the bottom. A t the upstream end of the flume is equipment to supply separately a constant and a variable discharge of fresh water. This makes it possible to reproduce tidal movements which can occur in flumes longer than 100 m. The variable discharge of fresh water is programmed according to one-dimensional tidal computations for flumes longer than 100 m. For a detailed description of the flume see Van Rees et. al. (1969) and Rigter (1973). Experiments used for verification Several tests, with plates (2 x 2 cm) on the bottom of the flume arranged in a diagonal pattern to obtain the desired roughness, are available. In this Paper two of these flume tests, which are used for verification of the numerical model, are represented. The tidal quantities in these two experiments are the same. However in test T22 there was no density difference between river- and sea-water, while in test TI80 this density difference was about 22 kg/m3 (see Table AI).

So the difference in the

tidal motion between these two tests is caused by effects of the density difference. TABLE A1 Boundary Conditions and Flow Parameters quality

symbol

test T22

test TI80

depth (averaged over T)

h

0.216 m

0.216 m 179.34 m

fictive length of flume

L

179.34 m

Chgzy-coefficient

C

19 m'/s

tidal period

T

I

tidal amplitude at sea fresh water-discharge density differences between river- and sea-water

AO

QL AP

1

19 m'/s

558.75 s

558.75 s

0.025 m

0.025 m

3 0.0029 m / s

3 0.0029 m / s

0

22 kg/m

3

121

APPENDIX I1

The f i n i t e d i f f e r e n c e equations f o r t h e c o n s e r v a t i o n of momentum i n t h e x-direction are:

n , j + ui - 1 , j

n -2.; I‘i+1 ,j

+

Ax

X

Un+l - n+ 1 i,j+l i,j-ll 2 Az

1 --

I

{PP+l,j

Pi,j

TFn 1.

.

-

Pi-1, j

2 Ax

+

1,J

+

The f i n i t e d i f f e r e n c e equation f o r t h e c o n s e r v a t i o n of mass i s : n+l

w. i,j

-

w

AZ

n+l i,j-l

=

n+l - Un+l (Ui , j i,j-I )

(

T F ~

2 ; , .J +

Az

TF;

i,j-l

) +

2 TF;

i

n+l (bi+2 ui+2,j

-

n+ 1 ) bi-2 ui-2,j 4 Ax

+

1

122

+

n+l (bi+2 "i+2,j-1

-

n+ 1

bi-2 u i - 2 , j - ~ ) 4 Ax

The f i n i t e d i f f e r e n c e equation f o r t h e p o s i t i o n of t h e f r e e s u r f a c e i s :

The f i n i t e d i f f e r e n c e e q u a t i o n s f o r t h e c o n s e r v a t i o n of t h e s a l t content a r e : -x c. - c n. 1,j L

.

~ J=

I

--

+ D

T

.

, ~ =

-

- bi-l

n+l n ui+l,j 'i+l,j

n+l ui-l,j

n

C i.

2 Ax

b.

T

c n. + l - cx . 1,j 1

{'i+I

n - 2 c n . C. I.J { i+I,j X

+cn i-19jl

Ax

n+ I n+ 1 C. - c 19j+; iZ i 7 j - 1 } n+l U.

i,j+i

TFY

+

i,j

n+l n+l - u i. , j - 1 i,j+i

C.

n+l i,j-I

C.

2 AZ n+l n+l { W .i , j + ~' i , j + l

-

n+l

w.1 , j - I

n+l

} TF~.+ 1-j

C1 . ,j-I)

2 Az

TFn

3i n+ 1

1 - c n+ . .)

Az

2 Az

+

- ~ , j }+

123 L I S T OF SYMBOLS

tidal amplitude at x

= 0

width concentration discretised c turbulent diffusion coefficient for salt in the x- and z-direction respectively transfer function

fb f (Ri)

ration between the mixing lengths for momentum in neutral and stratifiec conditions, depending on Ri acceleration due to gravity density profile at x

=

0

ration between the mixing lengths for salt in neutral and stratified conditions, depending on Ri h

water-dep th

H

transformed water-depth

U

velocity profile at x

= 0

velocity profile at x

= L’

uL’

roughness length

kS

L

fictive length of flume (Figure 3 )

L

length of flume (Figure 3 ) mixing length in neutral conditions

lm n

normal direction pressure

P

pressure at the free surface

PS

river-discharge

QL Ri

Richardson-number

t

time

T

tidal period

TF1,TF ,TF

transfer coefficients

U

velocity component in x-direction

2

n 1,j

3

U.

discretised velocity component u

W

velocity component in z-direction

n 1,j

W.

discretised velocity component w

X

longitudinal direction

Z

vertical direction

‘b Z

Ax,Az &X’&Z

position of the bottom coefficient of bottom roughness spatial steps in x- and z-direction respectively turbulent diffusion coefficient for momentum in the x- and z-direction respectively

124 K

Von Karman-coefficient

5

integration variable

P PO

0

density density of fresh water density differences

T

time step

5

position of the free surface

50

position of the free surface at x

=

0

REFERENCES

1

Abbott, M.R., 1960. Boundary layer effects in estuaries. Journ. Marine Research 18, no. 2: 82-100. 2 Abraham, G., Karelse, M. and Lases, W.B.P.M., 1975. Data requirement for one-dimensional mathematical modelling of salinity intrusion in estuaries. XVIth IAHR Congress, paper C32. 3 Boericke, R.R. and Hogan, J.M., 1977. An X-Z hydraulic/thermal model for estuaries, Journ. ASCE, HYI: 19-37. 4 Delft Hydraulics Laboratory, 1973. Computational methods for the vertical distribution of flow in shallow water. Report on literature study, project W 152, Delft. 5 Delft Hydraulics Laboratory, 1974. Momentum and mass transfer in stratified flows. Report on literature study, project R 880, Delft. 6 Delft Hydraulics Laboratory, 1975. Berekening dichtheidsstroom, keuze van de differentiemethode. Report R 897-1, Delft. 7 Delft Hydraulics Laboratory, 1976. Berekening van stroming in een getijrivier, het homogene deel. Report R 897-111, Delft. 8 Fischer, H.B., 1973. Longitudinal dispersion and turbulent mixing in open channel flow. In: Ann. Review Fluid Mechn. 5: 59-78. 9 Fischer, H.B., 1976. Mixing and dispersion in estuaries. In: Ann. Review Fluid Mechn. 8: 107-133. 10 Hamilton, P., 1973. A numerical model of the vertical circulation of tidal estuaries and its application to the Rotterdam Waterway. Univ. of Washington, Dept. of Oceanography, Seattle. 1 1 Hamilton, P., 1975. A numerical model of the vertical circulation of tidal estuaries and its application to the Rotterdam Waterway. Geoph. Journ. Royal Astron. S O C . 40: 1-21. 12 Harleman, D.R.F. and Thatcher, M.L., 1974. Longitudinal dispersion and unsteady salinity intrusion in estuaries. La Houille Blanche no. 1 / 2 : 25-33. 13 Hinwood, J.B. and Wallis, I.G., 1975a. Classification of model of tidal waters. Journ. ASCE 101, HYIO: 1315-1332. 14 Hinwood, J.B. and Wallis, I.G., 1975b. Review of models of tidal waters. Journ. ASCE 101, H Y 1 1 : 1405-1422. 15 Jamet, P., Lascaux, P. and Raviart, D.A., 1970. Une mEthode de rEsolution numsrique des Gquations de Navier-Stokes. Num. Math. 16. Springer-Verlag. 16 Kutler, P., Lomax, H. and Warming, R.F., 1972. Computation of space shuttle flow fields using non-centered finite difference schemes. AIM-paper no. 72-193. 17 Leendertse, J . J . , 1970. A water-quality simulation model for well-mixed estuaries and coastal seas; Vol. 1, Principle of computation. Memor. Rand Corp., RM-6230 RC, Santa Monica. 18 Van Rees, A.J. and Rigter, B.P., 1969. Flume study on salinity intrusion in estuaries. XIIIth IAHR Congress, paper C33. 19 Van Rees, A . J . , 1975. Experimental results on exchange coefficients for non-homogeneous flow, XVIth IAHR Congress, paper C36. 20 Richtmeyer, R.D. and Morton, K.W., 1967. Difference methods for initial value problems. New York, Interscience Publishers. 21 Rigter, B.P., 1973. Minimum length of salt intrusion in estuaries. Proc. ASCE 99, HY9: 1475-1496.

125 22 Roache, P . J . , 1972. Computational fluid dynamics. Albuquerque, Hermosa Publishers. 23 Vreugdenhil, C.B., 1970. Computation of gravity currents in estuaries. Delft Hydraulics Laboratory, publ. no. 86, Delft. 24 Vreugdenhil, C.B., 1974. Approximations in mathematical models for stratified flow. Delft Hydraulics Laboratory, Report S 114-IV, Delft.

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127

THE EFFECT OF METEOROLOGICAL FORCING ON THE CHESAPEAKE BAY: THE COUPLING BETWEEN AN ESTUARINE SYSTEM AND ITS ADJACENT COASTAL WATERS

ALAN J. ELLIOTT’ and DONG-PING WANG’ ’SACLANT ASW Research Centre, Viale San Bartolomeo 400, 19026 La Spezia, Italy 2Chesapeake Bay Institute, The Johns Hopkins University, Baltimore, Md., 21218

ABSTRACT Surface elevation and current meter records from the Potomac estuary are combined with elevation and wind stress data over the Chesapeake Bay to investigate the coupling between the Potomac, the Bay and the coastal ocean. The dominant sea level fluctuations in the Chesapeake Bay were found to be generated at the mouth of the Bay by the action of Ekman dynamics. Winds which blew parallel to the coast caused fluctuations in the mean sea level at the Bay mouth, these fluctuations then travelled northward up the Bay. Other fluctuations were due to an Ekman effect within the Bay itself and also due to longitudinal seiche motions. The oscillations within the Potomac were due, in part, to the local forcing but were also due to co-oscillation with the Bay. Low frequency sea level fluctuations within the Potomac were the result of the Ekman effects in the coastal water; these disturbances, which had originated at the mouth of the Bay, appeared to influence the entire estuarine system. The results suggest that future estuarine studies should include the coupling with the coastal ocean, both in modelling and in observational invesgations. The important time scales for the nontidal fluctuations extend at least to monthly and seasonal periods. Therefore long term (several months to several years) monitoring of the wind, sea level, density and currents is required before the forcing and response mechanisms will be fully understood.

128

Fig. 1

The Chesapeake Bay estuarine system. Tide stations: AN, Annapolis; SO, Solomons; DC, Washington: CO, Colonial Beach; LW, Lewisetta; GR, Grey Point; KP, Kiptopeake Beach

A

Wind towers : P , Patuxent River; N, Norfolk The position of the long-term current meter mooring in the Potomac is shown circled.

129

INTRODUCTION The Chesapeake Bay and its tributaries form one of the largest estuarine systems in the World. The Bay itself extends a total distance of about 1 7 0 nautical miles ( 3 1 5 km) from its mouth near Norfolk, Virginia, to its head where it meets the Susquehanna River (Fig. 1). The tidal wave (semidiurnal tide) takes slightly longer than 13 hours to travel from the mouth of the Bay to its head; as a result the next tidal wave enters the Bay before the head has been reached and so two high tides can simultaneously be contained within the Bay. The Bay is large enough for rotational effects to be important: in the lower and mid portions of the Bay the tide advances as a Kelvin wave and tidal range and velocities are significantly greater along the eastern shore. The Kelvin wave is absent from the region near the head of the Bay where friction and reflection cause the tide to resemble a standing wave. By assuming that the tide is a purely progressive wave along the entire length of the Bay, the estimated phase speed for long waves is 4 8 0 km/day. It is also possible to estimate that the period of a longitudinal seiche within the Bay will have a fundamental value of around 2-2.5 days. Previous studies within the Chesapeake Bay system have been directed towards an understanding of the density driven internal circulation (Pritchard; 1 9 5 4 , 1 9 5 6 ) and have consisted of short duration studies during calm weather. It is only in recent years that self-recording instruments have made it possible to collect fairly long records during varying weather conditions. The aim of this paper is to review our understanding of the response of the Chesapeake system to meteorological forcing in the light of some recent observations (Elliott and Hendrix, 1 9 7 6 ; Elliott, 1 9 7 7 ; Wang and Elliott, 1 9 7 7 ) and to present an overall picture of the mechanisms involved. YEAR-LONG OBSERVATIONS IN THE POTOMAC ESTUARY For a one-year period, from July 1 9 7 4 through July 1 9 7 5 , current measurements were made at three depths in the Potomac estuary (Elliott, 1 9 7 7 ) . Meteorological data were also collected and the complete data set was filtered and then further averaged with 2 4 hour non-overlapping blocks to produce a sequence of daily mean values. (A 25 hour box-car filter was used to remove the tidal signals; while not ideal this filter was considered to be sufficient in view of the large nontidal components and it had the further advantage of not causing an excessive l o s s of data at the ends of the

130

MODE 1

c

// / / / / / / / / / / /

/

/// / ////////

Fig. 2 Schematic representation of the two dominant modes observed in the response of the Potomac.

131

records.) The resulting low-pass signals showed marked current fluctuations with periods of 2-5 days and longer. The fluctuations -1 had r.m.s. values of around 4-6 cm.s , exceeding the long term mean values by a factor of about 2. When averaged over the year-long period the mean flow was directed seaward in the surface layers and landward at mid-depth and near the bottom, i.e. consistent with the circulation usually associated with a partially-mixed estuary.

This

'classical' circulation was observed for 43% of the time and had a mean duration of 2.5 days for each occurrence. However, a landward directed flow at all three levels (storage) and the reverse of the classical estuarine circulation (landward flow at the surface, outflow near the bottom) were the second most common flow patterns, each occurring for about 20% of the time and having mean durations of around 1.5 days. The data were first analyzed by methods which did not consider the frequency dependence of the response.

In particular, a multiple

regression analysis was made but it was found that only 50%-60% of the fluctuations in each of the current records could be related to local meteorological forcing. The poor results from the regression were in fact caused by the presence of two distinct modes of response, one of which was due to non-local forcing. The two modes were separated using EOF analysis in the time domain (Wallace and Dickinson, 1972) and the results are presented schematically in Fig. 2. The first mode was associated with the local wind forcing and contained 47% of the total velocity fluctuations. A downstream wind blew water out of the estuary causing a reduction in the mean water level and setting up a surface slope towards the mouth. The surface flow was directed seaward, while a return flow took place at middepth and near the bottom. This mode was reversible, i.e. an upstream wind stress could cause an increase in the mean water level with the mean surface sloping upwards toward the head. The surface flow would then be directed landward while the deeper water would flow seawards, i.e. the reverse of the circulation usually associated with a partially-mixed estuary. The second mode, which accounted for approximately 30% of the

total velocity fluctuations, was not related to the local wind (less than 1% of the wind fluctuations could be related to Mode 2, in contrast 80% of the wind fluctuations were connected with Mode 1). In addition, Mode 2 was not related to the local surface slope but was characterized by the rise and fall of the mean surface and by current fluctuations at mid-depth and near the surface.

Landward net

132

flow was associated with a rise in the mean elevation and, conversely, seaward net flow was accompanied by a reduction in the mean water level. The near-bottom flow was not influenced by this response, but for the upper portion of the water column the second mode was as significant as the local forcing (Mode 1). The results showed that, for the Potomac, a knowledge of the local wind field is not sufficient for predicting the net currents since only about 50% of the fluctuations can be explained in terms of local forcing. This is in contrast to the results obtained by Weisberg (1976) in Narragansett Bay, and suggests that the second mode (far-field response), which contaminated the effects of local forcing, was due to the interaction between the Potomac and the Chesapeake Bay. The above modal analysis was independent of frequency, i.e. the two modes were extracted from the observations but no time scales were assigned to them. Fig. 3 shows current, elevation and wind stress during the period of April 15 to May 1 8 , 1975. This particular month-long period contained two distinct regimes. During April 15 to 26, the local wind was relatively strong, having a peak value of -2 around 1 dyn-cm . The bottom current (U40) and the surface elevation (E3) appeared to be coherent with the downstream wind stress ( T ~ ) yet, ; while the current fluctuations seemed to be coherent at all three depths, there was the suggestion of an upward phase propagation. Visual inspection does not therefore suggest a simple Mode 1 type response although the surface current responded in part (and was in phase with) local wind. The dominant time scale appeared to be around 4-5 days. The second part of the record, May 4 to 18, was characterized by significantly weaker local winds. The currents and surface fluctuations, however, were comparable in magnitude to those observed earlier. The coherence between bottom current (U40) and elevation ( E j ) was fair, and there appeared to be no time lag in the response at the three depths. The mid-depth and bottom currents were in phase during this period while the surface flow appeared to be 180° out of phase. The fluctuations had a time scale of 2-3 days and part of the response seemed to be due to the local wind; however, the relatively large amplitudes suggest that the non-local contribution was significant at these higher frequencies.

133

-30'

30 I 15 "25

-I 5 -30

e.

,

-

-

,

.

) 1

A

.

^.

~

'

30 I

-30'

30 E3

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-30 --

DAYS

I

I

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15 APRIL

MAY

1975

Fig. 3 Example of the Potomac data (April 15 to May 18, 1975); -1 non-tidal velocities are in cm.s , E j is the non-tidal elevation at Lewisetta in cm, T~ and T~ are the down-stream and cross-stream components of wind stress in dyn.cm

-2

.

134

FREQUENCY DEPENDENT ANALYSIS To investigate the frequency dependence of the response and to study the coupling between the Bay and the Potomac, Wang and Elliott (1977) used spectral techniques and also EOF andlysis in the frequency domain. Data collected during the first two months of the year-long study were used (mid-July to mid-September, 1974), which includes wind, surface elevation and bottom current in the Potomac, plus a series of 3-day intensive current measurements (Elliott and Hendrix, 1976). In addition, the following discussion also includes surface elevation, slope and wind stress over the Chesapeake Bay (Fig. 1). For this analysis the data series were filtered using a Lanczos filter to remove the tidal and higher frequencies; the filter had a half-amplitude point of 34 hours and passed 95% of the energy at 50 hours. Wind Stress over the Bay As part of the year-long study, Elliott (1977) compared the wind stress at two locations near the Potomac estuary. One wind station (Patuxent River Naval Station) was located near the mouth of the Potomac, while the other (Quantico Marine Base) was situated about 100 km upstream from the mouth. The wind data from these two stations were found to be coherent, although the magnitude of the wind was higher at the more exposed Patuxent site. In contrast to the similarity between these two stations, wind data near the Atlantic coast at Norfolk (Fig. 1) was coherent with the wind at Patuxent River only for periods longer than five days. The poor coherence between the wind fluctuations at these two locations for shorter periods was probably a result of sea-breeze and other local effects which would be confined to the coastal region. Fig. 4 shows the spectra of the N-S and E-W components of wind stress at Norfolk and Patuxent River. At Norfolk the principal wind direction was N-S and the wind stress was significantly stronger than at the Patuxent site. The spectra for both locations displayed distinct peaks at periods of 20, 5 and 2.5 days. While the principal axis of the wind at Norfolk was in the N-S direction, the wind at Patuxent was oriented more in a NE-SW direction. However, at the 2.5 day period the wind energy was concentrated in the N-S direction at both locations.

Response of the Chesapeake Bay to Winds Non-tidal sea surface fluctuations can be the result of both atmospheric pressure variations and also the response to surface wind

135

stress. In the present study, however, it was found that the coherence between atmospheric pressure and mean sea level was low, suggesting that the wind fluctuations were the major driving force. The sea level spectra for four locations in the Chesapeake Bay are shown in Fig. 5. The marked similarity to the shape of the wind spectra (Fig. 4) confirms that the surface fluctuations were induced by the wind field. The 20 day fluctuations at the Bay mouth were the result of an Ekman flux in the adjacent coastal water, i.e., sea level increased at the Bay mouth when the coastal wind was blowing to the south, and decreased when the coastal winds blew to the north. These fluctuations were then damped in amplitude as they propagated up the Bay away from the mouth (Fig. 5). The 5-day fluctuations near the Bay mouth were driven by both the N - S and the E-W winds. The amplitude at this frequency increased slightly in the upper Bay (Fig. 5), which suggests local forcing. An increase of sea level in the upper Bay was associated with winds blowing to the west, a decrease with winds blowing to the east. This suggests that Ekman effects within the Bay itself were influencing the elevations in the upper Bay. The 2.5 day sea level fluctuations had higher amplitudes near the head of the Bay and they decreased seawards. The fluctuations were generated within the Bay by the local N-S winds. The presence of seiche oscillations is suggested by the increase in amplitude

towards the head of the Bay.

In addition, the period of oscillation,

2.5 days, is of the correct order for a longitudinal seiche within

the main portion of the Bay. Interaction between the Potomac and the Chesapeake Bay This section now returns to the discussion of the response of the Potomac and considers the frequency dependence of the response in the light of the results presented in the previous sections.

Fig. 6

shows the spectra and coherence functions calculated from the 2-month long records of surface elevation, surface slope and near-bottom current in the Potomac estuary. All three spectra had distinct peaks at periods of 20, 5 and 2.5 days. However, whereas the slope and elevation spectra were similar in shape to those obtained for wind and elevation within the Bay, the bottom current showed a relatively higher amplitude at the 2.5 day period.

Sea level and surface slopes

were significantly coherent at 20 and 2.5 days and, to a lesser extent, at 5 days.

In contrast sea level and bottom current were

coherent at 5 and 2.5 days - but not at 20 days.

136

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(days)

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S p e c t r a of t h e N-S a n d E-W Fig. 4 P a t u x e n t R i v e r and N o r f o l k .

0.4

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(cpd )

components of wind stress a t

137

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Fig. 5 Sea level spectra at four locations along the longitudinal axis of Chesapeake Bay.

138

The analysis of the year-long records had shown that the surface elevation was mainly associated with a Mode 2 response (exchange of surface water between the Potomac and Bay). In addition, we know that the 20 and 5 day sea level fluctuations were mainly generated at the mouth of Chesapeake Bay.

Therefore the fluctuations

which were observed in the Potomac appear to have been generated in the following manner: 1. The 20 day fluctuations were caused by the exchange of surface water with the Bay, i-e., they were basically a Mode 2 response. These fluctuations were the direct result of sea level variations at the ocean boundary caused by Ekman effects in the coastal ocean. The fluctuations were damped as they progressed up the Chesapeake Bay but they were still of sufficient amplitude when they reached the mouth of the Potomac to cause sea-level and surface current fluctuations within the estuary. They did not appear to influence the Potomac near-bottom currents. 2. The 5 day fluctuations in the Potomac are also believed to have originated, in part, at the mouth of the Bay and to have been caused by co-oscillation with the Bay (i.e., Mode 2). However, the increase of sea level in the upper Bay, and the significant correlation between bottom current and surface slope at this period (Fig. 6) suggest that local forcing was also important (i.e. Mode 1).

Thus,

the 5 day fluctuations consisted of both the Mode 1 and Mode 2 response. 3.

The 2.5 day fluctuations appeared to be due to the

seiche oscillations within the upper Bay. At this frequency the bottom current was coherent with the local surface slope, which suggests local forcing. On the other hand, the bottom current was also coherent with the surface elevation, which suggests that exchange between the two estuaries occurred over the whole water column. Thus, the 2.5 day motion in the Potomac had characteristics different from the Mode 1 or Mode 2 response. The time-domain EOF analysis fails to reveal the 2.5 day type response, presumably because the wind and sea level variations were dominated by the longer period fluctuations (Figs. 4 and 5).

INTERNAL ADJUSTMENT:

A FURTHER DRIVING MECHANISM

During September of 1974 an attempt wzs made to repeat a James River type of analysis (Pritchard; 1954, 1956) by using modern equipment to collect data in the Potomac estuary. Two cross-sections of the Potomac were occupied continually during two 5 day periods and

139

Period ( d a y s )

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F i g . 6 S p e c t r a and c o h e r e n c e f u n c t i o n s f o r near-bottom sea l e v e l and s u r f a c e s l o p e w i t h i n t h e Potomac.

current,

SEPT S E P -T 5

6

-

I

Fig. 7 Depth-time contour plot of the non-tidal current (cm.s ) in the Potomac estuary, September 5-6, 1974. (The time axis has its origin at 0000 hrs on September 3.)

-4

c

0

+J

Continuation of Fig. 7 during September 10-13, 1974.

V

Fig. 8

141

142

hourly samples were taken across each transect using vertically profiling current meters and CTD sensors (Elliott and Hendrix, 1976). During the analysis of the data, however, it was discovered that the circulation was more complex than the steady two-layered flow investigated 25 years earlier in the James. Figs. 7-8 show the vertical structure of the low-pass currents which were obse.rved at one of the transects. (The current data were first resolved into the longitudinal direction and then filtered with a 25 hour box-car filter.) At the beginning of September 5 (hour 49) the net circulation was in the correct sense for a partially mixed estuary. There was a seaward flow of 6-8 cm.s-’ near the surface, and a reverse landward flow of 16 crn.s-l near mid-depth. However, from hour 50 onward the bottom flow reversed its direction and started to flow seaward, reaching a maximum strength of 10-12 cm.s-’ by hour 85. There appears

to have been an upward phase propagation through the water column which resulted in the surface layers also reversing their direction of net flow. By hour 55 the surface water had started to flow landward and during hours 55-90 it had a negative velocity of 2-4 cm.s-*. During most of the first measurement period the residual flow was in the reverse sense to that usually associated with a partially mixed estuary. There was no sampling on September 7 and 8, coupled with the effects of the filter this caused a three day gap in the residual current records which continue in Fig. 8. By the end of September 9 (hour 168) there was a positive seaward flow throughout most of the water column. However, a three-layer flow then developed in which there was a landward flow near mid-depth. (This three-layered flow is in the reverse sense to the three-layered flow which can occur in tributary embayments, e.g. Baltimore Harbour, when a water of inter-

.

mediate density flows seaward at mid-depth) The three-layered flow intensified and reached a peak early on September 12 when the landward flow at mid-depth had a magnitude of 6-8 cm.s-’. Eventually, toward the end of the second period of observation, the flow reverted to the usual estuarine pattern. The time scale for the transient flow, from the initial reversal to the final recovery of the estuarine circulation, was of the order of 10 days. In general, the residual flow was highly coherent at the two transects and the profiling data agreed well with the results obtained from moored current meters. The intensive observations were deliberately made at locations which were near the long-term current meter mooring and the observations suggest a further mechanism which can cause current fluctuations in an estuary:

Prior to the start of the intensive measurements there had been a period of moderate down-stream winds in the Potomac (about 5 m.s-’). These winds were also present during the first 24 hours of the measurements and local forcing is thought to have caused the strong estuarine flow observed at the beginning of September 5 (this was a Mode 1 response). Following this wind stress, however, the horizontal salinity gradients were observed to be in the reverse sense to those usually associated with an estuary, i.e. the salinity decreased towards the mouth. This suggests that the reverse circulation may have been driven by the perturbed density field. The preliminary results from a numerical model which can include the effects of a wind stress (Elliott, 1976) have suggested that the salinity reversal was caused by the combined effects of surface water being blown out of the estuary and by higher salinity water, which would be advected into the estuary by a compensating return flow, up-welling within the estuary. Following the relaxation of the wind, the internal density would tend to drive the net currents in the opposite sense to that usually associated with a partially-mixed estuary. Observational evidence that the salinity gradient in an estuary can be reversed following a period of down-stream winds has been reported for other estuaries by Elliott (1976) and Weisberg (1976). Therefore the intensive observations are considered to have been taken during a period when the estuary was responding to adjustments in the density field: this adjustment had a time scale of around 10 days.

DISCUSSION The recent observational results, which have been summarized here, show that non-local forcing in the Chesapeake Bay due to the influence of the coastal ocean is an important mechanism which influences the distribution of non-tidal currents and elevations within the Chesapeake estuarine system. Previous work had concentrated on the analysis of the gravitationally driven internal circulation, based on data which had been collected for relatively short periods of time during calm weather. In contrast, the recent measurements show that the mean velocities associated with the meteorological forcing can be an order of magnitude larger than those associated with the gravitational circulation. Furthermore, the important time scales are now known to extend to at least 20 days which is significantly longer than was previously thought. to be the case. The intensive observations and preliminary results from a numerical study also suggest that the internal adjustment of a

144

perturbed density distribution, which has a time scale of the order of 10 days, may be an important mechanism influencing the estuarine circulation. The following comments can therefore be made on the need for future studies: 1. Is the Chesapeake Bay system typical? We have presented results which we believe to be valid for the Chesapeake Bay but it needs to be resolved whether these mechanisms are also important in other types of estuaries. 2. Numerical models need to include Ekman effects at the open coastal boundary. Effort should also be directed towards modelling the fluctuations in the non-tidal flow; this requires models that are efficient enough to be run repeatedly for simulations of around 100 tidal cycles in duration without excessive computer costs. The models should also be able to include the interactions between branching tributaries. (Three-dimensional models are capable of satisfying the interaction requirement, but are unlikely to be economically feasible when run for long simulations.) 3. The recent observations have shown that field studies of limited durations (say 5-10 days or less) in the Chesapeake Bay are unlikely to lead to a true understanding of estuarine dynamics and that much longer records are required to resolve the forcing mechanisms. It will be necessary to collect long records in several types of estuaries before such questions will be fully answered.

ACKNOWLEDGMENTS This study was made at the Chesapeake Bay Institute of The Johns Hopkins University and was supported by the National Science Foundation under grant OCE74 - 08463. We thank Dr D.W. Pritchard for the encouragement that he gave throughout the study.

REFERENCES Elliott, A.J., 1976. A numerical model of the internal circulation in a branching tidal estuary. Chesapeake Bay Institute Special Rept. 54, Ref. 76-7, 85 pp. Elliott, A.J., 1976. Response of the Patuxent estuary to a winter storm. Chesapeake Sci., 17:212-216. Elliott, A.J., 1977. Observations of the meteorologically induced circulation in the Potomac estuary. Est. Coastal Mar. Sci., (in press).

145

Elliott, A . J . and Hendrix, T.E., 1976. Intensive observations of the circulation in the Potomac estuary. Chesapeake Bay Institute Spec. Rept. 55, Ref. 76-8, 35 pp. Pritchard, D.W., 1954. A study of the salt balance in a coastal plain estuary. Jour. Marine Res., 13:133-144. Pritchard, D.W., 1956. The dynamic structure of a coastal plain estuary. Jour. Marine Res., 1 5 : 3 3 - 4 2 . Wallace, J.M. and Dickinson, R.E., 1972. Empirical orthogonal representation of time series in the frequency domain. Part I: Theoretical considerations. J . App. Meteor., 11:887-892. Wang, D-P and Elliott, A.J., 1977. Non-tidal variability in the Chesapeake Bay and Potomac River: evidence for non-local forcing. Submitted to J . Phys. Oceanogr. Weisberg, R . H . , 1976. The nontidal flow in the Providence River of Narragansett Bay: A stochastic approach to estuarine circulation. J. Phys. Oceanogr., 6:721-734.

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147

LONG-PERIOD, ESTUARINE-SHELF EXCHANGES IN RESPONSE TO METEOROLOGICAL FORCING NED P. SMITH Johnson Science Laboratory, Harbor Branch Foundation Fort Pierce, Florida 33450

ABSTRACT Recording current meter data from two approximately one-month periods are used to investigate non-tidal exchanges between Corpus Christi Bay, Texas, and the northwestern Gulf of Mexico. The net transport in May and June, 1975, is an outflow which appears to be driven primarily by slowly falling coastal water levels. Rising coastal water levels in July, 1976, appear to be responsible for a net inflow into the bay. In each study, non-tidal current variations, occurring over time scales on the order of three to six days, are superimposed onto the very long period net transport. Related studies in the same area suggest that these estuarine-shelf exchanges are in response to meso-scale meteorological forcing. Dominant processes include the set-up and set-down of coastal water levels by cross-shelf windstress, a cross-shelf Ekman transport produced by longshore windstress, and an inverse barometer effect. INTRODUCTION The exchange of water between estuaries and the inner continental shelf occurs over a wide range of time scales and in response to a variety of astronomical, thermodynamical and meteorological forces. The relative importance of these forcing mechanisms varies greatly both in space and time. An investigation of estuarineshelf exchanges should ideally extend over a time interval sufficiently long to include meteorological events lasting many days, and, in some areas, the fortnightly or monthly tidal periodicities associated with the principal tidal constituents cycling in and out of phase. Meteorological forcing of minor importance at times of spring or tropic tides may be relatively important under neap or equatorial tidal conditions. The northwestern Gulf of Mexico is a region of characteris-

148

tically low amplitude tides and strong winds.

Zetler and Hansen

(1970) have summarized available information on tidal harmonic constants for the Gulf of Mexico. For Port Aransas, Texas, in the northwestern corner (Site A in Figure l), the tide is mixed but principally diurnal. The 0 1 and K 1 diurnal constituents both have amplitudes on the order of 15 cm. The M 2 and S 2 partial tides are approximately 8 cm and 2 cm, respectively. Tidal ranges at times of tropic tides are generally less than 0.75 m.

Corpus Christi Bay

Fig. 1 . Study Sites A and B at the coast at Port Aransas, Texas, and at the entrance to Corpus Christi Bay, respectively. Insert shows the study area in the northwestern Gulf of Mexico. On the other hand, the National Weather Service lists the International Airport at Corpus Christi, Texas, as the third windiest in the country, with a multi-annual average wind speed of 23.5 km/hr. One might expect, therefore, that meteorological forcing is relatively important in exchanging water between the inner continental shelf and the intracoastal bays that lie just inside the northwestern rim of the Gulf of Mexico. The purpose of this paper is to document long-period exchanges occurring specifically between Corpus Christi Bay, Texas,

and the inner shelf of the northwestern Gulf of Mexico, using time series of direct current measurements; and to suggest some of the meteorological forcing mechanisms responsible for these exchanges. THE OBSERVATIONS Direct current measurements were obtained using a General Oceanics Model 2010 inclinometer recording current meter. Sampling was carried out just above the bottom in about 7.5 m of water at the edge of the ship channel connecting Corpus Christi Bay with the inner continental shelf of the northwestern Gulf of Mexico (Fig. 1) The first study was over an approximately 30-day period from 13 May through 12 June, 1975, at Site A on the coast at Port Aransas, Texas. The second study covered approximately 31 days from 29 June to 3 August, 1976. Measurements were made at Site B, at the entrance to Corpus Christi Bay, approximately 19 km from the coast along the Corpus Christi Ship Channel. Currents were measured at half-hourly intervals in both studies. The long-channel components of the recorded current velocity vectors were used in the analysis. Inclination angles were read to the nearest degree. Current speeds are non-linearly related to inclination angles, but within the speed range of 10 to 40 cm/sec the precision of the long-channel component current speeds is approximately 21 cm/sec. Water level measurements were obtained from a U . S . Army Corps of Engineers water level recorder located at Site A. Water level data were read relative to a datum of one foot (30.48 cm) below mean sea level, and have a precision of approximately 2 3 mm. RESULTS Since estuarine-shelf exchanges are largely driven by variations in coastal water levels, it is appropriate to begin by computing an energy density spectrum to determine over what time scales coastal water level variations occur. Figure 2 shows the spectrum of hourly water levels recorded at Site A between 29 January, 1974, and 5 March, 1975.

The energy density calculations

utilize a fast-Fourier transform technique, after the linear trend has been removed from the data (Fee, 1969). The principal diurnal and semi-diurnal tidal constituents stand out as clearly isolated spectral peaks and are consistent with the amplitudes reported by Zetler and Hansen (1970). Also evident are the overtides and compound tides, reflecting shallow-water effects, at periodicities of approximately eight and six hours.

150

t

Fig. 2. Energy density spectrum of hourly water levels measured at Site A between 29 January, 1974, and 9 March, 1975. Vertical axis is in cm2/c.p.h. Spectral resolution is 0.00042 c.p.h. Of particular interest here, however, is the rise of com-

puted energy density levels at periodicities longer than about two days. This may reflect many thermodynamic processes, and perhaps includes some of the long-period tidal constituents. However, it is felt that the increase in energy density levels in the longperiod part of the spectrum, and thus long-period shelf-estuarine exchanges, are due primarily to meteorological forcing in various forms . Two recent investigations have been carried out to.monitor shelf-estuarine exchanges directly with recording current meter data obtained between Corpus Christi Bay and the inner continental shelf of the northwestern Gulf of Mexico. The data provide information both on the relative importance of tidal and non-tidal exchanges, and on the characteristics of non-tidal exchanges occurring over a wide range of time scales. Figure 3 shows the half-hourly long-channel current components recorded from 13 May to 12 June, 1975, at Site A on the coast. Dominating the pattern is the approximately fortnightly cycle from tropic to equatorial and back to tropic tides, governed by the position of the moon in its orbit. Less apparent is the fact that the entire pattern is shifted into the ebb portion of the plot.

This is brought out clearly, however, when the tidal

151

Fig. 3 . Half-hourly, long-channel current components from Site A , in cm/sec, 1 3 May to 12 June, 1975.

Fig. 4. Filtered long-channel current components from Site A, in cm/sec, 14 May to 11 June, 1975. period components of the current record are removed by numerically filtering the raw data.

Figure 4 shows the numerically filtered

non-tidal long-channel components past Site A.

Two points are par-

152

ticularly noteworthy in this figure.

First, as noted above, non-

tidal currents are nearly consistently in the ebb portion of the plot. Second, there are, superimposed onto the net ebb, quasiperiodic variations in the long-channel current components over time scales on the order of three to six days. Non-tidal currents are generally less than 20 cmisec, and the average current past Site A over this time interval was an outflow of 8.4 cmisec. Although the estuarine-shelf exchanges occurring over intermediate time scales on the order of three to six days are of primary concern in this paper, a word regarding the very long period exchanges, explaining the net ebb recorded during this 30-day period, may be in order. Figure 5 shows the multi-year average monthly water levels computed for Galveston by the National Ocean Survey

60n

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J F M A M J J A S O N D J Fig. 5. Multi-year mean monthly water levels, relative to a datum one foot below mean sea level, for Galveston, Texas, 1958-1973. between 1958 and 1973.

The average annual cycle for the northwest-

ern rim of the Gulf of Mexico is comprised of a semi-annual rise and fall, with high water occurring during the months of May and September, and low water during January and July. The causes of the semi-annual variations in coastal sea level have been suggested

153

by Marmer (1954), Whitaker (1971) and Sturges and Blaha (1976), and appear to include both thermohaline and dynamic processes in the northwestern Gulf of Mexico. The spring run-off is primarily responsible for the relatively high coastal water levels in May. Seasonal warming and cooling produce a maximum expansion and contraction of the water column in September and January, respectively. Sturges and Blaha have suggested that the July minimum may be related to a temporary storage of water in a large anticyclonic gyre in the western Gulf maintained by the curl of the windstress. Of particular importance here is the decrease in coastal water levels between the May high and the July low. The net ebb recorded during the 30-day study period at Site A reflects a slow draining of the intracoastal bays as water levels fall at the coast. There may be the added effect of the outflow of freshwater entering the coastal bays from rivers draining South Texas, but river inflow into coastal bays is small in this semi-arid region. The relative importance of tidal and non-tidal exchanges past Site A and over this time interval can be investigated qualitatively by plotting the cumulative net displacements (Fig. 6). This is computed just like a progressive vector diagram, however since the calculations involve the scalar long-channel components

I-

!

2 a

4

Fig. 6. Cumulative net displacement, in km, past Site A, 13 May to 12 June, 1975.

154

of the current, the cumulative displacements can be plotted against time, rather than in x-y coordinates. Figure 6 shows clearly the transition from tropic to equatorial tidal conditions, but it is apparent that the tidal exchanges constitute little more than small perturbations on the much more important long-period exchanges. Dominating the plot is the net outflow that extends through the entire study period. The second study involved half-hourly direct current measurements at Site B (Fig. l) between 29 June and 3 August, 1976. Figure 7 is a plot of the long-channel current components at the edge of Corpus Christi Bay, approximately 19 km from the coast

-s

-ac U

5c

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a w

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> JUN

I

"

"

"

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"

'

"

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Fig. 7. Half-hourly, long-channel current components from Site B , in cm/sec, 29 June to 3 August, 1976. along the ship channel.

The same basic pattern appears, with a

well-defined transition from tropic to equatorial tidal conditions. Current speed ranges are somewhat diminished, reflecting the frictional and constrictional effects of the shallow waters and channels. The numerically filtered, non-tidal estuarine-shelf exchanges are shown in Figure 8. A markedly different pattern is apparent, however, contrasting with that recorded during May and June of the previous year.

The approximately 31 days of data from the mid sum-

mer, 1976, study shows a net inflow past Site B.

During the first

155

Fig. 8. Filtered long-channel current components from Site B, in cm/sec, 30 June to 3 August, 1976. two weeks of the study there was a net flood into Corpus Christi Bay. This resumed for the final two weeks, following a period of about a week during which there was a net outflow. Intermediate time scale non-tidal variations are apparent superimposed onto the very long period exchanges. Non-tidal currents during this time interval varied between approximately i-10 cm/sec. The average flow was a flood of just under 5 cm/sec. Figure 9 shows the cumulative net displacement past Site B during the 31-day study period. Again, the tidal motions appear as relatively small perturbations, with the most important exchanges occurring over much longer time scales. The dominant inflow during this time interval may be explained largely in terms of the very long period, semi-annual variations in coastal water levels. This study was conducted during the time of, and just following the July minimum, and as water levels began rising toward the October maximum. The net inflow thus reflects a slow flooding of the bays in response to rising coastal water levels. The brief period of net outflow may reflect the fact that the time period of the study fell so close to the time of lowest water that a quasi-steady rise had not yet begun. DISCUSSION In a recent paper, Weisberg (1976) discussed the need for sufficiently long current meter records, in view of non-tidal forcing occurring over time scales well in excess of the semi-diurnal

156

E

r:

v

0-

NET EBB ___-

NET FLOOO

30 -

GO I-

W z

2 90 W

120-

Fig 9. Cumulative net displacement, in km, past Site B, 30 June to 3 August, 1976. and diurnal tidal periodicities.

The results presented here sup-

port this suggestion, but indicate that in some areas at least seasonal variations in coastal water levels may result in correspondingly long period variations in the mean flow of water between estuaries and the inner continental shelf. In the northwestern Gulf of Mexico, with relatively small freshwater inflow into coastal bays from rivers draining South Texas, the semi-annual variations in coastal water levels appear to reverse the net estuarine-shelf transport over the same time scales. In other areas, the very long period variations may just alter the rate at which estuarine waters are exported onto the shelf. Superimposed onto the semi-annual variations in estuarineshelf exchanges, but occurring over time scales well in excess of tidal periodicities, are the quasi-periodic, meteorologically-forced exchanges, characteristically at time scales on the order of three to six days. Many other investigators have noted similar variations in current meter and water level records. Weisberg (1976) filtered out the tidal contribution to a 51-day current record from the Providence River, Rhode Island, and found quasi-periodic variations occurring over time scales on the order of four to seven days.

157

Beardsley, et a l . (1977) have reported coastal water level variations and sub-surface pressure fluctuations over time scales on the order of several days. Groves (1957) documented non-tidal water level variations over intermediate time scales at many coastal and island stations in the Atlantic and Pacific, and discussed some of the most probable meteorological forcing mechanisms. Two additional studies have recently been carried out in the northwestern Gulf of Mexico to investigate variations in coastal water levels and thus estuarine-shelf exchanges. In the first (Smith, 1977), regional pressure gradients were used to infer surface windstress over shelf waters. Statistically significant coherence-squared values were computed between variations in the volume of Corpus Christi Bay and variations in both the longshore and cross-shelf windstress components. Results indicated that the cross-shelf component of the windstress produces a set-up or setdown of coastal water levels over time scales on the order of two to four days, and thus forces a slow filling or draining of the bay. One may tentatively assign at least a part of the intermediate time scale variability noted in Figures 4 and 8 to the cross-shelf component of the surface windstress. Over longer time scales, the volume of the bay is more coherent with variations in the longshore component of the windstress, suggesting coastal water levels rise and fall in response to a cross-shelf Ekman transport of shelf waters. In a second study, just being completed, as yet unpublished data suggest that significant variations in coastal water levels may be forced by spatial variations in the surface pressure field. This inverse barometer effect seems to be particularly important over time scales on the order of two to six days, and water level variations estimated to be approximately ? 5 cm are exceeded by only three astronomical tidal constituents in the northwestern Gulf of Mexico. Analysis of non-tidal current or water level data and locally measured meteorological variables suggests that the estuarineshelf exchanges occurring over time scales on the order of a few days do not occur as a local response to meteorological forcing. Coherence spectra (not shown) computed from the long-channel current components at Site B and both the longshore and cross-shelf windstress components computed from coastal wind data indicated statistically insignificant values over time scales associated with meteorological forcing.

On the other hand, estuarine-shelf

158

exchanges were found to be statistically significant when windstress values were computed from regional pressure gradients (Smith 1977). Similarly, the theoretical inverse barometer relationship of -1 cm/mb was very nearly matched when cross-Gulf atmospheric pressure differences were compared with cross-Gulf water level differences. Yet a comparison of local atmospheric pressure and water level variations measured at Port Aransas, Texas, resulted in a relationship of -0.82 cm/mb. This suggests that the estuarine-shelf exchanges observed at some point along a coast may be more a response to meso-scale meteorological forcing than a purely locally driven process. CONCLUSIONS One may conclude that where tidal processes are small, such as in the Gulf of Mexico, or in many estuarine areas sufficiently removed from the coast, meteorological forcing over time scales on the order of several days may play a significant role in estuarineshelf exchanges. This is especially true in estuaries having little inflow of fresh water. Meteorological forcing may occur in several forms, with windstress and perhaps inverse barometric effects dominating. Studies repeated at various times of the year indicate that regions having substantial seasonal water level variations and small inflow of fresh water may undergo long-period reversals in the net transport between the estuary and the adjacent inner continental shelf lasting over periods of many weeks. The higher coherences between estuarine-shelf exchanges and regional meteorological forcing suggest that these exchanges do not occur at a response to purely local conditions. ACKNOWLEDGMENTS Mr. James C. Evans provided valuable help in the computer analysis of the current and water level data; Dr. J. S. Holland assisted in the installation and recovery of the recording current meters. Water level data used in the study were provided by Mr. D. T. Graham of the Army Corps of Engineers in Galveston, Texas. Harbor Branch Foundation, Inc., Contribution Number 75. REFERENCES Beardsley, R., H. Mofjeld, M. Wimbush, C. Flagg and J. Vermersch, Jr. 1977. Ocean tides and weather-induced bottom pressure fluctuations in the Middle Atlantic Bight. Journ. of Geophysical Res. 82 (21): 3175-3182.

159

Fee, E.

1969. Digital computer programs for spectral analysis of time series. Univ. of Wisconsin, Milwaukee, Center for Great Lakes Research, Special Report No. 6, 17 pages. Groves, G. 1957. Day to day variation of sea level. Meteorological Monographs 2(10):32-45. Marmer, H. 1954. Tides and sea level in the Gulf of Mexico. In: Gulf of Mexico, its origin, waters and marine life. Fishery Bulletin, Fish and Wildl. Serv. U. S. 55(89):101-118. Smith, N. 1977. Meteorological and tidal exchanges between Corpus Christi Bay, Texas, and the northwestern Gulf of Mexico. Estuarine and Coastal Marine Science 5(4):511-520. Sturges, W. and J. Blaha. 1976. A western boundary current in the Gulf of Mexico. Science 192:367-369. Weisberg, R. 1976. A note on estuarine mean flow estimation. Journ. of Marine Res. 34(3) :387-394. Whitaker, R. 1971. Seasonal variations of steric and recorded sea level of the Gulf of Mexico. Texas A & M University, Ref. 71-14T, 110 pages. Zetler, B. and D. Hansen. 1970. Tides in the Gulf of Mexico--a review and proposed program. Bulletin of Marine Sci. 20(1): 57-69.

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161

SURGE-TIDE INTERACTION I N THE SOUTHERN NORTH SEA

D.

PRANDLE and J. WOLF

I n s t i t u t e of Oceanographic S c i e n c e s , B i d s t o n O b s e r v a t o r y , M e r s e y s i d e , ENGLAND.

ABSTRACT O b s e r v a t i o n s o f s t o r m s u r g e s i n t h e R i v e r Thames show t h a t s u r g e p e a k s t c n d t o o c c u r on t h e r i s i n g t i d e and seldom, i f e v e r , o c c u r on h i g h t i d e .

This

t e n d e n c y h a s been a t t r i b u t e d t o t h e i n t e r a c t i o n between t i d e and s u r g e p r o p a g a t i o n

as d e s c r i b e d by t h e n o n - l i n e a r t e r m s i n t h e a s s o c i a t e d hydrodynamic e q u a t i o n s . A r e c e n t s t u d y by P r a n d l e and Wolf

( 4 ) examined t h e mechanics o f i n t e r a c t i o n

w i t h i n t h e R i v e r Thames and showed t h a t a n i m p o r t a n t component o f i t o r i g i n a t e s o u t s i d e o f t h e r i v e r ; t h i s component i s i n v e s t i g a t e d i n t h e p r e s e n t p a p e r . A method o f i d e n t i f y i n g i n t e r a c t i o n i n t h e s o u t h e r n N o r t h Sea i s d e v e l o p e d

i n v o l v i n g t h e u s e o f two hydrodynamic n u m e r i c a l m o d e l s , one s i m u l a t i n g t i d a l p r o p a g a t i o n and t h e o t h e r s u r g e p r o p a g a t i o n .

O p e r a t i n g t h e s e models c o n c u r r e n t l y ,

t h e c o u p l i n g between t i d e and s u r g e i s i n t r o d u c e d by p e r t u r b a t i o n terms which r e p r e s e n t t h e i n f l u e n c e i n e i t h e r model o f sea l e v e l s and v e l o c i t i e s computed by the other.

T h i s approach h a s been used t o s i m u l a t e t h e p a t t e r n o f i n t e r a c t i o n

which o c c u r r e d d u r i n g t h e d i s a s t r o u s s t o r m s u r g e o f 3 0 J a n u a r y t o 2 F e b r u a r y

1953.

I t i s shown t h a t i n t e r a c t i o n i n t h e s o u t h e r n N o r t h S e a r e s u l t s p r i m a r i l y

from the q u a d r a t i c f r i c t i o n term, d e v e l o p i n g s i g n i f i c a n t l y i n t h e c o a s t a l r e g i o n o f f LowestoEt as f a r s o u t h as t h e Thames e s t u a r y due t o t h e h i g h v e l o c i t i e s a s s o c i a t e d w i t h b o t h t i d e and s u r g e p r o p a g a t i o n i n t h a t area.

Changes i n t h e

s u r f a c e e l e v a t i o n o f t i d e and s u r g e due t o t h e e f f e c t s o f i n t e r a c t i o n may d e v e l o p r a p i d l y i n c e r t a i n l o c a l i s e d r e g i o n s s u c h a s t h e Thames e s t u a r y .

There

may a l s o b e l o n g e r p e r i o d c h a n g e s o f t h e o r d e r o f t h e d u r a t i o n o f t h e storm due t o a s y s t e m a t i c d i s p l a c e m e n t o f t h e M2 t i d a l regime.

1.

INTRODUCTION The o b s e r v e d t e n d e n c y o f s u r g e p e a k s t o o c c u r on t h e r i s i n g t i d e i n t h e

Thames h a s long i n t r i g u e d r e s e a r c h e r s i n t e r e s t e d i n t i d e and s u r g e phenomena.

I t i s a l s o o f m a j o r p r a c t i c a l i m p o r t a n c e s i n c e t h e e m p i r i c a l f o r m u l a e t h a t have been d e r i v e d t o p r e d i c t c o a s t a l f l o o d l e v e l s a r i s i n g from s t o r m s u r g e s must t a k e account o f t h i s t e n d e n c y .

Hence s u c h q u e s t i o n s a r i s e as t o w h e t h e r it i s

p o s s i b l e f o r s u r g e p e a k s t o o c c u r a t h i g h t i d e a n d , i f s o , u n d e r what c o n d i t i o n s . P r a n d l e and Wolf

( 4 ) have i n v e s t i g a t e d t h e dynamics o f t i d e - s u r g e i n t e r a c t i o n

162

OoE

1cig.1

I"E

3%

4'E

S c h e m a t i c representation o f the s o u t h e r n N o r t h Sea.

5OE

~n t h e R i v e r Thames and have t h e r e b y a t t e m p t e d t o p r o v i d e some e x p l a n a t i o n f o r t h e obsprved d i s t r i b u t i o n o f s u r g e s .

By a n a l y s i n g s t a t i s t i c a l l y t h e s o u t h w a r d

p r o p a g a t i o n o f a number o*f d i s c r e t e s u r g e p e a k s a l o n g t h e e a s t c o a s t o f B r i t a i n t h r y showed t h a t t h e p e a k s t e n d t o o c c u r on t h e r i s i n g t i d e i n t h e Thames

irrespective o f t h e p h a s e r e l a t i o n s h i p between t i d e and s u r g e i n t h e n o r t h e r n North Sea.

Using n u m e r i c a l models t o s i m u l a t e t i d e and s u r g e p r o p a g a t i o n a l o n g

t h e r i v e r from a s e a w a r d boundary a t Walton t h e o b s e r v e d i n c r e a s e i n i n t e r a c t i o n e f f e c t s alonq t h e r i v e r w a s reproduced.

However t h e predominance o f s u r g e p e a k s

o c c u r r i n c j on t h e r i s i n g t i d e w a s shown t o b e d u e , i n p a r t , t o i n t e r a c t i o n e f f f c t s which modify s u r g e p r o f i l e s p r i o r t o t h e i r a r r i v a l a t t h e mouth o f t h e Thames.

I n t h e i r s t a t i s t i c a l a n a l y s i s o f o b s e r v e d s u r g e s , P r a n d l e and Wolf

showed t h a t t h i s i n t e r a c t i o n , o c c u r r i n g o u t s i d e o f t h e r i v e r , s i g n i € i c a n t l y between L o w e s t o f t and Walton;

developed

it i s t h i s f e a t u r e o f i n t e r a c t i o n

t h a t i s of primary i n t e r e s t i n t h e present paper. The p a p e r c o n s i d e r s t h e p r o p a g a t i o n o f t i d e and s u r g e i n t h e s o u t h e r n North Sra w i t h t h e o b j e c t i v e o f e x a m i n i n g t h e m e c h a n i c s o f t h e i r i n t e r r e l a t i o n ship i n t h a t region.

The s t u d y i s l i m i t e d t o t h e s i m u l a t i o n o f a p a r t i c u l a r

s u r g e e v e n t , namely t h e m a j o r s u r g e o f 3 1 J a n u a r y t o 2 F e b r u a r y

(3)

1977;

Prandle

madr a c o m p r e h e n s i v e s t u d y o f t h i s s u r g e w i t h t h e a i d o f a n u m e r i c a l model.

The p r e s e n t a p p r o a c h employs two v e r s i o n s of t h i s same model o p e r a t e d c o n c u r r e n t l y and r e f e r r e d t o s u b s e q u e n t l y a s p a r a l l e l models. p r o p a g a t i o n and t h e o t h e r s u r g e p r o p a g a t i o n .

One model s i m u l a t e s t i d a l

N o n - l i n e a r i n t e r a c t i o n between

t i d e and s u r g e i s i n t r o d u c e d by p e r t u r b a t i o n terms which r e p r e s e n t t h e i n f l u e n c e o f t h e s u r g e on t h e t i d a l p r o p a g a t i o n a n d , l i k e w i s e , t h e i n f l u e n c e o f t h e t i d e on t h e s u r g c p r o p a g a t i o n .

I n t h i s way, it i s p o s s i b l e t o a c c o u n t f o r t h e

i n t e r a c t i o n between t i d e and s u r g e w h i l e r e t a i n i n g t h e s e p a r a t e i d e n t i t i e s o f

th? two plienornena.

It i s t h e n p o s s i b l e , f o r i n s t a n c e , t o d e t e r m i n e t h e way i n

which t h e M 2 t i d a l amphidromic s y s t e m i n t h e s o u t h e r n N o r t h S e a c h a n g e s d u r i n g t h i , c o u r s e of t h e s t o r m .

It i s a l s o p o s s i b l e t o v a r i o u s l y i n c l u d e o r omit

c e r t a i n o f t h e p e r t u r b a t i o n t e r m s so t h a t t h e i n t e r a c t i o n due t o a p a r t i c u l a r tcrm c a n be e x p l i c i t l y i d e n t i f i e d . The a p p l i c a t i o n o f t h e s e p a r a l l e l m o d e l s h a s d e m o n s t r a t e d t h a t t h e m a j o r s o u r c e o f i n t e r a c t i o n a r i s e s from t h e q u a d r a t i c f r i c t i o n t e r m .

I n consequence,

t h c a r c a s where t h i s i n t e r a c t i o n i s most e f f e c t i v e were i d e n t i f i e d by e x a m i n i n g t h r s p a t i a l d i s t r i b u t i o n of t h e v e l o c i t y f i e l d f o r b o t h t i d e and s u r g e p r o p a g a t i o n .

Throughout t h i s t e x t t h c word i n t e r a c t i o n i s u s e d t o d e n o t e b o t h a n a c t u a l p h y s i c a l p r o c e s s and i t s r e s u l t a n t e f f e c t o n s u r f a c e e l e v a t i o n ,

’

whcre t h e l a t t e r i s d e f i n e d as t h e d i f f e r e n c e i n s u r f a c e e l e v a t i o n between t h e v a l u e computcd from a s i m u l a t i o n o f t i d e p l u s s u r g e combined,

zc ,

of t h e s e p a r a t e components computed from s i m u l a t i o n s of t i d e a l o n e ,

surge a l o n e ,

Zs

;

i.e.

ZI = Z c

-

ZT - Z s

and t h e sum

ZT

,

and

SURGE HEIGHT (CMS)

140

-

120

-

100

-

80

-

I

4

--

20(/ 0

-100

Lerwick

-

Wlck

-

Aberdeen

N. Shlelds

1

-

Recorded s u r g e s t a t i s t i c s ;

I

ILowestoft I

0

-0

Walton

5% 20vo

Southend Tllbury Tower Pier

I

I

I

1

I

I

0.25%

I

-120-

Fig.2

0.25%

l i n e s c o n n e c t v a l u e s of

Z p as computed a t e a c h l o c a t i o n f o r P 2

= 0.25,

1,

5 a n d 20%.

165 2.

STATISTICS OF OBSERVED SURGES An analysis was made of observed surges at tlie following ports:

Lerwick,

Wick, Aberdeen, North Shields, Lowestoft, Walton, Southend, Tilbury and Tower Pier.

The locations of these ports are shown in figure 1 and extend from the

northern North Sea southwards along the east coast of Britain and thence into the Thames as far as Tower Pier. The observed data comprised 5 years of hourly recordings at each location for the years

1969 to 1973.

At each port, hourly residual heights were

calculated as follows:

Rt = Ot where R t

and

-

Pt - M

(1)

is the residual elevation, or surge (at time t 1;

Ot

the recorded elevation;

Pt

the predicted astronomical tide;

M

the annual mean of O t , calculated for each year separately. At each location, the surge data were analysed to compute probability

distributions expressed in terms of percentage exceedances of a particular surge The percentage of surges exceeding a value Z p

level. by

pz= n / N x 100, where

n is the number of surge values exceeding Z

a total number of hourly surge values period).

was denoted by PZ given

N

out of P (approximately 44,000 for the 5 year

The analysis was performed for positive and negative surges separately.

The results are shown in figure 2 in the form of the values of Zp corresponding to p z

=

0.25,

1,

5 and 2096 respectively for the various ports along the path of

propagation of the surge.

The horizontal scale in figure 2 represents the

distance between the ports.

The values of Zp for

pz = 0.25% and Pz

=

1%are

representative of peak surge levels at each location as might be observed during the propagation of a moderate to large storm surge. these two values of

pz,

The variations in Zp

,

for

show a steady and regular increase in amplitude between

Lerwick and Lowestoft and thereafter remain reasonably constant between Lowestoft and Tower Pier. The above analysis was then repeated but, rather than analyse the complete data set as a whole, the data was first separated into distinct subsets according to the timing of any particular observation relative to tidal high water. subset was then analysed separately as before.

Each

Figure 3 shows the mean surge

level for each port at four tidal phases namely; (a) rising tide, 3Q to 24 hours before high tide (HT); (b) high tide, HT -8h to HT +Qh; (c) falling tide, HT +2$n to HT +39h and (d) low tide, HT -6ih to HT -54h.

The divergence of the four

curves is a measure of the degree of interaction at each location.

The larger

values indicated by the curves for surges on the rising tide clearly illustrate the increase in interaction as surges propagate southwards.

The figure

166 MEAN SURGE HEIGHT

30-

I

(CYS)

28-

I

I

24

-

I

RISING T I D E

I I I

22 -

10

P , , , ",

I I I

26 -

/'

\,/' /---d

I I

-

I

8-

I I

6-

I

4-

I I I

I

I

2Lerrlck

Wlck

Aberdem

N. Shlelds

~LOl..l0fl

WO!IO"

-2 -4 -

-6

-

-8

-

-10

-

I I

I I

I I I

I I I

I

I I I I

-

0 t i o t i ~ o n l o l scale

Fig.3

'

100 200x)o

km

I I

I

I

Recorded mean s u r g e levels at f o u r tidal phases.

Southend

Tllbry

Tower Pler

i n d i c a t e s t h a t i n t e r a c t i o n can be d e t e c t e d as f a r northwards as Wick and it t h e n i n c r e a s e s c o n t i n u o u s l y as f a r as Tower P i e r .

An e x c e p t i o n t o t h i s c o n t i n u o u s

i n c r e a s e i s t h e s m a l l d e c r e a s e between North S h i e l d s and Lowestoft;

t h i s may

p o s s i b l y be a t t r i b u t e d t o d i s c o n t i n u i t i e s i n t h e t i d a l regimes between t h e c e n t r a l and s o u t h e r n North Sea and a l s o t h e d i s c o n t i n u i t y i n t h e c o a s t l i n e i n However, t h e p r e s e n t f e a t u r e of i n t e r e s t i s t h e

t h e r e g i o n of t h e Wash.

s i g n i f i c a n t i n c r e a s e i n i n t e r a c t i o n which o c c u r s between Lowestoft and Walton. The pronounced e f f e c t o f i n t e r a c t i o n clearly illustrated i n figure

4

a t p o r t s s o u t h of Lowestoft i s

which shows t i m e - s e r i e s o f observed surge and

p r e d i c t e d t i d e ( t h e l a t t e r t o one q u a r t e r of t h e v e r t i c a l s c a l e used f o r t h e s u r g e ) a t Lowestoft and Southend d u r i n g 1970.

The t i m e - s e r i e s shown were

considered t o be r e p r e s e n t a t i v e of c o n d i t i o n s a t t h e s e two p o r t s . of i n t e r e s t a r e :

The f e a t u r e s

( a ) a t Lowestoft, t h e r e i s an a p p a r e n t l y random d i s t r i b u t i o n

o f t h e timing of s u r g e peaks r e l a t i v e t o t h e phase of t h e t i d e , whereas ( b ) a t Southend, t h e surge peaks almost always o c c u r on t h e r i s i n g t i d e and never on high t i d e .

Hence f i g u r e

4

emphasises t h e e f f e c t s of i n t e r a c t i o n i n t h i s r e g i o n

a s p r e v i o u s l y e n u n c i a t e d from t h e s t a t i s t i c a l a n a l y s i s of recorded s u r g e s . Prandle and Wolf

(41

examined i n t e r a c t i o n i n t h e Thames using t h e

p a r a l l e l model approach ( $ 3 ) study.

;

figure

5

shows some of t h e r e s u l t s from t h i s

S t a r t i n g from a p r e s c r i b e d t i d e and s u r g e i n p u t a t t h e mouth of t h e

model ( W a l t o n - M a r g a t e ) t h e e f f e c t s of i n t e r a c t i o n a t Tower P i e r a r e i l l u s t r a t e d by t h e m o d i f i c a t i o n of t h e t i d e due t o t h e i n f l u e n c e of t h e s u r g e and, l i k e w i s e , t h e m o d i f i c a t i o n of t h e s u r g e by t h e i n f l u e n c e of t h e t i d e .

The f i g u r e

i l l u s t r a t e s t h a t t h e most s i g n i f i c a n t e f f e c t of i n t e r a c t i o n i s t h e r e d u c t i o n of t h e surge peaks through t h e i n f l u e n c e of t h e t i d e , i t was a l s o shown t h a t t h i s r e d u c t i o n of t h e s u r g e peaks w a s due t o t h e q u a d r a t i c f r i c t i o n term. examination of f i g u r e

5

shows t h a t t h e i n t e r a c t i o n s

s'- s

and

TI-T

r i v e r , a r e n o t r e s p o n s i b l e f o r t h e peak of t h e n e t r e s i d u a l , S'+T'-T on t h e r i s i n g t i d e .

within t h e

,

occurring

T h i s phenomenon must t h e r e f o r e be a f u n c t i o n of t h e s u r g e -

t i d e phase r e l a t i o n s h i p a t t h e mouth. e v i d e n t l y non-random

,

An

S i n c e t h i s phase r e l a t i o n s h i p i s

it may t h e n be concluded t h a t t h e timing of s u r g e peaks i n

t h e Thames i s dependent on i n t e r a c t i o n o c c u r r i n g seawards o f t h e l i n e between Walton and Margate.

3.

PARALLEL MODELS The b a s i s of t h e p r e s e n t modelling approach c o n s i s t s o f a combined model

of t h e s o u t h e r n North Sea and R i v e r Thames developed by P r a n d l e ( 2 ) m o d r l comprises two p a r t s dynamically i n t e r f a c e d ,

.

This

a one dimensional r e p r e s e n -

t a t i o n o f t h e River Thames t o g e t h e r w i t h a two dimensional r e p r e s e n t a t i o n of t h a t p a r t of t h e North Sea s o u t h of l a t i t u d e 53O20' extending westwards i n t o t h e E n g l i s h Channel a s f a r as t h e Greenwich meridian.

The model u s e s a n

168 SURGE lrn HEIGHT LOWESTOFT

I m TIDAL HElGHl

SOUTHEND

LOWESTOFT

SOUTHENO

LOWESTOFT

SOUTHENO

t

I

LOWESTOFT

SOUTHENO

291

1

292

1

293

I

294

I

295

I

296

1

297

,

298

1

299

I

3W

LOWESTOF T

SOUTHEND

Fig.4

T i m e - s e r i e s of p r e d i c t e d t i d e and r e c o r d e d s u r g e a t Lowestoft and Southend d u r i n g 1970.

metres.

(a) Mouth of the model : Walton-Margate. Fig.5

(b) Tower Pier.

P r o p a g a t i o n of t i d e a n d s u r q e i n t h e ‘l’hames computed hy parallel models (F’randle a n d WolI surge alone S t i d r m o d i f i r d by i n t e r a c t i o n T‘ , s u r g e modified by i n t e r a c t i n n S ’

.

.

{ 4 ) );

tide alone

T

I

170 explicit finite difference scheme for solving, by means of a forward timestepping procedure, the relevant equations of motion.

The schematic represen-

tation of the southern North Sea is shown in figure 1. Prandle (3) showed that this numerical model was able to accurately simulate the propagation of tide and surge throughout this region.

The present

objective is to gain an understanding of the properties of interaction by simulating tide and surge separately while introducing interaction between the two phenomena in the form of perturbations as already mentioned.

The applica-

tion of the parallel model approach to a one dimensional representation of the

,

Thames has been described by Prandle and Wolf ( 4 )

hence only the application

to the two dimensional representation of the southern North Sea will be described here. The relevant hydrodynamic equations may be expressed for space coordinates along lines of latitude and longitude :

% at

where

+

ax H

7 +

a

-

u v + H g - az + K U ( U * + V ~ ) ' / ~ - RV

ay H

ax

x , y are orthogonal axes positive to the east and to the north, t

time,

g

qravitational constant,

Z

elevation of the water surface above a horizontal datum,

D

depth of the bed below the same datum

u,v velocities (depth-averaged) in the respectively ,

and

= 0 ,

K

friction coefficient,

R

Coriolis parameter,

H = D+Z

, U

=

uH

,

V

=

X

and

y directions

vH.

Numerical simulation of tide and surge propagation in the southern North Sea (Prandle (3))

has indicated that interaction is largely insensitive to the

inclusion or omission of the convective terms (second and third terms in cquations ( 2 ) and ( 3 ) ) and hence, on this basis, it is justifiable, and also convenient, to omit these terms from further consideration. T h e concept of the parallel model approach requires that, for all

171 locations and at all times, the elevation and transports in the tidal model, Z

us

,

UT

and

andVT

vs

respectively, together with the corresponding values

zs ,

in the surge model must satisfy the relationships :

zC

= z T + z

uc

= UT

+

us

(6)

vc

= VT

+

vs

(7)

S

(5)

where the subscript C denotes values computed in a combined simulation of tide and surge.

Inserting ( 5 ) , ( 6 ) and

(7)into equations

(Z),

( 3 ) and ( 4 ) produces

the following equations for the combined propagation of tide and surge -a( U T + U S ) + ( D + Z S + Z T ) g - ( Z S + Za

at

a

at("+ZS)+-(U

a ax

T

+ u s )+-(Vaa y

)')'I2

) + K ( u s + u T ) ( ( u S + u T ) 2 +(vs+v

ax

~

T

T

:

T

+v ) = 0

s

I n the friction term in ( 8 ) and ( 9 ) it is assumed that the relationships

( 6 ) and (7)for transports in the

X

and

Y directions respectively also apply

for the associated depth-averaged velocities.

It may be shown that this

approximation is justifiable for small values o f

z/n .

The open-sea boundary conditions employed in the simulation of the propagation of tide, surge and tide plus surge involve the specification of the sea surface elevation

Z B ( t ) at every boundary grid square

B

.

Hence the boundary conditions to be satisfied in the parallel models are:

Z

B,C

= zB , T + zB , S

(11)

The assumption made in the use of parallel models is that equations ( 8 ) ,

( 9 ) and (10) may be separated into two parts as f o l l o w s :

aa t V T + ( D + Z + Z T S -

172

with boundary conditions

,?j'

=

'B,S

In operating the tidal model with ( 1 2 1 , (13) and (14), the surge parameters

zs,

Us

and

Vs

wliich appear in equations ( 1 2 ) and (13) are

cxvaluatcd from the simultaneous operation of the surge model, while in operating the surge model with (151, (16) and (17) the tidal parameters

ZT, U T and vT which appear in (15) and ( 1 6 ) are obtained from the concurrently-running tidal modcl.

Usinq this parallel model technique to simulate various surge events,

it was shown that the results from the separate simulations of tide and surge could be combined to give values in close agreement with results obtained from the simulation of tide and surge combined, thus satisfying conditions (51, ( 6 ) and ( 7 ) as required.

Hence, the additional terms underlined in equations

12) 1

(13), ( 1 5 ) an? ( 1 6 ) may be considered to represent the interaction between tide arid surge.

The magnitude of the shallow water interaction terms are

proportional to a product of surge amplitude and tidal amplitude.

However

the

magnitude of the interaction associated with the frictional terms is a more complex function involving products of the surge amplitude and the tidal amplitude with the respective powers of these amplitudes varying according to both the, instantaneous ratio of the tide and surge velocities and also their diffr,rence in direction. The interaction terms in ( 1 2 ) and (13) involving the value of

zs

are

subsequently referred to as the shallow water terms while the other interaction terms in (12) and (13) involving U s friction terms;

and

vs

are referred to as the quadratic

similar descriptions are used to refer t o the corresponding

terms in (15) and (16).

173

4.

STORM SURGE OF 3 l JANUARY

-

2 FEBRUARY 1953

The l o s s of l i f e and damage caused by t h e storm surge of 3 1 January

-

2

February 1953 r e p r e s e n t s one of t h e worst n a t u r a l d i s a s t e r s experienced i n r e c e n t h i s t o r y along t h e s h o r e s of B r i t a i n and t h e North Sea c o a s t of C o n t i n e n t a l Europe. The propagation of t h i s s u r g e i n t h e s o u t h e r n North Sea was s i m u l a t e d using t h e p a r a l l e l model t e c h n i q u e o u t l i n e d i n

3.

T i d e and s u r g e e l e v a t i o n s along t h e

o p e n - s e a boundaries of t h e models were p r e s c r i b e d from t h e d a t a used by Prandle

(3)

i n an e a r l i e r comprehensive examination of t h i s surge event.

In a d d i t i o n ,

t h e e f f e c t of l o c a l wind f o r c i n g w a s i n c o r p o r a t e d by adding f u r t h e r terms t o e q u a t i o n s (15) and ( 1 6 ) ; t h e form of t h e s e wind f o r c i n g t e r m s and t h e corresponding wind speed d a t a were a l s o e x t r a c t e d from t h e e a r l i e r s t u d y of Prandle. One l i m i t i n g f a c t o r p r e s e n t throughout t h e following d i s c u s s i o n of t h e r e s u l t s of t h i s s i m u l a t i o n i s t h e i m p l i c i t assumption t h a t t h e r e i s no i n t e r a c t i o n between t h e t i d e and s u r g e a t t h e open-sea b o u n d a r i e s of t h e model. While t h e a n a l y s i s o f recorded s u r g e d a t a d e s c r i b e d i n

5

2 showed t h a t some

i n t e r a c t i o n does o c c u r o u t s i d e of t h e model r e g i o n , i t i s c o n s i d e r e d t h a t t h e e s s e n t i a l f e a t u r e s of t h e r e s u l t s o b t a i n e d w i l l remain v a l i d . Computer r u n s were c a r r i e d o u t f o r ( 1 ) model of t i d e a l o n e ( T ) , (2)

model of s u r g e a l o n e ( S ) ,

(3)

model o f t i d e w i t h i n t e r a c t i o n from model ( 4 ) , below, due t o shallow w a t e r and q u a d r a t i c f r i c t i o n (TI),

( 4 ) model o f s u r g e w i t h i n t e r a c t i o n from model ( 3 ) , above, due t o shallow w a t e r and q u a d r a t i c f r i c t i o n ( S T ) ,

( 5 ) model o f t i d e p l u s surge combined (T+S 1. The purpose o f o p e r a t i n g model models ( 3 ) and

( 5 ) w a s t o confirm t h a t t h e v a l u e s from

( 4 ) s a t i s f i e d t h e r e q u i r e m e n t s o f ( 5 ) , ( 6 ) and ( 7 ) and hence

t h e r e s u l t s from t h i s model w i l l n o t be d i s c u s s e d f u r t h e r . ( a ) I n t e r a c t i o n a t Lowestoft and Southend

E l e v a t i o n s a t Lowestoft and Southend computed from t h e s e f o u r models a r e shown i n f i g u r e s 6 and

7

r e s p e c t i v e l y , v e r t i c a l l i n e s drawn on t h e s e

f i g u r e s i n d i c a t e t h e t i m e of h i g h t i d e a t t h e p a r t i c u l a r l o c a t i o n a s computed by model ( 1 ) .

The m o d i f i c a t i o n of t h e s u r g e through t h e i n f l u e n c e of t h e t i d e

i s shown b o t h by t h e divergence of t h e c u r v e s f o r S separate curve f o r

ST-s

,

and

s'

and a l s o by t h e

s i m i l a r l y t h e m o d i f i c a t i o n of t h e t i d e through t h e

i n f l u e n c e of t h e s u r g e i s shown by t h e d i v e r g e n c e of t h e c u r v e s f o r T and by t h e curve o f T'-T

.

and T'

The average magnitude of t h e i n t e r a c t i o n a t

Lowestoft i s about h a l f t h a t a t Southend, i . e .

approximately i n t h e r a t i o of

t h e magnitude of t h e r e s p e c t i v e t i d a l r a n g e s as suggested by t h e form of t h e i n t e r a c t i o n terms

( 5 3).

The t i m e - s e r i e s f o r

S'-Sat

Lowestoft shows l i t t l e

c o r r e l a t i o n w i t h t h e t i d a l phase whereas t h e corresponding t i m e - s e r i e s a t

174

J

O F 1 &i

'\

g ',I

'

-Im.

0.5m.

s'- S

& - _..-,--T'-T

G

w

an 1953

953

r

Fig.6

Computed values of tide, surge and interaction at Lowestoft.

175

3m. 2m.

I m. 0 -I m.

t

Im. 0 -1m.

0.5m.

&

0

-0.5m. 0.5m 0 -0-5m. -Im

Fig.7

$4 + s\- s +T-'

T

1953

-4 I

1st. Feb 1.953

.

2nd. Feb 953

Computed v a l u e s o f t i d e , s u r g e a n d i n t e r a c t i o n a t Southend.

176 Southend i s c l e a r l y i n f l u e n c e d by t h e t i d a l s t a g e .

The c u r v e s f o r T '

and T

at

both Lowestoft and Southend show t h a t t h e i n f l u e n c e of t h e s u r g e t e n d s t o reduce mean water l e v e l s f o r about

24 hours from 12.00 31 January t o 12.00 1 February.

This i s followed over t h e n e s t 24 hours by an a p p r e c i a b l e i n c r e a s e i n t h e t i d a l range a t Southend and a s i g n i f i c a n t phase d e l a y a t Lowestoft; an examination of t h e m o d i f i c a t i o n of t h e t i d a l regime throughout t h e a r e a of t h e model i s described i n

5 4(e).

The r e d u c t i o n i n mean t i d a l l e v e l a t b o t h l o c a t i o n s may be

a t t r i b u t e d t o t h e n e g a t i v e s u r g e l e v e l s preceding t h i s e f f e c t , t h e long d u r a t i o n of t h e s e changes t o t h e t i d a l regime a c c o r d s w i t h t h e s u g g e s t i o n made l a t e r (

§ k ( e ) ) t h a t a time c o n s t a n t of t h e o r d e r of a day o r more may be involved i n

t h e response of t h e t i d a l regime t o t h e i n f l u e n c e of t h e surge.

S'-S

andT'-T

The c u r v e s f o r

show, a t b o t h Lowestoft and Southend, a tendency t o c o u n t e r a c t

The n e t i n t e r a c t i o n c u r v e a t Southend, S ' - S + T ' - T , s h o w s

each o t h e r .

some

c o r r e l a t i o n with t i d a l phase, i n p a r t i c u l a r i t i l l u s t r a t e s t h e tendency f o r i n t e r a c t i o n t o i n c r e a s e s u r g e l e v e l s on t h e r i s i n g t i d e .

However, t h e n e t

i n t e r a c t i o n curve a t Lowestoft shows no obvious c o r r e l a t i o n w i t h t i d a l phase. ( b ) Components of i n t e r a c t i o n The model r u n s d e s c r i b e d i n t h e p r e v i o u s s u b s e c t i o n were r e p e a t e d but with t h e i n t e r a c t i o n between models

(3) and (4)l i m i t e d

f r i c t i o n only and (B) shallow w a t e r only.

t o ( A ) quadratic

The r e s u l t i n g i n t e r a c t i o n f o r t h e

( A ) and (B) and a l s o f o r t h e complete i n t e r a c t i o n examined i n t h e

c a s e s of

p r e v i o u s s u b s e c t i o n a r e shown i n f i g u r e 8 f o r f o u r l o c a t i o n s .

These r e s u l t s

show t h a t , a t a l l f o u r l o c a t i o n s , t h e m o d i f i c a t i o n of t h e s u r g e through t h e i n f l u e n c e o f t h e t i d e i n d i c a t e d by t h e c u r v e s f o r

s'-s

i s of t h e same o r d e r

a s t h e m o d i f i c a t i o n of t h e t i d e by t h e s u r g e as i n d i c a t e d by t h e c u r v e s f o r

TI-T

.

The tendency f o r t h e s e two i n t e r a c t i o n s t o oppose one a n o t h e r i s a l s o

e v i d e n t again.

The s e p a r a t e c u r v e s f o r i n t e r a c t i o n e f f e c t s of q u a d r a t i c

f r i c t i o n and shallow w a t e r r e s p e c t i v e l y show t h a t t h e q u a d r a t i c f r i c t i o n term

i s dominant throughout t h e a r e a and a c c o u n t s f o r almost a l l of t h e t o t a l interaction.

The i n t e r a c t i o n e f f e c t due t o shallow w a t e r of t h e t i d e on t h e

propagation of t h e s u r g e i s almost n e g l i g i b l e whereas t h e corresponding e f f e c t of t h e surge on t h e t i d a l p r o p a g a t i o n i s of some consequence p a r t i c u l a r l y a t Southend.

The t i m e - s e r i e s

f o r t h i s l a t t e r e f f e c t a t Southend i s h i g h l y

c o r r e l a t e d w i t h t h e t i d a l phase and c o n t r i b u t e s towards t h e c h a r a c t e r i s t i c i n c r e a s e i n surge l e v e l s on t h e r i s i n g t i d e .

The l a c k of any obvious s p a t i a l

coherence between t h e r e s u l t s a t d i f f e r e n t l o c a t i o n s i s d i s c u s s e d f u r t h e r i n t h e following s u b s e c t i o n . ( c ) S p a t i a l d i s t r i b u t i o n of i n t e r a c t i o n The s p a t i a l d i s t r i b u t i o n , a t 06.00

s' ,

t h e modified t i d e ,

T'

,

1 February 1953, of t h e modified s u r g e ,

t i d e p l u s s u r g e combined, ( T + S ) ( o r

(TI) +

(s'))

0.25rn.

s'-s

s'- s

T'-T

T\-T

s;s'

s'-s

T ~ T

T ~ T

LOWESTOFT

DOVER

HT.

s'- s

s'- s

T\-T

T\-T

s-' s

-

-

T ~ T 12,oo

24105

12,oo

24 00

HT.

s-' s T ~ T OSTEND 12100 24100

1st. Feb 1953.

Fig.8

HT.

Components o f i n t e r a c t i o n a t L o w e s t o f t , S o u t h e n d , Dover and O s t e n d . f r i c t i o n o n l y , --------- s h a l l o w w a t e r o n l y .

-

12100 1st. Feb 1953.

complete i n t e r a c t i o n ,

24 00

-q u a d r a t i c

178 t o g e t h e r with a l l o f t h e v a r i o u s components of i n t e r a c t i o n a r e shown i n f i g u r e s

9 and 10.

The v a l u e s shown f o r t h i s s p e c i f i c time may be regarded as

r e p r e s e n t a t i v e of v a l u e s o c c u r r i n g during a l a r g e s u r g e e v e n t .

The v a l u e s o f

i n t e r a c t i o n due t o q u a d r a t i c f r i c t i o n a r e shown t o be everywhere i n c l o s e aqreement with t h e v a l u e s f o r complete i n t e r a c t i o n .

The v a l u e s of i n t e r a c t i o n

due t o shallow water a r e shown t o be much s m a l l e r i n r e l a t i o n t o t h e i n t e r a c t i o n due t o q u a d r a t i c f r i c t i o n .

However, t h e shallow w a t e r i n t e r a c t i o n of t h e s u r g e

on t i d a l propagation i s s i g n i f i c a n t i n t h e r e g i o n of t h e Thames e s t u a r y and eastwards along a s e c t i o n from Ostend t o t h e mouth of t h e Rhine. a l s o c o i n c i d e s w i t h t h e maximum t o t a l i n t e r a c t i o n ;

This region

t h i s may be a t t r i b u t e d t o

13).

t h e l a r g e magnitude o f both t i d e and s u r g e e l e v a t i o n s i n t h i s r e g i o n (

The s p a t i a l d i s t r i b u t i o n of t h e v a r i o u s components of i n t e r a c t i o n appear An a t t e m p t w a s made t o f o l l o w t h e development through

t o be r a t h e r complex.

time of t h e s e s p a t i a l d i s t r i b u t i o n s , however an examination of s u c c e s s i v e d i s t r i b u t i o n s of t h e type shown i n f i g u r e 9 and 10 f o r v a l u e s a t h o u r l y i n t e r v a l s d i d not r e v e a l any c l e a r t r e n d s i n t h e changing d i s t r i b u t i o n s of interaction.

I t was concluded t h a t i n t e r a c t i o n i n t h i s r e g i o n does not develop

i n a slow and g r a d u a l f a s h i o n i n e i t h e r space o r time b u t , on t h e c o n t r a r y , develops r a p i d l y and o f t e n i n a l o c a l i s e d sense. ( d ) S p a t i a l d i s t r i b u t i o n of v e l o c i t y

S i n c e i t h a s been shown t h a t q u a d r a t i c f r i c t i o n i s t h e major cause of i n t e r a c t i o n i n t h e s o u t h e r n North S e a , it f o l l o w s t h a t t h e a r e a s where i n t e r a c t i o n develops most s t r o n g l y w i l l be t h o s e where b o t h t i d e and s u r g e velocities are largest.

F i g u r e 11 shows t h e s p a t i a l d i s t r i b u t i o n of mean

absolute v e l o c i t i e s f o r ( A ) t i d e alone, plus surge, ( T + s )

,

T

,

( B ) surge alone, S

,

and ( C ) t i d e

where t h e s e mean v a l u e s were o b t a i n e d by averaging v a l u e s

over a semi-diurnal p e r i o d from 14.30 1 February t o 03.00 2 February.

The

d i s t r i b u t i o n of t i d a l v e l o c i t i e s i n t h i s region i s f a i r l y well e s t a b l i s h e d and t h e r e s u l t s f o r f i g u r e l l ( a ) a r e i n good agreement w i t h t h e d i s t r i b u t i o n of maximum v a l u e s shown i n t h e a t l a s p u b l i s h e d by Seehydrographischer D i e n s t ,

1975 ( 5 )

Rostock

.

The d i s t r i b u t i o n f o r s u r g e a l o n e shows high v e l o c i t i e s

along t h e e a s t c o a s t of B r i t a i n extending i n t o t h e Dover S t r a i t and E n g l i s h Channel.

S i m i l a r l y t h e d i s t r i b u t i o n f o r t i d e p l u s s u r g e confirms t h i s

c o n c e n t r a t i o n of high v e l o c i t i e s .

Hence,

s i n c e almost a l l l a r g e s u r g e s which

occur i n t h e Thames e s t u a r y c o n t a i n a s i g n i f i c a n t component o r i g i n a t i n g i n t h e n o r t h e r n North Sea, t h i s component w i l l e x p e r i e n c e c o n s i d e r a b l e i n t e r a c t i o n as

i t p r o p a g a t e s i n t h e c o a s t a l r e g i o n o f f Lowestoft as f a r s o u t h as t h e Thames estuary

.

179

f S'

I50 100 5 0

. ..

> ..... "

Fig.9

T\-T

Computed t i d e , s u r g e , t i d e p l u s s u r g e a n d c o m p o n e n t s o f i n t e r a c t i o n ; i n s t a n t a n e o u s v a l u e s a t 06.00 1 F e b r u a r y 1953 in c m .

80

I-

i

........ 0 .....

PIO

s-s

........

s-s

r

T-T

I ( a ) Quadratic Friction Only

Fig.10

(b) Shallow Water Only

Computed components o f i n t e r a c t i o n due t o ( a ) q u a d r a t i c f r i c t i o n o n l y and ( b ) s h a l l o w w a t e r o n l y ; i n s t a n t a n e o u s v a l u e s a t 06.00 1 F e b r u a r y 1953 i n c m .

181

60

I

1

TIDE ALONE ( T I

SURGE ALONE ( S ) 2

TIDE PLUS SURGE (T+S)

Fig.11

D i s t r i b u t i o n o f mean v e l o c i t i e s computed f o r t i d e a l o n e , s u r g e a l o n e and t i d e p l u s s u r g e ; v a l u e s s h o w n a r e i n cms-1 averaged o v e r t h e p e r i o d 14.30 1 F e b r u a r y t o 03.00 2 F e b r u a r y 1953.

182 (el V a r i a t i o n i n t h e M2 d i s t r i b u t i o n o v e r t h e p e r i o d o f t h e s t o r m The s i m u l a t i o n s d e s c r i b e d i n

$ & ( a )w e r e r e p e a t e d w i t h t h e d i f f e r e n c e

t h a t t h e t i d a l d i s t r i b u t i o n s p e c i f i e d a t t h e open b o u n d a r i e s i n ( A ) model (1)

of t i d e a l o n e ; model

(5)

(C) in

o f t i d e p l u s s u r g e combined, w a s r e s t r i c t e d t o t h e s i n g l e M

2 I n t h e o r i g i n a l case t h e t i d e w a s composed o f t h e c o n s t i t u e n t s

constituent. f o r M2,

( 3 ) o f t i d e w i t h t h e i n f l u e n c e o f s u r g e and

( B ) model

S,,

K2, N2,

simulation of t h e

01,

K

1

and M k ( P r a n d l e

(2)

).

The r e s u l t s from t h i s

'53 s u r g e and t h e M2 t i d e showed t h a t t h e v a l u e s f o r

i n t e r a c t i o n o f ( A ) t i d e on s u r g e p r o p a g a t i o n ,

( B ) s u r g e o n t i d e p r o p a g a t i o n and

( C ) t h e c o m b i n a t i o n o f ( A ) and ( B ) w e r e , i n a l l c a s e s , a l m o s t i d e n t i c a l t o t h e v a l u e s o b t a i n e d f o r t h e s i m u l a t i o n o f t h e same s u r g e w i t h t h e more c o m p l e t e T h i s i n d i c a t e s t h a t t h e major surge t i d e i n t e r a c t i o n i n t h i s

t i d a l regime.

r e g i o n i s between t h e s u r g e and t h e M2 by Banks ( 1 )

t i d e , c o n f i r m i n g a s i m i l a r r e s u l t found

.

A p a r t i c u l a r advantage of r e s t r i c t i n g t h e simulation t o surge p l u s t h e t i d e o n l y i s t h a t it a l l o w s t h e d i s t r i b u t i o n o f t h e c o - p h a s e and c o - r a n g e 2 l i n e s a s s o c i a t e d w i t h t h e M2 t i d e t o b e e a s i l y d e t e r m i n e d a t a n y s t a g e d u r i n g

M

t h e surge event.

F i g u r e 1 2 shows s u c c e s s i v e d i s t r i b u t i o n s o f t h e M2 t i d e ,

e a c h r l ~ t e r m i n e dfrom a F o u r i e r a n a l y s i s o v e r o n e t i d a l c y c l e w i t h t h e t i m e o f t h e m i d - c y c l e as i n d i c a t e d , t h e i n t e r v a l between e a c h d i s t r i b u t i o n shown i s e q u a l t o t h e p e r i o d o f M2.

I n e a c h case, t h e l a t e s t d i s t r i b u t i o n i s s u p e r -

imposed o v e r t h e p r e v i o u s d i s t r i b u t i o n i n o r d e r t o i l l u s t r a t e r e l a t i v e d i s p l a c e m e n t s between s u c c e s s i v e t i d a l d i s t r i b u t i o n s .

The amphidromic s y s t e m

i s shown t o b e i n i t i a l l y d i s p l a c e d w e s t w a r d s and t h e n t o r o t a t e i n a n a n t i c l o c k w i s e s e n s e u n t i l it r e t u r n s t o t h e o r i g i n a l d i s t r i b u t i o n a f t e r approximately

3 days corresponding t o t h e d u r a t i o n o f t h e storm.

T h i s evidence

of a r e l a t i v e l y l o n g e r p e r i o d displacement of t h e t i d a l r e g i m e i s p a r t i c u l a r l y i n t e r e s t i n g s i n c e it o f f e r s t h e p o s s i b i l i t y of i n c l u d i n g a s y s t e m a t i c c o r r e c t i o n t o t h e predicted t i d e i n t h e course of a l a r g e surge event.

However, t h e

d i s p l a c e m e n t s shown by t h e p r e s e n t model a r e s e v e r e l y r e s t r i c t e d by t h e assumption o f f i x e d boundary c o n d i t i o n s .

An e q u i v a l e n t s i m u l a t i o n u s i n g a model

o f t h e whole o f t h e N o r t h S e a s h o u l d p r o v e e x t r e m e l y i n t e r e s t i n g .

5.

CONCLUSIONS

1.

An e x a m i n a t i o n o f s t o r m s u r g e s r e c o r d e d i n t h e R i v e r Thames h a s shown t h a t

s u r g e p e a k s t e n d t o o c c u r on t h e r i s i n g t i d e .

This effect is attributed t o

i n t e r a c t i o n between t i d e and s u r g e a s d e s c r i b e d by t h e n o n - l i n e a r terms i n t h e r e l e v a n t hydrodynamic e q u a t i o n s .

,

Fig.12

Variation in the M2 tidal regime over the period of the '53 s t o r m ; continuous lines show the distribution at the times stated, dashed lines show distribution one (MZ) period earlier. Co-range lines show amplitude in cm.

183

184 2.

A method of i d e n t i f y i n g t h e mechanics of i n t e r a c t i o n i n t h e s o u t h e r n North

Sea has been developed i n v o l v i n g t h e use of two numerical models, one s i m u l a t i n g t i d a l propagation and t h e o t h e r s u r g e propagation.

The two models a r e o p e r a t e d

c o n c u r r e n t l y w i t h c r o s s l i n k a g e from p e r t u r b a t i o n t e r m s which i n t r o d u c e t h e i n f l u e n c e of t h e s u r g e i n t o t h e model of t i d a l propagation.and t h e i n f l u e n c e of t h e t i d e i n t o t h e s u r g e model.

The magnitude of t h e s e i n t e r a c t i o n terms

were shown t o be a f u n c t i o n of s u r g e amplitude and t i d a l amplitude with t h e r e s p e c t i v e powers of t h e s e a m p l i t u d e s , i n p a r t , dependent on i n s t a n t a n e o u s flow conditions.

3.

T h i s modelling approach w a s used t o s i m u l a t e t h e i n t e r a c t i o n o c c u r r i n g

during t h e d i s a s t r o u s storm of

30 January t o 2 February 1953.

I t w a s shown t h a t

i n t e r a c t i o n i n t h e s o u t h e r n North Sea r e s u l t s p r i m a r i l y from t h e q u a d r a t i c f r i c t i o n term and t h a t t h e m o d i f i c a t i o n of t h e s u r g e p r o p a g a t i o n by t h e t i d e i s of a s i m i l a r o r d e r of magnitude a s t h e m o d i f i c a t i o n of t h e t i d a l propagation by t h e surge.

The i n t e r a c t i o n from shallow w a t e r terms i s g e n e r a l l y r e s t r i c t e d t o

t h e m o d i f i c a t i o n of t i d a l propagation by t h e surge and is only of s i g n i f i c a n c e i n t h e Thames e s t u a r y and t h e r e g i o n e a s t of t h e e s t u a r y between Ostend and t h e mouth of t h e Rhine.

However, t h e t i m e - s e r i e s f o r t h e shallow water i n t e r a c t i o n

i n t h e Thames shows t h a t t h i s term c o n t r i b u t e s t o t h e i n c r e a s e i n s u r g e h e i g h t s

on t h e r i s i n g t i d e i n t h e r i v e r .

k.

A n examination of t h e s p a t i a l and temporal developments o f t h e v a r i o u s

components of i n t e r a c t i o n s u g g e s t s t h a t changes i n w a t e r l e v e l due t o i n t e r a c t i o n can develop r a p i d l y i n time and may be l o c a l i s e d i n space.

5.

A s t u d y of t h e s p a t i a l d i s t r i b u t i o n s o f v e l o c i t y f o r both t i d e and s u r g e w a s

made s i n c e t h e importance of t h e q u a d r a t i c f r i c t i o n term s u g g e s t s t h a t i n t e r a c t i o n w i l l develop most e f f e c t i v e l y i n t h o s e r e g i o n s where t h e v e l o c i t i e s a s s o c i a t e d w i t h b o t h t i d e and s u r g e p r o p a g a t i o n a r e g r e a t e s t .

These s p a t i a l

d i s t r i b u t i o n s showed t h a t t h e c o a s t a l r e g i o n around Lowestoft as f a r s o u t h a s t h e Thames e s t u a r y i s an a r e a of h i g h v e l o c i t i e s f o r both t i d e and s u r g e and hence t h i s a c c o r d s w i t h t h e important o b s e r v a t i o n t h a t i n t e r a c t i o n develops r a p i d l y between Lowestoft and t h e Thames.

6.

A s i m u l a t i o n of t h e

'53 s u r g e w i t h t h e M2 t i d e only,showed t h a t almost

of t h e s u r g e - t i d e i n t e r a c t i o n may be accounted f o r by t h i s c o n s t i t u e n t .

all

This

s i m u l a t i o n a l s o enabled t h e displacement o f t h e M2 t i d a l regime by t h e surge event t o be t r a c e d .

The displacement w a s found t o c o n s i s t of an o r d e r l y a n t i -

clockwise r o t a t i o n of t h e amphidromic system w i t h an a s s o c i a t e d t i m e - c o n s t a n t of about

3 days o r , e f f e c t i v e l y , t h e t o t a l d u r a t i o n of t h e storm.

185 ACKNOWLEDGEMENTS The work described in this paper was funded by a Consortium consisting of the Natural Environment

Research Council, the Ministry of Agriculture

Fisheries and Food, and the Departments of Energy, Environment, and Industry.

REFERENCES

J. E. Banks, Phil. Trans. R. SOC. Lond., A , 275 (1974) 567-609. Prandle, Institute of Oceanographic Sciences, Bidston, Merseyside, England, 4 (1974). 3 D. Prandle, Proc. R. SOC. Lond., A, 344 (1975) 509-539. 4 D. Prandle and J. Wolf, "The Interaction of Surge and Tide in the North Sea and River Thames" (in press). 5 "Atlas der Gezeitenstrome fur die Nordsee, den Kana1 und die Irische See" Uritte, verbesserte Auflage Seehydrographischer Dienst, der Deutschen Demokratischen Republik, Rostock 1975. 1 2

D.

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187

RESIDUAL PHENOMENA IN ESTUARIES, APPLICATION TO THE GIRONDE ESTUARY R. BONNEFILLE Electricit6 de France, Chatou (France)

ABSTRACT Integration of equation of residual phenomena (velocity and salinity) in the case of an estuary with constant width and depth, shows the possibility to have some closed residual streamlines near the bottom. In the general case, integration is more complicated, but the conclusion is the same. Numerous data about the Gironde estuary are used to estimate the value of the three more important new coefficients introduced by the theory of residual

phenomena : the longitudinal and vertical mixing coefficients of salinity and the vertical mixing coefficient of momentum. BRIEF REVIEW OF RESIDUAL PHENOMENA The theory of residual phenomena (Pritchard, 1956) is based on the division of physical events in two elements. The first one i s independant of the time, there is the residual part ; it is the time-averaged value of the considered evenLs during a given period, at the minimum a tide-period. The second element depends on the time ; it represents the fluctuation of the physical event, induced by the tide,relatively their mean values. For example let us considering the velocity vector

6; and

the salinity s

-f

and V, and sm are the fluctuante components, the time-averaged values of those are n u l l . If, we introduce these fonctions in the momentum and diffusion equations ; then, i f this equations are time-averaged, as in the conventional theory of the turbulence, new terms are appearing ; they are introduced bv the products of -f

fluctuantes components Vm and .,s

This terms are modelised by using mixing

188 coefficients. The horizontal components of these coefficients represent the dispersion by tide currents-on areas the dimensions of whose are from 100 m to I0 km. Of course, the mixing coefficients are greather than the turbulent

coefficients, about 100 times more, for example from 100 to 1000 m2/s. Nevertheless the vertical mixing coefficients have about the same value of vertical coefficients of turbulence ( I to 10 cm2/s). The same method is used to define the width-averaged residual quantities + from the time-averaged residual events ; for example V and defined by the

s

relationships. + vo = 3(x,z) + +V’(x,y,z) so = s ( x , z ) + s’(x,y,z) with

on which b(x,z) i s the width of the estuary and Ox, Oy, Oz the longitudinal,

lateral and vertical axes of coordinates. Their averaging operation over the width of the estuary introduces new terms due to the fluctuations of the velocity and salinity from a side to the other side of the estuary. These new terms are also modelised by new mixing coefficients (Ronday, 1975). But by the effects of the sides of the estuary the equation transformations are more complicated, and it is necessary to do some assumptions about the values of residual quantities on the sides. A preliminary study showed that it is usefull to assume that the residual velocity is null on the side : Vo(x,*b/2,z)

=

0

It is now possible to modelise the side effects as the mixing effects, and

define the general mixing coefficients : Kdx, Kdz for the momentum, from the momentum equation, Ksx,

Ksz

for the salinity from the diffusion equation.

After elimination of negligeable terms, the equation are reduced at the following forms (Chatwin, 1976) :

- momentum equation :

-

continuity equation :

a (bw) axa (bu) +x

=

0

- diffusion equation :

189 where x

is longitudinal seawards, z vertical upwards,

g

is the gravity acceleration,

p

is the specific mass of water,

+ u a n d w are the horizontal and vertical components of the residual velocity V. The continuity equation suggests to search a stream function Y o , that is obtained easily if we do the following assumptions :

-

the depth of the estuary is constant : ho doesn't depend of x or z , the width b is constant in a cross-section : b(x,z) becomes b(x), the residual discharge

QR =

dz

is constant along the estuary,

-

the mixing coefficient of momemtum Kdz is independant of z, the residual velocity is null on the bottom, as on sides,

u(x,-ho)

-

=

w(x,-ho)

=

0

the water specific mass p depends linearly on the residual salinity :

p = p0(l+aS)

a is a constant, its value is approximatly 0,76 if the salinity is expressed in gr/liters,

-

-

the residual salinity is assumed to be constant on the vertical ; we put

s = S,(X)

where S s is the superficial width and time-averaged salinity ; this assumption consists on the first order solution of the diffusion equation. Let we us define the non-dimensional stream-function Yo(x,Z) by :

we obtain :

Yo

=

1 (22-1) (2+1)2Z 2A

1

+ - (Z2-1)2 2

with

and for u and w

w(x,Z)

=

h d - (bus) (22-1) (1+2)'2 b dx

where there are introduced :

- the vertical-averaged horizontal residual velocity

190 - the horizontal residual velocity due to the salinity gradient us(x)

aghd dSs -48Kdz dx This solution shows that the horizontal residual velocity u contains two

=

proportional parts to um and us. The first one flows seawards ;which is more important near the mouth and near the end of the estuary in the zone where the variation of salinity is small. The second part is seaward near the free surface and upward near the bottom ; it is important on the central zone of the estuary near the inflexion point of the longitudinal distribution of salinity. This opposition between the two parts of the velocity u gives to the vertical distribution of the residual velocity its specifical aspect, particularly the change of direction near the bottom in the zone called the "neutral point" (Hansen and Rattray, 1965). As Yo is function of x by the parameter A, it is easy to trace the stream lines, defined for the constant values of Y o , by the equation : 24

3+A + -z3

2

1 +3A - -z2 2

- ~y~

= 0

The figure 1 shows that the neutral point doesn't exist if A

I 1

only ; that is

happening on the mouth and the end of the estuary. On the contrary for A < 1 the direction of the residual velocitycbanges near the bottom. In this case we could have closed residual trajectories as it is shown in the figure 2, for which the stream lines are computed for a possible distribution of A ( x ) along the estuary. Using this results it is easy to integrate the diffusion equation at the second order approximation ; it appears a small variation of the salinity along the verticale

s(x,Z)

=

usho dSs [ (SZ3+15Z2-10)z2 SS(X) - __20Ksz dx

-3(2-Z2) Z2] 2%

EXTENSION TO THE ESTUARIES WITH VARIABLE DEPTH Taking in account that the depth h is function of x and y , brings a set of complications in the equations. But it is possible to use the same method : let we us define the functions q(x,y),

5(x) and the mean depth ho(x),

by the

relationship :

It is also possible to obtain results like with a constant depth ; if we introduce

the function G(x) :

191 which is used in the expression of us :

In the case of an estuary with a flat bottom, we have f3

=

0, and we find

again the same expression for u(x,Z) except that h, depends on x. The expression of the vert cal component w is more complicated by the fact of the slope of the bottom : W(X,Z)

=

4

1

is the mean elevation of the free surface and hl(x)

where h2(x)

the elevation of

the flat bottom, or ho

h2 - hi

=

APPLICATION TO THE GIRONDE ESTUARY The numerous measurements of velocities and salinities made in Gironde (Bonnefille, 1971) (17 surveys from 1965 to 1975, with sometimes simultaneous explorations on 5 verticales) allowed us trying to confirm the theories of residual phenomena and to estimate the new mixing coefficients Kdz, Ksx and Ksz. The figures3 and 4 show the mean characteristics of the estuary S,, b and ho. The figure 5 gives an example of comparisons of theoritical and measured distributions of residual velocity u(Z) and salinity s ( Z ) for differents sections of the estuary. It has been also possible to compute for each set of measurements, respectively um, us, Kdz, K,,

and finally QR.

The analysis of data shows an interesting first result. The residual depth ho(x) and the horizontal gradient of salinity dSs have about always the dx same value ; this fact increases the interest of residual phenomena, because

-

the determination of A(x) becomes very easy. The analysis of results carrier on the following conclusions. Residual discharge QR (figure 6) Theoretically, if f3 is null, QR is constant ; this assumption is not well confirmed by the results. Nevertheless a more interesting hypothese would be that the rapport QR/h is nearly non-dependant on x. In this case we have UmSs

=

Ksx

dSx

dx

expression giving easily Ksx Longitudinal mixing coefficient of salinity Ksx (figure 7) K,,

-

This coefficient could be considered as constant 4000 m2/s

192

Vertical mixing coefficient of salinity Ksz (figure 8) The distribution of Ksz along the estuary is not simple ; but this coefficient is not important because it appears only on the vertical distribution of salinity. That is a second order phenomena. We can admit K,, Ks,

-

-

3-6 cm2/s for ho 17 cm2/s for ho

-

7 m 11 m

Vertical mixing coefficient of momentum Kdz (figure 9) This coefficient increases with the size o f the flow, and mainly wi h the size of the aera of the cross-section. Kdz is very important because it is a main parameter on A(x) which determines the general form of the residual circulation. The mean value of Kd, is Kdz

-

1

to 20 cm2/s

for the cross-section areas 30000 to I00000 m2. REFERENCES Bonnefille, R., 1971. Remarques sur les gcoulements moyens 1 l'aval de la Gironde. AIRH, Paris, 4 : 229-233. Chatwin, D.C., 1976. Some Remarks on the Maintenance of the Salinity Distribution in Estuaries. Estuarine and Coastal Marine Science, 4 : 555-566. Hansen, D.V. and Rattray, M., 1965. Gravitational circulation in straits and estuaries. Journal of Marine Research, 23 : 104-122. Ronday, F.C., 1975. Etude de l'envasement et de la variation longitudinale du coefficient de dispersion dans les estuaires partiellement stratifigs. Annales des Travaux Publics de Belgique, 4 : 1-18. Pritchard, D.W., 1956. The dynamic structure of a coastal plains estuary. Journal of Marine Research, 1 5 ( 1 ) : 33-42.

193 2 0

-

a6

-1

0

F i g . 1 - Residual streamlines of a partially stratified estuary.

Z 0

- 0.5

-1

Pig.2

7 w

Example of residual streamlines in an estuary

-

0

20

Mkm

<

c 4

<

20

Fig.3

The Gironde Estuary

I

40

I

I

50

60

dS

Fig.4 Values of h o , b, -2 along the dx Gironde Estuary

$0

km

194

-1

-----

measured distribution theoritical distribution

-1 -0.2

Fig.5

0

to14 M.6 cm/rcc Horizontal residual velocity u

Example of measured and theoritical residudil salinity and velocity at km 89 the 27 june 1968

195

4 2

10'

I

9

4

I

!

i I.

2

10'

4 0

I

LO 50 60 Fig.6 Residual discharge Qr

50 60 #Okm LO Fig.7Mixing coefficient Ksx

K~~ (cm2/s)

I 100 ,

l

1

100

I 70 krn

40

40

20 20

10

10

4

4

2

1

1 T

0.

I

I

50

I

I

60

i 0.4 0 krn

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Reprinted from: Hydrodynamics of Estuaries and Fjords, edited by J.C.J. Nihoul 0 1978 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

197

ASYMMETRY AND ANOMALIES OF CIRCULATION AND VERTICAL M I X I N G I N THE BRANCHING OF A LAGOON-ESTUARY

Y. GALLARDO O.R.S.T.O.M.,

C e n t r e Ocganologique d e B r e t a g n e , Brest ( F . )

ABSTRACT

The lagoon E b r i 6 , i n I v o r y C o a s t , i s formed o f d i f f e r e n t bays and b r a n c h e s b e f o r e communicating by a n a r t i f i c i a l c a n a l w i t h t h e Gulf of Guinea. The two p r i n c i p a l l a t e r a l b r a n c h e s a r e q u i t e d i f f e r e n t : t h e w e s t e r n c h a n n e l forms a n a t u r a l p r o l o n g a t i o n of t h e c e n t r a l c h a n n e l w h i l e t h e e a s t e r n c h a n n e l b e g i n s w i t h a c o n s t r i c t i o n . Observed a t t h e e a s t e r n and w e s t e r n e n t r a n c e s o f t h e lagoon, t h e

c i r c u l a t i o n i s s t a t i s t i c a l l y d i f f e r e n t , p a r t i c u l a r l y i n t h e upper l a y e r and d u r i n g ebb-tide.

The e a s t e r n c h a n n e l shows, a t t i m e s , a n o m a l i e s o f r e s i d u a l v e l o c i t y

p r o f i l e s which d e t e r m i n e t h e r e l a t i v e asymmetry : t h e r e i s o f t e n a seaward j e t i n t h e mid l a y e r . The s t r o n g e r r e s i d u a l a n o m a l i e s a r e connected w i t h s e n s i b l e d e p a r t u r e s from t h e s e m i - d i u r n a l p e r i o d , i n v o l v i n g t h e e x i s t e n c e of b e a t s between t h e t i d e s and o t h e r s u b t i d a l f r e q u e n c i e s . A d e c r e a s e o f R i c h a r d s o n number o c c u r s d u r i n g t h e anomalous p r o f i l e s . The v e r t i c a l m i x i n g , i t s asymmetry and a n o m a l i e s could b e e x p l a i n e d by a c r i t e r i o n f o r t h e m a i n t e n a n c e o f t u r b u l e n c e , depending on t r a n s i e n t s t a g e s o f r i v e r d i s c h a r g e s and on t h e wind a t t h e s u b t i d a l f r e q u e n c i e s

IVORY

Fig.

COAST

I - LAGOON EBRIE - L o c a t i o n s o f t e m p e r a t u r e s - s a l i n i t i e s - c u r r e n t s o b s e r v a t i o n s

198 INTRODUCTION

The s t u d y of h o r i z o n t a l t r a n s p o r t and o f v e r t i c a l m i x i n g a t t h e p r i n c i p a l e a s t e r n and w e s t e r n g a t e s of t h e l a g o o n E b r i S ( f i g . 1 ) w a s i n c l u d e d i n a l a r g e r and p r e v i o u s l y begun e n v i r o n m e n t a l program a t t h e "Centre d e Recherches OcSanographiques d'Abidjan" i n Ivory Coast. Geophysical and p h y s i c a l d e s c r i p t i o n s (TASTET, 1974) of t h e whole l a g o o n have i n d i c a t e d i t s morphology, t h e monthly f r e s h w a t e r i n f l o w s , t h e t i d e s and t h e i r c u r r e n t s . Because of t h e c o m p l i c a t e d topography w i t h i t s s i l l s , c o n s t r i c t i o n s and bays and b e c a u s e o f t h e v a r i a b l e f r e s h w a t e r i n f l o w , t h e t i d e s i n t h e l a g o o n a r e much damped down and o f t e n o u t o f phase w i t h r e s p e c t t o t h e o c e a n i c t i d e s . TASTET (1974, p .

18) o b s e r v e s t h a t t h e s e p h a s e d i f f e r e n c e s may r e a c h r e s p e c t i v e l y 40

m i n u t e s and 2 . 6 h o u r s i n t h e C e n t r a l Channel, n o t f a r from t h e e n t r a n c e o f t h e e a s t e r n Channel. The seaward f l o w measured i n t h e c a n a l , d u r i n g a t i d a l c y c l e of 3 -1 , v a l u e which c o r r e s p o n d s w e l l t o t h e mean f r e s h j u n e 1966 w a s more t h a n 400 m s w a t e r i n f l o w of J u n e 1970, 1971, 1972 (TASTET, 1974, t a b . 2 , 3 , 4 ) .

I

GUINEA GOLF

7

'?Km

I

F i g . 2 - Geography and i s o b a t h s o f t h e i n n e r e s t u a r y

(from J . P .

TASTET,

1973)

199 From January to July 1976, current, salinity and temperature measurements were made, every month, at the locations B and Y of fig. 2 : until April, EKMAN current-meters, sampling five minutes, every two hours, were completed with a mooring of two AANDEFAA currentmeters recording integrated velocity, direction, temperature and depth every ten minutes. From August

to

December, the program

was modified and concentrated in the

eastern channel, in order to obtain some tangible results between the circulation, averaged over several tidal cycles, and the estimations of fresh-water inflow. EAST-WEST

ASYMMETRY

A glance at the fig. 2 indicates the complexity of the estuarine morphology ;

however, the net flow coming from the canal must reach the area of branching without sensible l o s s because of the large and deep central channel. Since both branches represent approximately equivalent areas, and under the assumption

that the water

and Vb surface remains horizontal, it could be supposed that the velocities V Y would be inversely proportional to the surface of the vertical sections Sy and Sb ; that would lead to the following relationship

v

Y

=

(S

b

/s

y

) Vb

=

0.53 Vb

Vy, Sy, Vb, Sb, are the tidal velocities and the vertical sections at the locations y and b. The observed velocities, summed over ebb or flow periods, are distributed along a principal axis expressed by

-

Y - 7 = x - x S

S

X

_ _ where x, y

are the means, s x

2

sy

the variances of the summed velocities

x and y. Fig. 3a, b, c, d, show our results at the depths 2 m and 4 m, for ebb (downstream velocities) and flow (upstream velocities) compared to the simple model described above ; the upper layer deviates eastwards during flowing tide (fig. 3 a) ; during ebb-tide there is an excess of velocities in the upper layer and a l o s s in the mid layer (fig. 3 b, d) of the western section ; the mid flow (fig. 3 c) is far from a linear partition and presents important fluctuactions in January, February and April. When tidal amplitude is the highest (April) and when the fresh water inflow is maximum (June),

the upper layer follows better the theoretical branching both

during ebb and flow. When the tidal amplitudes are the lowest (March and May), the asymmetry seems to be the greatest. Nevertheless, the strong asymmetry of February has no obvious cause. On the contrary, in the mid layer, the asymmetry

200

is great for the highest tides (April) ; however the asymmetries in January and February are obscure. Downstream 2m. depth veloclties a t Y

Upstream 2 m . depth velocitlri a t Y

100-

100-

,

ream 2 m d e p t h ' velocities a t B 'I 0

4 m . depth

0

-

/

,

I

I

I

I

I

100

I

.

Downstream 2 m veiacltlrs

.

200

4 m. depth downstream

upstream

100-

4 m. depth B 100

200

Fig. 3 . Theoretical and observed partitions of

the summed velocities of the

channels B and Y. Transverse motions, computed from AAnderaas during the period January-April, represent 20-25 Z of the axial current in the West and only 10-15 % in the East. The effect of SW wind on the transverse motion is possible in the wide western estuary. Richardson number too, computed between 2 m and 4 m depth at maximum ebb and flow are statistically different : greater in the eastern branch during ebb-tide but in the western branch during flow. On the whole, over a complete tidal cycle the Ri values are not significantly different in both branches. The mean circulation observed during ebb and flow in each layer of B should -1

correspond to a landward motion of the upper layer (about 5 cm s -1

seaward current in the mid layer (about 12 cm s

) and to a

) : the spatial asymmetry

existing from January to July, for variable tidal amplitudes, must be connected to a great anomaly of circulation. These results are confirmed and completed by the sequences of residual (i.e. averaged on a semi-diurnal tidal cycle) velocities and salinities in the eastern channel.

201 ANOMALIES OF CIRCULATION Sequences of eight and six tidal cycles, respectively in August and October, were realized in the eastern channel. Tidal amplitudes were practically constant during each cycle. Constriction of the eastern channel greater tidal currents and the eastern origin of the fresh water inflow during the second part of the year, have fixed the choice of longer observations in this channel to the detriment of western channel. Profiles of residual velocities (fig. 4 , 5 ) are variable and point out, at times, a persistent anomaly of the profile similar to those inferred above from the asymmetry : a maximum of seaward circulation at mid depth gives a jet profile.

0

2

4 6

4.d n).t

Fig. 4 - Sequence of 8 tidal cycles in August 1976 : residual velocities, deviations temperatures and salinities from their mean profile. Numbers indicate the cycles. -

oF

CYCLES

Fig. 5 -

@

tY/iO/T6

Sequence of 6 tidal cycles in October 1976.

2715-

23-

-

21 *9

-

47-

Fig. 6 - Diurnal oscillations of residual salinities during the sequence of August. Number indicate the cycles. The comparison of fig. 4 with fig. 6, showing diurnal oscillations of residual

202

salinities, indicates that strong changes of salinity occur with the appearance or disappearance of the jet profile, respectively between the cycles 1-2 and 4-5

;

moreover a landward jet, existing during the seventh cycle, disappears with a decreasing salinity. The diurnal oscillation is not obvious on the horizontal circulation of fig. 4 ; however it becomes clear on the residual, relative, vertical motion of the maximum vertical salinity gradient. The diurnal oscillation of fig. 6 is still well marked in the maximum velocities during flow. During October, we observe increasing periods between the appearances of the maximum flow velocities which coincide with the residual seaward anomalies. On the contrary, when the period decreases below that of the semi-diurnal tide, the residual anomaly is landwards in the mid layer. The frequency of maximum ebb velocities remains quasi-constant, with a slight tendency below the semi-diurnal. In short, those results indicate that oscillations existing in a frequency range lower than the diurnal, may give rise to beats, from which arise the anomalies of circulation. In fact, the diurnal tidal oscillation in August, seems to reduce the lagging and, consequently, the anomalies with respect to the strong anomalies of October. ANOMALIES OF VERTICAL MIXING Is the gradient Richardson number a good indicator

of vertical mixing or not ?

generally the small tides (March and May) generate values o f Ri frequently greater than 2, while high tides give numerous values lower than 2. However, the effects of vertical mixing for the same tides are different in the two branches : a glance at fig. 7 indicates the habitual stronger stratification in the western channel (segments Y are larger than segments B). But, on the whole, the differences between the eastern and western

Fig. 7 - Temperature

-

Ri values are not significant.

Salinity - Depth diagrams (numbers indicate months)

.

+ and = 2, 4 , 6 m depth = 0, 10, 20 m sea depth Lagoon observations averaged on a tidal cycle

Gs and GTare the standard deviations on 8 successive tidal cycles.

203

We ha1.e pointed out a large time scale variability of velocity profiles in the eastern channel, with the appearance of jet profiles. We observe a decreasing tendency of the Ri values in presence of these anomalous profiles : TABLE 1 Richardson's numbers at maximum velocities AUGUST

tidal cycle no FLOOD upper lower

EBB

upper lower

averaged Ri

1

2

3

4

0.5

1.3

5.4

7

11

0.6

0.2

0.6

12

10

1.8

0.6

58-0.7

1.8 59-5.3

5

7

6

556-2.5

8

2600-3.7

0.5

0.5

0.9

0.0

11

1.3

15

4.5

4.2

2.2

142

18

2

3

4

0.2

-

4.1

4.8 3.8 4.1

OCTOBER I

(FLOOD upper lower EBB

upper lower

averaged Ri

49-0.5

5

123d5.3-0.8

6 22

4.9

0.2

7.0

6.4

1.9

21

3.0

6.4

4.7

1.7

1.1

14

0.0

1.0

0.2

56-0.0

0.4

0.0

0.4

13

0.8

31

1.5

To the anomalies of cycles 2, 3, 4 in August and cycles 2, 4, 5, 6 in October

often correspond averaged Ri significantly lower than in the other events ; more precisely, the process of destabilization between two consecutive cycles occurs in the upper layer, as indicated by the arrows on the table 1 .

To our mind, the process of destabilization which appears at frequencies lower than the semi-diurnal tide, contributes moreover to the vertical mixing by changing the conditions of maintenance of turbulence. The ratio A z / K z may define a critical value of the Richardson number (TAYLOR, 1931, PROUDMAN, p. 101, 1953) below which turbulent energy is supplied from the mean motion. Kz and A z , the coefficients of eddy-diffusion and of eddyviscosityare computed from the residual velocity and salinity fluctuations in August and October ; fluctuations of vertical velocities are estimated from the vertical oscillations of the salinity around its mean value at the depth 3 m. We obtain two different critical values, 2.1 in August and 6.5 in October ; that result suggests a variability of the turbulence which could explain some apparent anomalies of vertical mixing and the better mixing in the eastern

-

estuary ; more precisely, the mean product w's' of vertical velocities and salinities fluctuations is the essential factor of the variability between August and October : when that product increases, the critical value decreases.

204

-

w's' increases tenfold in August with respect to October.

Lastly Kz is found larger in August than in October : their relative magnitude equals

8. The ratio Az/Kz, estimated at a smaller time-scale, from the temperature and current AANDERAA measurements every ten minutes,gives the following results : during the lowest tidal amplitude (March) its values are 4 . 6 at location B and 3.0 at location Y ; during the highest tidal amplitude (April) its values are 0.10 and 0.11.

For these very different tidal velocities (multiplicative factor of 3) the

coefficient of eddy-diffusion Kz is in the range 0.01

*

and 5 - 50 cm

s-I

-

1 cm2 s-I

(low amplitude)

(high amplitude) : the highest value is found at Y for low

amplitude, at B for high amplitude. The principal theoretical and experimental results quoted by WELANDER (1968, p. 22-26)

indicate that turbulence can be sustained when the flux RICHARDSON number,

defined as Rf

=

(Xz/Az). Ri, lies generally below the mean value 0.3. From the

local Ri observed during March, April, August and October it appears that Rf is about a few unities. Theoretically, and that is observed by comparison of March, August and October, the ratio A z / k z does not depend on the scales. The too large values o f the observed R arises, to our mind, principally from the vertical scale f of the local Ri observed : the vertical gradients should be estimated every 30 cm o n the vertical, in order to obtain realistic values of Ri and hence of R

cm/a - 2 4

-20

-16

-12

-8

-4

0

t4

f'

t8

Fig. 8 - Means of the velocity profiles in the eastern channel DISCUSSION The evolution of transports on fig. 8 is coherent with the habitual monthly fresh water inflows (TASTET, 1974) : the transport of about 600 m3

s-'

in June

corresponds well to the strong rainfalls in 1 9 7 6 . The anomalous profiles of August and October may represent transient stages of the river discharges for which the mean wind drift is opposite. The T-S diagrams of January (fig. 7) indicate more mixing than the consecutive months of the dry season : It is well known that atmospheric circulation is particular in January, with a seaward wind whichgives rise to a coastal upwelling. COLIN (personal communication, 1977) shows a significant

205 diurnal pike and an important variability around 4-6 days, for the annual wind spectrum in Abidjan. These scales correspond well to the changes of residual salinities and velocities observed in the eastern channel. WEISBERG (1976) demonstrates the effect o f the wind variability on the estuarine circulation, and the necessity of measuring numerous tidal cycles, in order to obtain the "mean" circulation.

Obviousness, the habitual S W wind has a very different effect on the residual circulation of the eastern and western channels : fig. 9 a indicates that anomalous profiles are often generated in the eastern channel, because the wind drives the circulation landward. On the contrary in the western channel, (fig. 9 b), the seaward circulation is favoured in the upper layer and, consequently, the typical estuarine circulation appears better.

Fig. 9 - a b

-

unsteadiness of the residual velocity profiles at station B. steadiness of the residual velocity profiles at station Y. (extrapolated from 1 or 2 meters depth to the surface).

CONCLUSION

We have observed a great variability of the residual circulation in a branching lagoon estuary. That variability may give rise to asymmetries between the eastern and western channels. The coefficient of eddy-diffusion Kz presents too a high range of variability which could be estimated from the dimensionless ratio Az/Kz. With

206

respect to the general theoretical and experimental .resultswhich give a flux RICHARDSON number in the range 0.1-0.5,

it appears that the gradient RICHARDSON

numbers should be observed with a vertical distance of about 30 cm. The effect of the wind direction and velocity on the asymmetry is pointed out. REFERENCES PROUDMAN, J., 1953. Dynamical Oceanography. Methuen, London, 4 0 5 pp. TASTET, J . P . , 1974. L'environnement physique du systsme lagunaire Ebrig. S6rie doc. depart. sciences de la terre. Universitg d'Abidjan, I 1 ; 2 8 pp, 58 fig., 4 cartes hors texte. TAYLOR, G.I., 1931. Effect of variation in density on the stability of superposed streams of fluid, Proc. Royal SOC. (A), 1 3 2 , London, 4 9 9 pp. WEISBERG, R.H., 1976. The nontidal flow in the Providence River of Narragansett Bay : a stochastic approach to estuarine circulation. J. Phys. Oceanogr., 6, 721-734. WELANDER, P . , 1968. Theoretical forms for the vertical exchange coefficients in a stratified fluid with application to lakes and seas. Acta R. SOC. Sci. Litt. Gothoburgensis Geophys., I , 1-26.

Reprinted from: Hydrodynamics of Estuaries and Fjords, edited by J.C.J. Nihoul 0 1978 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

207

DISPERSION BY TIDE-INDUCED RESIDUAL CURRENT VORTICES

J.T.F. ZIMMERMAN Netherlands Institute for Sea Research, Texel, The Netherlands

ABSTRACT It is shown that the nonlinear interactions of a tidal wave propagating over a complicated bottom topography produce a time-independent residual current velocity field which is a quasi-random function of space. The superposition of this Eulerian residual velocity field and the oscillatory tidal current gives rise to a dispersion process in the Lagrangian sense, the "diffusion coefficient" of which is related to the relevant parameters of both Eulerian components of the total velocity field. For diffusion time intervals small as compared with the Lagrangian velocity correlation time-scale, it appears that the gross transport of dissolved substances in a tidal channel should be described by the telegraph equation which takes into account the finiteness of the random Lagrangian particle velocities.

INTRODUCTION Longitudinal dispersion (along the channel axis) in tidal inlets can be controlled by a manifold of physical processes (Fischer, 1976). If the effective longitudinal diffusion coefficient (K) is derived empirically, for instance from the salinity distribution, it is often difficult to say which of these processes is the dominant one. This impedes the expression of the diffusion coefficient in relevant physical parameters. Wowever, for areas which are dominated by tidal currents, by first guess, Arons and Stommel (1951) expressed K in the amplitude of tidal velocity (U1) and tidal displacement (11): K = blUlll

It must be kept in mind, of course, that ( 1 ) arises primarily from dimensional reasoning and that, consequently, all of the relevant physics is buried in the factor bl. In the present paper we shall deal with a particular phenomenon which may be 3 2 -1 responsible for rather large diffusion coefficients (order 10 m S in tidal inlets, viz. the existence of a field of quasi-two dimensional (horizontal) irregularly distributed residual currents. These residual currents arise from nonlinear interactions of a tidal wave propagating over an area of complicated topography. They can be understood from the vorticity transfer between the tidal

(1)

208 current and the mean flow (Sugimoto, 1975; Zimmerman, 1977), the irregular topography acting as a catalyst. Their pronounced existence in many areas, mainly in the form of vortices, has been demonstrated from field measurements (Sugimoto, 1975; Zimmerman, 1976) and by numerical (Tee, 1976) or hydraulic (Yanagi, 1976) modeling. By their very existence residual eddies contribute to the longitudinal dispersion. Conceptually their dispersive action may be conceived in two different ways. First, the eddies produce an irregular distribution of residual current shear. It is well known that any shear, in collaboration with smaller scale (tideinduced) turbulence, enhances diffusion in the direction perpendicular to the shear direction. If regarded as such (Sugimoto, 1975) the longitudinal diffusion coefficient can be expressed in velocity (uo) and length (10) scales connected with the residual eddy velocity field:

Here, again, it is the factor of proportionality (bg) that contains the relevant physics. There is, however, another way in which a field of randomly distributed residual eddies produces dispersion on a large scale, even without the concurrent existence of smaller scale turbulence. This second effect arises from the fact that the Eulerian superposition of an oscillatory (tidal) motion on a field of residual currents, distributed randomly in space, can be transformed in a Lagrangian motion which is partly random in time. This effect is the subject of the present paper.

GENERATION OF RESIDUAL EDDIES The large scale, quasi-two dimensional, flow produced by a tidal wave propagating through a non-rotating basin of uniform depth can be considered to be vorticity-free, the longitudinal velocity being at each moment only a function of the (x)-coordinate along the channel axis. Vorticity can, however, be produced by friction along the sides of the channel as well as by depth differences of the channel bottom. We shall concentrate here on the latter effect and outline the consequent production of residual vorticity, which is illustrated in Fig. 1. A tidal wave is running through a channel of depth H, producing an oscillatory

current velocity U1 along the channel axis. If now the bottom topography contains a depth perturbation h (x,y), the tidal velocity field will contain an oscillatory velocity perturbation { u l (x,y), v1 (x,y)1 which, in general, will no longer be vorticity free. Let the associated vorticity be given by

It may then be shown (Zimmerman, 1977) that, in linear approximation, in case of considerable bottom friction (small depth), to1 obeys the equation:

209

a

T

flood

-

T T

ebb

1 r----4

'3

'li]

+

-

t

+

-

t

-

-

t

+

+

11 I

I

b

Fig. 1. a. Signs of various tidal parameters during the flood and ebb stages of the tide in a channel in which a shoal is present, shown by the dashed lines in left hand portion of the figure; the advective residual vorticity production term is outlined in the outermost column on the right. b. Residual circulation over the shoal due to the production of residual vorticity.

210

i.e. vorticity is produced by the bottom friction torque in the second term on the righthand side, k denoting the bottom friction coefficient, relating the bottom stress ?ib

(T )

b

to the depth mean velocity by:

= -kpU

(5)

1

A localised depth perturbation, in Fig. 1 assumed to be a shoal, reduces

the

tidal velocity over the shoal by an increase in bottom stress, thereby creating vorticity. The sign of the vorticity as well 3 s its longitudinal gradient reverses sign during the tidal cycle, as shown in Fig. 1. Hence, in linear approximation the tidal mean vorticity vanishes. However, residual vorticity can be produced if nonlinear advection terms are retained in the vorticity equation (Zimmerman,

1977). In that case, if U1>>u1, we get:

Averaging (6) over the tidal cycle gives:

where w o stands for the residual vorticity. As is shown in Fig. 1, the left hand side of (7) is of opposite sign on both sides of the shoal, but does not reverse sign during the tidal cycle. Hence, a cyclonic and an anticyclonic vortex are produced near the shoal (Fig. lb). Now, in the real situation of many shallow tidal areas, the perturbations in water depth have a quasi random character, thereby randomizing the residual current velocity field. The latter, therefore, can be thought of as to exist of a random distribution of eddies of different strenght and size. Assuming now an ensemble of such random residual current fields, we may construct the spatial covariance , from which a 2 %

representative velocity scale < u g > = O

7 0

and length scale:

dr



are derived.

LONGITUDINAL DISPERSION By definition the Eulerian residual current velocity field is a random function of space but time-independent. In a qualitative way it may now be shown that the Lagrangian residual velocity of a single waterparcel is a random function of time, if an oscillatory tidal current is added to the Eulerian residual current velocity field. The random structure in space of the residual current velocity field is shown in Fig. 2, together with the path of a particular waterparcel.

211 Starting at position 0 at the beginning of the flood tide, the parcel arrives at A' at slack water. If only the tide were acting it should arrive at A at that

time. Hence the displacement A A' is due to the residual current velocity field. The same way of reasoning applies during its backward (ebb) motion from A' to B'. Here BB' is the displacement due to the residual current velocity field. Thus, during one tidal cycle the particle experiences a residual displacement OB'. The procedure may be repeated for all tidal cycles here after. Because of the random

Fig. 2. Path of a particle during a tidal cycle shown by a solid line. The residual current velocity field is represented by dashed streamlines. The tidal motion proper, during the flood and ebb stages of the tide each, is shown by the thick straight dashed lines.

212

character of the residual current velocity it will be obvious that successive displacements AA', BB',

._....form

a random vector series, i.e. the particle

experiences a "tidal random walk". Of course, neighbour vectors may be correlated, the magnitude of the correlation depending on the ratio of the r.m.s. step length of the tidal random walk and the integral length scale (lo) of the residual current velocity field. Considering now the problem of longitudinal particle dispersion as a random walk problem, we may use Taylor's (1921) classical theory of turbulent diffusion to express the effective longitudinal diffusion coefficient in the r.m.s. 2

longitudinal step length, , and the correlation coefficient of neighbour steps, c, provided that the successive displacements constitute a first-order Markov process. If now y(") denotes the displacement due to the residual current velocity field during the n'th step and

E

is the time-interval of each step, assumed to

be half a tidal period here, then the mean-square Lagrangian residual velocity 2

during each step is defined by 2

< v > = 7 2

(9)

E

whereas the correlation coefficient between the n'th and the m'th step is given by (n) (m+m)>

>

Then, in the limit E+O,

assuming lim

to approach a finite value ( ' r ) , the I-c mean-square particle displacement after time interval t reads: O'E

from which an effective longitudinal diffusion coefficient can be defined by:

For t>>T, K attains the well-known form: 2

K =

T

2

up to here K is expressed in Lagrangian quantities and T. Since, in

general, our information about the residual current velocity field is of Eulerian character we have to express the former quantities in the characteristics of the Eulerian velocity field. The latter is now supposed to consist of a homogeneous and normal random distribution in space of characteristic eddies represented by a stream function of Gaussian form (Zimmerman, 1976) upon which is superimposed an oscillatory tidal current. The Eulerian velocity field is then represented by the following set of parameters: tidal m.s. velocity length scale

residual

ratio tidal/res.

"1

2 QO>

U

.l1

10

A

2

213 2 %

By using the assumption U1>> in a perturbation procedure for the EulerLagrange transformation (Zimmerman, 1976) it can be shown that the effective longitudinal diffusion coefficient may ultimately be expressed by : K = b

(U,X)

(14)

Ulll

where b ( u , X ) is a complicated function of the energy density spectrum of the residbal current velocity field, weighted by functions which depend on U and A. Note that (15) resembles ( 1 ) . However, here b

(U,X)

is not an empirical factor Of

proportionality but is theoretically related to the characteristics of the components of the Eulerian current velocity field. An equivalent expression, similar to (21, for K is: 2 4

K

=

c (U,h)



(15)

10

Here again c (u,A) is a complicated function of u and A. Formulas ( 1 5 ) and (16) show that K can neither be described by either only the tidal parameters (U1, 11) or only the parameters (u0,lo) of the residual current velocity field. Although

dimensionally the products of both sets of parameters produce a diffusion coefficient, it is shown by the dependence of the factors b and c on u and A, that it is the interaction of both field which does give rise to a dispersion process.

AN APPLICATION OF THE TELEGRAPH-EQUATION The equation (11) for the second moment of the particle position corresponds to a transport equation of the form (Monin and Yaglom, 1971; Corrsin, 1974):

where c is the cross-sectional mean concentration of the transported dissolvent This equation is the Telegraph equation which has a "wavelike" character for t<>-r.The deviation from a "normal" diffusion equation for small t, due to the first term on the left hand side of ( 1 6 ) , is a consequence of the finiteness of the particle velocity at all times. Since, in tidal areas,

7

may be of the order of several tidal

periods, the consequences of "diffusion with a finite velocity" in such areas should be briefly mentioned here. In Fig. 3 we have shown the spread of a substance initially confined to a narrow band in the longitudinal direction, but homogeneously distributed over the channel cross-section. Assuming now that the residual current velocity field in the channel consists of a random distribution of residual eddies, the crosssectional mean concentration of the substance develops according to (16). In Fig. 3 the cross-sectional mean concentration after one tidal period ( T ) is shown on the assumption that T<
214

a

b

tt

Fig. 3. a. The splitting of a band of dissolved substance in a tidal channel by the existence of a pair of large residual eddies. Both the initial (shaded) distribution and the distribution after one tidal cycle are shown. b. The distribution of cross-sectional mean concentration of the initial band (shaded) and after one tidal cycle (solid line) in the presence of residual eddies, the largest ones of which are shown in a. The dashed line shows the distribution of the cross-sectional mean concentration if, from the outset, the band-width of the substance cloud should have been large as compared with the scale of the residual eddies. velocity is not taken into account. In this simplified example, the occurrence of two concentration maxima after a small time interval has a simple explanation. As the longitudinal dimension of the substance cloud is initially small as com-

pared with the length-scale of the residual current vortices, the main effect of the large eddies is the splitting of the substance cloud in parts which are transported in opposite directions relative to their initial position (Fig. 3a). In the meantime, these parts are diffused to some degree by smaller eddies. For the cross-sectional mean concentration the result after one tidal period is a smoothed distribution with two maxima (Fig. 3b), the positions of which are

?i

located approximately at a distance of ? T from the initial position of the substance band. It might be that the complex patterns of tracer distribution in

tidal areas with frequent occurrence of cloud splitting, as described for instance by Talbot and Talbot ( 1 9 7 4 ) , should be interpreted in this way.

A NUMERICAL EXAMPLE

To give an idea of the order of magnitude of parameters discussed before, we present here some values which are representative for the western Dutch Wadden Sea, a tidal area of about 1 3 0 km2 in extent (for details see Zimmerman, 1 9 7 6 ) . T'dal

current velocities in this area have a mean amplitude of 0 . 8 5 ms-',

producing

a mean amplitude of tidal displacement of 5 . 9 km. A representative scale velocity of residual eddies, which occur frequently in this area, is about 0 . 1 2 ms-l. These eddies have an average length scale of 2 . 2 km. Hence the magnitude of the ratios u and A is, respectively, 2

and 0 . 4 .

This gives a correlation

coefficient for neighbour steps (eq. 10, m = I) of 0 . 8 5 . Then the Lagrangian velocity correlation time scale (T) amounts to 3 . 5 tidal periods. For the longitudinal diffusion coefficient for t>>.c,expressed by ( 1 4 ) we get a factor of proportionality, b(u,A), of 2 . 9 .

Hence, an average value of the longitudinal

diffusion coefficient is about 8 0 0 m 2 s - ' . lation time scale

( 7 )

Note, that the magnitude of the corre-

is such that for small time intervals the particular effects

of "diffusion with a finite velocity" as described above should be taken into

account in this area.

REFERENCES

Arons, A.B. and H. Stommel, 1951. A mixing length theory of tidal flushing. Trans.

Am.

geophys. Un., 3 2 : 419-421.

Corrsin, s . , 1 9 7 4 . Limitations of gradient transport models in random walks and in turbulence. Adv. Geophys.,l8A: 25-60. Fischer, H.B., 1 9 7 6 . Mixing and dispersion in estuaries. Ann. Rev. Fluid Dyn., 8 : 107-133.

Monin, A.S. and A.M. Yaglom, 1 9 7 1 . Statistical fluid mechanics, vol. 1. MIT Press, Cambridge, Mass., 7 6 9 pp. Sugimoto, T., 1 9 7 5 . Effect of boundary geometries on tidal currents and tidal mixing. J. Oceanogr. SOC. Japan, 3 1 : 1 - 1 4 . Talbot, J.W. and G.A. Talbot, 1974. Diffusion in shallow seas and in English coastal and estuarine waters. Rapp. P.-v. Cons. perm. int. Explor. Mer, 1 6 7 : 93-110. Taylor, G.I., 1 9 2 1 . Diffusion by continuous movements. Proc. London math. SOC., ( 2 ) 20: 196-212.

Tee, K.T., 1 9 7 6 . Tide-induced residual current, a 2-D nonlinear numerical tidal model. J. Mar. Res., 3 4 : 603-628. Yanagi, T., 1 9 7 6 . Fundamental study on the tidal residual circulation I. J. Oceanogr. SOC. Japan, 3 2 : 1 9 9 - 2 0 8 .

Zimmerman, J.T.F., 1976. Mixing and flushing of tidal embayments in the western Dutch Wadden Sea-11: Analysis of mixing processes. Neth. J. Sea Res., 10: 397-439. Zimmerman, J . T . F . ,

1977. Residual vortex formation by oscillatory (tidal) currents

Geophys. Astrophys. Fluid Dyn. (in press).

Reprinted from:Hydrodynamicsof Estuaries and Fjords,edited by J.C.J.Nihoul 0 1978 Elsevier Scientific Publishing Company,Amsterdam - Printed in The Netherlands

217

CORIOLIS, CURVATURE AND BUOYAK'U'CY EFFECTS UPON DISPERSION IN A NARROW CHANNEL RONALD SMITH Department of Applied Mathematics and Theoretical Physics, University of Cambridge

ABSTRACT The work of Smith (1976) concerning buoyant contaminants in shallow channels is extended to include allowance for curvature of the channel and for the earth's rotation. For weak buoyancy the Coriolis and centrifugal effects lead to a transverse circulation and therefore reduce the longitudinal dispersion relative to that in a straight non-rotating channel. The response of the more dense fluid to these effects is inhibited by the proximity of the channel bed. Thus, it is the less dense fluid that tends to be moved to the outside curve of bends and to the right of the flow direction in the Northern Hemisphere. For stronger buoyancy the transverse density variations reduce or even reverse the circulation and can lead to an increase in the dispersion. Numerical results for the dispersion coefficient are presented for channels of parabolic cross-section. INTRODUCTION Much of engineering practice leans heavily upon the search for analogies to suggest empirical equations whose coefficients and range of applicability to the case in hand can be determined by laboratory experiments and field observations. For the dispersal of pollutants in a fluid environment the obvious analogy is with gaseous diffusion (Csanady 1973). Thus, if the dispersion is uni-directional and if the contaminant is conserved, then the appropriate model equation is

Here A

-

-

is the cross-sectional area, c the concentration, u

the bulk

velocity, and D is the effective diffusivity coefficient. There is a wealth of evidence to show the efficacy of the diffusion model. One noteworthy fact is that when there are currents D

can be several orders of magnitude greater

than the turbulent eddy-diffusivity (Fischer 1973). The explanation of this enhanced dispersion and an accurate theoretical prediction of the dispersion coefficient for neutrally-buoyant contaminants in laminar pipe flow were given by G . I . Taylor in 1953. If equation (1) is compared with the cross-sectionally averaged diffusion

218 equation, then D

D

=

K1

+

can be formally identified:

7 7 i a j

-

Here ~1 is the longitudinal laminar or turbulent diffusivity, denotes cross-sectional averaging, and ' indicates departures from the average. Thus, to determine the shear-dispersion contribution to D

it suffices to calculate

the departures of the longitudinal velocity and the concentration from their cross-sectional mean values.

Taylor's contribution was to recognise that

after a sufficient length of time the steepening of transverse gradients due to the velocity shear would be approximately balanced by transverse diffusion ( s e e Figure 1). In this asymptotic state it is comparatively easy to calculate c' and Taylor (1953) derived the result

miax; = a

2 ~ - 2 ~ m, ~ 2

(3)

where a is the pipe radius and inverse dependence upon

K~

K2

the transverse diffusivity.

It is the

which enables the shear-dispersion term to dominate

the longitudinal diffusion in equation (2)

Fig. 1

Steepening of transverse concentration gradients by the velocity shear.

There are many ways in which Taylor's work has been extended. For open channel flow there is a preliminary asymptotic balance between vertical shear and vertical mixing, and Elder (1959) derived an equation of the same dimensional form as (3) but with the water depth h taking over the role of the pipe radius. At a later stage there is an asymptotic balance between lateral shear and lateral mixing, and the shear-dispersion is dramatically increased due to the channel breadth b

replacing h

(Fischer 1967).

Holley, Harleman & Fischer (1970)

demonstrated that for estuary flows the oscillatory nature of the flow interferes with the second asymptotic balance unless the time scale of lateral mixing is less than the period of oscillation.

219

Another class of extensions to Taylor's work is the inclusion of extra

.

Erdogan & Chatwin physical effects which modify one or both of u' and c' (1967) have shown that weak curvature and buoyancy effects can separately be accounted for by adding further terms to Taylor's formula ( 3 ) for shear-dispersion. Proceeding from the relationship (2) Fischer (1972) has shown that a temporal (i.e. mean and fluctuating) and a spatial (i.e. transverse and vertical) decomposition of u'

and

c'

leads to a corrwponding additive decomposition of

the shear-dispersion. When there are several different physical effects modifying the flow, it would be extremely convenient if a phenomenological decomposition likewise led to an additive result for D

.

The present paper shows

that this is not the case. The chosen problem is an attempt to model the dispersion, on time scales of the order of one tidal period, of heat or salt in a narrow river or estuary. The physical effects accounted for are buoyancy, channel curvature, and the earth's rotation. Numerical results are presented to show the interactions between these effects.

Studies which have considered one of the first two

effects in the context of open channel flow include: Fischer 1969, Harleman

&

Thatcher 1974, Imberger 1976, Smith 1976. The inclusion of the Coriolis effect stems from a remark made in a review paper by Fischer (1976) noting the absence of any analyses of this potentially relevant effect. In the present paper confirmation is given of Fischer's speculation that, in the Northern Hemisphere, the earth's rotation tends to enhance the salinity to the left of the direction of flow, as observed by Pritchard (1952). LIST OF HYPOTHESES Fluid motion in rivers and estuaries is almost inevitably turbulent. As the underlying mathematical mode for the flow we represent the turbulent transports of mass and momentum in terms of eddy-diffusivity tensors with principal axes in the longitudinal, transverse and vertical direction. A more easily justified aspect of the mathematical modelling is the use of the Boussinesq approximation, in which we include the buoyancy effect due to density variations but neglect the corresponding inertia variations.

A l s o , we shall neglect small-

scale irregularities in the channel shape (Okubo 1973). From this complicated starting point we seek those simplifications implied by the physical assumptions. These assumptions are: i) Cross-sectional mixing takes place in much less than the tidal period.: ii) The dispersion time scale is comparable with the tidal period; iii) Buoyancy, curvature and Coriolis effects all modify the dispersion at leading order; iv) The channel is much shallower than it is wide.

220

The present work extends that of Smith (1976) in its more comprehensive nature of the third assumption. For ease of reference the author's earlier paper is henceforth designated as NARROW CL-

S

.

APPROXIMATION

Assumptions (i) and (ii) together imply that the longitudinal length scale of the pollutant distribution greatly exceeds a typical channel breadth scale

L

Accordingly, the first step in S

B. E

=

was to introduce the small parameter.

BIL

and to specify the E-ordering of the many terms in the equations of motion in relation to the basic dimensional quantitites B typical longitudinal velocity.

and U

, where

U

is a

In the present paper the new assumption (iii)

leads to a new scaling.

C'

El-

P

Trans. Drag €*@

-

Trans. Vel. EP

o ( Fig. 2. Derivation of the scaling relationships from the physical assumptions.

221

To derive this scaling we follow through the mathematical consequences of the eddy diffusivities being of order E$ , where $ is to be determined. For brevity the full eqautions of motion are not presented, but use may be made of the simplified equations (4.5) given below.

From the diffusion equation the assumed

order of the diffusivities implies that c' is dispersion time scale is of order E ~ - ~ The c' if the transverse velocity is of order E' diffusivities). The associated transverse drag

.

.

of order E - ~ and that the transverse circulation can modify (i.e. of the same order as the due to the turbulent stresses

is of order E~~ By definition the longitudinal velocity is of order unity. Thus, Coriolis and centrifugal effects lead to transverse driving forces of the appropriate magnitude if the Coriolis parameter is of order E~~ and the radius of curvature is of order E - ~ ~Since the transverse density variations are

.

only of order el-' be of order E38-1

, the reduced gravity associated with density variations must , if the buoyancy forces as well are to contribute significant-

ly to the transverse circulation. In geophysical applications the tidal frequency is of the same order as the Coriolis frequency. To satisfy assumption (ii) we This can be satisfied only if $ = 2 1 3 . need the dispersion time scale sBw2 Because the contaminant is well-mixed across the channel it follows that the entire contaminant distribution tends to be carried along the channel with the flow at the bulk velocity u In order to focus attention upon less trivial

.

.

aspects o f the motion it is mathematically convenient to use axes moving at the bulk velocity. Proceeding as in S , a regular perturbation expansion

leads to a preliminary simplification of the equations of motion. The main feature is that the transverse circulation can be described in terms of a stream function (see figure 3 ) . v(O) =

aZ$

, w

=

-a YJ,

Having derived simplified equations by the expedient of using moving axes, it is valid to transform the result back into the more familiar stationary axes:

222

Z

I

-Y

Fig. 3.

Streamlines for the transverse circulation

Here

and

K~

contaminant, 5 ~

are the transverse and vertical eddy diffusivities for the

~3 v l 2

and

v l 3

are eddy viscosities for the longitudinal motion,

is the height of the free surface, g the gravitational acceleration, 2

3and

v33

are eddy viscosities for the transverse circulation,

reduced gravity (positive for a buoyant contaminant),

V22

is the

is the radius of

R

curvature of the channel, f the Coriolis parameter and h Without loss of generality c(O)

ag

has been identified with

-

the water depth. u (0) is

c , and

required to have cross-sectional average u. For any given channel it is not to be expected that the buoyancy, curvature

.

and Coriolis effects should all significantly modify D Thus, the specific c-scalings derived and used above would not necessarily be appropriate. Fortunately, the final results (5.4) have a much wider range of validity than might be inferred from their derivation. The reason for this is that by retaining as many physical effects as possible the analysis has tended to retain negligibly small terms rather than to neglect potentially large terms. A careful reappraisal of the approximations enables us to show that if

223

then it is necessary that at least one of the exponents should lie in the regions shown in figure 4 , and the other exponents can lie to the right of the regions. The permitted range of the reduced gravity is more restrictive than that in

S

due to the present neglect of the longitudinal buoyancy-driven current.

f'-

Fig. 4 .

Range of €-exponents of (a) the Coriolis parameter, (b) the channel

curvature, and

(c)

the reduced gravity, for the validity of equations (4,5).

SHALLOW WATER APPROXIMATION Continuing as in

S

, we

make a final simplification of the equations of

224

motion by invoking assumption (iv).

To do this we introduce a second small

parameter

6

=

H/B

and we define a new vertical coordinate

where H

is a typical channel depth. The 6-scalings which are compatible with

assumptions (ii) and (iii) are

-1 * f = 6 f ,

A regular perturbation expansion

leads to the results

Here ":u

-

is required to have cross-sectional average

constant J,(x,y,t) is chosen Y?

so

that J,o

, the integration

is zero at the free surface, and

denote the two sides of the channel.

QUALITATIVE ASPECTS OF THE FLOW Equations (6a,b,c) admit of straightforward physical interpretations. The longitudinal current is driven by the pressure gradient associated with the slope of the free surface, and any influences of the three physical effects are relegated to higher order correction terms.

In contrast, the expression (6b) for

the transverse circulation is a linear sum of buoyancy curvature and Coriolis terms. Near the free surface the direction of these contributions to the circulation is respectively from less dense to more dense fluid, towards the outside curve of bends, and to the right of the flow direction. Near the channel bed there is a slower but thicker region of return flow. Finally, the variation in the concentration across the channel is determined by a balance

225

between transverse diffusion as augmented by the transverse circulation and the steepening of the transverse gradients due to the velocity shear. AS was noted by Fischer (1969), the augmented transverse diffusion agrees with Elder's (1959) results for shear dispersion in a transverse rather than a longitudinal context. Our intention is to derive an equation involving only derivatives. Thus,

aY c(l) O

-

c and its partial

is to be regarded as a derived quantity and, unless

buoyancy is negligible, the linearity of equation (66) for the transverse circulation becomes illusory. Indeed, if the expression for $o is substituted into equation (6c) and the integrations are performed, then ' ( c , This means that a ccl) and hence for 2 ) Y O Y O In extreme nonlinear way upon f, R and a c X have multiple roots, but strictly this is beyond

.

there results a cubic equation $o depend in a strongly circumstances the cubic can the range of the present

analysis which implicitly assumes continuous dependence of all flow variables. The stronger the transverse circulation the weaker is the transverse concentration gradient. This fact enables us to make several qualitative deductions concerning the transverse density variations. When buoyancy and rotation are the main effects then, in the Northern Hemisphere, concentration gradients are most pronounced when the contaminant is more dense to the left of the flow direction. This asymmetry tends to lead to the density being greater on the left bank. Similarly, when buoyancy and curvature are the main effects the are in opposition to each other if the lighter fluid is contributions to Q0

towards the outer curve of the bend. Thus, the tendency is for the density to be greater on the inner curve, Dyer (1977) presents arguments to suggest that when there is significant vertical stratification the opposite occurs. The remaining combination, of rotation and curvature, has most pronounced concentration gradients for right hand bends.

However, if the channel depth profile is

symmetric then the density distribution is likewise symmetric. DISPERSION COEFFICIENT

c

Substituting the result (6c) into equation (S), we find that does The appropriate value of the dispersion coefficient

indeed satisfy equation (1). is given by the formula

A2

D = +ly+{/y y- Y- dy'l-h(u:o)

-

;)dd2{1ZhK2dz

+ 1sh;dz)-l

dY

(7) Two points are of immediate note. First, any transverse circulation reduces the dispersion relative to that of a neutrally-buoyant contaminant in a straight m n rotating channel (this explains the drop in D along an estuary where the salinity gradient is largest). Secondly, even if buoyancy is negligible the way that JI, occurs in equation (7) precludes D from being decomposed into separatecontributions

226

associated with curvature and rotation. If buoyancy is not negligible then the complexity is compounded by the nonlinear dependence of $, upon the physical effects. Eddy diffusivities are at best an approximation to actual turbulent transports. Thus, there is necessarily some arbitrariness in the functional forms of the four eddy diffusivities KZ, ~ 3 v13, , VI2. Fortunately, D is given as an integral and is not sensitive to the details of the modelling of the turbulence. A particularly simple, and fairly realistic, model is given by the formulae Ki =

Kf lfiLo) I (h + 5)

vij

Y

= Vfj

r

(0)

Iuo

I(h

+

5) ,

(8)

where fi is the velocity at the free surface, and K ! are empirical vfj constants (Talbot & Talbot 1 9 7 4 ) . Using equation (6a) we find that an equivalent model is Ki =

%:

(h + 5 ) 3 / 2 A/.f;:(h

+

5 ) 3 / 2 dy

.

A similar model in which the eddy diffusivities are proportional to the local depth was used by Fischer (1972). For definiteness we restrict attention to channels of parabolic cross section: C = O

, h=H[l-(~/B)~l ,

-B
and we specify the parameter values Ki = -02

K; = V&

9

= .005.

The appropriate non-dimensional measures for the longitudinal dispersion coefficient and for the curvature, buoyancy and Coriolis effects are DH/B21ul

,

H/R

, G

=

agHBa,c/l;l;

,

fH/;.

Figures (5a-e) show the dependence of the longitudinal dispersion coefficient upon both buoyancy and rotation for various values of the curvature. To obtain results for curvature of negative sign it suffices to reverse the sign of fH/; For

.

.

non-symmetric depth profiles it would be necessary to reverse the sign of G It is noteworthy that while each effect separately reduces the longitudinal dispersion their interactions can increase the dispersion. For example, in figure (5a) with fH/; = .03 the effect of increasing the buoyancy gradient is first to increase D and subsequently to reduce D

.

This, and several other qualitative

features of figures (5a-e), can be explained in terms of the contributions of the From equation (6c) we see that the transverse three physical effects to q0 density gradient reverses sign at the centre of the channel. Thus, buoyancy tends

.

to lead to two circulation cells, as sketched in figure ( 2 ) .

This is unlike

221

rotation and curvature which tend to lead to a single circulation cell. The effect of increasing buoyancy from zero is to increase $o at one side of the channel and reduce it at the other side. Whether this increases or decreases D depends upon whether the circulation or turbulence term dominates in the formula (7). Thus, in figure (5a) buoyancy decreases the longitudinal dispersion when the Coriolis effect is weak, but buoyancy increases the dispersion when the Coriolis effect is strong (e.g. with fH/; = .03). In figures (5b-e) the centrifugal effect is progressively stronger and the region of weak circulation is moved upwards (i.e. in the upper half plane of figures (5b-e) curvature and rotation are in opposition to each other).

(5a)

H/R = 0

04

. I 1 3

Lt

0

-.

04

Fig. 5 Contours of the non-dimensional dispersion coefficient DH/B21;1 straight and curved channels of parabolic cross-section.

for

228

.04

t 1 3 \

I Lc

0

- ,04

229

(5d)

H I R = ,015

04

T 0

- .04 (5e)

04

0

- . 04

HIR =-02

230 USE OF FIGURES 5a-e

If the above parameterisation ( 8 . 9 ) of the turbulence is accepted, then it is straightforward to use figures (5a-e) to get quantitative predictions of the dispersion coefficient. First, the depth-to-radius-of-curvature ratio H/R determines which of the figures is relevant. For example, with H=10m

,

R=500m

it is figure (5e) that must be used.

Secondly, the buoyancy gradient and the flow

.

For example, with rate determine the appropriate position on the figure -4 0 -1 -1 -4 -1 a,c=lO C m , B=lOC~n,u=.lms , f = l O s , the coordinate position in figure (5e) is G = .01

, fH/;

=

.01

2 -1 Thus, for a right-handed bend the longitudinal dispersion coefficient is 5.8 m s

.

For a left-handed bend the sign of the Coriolis term has to be reversed and the dispersion coefficient is 3.9 m'8-l.

In a tidal flow the two different values

of D would correspond to the two flow directions for the same bend. that is in If it is only the choice of the empirical constants K;, K;, V;3 question, then it is still possible to use figures (5a-e). The curvature varameterisation of the figure is with respect to [2

X

10' Vi3(KiK;)']-'

H/R

The contours are the values of E O K ~ D H / liil B~

,

and the coordinate axes are in terms of [ l o 6 K ; V ~ ~ ( K ; K ~ ) ' ] - ' C ( g a x ~ B ~ / ~ ~,[2 ~~ X

1O4V23(K;K$)']-'fH/;

.

For more complicated models of the eddy diffusivities it would be necessary to calculate the dispersion coefficient directly from equations (6,7). CONCLUDING REMARKS A systematic derivation has been given of the flow and of contaminant dispersion in narrow channels. and buoyancy effects.

Allowance has been made for Coriolis, curvature

It has been found that there is a tendency for the fluid

to be slightly more dense on the inner curve of bends and in the Northern Hemisphere, to the left of the flow direction. Although the three physical effects contribute additively to the flow the same is not true of the dispersion coefficient. For a parabolic channel, with a simple model of the turbulent diffusivities, numerical results have been calculated. These results permit quantitative predictions of

231 t h e dispersion coefficients and show t h a t a l l t h r e e e f f e c t s can s i g n i f i c a n t l y modify t h e d i s p e r s i o n .

ACKNOWLEDGEMENT The a u t h o r would l i k e t o thank C.E.G.B.

f o r f i n a n c i a l support.

232

REFERENCES Csanady, G.T.

1973:

Turbulent diffusion in the environnent. D. Reidel,

Dordrecht-Holland. Dyer, K.R. 1 9 7 7 : Lateral circulation effects in estuaries. N.A.S. Symposium on Geophysics of Estuaries (In press). Elder, J.W.

The dispersion of marked fluid in turbulent shear flow. J.

1959:

Fluid Mech. 5 , 544-560. Erdogan, M.E.

&

Chatwin, P.C.

1967:

The effects of curvature and buoyancy

upon the laminar dispersion of solute in a horizontal tube.

J. Fluid Mech.

2 9 , 464-484.

Fischer, H.B. 1 9 6 7 : The mechanics of dispersion in natural streams. J. Hydraul. Div. A.S.C.E. 9 3 , 187-216. Fischer, H.B. 1 9 6 9 :

The effect of bends on dispersion in streams. Water Resour.

Res. 5 , 496-506. Fischer, H.B.

1 9 7 2 : Mass transport mechanisms in partially stratified estuaries. J. Fluid Mech. 5 3 , 671-687.

Fischer, H.B. 1 9 7 3 : Longitudinal dispersion and turbulent mixing in open channel flow. Ann. Rev. Fluid Mech. 5 , 59-78. Fischer, H.B.

1976:

Mixing and dispersion in estuaries. Ann. Rev. Fluid Mech.

8 , 107-133.

Harleman, D.R.F.

&

Thatcher, M.L. 1 9 7 4 :

Longitudinal dispersion and unsteady

salinity intrusion in estuaries. Houille Blanche 2 9 , 25-33. Holley, E.R., Harleman, D.R.F.

&

Fischer, H.B.

estuary flow, J. Hydraul. Div. A.S.C.E.

1970:

Dispersion in homogeneous

9 6 , 1691-1709.

Imberger, J. 1 9 7 6 : Dynamics of a longitudinally stratified estuary. Proc. 15th Int. Conf. Coastal Eng. Hawaii (to appear). Okubo, A.

1973:

Effect of shoreline irregularities on streanwise dispersion

in estuaries and other embayments. Neth. J. Sea Res. 6 , 213-224. Pritchard, D.W. 1 9 5 2 : Salinity distribution and circulation in the Chesapeake Bay estuarine system. J. Mar. Res. 11, 106-123. Smith, R.

1976:

Longitudinal dispersion of a buoyant contaminant in a shallow

channel. J. Fluid Mech. 7 9 , 677-688. Talbot, J.W.

&

Talbot, G.A.

1974:

coastal and estuarine waters.

Diffusion in shallow seas and in English Rapp. P.-v. Reun. Cons. int. Explor. Mer.

1 6 7 , 93-110.

Taylor, G.I.

1953:

through a tube.

Dispersion of soluble matter in solvent flowing slowly Proc. Roy. SOC. A 2 1 9 , 1 8 6 - 2 0 3 .

233

HOW SOME NElJ FUNDAMENTAL RESULTS ON RELATIVE TURBULENT DIFFUSION CAN BE RELEVANT I N ESTUARIES AND OTHER NATURAL FLOvJS

P.C.

CtiATWIN and PAUL J . SULLIVAN*

Department of Applied Mathematics and T h e o r e t i c a l P h y s i c s , U n i v e r s i t y of L i v e r p o o l , L i v e r p o o l , England.

ABSTRACT

T h i s p a p e r d e a l s w i t h t h e r e l a t i v e t u r b u l e n t d i f f u s i o n o f a c l o u d of p a s s i v e contaminant, and i n p a r t i c u l a r w i t h t h e ensemble mean c o n c e n t r a t i o n CQ,t) t h e s t a t i s t i c s of t h e f l u c t u a t i o n o f c o n c e n t r a t i o n c ( y , t ) .

and

Recent work b y t h e

a u t h o r s (Chatwin and S u l l i v a n , 1978a), u s i n g t h e fundamental e q u a t i o n s ( w i t h o u t eddy d i f f u s i v i t i e s ) , h a s l e d t o r e s u l t s which a r e i n many ways q u i t e d i f f e r e n t from t h o s e p r e s e n t e d e . g .

i n Csanady ( 1 9 7 3 ) .

-Among

t h o s e r e s u l t s summarized h e r e

a r e new e s t i m a t e s of t h e magnitudes of C and c 2 t h r o u g h o u t t h e c l o u d , a p r o p o s a l

-

t n a t t n e r e i s a s m a l l c o r e r e g i o n s u r r o u n d i n g t n e c e n t r e i n which b o t h C and c 2 a r e much l a r g e r t h a n e l s e w h e r e , a p h y s i c a l e x p l a n a t i o n of how t h e d i s t r i b u t i o n s

-

of C , c 2 and c e r t a i n c o r r e l a t i o n s and s p e c t r a c a n b e s e l f - s i m i l a r

outside the core

i n i s o t r o p i c t u r b u l e n c e , and some s p e c u l a t i o n s a b o u t t h e profound r o l e of m o l e c u l a r diffusion.

Some e x p e r i m e n t a l c o n f i r m a t i o n of t n e p h y s i c a l p i c t u r e l e a d i n g t o t h e s e

r e s u l t s has a l s o been o b t a i n e d (Chatwin and S u l l i v a n , 1 9 7 8 b ) .

The main p a r t of

t h e p r e s e n t paper i s a n e x a m i n a t i o n of how f a r t h e s e r e s u l t s c a n b e used i n t h e c o m p l i c a t e d v e l o c i t y f i e l d s o c c u r r i n g i n e s t u a r i e s and o t h e r n a t u r a l f l o w s .

In

p a r t i c u l a r , c e r t a i n t i m e s c a l e s a r e o b t a i n e d which measure d e p a r t u r e s of t h e v e l o c i t y f i e l d i n t h e neighbourhood o f t h e c l o u d from i s o t r o p y , homogeneity and statistical stationarity.

I t i s a r g u e d t h a t f o r t i m e s s i n c e release l e s s t h a n

t h e smallest of t h e s e , t h e e f f e c t s of a n i s o t r o p y , inhomogeneity and s t a t i s t i c a l u n s t e a d i n e s s of t h e v e l o c i t y f i e l d on t h e d i s p e r s i o n o f t h e c l o u d c a n b e i n c o r p o r a t e d i n t h e d e s c r i p t i o n summarized above.

For t h e Tay e s t u a r y t h i s minimum

t i m e i s about 3 hours.

1. INTRODUCTION Suppose t h a t a t a f i x e d t i m e i n a n a t u r a l f l o w , s p e c i f i e d f o r example i n a n e s t u a r y by t n e i n t e r v a l t

s i n c e t h e l a s t h i g h t i d e , a q u a n t i t y M of p a s s i v e

*Permanent a d d r e s s : Department o f A p p l i e d Mathematics, U n i v e r s i t y o f Western O n t a r i o , London, Canada.

234 contaminant i s r e l e a s e d .

Suppose f u r t h e r t h a t t h e c e n t r e of mass of t h e cloud of

contaminant i s i n i t i a l l y a t x x

-0

.

-0

and t h a t i t i s d i s t r i b u t e d i n a s p e c i f i e d way about

Let t h e d i s t r i b u t i o n of c o n c e n t r a t i o n a t any time t > t be r ( 2 , t ) .

the v e l o c i t y f i e l d i s t u r b u l e n t the value of p r o p e r t i e s depend both o n 2 and t . on x -0

r

Because

i s a random v a r i a b l e whose s t a t i s t i c a l

Indeed t h e s e p r o p e r t i e s a l s o depend e x p l i c i t l y

and t o , b u t , f o r b r e v i t y , t h e n o t a t i o n i n t h i s paper w i l l n o t i n d i c a t e such

dependence. The s t a t i s t i c a l p r o p e r t i e s of

r

considered i n t h i s paper w i l l a l l be ensemble

averages over the s e t of r e a l i z a t i o n s of the d i s p e r s i n g cloud, i n each one of which t h e r e l e a s e c o n d i t i o n s , i n c l u d i n g t h e time and p o s i t i o n of r e l e a s e , a r e identical.

I n each r e a l i z a t i o n axes a r e taken whose o r i g i n coincides w i t h t h e

c e n t r e of mass of t h e cloud throughout t h e r e a l i z a t i o n .

When t h e s e axes a r e

employed, t n e r e l e v a n t flow dynamics i s contained i n t h e v e l o c i t y L Q , t )

of a

p o i n t r e l a t i v e t o t h a t o f t h e c e n t r e of mass, w h e r e 2 measures the p o s i t i o n of t h e p o i n t r e l a t i v e t o t h e c e n t r e of mass.

The v e l o c i t y f i e l d i s determined by t h e

momentum equations, which a r e independent of t h e contaminant when t h i s i s p a s s i v e . * The p h y s i c a l processes r e s p o n s i b l e f o r t h e d i s p e r s i o n of a cloud a r e summarized i n s e c t i o n 2. diffusion.

With t h e axes used, t h i s d e s c r i p t i o n i s i n t h e framework of r e l a t i v e

Although some of t h e r e s u l t s apply when the l e s s p r a c t i c a l l y important

framework of a b s o l u t e d i f f u s i o n i s used, t h e l a t t e r i s more d i f f i c u l t t o d i s c u s s i n d e t a i l because i t i s a f f e c t e d by t h e a n i s o t r o p i c l a r g e eddies of t h e turbulence whose s t r u c t u r e i s not governed by t h e i n e r t i a l subrange.

With r e l a t i v e d i f f u s i o n

nowever t h e s e l a r g e eddies a r e normally i n s i g n i f i c a n t , a t l e a s t i n i t i a l l y .

The

r e l a t i v e d i f f u s i o n o f a cloud of contaminant i s of b a s i c p h y s i c a l i n t e r e s t and importance, not l e a s t because o f t h e l i g h t i t throws upon t h e s t r u c t u r e of t h e velocity f i e l d .

It i s a l s o of g r e a t p r a c t i c a l s i g n i f i c a n c e .

One example i s t h e

a c c i d e n t a l s p i l l a g e of a t o x i c chemical when 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 f, including i t s f l u c t u at i o n s , s

t be analyzed w i t h i n t h e framework of r e l a t i v e

d i f f u s i o n ( t o avoid the a r t i f i c i a l smearing which occurs w i t h a b s o l u t e d i f f u s i o n ) . These p r o p e r t i e s determine t h e p r a c t i c a l p r o b a b i l i t i e s t h a t a r e needed, f o r example t h e p r o b a b i l i t y t n a t , f o r a given period of time, the maximum c o n c e n t r a t i o n exceeds a given l e t h a l v a l u e .

The importance of t h e f l u c t u a t i o n s i n such a c a s e i s

e v i d e n t s i n c e t h e i r r o o t mean square v a l u e i s normally a t l e a s t of t h e same order a s t h e ensemble mean c o n c e n t r a t i o n and o f t e n much g r e a t e r .

A different practical

problem i s t h e way i n which sound and l i g h t a r e absorbed and t r a n s m i t t e d by t h e cloud, and t h i s i s n a t u r a l l y analyzed i n terms of t h e s p e c t r a l decomposition of (Batchelor, 1 9 5 9 ) .

of a patch of plankton, f o r example.

*

r

Such f a c t o r s may s i g n i f i c a n t l y a f f e c t t h e b i o l o g i c a l growth I n view of t h e widespread occurrence of such

Altnough some of t h e t h e o r e t i c a l r e s u l t s apply a l s o when i t i s not p a s s i v e .

235 i m p o r t a n t p r a c t i c a l problems, s e c t i o n 3 o f t h i s p a p e r a n a l y z e s c o n d i t i o n s under which t h e d e s c r i p t i o n g i v e n i n s e c t i o n 2 c a n b e used i n t h e c o m p l i c a t e d f l o w s existing i n nature.

2. THE EVOLUTION OF A CLOUD I N INCOMPRESSIBLE FLOW The e q u a t i o n g o v e r n i n g

where

2

= z(1.t)

r

i n each r e a l i z a t i o n is

i s t h e r e l a t i v e v e l o c i t y f i e l d d e f i n e d i n s e c t i o n 1, and s a t i s f i e s

the incompressibility condition

v.r=0 . In (l),

(2)

i s t h e molecular d i f f u s i v i t y .

K

A n immediate, and o t h e r w i s e o b v i o u s ,

consequence of (1) and (2) i s t h a t t h e t o t a l amount of c o n t a m i n a n t i s c o n s t a n t , i . e .

/ r ( z t)

d3y

=

M.

(3)

I t i s now u s e f u l t o f o l l o w t h e normal p r a c t i c e and w r i t e b o t h 2 and of t h e i r ensemble means,

-u = ~ ( y , t )and -T

=

r

= C + c ;

g +2

;

as t h e sum

= l ( y , t ) and C = C ( y , t ) r e s p e c t i v e l y , and f l u c t u a t i o n s ,

c = c(y,t) respectively.

overbar,

r

-2 = 3

Thus, d e n o t i n g a n ensemble mean by a n

-r = c ;; HF = O0. . =

From ( 1 ) t o ( 4 ) t h e r e f o l l o w i n t h e u s u a l way: i C d ~ = M ; ~ c d 3 ~ = O ;

aca t + l . v c + v.(s)Kv2C =

;

E q u a t i o n s (6) and ( 7 ) a r e c o u p l e d tnrough t h e terms i n

E,

i n particular.

In ( 7 ) ,

#

t h e second and t h i r d terms on t h e l e f t hand s i d e a r e t h e e f f e c t of t u r b u l e n t t r a n s f e r w h i l e t h e f o u r t h term r e p r e s e n t s p r o d u c t i o n by f e e d i n g from t h e f i e l d of C.

Of t h e two m o l e c u l a r terms on t h e r i g h t hand s i d e , t h e f i r s t i s o r d i n a r y

d i f f u s i o n b u t t h e second i s e v e r y w h e r e n e g a t i v e and r e p r e s e n t s d i s s i p a t i o n .

X e s u i t s when

K =

0 - a c l o u d of marked f l u i d p a r t i c l e s

C o n s i d e r f i r s t a c a s e when t h e e f f e c t s o f

K

a r e n e g l i g i b l e , which may b e a

good a p p r o x i m a t i o n i n t h e e a r l y s t a g e s of t h e d i s p e r s i o n .

With

K

= 0,

i t follows

from (1) and (2) t h a t

s2 0 so that / Dt Thus, u s i n g ( 4 ) , M2 1 C 2 d k + / c 2 d 3-y Lo3 2@

Dt

=

=

=

where L

r 2 d k = constant.

(8)

'

i s a l e n g t h d e t e r m i n e d e n t i r e l y by t h e i n i t i a l c o n d i t i o n s .

Now s u p p o s e

t h a t a t time t t h e c l o u d i s , on t h e a v e r a g e , s p r e a d o v e r a volume o f o r d e r L 3 ( t ) ,

236 where

BY mass c o n s e r v a t i o n , as e x p r e s s e d by t h e f i r s t e q u a t i o n i n ( 5 ) , i t f o l l o w s t h a t ,

a t l e a s t i n most of t h e c l o u d , C i s of magnitude o f o r d e r M / L 3 ( t ) . obvious t h a t L(t)* from (9) t h a t

as t*

i t follows t h a t f C 2 d 3

1 7d 3 y + M 2 / L o 3

as t

+

-

-.

-f

0 as t

Since it i s

Thus i t f o l l o w s

+ m.

Hence c 2 c a n n o t , i n most o f t h e c l o u d ,

-

have a magnitude o f o r d e r M2/L6 as m i s t a k e n l y assumed by Csanady (1973, p.236) and others.

R a t h e r (Chatwin and S u l l i v a n , 1 9 7 8 a ) , t h e magnitude of c 2 i n most o f t h e

cloud i s of o r d e r M2/L

3L3(t), consistent with (9).

-0

magnitudes o f C and c 2 , i t f o l l o w s t h a t ,

if

Granted t h e s e e s t i m a t e s of t h e

t h e i r d i s t r i b u t i o n s are s e l f - s i m i l a r ,

they must b e o f t h e form (Chatwin and S u l l i v a n , 1978a) :

The c o n d i t i o n s under which s e l f - s i m i l a r i t y c a n b e e x p e c t e d a r e d i s c u s s e d i n Chatwin and S u l l i v a n (1978a) and, from a d i f f e r e n t v i e w p o i n t , i n s e c t i o n 3 o f t h i s p a p e r . I t i s however i m p o r t a n t t o emphasize t h a t , even when t h e r e i s no s e l f - s i m i l a r i t y ,

t h e magnitudes o f C and

2 are

as i n (11) and (12) o v e r most of t h e c l o u d .

However t h e s e magnitudes c a n n o t h o l d o v e r t h e whole c l o u d .

-

There is a core

r e g i o n of s m a l l volume surroundingthe c e n t r e w i t h i n which t h e magnitudes o f C and c 2 a r e a t l e a s t one o r d e r g r e a t e r t h a n t h o s e i n (11) and ( 1 2 ) .

Within t h i s core the

p r o b a b i l i t y p e r u n i t volume of f i n d i n g marked f l u i d i s much g r e a t e r t h a n o u t s i d e , and t h e p r o d u c t i o n t e r m i n ( 7 ) i s comparable w i t h t h e o t h e r terms o n t h e l e f t hand

-

s i d e , i n c o n t r a s t w i t h what happens o u t s i d e t h e c o r e when, a c c o r d i n g t o (11) and ( 1 2 ) , p r o d u c t i o n of c 2 i s n e g l i g i b l e compared w i t h i t s t r a n s f e r .

The d e t a i l e d s t r u c t u r e

o f t h e c o r e r e g i o n depends o n t h e s m a l l s c a l e s t r u c t u r e of t h e t u r b u l e n c e b u t i n

-

one c a s e , examined i n d e t a i l i n Chatwin and S u l l i v a n ( 1 9 7 8 a ) , t h e magnitudes o f C and c 2 w i t h i n t h e c o r e w e r e of o r d e r M/L

r a d i u s w a s of o r d e r L 3 / L 2 .

and M2/LE

respectively, while its

It seems l i k e l y t h a t t h e c o r e r a d i u s w i l l always b e

f a r t o o s m a l l t o measure d i r e c t l y i n e x p e r i m e n t s , whatever t h e s m a l l s c a l e s t r u c t u r e of t h e turbulence.

However e x p e r i m e n t s on a plume e m i t t e d from a s t e a d y

s o u r c e (Chatwin and S u l l i v a n , 1978b), w h i l e n o t b e i n g a b l e t o e x p l o r e t h e c o r e d i r e c t l y , showed t h a t t h e d i s p e r s i o n w a s dominated by t h e c e n t r a l p o r t i o n of t h e plume, c o n s i s t e n t w i t h t h e e x i s t e n c e of a c o r e , and t h a t t h e dominant mechanism causing spreading was t r a n s f e r

f r o m and t o t h e c e n t r a l r e g i o n by e d d i e s of t h e

v e l o c i t y f i e l d w i t h a l e n g t h s c a l e o f o r d e r L ( a wave number of o r d e r L

-1

).

The e f f e c t s o f m o l e c u l a r d i f f u s i o n The dominant t r a n s f e r mechanism by e d d i e s w i t h a l e n g t h s c a l e of o r d e r L

i s p a r t of a c a s c a d e p r o c e s s i n which i n t e r a c t i o n s between e d d i e s of t h e v e l o c i t y

237 f i e l d w i t h components of t h e f l u c t u a t i n g c o n c e n t r a t i o n o f comparable s i z e produce s m a l l e r s c a l e components a n d . s o on.

Thus, as t i m e e v o l v e s , more and more of t h e

f l u c t u a t i n g c o n c e n t r a t i o n f i e l d i s c o n t a i n e d i n components of s m a l l l e n g t h s c a l e ( h i g h wave number). with t i m e .

One r e s u l t of t h i s p r o c e s s i s t h e s h r i n k i n g of t h e c o r e r a d i u s

However t h i s p r o c e s s a l s o r e s u l t s i n t h e i n c r e a s e o f t h e s u r f a c e a r e a o f

t h e c l o u d and t h e i n c r e a s e of t h e g r a d i e n t s of of m o l e c u l a r d i f f u s i o n w i t h t i m e .

r,

and t h u s a n i n c r e a s e i n t h e e f f e c t

E v e n t u a l l y a b a l a n c e i s s e t up i n which m o l e c u l a r

d i f f u s i o n e f f e c t i v e l y p r e v e n t s t h e e x i s t e n c e of components of t h e f l u c t u a t i n g c o n c e n t r a t i o n f i e l d w i t h l e n g t h s c a l e s smaller t h a n t h e c o n d u c t i o n c u t - o f f

length

I

(K*v/E)',

where v i s t h e k i n e m a t i c v i s c o s i t y and

k i n e t i c energy per u n i t m a s s (Batchelor,

1952b).

E

i s t h e r a t e of d i s s i p a t i o n of

-

This balance occurs throughout t h e

c l o u d . i n each r e a l i z a t i o n , b u t , when ensemble means l i k e C and c 2 are c o n s i d e r e d ,

i t s e f f e c t i s s i g n i f i c a n t l y g r e a t e r i n t h e c o r e t h a n e l s e w h e r e b e c a u s e of t h e f a r g r e a t e r p r o b a b i l i t y p e r u n i t volume of f i n d i n g marked f l u i d t h e r e .

t o r e d u c e t h e magnitude of c 2 t h e r e .

e f f e c t of be

K

Thus t h e p r i m a r y

w i l l b e f e l t i n t h e c o r e , and t h r o u g h t h e d i s s i p a t i o n t e r m i n ( 7 ) ,

will

The r e s u l t o f t h i s e f f e c t i s t h e n t r a n s f e r r e c

outwards t o t h e r e g i o n o u t s i d e t h e c o r e .

I n t h i s region the d i r e c t e f f e c t s of

K

a r e n e g l i g i b l e , and t h e r e i s no r e a s o n why t h e s h a p e s o f t h e d i s t r i b u t i o n s of C and

-

c 2 t h e r e s h o u l d b e a f f e c t e d by i t .

Nor, by mass c o n s e r v a t i o n , c a n t h e magnitude of

C i n most of t h e c l o u d b e d i f f e r e n t from M / L 3 .

of

-

K

Thus, i f (11) h o l d s when t h e e f f e c t s

a r e n e g l i g i b l e , so does i t a l s o when t h e y are n o t .

c 2 c a n no l o n g e r be of o r d e r M2/L

3L3;

the effect of

the core causes i t t o decrease f a s t e r than t h i s .

K

However t h e magnitude of t r a n s f e r r e d outwards from

Thus, i f (12) h o l d s when t h e

e f f e c t s of

K a r e n e g l i g i b l e , i t must b e r e p l a c e d by M2 c2 = -3g(y) n(t,K,v,&), Lo

-

when t h e s e e f f e c t s a r e i m p o r t a n t . s t r o n g l y on

K.

I n (13),

n

t e n d s t o z e r o as t

-f

m

and depends

Evidence t o r e i n f o r c e t h i s d e s c r i p t i o n i s g i v e n i n Chatwin and

S u l l i v a n (1978a, 1978b).

U s e f u l c o r r e l a t i o n s and t h e i r a s s o c i a t e d s p e c t r a I t i s p a r t i c u l a r l y v a l u a b l e t o u s e c o r r e l a t i o n f u n c t i o n s and t h e i r s p e c t r a

t o d e s c r i b e 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 n g c o n c e n t r a t i o n f i e l d n o t l e a s t b e c a u s e of t h e p r a c t i c a l a d v a n t a g e t h a t e x p e r i m e n t a l n o i s e c a n be f i l t e r e d o u t from t h e measured c o n c e n t r a t i o n r e c o r d s .

Two n a t u r a l c o r r e l a t i o n s , b o t h

t h e o r e t i c a l l y and p r a c t i c a l l y , a r e R ( y , t ) and Q(r, t ) d e f i n e d by

R(y,t )

=c(_o, t > c ( y , t >

(14)

Thus R i s a measure of t h e c o h e r e n t s t r u c t u r e o f t h e f l u c t u a t i n g c o n c e n t r a t i o n f i e l d i n r e l a t i o n t o t h a t a t t h e c e n t r e w h i l e Q i s a measure o f c o h e r e n t s t r u c t u r e

238 throughout t h e whole c l o u d .

The c o r r e l a t i o n s c o r r e s p o n d i n g t o R and Q i n a plume

were shown i n Chatwin and S u l l i v a n (197813) t o b e s e l f - s i m i l a r , same form, c o n s i s t e n t w i t h t h e e x i s t e n c e o f a c o r e r e g i o n .

and of a l m o s t t h e

From t h e second o f

( 5 ) , i t f o l l o w s immediately t h a t /R(Y, t)

dk =

jQ(y,t) d k = 0,

(16)

and t h e c o r r e s p o n d i n g r e s u l t s f o r a plume were v e r i f i e d e x p e r i m e n t a l l y i n Chatwin and S u l l i v a n (1978b)

. -

JQ d3y a r e f i n i t e s o t h a t R and Q

I t c a n b e shown t h a t b o t h j R d 3 y and

have F o u r i e r t r a n s f o r m s , I$(&,

I)(&t,)

t ) and

I$@,t) = j R ( y , t ) e x p ( - i k . y )

r e s p e c t i v e l y , d e f i n e d by

d3y

(17)

d3y.

(18)

and $&,t)

= j Q ( y , t ) exp(-ik.y)

-

From t h e i n v e r s e o f (17) i t f o l l o w s t h a t c 2 ( o , t ) = ( Z x 3 jI$(lc,t) d3& = R ( g , t ) ,

s o t h a t Iq5&,t)

1

(19)

- .

i s t h e magnitude o f t h e F o u r i e r components r e p r e s e n t i n g R ( y , t )

S i m i l a r l y , from t h e i n v e r s e of ( 1 8 ) , J 2 ( y , t ) d 3 y = ( Z T ) - ~ j $ ( & , t ) d315 = Q ( 2 - t ) . Since, i n each r e a l i z a t i o n j c ( y ' , t )

(20)

c(y+y', t ) d 3 r ' i s a normal c o r r e l a t i o n f u n c t i o n

w i t h a F o u r i e r t r a n s f o r m which p r o v i d e s a n e n e r g y s p e c t r u m , i t f o l l o w s by l i n e a r i t y t h a t $(&,t) is t h e average energy spectrum.

Both I$(k,t) and

$&,t)

contain f a r

more d e t a i l t h a n c o u l d r e a s o n a b l y b e measured i n a n e x p e r i m e n t , when one would p r o b a b l y a t t e m p t t o f i n d t h e i r v a l u e s on t h e s p h e r e s spectra corresponding t o

I$

lkl

= constant.

However t h e

and $b f o r a plume were measured i n Chatwin and

S u l l i v a n (1978b) and found t o b e s e l f - s i m i l a r and n e a r l y G a u s s i a n . An i m p o r t a n t a p p l i c a t i o n of Q and j i i s t o t h e complete s p e c t r a l decompositionof t h e c l o u d which i s o b t a i n e d by t a k i n g t h e F o u r i e r t r a n s f o r m of t h e r e l a t i o n ( s e e (13) o f S u l l i v a n ( 1 9 7 5 ) ) : r(y',t)

r(y+y',t)

d y ' = jCb',t)

C(y+y',t)

d3y' + Q(y,t).

T h i s e q u a t i o n i s e x t r e m e l y i m p o r t a n t f o r p r o c e s s e s , l i k e a b s o r p t i o n and t r a n s m i s s i o n

of l i g h t and sound, t h a t depend o n t h e i n s t a n t a n e o u s c o n c e n t r a t i o n s , n o t o n l y t h e fluctuations.

3 . THE EFFECTS OF ANISOTROPY, INHOMQGENEITY AND UNSTEADINESS Many f e a t u r e s of t h e d e s c r i p t i o n j u s t g i v e n seem c e r t a i n t o a p p l y i n any

-

t u r b u l e n t v e l o c i t y f i e l d , i n p a r t i c u l a r t h e e s t i m a t e s of t h e magnitudes of C and

c 2 , t h e e x i s t e n c e and i m p o r t a n c e o f t h e c o r e r e g i o n , and t h e e f f e c t s o f

-

K.

However t h e p a r t i c u l a r form o f t h e v e l o c i t y f i e l d d e t e r m i n e s t h e s h a p e s o f t h e d i s t r i b u t i o n s of C and c2, and how L ( t ) v a r i e s w i t h t i m e .

For t h e s p e c i a l case

of a s t a t i s t i c a l l y s t a t i o n a r y E u l e r i a n v e l o c i t y f i e l d a t h i g h Reynolds number t h e i n e r t i a l s u b r a n g e may e x t e n d from metres t o t e n s of k i l o m e t r e s ( R i c h a r d s o n , 1926; Ozmidov, 1960).

P r o v i d e d L ( t ) l i e s w i t h i n t h e i n e r t i a l s u b r a n g e and i s s i g n i f i -

239 c a n t l y s m a l l e r t h a n l e n g t h s c a l e s t y p i c a l of i n h o m o g e n e i t i e s i n t h e v e l o c i t y f i e l d ( e . g . t h e d e p t h and w i d t h o f a n e s t u a r y ) , t h e d i s p e r s i o n i s governed by e d d i e s (of l e n g t h s c a l e s o f o r d e r L ( t ) ) whose s t a t i s t i c a l s t r u c t u r e i s i s o t r o p i c and homogeneous, and d e t e r m i n e d e n t i r e l y by

7o u t s i d e

d i s t r i b u t i o n s of C and

E.

Under t h e s e c i r c u m s t a n c e s t h e

t h e c o r e must have t h e s e l f - s i m i l a r

and (13), and L ( t ) w i l l be g i v e n a p p r o x i m a t e l y by ( B a t c h e l o r ,

L(t)

= a

4s

(t-t ) 3

forms (11)

1952a):

,

(21)

where a i s a u n i v e r s a l d i m e n s i o n l e s s c o n s t a n t of o r d e r u n i t y .

But i n many n a t u r a l f l o w s t h e s e r a t h e r s p e c i a l c o n d i t i o n s do n o t h o l d s o t h a t i t i s p e r t i n e n t t o e n q u i r e w h e t h e r t h e r e s u l t s t h a t t h e y g i v e can b e e x t e n d e d to a p p l y i n such f l o w s . r e l e a s e which a r e section.

not

The e x t e n s i o n s c o n s i d e r e d h e r e a p p l y f o r t i m e s a f t e r

l a r g e compared w i t h t h e time t a k e n f o r mixing o v e r t h e c r o s s -

T h e r e f o r e t h e y do

=

l e a d to t h e d e s c r i p t i o n i n terms o f a n e f f e c t i v e

l o n g i t u d i n a l c o e f f i c i e n t which w a s p r o v i d e d f o r a b s o l u t e d i f f u s i o n by T a y l o r ( 1 9 5 4 ) .

Zn p a s s i n g i t i s worth m e n t i o n i n g t h a t i n many n a t u r a l flows i t i s d o u b t f u l i n any c a s e w h e t h e r t h i s d e s c r i p t i o n i s e v e r p r o p e r l y j u s t i f i e d (Chatwin, 1975, 1976; Dewey and S u l l i v a n , 1977). The e s s e n t i a l p o i n t u n d e r l y i n g t h e e x t e n s i o n s t o b e d i s c u s s e d h e r e i s t h a t €rom t h e p o i n t of view of a n o b s e r v e r moving w i t h t h e c l o u d , t h e c o r r e c t v i e w p o i n t f o r r e l a t i v e d i f f u s i o n , a l l t h e p o s s i b l e c o m p l i c a t i o n s o c c u r r i n g i n n a t u r a l flows a r e o b s e r v e d as s t a t i s t i c a l u n s t e a d i n e s s i n t h e r e l a t i v e v e l o c i t y f i e l d

x(1.t).

There a r e f o u r p r i n c i p a l s o u r c e s of s u c h u n s t e a d i n e s s which a r e i n g e n e r a l n o t independent. (i)

These a r e :

u n s t e a d i n e s s i n t h e mean E u l e r i a n v e l o c i t y f i e l d L ( 5 , t ) l i k e t h a t due t o t h e t i d a l e f f e c t i n e s t u a r i e s - h e r e x i s measured r e l a t i v e to a f i x e d o r i g i n ;

( i i ) changes t h a t a r i s e from t h e c l o u d b e i n g c o n v e c t e d i n t o r e g i o n s w i t h d i f f e r e n t topography and d e n s i t y d i s t r i b u t i o n s ; ( i i i ) t h e s t a t i s t i c a l s a m p l i n g by t h e c l o u d of r e g i o n s of d i f f e r e n t t u r b u l e n c e s t r u c t u r e o v e r t h e f l o w c r o s s - s e c t i o n which i s caused by c o n v e c t i o n by l a r g e eddies; ( i v ) t h e c o n v e c t i o n of t h e c l o u d t h r o u g h r e g i o n s of a p p r e c i a b l e s h e a r i n A time s c a l e c a n b e a s s o c i a t e d w i t h e a c h of

I(&t, ) .

( i ) to ( i v ) , a n d , p r o v i d e d each o f

t h e s e i s l a r g e compared w i t h t h e r e l a t i v e d i f f u s i o n t i m e s c a l e T , g i v e n by

then i t i s reasonable t o suppose t h a t t h e e f f e c t s of ( i ) t o ( i v ) can be incorp o r a t e d i n t h e s c a l e s L ( t ) and E ( t ) , b u t t h a t o t h e r w i s e t h e d i s p e r s i o n i s as i n i s o t r o p i c t u r b u l e n c e p r o v i d e d t h e Reynolds number i s s u f f i c i e n t l y h i g h f o r t h e i n e r t i a l s u b r a n g e to e x t e n d o v e r l e n g t h s c a l e s l a r g e r t h a n L ( t ) .

This proposal

seems p a r t i c u l a r l y p r o m i s i n g s i n c e o b s e r v a t i o n s of i n e r t i a l s u b r a n g e p r e d i c t i o n s l i k e (21), r e p o r t e d by R i c h a r d s o n ( 1 9 2 6 ) , Ozmidov (1960) and o t h e r s , have t a k e n

240 p l a c e i n flows where t o some d e g r e e a l l of e f f e c t s ( i ) t o ( i v ) a r e p r e s e n t .

Under

t h e s e c i r c u m s t a n c e s t h e d i s t r i b u t i o n s o f C and c 2 w i l l h a v e t h e s e l f - s i m i l a r

forms

-

(11) and ( 1 3 ) , where L ( t ) i s d e t e r m i n e d ( B a t c h e l o r 1952a) by

dL = 3 a % ( t ) ( t - t o ) 2

,

(23)

dt which i s c o n s i s t e n t w i t h ( 2 1 ) .

When (23) h o l d s i t f o l l o w s a p p r o x i m a t e l y from (22)

that

T = I3( t - t 0 ) .

(24)

Time s c a l e s a s s o c i a t e d w i t h e f f e c t s ( i ) t o ( i v ) The t i m e s c a l e T1 a s s o c i a t e d w i t h e f f e c t ( i ) can b e d e f i n e d by

alqjp

= (”

/I/a t

and i n many c a s e s , i n c l u d i n g e s t u a r i e s , i t w i l l b e l a r g e w i t h r e s p e c t t o t h e time s c a l e T 2 a s s o c i a t e d w i t h ( i t ) .

The s i m p l e s t way of d e f i n i n g T 2 i s i n terms

of a l e n g t h s c a l e l ( 2 , t ) which measures t h e s t r e a m w i s e d i s t a n c e o v e r which e i t h e r t h e topography o r t h e d e n s i t y d i s t r i b u t i o n changes a p p r e c i a b l y . the value of

In d e f i n i n g 1,

r e f e r s t o t h e p o s i t i o n of t h e c e n t r e of mass of t h e c l o u d and 1 may

depend e x p l i c i t l y on t t h r o u g h e f f e c t ( i ) .

Having d e f i n e d 1, i t i s n a t u r a l t o

d e f i n e T 2 by

where V i s t h e component o f v i n t h e s t r e a m w i s e d i r e c t i o n . A t i m e s c a l e a s s o c i a t e d w i t h e f f e c t ( i i i ) , s a y T3, i s t h a t t a k e n f o r a n element o f c o n t a m i n a n t t o wander, by c o n v e c t i o n w i t h l a r g e e d d i e s , o v e r t h a t p a r t of t h e c r o s s - s e c t i o n c o n t a i n i n g a p p r e c i a b l e changes i n t h e t u r b u l e n c e s t r u c t u r e .

The s h o r t e s t such t i m e i s r e l e v a n t f o r t h e t y p e of a p p l i c a t i o n b e i n g c o n s i d e r e d h e r e and i s o b t a i n e d , i n many c a s e s i n c l u d i n g most e s t u a r i e s , by d e t e r m i n i n g t h e t i m e t a k e n f o r a n element of c o n t a m i n a n t t o wander o v e r t h e

depth of t h e f l o w

I f t h e l o c a l d e p t h i s d k , t ) t h i s g i v e s ( S u l l i v a n , 1971b) a n e s t i m a t e of T

3 as

where uA i s t h e ( l o c a l ) v a l u e o f t h e f r i c t i o n v e l o c i t y which may a l s o v a r y w i t h

x

and t . Although e f f e c t ( i v ) , t h a t o f s h e a r i n t h e mean E u l e r i a n v e l o c i t y f i e l d , i s c l e a r l y coupled w i t h o t h e r e f f e c t s , p a r t i c u l a r l y ( i i i ) , i t h a s b e e n i n c l u d e d s e p a r a t e l y b e c a u s e i t p r o v i d e s a mechanism f o r t h e growth of L ( t ) t h a t i s r a t h e r d i f f e r e n t from t h a t a c t i n g i n i s o t r o p i c homogeneous t u r b u l e n c e which w a s c o n s i d e r e d e a r l i e r , v i z . t h e random e n c o u n t e r s o f t h e c l o u d w i t h e d d i e s whose l e n g t h s c a l e s

are of o r d e r L ( t ) .

For i n t h e p r e s e n c e o f mean s h e a r t h e r e w i l l b e , on t h e a v e r a g e ,

a velocity difference

A 1 over

t h e c l o u d g i v i n g a ( v e c t o r ) r a t e of e x t e n s i o n o f t h e

c l o u d p r o p o r t i o n a l t o Av. C o n s i d e r i n g f o r s i m p l i c i t y t h e component AV of A 1 i n

241 t h e s t r e a m w i s e d i r e c t i o n g i v e s L/AV as a n e s t i m a t e o f T4,

the relevant t i m e scale.

I n many c a s e s t h e s h e a r w i l l b e g r e a t e s t i n t h e v e r t i c a l d i r e c t i o n . a x i s i n t h e v e r t i c a l a l l o w s AV t o b e e s t i m a t e d by AV

1

Taking t h e z

L(aV/az) p r o v i d e d second

d e r i v a t i v e s of V w i t h r e s p e c t t o z a r e a p p r o p r i a t e l y s m a l l , t h a t i s p r o v i d e d Using (21) t h i s r e q u i r e s * ( t - t ) << i ( Z V / a z I / a ~ ” ~ / 2 . ~ v / a z ~ I i ~ ” .

/a2V/az21L < < l a V / a z / .

When t h i s c o n d i t i o n h o l d s , T4 = L/AV c a n b e e s t i m a t e d by T~

lav/azI-l.

Mean s h e a r a l s o a f f e c t s t h e

d i s t r i b u t i o n of c 2 ,

(28) f o r c o n v e c t i o n of t h e c l o u d

by t h e l a r g e s t e d d i e s ( e s p e c i a l l y t h e e n e r g y c o n t a i n i n g e d d i e s ) i n r e g i o n s of s u b s t a n t i a l mean s h e a r p r o d u c e s f l u c t u a t i o n s i n c o n c e n t r a t i o n a t a wave number o f o r d e r 1AV.T’ 1-l where T ’ i s a “ t u r n o v e r ” t i m e of t h e s e e d d i e s .

Of c o u r s e , i n t h e

absence of mean s h e a r , t h i s e f f e c t i s n o t p r e s e n t and t h e l a r g e s t e d d i e s a f f e c t t h e c l o u d o n l y through t h e r o l e s t h e y p l a y i n e f f e c t s ( i i ) and ( i i i ) .

The v a l u e s of t h e t i m e s c a l e s i n some s p e c i f i c cases F i s c h e r (1972) q u o t e s measurements by Bowden i n t h e Mersey o n t h e b a s i s o f which h e c o n s i d e r s t h a t p a r t of t h e e s t u a r y c a n b e c o n s i d e r e d as a u n i f o r m c h a n n e l

9 km long w i t h a maximum d e p t h o f a b o u t 20 m and a r o o t mean s q u a r e t i d a l v e l o c i t y of a b o u t 1 m s

-1

.

comparable d a t a . -1

was a b o u t 2 m s

A s t u d y of t h e Tay e s t u a r y by W i l l i a m s and West (1973) p r o v i d e s

.

The maximum d e p t h w a s 25 m and t h e maximum t i d a l c u r r e n t v e l o c i t y Over a d i s t a n c e of a b o u t 10 km between F l i s k and Newport t h e

c r o s s - s e c t i o n a l a r e a o f t h e Tay v a r i e s l i n e a r l y between 5000 m2 and 16000 m2. d e f i n e d i n ( 2 5 ) , i s of c o u r s e t h e same i n b o t h e s t u a r i e s

The v a l u e of T1,

s i n c e i t i s due t o t h e t i d e . T1

2

12 hrs

4 x

1

Thus

lo4,.

(29)

For t h e Mersey t h e v a l u e o f T2, d e f i n e d i n ( 2 6 ) , i s i n f i n i t e b e c a u s e t h e r e i s no 10 km and V = 1 m s - l ,

v a r i a t i o n i n t h e c r o s s - s e c t i o n , w h i l e f o r t h e Tay, t a k i n g 1 the estimate obtained i s T~

104s.

L-

(30)

TO

estimate T3 i n t h e a b s e n c e of more d e t a i l e d d a t a from e i t h e r e s t u a r y t a k e

uj,

^.

0.05V = 0.05 m s - l T~

--

4

x

and d

2

20 m.

From (27) t h i s g i v e s

lo%,

f o r both e s t u a r i e s .

(31) U n f o r t u n a t e l y i t i s n o t p o s s i b l e t o e s t i m a t e T4 f o r e i t h e r

e s t u a r y w i t h o u t d e t a i l e d knowledge of t h e mean v e l o c i t y p r o f i l e s .

But assuming

T4>T3, t h e s m a l l e s t o f t h e t h r e e times i n (29) t o (31), i t f o l l o w s from (24) t h a t i n t h e Tay t h e d i s p e r s i o n o f t h e c l o u d s h o u l d b e e s s e n t i a l l y as i n i s o t r o p i c t u r b u l e n c e , e x t e n d e d i n t h e way d e s c r i b e d above, p r o v i d e d T << T 3 , i . e . (t-to)

provided

<< 12 x 103s = 3l/, h r s .

*Note t h a t

E

-7

i s t y p i c a l l y of o r d e r 10

m2s

-3

( S u l l i v a n , 1971a).

242 F i n a l l y i t s h o u l d be s t r e s s e d t h a t t h e estimates i n (29) t o (31) a r e v e r y rough and b a s e d o n i n a d e q u a t e d a t a .

In o r d e r t o a p p l y t h e i d e a s o f t h i s s e c t i o n i n

any p a r t i c u l a r c a s e t h e t i m e s c a l e s must b e e s t i m a t e d t a k i n g i n t o a c c o u n t b o t h t h e d e t a i l s of t h e p a r t i c u l a r e s t u a r y and t h e h i s t o r y s i n c e r e l e a s e o f t h e c l o u d .

REFERENCES 1952a. D i f f u s i o n i n a f i e l d o f hoaogeneous t u r b u l e n c e . 11. The B a t c h e l o r , G.K., r e l a t i v e m o t i o n o f p a r t i c l e s . P r o c . C a b . P h i l . SOC., 48: 345-562. Batchelor, G.K., 1952b. The e f f e c t of homogeneous t u r b u l e n c e o n material l i n e s and s u r f a c e s . P r o c . Roy. SOC. Lond., A213: 349-366. B a t c h e l o r , G . K . , 1959. S m a l l - s c a l e v a r i a t i o n o f c o n v e c t e d q u a n t i t i e s l i k e temperature i n t u r b u l e n t f l u i d . P a r t 1. G e n e r a l d i s c u s s i o n and t h e case o f s m a l l conduct i v i t y . J . F l u i d Mech., 5: 113-133. Chatwin, P.C., 1975. On t h e l o n g i t u d i n a l d i s p e r s i o n o f p a s s i v e c o n t a m i n a n t i n o s c i l l a t o r y f l o w s i n t u b e s . J . F l u i d Mech., 71: 513-527. Chatwin, P.C.,1976. Some remarks o n t h e m a i n t e n a n c e o f t h e s a l i n i t y d i s t r i b u t i o n i n e s t u a r i e s . E s t . & C o a s t a l Mar. S c i . , 41: 555-566. Chatwin, P.C. and S u l l i v a n , P . J . , 1978a. The r e l a t i v e d i s p e r s i o n of a c l o u d of passive contaminant i n incompressible t u r b u l e n t flow. Submitted t o J . F l u i d Mech. Chatwin, P.C. and S u l l i v a n , P . J . , 1978b. Measurements of c o n c e n t r a t i o n f l u c t u a t i o n s i n r e l a t i v e t u r b u l e n t d i f f u s i o n . S u b m i t t e d t o J. F l u i d Mech. Csanady, G . T . , 1973. T u r b u l e n t D i f f u s i o n i n t h e Environment. D. Reidel Publishing Company, D o r d r e c h t , 248 p p . Dewey, R. and S u l l i v a n , P . J . , 1977. The a s y m p t o t i c s t a g e of l o n g i t u d i n a l t u r b u l e n t d i s p e r s i o n w i t h i n a t u b e . J. F l u i d Mech., 80: 293-303. F i s c h e r , H.B., 972. Mass t r a n s p o r t mechanisms i n p a r t i a l l y s t r a t i f i e d e s t u a r i e s . J . F l u i d Mech., 53: 671-687. Ozmidov, R . V . , 1960. The i n v e s t i g a t i o n o f medium-scale h o r i z o n t a l t u r b u l e n t exchange i n t h e o c e a n u s i n g r a d i o l o c a t i o n o b s e r v a t i o n s o f a f l o a t i n g buoy. Akad. Nauk SSSR, Dokl. E a r t h S c i . 126: 536-538. R i c h a r d s o n , L.F., 1926. Atmospheric d i f f u s i o n shown on a d i s t a n c e - n e i g h b o u r g r a p h . P r o c . Roy. SOC. Lond., A 1 1 0 : 709-737. S u l l i v a n , P . J . , 1971a. Some d a t a o n t h e d i s t a n c e - n e i g h b o u r f u n c t i o n f o r r e l a t i v e d i f f u s i o n . J . F l u i d Mech., 47: 601-607. S u l l i v a n , P . J . , 1971b. L o n g i t u d i n a l d i s p e r s i o n w i t h i n a two-dimensional t u r b u l e n t s h e a r f l o w . J . F l u i d Mech., 49: 551-576. S u l l i v a n , P . J . , 1975. "The 4/jxds law o f r e l a t i v e d i f f u s i o n " . Mem. SOC. Roy. S c i . Li\ege, 6e sdrie, 7: 253-260. Taylor, G . I . , 1954. The d i s p e r s i o n o f m a t t e r i n t u r b u l e n t f l o w t h r o u g h a p i p e . P r o c . Roy. Soc. Lond., A223: 446-46%. W i l l i a m s , D . J . A . , and West, J . R . , 1973. A one-dimensional r e p r e s e n t a t i o n of mixing i n t h e Tay e s t u a r y . I n A.L.H. Gameson ( E d i t o r ) , M a t h e m a t i c a l and h y d r a u l i c m o d e l l i n g of e s t u a r i n e p o l l u t i o n . Uater P o l l . R e s . Tech. P a p e r No. 13, D e p t . E n v i r o n . , London, pp. 117-125. Professor Sullivan acknowledges the financial support of the National Research Council of Canada during the period when this work was done.

243

A ONE-DIME"JSIONAL TIDAL MDDEL FOR ESTUARINE

"bDm

Langley R. Pluir Canada Centre for Inland Waters, Burlington, Ontario, Canada

INTRODUCTION This paper is concerned with one-dimensional numerical mdelling of unsteady flow in networks of canals, rivers, and estuaries. The mdel will allow the conputation of water elevation and velocity in any netsmrk of open channels to which the following assmptions are applicable: (1) flow is physically possible; (2) flow is entirely subcritical (i.e. the Froude number is less than 1.0); (3) flow is one-dimensional in space (i.e. laterally and vertically homgeneous) ; (4) appropriate boufidary conditions are available; and (5) the section geometry of the channel is fixed (i.e. no deposition or scouring occurs). Flow in open channels can be described by two equations, one expressing the conservation of mass (the continuity equation) and one expressing the conservation of momntum in the longitudinal direction (the mmertum equation). In general terms, these equations form a set of non-linear partial differential equations. Depending upon the assqtions made, there are various methods available for the solution of these equations; but, since the developmt of the digital computer, nmrical methods have generally been used. This papr develops all of the theory necessary to construct a nmrical mdel for simulating unsteady flow conditions in networks of open channels. The first section gives a form1 method for describing the flow relationships in any netmrk. The inp>licit finite difference method is then described and extended for use in networks. The equations of motion are given, with a description of a generalized equation solver, and the properties of the finite difference scheme are discussed. The final section describes one of many applications of the numerical model to show that the method does work.

THE ""XIRK AS A G W H The essential features of open channel flow in a netwrk may be illustrated by considering an open channel network as if it were a mathematical entity known as a graph. A graph, in mathematical terms, m y be defined as a connected set of lines on a plane surface. The pints at which various lines meet or cross

244

are known as nodes and, if direction m y be determined, the graph is said to be directed. The relationships in the graph are purely topological in that distance relationships are not preserved. The graph representation of a network of open channels consists of a number of lines called branches representing the elementary reaches of the open channel network and a certain n of nodes, each of which identifies the location at which two or m r e branches intersect. To make the graph m r e general and at the same tire m r e applicable to river or estuarine systems, additional nodes are allowed on the boundaries of the graph or at arbitrary locations on the graph. For example, the river system shown in Fig. l ( a ) m y be schematized into the graph shown in Fig. l(b). In the figure, the branches of the graph are shown as lines and the nodes are shown as dots.

(a)

Fig. 1. A river network and its graph representation.

245

In nodelling unsteady flow situations, each branch is considered equivalent to an elerrentary reach in the channel network. The nodes are placed at locations where flow properties are required or are k n m . At each node, it is necessary to know the cross-sectional geometry of the open channel and, during the solution of the equations of mtion of the system, it is assumed that the system parmters will vary linearly between adjacent nodes. Therefore, in the schemtization of an open channel network, it is necessary to take into account the physical param eters of a system, and these considerations are reflected in the additional nodes selected for inclusion on the graph. It is possible to rearrange the graph so that there are, at mst, thee branches connected to each node. If m r e than three branches are connected to a particular node, as in Fig. 2(a), then the offending node m y be split into two or m r e nodes, each of which has three branches, as in Fig. 2(b). The physical distance between the new nodes will be zero. The pwpose of this schemtization is to identify flow relationships and to sirrplify these relationships to an extent which allows s-le

bookkeeping for a computer program.

Fig. 2. Splitting of the junction node. This splitting of nodes allows the identification of exactly four separate types of nodes. Each network can be schemtized using only these four types of nodes. They are: (1) a bounding node which is connected to only one branch:

(2) an interior node which is connected to exactly two branches: ( 3 ) a convergent node which has two branches entering and one branch leaving; and ( 4 ) a divergent node which has one branch entering and two branches leaving.

Types 3 and 4 are known collectively as junction nodes. In an unsteady flow problem, any given junction node could be either type 3 or type 4 depending upon the direction of flow. This situation m y be automatically resolved by the cow puter program, if a node is classified according to its type when all flow is in the positive direction. Making this distinction between convergent and divergent nodes eases the computer programing difficulties without making the theoretical treamnt any m r e difficult.

246

FINITE DIFFERENCE METHODS

These methods use finite difference approximations to the partial derivatives appearing in the equations of mtion to transform the set of partial differential equations into a set of algebraic equations. It is the direct finite difference methods that hold the m s t promise for the solution of engineering problems when a digital computer m y be used to solve large systems of algebraic equations. In this case, "direct" means the conversion of the partial differential equations to a finite difference formulation without using the characteristic equations. T b c recent reviews of the various available methods are given in Price (1974) and

Strelkoff- (1970). There are two finite difference methcds, and the distinction between them lies in the method in which the finite differences are formulated and the resulting rrethds for the solution of the equations. The implicit method requires that all of the equations be solved simultaneously in order to advance the solution one time step. The explicit method proceeds down the open channel solving only one equation at one time. There are a large nmnber of finite differencing schexes available for use with each method. Because of the bookkeeping and equation solving requir-ts

of the implicit

scheme, the explicit method is much simpler to use. However, the explicit schemes are restricted in the size of the computational time step required to ensure a stable corrputational procedure. Nwrical stability is achieved when smll errors introduced in the computation diminish rather than increase in m g nitude with succeeding computations. If too large a time step is used, the true solution to the equations m y well be completely msked by the errors. The restriction is given by the well-born Courant condition, which is:

Ax +

At u

M

where : B = width of water surface; A = cross-sectional area; g = acceleration due to gravity; u Ax

= =

velocity; and the distance interval used.

If friction is important, Garrison et al. (1969) have shown that the maximum At m y be further limited by the following stability criterion: At 5 [

u

+

JgA/B

2.2 (A/B)

247

where n is the Manning friction factor. Examination of the stability criteria will show that, for typical river applications, time steps on the order of a few seconds m y be required. For problems in large river systems which m y involve tidal cycles or input hydrographs extending over several days, these smll time steps cause the explicit method to be very wasteful of computer t i n e . The mjor restriction on the explicit ~ t h o dis the difficulty in handling flow in networks. As the amputations proceed downstream and reach a junction, some arbitrary decision must be m d e abut how much of the flow enters each branch. The profiles in each branch must then be computed separately and the results, at the end of the branches, compared. If these results are incorrptible, then the computation must start again at the top of the branch and iterate i n method, tl$s some fashion until the flaws mtch properly. In the -licit problem does not arise since all of the equations are solved simultaneously. THE IMPLICIT MFTHOD Of the two types of finite difference schemes available for the solution of the equations of mtion, the implicit scheme has been chosen for reasons which have been discussed briefly and which will become clearer by the end of this section.

+R

I

t

OAtJ

xi+l

Fig. 3. The weighted four-pint finite difference scheme.

Consider a rectangular grid, not necessarily uniform on the x-t plane as shown in Fig. 3 . The function value and partial derivatives of a function at the pint x = xi + 2 and t = t. 3 + 0At are given by:

If tQs above are applied to a single-branched river o r channel which contains only bounding or interior nodes, such as in Fig. 4, then at tire = t,+l, there are two unknowns for each node or grid pint. A river system containing N branches, corresponding to N+1 nodes, has 2(N+1) unknowns at time t,+l. For each pair of adjacent nodes at tire t,+l, a continuity equation and a m m t u m equation m y be written in finite difference form relating the unknown values at tiw t. to the known values at time tj. Since there are two equations between 3+1 each pair of adjacent nodes, there are a total of 2N equations in 2(N+1) unknowns. The addition of any two additional independent equations relating unknowns will produce a set of 2N+2 equations with 2N+2 unknowns which m y be solved s h l taneously to produce the solutions at time tj+l.

Fig. 4. An interior node. The implicit method my easily be extended for use in a network of channels by ansidering the interior node shown in Fig. 4 and the junction node shown in Fig. 5(a). In Fig. 4, there are three nodes associated with the interior node, two bounding nodes and the interior node itself. The system is comprised of two branches for which four equations m y be written. The two boundary conditions supply the other two equations necessary f o r the computation of the six unknown quantities. 0

Fig. 5. A junction node.

0

249

Consider the junction node shown in Fig. 5(a) and split this junction node into three nodes as shown in Fig. 5 ( b ) . There are now six nodes associated with the original junction ncde and hence twelve unknowns. The three branches provide six equations and the three boundary conditions provide an additional three equations, giving a total of nine equations for the twelve unknowns. The three remining equations necessary for a solution are provided fairly easily, since splitting the junction ncde into three parts has placed the actual junction itself between nodes where it m y be treated as a simple input-output system and we m y ignore the details of the physical mechanisms operating inside

the junction. At any junction, a continuity equation will relate the amunt of water entering the junction to the a m m t of water leavinq the junction. This continuity equation provides one equation. Noting that the actual distance between the junction nodes is zero will provide two additional equations, since the water level at the three "junction nodes" must be equal. Hence, there are a total of twelve equations in twelve unhowns and the systm of equations is solvable. It m y be sham, by s-le mthemtical induction, that the *licit method may be used in any arbitrary network of open channels that can be schemtized as a plane graph. The major advantage of the implicit method is that, by solving all of the equations simultaneously,the flaw in the various channels is computed autmtically and no complicated iterative techniques are required to balance the various flows. The only disadvantage in the implicit method is the necessity of solving a large number of equations simultaneously. EQUATIONS OF NTION Although both the bookkeeping system suggested by the graph methcd of describing the flow relationships and the equation solver m y be programned to be independent of the form of the equations used in the d e l , a choice of equations is necessary in order to proceed. The most generally useful set of equations to employ for describing flow in open channels is the non-linear St. Venant equations. If an open channel m y be schematized as shown in Fig. 6, then the St. Venant Equations m y be written as:

250

Z+Y W

Fig. 6. Definition sketches for an irregular channel.

251

where: Q = cross-sectional discharge; u = cross-sectional mean velocity; t =time; x = distance along the channel;

q

acceleration due to gravity; A = cross-sectional area; z = water level above a horizontal datum; =

K = friction coefficient, where K = 1/C2 (Chezy friction), or (Manning's friction); K = nz/2.22(h*) h* = total depth of watex, or hydraulic radius; b = width of conveyance section of channel; bs = width of storage section of c b e l ; and q = lateral inflow per unit length of channel. In these equations, it is assthat neither the water in the storage section of the channel nor the lateral inflow affect the m m t u m . A term involving the wind stress could easily be added to the m m t m equation, and it will be seen that this muld not affect the equation solver in any way. The junction equations are fairly simple. Continuity requires that the inflow to a junction must equal the outflow and, since the length of the junction is effectively zero, no storage takes place in the junctions. Therefore, if nodes i, i+l, and k are the nmkers associated with the junctions, the continuity equation is:

The junction energy equations are simply that the water levels inside the junction are equal, or:

All of the various types of bundary conditions m y be expressed in terms of the discharge and water level elevation at the node at which the bounddry conditions are given. By substituting the finite difference scheme into the equations of mtion, we obtain a set of algebraic equations relating the properties of the system at t k step j to the properties at tirre step j+l. In these equations, the only unknowns

252

are the discharge and water level at the time j+l, and all of the equations m y be written in the form:

where the functions fl to f7 are in terms of the section properties and the discharges and water levels at tine j and at nodes i, i+l, and k. Since the m m t u m equation is non-linear, some of these functions fl to f7 are not constant but are functions of the unknown Q or z or both. For example, in the m m t u m equation, one of the non-linear terms is uaa/ax. In the solution technique, this term is quasi-linearized to UaQ//ax where U is the velocity computed from the last pass through the equation solver. The system of equations is now in linear form and m y be expressed as a matrix equation of the form:

where X is a solution vector. The mtrix, W, is reduced in three steps: (1) Zero diagonal entries in the mtrix are filled. Any row, i, that has a zero diagonal entry has added to it a row, k, that has a non-zero entry in the ith column, where i
fill the solution vector. Each t h the mtrix, W, is reduced, the values for Q and z , extracted from the solution vector, are used to recompute the coefficients. The mtrix consisting of the new coefficients must now be solved. This process is repeated until convergence to a predetermined nunker of significant digits is achieved in the Q and z values. Convergence to four significant digits usually takes less than 10 iterations. Significant reductions in computing.requirements can be m d e by avoiding storing or operating on the zero coefficients of the netwDrk mtrix. This mthcd, known as a sparse mtrix technique, can easily be combined with Gaussian elimination to prcduce a reasonably efficient equation solver. The method used in this study is similar to that used by Chandrashekar et al. (1975, 1976) and is given in detail by Budge11 (1977a). For a banded matrix, execution t k increases linearly with the number of equations. One of the m s t useful features of the equation solver, and of the bkkeeping scheme suggested by the graph method of describing the flow relationships, is

253

that the numerical rrcdel m y very easily be programmed to be completely independent of the exact form of the equations used. This can allow further saving in computer t h if equations other than the full non-linear St.s n o i t a u q e , t ' V m y be used. Alternatively, the mdel m y be easily expanded to take into account such complexities as m r e complicated junction equations, wind stress, or dispersion of a conservative tracer with m i n h u m of programning changes. PROPEHTIFS OF THE FINITE -D

SCHEME

The set of algebraic finite difference equations is formally equivalent to the set of partial differential equations, describing the flow properties of the open channel network, only in the limit, as the time and distance steps approach zero. In any application of finite difference n-ethods, it is necessary to examine the stability, computational error, and conservation properties of the finite difference scheme. If the weighted, implicit four-point finite difference scheme given by (1) is applied to the linearized shallow water equations, an examination of the properties of the finite difference equation m y be undertaken (Muir,1975). The results are that: (1) The growth factor for the propagation of error waves through tine is given by:

(XI=

1 + (2a-2)2a + (8-l)b 1 + 4a2a + 8b

where : 0 = weighting factor from the finite difference schene; UAX a = gh (at)' tan2 ( 7); ax b = KAt; h = mean depth of water; u = frequency of wave; u = man flow velocity; and

n = kkmning's roughness coefficient. Hence, if 0.5<8<1.0, the schem is unconditionally stable and, if 05W0.5, the scherre is conditionally stable depding on the relative magnitudes of Ax and At. (2) Qualitatively, the error involved in the approximation of the partial differential equation to the finite difference equation is of the form:

E

=

(28-1)0(At) + O(At2) +

O(AX2)

254

where : 0 = "the order of"; so when: 0 = 0.5, the schew has second-order accuracy, and the accuracy of the

solution is independent of the relationship between Ax and At. (3) The s c h e anserves mss over one t h e step if 6 = 0.5. one time step:

In general, over

when the superscripts refer to tine and the subscripts refer to the two ends of a branch. Hence, the departure from conservation is usually small, even if 0 varies from 0.5. All of the above results are obtained from an analysis of the linearized equation, but nmrical experiments have sham that the results are also applicable to the f u l l non-linear system of equations for a large number of practical examples. APPLICATION The theory presented in this paper has been programed in FoF?T.RAN IV, and the resulting computer program has been tested on a number of examples. Only one of these examples will be given here. The St. Clair River joins Lake Huron and Lake St. Clair and is approximately 45 kilmtres long, about 610 metres wide, and approximately 10 metres deep. The river has been extensively dredged and is a mjor navigational waterway. The lower end of the river, where it empties into Lake St. Clair, is a amplicated delta region with a very large m u n t of swampy ground and very shallow channels. Historically, the flow has varied. between 2830 and 6795 m3/sec with a 50 per cent exceedence level of 5210 m3/sec. The U.S. Army Corps of Engineers has maintained an extensive water level gauging network on the river for a number of years (P. Cox, personal cormmication, 1973). A m p of the river, giving the node numbering, is shown in Fig. 7. The geemtrical data used in the model are given in Table 1. Exarmna ' tion of the hydrographic field sheets shows that the cross-sectional shape of the river m y be approximated by a very wide rectangle and, as a result, the hydraulic radius (cross-sectionalarea/surface width) may be expressed as the mean depth. The river contains one mjor island, Stag.Island,which is included in the &el. The upper limit of the mdel is at Fort Gratiot and the lower limit is taken at AlgOMC, at the top of the delta region. No attempt was made to &el

the flow

255 TABLE 1.

Geometrical data for St. Clair River system. Datm is 173.74 M ( I m ) Node Number

Topwidth (m)

7 8 9 10 11 12 13 14 15 16 17 18 19 20

304.8 396.2 874.8 670.6 594.4 627.9 813.8 670.6 670.6 493.8 518.2 688.8 847.3 1005.8 868.7 688.8 682.8 841.2 457.2 466.3

Invert Level Below Datum (m)

Reach Length (m)

Manning’s n

8.59 8.63 4.97 7.61 6.68 7.01 7.51 4.97 6.19 7.43 8.78 7.04 6.18 5.08 6.48 8.03 8.02 6.58 3.09 3.13

2200 1166 1280 3871 3734 1189 6187 31 5547 1844 2743 1890 2896 3338 2804 2758 1859

0.033 0.028 0.023 0.023 0.023 0.020 0.020 0.020 0.020 0.025 0.025 0.025 0.021 0.023 0.023 0.025 0.022

3109 1920

0.020 0.020

-

-

Water Level Gauge Name Fort Gratiot

MBR Dry Dock Marysville

St. Clair

Marine City Roberts Landing Algonac

in the delta region, since the section geometry of this region completely invalidates the hypothesis of one-dimensional flow and since there is not enough water level data to properly calibrate a mdel. A mjor problem in delling a natural river lies in the choice of a friction factor. In the case of the St. Clair River, a great deal of information is available which allows for a very fine calibration. A useful pint in the calibration process is that a change in the friction factor in a given reach affects only the flow upstream of that reach. The calibration period chosen for the St. Clair River was 19 June, 1973. O n this date, discharge measurerents were being taken in the river by the U.S. Army Corps of mgineers at St. Clair, Michigan. The mean flow on this day was 6987 m3 Isec.

F X th ~ i s start,tke Ca’LihratLon procedure was: (1) use a discharge of 6087 m 3 / s e c and a h c m n w a t e r level a t Alqonac:

(2) start with a Manning’s friction factor of 0.020 which, from practical experience, is too low for this river; (3) make a cquter run and compare the calculated water levels with the observed water levels upstream; (4) adjust the friction factor in the lowest sections of the rodel until the calculated water level at the second gauge upstream agrees with the observed

256

LAKE HURON

AWATER LEVEL RECORDING STATION

/

Fig. 7 .

The St. Clair River.

257

water levels; (5) nwve upstream in this fashion, adjusting friction factors for one section at a time until all of the computed water levels agree with the observed water

levels within a certain predetermined accuracy; and (6) using two observed water levels and the determined friction factors, IMke another cquter run to check that the discharge is correct. If it is not, adjust all friction factors up or dawn by the same relative m u n t until the camputed and observed discharges mtch properly. Using this procedure, the mdel was calibrated for 19 June, 1973, to within k0.02 Etres as the mimm error in water level. This accuracy is certainly within the expected limits of any one-dirrwsionalmdel, assuming steady state flow conditions and considering the practical accuracy of discharge reasurmts. The calibration process used about 20 computer runs to mtch up the 9 water level gauges on the river. Although the process of calibration could easily be incorporated into the computer program, "intelligentguesses" allow a great saving in computer t k . After the rrodel was calibrated for 19 June, 1973, two additional runs were m d e to show that the &el would predict other flow conditions properly. The discharge for 19 June, 1973, was approhtely 6087 m3/sec, and this corresponds to an extremely high flow condition. The additional runs were: 18 July, 1968, which corresponds to a discharge of approxktely 5181 m3/sec, a "rredium" flow condition; and 13 November, 1964, a "low" flow condition with a discharge of approximately 4162 m3/sec. The observed water levels, computed water levels, and differences are shown in Table 2 . Unfortunately, not all of the water level gauges were in operation all of the tine. However, a good indication of the accuracy of the &el is given by this table. To show that the d e l properly predicts transients, it was run for a period during which a large transient passed through the St. Clair fiver. During 3-5 December, 1970, a storm was in progress, and this storm resulted in a one-metre storm surge on Lake Huron at Fort Gratiot. At that t k , three gauges were in operation in the St. Clair River: one at Fort Gratiot, one at St. Clair, Michigan, and one at Algonac. Data are available f r o m these gauges at hourly intervals. A run was m d e using the observed water levels at Fort Gratiot and at Algonac as boundary conditions. The schematization was revised to use only 13 nodes, and all data from the steady state runs were retained. Fig. 8 shms the observed values campared with the calculated values for node 10 (St. Clair, Michigan). The agreement is quite close, considering that only hourly water levels are available. This time step is clearly too long to give adequate resolution in the phase of the transient. Aside from the examination of transient flaw in the St. Clair River, a very

258

578. C

t

l'i I'

I'

577.5 (176.0m:

I

--

I1 I1

recorded water levels computed water levels

'

I I +-'

al

L t

.-=

577.c

?

3 &

. I -

(175.75 m 576.5

576.1 I

06:oo

18:OO

Dec. 3,1970 Fig. 8.

I

06:oo

I

I

18:00

Dec. 4, 1970

Water levels at St. Clair, Michigan.

I

I

06 :oo

I

259

TABLE 2. Observed and camputed water levels for the St. Clair River under various discharges. Datum is 173.74 M (IGLD) Node Number

19 June, 1973 18 July, 1968 13 November, 1964 Obs. (m) Computed (m) Obs. (m) Computed (m) Obs. (m) Computed (m) 3.11

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2.95 2.84 2.70

2.48

2.24 2.11 2.06 2.70

computed Discharge

3.11 2.97 2.96 2.94 2.83 2.73 2.73 2.65 2.65 2.48 2.44 2.34 2.33 2.26 2.19 2.13 2.09 2.06 2.73 2.65 6087

2.41 2.29 2.16 2.05

1.84

1.51 1.46 2.05

2.41 2.28 2.27 2.25 2.15 2.06 2.06 1.99 1.99 1.84 1.80 1.74 1.71 1.64 1.57 1.52 1.48 1.46 2.06 1.99 5181

1.52 1.44 1.21

1.01

0.64 1.21

1.52 1.41 1.39 1.37 1.29 1.21 1.21 1.14 1.44 1.01 0.97 0.92 0.88 0.81 0.75 0.70 0.67 0.64 1.21 1.14 4162

extensive study of the tidal propagation in a large Arctic estuary has been carried out by Budge11 (1977a). A part of the analysis consisted of an examination of the generation of shallow water tidal constituents using Power Spectral Analysis techniques. The results of t h i s analysis were excellent and have been published by Budge11 (1977b). SU1MARY AND CONCLUSIONS By

describing a network of open channels in terms of graph theoretic concepts,

t h e flow relationships in any network m y be expressed in terms of only four types of nodes. This allows a ccanputer program to be written which is totally method independent of any particular network toplogy. By using the -licit and a sparse matrix equation solver, the numerical msdel is independent of both the netmrk topology and the particular form of field equations. The m e 1 has been applied to a number of examples such as steady state river flow, transient river flow, and tidal propagation in large estuaries with excell-

m y be rewritten to take into account ent results. With minimum effort, the &el the dispersion of a conservative tracer and, hence, could easily be used for water quantity studies as well as flow studies.

260

REFEREXCES 1977a. Tidal Propagation in Chesterfield Inlet, N.W.T. Manu. Budgell, W.P. Rep. Ser.: 3, Ocean and Aquatic Sciences, Canada Centre for Inland Waters, Burlington, Ontario. Budgell, W.P. 1977b. Numerical Simulation of the Tides in Chesterfield Inlet. In: Proc. XVII Congress of the I.A.H.R., Baden-Eaden, Germany. Chandrashekar, M., L.R. Muir, and T.E. UMY. 1975. A Numerical Two-Dimensional Flow W e 1 for River Systems. In: Proc. Modelling 75, Am. S o c . Civ. Eng. Specialty Conference, San Francisco, 407-425. Chandrashekar, M., L.R. Mukc, and T.E. UMY. 1976. A General Two-Dimensional River Simulator. In: Proc. Int. Symp. on Large Engineering Systems. The University of Manitoba. (in press). Garrison, J.M., J.P. Granju, and J.T. Price. 1969. Unsteady Flow Simulation in Rivers and Reservoirs. J. Hydraul. Div. Am. S o c . Civ. Eng., 95:5:1559-1576. Muir, L.R. 1975. Unsteady Flow in Networks of Open Channels. Manu. Rep. Ser.: 1, Ocean and Aquatic Sciences, Canada Centre for Inland Waters, Burlington, Ontario. Price, R.K. 1974. Comparison of Four Numerical Methods for Flood Routing. J. Hydraul. Div. Am. S o c . Civ. mg., 100:7:879-899. Strelkoff, T. 1970. Numerical Solution of Saint-Venant Equations. J. Hydraul. Div. Am. Soc. Civ. Eng., 96:1:223-252.

261

A NEW APPROACH TO THE COMPUTATION OF TIDAL MOTIONS IN ESTUARIES B.M. JAMART and D.F. WINTER Department of Oceanography, University of Washington, Seattle, WA 98195 ABSTRACT In this paper we present a summary of a new procedure for computing periodic tidal motion in estuaries with irregular boundaries and variable depth. The new method, described by Pearson and Winter (1977), appears to provide both computational speed and numerical accuracy in a wide variety of problems of practical interest. The starting point of the analysis is the standard set of vertically-integrated, time-dependent equations expressing conservation of horizontal momentum and mass in two dimensions. Since the motion in the estuary is assumed to be periodic, the dependent variables are Fourier decomposed. However, in order to avoid awkward Fourier decomposition of nonlinear terms and to decouple the different modes, advection and friction are evaluated by an iterative procedure. The time-dependent equations of motion are replaced by an equivalent set of modal equations, and it is shown that the boundary value problem consisting of the modal equations and appropriate boundary conditions can be rephrased in terms of a variational principle. The variational principle is used together with a finite element method to solve for the unknown flow variables.

We describe here an application of the method to a

segment of Hood Canal, Washington, where there is an interest in pollutant dispersal by tidal currents.

INTRODUCTION Pearson and Winter (1977) recently described a new approach to the computation of periodic tidal motion in homogeneous estuaries and embayments with irregular shorelines and variable depth. In this paper we set forth the main features of the approach and describe some preliminary results of an application of the method to a segment of Hood Canal, a deep, weakly stratified inlet in Puget

Sound, Washington, U.S.A.

The highlights.of the method of Pearson

and Winter can be summarized as follows:

since shallow water theory

is assumed to apply, the governing equations are the standard vertically-integrated, time-dependent equations expressing conservation of horizontal momentum and mass in two dimensions [see, for example, Hess and White ( 1 9 7 4 ) l . The momentum equations include convective acceleration, Coriolis acceleration, and terms describing the effect of wind stress and bottom friction. Boundary conditions are specified along the shoreline boundaries, across the mouth of the estuary, and along other open boundaries.

(The appropriateness

of various boundary conditions is discussed by Pearson and Winter.) Since the motion in the estuary is assumed to be periodic, the dependent variables (water surface height and depth-averaged velocities) are Fourier decomposed. Consequently, the time-dependent equations of motion are replaced by an equivalent set of modal equations, with only x and y as independent variables. In order to circumvent the coupling between modes arising from the Fourier decomposition of the nonlinear terms, the latter are treated by an iterative procedure, similar to the one described below. Next, the boundary value problem consisting of the modal equations and appropriate boundary conditions is rephrased in terms of an equivalent variational principle. The variational principle is then used together with a finite element method to solve for the unknown variables, i.e., the coefficients of the Fourier series representations of the free surface elevation and of the depth-averaged velocities throughout the estuary. The actual space- and time-dependent currents and heights can then be reconstructed by Fourier synthesis. The present discussion will be restricted to a special case of more general problem statements discussed by Pearson and Winter (1977); the reader is referred to that paper for further details. In the following section, a somewhat simplified version of the governing equations is set forth and a review of the analytic procedure is presented. The next section is devoted to the numerical solution of the boundary value problem by a finite element technique. Preliminary results of tidal computations in Hood Canal, Washington, are presented in the last section. FORMULATION OF THE PROBLEM Denote by A the domain of interest in a homogeneous estuary bounded by an open section rl, along which the periodic free surface height is prescribed, and a solid shoreline section r 2 along which

263

the normal flow component vanishes.

It is assumed that the effect

of bottom friction on the horizontal momentum balance can be represented by a term proportional to y ‘ D ,

where 5 denotes the vertically

averaged velocity and D(x,y) represents the time-mean depth. For convenience, in this analysis we neglect convective accelerations and wind stress. Since the estuary to which we intend to apply the analysis is deep and narrow, we neglect Coriolis acceleration and horizontal diffusion, and we linearize the equation for conservation of mass. to

With these simplifications, the equations of motion reduce

+

u -t

gVh

ht + V.

U

=

- E =

(uD)

=

D 0

,

(2)

where h(x,y,t) is the elevation of the water above mean sea level, the subscripts denote derivatives, V is the nabla operator, and E denotes a (constant) friction coefficient. The boundary conditions are

h

h*(x,y,t) on

=

_ u.n

=

0

on

rl

T2

(3)

(4)

where h* is a known function and g is the outward-directed unit normal vector. Let w be the fundamental circular frequency of the periodic motion and let N be the number of modes required to represent the temporal variation of the dependent variables. In the absence of a “stationary” forcing function (as might be produced by persistent wind or river discharge), and since some of the nonlinear terms have been linearized and others neglected, there is no need to include a zeroth-order mode in the Fourier expansions of thedependent variables and forcing term. We can write, therefore,

N =

Re C n=l

gn

(x,y)e -inwt

(5)

264

Substitution of (5) and (6) into (1)- (4) leads to coupled modal equations o f the form

En -

inw

gVHn

inw Hn - V.

=

EU D -n

(EnD) =

0

with boundary conditions Hn

=

rl

on

Hn*

(10)

Clearly, there is no coupling between modes in the linearized problem.

Elimination of LJn between (8) and (9) yields an equation

for Hn that can be written as

V. (DVH,)

+

E2

~

+

inwED

2VD.VH

~

+

+

n n w

n2 w 2 D

+

~

9D

inwE Hn

=

0

(12)

The boundary condition (11) becomes -aHn - -

0

(13)

an Since we wish to compute Hn at a large number of points in the estuary, it is desirable for computational reasons (core memory limitation, essentially) to solve separately for the real and imaginary parts of H However, it can be seen from Eq. ( 1 2 ) that the n' real and imaginary parts of H are coupled through two of the n frictional terms (physically, this corresponds to a modification of the wave phase due to bottom drag). In order to affect the separation o f the real and imaginary parts of Hn, we employed an iterative procedure in which the frictional terms responsible for the coupling are first placed on the right-hand side of the equations. Moreover, in order to guarantee the existence of a relatively simple equivalent

26 5

variational principle, we also put uncoupled frictional terms on the right-hand side and treat them all in the global iterative scheme. Thus, we set Hn = hn +isn in Eqs. (12) and (13), and assume that the dominant forcing excites hn (i.e., hn* # 0 and Isn*l /hn*l). Using superscripts to index the successive steps of iteration, we write

where

The boundary conditions for (14) and (15) are, respectively, hn*

on

rl

n*

on

rl

hn

-

sn

= s

-asn _ an

0

on

r2

The iterative scheme proceeds as follows: as a first step (k = O ) , compute h (’) by solving (14) subject to (18) with fl = 0; then n in (17), making f2 !j 0, and thereby ensuring the substitute hn (’) existence of a nontrivial sol.ution sn (2) for the pair (15) - (19)

* = 0. The next step (k = 1) is the computation of hn ( 3 ) n using hn(l)and sn (2) in (16), followed by a calculation of s,(~) based on hn(3) and s (2). The procedure is then repeated until n sufficient convergence is achieved. The problem formulation for hn, i.e., solving (14) subject to even if s

(18) with known fl, is equivalent to the variational condition 6J = 0 for a l l 6h

n

vanishing on

rl, where (20)

266

on the (The requirement that the normal derivative of hn vanish variationally free boundary (r2) is the natural boundary condition for this well-known variational principle.) A similar variational principle holds for sn. FINITE ELEMENT SOLUTION Although the use of the finite element method (F.E.M.) in hydrodynamics is a fairly recent development, there are now available a number of texts and papers which set forth the theoretical basis of the method and examples of its use; presentations by Gallagher, Oden, Taylor, and Ziemkiewicz (1975), Connor and Wang ( 1 9 7 3 ) , Gray, Pinder and Brebbia (1977) are representative. We shall only briefly review the method as it pertains to the problem at hand, and refer the reader to the literature for more detailed accounts. The domain A is first divided into a number of finite elements (of triangular shape in this particular application) and we seek to compute the values of h and sn at the element vertices. n Let M denote the total number of vertices (or nodes), ml of which are located on the open boundary.

The unknown variable (say, hn) and parameters (e.g., D) are then approximated over each element by some interpolative representation in terms of their nodal values (hni, Di): we use the simplest possible basis (or shape) functions,i.e., linear interpolators and, as a consequence, the function J in Eq.(20),expressed as the sum of the integrals over each element, is a quadratic function of the hni values. variational condition requires that we set

The

for all nodes except those belonging to rl. This yields a set of (M-ml) linear algebraic equations in M unknowns. This set, together with the ml equations expressing the boundary condition on

rl,

completely determines the problem. The numbering of the nodes can be done so as to minimize the

bandwidth of the coefficient matrix, because the partial derivative of J with respect to a particular hni involves only integrals over those elements having.hni as one of their nodes. The integration over each element is carried out either analytically (as for the contribution to the coefficient matrix of the first two terms of (20)) or numerically (as is necessary for the right-hand side

267

vector, since fl is a nonlinear function of position). The L-U decomposition of the banded coefficient matrix was performed by an algorithm that utilizes row equilibration and partial pivoting. Since the matrix is the same for both variables hn and sn and does not change during the iteration process, its triangular decomposition need be done only once. Finally, after the calculation of hn and sn has converged, the Fourier coefficients U of the velocities are computed according to -n Eq. (8). The differentiation of the Hn field, that is linearly interpolated over each triangle, gives constant element values that are taken as representative of the velocity coefficients at the centroid of each element. APPLICATION TO HOOD CANAL In an effort to ascertain the potential of the method for resolving efficiently the intricacies of natural tidal flows, we selected a segment of Hood Canal, Washington,as a test region. Hood Canal is the westernmost arm of Puget Sound (Fig. l), which is a complex network of fjords and channels that communicates with the Pacific Ocean through the Strait of Juan de Fuca. The construction of a base for nuclear submarines at Bangor on the shore of Hood Canal makes the region of particular environmental interest and concern. Qualitative information concerning tidal currents in Hood Canal has been obtained from the Puget Sound hydraulic model and is available in various publications and technical reports from the University of Washington (e.g., Rattray and Lincoln, 1955; McGary and Lincoln, 1977). Although Hood Canal is not a truly homogeneous estuary, the intensity of freshwater runoff is usually low, so that the estuarine circulation mode does not modify substantially the barotropic tidal mode. The test region has three open boundaries. The driving boundary condition (tidal height) is imposed along the northernmost boundary which is located off South Point on the seaward side of the sill. Rectangular basins of uniform depth whose volumes are roughly equivalent to those of Dabob Bay and the southern portion of Hood Canal are appended to the other two open boundaries, denoted r 3 and r4, where a simple analytical solution is matched to the interior solution. In order to simulate the influence of Dabob Bay and southern Hood Canal on tidal currents in the Bangor region, we approximated water motion in the appendages by linear, frictionless long waves which are perfectly reflected at the closed ends of the

268

Figure 1.

Map of Puget Sound. is the test region.

The hatched segment of Hood Canal

269

rectangular basins. The matching condition relates the normal surface slope to the elevation on those boundaries. Such a condition can easily be incorporated in the variational principle. Alternatively, it can be implemented by direct manipulation of the algebraic system of equations in the following way: first, assume that the boundary elements along r 3 and r4 are part of the domain in which the analytical solution is applicable; then replace theequations generated by the variations of the variable along such boundaries (e.g., aJ/ahnk = 0, h n k E r 3 or

r4

)

by expressions that

relate the solution at each boundary node to that at a corresponding interior point in the direction of propagation of the one-dimensional wave. We performed numerical experimentation with both the methods for the case of a rectangular basin and each was found satisfactory. Weusedthe latter (direct manipulation of the equations) in the Hood Canal production runs.

The finite element grid shown in Fig. 2 was prepared manually, and consists of 1495 triangles (853 nodes) of gradually varying area. The mesh resolution is finest in the region of principal interest between the seaward boundary and Hazel Point. Also, the resolution is finer along the sides of the channel as required for a fair approximation of the bottom topography (Fig. 3). The shallowest depth along the shoreline boundary is taken as 5 fathoms (about 9 m) so as to avoid unrealistically high friction coefficients and to make certain that linearization of the mass conservation equation remains a valid approximation. The bandwidth of the coefficient matrix is equal to twice the largest difference between node numbers of an element plus one, and in this case is 51: its minimization is hampered somewhat by the bifurcation of the channel at the southern end. The next seven figures display selected results of a calculation corresponding to the hypothetical case of single mode forcing of period T = 12.5 hours , h? = 2 m on r l , and E = 3 . 1 0 - 3 m sec-l. P Fig. 4 shows contours of the solution for h (x,y), the cosine com1 ponent of the elevation. Convergence within a tolerance of 0.5% for both hl and s1 was achieved after the sixth iteration. The next figure shows the velocity field at the time of maximum flood, when the sole effect of friction is to slow the flow along the (shallow) shoreline. In this and subsequent figures, a denotes nondimensional time t/T . The velocity field is more varied during P the transition from flood to ebb, i.e., around a = 0 . The crosschannelvariationsof phase due to "bottom" friction are such that,

270

l +

d

~

N -

+

I

I

I

0

I

1.

~

-...

~

. -1 I

/

0

r

.

W

W

b

' M NIW Z P ' 3 3 0 221 40 lS3M SU313W

4

~

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f

F i n i t e element g r i d .

3

0 0 0 L

-

W -

Figure 2.

0

0

0 0 0 0

0

N - - N I I

a

0 0 0

0

N

0

0 0

0)

0

-

0 0

m

c

0 0

0

P

0

-

- 0

zo

N

z . -

1a 0

00 0

P -

C

C

.. .. .

.

1

0

Q

--I

z

H

a

..I:.:::..

7

0

0

0

c

0

0

0

:.: :

0

- N 0 0 P

0 0 0

W

0 0 0

0 0 0

~

0 0 0

~

0 0 0

.

0 0 0

J

J

W

O

-

N

W

*c

P

~

axn6Td

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

@

METERS WEST OF 122 DEG. 42 M I N . W .

' u o ~ b a xq s a q 3 0 KqdpxbodoL

~

as observed in nature, the tide turns first along the shores. This topographical effect produces a shear that generates eddy-like structure during the transition period. Because of the several simplifications introduced in the equations and forcing function considered in this example, we have not made quantitative comparisons between our preliminary results and available hydraulic model data.

However, it can be said that so far as their scale and

position are concerned, the eddy patterns shown in these figures are similar to those observed in the Puget Sound hydraulic model. For programming convenience, we used a CDC 7600 computer that has a large core memory. The program solving the problem described here utilizes a mass storage of about 125,000 words, is compiled in less than 3 CP seconds and executed in 6.5 CP seconds. DISCUSSION In this paper we have described preliminary steps and results of a study intended to demonstrate the feasibility of the method proposed by Pearson and Winter (1977) for the computation of tidal motion in homogeneous estuaries. The new method derives its efficiency, as compared with time-stepping procedures, from a combination of harmonic decomposition and an iterative scheme to treat nonlinearities. In solving the resulting elliptic problem, the method takes advantage of the superior flexibility introduced by the finite element approach.

Lastly, the reformulation of the

problem in terms of a variational principle, although not a necessary condition for the applicability of a finite element procedure, decreases the order of the highest derivatives to be approximated and simplifies the implementation of the boundary conditions. Finite element methods are becoming increasingly popular among tidal modelers (e.g., Grotkop, 1973; Connor and Wang, 1973; Taylor and Davis, 1975; Brebbia and Partridge, 1976; Sundermann, 1977). Some workers, however, contend that they are economically unattractive, in comparison to finite difference methods, for practical calculations (e.g., Weare, 1976).

Our experience with the present

approach leads us to assert that the combination of harmonic decomposition and iteration is likely to provide computational efficiency in a wide variety of practical problems. We were pleased to learn at the Ninth Liege Colloquium that another group of researchers (Le Provost and Poncet, 1977) is currently working along similar lines.

0 0

0 0

0 0

0 0

z

H

+ 0

a_ H _J _J

0 0

a

0 0

0 0

'2-

a

-

m m U m

a

0

0 0

0 0

0 0

0 0

m 0 m 0 P 0 ( 0D

0 0 0 0 0 0 0 0 0 0 0m 0d 0 m 0 N z

Isopleths of hi, values in meters.

' M ' N I W ZP '33R 221 =ID lS3M SU313W

0 0 0 0 d 0 O 0 2 2 0 z z 2 d d

F i g u r e 4.

0

I -I [1L

z

0

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0 0

2I EI

-

273

3 3 3 3 N 3

3 3

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D 4

3 3 3

d 4

3 3 *

z

3 2 N +

2E

Ez 00 o w d

0 0 CD P

0 O L

I-

1

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o=

g z

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I

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0 0

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0 0

274

0 ~

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0

F i g u r e 5.

0

0 0

~

c

z H

0 LL H

-J

0 0

a

0 0

~0

0 0

~ ~

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0 0

a 0

= -0.25.

0

z

-I H

€3

a. I

0 0

0 0

zI~ *EI

t3 0

0 m

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0 0 c\J

0 0 0 03

+

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*

0 0 0 +

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Yl

m

.

z

r*

1 t-

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-=I1

3 3 3

N 1

0

0

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c\J=

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oz

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0-

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oz

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m a n

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' M 'NIW ZP '33a 221 40 lS3M S E U W Velocity f i e l d at t i m e

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>

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2Zl A0 l S 3 M S8313W V e l o c i t y f i e l d a t t i m e a = -0.04.

' M 'NIW ZP '3311

~ 0 y0 y0 0 z

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Figure 6.

275

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D

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m

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@

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b

ZZI & l SI 3M

Velocity field at time a = -0.01.

' M 'NIW ZP ' 3 3 0

a m + d d 2 2 dd - o4m

Figure 8.

W

>

?

I

0 0 0

0 1 ) -

0

I

~ .+@J

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277

218

Figure 9 .

+ z

H

0 a_ H

_1

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a

t-

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d ? 2 + d O d dN &

Figure 10.

~

0 0 0

P

0 0 0

= +0.01.

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ci

0 0

0 W (I)

II

\

0 0

ci

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280

ACKNOWLEDGMENTS The research reported here was supported in part by the Oceanography Section, National Science Foundation, under Grant OCE76-00406, and in part by the Washington Sea Grant Program, which is maintained by the National Oceanic and Atmospheric Administration, U.S. Department of Commerce.

We are grateful to

Ms. Jane M. Glass and Dr. Lawrence Lewin for their assistance in preparing the finite element grid of Hood Canal and the visual display of computer results. REFERENCES Brebbia, C.A., and Partridge, P.W., 1976. of water circulation in the North Sea. Modelling, l(2) : 101-107.

Finite element simulation Applied Mathematical

Connor, J.J., and Wang, J.D., 1973. Mathematical models of the Massachusetts Bay. Part I: Finite element modeling of twodimensional hydrodynamic circulation. Report No. MITSG 74-4, M.I.T., Cambridge, 57 pp. Gallagher, R.H., Oden, J.T., Taylor, C., and Zienkiewicz, O.C., Eds., 1975. Finite Elements in Fluids, Vol. 1, Wiley Interscience, London, 290 pp. Gray, W.G., Pinder, G.F. and Brebbia, C.A., Eds, 1977. Finite Elements in Water Resources, Pentech Press, London, 1008 pp. Grotkop, G., 1973. Finite element analysis of long-period water waves. Computer Methods in Applied Mechanics and Engineering, 2: 147-157. Hess, K.W., and White, F.M., 1974. A numerical tidal model of Narragansett Bay. Marine Tech. Rept. No. 20, University of Rhode Island, Kingston, 141 pp. Le Provost, C., and Poncet, A., 1977. Finite element method for spectral modelling of tides. In preparation. ( L e P r o v o s t ’ s a f f i l i a t i o n i s w i t h t h e I n s t i t u t de M e c a n i q u e , G r e n o b l e , F r a n c e . ) McGary, N., and Lincoln, J.H., 1977. Tide Prints: Surface Tidal Currents in Puget Sound. Washington Sea Grant Publication distributed by University of Washington Press, Seattle, Washington, 51 pp. Pearson, C.E., and Winter, D.F., 1977. On the calculation of tidal currents in homogeneous estuaries. Journal of Physical Oceanography, 7 (4): 520-531. Rattray, M., Jr., and Lincoln, J.H., 1955. Operating characteristics of an oceanographic model of Puget Sound. Transactions, American Geophysical Union, 36: 251-261. Skdermann, J., 1977. Computation of barotropic tides by the finite element method. In: Finite Elements in Water Resources, Pentech Press, London, pp. 4.51-4.67

281

Taylor, C., and Davis, J . M . , 1975. Tidal propagation and dispersion in estuaries. In: Finite Elements in Fluids, Vol. 1, Wiley Interscience, London, pp. 95-118. Weare, T.J., 1976. Finite element or finite difference methods f o r the two-dimensional shallow water equations? Computer Methods in Applied Mechanics and Engineering, 7: 351-357. Contribution No. 1002, Department of Oceanography, University of Washington, Seattle, WA.

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283

A NUMERICAL MODEL OF BARATROPIC MIXED TIDES BEmEEN VANCOUVER ISLAND AND THE

MAINLAND AND ITS RELATION TO STUDIES OF THE ESTUARINE CIRCULATION

P.B.

CREAN

I n s t i t u t e of Ocean S c i e n c e s , Department of F i s h e r i e s and the Environment, Sidney,B.C.

INTRODUCTION

The waters between Vancouver I s l a n d and t h e mainland c o a s t s of B r i t i s h Columbia and the S t a t e of Washington c o n s t i t u t e a complex e s t u a r i n e system i n which t h e dominant s o u r c e o f f r e s h w a t e r i s t h e F r a s e r River, Fig. 1.

Before

any r e a l i s t i c e c o l o g i c a l models r e l a t i n g t o t h e major f i s h e r i e s of t h e region, i n c l u d i n g p o s s i b l e e f f e c t s o f domestic and commercial e f f l u e n t s o r o f o i l s p i l l s , can be undertaken it i s necessary t o achieve a q u a n t i t a t i v e understanding of the p h y s i c a l p r o c e s s e s t h a t determine t h e e s s e n t i a l c h a r a c t e r of t h e flow f i e l d and d i s t r i b u t i o n s o f s c a l a r p r o p e r t i e s .

This i n v o l v e s an extended series

of s t u d i e s and i t i s t h e purpose o f this paper t o review b r i e f l y the p r e s e n t s t a t u s of t h i s work w i t h emphasis on t h e r o l e s played by the t i d e s and t i d a l

streams, then t o d e s c r i b e , and p r e s e n t some r e s u l t s from, a numerical t i d a l model and l a s t l y t o i n d i c a t e i n g e n e r a l terms how t h e s e r e s u l t s are p r e s e n t l y being used t o i n t r o d u c e the e f f e c t s o f t i d e s i n t o a n experimental "upper l a y e r " model used t o s t u d y t h e flow o f f r e s h w a t e r from t h e F r a s e r River over t h e s a l t

w a t e r i n the S t r a i t o f Georgia. The b a s i c geographical f e a t u r e s o f the r e g i o n , shown i n Fig. 1, c o n s i s t e s s e n t i a l l y of a n o u t e r s t r a i t (Juan de Fuca S t r a i t ) 140 km long and an i n n e r

s t r a i t ( S t r a i t o f Georgia) of l e n g t h 2 2 0 km.

A complex system of i s l a n d s

(San Juan and Gulf I s l a n d s ) and passages i s l o c a t e d between t h e i n n e r and outer straits.

The i n n e r s t r a i t i s f u r t h e r connected t o t h e open s e a thruugh

a network of narrow channels, c o n t a i n i n g r e l a t i v e l y shallow s i l l s , which l e a d i n t o Johnstone S t r a i t , a s i n g l e passage between Vancouver I s l a n d and t h e mainland. Bottom topography i n t h e region o f the model i s i l l u s t r a t e d i n Fig. 2 u s i n g depth contours i n fathoms taken from e x i s t i n g hydrographic c h a r t s . I n the o u t e r s t r a i t maximal depths decrease g r a d u a l l y from some 300 m near t h e e n t r a n c e t o about 100 m i n the channels between t h e s h o a l s i n i t s i n n e r end.

I n t h e region o f the San Juan and Gulf I s l a n d s , two main conveying channels between the i n n e r and o u t e r s t r a i t s may be d i s t i n g u i s h e d , Haro and Rosario Straits.

Of t h e two Haro S t r a i t c o n t a i n s t h e d e e p e s t channel, c h a r a c t e r i z e d

284 by depths of o r d e r 180 m - 250 m. a r e generally of o r d e r 150 m

-

Maximal depths along t h e S t r a i t of Georgia

400 m.

A topographical f e a t u r e o f p a r t i c u l a r

i n t e r e s t i n t h e p r e s e n t c o n t e x t i s t h e l a r g e l o n g i t u d i n a l change i n crosss e c t i o n a l a r e a of t h e s t r a i t i n t h e v i c i n i t y o f t h e e x t e n s i v e shallow bank t h a t d e l i n e a t e s the seaward l i m i t s of t h e F r a s e r River d e l t a .

128"

50'

50°

PACIFIC

OCEAN

48"

18"

128'

124"

126'

Fig. 1. The waters between Vancouver I s l a n d and t h e mainland c o a s t . l i n e s denote t h e boundary of t h e numerical model g r i d scheme.

I

The d o t t e d

A c o n s i d e r a b l e number o f s t u d i e s p e r t i n e n t t o t h e p h y s i c a l oceanography of

t h e system have been made over the p a s t f i f t y y e a r s .

Hutchinson and Lucas (1931)

have described t h e water p r o p e r t i e s , n u t r i e n t c o n c e n t r a t i o n s and phytoplankton populations i n t h e S t r a i t of Georgia.

The d i s t r i b u t i o n and o r i g i n s of t h e

primary water masses w i t h i n t h e Juan de Fuca S t r a i t and the S t r a i t of Georgia were described by Redfield (1950a).

Waldichuk (1957) and a l s o Tully and

Dodimead (1957) have d i s c u s s e d t h e p h y s i c a l oceanography o f t h e S t r a i t of Georgia.

285

Fig. 2 .

Contoured depths i n fathoms i n the modelled region.

Oceanographic f e a t u r e s o f Juan de Fuca S t r a i t have been p r e s e n t e d by Herlinveaux and Tully ( 1 9 6 1 ) , and more g e n e r a l l y , o f the passage between Vancouver I s l a n d and t h e mainland by Herlinveaux and Giovando ( 1 9 6 9 ) .

Measurements of s u r f a c e

flows a s s o c i a t e d w i t h the F r a s e r River discharge have been made by Giovando and Tabata ( 1 9 7 0 ) , while a study of sub-surface c u r r e n t r e c o r d s , each of about one y e a r d u r a t i o n , from a c r o s s - s e c t i o n a l

a r r a y o f c u r r e n t meters l o c a t e d i n t h e

approximate v i c i n i t y o f the F r a s e r d e l t a i n t h e S t r a i t of Georgia h a s been r e p o r t e d by Chang, Pond, Tabata ( 1 9 7 6 ) .

Good p r e d i c t i o n s o f t i d a l e l e v a t i o n s

along t h e F r a s e r River under d i f f e r e n t d i s c h a r g e r a t e s have been o b t a i n e d using

a one-dimensional numerical model (Ages and Woollard, 1976).

Currents and

s c a l a r p r o p e r t i e s i n Johnstone S t r a i t , t h e s i n g l e channel through which any exchanges of water between t h e n o r t h e r n p a r t of the S t r a i t of Georgia and t h e

open s e a must occur, have been d i s c u s s e d by Thomson (1976).

Extensive d a t a on

water e l e v a t i o n s and c u r r e n t s have been g a t h e r e d by t h e Canadian Hydrographic Service.

Data records from c u r r e n t meters moored over c r o s s - s e c t i o n s i n Juan

de Fuca S t r a i t , S t r a i t o f Georgia and Johnstone S t r a i t a r e a v a i l a b l e , e t al.

1970,a,b,c,

1972,a,b,c,

Tabata and S t i c k l a n d 1972,a,b,c,d,

1973, Huggett e t a l . 1976a,b).

(Tabata

Department o f Environment

A d d i t i o n a l v e l o c i t y measurements

fiom v e s s e l s anchored i n t h e g e n e r a l v i c i n i t y of the F r a s e r River plume have a l s o been r e p o r t e d (Tabata e t a l . 1970).

A n e x t e n s i v e survey of t i d a l c u r r e n t

measurements i n Juan de Fuca and Rosario S t r a i t s has been p r e s e n t e d by Parker (1977).

Data on t h e d i s t r i b u t i o n o f s c a l a r p r o p e r t i e s , has a l s o been o b t a i n e d ,

p a r t l y w i t h a view towards t h e subsequent three-dimensional of processes dominating the e s t u a r i n e c i r c u l a t i o n .

numerical s i m u l a t i o n

Thus, i n t h e course o f

twelve oceanographic c r u i s e s over a y e a r , the b a s i c s e a s o n a l changes i n w a t e r p r o p e r t i e s throughout the S t r a i t o f Georgia, t h e v i c i n i t y o f t h e San Juan I s l a n d s and Juan de Fuca S t r a i t were determined (Crean and Ages, 1971).

F u r t h e r scalar

d a t a i n Juan de Fuca S t r a i t , where t h e seaward boundary o f a three-dimensional model might p l a u s i b l y be l o c a t e d , and over a p e r i o d when an e x t e n s i v e a r r a y o f moored c u r r e n t meters were i n p o s i t i o n over a c r o s s - s e c t i o n of t h e S t r a i t , have a l s o been o b t a i n e d (Crean and Miyake, 1976, Crean and Lewis, 1976).

The f i r s t

of t h e s e d a t a sets i n c l u d e s simultaneous time-series o b s e r v a t i o n s by two v e s s e l s i n c o l l a b o r a t i o n w i t h the Department o f Oceanography of the U n i v e r s i t y of Washington.

I n a d d i t i o n t o t h e above, numerous l o c a l oceanographic s t u d i e s

have been made of t h e v a r i o u s mainland i n l e t s .

The dominant e s t u a r i n e f e a t u r e s o f t h e system may be i l l u s t r a t e d by w i n t e r and summer s a l i n i t y d i s t r i b u t i o n s over v e r t i c a l s e c t i o n s taken through t h e median l i n e s along t h e S t r a i t of Georgia, Haro and Juan de Fuca S t r a i t s , Fig. 3 (Crean and Ages, 1971).

The major source of s a l t water i s l o c a t e d a t depth

a t the e n t r a n c e of Juan de Fuca S t r a i t .

Between w i n t e r and summer t h e r e occurs

a marked i n c r e a s e i n t h e s a l i n i t y of t h e deep w a t e r i n Juan de Fuca S t r a i t as shown by t h e i n c r e a s e d shaded a r e a , r e p r e s e n t i n g w a t e r having a s a l i n i t y i n excess o f 32.5%, which i s thought t o be a s s o c i a t e d w i t h upwelling p r o c e s s e s on the continental s h e l f .

The major s i n g l e source o f f r e s h water e n t e r i n g t h e

system i s t h e F r a s e r River which i s e s t i m a t e d t o account f o r some 70-75% o f t h e t o t a l f r e s h water i n p u t t o t h e i n s h o r e passage between Vancouver I s l a n d and t h e mainland (Herlinveaux and T u l l y , 1961).

There e x i s t s a s t r o n g s e a s o n a l

v a r i a t i o n , determined by the thaw i n t h e mountainous h i n t e r l a n d , i n t h e s u r f a c e d i l u t i o n i n t h e g e n e r a l v i c i n i t y of t h e r i v e r mouth.

Of p a r t i c u l a r i n t e r e s t i s

t h e region of t h e " s i l l s " between t h e i n n e r and o u t e r s t r a i t s where the h o r i z o n t a l s a l i n i t y g r a d i e n t s are i n d i c a t i v e o f s t r o n g mixing p r o c e s s e s .

The

i n t e r m e d i a t e s a l i n i t y w a t e r which r e s u l t s can e i t h e r move seaward a t t h e s u r f a c e i n Juan de Fuca S t r a i t o r s i n k t o form deep w a t e r i n the S t r a i t o f Georgia.

As

w i l l be shown below, the t i d a l streams i n this region are s t r o n g , and c l e a r l y

e x e r t a major i n f l u e n c e on the fundamental e s t u a r i n e c i r c u l a t i o n . Annual v a r i a t i o n s i n the water s a l i n i t i e s a t depth 100 m are e v i d e n t throughout these c o a s t a l waters (Pickard, 1 9 7 5 ) .

Following the summer s a l i n i t y maximum i n

Juan de Fuca S t r a i t there i s evidence of a f a l l maximum i n Boundary Passage followed by a w i n t e r maximum i n the S t r a i t of Georgia. annual mean values a r e s m a l l .

Long-term v a r i a t i o n s of

Reference below t o a three-dimensional numerical

s i m u l a t i o n o f t h e main f e a t u r e s of the e s t u a r i n e c i r c u l a t i o n w i t h i n the system may be taken t o imply t h e s e a s o n a l v a r i a t i o n s about t h e normative f e a t u r e s o f the w i n t e r and summer s a l i n i t y d i s t r i b u t i o n s a s s e c t i o n a l l y shown i n Fig. 3 . I t is of i n t e r e s t t o n o t e t h a t i n r e c e n t y e a r s , t h e p o s s i b i l i t y o f

c o n t r o l l i n g t h e F r a s e r River discharge has been d i s c u s s e d .

I n such a c a s e , t h e

major i n t r u s i o n of s a l t water p r e s e n t a t depth i n Juan de Fuca S t r a i t during t h e summer would n o t be accompanied by maximal d i l u t i o n of t h e s u r f a c e waters i n the S t r a i t of Georgia.

Any s i g n i f i c a n t a s s o c i a t e d change i n t h e e s t u a r i n e

c i r c u l a t i o n might e f f e c t primary food production c r i t i c a l t o the major salmon f i s h e r y a s s o c i a t e d w i t h t h e F r a s e r River.

Thus one of t h e o b j e c t s of t h e s e

s t u d i e s concerns the p r e d i c t i o n of any such change l i k e l y t o r e s u l t from the imposition of such a c o n t r o l . Both t i d e s and winds e x e r t a major i n f l u e n c e on the motions o f f r e s h water e n t e r i n g t h e S t r a i t o f Georgia from the F r a s e r River.

S a t e l l i t e pictures

such as t h a t shown i n Fig. 4 , can sometimes g r a p h i c a l l y i l l u s t r a t e the g e n e r a l e x t e n t o f t h e shallow r i v e r plume t h a t forms i n t h e s o u t h e r n p a r t o f the S t r a i t o f Georgia when s h a r p c o n t r a s t s occur between the s i l t y w a t e r i n t h e plume and s e a water i n t h e S t r a i t .

Reference t o F i g s . 1 and 2 p e r m i t ready r e c o g n i t i o n

of such f e a t u r e s as the S t r a i t o f Georgia, the Gulf and San Juan I s l a n d s , and t h e F r a s e r River D e l t a , j u s t t o t h e n o r t h o f which l i e s t h e g r e a t e r d i s t r i c t of Vancouver.

The Main A r m o f t h e F r a s e r River, which carries about 80% of the

t o t a l d i s c h a r g e , i s c l e a r l y d i s t i n g u i s h e d and t h e southward v e e r i n g flow i s p a r t l y brought about by an angled t r a i n i n g w a l l , n o t e v i d e n t i n t h e p i c t u r e , which extends o u t t o t h e edge o f the deep w a t e r . Unfortunately such p i c t u r e s do n o t a f f o r d a ready means o f following t h e changing geographical e x t e n t of t h e plume, s i n c e c l e a r s k i e s and such c l a r i t y

of c o n t r a s t a r e g e n e r a l l y a s s o c i a t e d w i t h north-westerly winds.

The p r e s e n t

p i c t u r e w a s taken d u r i n g a p e r i o d o f 10-13 m / s e c winds from the northwest and coincided w i t h a lower-low w a t e r i n the S t r a i t . Based p a r t l y on o b s e r v a t i o n s , which have been taken r e c e n t l y i n connection

w i t h an "upper l a y e r " model of the plume d e s c r i b e d b r i e f l y below, a p r o v i s i o n a l g e n e r a l d e s c r i p t i o n o f the formation of the plume under moderate river-flow c o n d i t i o n s i s suggested as follows.

A s the water l e v e l f a l l s a t the r i v e r mouth

and t h e t i d a l streams are ebbing i n the S t r a i t the s a l t wedge i n the r i v e r

288

BOUNDARY -JUAN

DE FUCA

0

\

.

100

.

1

.

.

E I a 200 k W

0

300

4 - 8 DEC. 1967

0

100

E I k

a a

200

W

30C

1 - 6 J U L . 1968

400

Fig. 3 . Winter and summer contoured v e r t i c a l s e c t i o n s of s a l i n i t i e s through t h e median l i n e s of Juan de Fuca S t r a i t , Haro S t r a i t , Boundary Passage and the S t r a i t of Georgia (Crean and Ages, 1971).

289 r e t r e a t s and the rate o f d i s c h a r g e from the r i v e r i n c r e a s e s .

The s i l t y w a t e r

spreads o u t over the s a l t water t o form a southward v e e r i n g plume. hour before lower-low water the d i s c h a r g e r a t e a t t a i n s i t s maximum.

Roughly an

As t h e

streams t u r n t o f l o o d and l e v e l s rise i n the S t r a i t the discharge r a t e d e c r e a s e s , t h e s a l t wedge i n t r u d e s i n t o t h e r i v e r and the shallow plume j u s t formed, moves northward and outward away from t h e r i v e r mouth.

The p r o c e s s is then r e p e a t e d

w i t h new plume forming over the remnants o f i t s predecessors except i n such a

case a s shown i n Fig. 4 , where winds have swept such remnants southward towards t h e s t r o n g tidal-mixing region o f the S a n Juan I s l a n d s .

Fig. 4. S a t e l l i t e p i c t u r e (LANDSAT-1 imagery from 1100 km) o f t h e s o u t h e r n S t r a i t of Georgia and the region of t h e San Juan and Gulf I s l a n d s showing t h e F r a s e r River plume. The land m a s s i n t h e lower l e f t h a n d c o r n e r i s Vancouver Island. Under f r e s h e t c o n d i t i o n s , i t appears t h a t t h e r i v e r flow h o l d s i t s e n t r a n t d i r e c t i o n i n a s t r o n g narrow stream f o r a c o n s i d e r a b l e d i s t a n c e a c r o s s the S t r a i t and then t u r n s northward.

This s t r e a m can maintain i t s i d e n t i t y o v e r a

d i s t a n c e of some 90 km from t h e r i v e r mouth, e v e n t u a l l y following t h e l i n e of t h e mainland c o a s t and then d i s p e r s i n g (Giovando and Tabata, 1970).

Clearly

t h e r e e x i s t s much d i v e r s i t y i n t h e p o s s i b l e motions of t h e 'plume' d e r i v i n g from v a r i a t i o n s i n t h e r a t e o f d i s c h a r g e from t h e r i v e r and t h e mixed n a t u r e of the t i d e s .

Over and above t h e s e c o n s i d e r a t i o n s t h e r e remain t h e e f f e c t s of

wind stress a c t i n g over the s u r f a c e o f t h e 'plume'. I t i s e v i d e n t t h a t t h e t i d e s and t i d a l streams p l a y an important r o l e i n

determining t h e e s t u a r i n e c i r c u l a t i o n w i t h i n t h e system, n o t only with r e s p e c t t o mixing p r o c e s s e s , b u t a l s o w i t h r e g a r d t o t h e r e l a t i v e p r o p o r t i o n s of t h e F r a s e r River discharge f i n d i n g seaward e g r e s s r e s p e c t i v e l y through Juan de Fuca o r Johnstone S t r a i t s .

I t i s a l s o p o s s i b l e t h a t t h e r e may e x i s t s i g n i f i c a n t

re9i'ihual flows induced by t h e b a r o t r o p i c t i d e . Basic f e a t u r e s o f the main semi-diurnal (M ) and d i u r n a l (K1) t i d e s are 2 i l l u s t r a t e d by cophase and corange diagrams o b t a i n e d from an e a r l i e r model described b r i e f l y below, Fig.

5.

Of p a r t i c u l a r i n t e r e s t i s t h e semi-diurnal

degenerate amphidromic system i n t h e i n n e r p a r t of Juan de Fuca S t r a i t and r a p i d changes i n amplitude and phase through t h e region o f t h e San Juan I s l a n d s . Tides a r e of t h e mixed type and i t w i l l be noted t h a t both c o n s t i t u e n t s are of s i m i l a r amplitude i n the n o r t h e r n p a r t o f the S t r a i t o f Georgia. The e a r l i e s t t h e o r e t i c a l work on t i d e s i n t h e system w a s undertaken by Redfield (1950b).

Assuming a l i n e a r s u p e r p o s i t i o n o f damped incoming and

r e f l e c t e d waves he concluded t h a t t h e f r i c t i o n a l d i s s i p a t i o n w a s considerably g r e a t e r than t h a t found i n applying t h e same technique t o Long I s l a n d Sound and t h e Bay o f Fundy.

S t u d i e s using an o v e r a l l one-dimensional model o f a l l t h e

major conveying channels i n t h e system, o p e r a t e d by p r e s c r i b i n g e l e v a t i o n s a t t h e e n t r a n c e t o Juan de Fuca S t r a i t and n e a r t h e n o r t h e r n end of Vancouver I s l a n d , s a t i s f a c t o r i l y reproduced t h e main f e a t u r e s o f t h e M2 and K1 c o n s t i t u e n t s , b u t r e q u i r e d an unusually l a r g e f r i c t i o n a l d i s s i p a t i o n t o o p t i m i s e the agreement between t h e c a l c u l a t e d and observed t i d a l c o n s t a n t s around t h e i n t e r i o r o f t h e modelled r e g i o n .

I t w a s concluded t h a t t h e i n c l u s i o n of t h e C o r i o l i s e f f e c t

was e s s e n t i a l before any f u r t h e r p r o g r e s s could be made ( C r e a n , 1969).

A two-

dimensional, v e r t i c a l l y - i n t e g r a t e d numerical t i d a l model w i t h m u l t i p l e l a t e r a l boundary openings s i m u l a t i n g connections with t h e ocean and the e n t r a n c e s t o t h e various mainland i n l e t s was developed.

E x c e l l e n t agreement was obtained

between t h e observed and computed M2 and K 1 t i d a l harmonic c o n s t a n t s u s i n g an o v e r a l l c o e f f i c i e n t of f r i c t i o n of .0025 as commonly a p p l i e d i n work on c o a s t a l seas.

Flows a t t h e boundary openings i n t h e mainland c o a s t of t h e S t r a i t of

Georgia proved, however, t o be q u i t e u n r e a l i s t i c .

The a d d i t i o n of f u r t h e r one-

dimensional models t o t h e e x i s t i n g two-dimensional scheme reduced t h e number of e x t e r n a l openings t o two, each s i m u l a t i n g connections w i t h t h e open ocean (Crean 1976).

Numerical experiments with t h i s model p e r m i t t e d a proper

291

Fig. 5 . Corange (cms) and cophase diagrams o f the main semi-diurnal (Ma) and d i u r n a l (K1) t i d a l c o n s t i t u e n t s d e r i v e d f r o m earlier s t u d i e s employing an o v e r a l l numerical model o f the system.

292 d i s t r i b u t i o n o f f r i c t i o n t o be determined which i n t u r n r e v e a l e d s i g n i f i c a n t non-linear

i n t e r a c t i o n s between t h e major t i d a l c o n s t i t u e n t s i n t h e region of

t h e San Juan I s l a n d s .

F u r t h e r , i t was shown t h a t t h e 4-km mesh s i z e of t h e

two-dimensional p a r t o f t h e model could n o t adequately s i m u l a t e t h e propagation of long waves through Haro S t r a i t unless a l i n e o f contiguous meshes having depths corresponding t o maximum channel depth, r a t h e r than t o t h e averaged depth over each mesh, was included.

A summary o f t h i s work i s being submitted

f o r publication.

A t an e a r l y s t a g e of the work d e s c r i b e d above, a c o n f i g u r a t i o n o f t h e c o a r s e ( 4 km) g r i d model w i t h boundary openings conforming t o t h o s e employed i n the p r e s e n t f i n e g r i d ( 2 km) mesh model was attempted.

This a t t e m p t w a s e v e n t u a l l y

abandoned because of the h i g h l y u n r e a l i s t i c flow f i e l d a s s o c i a t e d with t h e open boundary i n the S t r a i t o f Georgia.

FUNDAMENTAL HYDRODYNAMICAL EQUATIONS

The v e r t i c a l l y i n t e g r a t e d e q u a t i o n s of c o n t i n u i t y and momentum a p p l i c a b l e t o t i d a l motions i n a f l a t r o t a t i n g s e a are:

+

g (<+h)

a

(5)

+

kUJU2+V2 (5+h)

=

0

wnere t h e n o t a t i o n i s a s follows: X,Y

C a r t e s i a n c o o r d i n a t e s i n t h e p l a n e of t h e undisturbed sea surface

t

Time

5 (X,Y I t)

E l e v a t i o n of t h e sea s u r f a c e

u (X,Y I t) V(x,y,t)

Components of v e r t i c a l l y i n t e g r a t e d v e l o c i t y i n d i r e c t i o n s x and y

h(x,Y)

Depth o f water below the undisturbed s u r f a c e

f

C o r i o l i s parameter, assumed c o n s t a n t over the r e g i o n

293 k(x,y)

Coefficient of f r i c t i o n

g

A c c e l e r a t i o n due t o g r a v i t y

E

H o r i z o n t a l eddy v i s c o s i t y

I n t h e d e r i v a t i o n o f t h e s e e q u a t i o n s i t i s assumed t h a t t h e v e l o c i t i e s are e s s e n t i a l l y independent of depth. Equations (l), ( 2 ) and ( 3 ) are s o l v e d numerically f o r the sea a r e a .

The

f i n i t e d i f f e r e n c e scheme i s s p a t i a l l y s i m i l a r t o t h a t used by Hansen (1961) b u t a modified numerical procedure i n t i m e i s employed ( F l a t h e r and Heaps, 1 9 7 4 ) . The a r e a i s r e p r e s e n t e d on a system o f orthogonal g r i d l i n e s . n o t a t i o n i s shown i n Figure 6 .

The g r i d

Each mesh i s of s i d e 2 km and i s c h a r a c t e r i z e d

by a c e n t r a l e l e v a t i o n p o i n t while t h e normal components of v e r t i c a l l y i n t e g r a t e d v e l o c i t y are e v a l u a t e d a t the mid-points of t h e s i d e s . To c a l c u l a t e t h e e l e v a t i o n 5 . ( t + A t )

a t t h e c e n t r e of a mesh equation (1) i s

replaced by:

Fig. 6.

Notation used i n t h e computational g r i d .

294

The m o m e n t u m equations

I=

U.(t+At)-Ui(t)

J

1=

(2)

and ( 3 ) a r e replaced by:

fTi(t)

A t

-

fE. ( t + A t ) - g f i X - n All

r

ci(t+At)

-
I

295 where t h e averaging of a g r i d v a r i a b l e w i t h the neighbouring v a r i a b l e of t h a t type i n t h e d i r e c t i o n x o r y i n c r e a s i n g i s denoted r e s p e c t i v e l y by expressions such a s : -X

u.

=

3 (U.+U. 1

1+1)

’

To denote t h e value a t a p o i n t o f a v a r i a b l e o b t a i n e d by averaging t h e four values of t h a t v a r i a b l e surrounding t h e p o i n t , the n o t a t i o n i s :

E,

= h

Gi

=

k

where H = Ci i

( H I+Hi-n+Hi-n+l+Hi+l)

(Ui+Pi+n+Ui+n-l+Ui-l

1

5,

=

+ hi

‘a (Vi+Vi-n+?/i-n+l+Vi+l)

DESCRIPTION O F THE MODEL

The g r i d scheme used i n t h e model i s shown i n Fig.

7 where each square

corresponds t o t h e f o u r s i d e s of a mesh a s i l l u s t r a t e d i n Fig. 6.

Coastlines

a r e simulated by mesh s i d e s where the normal component of v e l o c i t y i s s e t equal t o zero.

S i m i l a r l y , narrow i s l a n d s o r land b a r r i e r s a r e d e p i c t e d by

closed mesh s i d e s w i t h i n t h e i n t e r i o r o f t h e s e a .

I n some i n s t a n c e s ambiguities

e x i s t as t o which s i d e of a mesh b e s t r e p r e s e n t s a p a r t o f t h e c o a s t l i n e .

Such

ambiguities are r e s o l v e d by e n s u r i n g t h a t r e g i o n a l s u r f a c e a r e a s i n t h e model accord w i t h those o b t a i n e d from hydrographic c h a r t s . The average depth over a mesh i s s p e c i f i e d a t each e l e v a t i o n p o i n t w i t h i n t h e i n t e r i o r of t h e sea.

I n r e g i o n s where banks may be uncovered a t low w a t e r ,

notably near t h e mouth of the F r a s e r River, an a r b i t r a r y depth o f 5 m has been assigned. model.

This i s considered adequate a t this s t a g e of development of t h e

Any r e a l i s t i c numerical s i m u l a t i o n o f flows over t h e shallow banks

near the F r a s e r River mouth must a l s o t a k e account o f t h e r i v e r d i s c h a r g e and

i t s v a r i a t i o n s , and a l s o the motion o f the s a l t wedge. A number of p a s s e s a r e included w i t h i n the modelled r e g i o n such as, f o r

example, those l e a d i n g from t h e region o f t h e Gulf I s l a n d s i n t o the S t r a i t of Georgia.

I n some c a s e s t h e s e have been s i m u l a t e d by one-dimensional channels

i n s e r t e d i n t o t h e two-dimensional

scheme.

E a r l i e r work w i t h the o v e r a l l t i d a l

model showed t h a t t h e comparatively s m a l l flows through these p a s s e s a r e n o t c r i t i c a l t o the proper s i m u l a t i o n of the t i d a l streams i n the main conveying channels of t h e system, though they do p o s s e s s c o n s i d e r a b l e l o c a l importance. These f e a t u r e s of the model w i l l n o t be d i s c u s s e d h e r e .

D e t a i l e d adjustments

of t h e flows through t h e s e p a s s e s w i l l be undertaken when p e r t i n e n t f i e l d o b s e r v a t i o n s , p r e s e n t l y planned, are completed.

296

F

E

A VA NCOU VE R MESH SIZE 2KMS

Fig. 7 . The numerical model g r i d scheme. Rows of meshes where e l e v a t i o n s a r e p r e s c r i b e d on open boundaries a r e denoted by AB, CD and EF.

The value of t h e f r i c t i o n c o e f f i c i e n t used i n t h e S t r a i t of Georgia and i n Juan de Fuca S t r a i t i s 0.003.

I n the r e g i o n o f t h e channels between the San

Juan and Gulf I s l a n d s t h i s value w a s i n c r e a s e d t o 0.03.

The l a t t e r value

appears unusually high when compared w i t h those normally employed i n work on c o a s t a l s e a s , f o r example F l a t h e r and Heaps ( 1 9 7 4 ) .

The b a s i c argument f o r

the use of such high a value i n t h e i s l a n d r e g i o n d e r i v e s from e a r l i e r work

with t h e o v e r a l l model o f t h e waters between Vancouver I s l a n d and the mainland. I t was found t h a t s a t i s f a c t o r y agreement between the observed and computed

t i d a l e l e v a t i o n s around t h e i n t e r i o r o f t h e modelled sea could only be achieved by t h e use o f such high f r i c t i o n a l d i s s i p a t i o n , i n p a r t i c u l a r with r e s p e c t t o t h e proper p r o p o r t i o n i n g of t h e t i d e s between t h e S t r a i t o f Georgia

297 and t h e Puget Sound system.

A c o r r o b o r a t i v e f e a t u r e of t h e argument concerns

the e x i s t e n c e o f a s i g n i f i c a n t non-linear d i u r n a l (K1) and semi-diurnal

i n t e r a c t i o n between t h e primary

(M2) c o n s t i t u e n t s which was shown t o be due t o

the high f r i c t i o n a l d i s s i p a t i o n i n the topographically-complex channels o f t h e i s l a n d region. When t h e model i s o p e r a t e d i n t h e absence o f l a t e r a l s t r e s s e s small g r i d s c a l e f l u c t u a t i o n s i n t h e d i r e c t i o n s of v e l o c i t y v e c t o r s ("noodling") occur i n regions where t h e a d v e c t i v e a c c e l e r a t i o n s a r e important.

The value of t h e

h o r i z o n t a l eddy c o e f f i c i e n t used t o e l i m i n a t e this f e a t u r e of the model was

lo6

cm2/sec, which, i n view of t h e mesh s i z e and timestep,

implies a r e l a t i v e l y

small degree of l a t e r a l averaging (Icuipers and Vreugdenhil, 1973).

The

problem of numerically e v a l u a t i n g second d e r i v a t i v e s n e a r c o a s t a l boundaries was avoided by s e t t i n g t h e c o e f f i c i e n t e q u a l t o zero i n such circumstances. The implied assumption i s t h a t g e n e r a l l y the bottom s t r e s s a c t i n g on t h e w a t e r i n a r e l a t i v e l y shallow mesh contiguous t o the c o a s t i s much g r e a t e r than t h e lateral stress. The maximum p e r m i s s i b l e t i m e s t e p y i e l d i n g s t a b l e s o l u t i o n s i s obtained from t h e Courant-Friedricks-Lewy

A t 5

criterion,

.

A9.

The t i m e s t e p t h a t r e s u l t s f o r the p r e s e n t scheme i s 2 3 s e c s . Elevations were p r e s c r i b e d along the t h r e e open boundaries of t h e model shown by t h e l i n e s , AB, C D , EF i n F i g . 7 , by i n t e r p o l a t i o n of values from t h e o v e r a l l model of t h e system which used a 4 km mesh.

The boundary c o n d i t i o n s

i n t h e c a s e of t h e l a t t e r were p r e d i c t e d t i d e s based on 6 1 harmonic constituents.

A s noted above, e a r l i e r a t t e m p t s t o o p e r a t e p a r t o f t h e 4 km

mesh model by p r e s c r i b i n g e l e v a t i o n s a t t h e s e l o c a t i o n s f a i l e d , f i r s t l y , because of t h e i n s u f f i c i e n t s e n s i t i v i t y of t h e model t o p e r m i t proper adjustment o f t h e f r i c t i o n a l d i s s i p a t i o n by o p t i m i s i n g computed and observed e l e v a t i o n s around the i n t e r i o r of the modelled sea and secondly because of t h e i n a b i l i t y of t i d e gauges a t e i t h e r end o f a long open boundary t o adequately r e s o l v e t h e s l o p e s of the w a t e r s u r f a c e r e q u i r e d t o balance t h e geotrophic a c c e l e r a t i o n of t h e t i d a l streams moving normal t o t h e open boundary.

To overcome t h e s e d i f f i c u l t i e s i n t h e p r e s e n t c a s e , t h e d i s t r i b u t i o n

of f r i c t i o n a l d i s s i p a t i o n and the e l e v a t i o n s o f t h e water s u r f a c e p r e s e n t e d a t open boundaries, w e r e taken from t h e o r i g i n a l o v e r a l l model of t h e system. Subsequent comparisons of t h e v e l o c i t y v e c t o r f i e l d s o b t a i n e d a t comparable i n s t a n t s from t h e two models showed e x c e l l e n t agreement e x c e p t where modified by t h e changed r e p r e s e n t a t i o n o f t h e c o a s t a l boundaries.

RESULTS

I l l u s t r a t i v e o f t h e performance o f t h e model, some comparisons o f computed e l e v a t i o n s with t h o s e p r e d i c t e d on t h e b a s i s of 6 1 harmonic t i d a l c o n s t i t u e n t s over a seven-day p e r i o d are now p r e s e n t e d f o r r e p r e s e n t a t i v e l o c a t i o n s around t h e i n t e r i o r of t h e modelled s e a , Fig. 8.

VANCOUVER ISLAND CURRENT M E T E R S TIDE GAUGES 7080 PEDDER BAY 7330 FULFORD HARB 7510 TUMBO CHANNE 7564 FERNDALE

+

Fig. 8. and 1 0 .

Locations o f c u r r e n t meters and t i d e gauges r e f e r r e d t o i n F i g s . 9

A t Tumbo Channel (7510) and Ferndale (7564) t h e e x c e l l e n t agreement o b t a i n e d

i s t y p i c a l of t h e r e s u l t s o b t a i n e d throughout t h a t p a r t o f t h e S t r a i t o f Georgia included w i t h i n the model, Fig. 9 . The most d i f f i c u l t a r e a of t h e model i n which t o o b t a i n good agreement i s i n the region of Haro S t r a i t and i n the degenerate semi-diurnal amphidromic region i n the i n n e r p a r t o f Juan de Fuca S t r a i t , regions c h a r a c t e r i z e d by r a p i d changes i n t h e s p a t i a l d i s t r i b u t i o n of the harmonic c o n s t a n t s Fig. 5. A comparison of t h e p r e d i c t e d e l e v a t i o n s with those computed by t h e model

f o r Fulford Harbour (7330) shows good agreement though t h e curves from t h e model

299

Comparisons of t h e e l e v a t i o n s computed by the model (continuous l i n e ) f o r some t y p i c a l l o c a t i o n s w i t h t h o s e p r e d i c t e d (dashed l i n e ) on t h e b a s i s of t i d a l harmonic c o n s t a n t s .

Fig. 9.

300 show a s m a l l phase l e a d throughout the r e c o r d and their e x i s t s a modest discrepancy i n t h e t i d a l ranges.

More d e t a i l e d adjustments o f f r i c t i o n a l

d i s s i p a t i o n i n t h i s p a r t o f the model, i n c l u d i n g t h a t i n t h e p a s s e s l e a d i n g i n t o t h e S t r a i t o f Georgia from t h e r e g i o n o f t h e Gulf I s l a n d s , could probably improve t h e comparison.

Such adjustments are p r e s e n t l y cqnsidered of secondary

importance t o t h e s u c c e s s f u l reproduction of t h e t i d a l streams i n t h e main conveying channels o f t h e system and w i l l n o t be attempted u n t i l t h e acquisition of f u r t h e r f i e l d data. The Pedder Bay ( 7 0 8 0 ) gauge i s l o c a t e d near t h e semi-diurnal amplitude minimum.

The agreement o b t a i n e d i s good though t h e r e i s evidence of more

accentuated h i g h e r t i d a l harmonics i n the values from t h e model. The primary o b j e c t of t h i s numerical study concerns t h e r e l i a b l e p r e d i c t i o n of t h e b a r o t r o p i c t i d a l streams.

During t h e i n t e r v a l of time s e l e c t e d f o r

t h i s t r i a l of t h e model an e x t e n s i v e a r r a y o f c u r r e n t meters w a s over a c r o s s - s e c t i o n o f Juan de Fuca S t r a i t . shown i n Fig. 8.

i n position

The l o c a t i o n s o f t h e moorings are

Four Aanderaa r e c o r d i n g c u r r e n t meters were suspended a t

each mooring i n t h e deeper p a r t o f t h e channel.

Comparisons r e p r e s e n t a t i v e of

the agreement between the v e r t i c a l l y averaged v e l o c i t i e s p r e d i c t e d by t h e model and measured t i d a l v e l o c i t i e s i n this s e c t i o n a r e shown i n F i g . 1 0 . The measured v e l o c i t i e s are resolved i n t h e d i r e c t i o n of the model axes and

a t t h i s l o c a t i o n may be considered as t h e l o n g i t u d i n a l and cross-channel components o f flow.

(For purposes of t h e comparison t h e low frequency non-

t i d a l p a r t o f the measured v e l o c i t i e s has been removed by f i r s t s u p p r e s s i n g 2 2 A /24.25 f i l t e r (Godin 19721, and t h e n the t i d a l s i g n a l by means o f an A 24' 25 s u b t r a c t i n g t h e r e s i d u a l from t h e o r i g i n a l record.) The comparisons o f t h e observed and p r e d i c t e d longikudinal V components show e x c e l l e n t agreement with r e s p e c t t o phase.

The range o f the observed v e l o c i t i e s a t 15 m and 50 m

i s appreciably g r e a t e r than t h a t p r e d i c t e d by t h e model.

This f e a t u r e ,

however, accords w e l l w i t h e x p e c t a t i o n based on observed p r o f i l e s i n open channel flow.

Unfortunately c u r r e n t meter malfunctions precluded t h e

p r e s e n t a t i o n o f a f u r t h e r comparison c l o s e r t o t h e sea bottom.

Such

comparisons i n t h e base of the more extended r u n of the o r i g i n a l o v e r a l l c o a r s e g r i d model show t h a t indeed a t some 5-6 metres o f f t h e s e a bottom t h e p r e d i c t e d , vertically-averaged,

t i d a l v e l o c i t i e s a r e considerably g r e a t e r than those

observed. The smaller measured cross-channel components of v e l o c i t y show much more v a r i a b i l i t y than t h o s e d i r e c t e d l o n g i t u d i n a l l y .

Occasional i n t e r v a l s of good

agreement between the values p r e d i c t e d by t h e model and those observed a r e thought t o conform e s s e n t i a l l y t o b a r o t r o p i c t i d a l o s c i l l a t i o n s , while the more i r r e g u l a r motions are a s s o c i a t e d w i t h t h e i n t e r m i t t e n t e x c i t a t i o n o f i n t e r n a l modes.

I t w i l l be r e c a l l e d , F i g . 4 , t h a t the region where these

301

!I(!

1 5 r N 2 D E P I H 050

103,

5TN 2

DEPTH

:03

Fig. 1 0 . Comparisons of t h e computed cross-channel (U) and l o n g i t u d i n a l (V) t i d a l v e l o c i t y components (smooth curves) w i t h t h o s e observed ( i r r e g u l a r curves) a t t h r e e d i f f e r e n t depths ( 1 5 , 50 and 100 m ) a t S t a t i o n 2 i n Juan de Fuca S t r a i t (Fig. 8 ) .

observations were made i s s t r a t i f i e d .

During t h e p e r i o d of t h e measurements,

time s e r i e s STD o b s e r v a t i o n s (Crean and Miyake, 1976) i n d i c a t e d displacements i n t h e depths o f isopycnals near t h e c o a s t l i n e s of up t o 50 m which were approximately o f semi-diurnal p e r i o d . I n a d d i t i o n t o t h e above, comparisons were a l s o made between t h e e l e v a t i o n s and v e l o c i t i e s o b t a i n e d from t h e o v e r a l l coarse g r i d and l i m i t e d - a r e a f i n e g r i d models r e s p e c t i v e l y .

N o s i g n i f i c a n t d i f f e r e n c e s were noted between t h e r e s u l t s

obtained from each model e x c e p t i n those p a r t s o f t h e v e l o c i t y f i e l d s which a r e s t r o n g l y i n f l u e n c e d by l o c a l c o a s t a l boundaries. Fig. 11 shows t h e t i d a l e l e v a t i o n s and t h e l o n g i t u d i n a l component o f t h e t i d a l streams, as o b t a i n e d from t h e model f o r t h e p e r i o d 1 2 0 0 , 1 5 March - 1 6 0 0 , 1 6 March 1 9 7 3 , a t a c e n t r a l l o c a t i o n i n t h e S t r a i t of Georgia o f f t h e mouth o f

t h e F r a s e r River, denoted by t h e l e t t e r M i n F i g . 8.

The range between higher-

high water and lower-low water conforms approximately t o t h e range contained between mean higher-high water and mean lower-low water a t t h i s l o c a t i o n . Corresponding v e c t o r p l o t s o f t h e ebb and f l o o d v e l o c i t y f i e l d s when the streams a r e maximal a r e shown i n F i g s . 1 2 and 1 3 .

Small c r o s s e s denote t h e

c e n t r e of a mesh where t h e v e l o c i t y components, r e s p e c t i v e l y averaged over those obtained from o p p o s i t e mesh s i d e s , are p l o t t e d .

When t h e l i n e denoting

t h e v e l o c i t y magnitude and d i r e c t i o n exceeds 30 cms/sec,

the vector i s

represented by two p a r a l l e l l i n e s of e q u a l l e n g t h , and s o on f o r succeeding m u l t i p l e s of t h a t speed, t o avoid confusion due t o excessive overlapping. Throughout most of Juan de Fuca S t r a i t and t h e S t r a i t of Georgia t h e t i d a l streams a r e e s s e n t i a l l y r e c t i l i n e a r .

I n t h e i n n e r p a r t of Juan de Fuca S t r a i t

t h e r e occurs some r o t a t o r y c h a r a c t e r t o t h e streams though proper analyses i n terms of t h e t i d a l c o n s t i t u e n t s awaits a t r i a l of t h e model of s u f f i c i e n t d u r a t i o n t o p e r m i t s e p a r a t i o n of t h e major c o n s t i t u e n t t i d e s by harmonic analysis. The v e l o c i t y v e c t o r s s u g g e s t some i n t e r e s t i n g a s p e c t s of the t i d a l flows near c o a s t a l f e a t u r e s o f f t h e southernmost p a r t o f Vancouver I s l a n d where l a r g e speeds and marked changes i n flow d i r e c t i o n occur.

The formation of back

eddies changing i n l o c a t i o n from ebb t o f l o o d i s c l e a r l y e v i d e n t .

Examination

of a time sequence of such v e c t o r p l o t s i n d i c a t e s t h a t t h e eddy begins t o form when t h e stream a t t h a t p a r t i c u l a r l o c a t i o n has a t t a i n e d s u f f i c i e n t magnitude; the eddy then i n c r e a s e s i n s p a t i a l e x t e n t u n t i l t h e stream t u r n s .

A further

p o i n t of i n t e r e s t concerns t h e d i f f e r e n c e s between s t r e n g t h and d i r e c t i o n o f t h e ebb and f l o o d streams i n Haro S t r a i t , a g a i n emphasizing t h e importance of l o c a l topography.

I t i s of i n t e r e s t to n o t e t h a t t h e s e flow f e a t u r e s l i e along

what i s probably t h e primary r o u t e followed by w a t e r from t h e F r a s e r River moving seaward through Jclan de Fuca S t r a i t .

Unfortunately,

t h e 2 km g r i d s c a l e

used i n t h i s model i s too c o a r s e t o p e r m i t o t h e r than q u a l i t a t i v e i n d i c a t i o n s

303 of t h e s e dynamical f e a t u r e s , which a r e p e r t i n e n t t o l o c a l navigation and which a r e h e l d by some informants f a m i l i a r w i t h t h e s e waters t o be real.

Field

observations a r e planned t o t e s t their a u t h e n t i c i t y .

TIDE

VERSUS

TIME (HRS)

- ELEVATION .___

VELOCITY

1

V-COMPONENT

50.0

100.0

-

-

0 W

tn \

E

I 0.0

0.0

2

u

>

+

t

t

0

0 _I

w

w

1

-50.0

- 100.0

-200.0 HRS 12 a

20

00

04

08

IE

. ^-100.0 10

L

15 M A R C H , 1 9 7 3

16 M A R C H . 1 9 7 3

Fig. 11. T i d a l e l e v a t i o n s and t h e l o n g i t u d i n a l component of v e l o c i t y computed by t h e model f o r a l o c a t i o n i n m i d - s t r a i t o f f t h e mouth o f t h e F r a s e r River (M i n F i g . 8) The words EBB and FLOOD denote the i n s t a n t s a t which t h e v e l o c i t y v e c t o r diagrams i n F i g s . 1 2 and 1 3 o b t a i n .

.

Of p a r t i c u l a r relevance t o the movement o f the F r a s e r River discharge i s t h e c h a r a c t e r of t h e t i d e s and streams i n the s o u t h e r n p a r t o f t h e S t r a i t of Georgia.

Considering t h e flow f i e l d i n the g e n e r a l v i c i n i t y o f t h e i r r e g u l a r

l i n e of c l o s e d mesh s i d e s denoting t h e F r a s e r River t r a i n i n g w a l l which extends over t h e shallow banks t o t h e edge of t h e deep water i n t h e s t r a i t , it i s e v i d e n t t h a t streams decrease markedly i n magnitude w i t h n o r t h e r l y p r o g r e s s i o n along the l o n g i t u d i n a l a x i s of t h e s t r a i t , and a l s o undergo a s i g n i f i c a n t change i n d i r e c t i o n .

More l o c a l l y the v e l o c i t y v e c t o r s show t h e streams moving

o f f o r onto the shallow banks.

( N o a t t e m p t has been made a s y e t

t o include the

F r a s e r River d i s c h a r g e i n this model s i n c e t h i s i s s m a l l compared t o t h e t i d a l f l u x e s o c c u r r i n g i n t h e S t r a i t i t s e l f and cannot be adequately simulated i n a

.,...,,,.

I: : : :4: : - . , ,, , ,,,,,

I

I

Fig. 1 2 .

Ebb v e l o c i t y v e c t o r s o b t a i n e d from the model.

A number o f i n t e r e s t i n g dynamical i n f e r e n c e s may be made from s l o p e s of t h e

water s u r f a c e as i l l u s t r a t e d by three-dimensional p l o t s such as those shown i n Fig. 14.

I n crude terms, a l l l a n d h a s been removed above a base p l a n e

corresponding t o t h e lowest e l e v a t i o n o f the w a t e r s u r f a c e computed i n t h e modelled s e a a t t h e p a r t i c u l a r i n s t a n t i n q u e s t i o n . r e p r e s e n t e d as ' h o l e s ' i n t h e sea s u r f a c e .

Thus, i s l a n d s are

The i n t e r s e c t i o n of two l i n e s i n

t h e s e a a r e a corresponds t o t h e c e n t r e o f a mesh i n t h e model where e l e v a t i o n s

are c a l c u l a t e d while a s i n g l e l i n e i n d i c a t e s the e l e v a t i o n s along a channel represented i n t h e model by an i s o l a t e d sequence of contiguous meshes.

The

f i e l d s of e l e v a t i o n c o i n c i d e i n time w i t h t h e ebb and f l o o d v e l o c i t y f i e l d s

305 shown i n F i g s . 1 2 and 1 3 r e s p e c t i v e l y .

.

Fig. 13.

I

I1

VANCOUVER

n

r

Flood v e l o c i t y v e c t o r s o b t a i n e d from t h e model.

A s noted above, the proper p r o p o r t i o n i n g o f t i d a l e l e v a t i o n s and times o f

high w a t e r between the S t r a i t of Georgia and t h e Puget Sound system r e q u i r e d t h e i n t r o d u c t i o n of h i g h f r i c t i o n a l d i s s i p a t i o n i n t h e region o f the San Juan Islands.

The marked s l o p e s o f t h e water s u r f a c e through this r e g i o n , which a r e

more d r a m a t i c a l l y i n s t a n c e d i n t h e case o f t h e f l o o d t i d e f i g u r e , a r e c l e a r l y evident.

E l e v a t i o n s i n t h e a r e a contained between Vancouver I s l a n d and t h e

Gulf I s l a n d s are s i m i l a r t o t h o s e i n t h e n o r t h e r n p a r t o f Haro S t r a i t and can d i f f e r markedly from t h o s e i n t h e S t r a i t o f Georgia.

The r e s u l t i n g s t r o n g

t i d a l flows through t h e p a s s e s l e a d i n g i n t o the S t r a i t of Georgia a r e w e l l known t o l o c a l navigators.

306

,$Y

MAXIMUM FLOOD

A GULF ISLANDS B SAN JUAN ISLANDS C FRASER RIVER WA

-90

~MAXIMUM EBB I8 5 hrs 15 MARCH, 1973 -BASE P L A N E ELEVATION -149

4

Fig. 1 4 . Three-dimensional diagrams showing t h e shape of t h e water s u r f a c e a t t h e same i n s t a n t s r e s p e c t i v e l y as t h e p r e c e d i n g diagrams of ebb (Fig. 1 2 ) and flood ( F i g . 13) v e l o c i t y v e c t o r s .

307 I n t h e S t r a i t of Georgia, there are cross-channel s l o p e s of t h e water s u r f a c e which balance the g e o s t r o p h i c a c c e l e r a t i o n s due t o the t i d a l streams moving p a r a l l e l t o the major a x i s o f t h e s t r a i t , while i n t h e v i c i n i t y of the r i v e r mouth

l o c a l s l o p e s are a s s o c i a t e d w i t h t h e motion of water r e s p e c t i v e l y o f f , o r

onto, the shallow banks. A f u r t h e r f e a t u r e o f t h e s e diagrams concerns secondary undulations o f t h e

water s u r f a c e i n t h e inner p a r t of Juan de Fuca S t r a i t .

These are more

s t r o n g l y i n evidence when t h e streams are running f u l l than a t s l a c k water and w i l l be i n v e s t i g a t e d f u r t h e r .

A t p r e s e n t i t appears that they may be

a s s o c i a t e d w i t h d e p r e s s i o n s i n the water s u r f a c e o f some 10-20 c m near t h e c e n t r e of t h e l a r g e e d d i e s near c o a s t a l f e a t u r e s which have been r e f e r r e d t o above.

PRESENT AND FUTURE WORK

I n c o l l a b o r a t i o n w i t h M r . S . Huggett of t h e Canadian Hydrographic S e r v i c e , v e l o c i t y f i e l d s d e r i v e d from the model a r e being used a s the b a s i s f o r an a t l a s of t i d a l c u r r e n t s i n Juan de Fuca S t r a i t and i n t h e region of t h e San Juan I s l a n d s .

S p e c i f i c d e t a i l s o f the t i d a l streams a s s o c i a t e d w i t h c o a s t a l

f e a t u r e s , which a r e p o o r l y resolved by the 2 km mesh s i z e o f t h e model, a r e being i n d i v i d u a l l y checked by t r a c k i n g s u r f a c e f l o a t s from an a i r c r a f t equipped with a s p e c i a l s i g h t i n g d e v i c e , i n e r t i a l n a v i g a t i o n system, r a d a r a l t i m e t e r , and magnetic t a p e d a t a r e c o r d e r (Grasty e t a l . 1 9 7 7 ) . An important a s p e c t of t h i s work concerns r e s i d u a l c u r r e n t s induced by t h e b a r o t r o p i c t i d e and their p o s s i b l e s i g n i f i c a n c e t o t h e e s t u a r i n e c i r c u l a t i o n . I t is e v i d e n t from t h e computed v e l o c i t y f i e l d s t h a t r e l a t i v e l y l a r g e n e t

flows occur i n r e g i o n s c h a r a c t e r i s e d by s t r o n q t i d a l streams and complex topography.

Using t h e v o r t i c i t y equation d e r i v e d from t h e depth mean flow

momentum and c o n t i n u i t y e q u a t i o n s , t h e terms g e n e r a t i n g such secondary flows have been discussed by Kuipers and Vreugdenhil (1973).

A s h o r t review o f t h e

problem of r e s i d u a l flows h a s been p r e s e n t e d by Nihoul (1975).

I n the c o n t e x t

of t h e p r e s e n t work t h e r e e x i s t s t h e p o s s i b i l i t y o f examining r e s i d u a l flows o b t a i n e d from models of 2 km and 4 km mesh s c a l e s of t h e same complex geographical r e g i o n .

Analyses of t h e s e r e s u l t s and comparisons w i t h observed

r e s i d u a l flows o b t a i n e d from moored c u r r e n t meters a r e n o t as y e t completed. A s d e s c r i b e d above, t h e e v e n t u a l o b j e c t of t h i s work concerns a r e s o l u t i o n

o f the main dynamical f e a t u r e s of t h e e s t u a r i n e c i r c u l a t i o n .

The q u e s t i o n

a r i s e s as t o how b e s t t h e r e s u l t s o b t a i n e d from t h e ’ b a r o t r o p i c model p r e s e n t e d above may be employed t o f u r t h e r this end.

C l e a r l y a major concern

i s t h e motion of f r e s h w a t e r i n t h e s o u t h e r n p a r t of t h e S t r a i t of Georgia, which may move southward i n t o the s t r o n g t i d a l mixing a r e a of t h e San Juan I s l a n d s o r northward i n t o t h e S t r a i t of Georgia where t h e t i d a l flows a r e

308 r e l a t i v e l y weak.

In t h e l a t t e r c a s e , it may e v e n t u a l l y move seaward through

Johnstone S t r a i t o r t h e accumulated discharge over a lengthy p e r i o d may be returned southward by t h e a c t i o n of winds.

The problem i s immensely complex.

The thickness o f the plume i s commensurate with t h e t i d a l range a t the r i v e r mouth.

There e x i s t extensive a r e a s o f drying banks.

The i n t r u s i o n of t h e

s a l t wedge i n t o the F r a s e r River i s r e l a t e d t o the s t a t e o f the t i d e and t o v a r i a t i o n s i n the s t r e n g t h of t h e r i v e r discharge.

There i s a l s o the problem

of simulating propagating shallow f r o n t s between r i v e r water and sea water, o r the remains of preceding "plumes", i n t h e S t r a i t .

Any attempt t o include a l l

these f e a t u r e s i n a s i n g l e three-dimensional model appears l i k e l y a t p r e s e n t merely t o p r o l i f e r a t e dynamical confusion a t h e r o i c expense.

The s i m p l e s t

approach holding promise of simulating t h e dominant a s p e c t s of t h e shallow'upper l a y e r motions i n t h e S t r a i t o f Georgia, and which u t i l i z e s d a t a from t h e above b a r a t r o p i c t i d a l model, w i l l be summarized b r i e f l y .

This i s a c o l l a b o r a t i v e

p r o j e c t with co-workers a t t h e I n s t i t u t e of Oceanography o f The University o f B r i t i s h Columbia, (Stronach, 1978). Consider i n i t i a l l y a s y s t e m i n which a l i g h t e r l a y e r of f l u i d i s superimposed upon a denser lower l a y e r and i n which t h e motions w i t h i n each l a y e r may be described by the v e r t i c a l l y i n t e g r a t e d equations of motion and c o n t i n u i t y a s applied t o f l a t r o t a t i n g s e a (Proudman 1952).

I n t h e p r e s e n t c a s e , the top

l a y e r i s confined t o t h e upper few metres of t h e water column which i s g e n e r a l l y some 150 - 300 m deep.

Consistent with observations from moored c u r r e n t

meters over a cross-section

of t h e s t r a i t near t h e r i v e r mouth, t h e presence

of t h i s upper l a y e r does n o t s i g n i f i c a n t l y a f f e c t t h e motion i n the lower p a r t of t h e water column which t o a f i r s t approximation i s dominated by t h e b a r o t r o p i c t i d a l streams.

I f now t h a t p a r t of t h e p r e s s u r e g r a d i e n t i n the

lower l a y e r which d e r i v e s from t h e presence of t h e upper l a y e r , is s e t equal t o zero, equations of motion and c o n t i n u i t y applying t o a s i n g l e upper l a y e r may be obtained i n which the thickness of t h e l a y e r now r e p l a c e s the e l e v a t i o n of the water s u r f a c e a s one of the dependent v a r i a b l e s , and i n which a d d i t i o n a l terms i n the momentum equations e f f e c t i v e l y induce "buoyant spreading". Empirical t e r m s a r e a l s o included t o approximate entrainment of water from t h e lower l a y e r i n t o the upper.

A p a r t i c l e o f water i n t h i s upper l a y e r w i l l

experience a p r e s s u r e g r a d i e n t a s s o c i a t e d w i t h those s l o p e s of t h e water s u r f a c e giving r i s e t o the b a r o t r o p i c t i d a l motions i n t h e much t h i c k e r lower l a y e r . A t any p o i n t t h e r e w i l l a l s o e x i s t a

stress a s s o c i a t e d with t h e d i f f e r e n c e

between t h e upper l a y e r v e l o c i t y a t t h a t p o i n t and the b a r o t r o p i c t i d a l v e l o c i t y underneath.

Both the components of t h e f r e e s u r f a c e b a r o t r o p i c t i d a l s l o p e and

underlying v e l o c i t i e s a r e known throughout t h e s t r a i t from t h e c r i g i n a l t i d a l model.

The a d d i t i o n of a s a l t conservation equation, with an empirical expression

t o include entrainment, permits computation of t h e s p a t i a l d i s t r i b u t i o n s o f upper

309 layer s a l i n i t y i n the southern strait. t o approximate the f i e l d o f d e n s i t y .

These s a l i n i t i e s may then be used There i s now a t hand a simple and

r e l a t i v e l y inexpensive system which p e r m i t s numerical experiments i n connection w i t h t h e changing s u r f a c e s l o p e s of the b a r o t r o p i c t i d e , varying r i v e r d i s c h a r g e ,

entrainment, mixing, v e r t i c a l and l a t e r a l stresses, wind stresses, f i e l d a c c e l e r a t i o n s and t h e formulation of adequate boundary c o n d i t i o n s .

A major

i n t e n t o f this study i s t h e e s t a b l i s h m e n t o f s i g n i f i c a n t q u a n t i t a t i v e r e l a t i o n s between the r e l a t i v e volume t r a n s p o r t s of water l e a v i n g t h e S t r a i t o f Georgia, r e s p e c t i v e l y through t h e n o r t h e r n and s o u t h e r n openings and t h e r i v e r d i s c h a r g e

rates, winds and t i d e s . An important f e a t u r e of this approach i s t h e r e l a x a t i o n o f t h e s t r i n g e n t

l i m i t a t i o n on the time s t e p imposed by the propagation speed of f r e e s u r f a c e waves i n a r e g i o n where a small mesh s i z e i s r e q u i r e d and the w a t e r i s r e l a t i v e l y deep.

This work, t o g e t h e r w i t h an a p p r o p r i a t e program of f i e l d

measurements, which i n c l u d e s STD c a s t s , v e l o c i t y p r o f i l i n g , drogue t r a c k i n g , and a l s o t h e r e c o r d i n g of s u r f a c e v e l o c i t i e s a t the r i v e r mouth, i s p r e s e n t l y i n progress.

The r e s u l t s o b t a i n e d so f a r are most encouraging.

Though, t o d a t e , t h e model assumes dependence o f the v e l o c i t y components and s a l i n i t y i n t h e upper l a y e r , it i s proposed t o t a k e account o f t h e v e r t i c a l g r a d i e n t s , which are i n f a c t an important f e a t u r e of t h e plume, a t l e a s t near

the r i v e r mouth, by using t h e s i m i l a r i t y t h a t e x i s t s i n the d e p t h - p r o f i l e s a t i n c r e a s i n g d i s t a n c e s from the r i v e r mouth. W i t h r e g a r d t o f u t u r e work, two s t u d i e s a r e planned.

The f i r s t of t h e s e i s

a n o b s e r v a t i o n a l program t o determine t h e c h a r a c t e r of deep s a l t water i n t r u s i o n s i n t o t h e s o u t h e r n S t r a i t o f Georgia and p o s s i b l e r e l a t i o n s t o external forcing.

The second i n v o l v e s the i n c e p t i o n o f three-dimensional

numerical model s t u d i e s w i t h p a r t i c u l a r emphasis on maximal u t i l i z a t i o n of d a t a d e r i v i n g from t h e two-dimensional b a r o t r o p i c t i d a l and upper l a y e r buoyant spreading models r e f e r r e d t o above. I t would p r e s e n t l y appear t h a t these s t u d i e s w i l l culminate i n a three-

dimensional numerical s i m u l a t i o n of t h e major e s t u a r i n e motions and which might include such l o c a l p r o c e s s e s on the c o n t i n e n t a l s h e l f as could be shown t o s i g n i f i c a n t l y a f f e c t the inflow of s a l t w a t e r .

This model could, i n

p r i n c i p l e , be coupled to an atmospheric model s i m u l a t i n g t h e meso-scale orographic m o d i f i c a t i o n s of t h e g e o s t r o p h i c wind f i e l d by t h e mountains on t h e mainland and Vancouver I s l a n d .

The r e a l i z a t i o n o f such an o v e r a l l model,

however, w i l l r e q u i r e an extended program of f i e l d o b s e r v a t i o n s and numerical experiments t o s u f f i c i e n t l y r e s o l v e the e s s e n t i a l dynamical f e a t u r e s of t h e s y st e m

.

ACKNOWLEDGEMENTS The computer programming was c a r r i e d o u t by M r . P . J . D r . S . Pond and D r . R.

Richards.

Thomson for c r i t i c a l l y reviewing t h e paper.

I thank

311 REFERENCES Ages, A . and Woollard, A . , 1976. The t i d e s i n t h e F r a s e r River. P a c i f i c Marine Science Report 76-5: 100 pp. I n s t i t u t e o f Ocean Sciences, P a t r i c i a Bay, V i c t o r i a , B.C. Chang, P . , Pond, S . , and Tabata, S . , 1976. Subsurface c u r r e n t s i n t h e S t r a i t of Georgia, west of Sturgeon Bank. J. F i s h . Res. Board Can., 33: 2218-2241. Crean, P . , 1969. A one-dimensional hydrodynamical numerical t i d a l model of t h e Georgia-Juan de Fuca S t r a i t system. Fish. R e s . Board Can. Tech. Rep. No. 156: 32 PP. Crean, P . , 1976. Numerical model s t u d i e s o f t h e t i d e s between Vancouver I s l a n d and t h e mainland c o a s t . J. Fish. Res. Board Can., 33: 2340-2344. Crean, P . and Ages, A . , 1971. Oceanographic r e c o r d s from twelve c r u i s e s i n t h e S t r a i t of Georgia and Juan de Fuca S t r a i t , 1968. Dep. Energy, Mines and 1-5: 389 pp. Resources. Mar. S c i . Branch. Crean, P.B. and Lewis, A.G., 1976. Monthly s e c t i o n s - S t r a i t of Juan de Fuca, June-Dec. 1973. Data Report 39 ( P r o v i s i o n a l ) I n s t i t u t e of Oceanography, University o f B r i t i s h Columbia. Crean, P.B. and Miyake, M . , 1976. STD s e c t i o n s S t r a i t of Juan de Fuca, MarchA p r i l 1973. Data Report 38 ( P r o v i s i o n a l ) I n s t i t u t e of Oceanography, University of B r i t i s h Columbia. Department of t h e Environment, 1972a. Data record of c u r r e n t o b s e r v a t i o n s , S t r a i t of Georgia, s e c t i o n 2 , Cape Lazo t o Grief P o i n t , 1970. Water Manage. Serv., Mar. S c i . D i r . Pac. Reg., Manuscr. Rep. S e r . 9: 88 p. Department of t h e Environment, 197233. Data record of c u r r e n t o b s e r v a t i o n s , S t r a i t of Georgia, s e c t i o n 4 , Gabriola I s l a n d t o Gower P o i n t , 1969-1972. Water Manage. Serv., Mar. S c i . D i r . Pac. Reg., Manuscr. Rep. S e r . 1 0 : 153 p . Department of t h e Environment, 1972c. Data record of c u r r e n t o b s e r v a t i o n s , S t r a i t of Georgia, s e c t i o n 5 , P o r l i e r Pass t o Sand Heads, 1969-1972. Water Manage. Serv., Mar. S c i . D i r . Pac. Reg., Manuscr. Rep. S e r . 11: 124 p . Department of t h e Environment, 1973. Data r e c o r d o f c u r r e n t o b s e r v a t i o n s , S t r a i t o f Georgia, s e c t i o n 6 , Samuel I s l a n d t o P o i n t Roberts, 1969-1970. Water Manage. Serv., Mar. S c i . D i r . Pac. Reg., Manuscr. Rep. S e r . 1 2 : 96 p . F l a t h e r , R.A. and Heaps, N.S., 1975. T i d a l computations f o r Morecambe Bay. Geophys. J. R. A s t r . SOC. 42: 489-517. Giovando, L. and Tabata, S . , 1970. Measurements of s u r f a c e flow i n t h e S t r a i t of Georgia by means of f r e e - f l o a t i n g c u r r e n t f o l l o w e r s . F i s h . Res. Board Can. Tech. Rep. No. 163: 69 pp. Godin, G . , 264 pp.

1972.

Grasty, R.L. recovery.

The a n a l y s i s of t i d e s .

University of Toronto P r e s s , Toronto,

and Gower, J.F.R., 1977. I n e r t i a l n a v i g a t i o n f o r f l i g h t p a t h Geol. Survey Can. 76-30, 1 2 pp.

Hansen, W. , 1962. Hydrodynamical methods a p p l i e d t o oceanographic problems. Proc. Symp. Math. Hydrodynamical Methods o f Phys. Oc. 1961. I n s t i t u t fiir Meereskunde d e r U n i v e r s i t a t Hamburg. 1962: 25-34. Herlinveaux, R.H. and T u l l y , J . P . , 1961. Some oceanographic f e a t u r e s o f Juan de Fuca S t r a i t . J. F i s h . R e s . Board Can. 18: 1027-1071. Herlinveaux, R.H. and Giovando, L.F., 1969. Some oceanographic f e a t u r e s of t h e i n s i d e passage between Vancouver I s l a n d and t h e mainland of B r i t i s h Columbia. F i s h . Res. Board C a n . Tech. Rep. N o . 142: 48 pp.

312 Huggett, W.S., Bath, J . F . and Douglas, A . , 1976. D a t a r e c o r d of c u r r e n t o b s e r v a t i o n s , Juan de Fuca S t r a i t , 1973. I n s t i t u t e of Ocean Sciences, P a t r i c i a Bay, V i c t o r i a , B.C. Huggett, W . S . , Bath, J . F . and Douglas, A . , 1976. D a t a r e c o r d of c u r r e n t o b s e r v a t i o n s , Johnstone S t r a i t , 1973. I n s t i t u t e o f Ocean Sciences, P a t r i c i a Bay, V i c t o r i a , B.C. Hutchinson, A . H . , and Lucas, C . C . , 1931. The e p i t h a l a s s a of t h e S t r a i t o f Georgia: s a l i n i t y , temperature, pH, and phytoplankton. Can. J. R e s . , 5: 231-284. Kuipers, J . , and Vreugdenhil, C . B . , 1973. C a l c u l a t i o n s o f two-dimensional h o r i z o n t a l flow. Report on b a s i c r e s e a r c h S 163, P a r t 1, D e l f t Hydraulics Laboratory. Nihoul, J . C . J . , 272 pp.

1975.

Modelling of marine systems.

E l s e v i e r , Amsterdam,

Parker, Bruce B . , 1977. T i d a l hydrodynamics i n t h e S t r a i t o f Juan de Fuca S t r a i t of Georgia. U.S. Dept. o f Commerce, NOAA Technical Report NOS 69. Pickard, G . L . , 1975. Annual and l o n g e r term v a r i a t i o n s of deepwater p r o p e r t i e s i n t h e c o a s t a l waters o f southern B r i t i s h Columbia. J. Fish. Res. Board Can. 32: 1561-1587. Proudman, J . , 1953.

Dynamical Oceanography.

Methuen, London.

409 pp.

Redfield, A . C . , 1950a. Proceedings of the colloquium on t h e f l u s h i n g of e s t u a r i e s . Woods Hole Oceanographic I n s t i t u t i o n Ref. N o . 50-37: 175-177. Redfield, A.C., 1950b. The a n a l y s i s of t i d a l phenomena i n narrow embayments. Papers i n p h y s i c a l oceanography and meteorology, Vol. X I , No.4, Massachussetts I n s t i t u t e of Technology and Woods Hole Oceanographic I n s t i t u t i o n . Stronach, J . , 1978. Observational and modelling s t u d i e s o f the F r a s e r River plume. Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia. 1972a. Summary o f oceanographic records Tabata, S . , and S t i c k l a n d , J . A . , obtained from moored i n s t r u m e n t s i n the S t r a i t of Georgia -- 1969-1970: c u r r e n t v e l o c i t y and seawater temperature from s t a t i o n H-06. Dep. Environ. Water Manage. S e r v . , Mar. S c i . Branch, Pac. Reg. Pac. Mar. S c i . Rep. 72-7: 132 p . Tabata, S . , and S t i c k l a n d , J . A . , 197233. Summary o f oceanographic records obtained from moored instruments i n t h e S t r a i t of Georgia -- 1969-1970: c u r r e n t v e l o c i t y and seawater temperature from s t a t i o n H-16. Dep. Environ. Water Manage. Serv. Mar. S c i . Branch, P a c . Reg. Pac. Mar. S c i . Rep. 72-8: 144 p . Tabata, S . , and S t i c k l a n d , J . A . , 1972c. Summary of oceanographic records obtained from moored i n s t r u m e n t s i n t h e S t r a i t of Georgia -- 1969-1970: c u r r e n t v e l o c i t y and seawater temperature from s t a t i o n H-26. Dep. Environ. Water Manage. Serv., Mar. S c i . Branch, Pac. Reg. Pac. Mar. S c i . Rep. 72-9: 1 4 1 p. Tabata, S . , and S t i c k l a n d , J . A . , 19728. Summary of oceanographic records obtained from moored i n s t r u m e n t s i n t h e S t r a i t of Georgia -- 1968-1970. Current v e l o c i t y from s t a t i o n s F-11, M-10 and 1-31. Dep. Environ. Water Manage. S e r v . , Mar. S c i . Branch, P a c . Reg. Pac. Mar. S c i . Rep. 72-10: 2 2 p. and Wong, J . , 1970a. Current Tabata, S . , Giovando, L.F., S t i c k l a n d , J . A . , v e l o c i t y measurements i n t h e S t r a i t o f Georgia -- 1967. F i s h . Res. Board Can. Tech. Rep. 169: 245 p . Tabata, S . , Giovando, L.F., S t i c k l a n d , J . A . , and Wong, J . , 1970b. Current v e l o c i t y measurements i n t h e S t r a i t o f Georgia -- 1968. Fish. Res. Board Can. Tech. Rep. 178: 1 1 2 p .

313

Tabata, S . , Giovando, L.F., S t i c k l a n d , J . A . , and Wong, J . , 1970c. Current v e l o c i t y measurements i n the S t r a i t of Georgia -- 1969. Fish. Res. Board Can. Tech. Rep. 191: 7 2 p . Tabata, S . , S t i c k l a n d , J . A . , and D e Lange Boom, B.R., 1971. The program o f c u r r e n t v e l o c i t y and w a t e r temperature o b s e r v a t i o n s from moored i n s t r u m e n t s i n t h e S t r a i t of Georgia -- 1968-1970 and examples of records obtained. F i s h . Res. Board Can. Tech. Rep. 253: 2 2 2 p Thomson, R.E., 1976. T i d a l c u r r e n t s and e s t u a r i n e c i r c u l a t i o n i n Johnstone S t r a i t , B r i t i s h Columbia. J. Fish. Res. Board Can. 33: 2242-2264. Tully, J . P . and Dodimead, A . J . , 1957. P r o p e r t i e s of t h e water i n t h e S t r a i t J . Fish. R e s . Board of Georgia, B r i t i s h Columbia, and i n f l u e n c i n g f a c t o r s . Can. 14: 241-319. Waldichuk, M . , 1957. P h y s i c a l oceanography of t h e S t r a i t of Georgia, B r i t i s h Columbia. J . Fish. R e s . Board Can. 1 5 : 1065-1102.

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315

NUMERICAL INVESTIGATIONS OF THE INFLUENCE OF COASTAL STRUCTURES UPON THE DYNAMIC OFF-SHORE PROCESS BY APPLICATION OF A NESTED TIDAL MODEL H.-G. RAMMING Institut fiir Meereskunde der Universitlt Hamburg

SUMMARY By application of a TIDAL NORTH SEA MODEL with high resolution in costal areas, several numerical investigations were carried out dealing with predictions on dynamical processes in off-shore shallow-water regions as subsequence of structural measures. One of these investigation series and their results will be presented and discussed here in the following. The subject deals with the effect of the structural change of an existing breakwater and about the effect of a wash-up of a sand deposit in the shoal area upon the water levels, velocities, tidal residual currents and the mean transports of one tidal cycle in the off-shore range. The size of the investigation area is determined by the approximate effect in the remote region. Here it must be ensured that the boundary conditions of the selected part area obtained from the NESTED NORTH SEA MODEL (RAMMING, 1975) will not be influenced by the structural changes. The smallest mesh size of the applied model is 51 m. The areas of a differing mesh size are in an interaction to each other. In,thecourse of this article certain technical, practical details of the numerical treatment of the dynamic processes in shallow water with a very complex morphology will be stated so as used by the author. Numerical models are only possibility for prognosticating alteration tendencies after structural measures in the coastal area. Numerical models have certainly great advantages in comparison to other investigation methods, but they also have their limits, however which should not be misunderstood. The obtained results must be particularly critical evaluated in respect of their physical importance. The numerical investigations exclude a simultaneous assertion about a possible transport of solids since it was started from an unchangeable morphology apart from structural measures given into the model. Therefore, only forcasting for short periods of time will be possible. Long-time approximations with the aid of numerically determined residual currents and their alterations are only conditionally permissible.

316

INTRODUCTION In coastal research work as well as in coastal engineering, numerical models for reproduction and estimation of dynamic processes are used additionally in an increasing number for some years already apart from the hydraulic models. In both model types and investigation methods, the preparation o f a similarity to nature will be a necessary condition for further investigations. This conformity between the process of nature - mostly determinded by measurements and the obtained model results is limited in the case of a hydraulic model by the technique - as well as in the case of the numeric model

-

-

a.0.

a.0. by the

approximation of the topography and by the given solution methods. The recognition and knowledge of these borders should be included in the considerations when evaluating and judging the results. The application of numerical methods for the reproduction of the movement processes in various depths of the sea by the use of at first course-mesh screen grids has led to remarkable results, and has given considerable impulses to hydrographical work. Therefore it was quite obvious to develop a model for the coastal range by a screen-grid solution under observation of the interaction between the areas of different mesh widths which could also be useful for coastal engineering. The prediction of an lateration of the dynamical processes as subsequence of building-constructional measures within the coastal range (breakwaters, wash-ups, deepenings of waterways, etc.) is a task for which a numerical model with a high screen-grid solution, when expertly used, can be at least indicate possible tendencies. The depth distribution will go into the numerical model. The topography as well as the coast line will be all the better approximated if the screen-grid distance is less. The applied method permits such a screen-grid solution and, thus, represents once more its flexibility for investigations in the coastal range. For the numerical investigations, the method developed by HANSEN (HANSEN 1956, 1961) was used and as a base model, the already classically North Sea model

was applied. Here it may be permitted to abstain from a detailed description of the method, the discretisation, etc. in this connection and to refer to the appropriate literature (BRETTSCHNEIDER 1967; HANSEN 1956,

SUNDERMA"

1961; RAMMING 1966,

1970,

1972, 1973;

1966).

RAMMING (1975) gives further methodical hints towards the peculiarity of the method in regard to the screen-grid dissolution in the coastal range. The following vertically integrated differential equations were used:

317 SOME REMARKS ON NUMERICAL HANDLING OF DYNAMICAL PROCESSES IN SHALLOW WATER AREAS

In previous models with course resolution and traditional quadratic bottom friction the simulation of physical processes producing water levels and velocities in extremely shallow water near the coast was not sufficiently accurate enough for various practical purposes.

It is well known that well established in the meanwhile vertically integrated equations of motion are not valid for describing the dynamical processes in shallow water areas. The refined grid near the coast and in the estuaries permits a better approximation of the coast, the islands and the morphology in general. It has to be considered that as a consequence the agreement between computed and observed water levels and velocities at selected points has improved considerably. Moreover one has to consider the fact that sands and flats fall dry for some hours during a tidal cycle. It must also be considered that the main transport of water masses takes place in narrow channels on the flats. The refined model considers these facts explicitly but still the accuracy seems not to be good enough to encourage predictions, e.g. on effects of off-shore constructions. In particular the results show unsatisfactory phase lags at grid points located in areas with a very small water depth. Since the work of TAYLOR (1920) and DEFANT (1929) it is known that in numerical models the phase can be influenced by variations of the

bottom friction. The results have been applied for the present investigations. The application of the usual friction term is sufficient for water depths of more than 10 m. But in extremely shallow water areas with actual water depths below 3 or 4 m there may be some problems with numerical stability as a consequence of the numerical handling of the difference equation of motion.

Let u s consider the usual vertically integrated hydrodynamical equations in a simple one-dimensional case:

From equation ( 4 ) follows, partially written in difference from

Let u s only consider the first part of the equation (6) which describes the energy dissipation due to bottom friction. This term with paranthesis is responsible for numerical stability, because it exists the following inequality (7)

1 - 2 At r lul H

>

0

or

(8) H

> 2 At r

lul

otherwise one wins energy in a numerical way. It is obvious that this condition is very important for numerical experiments in shallow water and one has to find a solution to modify the difference equation in such a way that a numerical

318

instability will not occur and for all that a good agreement between observations and numerical results can be expected. In other words: the difference equations must be applicable for shallow water without violation of physical laws. In past experiments the equation (6) had been reduced to (9)

U'

= 2 At g

la;; ac

whenever the condition (7) had been violated. By means of the following arguments one can show that this neglect leads to a non-acceptable result: the neglect of the whole term is not correct, because in that case the velocity without consideration of the is only determinated by the gradient term g

acX

velocity in the time step before. if we consider the equation of continuity (5) and substitute u by g

acX

then it

follows

After some simple transformations and neglects of terms which have a second deviation it comes out the equation

The variation of depth at the flats and sands is very small that is equivalent to h

-

0 and one can say as a first approximation

Tbis means a stop of the water motion at those areas or in those discrete points of the model where the bottom friction term had been neglected because of the small actual water depth. The sands and flats do not fall dry because the given condition is not correct. The results which are coming out are not satisfactory. In nature these water masses flow with relative high velocity into the narrow channels - so called "Priele". In the numerical model these water masses are missing, they are not present at other discrete points which now have a phase displacement (see fig. 5 ) .

It is possible to solve this problem by means of varying the time step because the condition (7) depends also on t, but this method seems to be very complicated. One can avoid these complications if one varies the quotient

1 H of

the

friction term in such a form that numerical stability will not be voilated. The following treatment has been selected:

H + H e-pH (13)

with H

R =

(H

+

HI)

0

=

H

1

= 1

and

p = 0.5; I ; 2; 3.....10

The investigations with the aim to prove this treatment have been made in a simple tidal model for an open channel with the following geometry:

319

00 1

I0

05

00

15

00

00

05

10

15

25

20

30

35

DEPPH IN M

3)

FIG.

1

' X ( H + H ~ ~ - P H with ) - 13) (H+HlP

HI = Ho=1 = 0.5; li

COMPARISON OF D I F F E R E N T BOTTOM F R I C T I O N TERMS

_.......10

320

- COMPUTATION WITH --_ COMPUTATION WITH

RE

fi

9

Ri

(HIHI)

,'1 -0.5 ~

I

I

0 1 2 3 4 5 6 7 8 9 1011 12 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 h FIG. 2

COMPUTED WATER LEVEL VERSUS T I m AT DISCRETE POINTS 1 - 8

a) constant depth

h

=

0.60 m

b ) constant width

c) total length of the channel: 4 000 m d) 8 inner points and the boundary conditions: a) at the open boundary

5

0.5 cos (2 n/T) t

=

b ) at the closed boundary:

The convective term u

u

a,

=

with T

=

43 200 s

0

has been considered.

At fig. 1 the curves versus h with different factor p are presented. It is easy to see that the function 1 / H is not suitable for shallow water. The best results have been obtained with p = 1 . The advantage of this treatment is the convergence to the traditional bottom friction term with increasing water depth. F t g . 2 and fig. 3 show the results of the investigations with different friction

terms.

321

- COMPUTATION ---

WITH

R=

3

COMPUTATION WITH R = r/H+Ho

IH+H~P

UmI

T= 0h

0.

1

T= Sh

-0.4

-------I

PNr: 1 FIG. 3

I

I

2

3

I

4

1

5

I

I

6

7

1

8

T= gh

1

F-.

.rq.

3

COMF'UTED SURFACE SLOPE VERSUS X

It can be definitely stated:

a) a phase lag from 30' at point 2 up to 50' at the points 7 and 8, b) the water level at low water time is lower (about 14 cm) by means of the

special treatment against the usual, c) the water level at high water time decreases in the same case about 4 cm,

d) the smaller friction term allows an increasing velocity in shallow water. Fig. 4 shows the computed bottom friction energy versus time of this discussed one-dimensional tidal channel. Three computations with different bottom friction treatments have been made.

322

E&=[m3/=?l

3500 3000 2500 2000

15OG

10oc

50C I 1

1

7

0

1

~

2 3

1

4

'

Computation with

.........

1 R = -r

_ _ _ _ _

R=-

H

lul u

'

H + HI

r I ~ uI

H + H e-H R =

O

(H FIG. 4

r

IUI

'

'

5 6 7 8

u

+ HI)

ENERGY OF BOTTOM F R I C T I O N VERSUS TIME

'

'

'

1

1

9 1 0 1 1 12h

FIG.

5 COMPARISON BETWEEN MEASURED AND COMPUTED WATER LEVEL AT TWO SELECTED

P O I N T S I N THE GERMAN BIGHT FOR TWO D I F F E R E N T TREATMENTS OF BOTTOM F R I C T I O X

Ctml 1

CCml

WANG EROOGE / WEST

LEUCHTTURM MELLUM PLATE

1

I

0 1 2 3 4 5 6 7 8 9 10 11 12

I

I

I

-

MEASUREMENT COMPUTATION WITH R = ff

-----

I

I

I

I

I

I

I

I

0 1 2 3 4 5 6 7 8 9101112h

COMPUTATION WITH

R = r(H+H@) IH+H#

324

Fig. 5 shows a comparison between observed and differently computed water levels. These two examples are selected out of 80 points. This reproduction is not the best one but it is to comprehend a statistically average. In conclusion it can be stated that by means of the special bottom friction treatment the vertically integrated hydrodynamic equations in difference form can be modified to be applicable for shallow water without violation of physical laws and one can achieve good results in a tidal model for an open channel and in a two-dimensional model with shallow waters areas. Naturally, this equation is an empiric one which containes a numerical part as well as a physical part. The advective terms as an important part of the description of the dynamical processes in shallow water in the numerical model can not be neglected. Unfortunately, they cause disturbances of a numerical type in the model which can be stabilised by using a time-step and screen-grid dependent coefficient Ah, which describes the horizontally turbulent eddy viscosity. Shoal areas are often passed through by narrow channels with depths up to

5 m and more. On the grid points on the edges of these channels as well as on point$ where the shallow water area borders on navigable channels or waterways respecah tively, occur instabilities on account of the large Here also, the above

r. X

mentioned coefficient will have a stabilizing effect. I f one considers the following equation (14)

Ah =

I - a (2 AI)’ 7 .2 At

(BRETTSCHNEIDER, 1967)

so it can be generalized by a dependency to the bottom slope to

- -Ah (15)

4,

=

1-ae 4

A1

. (2 Al)’ 2 At

and simplifying to Ah

I - a ( 1 -=)

(16)

4,

(2 A1 A

*-

= =

(2 Al)’ 2 At

repective grid distance in x- or y-direction)

In the submitted investigations, the smoothing factor a = 0.98 was set, 2 takes up values between 2.5 to 170 m / s according to grid distance. It is

h easily recognisable that by medium formation at a strong bottom slope a stronger

dampening will occur. The advective terms u

a, v 5, a, v 5 a, and u a, r, X

will be treated on the

boundaries of the numerical models as follows: on the open boundary applies

a, or r a, = 0; r

on the closed boundaries the tzrms

v

a,

and u

a,

2

and v

a, q

cause no difficulties.

are treated in the same way as in the center of the area since the

components of velocity will disappear at the respective boundaries in accordance to the presuppositions, therefore, they will also be outside of the area equal to 0 (see fig. 6).

325

+

x

+

+

i

x

+

x-

u=o +

FIG. 6

x

I

+

x

THE GRID

The mean velocities of a tidal cycle will be determined as follows: T

v dt ; u and v are middled on the central S-point and form 0 T o the components of the mean velocity u (MIm = J + G2 in the point M with the -

u

=

-I T

-

u dt;

v

=

1

r

-

direction cp

=

V

arc tan = U

cz

.

The components of the residual currents of a tidal cycle and its direction are defined as

T Hu dt

r

= o and T

v

r

H dt

=

T .f Hv dt 0 T { H dt

.

These components are also middled on the central S-point. The residual current velocity in point M is the u (M)R

=

J

u:

+ v 2 and the direction cp

=

arc tan

". .

ur The residual current circulation is superimposed to the harmonica1 movement and enables a detailed insight into the long-time movement process which can be very informative for the judgement of the effectiveness of a structural measure.

326

Since overflowings of sandbanks as well as dry-falling sands occur in the shallow-water areas, these movement processes must also be reproduced by numerical methods which are physically interpretationable.

It is necessary to distinguish all thinkable cases, because it is possible that from time to time in some points there will be no water. One has to check the neighbourhood of each gridpoint from time step to time step with regard to the actual water depth, the depth distribution and the physical possibility of transports and directions of transports. It is also necessary to pay attention to the velocity of the overflow and the draining processes of extreme shallow water in order to make sure that the process in nature is in a good agreement with the process in the hydrodynamical-numerical

model. The covered distance of water is

computed and it will be checked if the next grid point has been reached. With this simple method one can find out the motion of the water line. During the treatment of the continuity equation it should be checked whether

5 is 5 0. It will be sensible to select 0.02 m as a coefficient data. If h + 5 is 5 0 . 0 2 , then the area represented by this point shall be considered as dry and 5 = -h is applied. The values h + 5 5 0.02 will be stored by points and are then again transported to the point if h + 5 is > 0.02, i.e. if the actual water h +

depth exceeds this value. Thus it will be possible that mass deviations may occur in time steps, but over a tidal cycle, the masses will be maintained. THE EFFECT OF THE STRUCTURAL CHANGE OF A BREAKWATER ON MOVEMENT PROCESSES The breakwater has a length of approx. 1 400 m and a present height of

NN - 1.40 m. The following three cases were investigated: Model

I

Model

I1

Model I11

breakwater, actual state breakwater, height NN + 1.50 m breakwater, actual height, half length

These marks are also valid for all figures within this paper. The model Before the beginning of the investigations it had to be checked at first: a) how large should the model area be selected? b) Where is a further screen-grid dissolution to be effected in accordance to tasks to be handled? c) How far can or should a reduction of the screen-grid distance be carried out so

that the peculiarities of the morphology and the breakwater can be well

approximated? d) How great will be the time step depending on the screen distance and the depth? Will the, thus, caused computation expenditure be supportable? e) What tide should be selected for the investigations and what boundary values will be available?

327

to a) The size of the model area to be selected will depend at first on the numerical investigations to be carried out and upon given questions. In the present case, the influence of a building-construction upon the dynamical processes must be investigated. It was also to be checked, which alterations of these vertical and horizontal movement processes will have to be expected in case of a possible constructional interference. Therefore, the size of the investigation area should be so selected that the open boundary of the numerical model can not be influenced by the changes of processes of movement to be expected. Based on the available floating measures and the morphology North of Wangerooge and Spiekeroog, the model area was selected as shown in figures 7 and 8. From the later obtained results it can be seen that the open borders are sufficiently far away form the range in influence. to b) The selection of the screen grid and the degree of solution depends on the morphology and to the same extent upon the necessary accuracy of reproduction of the changes of movement to be expected. Available were a part model with a mesh size of 457 m. The area North of Wangerooge offers itself for a further screen-grid solution. On the one side there are float measurements available and on the other side, changes in the processes of movements can be expected for certain in the socalled remote range in relation to the breakwater because of building-constructional measures which had to be investigated. A screen distance of 153 m was selected in this part area. In this manner, the topography could be reproduced in a fineness which conforms to the available maps. This part area is illustrated in the figures 7 and 8. A further solution

- the one-third-refinement method - was naturally required in the close range of the breakwater. A screen distance of 51 m between points of equal importance

(c/c,

u/u, v/v) has for the approximation of the morphology in the numerical model

the subsequence that here a depth is indicated every 25.5 m since the

5- as well

as the u- and v-points are available. Unfortunately the breakwater does not run parallel to the screen. The necessary deviations in the reproduction of the location in the numerical model should be considered as unimportant. to c) From the explanations under b) it can be concluded that the solution is an optimum one. The results which will be later explained add to this a multiple number of hints. In this connection it should also be mentioned, that to my knowledge, for investigations of this principal importance for the first time such extreme screen solutions have been applied for the first time and, apart from this, an interaction between areas of different grid distance were ensured.

8ZE

. FIG. 7

MAP AND THE NESTED MODEL I N THE AREA OF THE BREAKWATER

329

. . . . . . . . . . . . ' - -, . . . :-\.-,. . . . . . . . . . . . . . . . . . . . . . . , .\ ,., . . . . 2 , ', . . . . . . . \. '* '\. . . . . . . . . . . ... \

\

.

'

\

\

I

.

.

.

.

.

-

a

*

.b

:.,

_I

.

. .

FIG. 8

\

\

. ,.-.

. .

AREAS A, B AND C WITH DIFFERENT GRID DISTANCES

. . . . . . \

330

to d) The area with the lowest screen distance is bordered in the West by the breakwater's head. The further in the West situated deep channel has already a grid distance of 153 m. A further screen solution also in this range would lastly bring no other results and would led to a not anymore supportable computation expenditure. The condtion for the numerical stability to Courant-Friedrichs-Lewy ( 1 9 2 8 ) 2 At (time step)

<2 J

A 1 (smallest grid distance) 2 g %ax

(greatest depth)

leads through the great depths in the above mentioned case to a very short time step. With the given screen refinements and the given morphology follows the necessity for the numerical model to use a time step of 3 seconds. This means for h a tidal period of 44700 s = 12 25 min = 14900 time steps. This computation expenditure is considered as supportable. The periodical stationarity was obtained after two computed periods, this however, only because an approximated initial distribution were at our disposal. This model will be available for other possible investigations and is programmed to an optimum. The obtained results justify this computation expenditure. to e) An extensively undisturbed i.e. a not by wind influenced tidal period was selected for the investigations. The author had water-level observations of various levels out of an investigation series from the June 2 0 , 1969 in hand. Naturally, any other tide can be used, but it should as free as possible from wind influences and the measuring period should not be in the range of the nippor spring time. Since this model area is a part of the North sea model, by the aid of which the movement processes in the Weser-Jade-Regime were determined to a tast-orientated screen solution, the necessary water level values could be taken in the points at the open model boundary as f(t) from the available model. After completing the timely, linear interpolation and the subsequent 5-time overlapping middle structure for smoothening the curvature (this is absolutely permissible because of the very short time step) the water level values for the intermediate points were also obtained by linear interpolation

-

this is also permissible because of

the short screen distance. Natural similarity The conformity of the determined velocity- and water level values to measurements and ovservations is an unalterable condition for prognostical investigations. This applies to the hydraulical model as well as to the numerical model. Measurements are mostly only available in a few discrete points. If the

331

numerical model renders the timely course of the horizontal and vertical movements in these points or if the average deviations remain within the scope of the numerically possible accuracy, then it can be supposed that the reproduction of the processes of movement has succeeded within the scope of permissible error limits. This will not exclude, however, the critical areas will always exist which require a more close investigation and examination. The figures 10, 11, 12 show the agreement between observations and computations of water levels in the basic model. Within the scope of the investigations which are the subject of this report, the numerically determined results were compared with the water level values on the tide gauge in Wangerooge West (June 20, 1969) and, thereby, it was determinded: I . High water (model)

+ 122 cm

High water (nature)

+ 126 cm

Difference model/nature

-

2. Low water (model)

4 cm

- 130 cm

Low water (nature)

-

138 cm

Difference model/nature

+

8 cm

+

20 min

3. Setting-in time high water (model) - Setting-in time high water (nature)

Since the tide gauge Wangerroge West is extremely unfavourable situated, the deviations must be considered as negligible and absolutely permissible. It can be supposed that the screen solution with all its consequences is the cause of the result from the reproduction of these movements. Unfortunately the author had not other observations from this period for the model range in hand for comparing the natural similarity further. For the reproduction of the movements in extremely shallow water with periodically dry-falling sands and shoal areas as well as water bearing channels, the treatment for bottom friction was well proved during the previous investigations as used again. THE RESULTS Velocities Model I Here the remark should be permitted that the solution of the grid within this area

-

as far as illustrated in the area A - is insufficient. With a point

distance of 4 5 7 m, the locally extremely complicated current movements can not be reproduced satisvactorily. The illustrations of these processes in the areas B ( 1 5 3 m) and C (51 m) will provide a much better information over the in- and

overflowing process.

332

1.

WITTDUN

2. CUXHAVEN 3. WANGEROOGE/WEST 4 . MELLUM PLATE FIG. 9

20.6.69 20.6.69 20.6.69 20.6.69

5. SCHILLING

6. VOSLAPP 7. VAREL 8. DANGASTER SIEL

16.9.65 16.9.65 16.9.65 16.9.65

MAP OF SPECIAL POINTS IN THE NORTH SEA MODEL IN WHICH THE COMPUTED AND OBSERVED TIDAL CURVES HAD BEEN COMPARED

333

1 WlTlWN

cM 2.0

I 5

1.0

0.5

0

-0.5

-1.0

-1.5

C -2.0

i

i

i

i

i

3

4

WANGEROOGE/ WEST

MELLUM RATE

s

i

o

n

u

w

cw I.0

1.5

1.0

0.5

0

-0.1

-1.0

-1.1

5” ME2 20.6 69 -2.0

o

I

I

J

4

5

MEASUREMENT FIG.

10

4

i

i

s

10

11

U Y I

RESULT OF THE NUMERICAL MODEL

COMPARISON BETWEEN OBSERVED AND COMPUTED TIDAL CURVES

.

334

5

6

SCHILLIG

VOSLAPP

m 2.0

1.1

1.0

0.1

0.0

-0.S

-1.0

-1.1

-2.0

RESULT OF THE NUMERICAL MODEL

MEASUREMENT

FIG. 1 1

CONPARISON BETWEEN OBSERVED AND COMPUTED TIDAL CURVES The duration of the tidal current vf is normally longer then the duration

of the low-tide current v

(see also mean transports of a tidal cycle, illustrated

in table 2). The high tidal current velocities with v = approx. 1.8 m/s in the max southern part of the Harle are noteworthy. Model I1 The velocities at the breakwater head are greater and retain this order of magnitude also over a longer period in comparison to model I. The flow-in time is also longer. Both of the afore mentioned processes are of importance for the interpretation of the numerically determined average transports of a tidal cycle

-

and

they will find here their confirmation. Extremely high tidal current velocities ( 1 . 5 / 1 . 7 5 / 2 . 1

m/s) occur in the

Harle at the south of the breakwater, whereas the low-tide velocities will be low. These high velocities can

-

when the water i s running up on the flat, i.e. with

335

7

8

VAREL

DANCASTER SIEL

I

2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

-l.s

H E 2 16.9. 65 -?..I t

0

I

t

8

b

I

8

1

*

8

1 0 1 1 I t

I

0

.

2

.

S

.

b

.

5

.

8

.

1

.

.

8 - 8

.

10

.

11

.

I t

RESULT OF THE NUMERICAL MODEL

MEASUREMENT

FIG. 1 2

.

1

COMPARISON BETWEEN OBSERVED AND COMPUTED TIDAL CURVES

the occurrence of morphological barriers - lead to strong eddy formations and may change the morphological structure. The water level changes in the points of the investigation area at the north of the breakwater and remain, when compared to model I, in the centimeter range. The high water, however, lies in the Harle and south of the breakwater at a phase difference of approximately 30 minutes clearly lower - about 25 to 30 cm.

This influence is effective up to 3 100 m south of the breakwater. Model 111 Apart from the changes which results solely on account of conditions

-

shortening of the breakwater by 50 % - there is a strong similarity between the

results oE the model I and the model 111. This local change is obviously of no special importance for the total process

-

with the exception of a phenomen which

is of a considerable importance in my opinion.

336

* * * * - * *.""

...... ......

t

l

'

.

.

'

.*"'

.*.'.. 0

0

.

"

.

I

...... 0

.

.

I

0

.

.

0

.

.

.

-

* * * *

. * * ' . - * * * * * * * ' . *

"'..' '

0

.

.

' * * ' * '

..*'*.

.+-c - 4 - c

............ . .. .. .. .. .. .. .. .. .. .. .. . ............... ............... ............... ............... ............... .......

...........

.

r

.

.

*

*

+

t

.

r

r

r

.

r

*

I1

,..*.+..

.......... r .......... .......... T

-

.

*

*

*

r

r

~

....... ........ 1 " " " ' ........

t T

~

I * * - - * * *

r

r

............. +

.

r

r

r

r

~

r

...... 4

r

r

r

/

~

.

-

~

r

~

.

r c.-.-.\,

, - r r r r r - - . - , \ , . + . C , . r c A - . . - v

. .. .. .. .. .. .. .. ........ . .. .. ... .................................... r

.

............... ............... ....... .

+

k \ L

- *

.

+

+

*

?

+

.

7

T

.

.

+

+

+

I11

.

- - - - - \ \ - d ? - - L P

...... --. ,..4*,,,,-Lrr -

A

4

L

c c c ---+-_._---+c-.v

c

2

a

4

+

t

*

----A

...... ........... ........ ....... ........ I, "." ".' . . . . . 4

d

,,,c.f

, . . , , , . . - c r t

* . . * - * * .

. . - , . . , , . * T T

SCALE

-

1 m/s

FIG. 13 AREA A - DISTRIBUTION OF VELOCITIES 6 HOURS AFTER MOON'S TRANSIT THROUGH THE MERIDIAN OF GREENWICH (20.6.69 - 11.02 h )

337

SCALE

-

I m/s

FIG. 1 4 AREA B - DISTRIBUTION OF VELOCITIES 7 HOURS AFTER MOON'S TRANSIT THROUGH THE MERIDIAN OF GREENWICH (20.6.69 - 12.02 h )

H

-.

.

H

H

*P ......... .. .. .. .. .. .. .... .. .. .. . .. .. .. .. . ... ... ... ... .............. .

.. .. .. .. .. .. .. .. .. ........ .. .. ............. .. .. .. .. .. ........

......... .. .. .. .. .. .. .. ..

H H

H

I

v)

I

e

E

.

v ) .

--

0

N

4-2

h

AREA C - DISTRIBUTEON OF V E L O C I T I E S 6 HOURS AFTER MOON'S TRANSIT THROUGH THE MERIDIAN OF GREENWICH (20.6.69 - 1 1 . 0 2 h )

F I G , 15

I m/s

-.

SCALE

339

I

I1

I11

SCALE

-1

FIG. 16

m/s

AREA C - DISTRIBUTION OF VELOCITIES 9 HOURS AFTER MOON'S TRANSIT THROUGH THE MERIDIAN OF GREENWICH (20.6.69 - 14.02 h )

(rI Z O ' L L - 6 9 ' 9 ' 0 Z ) H3IMN3X83 60 NVIaI8El4 3Hd H3flOXHd LISNWL S,NOOI.I 83LdV SInOH Z I S3IdI3OT3A 60 NOIdflB18dSIa - V VX8V

s p

I

LI

-

'3Id 3TV3S

1

I11

I1

I

OPF

341

I

a

co7-T WANGEROOGE

111

0

( , ,I

WANGEROOGE

F I G . 18 DISPLACEMENT OF DIYERGENCE

342

The through-current flows more uniformly as has been expected but the turn-in of the current takes place farther west in comparison to the present condition of the breakwater. The current splitting area The effect of the alteration of the breakwater towards the position of the current dividing area is of special interest. From the lines of the directions of maximum velocity per each model investigation, the point can be found in which in the near vicinity before the island, the turning point can be located of the current path. A l l the further this point is wandering to the east, all the greater are the strand-parallel running tide and low-tide velocitis which cause simultaneously the transport of solid particles. The following result is on hand: By a rise of the breakwater to NN + 1.50 m, the turning point is dispiaced by 3 5 0 0 m to the east and by a shortening of the breakwater by 50 % at the present height of NN

-

1.40 m it will be displaced by 1 000 m to the west.

Water level comparisons In connection to the evaluation of the results, the water levels at tF.e north and the south of the breakwater were middled locally at hourly intervals. Thereby, it was determined in the present condition as well as in case of a shortening of the breakwater by 50 %

-

with deviations in the centimeter range -

the water levels at the north and south of the structure will be equal. If the breakwater is raised, however to NN + 1.50 m

-

i.e. free of tidal high water -

striking differences will occur at low water. The average water levels at the south of the breakwater are as follows: 10.02 h

7 cm

11.02 h

9 cm

12.02 h

38 cm

13.02 h

4 2 cm

14.02 h

41 cm

15.02 h

34 cm

16.02 h

21 cm

17.02 h

7 cm lower.

A s stated in table 1 , only the area C was drawn in, i.e. the listed values

indicate only the local effect. If the water level values are middled over the tidal period, then it can be clearly seen that with a rise of the breakwater to NN + 1.50 m in all three areas A, B and C, i.e. in the coastal as well as in the remote range, the mean water level of a tidal period south of the breakwater will be 13 cm lower than north of the structure.

343

TABLE 1 Mean water levels every hour in the Northern and Southern area of the breakwater (in cm) Time

Breakwater Actual State

Breakwater Height NN + 1 . 5 0 m

Breakwater Half Lenght

June 2 0 , 1969

North

North

North

05.02 06.02 07.02 08.02 09.02 10.02 11.02 12.02 13.02 14.02 15.02 16.02 17.02

South

32

33

- 17

- 17

-

-

60

- 94

- I 12 -1 12 - a2 - 15 47 90 104 93

- I 12 - I 14 - a3 - 24 45 a7 102 93 54

54

31

30

27

- 18 - 60

- 18 - 60 - 93

-I 10 -1 i a - 99 - 50

53 93 106 94 54

11

-113 - 83 - 17 46

52 72 73 47

103 93 54

-I -1

-

South

- 17 - 59 - a9

31 18 61 94 12 11 a0 12

-

60

- 94

South

-

93

- 1 12

-112 -113 - a3 - 17 45 a7 102 92 54

aa

TABLE 2 Mean water level o f one tidal cycle (June 20, 1969) in the northern and southern area of the breakwater (in cm) ~

Breakwater Actual State

Breakwater Height NN + 1.50 m

Breakwater Half Length

Area A North South

-

-

5 22

- 5 - 35

-

6

-

-

-

5 22

Area B North South

- a

6

-

6

- 22

- a

-

-

7 9

Area C North South

-

7

-

- a

6 22

Average transports of a tidal period

I T I T If the expressions - J Hu dt and - J Hv dt with H = h + T o

T o

6

and T the period

time during an investigation are formed one will obtain at the end of a period the average mass transports at the discrete u- and v-points with % and &. These values are middled to the enclosed 6-points V

u Jr; u x ” + + x

-f V

344

I1

SCALE

-1

m

2

/s

FIG. 19 AREA B

-

MEAN MASS TRANSPORT OF ONE TIDAL CYCLE

345

1

I I \

l l I l l l l l l [

I

\

\

\

\

\

'

, , . .

1

I

1

"

"

\ \ \ \ ' \ I \ \ \ \ \ \ I ' * ' * -

i I

/

,

,

r

r

r

r

r

-

-

.

r I ................... . ............................ .......................... {- , . . . . . .

ih +

5

,,

\\'\ ' \' I '

LL

I

,.

I1

. . . . . . , . . . . .

' ' . - - -

,. J I11

\ \ \ \ \ \ SCALE

-

I m'/s

FIG. 20 AREA C

-

ME&? MASS TRANSPORT OF ONE TIDAL CYCLE

346

and by means of tg cp

=

--

Hv/Hu the direction of the average transport will be gained

simultaneously. The figures 19, 20, 21 will disclose the average trnasport procedure and confirm all together the hitherto described results. In table 2 are the masses determined for all three tasks and for the three partial sections between the breakwater and Spiekeroog, the western and the eastern half of the breakwater which flow per second into the shoal area in a generally southern direction. The illustration is very informative. It shows in detail: a) the general conformity of the amount of the numerically determined average transports with the models I and 111, b) an almost doubling of the transport in the Harle at a rise of the breakwater to NN + 1.50, c) a great increase of the transports in the western half of the breakwater in case of a shortening by 50 %. Conclusion a) The screen-grid solution increases the assertion ability of a numerical model on questions from coastal engineering. Apart from this, the numerical model

offers the possibility to compute a multiple number of model representations and to determine functional interconnections by comparative observations. b) The breakwater is a neuralgical point of the investigation area. The existence of this structure, its height and length as well 3 s its direction determine considerably the course of the motive processes at the north of Wangerooge, in the Harle as well as in the wadden area behind of Spiekeroog and Wangerooge. c) An inclusion of the shoal region in the eastern direction up

,to

the Minsener

Oog and further to the west behind of Spiekeroog as well as the total surroundipgs of the island Wangerooge up to the Blaue Balje seems to be sensible in order to explain suspected influencing of the transport proceedings. d) The Blaue Balje

-

particularly after heaping-up the Minsener Oog - seems to

win importance for the transport behind WangerAoge since the average transports z of a tidal cycle predominate clearly in the direction of the shoal area. e) In a comparison of the average transports of a tidal cycle in the three cases to the average transports of a tidal cycle in the three cases to be investigated, the amounts should also be checked by measurements. Important for the judgements of the results, however, should be the relations. f) The distribution of tidal currents with all three investigations permits the determination that in case of a rise of the breakwater to NN + 1.50 m, the water masses before Wangerooge will shift at first very far to the east

- further than at the present condition of the breakwater

-

and the partake in

347

I

I II 111

UI

I

111

8.6 16.4 Z9 1 (lo3 m3

s-l)

AREA BETWEEN THE BREAKWATER AND THE I S L E OF SPIEKEROOG

I

I

I

I

I (103

(lo3

m3 s-1 1

WESTERN HALF OF THE BREAKWATER

m3 s-I>

EASTERN HALF OF THE BREAKWATER

MEAN MASS TRANSPORT I N ONE TIDAL CYCLE I N TOTAL:

I: 11: 111:

F I G . 21

8 . 6 + 10.9 + 7.5 1 6 . 4 + 0.0 + 0.0 7 . 9 + 1 5 . 2 + 6.1

3

3

-1

27.0 x lo3 m3 s-] 16.4 x 10 m s - ~ 2 9 . 2 x 103 m3 s

COMPARISON OF THE MEAN MASS TRANSPORT OF ONE TIDAL CYCLE BETWEEN THE THREE DIFFERENT INVESTIGATIONS

the dynamical processes before the breakwater with large streams directed parallel to the strand and in the Harle. g) A weakening of the strong coat-parallel currents shows the numeric model at a shortening of breakwater to 50 % of the present length. h) It should be mentioned once more that possible morphological changes were not taken into consideration with these investigations.

348 Nomenclature acceleration o f gravity

m s

depth from the undisturbed sealevel to the bottom

m

5)

m

total water depth (h +

deviation from the undisturbed sealevel

-2

m

additional depths in the bottom friction term

m

components of the vertically integrated velocities

m s

space coordinates

m

time coordinate

S

-1

-1

Coriolis parameter constant bottom friction coefficient with out dimension (0.0025) components of the bottom friction force Laplace-operator coefficient of horizontal diffusivity

m

2

s

-1

half time step REFERENCES Brettschneider, G. Anwendung des hydrodynamischen Verfahrens zur Ermittlung der M2-Mitschwingungsgezeit in der Nordsee. Mitteilungen des Instituts fiir Meereskunde der Universitzt Hamburg VII, 1967. Courant, R., Friedrichs, K., Lewy, H. uber die partiellen Differentialgleichungen der mathematischen Physik, Math. Annalen, 100, 1928. Defant, A. Dynamische Ozeanographie, Berlin 1929. Hansen, W. Theorie zur Errechnung des Wasserstandes und der Strijmungen in Randmeeren nebst Anwendungen, Tellus, VIII, 1956. Hansen, W. Hydrodynamical Methods Applied to Oceanographic Problems, Mitteilungen des Instituts fiir Meereskunde der Universitat Hamburg I, 1962. Ramming, H.-G. Shallow Water Tides, Mitteilungen des Instituts fiir Meereskunde der Universitat Hamburg X, 1968. Ramming, H.-G. Investigations of Motion Processes in Shallow Waters, Report on the Symposium on Coastal Geodesy, Miinchen 1970. Ramming, H.-G. Reproduction of Physical Processes in Coastal Areas. Proc. of the 13th Conference on Coastal Engineering, Vancouver/Canada 1972. Ramming, H.-G. Reproduktion physikalischer Prozesse in Kiistengebieten (in wesentlichen Teilen veranderte Fassung in deutscher Sprache, Die Kiiste, Heft 22, 1973. Ramming, H.-G. A Nested North Sea Model with Fine Resolution in Shallow Coastal Areas, Mbmoires Soci6t; Royale des Sciences de LiGge, 6e sgrie, tome X, 1976, pp 9-26. Scndermann, J. Tidal Waves in Schematic Estuaries, Proc. 12th Conference of Coastal Engineering, Washington/USA, 1971. Taylor, G. J. Tidal Oscillations in Gulfs and Rectangular Basins, Proc. London Math. SOC., 1920.

349

APPLICATION OF A FINITE ELEMENT HYDRODYNAMIC MODEL TO THE GREAT BAY ESTUARY SYSTEM, NEW IIAMPSHIRE, U . S . A .

RONNAL P .

REICHARD and BARBAROS CELIKKOL

Mechanical Engineering Department, University of New Hampshire, Durham, N . H .

03024

ABSTRACT A vertically integrated, one layer, hydrodynamic model developed by Connor

and Wang for Massachusetts Bay has been adapted to the Great Bay Estuary, New Hampshire.

This finite element model was found to be well suited to the complexities

of the estuary.

Initial model development and testing was carried out.

The bottom

friction coefficient was the dominant parameter in the calibration process.

A method

for selection of the bottom friction coefficient in the calibration process was developed. cluded.

Eddy viscosity terms, which primarily act as numerical damping, were in-

The model was calibrated using an extensive set of tidal sea level and

current data recently collected by the University of New Hampshire and the National Ocean Survey (NOS). The data sets used in the calibration and validation process include sea level data used to specify open boundary conditions in the model. Validation results show good agreement between tidal elevation measurements and model predictions.

Predicted currents have a different scale of resolution than the

current measurements, making direct comparisons difficult to interpret.

A program

for processing current data to make more meaningful comparisons is presented briefly.

INTRODUCTION

Increased environmental awareness has created a demand for analysis of estuarine processes and understanding how these processes are affected by development of adjacent coastal land.

The foundation of a comprehensive analysis program

is understanding the hydrodynamic behavior of the estuary.

The work presented

here, the implementation of a predictive, computer-based numerical hydrodynamic model, is the first step in establishing a program for analysis of the Great Bay Estuary system. Hansen (1956) formulated an approach to numerical hydrodynamic modeling used by most modern investigators. Leendertse (1967) made significant contributions in the area of numerical stability and accuracy while developing a two-dimensional vertically averaged numerical hydrodynamic model, utilizing the finite difference method to solve the differential equations.

This model has been applied to severai

estuaries, including Jamaica Bay by Leendertse (1972), and Narragansett Bay by

350 Hess and White (1974). Connor and Wang (1973) subsequently developed a similar model, using the finite element method, and applied it to Massachusetts Bay.

Three-

dimensional models have been developed by Leendertse (1973) (multi-layered finite difference) and Connor and Wang (1974) (two-layer finite element), but these models are presently very difficult and expensive to use. In the work presented here Connor and Wang's two-dimensional finite element model was selected because it is ideally suited to the particular hydrodynamic and topographic conditions found in the Great Bay Estuary system.

Initial application

has been carried out, including a series of tests used to develop a numerically stable grid.

Field data indicates that the surface slope induced pressure gradient is

balanced by bottom friction, which is confirmed by an order of magnitude analysis of the Navier-Stokes equation for one-dimensional channel flow.

This balance is

used as a quantitative aid in selecting new bottom friction coefficients in the calibration process.

Boundary conditions for model calibration and validation

were specified directly from sea level data, while both sea level and current data were compared with model predictions. and a different set for validation.

One set of field data was used for calibration, Tidal elevation predictions compare favorably

with measured values, but scale of resolution differences between measured and predicted currents are apparent in the current comparisons.

Plans for obtaining current

measurements on an appropriate scale are presented, which will allow further refinement of the model.

A PHYSICAL DESCRIPTION OF THE ESTUARY The Great Bay Estuary system (Figure l), located in southeastern New Hampshire and southwestern Maine, consists of 45 square kilometers of tide water, and its tributaries drain the 2,410 square kilometers of the Piscataqua River Basin.

The

estuary is a complex combination of tidal flats, and channels which average about 10 meters in depth. only seaport.

The mouth of the estuary is at Portsmouth, which is New Hampshire's

This segment of the estuary is dominated by a main channel, which is

bordered by islands.

The surrounding area is heavily developed, with residen-

tial, commercial, and industrial uses, including the Portsmouth Naval Shipyard. Farther up the estuary is the lower Piscataqua River, a tidal channel which has been dredged (minimum depth 10 meters) to facilitate shipping.

The upper Piscataqua

River is a shallow tidal river, formed by the convergence of the Cocheco and Salmon Falls Rivers in Dover.

Joining the Piscataqua River at Dover Point is Little Bay,

an L-shaped body of water with two tributaries, the Bellamy and Oyster Rivers.

The

Great Bay is a large, shallow bay characterized by tidal flats with a network of channels.

A small channel from the Winnicut River and a larger one from the

Squamscott and Lamprey Rivers join to form the main channel, which continues through Furber Strait into Little Bay. used primarily for recreation.

Great Bay and Little Bay are undeveloped, and are

SQUAMSCOTT RIVER

Figure 1. The Great Bay Estuary system, New Hampshire, USA. The location of sea level stations (triangles) I and current stations (circles) refered to throughout the text are identified.

352 The c u r r e n t flow i n t h e e s t u a r y i s p r e d o m i n a n t l y t i d a l , as t h e f r e s h w a t e r component i s l e s s t h a n one p e r c e n t of t h e t i d a l prism.

The t u r b u l e n c e a s s o c i a t e d

w i t h t h e t i d a l flow i n t h e e s t u a r y p r o d u c e s a v e r t i c a l l y w e l l mixed w a t e r column throughout t h e e s t u a r y .

To i l l u s t r a t e t h i s , s u r f a c e and bottom v a l u e s of t e m p e r a t u r e

arid s a l i n i t y a r e p r e s e n t e d i n F i g u r e ( 2 a ) and (2b) f o r p e r i o d s of h i g h low (August) f r e s h w a t e r d i s c h a r g e i n t o t h e e s t u a r y .

(March) and

The l a r g e d e v i a t i o n i n t h e

March s a l i n i t y c u r v e s a t km 18 c o r r e s p o n d s t o a measurement t a k e n a t t h e mouth of I t s h o u l d b e n o t e d t h a t t h e s t r a t i f i c a t i o n i s l o c a l , and g i v e s

t h e Oyster River.

a n i n d i c a t i o n o f t h e d i s t a n c e s r e q u i r e d f o r t o t a l mixing. The t i d a l r a n g e i n t h e e s t u a r y i s g r e a t e s t a t Portsmouth (2.5 m a v q . ) , d e c r e a s e s t o Dover P o i n t River

(2.1 m a v g . ) .

(2.0 m a v g . ) , and t h e n i n c r e a s e s s l i g h t l y t o t h e Squamscott

T i d e p h a s e v a r i e s s i m i l a r l y , i n c r e a s i n g from Portsmouth Harbor

t o Dover P o i n t , and t h e n i n c r e a s i n g o n l y s l i g h t l y t o t h e Squamscott R i v e r .

Tide

d a t a a t s i x l o c a t i o n s i n t h e e s t u a r y f o r a s p r i n g t i d e o c c u r r i n g on J u l y 11, 1 9 7 5 a r e p r e s e n t e d i n s e v e r a l forms t o i l l u s t r a t e t h e t i d a l c h a r a c t e r i s t i c s of t h e e s t u a r y . 'Table (1) p r e s e n t s t i m e and e l e v a t i o n a t h i g h and l o w t i d e s .

F i g u r e (3a) i s a t i d a l

e l e v a t i o n t i m e s e r i es f o r each of t h e f i r s t f o u r s t a t i o n s i n T a b l e (1). The d e c r e a s i n g a m p l i t u d e and i n c r e a s i n g phase from Portsmouth Harbor t o Dover P o i n t i s r e a d i l y apparent.

Figure

( 3 b ) i s a t i d a l e l e v a t i o n t i m e s e r i e s f o r Portsmouth Harbor and

each of t h e l a s t t h r e e s t a t i o n s i n T a b l e (1). R e l a t i v e l y l i t t l e p h a s e and a m p l i t u d e v a r i a t i o n i s s e e n i n t h e h a l f of t h e e s t u a r y from Dover t o t h e Squamscott R i v e r . The t i d a l e l e v a t i o n d a t a , coupled w i t h t h e geometry of t h e e s t u a r y , l e a d s q u i t e n a t u r a l l y t o c h a r a c t e r i z a t i o n of t h e e s t u a r y as a t i d a l c h a n n e l ( P i s c a t a q u a R i v e r ) connecting a t i d a l r e s e r v o i r

( L i t t l e Bay and G r e a t Bay) t o t h e o c e a n .

The d i s -

t i n c t i o n between t h e c h a n n e l and r e s e r v o i r c a n b e s e e n c l e a r l y i n F i g u r e (4), which p r e s e n t s h o u r l y t i d a l e l e v a t i o n a l o n g t h e l e n g t h of t h e e s t u a r y .

From Portsmouth

Harbor t o Dover P o i n t (0 t o 1 5 km) t h e s u r f a c e s l o p e s are r e l a t i v e l y l a r g e .

At

Dover P o i n t , e x t r e m e l y l a r g e g r a d i e n t s o c c u r , i n d i c a t i n g a b l o c k a g e o r chokinq of the t i d a l flow.

Beyond t h i s a r e a , s u r f a c e g r a d i e n t s d e c r e a s e r a p i d l y .

High Tide Location

Elevation (cml

Time

(hr) -0.03

Low Phase (hr)

Tide

Elevation (cm)

Time (hr)

-

-166

6.32

-140

6.80

Phase (hr)

-

Tidal Range

(cm) 356

Portsmouth Yacht Club

190

A t l a n t i c Iieiyhts

165

0.38

+0.41

I I i l t o n Park

146

1.50

1.53

-111

7.82

1.50

257

Be l l a m y R i v e r

138

2.00

2.03

-101

8.27

1.95

239

Adams I'uint

140

2.30

2.33

-102

8.98

2.66

242

Squamscott R i v e r

150

2.27

2.30

-105

9.00

2.68

255

Table 1

0.48

T'idal c h a r a c t e r i s t i c s of t h e G r e a t Bay

E s t u a r y system from sea l e v e l d a t a c o l l e c t e d J u l y 11, 1975

305

353 23.0

20.5

AUGUST

-Y -w

18.0

‘L

--

-

‘\

155-

\

[L

3

I-

a

5 13.0 a

z w I-

-

10.5

8.0

MARCH

5.5

-

---__--

-\--

I

30

I

I

ESTUARY

F i g u r e 2a.

S u r f a c e (-)

___-__

- _ _ _ _ - _ / - - -

I

I

I

5

0

LENGTH (KM)

temperature a t l o w t i d e

and b o t t o m ( - - - - - )

i n t h e estuary.

2 4.5

f a v)

.^

MARCH

r

30

20

25

ESTUARY

F i g u r e 2b. t h e estuary.

S u r f a c e (-

)

10

15

LENGTH

and b o t t o m (-----)

5

(KM) s a l i n i t y a t l o w t i d e in

354

-I

0

I-

I

9 w

I W

F!

I-

-200 0

I

I

I

I

1

1

I

I

1

I

I

2

4

6

8

10

12

14

16

18

20

22

1

T I M E (FIRS)

Figure 3a.

Tidal elevation for July 11, 1975 at four locations in the estuary:

Portsmouth Yacht Club;----Squuamscott

Atlantic Heights;---

Hilton Park;

River.

200,

-200 0

I

I

I

I

I

I

1

I

I

I

I

2

4

6

8

10

12

14

16

18

20

22

2

TIME ( H R S )

Figure 3b.

Tidal elevation for July 11, 1975 at four locations in the estuary:

-Portsmouth

Yacht Club; -.-

--Squamscott River.

Bellamy River;

---

Adams Point;

355 Current v e l o c i t i e s vary considerably throughout t h e e s t u a r y , b u t c h a r a c t e r i s t i c v e l o c i t i e s can be i d e n t i f i e d f o r s e c t i o n s of t h e e s t u a r y . Location

Velocity

A t l a n t i c Heights

(M/S)

2.00

Piscataqua River

1.50

Dover P o i n t

2.00

L i t t l e Bay

0.75

Adams P o i n t

1.00

Great Bay

0.50

Current phase changes very l i t t l e throughout t h e e s t u a r y , with a maximum of approximately one-half

hour between Portsmouth Harbor and t h e Squamscott River.

There i s a c o n s i d e r a b l e d i f f e r e n c e between t i d e e l e v a t i o n phases and c u r r e n t In t h i s e s t u a r y t h e s u r f a c e s l o p e i s balanced p r i m a r i l y by bottom f r i c t i o n

phases.

r a t h e r than a c c e l e r a t i o n s .

Therefore, s u r f a c e s l o p e i s t h e primary c u r r e n t d r i v i n g

f o r c e , and s l a c k water occurs when t h e s u r f a c e s l o p e i s very small.

I n L i t t l e Bay

and Great Bay, s l a c k water c o i n c i d e s w i t h high and low t i d e , while i n Portsmouth Harbor i t occurs about two hours a f t e r high and low t i d e .

Current v e l o c i t y ,

tidal

e l e v a t i o n , and s u r f a c e s l o p e time s e r i e s a r e p r e s e n t e d f o r two l o c a t i o n s i n t h e estuary.

Figure (5a) shows t h e two hour d i f f e r e n c e i n phase between c u r r e n t

v e l o c i t y and t i d e h e i g h t i n Portsmouth Harbor.

-I

There i s about an hour phase d i f f e r -

50-

0

I-

= '3

0-

w

I

-

-50-

I-

-

-100

t

-I5O

30

25

20

15

ESTUARY LENGTH

Figure 4 .

10

5

0

(KM)

Hourly s u r f a c e e l e v a t i o n along t h e l e n g t h of t h e e s t u a r y f o r 03:OO t o

15:OO J u l y 11, 1975.

S o l i d l i n e s i n d i c a t e ebb t i d e , dashed l i n e s f l o o d t i d e .

356

-200

I

I

2

4

1 6

I

I

I

I

I

I

I

1

8

10

12

14

16

18

20

22

4

TIME (HRS)

Figure 5a.

Current velocity (-;cm/s),

surface slope (---;xlO

-6

and

tidal elevation (--;cm),

for July 11, 1975 in Portsmouth Harbor.

)

2 I50

I

I

I

I

I

I

1

I

I

I

I

Current velocity ( ;cm/s) , tidal elevation ( -- ;cm) , and -6 ) for July 30, 1975 in Little Bay. surface slope (--.-;xlO

Figure 5b.

357 ence between c u r r e n t v e l o c i t y and s u r f a c e s l o p e i n d i c a t i n g t h a t o t h e r f o r c e s must F i g u r e ( 5 b ) shows s l a c k w a t e r c o i n @ d i n g w i t h h i g h and low

be t a k e n i n t o a c c o u n t .

(ib

t i d e , and l i t t l e o r no p h a s e d i f f e r e n c e between s u r f a c e s l o p e and c u r r e n t v e l o c i t y i n L i t t l e Bay.

THE HYDRODYNAMIC MODEL

C i r c u l a t i o n A n a l y s i s by F i n i t e E l e m e n t s (CAFE) i s a t w o d i m e n s i o n a l , v e r t i c a l l y a v e r a g e d , n u m e r i c a l hydrodynamic model d e v e l o p e d by Connor and Wang. model s o l v e s a s i m p l i f i e d form of t h e N a v i e r - S t o k e s

The

and c o n t i n u i t y e q u a t i o n s u s i n g

a f i n i t e element t e c h n i q u e . The e q u a t i o n s g o v e r n i n g f l o w i n a n e s t u a r y are t h e N a v i e r - S t o k e s e q u a t i o n :

a at

-

+

~

(pu.) 1

a ap +7 (pu.u.) = - axj 1 7 ax.

a Uk a (5 1 ax. ax k

and t h e c o n t i n u i t y e q u a t i o n :

*a +t

~

a axi

(pu.) = 0 1

The e q u a t i o n s a r e s i m p l i f i e d by assuming i n c o m p r e s s i b l e f l o w , c o n s t a n t d e n s i t y , c o n s t a n t eddy v i s c o s i t y , and t h a t t h e second d e r i v a t i v e of v e l o c i t y w i t h r e s p e c t t o perpendicular coordinates i s s m a l l .

The e q u a t i o n s a r e t h e n ensemble a v e r a g e d t o

obtain:

-

a

mi) =

0

1

Where t h e o v e r b a r d e n o t e s a n ensemble a v e r a g e , and t h e p r i m e a f l u c t u a t i o n a b o u t t h e ensemble a v e r a g e .

Assuming v e r t i c a l v a r i a t i o n s of t h e v a r i o u s p a r a m e t e r s are

s m a l l , t h e e q u a t i o n s may b e v e r t i c a l l y a v e r a g e d w i t h o u t l o s s of meaning:

a

(Ix +

a ax

-

a

d a z % + ; - ( V q )

a

( P I % ) + - qx dx

=

-

+

a 3y

qy

=

91

With t h e f o l l o w i n g d e f i n i t i o n s :

qx

=

pgH

an

ax + f q

a an + - ( V % ) = - p g H r - f q ay Y

dx

a

3)

a y (U (US ) + -

u d z = pUH

l:h

And t h e c o n s t i t u t i v e r e l a t i o n s :

a a P + ax xx + ay yx

+ -F

a

x

+-F

ax

a

+--F XY

aY

YY

Tb

+

Ts

+Tb+TS Y Y

358

Where U and V a r e v e r t i c a l l y averaged h o r i z o n t a l v e l o c i t y components,

n

t h e surface

e l e v a t i o n above mean water l e v e l , f t h e C o r i o l i s parameter and Cf t h e bottom f r i c t i o n coefficient.

The product of v e l o c i t y f l u c t u a t i o n s (with r e s p e c t t o t h e v e r t i c a l

average v e l o c i t y ) a r e i n t e r n a l s t r e s s e s

T xx,

T

~

T~~ ~

which, , t o g e t h e r with h o r i z o n t a l

Reynolds s t r e s s , a r e v e r t i c a l l y averaged and then approximated by eddy v i s c o s i t y The s u r f a c e and bottom xx' &yyT €xy. r e s u l t from v e r t i c a l l y averaging t h e v e r t i c a l Reynolds shear

terms, with eddy v i s c o s i t y c o e f f i c i e n t s s t r e s s , ,cS and

T~

E

s t r e s s , and a r e approximated by q u a d r a t i c f u n c t i o n s of wind and c u r r e n t v e l o c i t y respectively. Boundary c o n d i t i o n s a r e t r e a t e d by s p e c i f y i n g one component of flow o r t h e surface elevation.

Flow normal t o t h e boundary i s s p e c i f i e d a s zero f o r land

boundaries, and s e t equal t o t h e flow r a t e a t r i v e r boundaries. boundaries a r e t r e a t e d by s p e c i f y i n g s u r f a c e l e v e l e l e v a t i o n .

Open o r ocean The equations,

in-

cluding t h e a p p r o p r i a t e boundary c o n d i t i o n s , a r e w r i t t e n a s v a r i a t i o n a l s t a t e m e n t s , which s e r v e a s t h e b a s i s f o r t h e f i n i t e element method. The f i n i t e element method approximates t h e s o l u t i o n of a boundary v a l u e problem with a f u n c t i o n of piece-wise continuous polynomials.

This involves d i s -

c r e t i z a t i o n of t h e continuum i n t o an e q u i v a l e n t system of f i n i t e elements.

Connor

and Wang s e l e c t e d t h e s i m p l e s t c o n f i g u r a t i o n , t r i a n g l e s with nodes a t t h e v e r t i c e s . The values of t h e v a r i a b l e s w i t h i n t h e element a r e assumed t o be a l i n e a r f u n c t i o n of t h e values a t t h e nodes.

The equations a r e transformed f o r a p p l i c a t i o n t o an

element using t h i s l i n e a r polynominal r e p r e s e n t a t i o n .

Treatment of t h e e n t i r e

continuum i s accomplished through summation of t h e c o n t r i b u t i o n s of each element. Each nodal v a l u e i n f l u e n c e s a l l of t h e elements c o n t a i n i n g t h a t node, and each element value i n f l u e n c e s t h e t h r e e nodes of t h e element.

Depth i s s e l e c t e d a t

each node p o i n t , while bottom f r i c t i o n and eddy v i s c o s i t y a r e s e l e c t e d f o r each element.

Solutions f o r q

,

4..

and

n

a r e obtained a t each node.

This model i s s i m i l a r t o t h e f i n i t e d i f f e r e n c e model developed by Leendertse. The p r i n c i p l e d i f f e r e n c e s a r e i n c l u s i o n of t h e eddy v i s c o s i t y t e r m s which Leendertse n e g l e c t s , and t h e s o l u t i o n technique.

Properly formulated, t h e method of s o l u t i o n

should have l i t t l e e f f e c t on t h e model r e s u l t s . disadvantages t o each method.

There a r e , however, advantages and

The f i n i t e d i f f e r e n c e method i s more e a s i l y under-

stood and a p p l i e d , and has w e l l developed s t a b i l i t y c r i t e r i a .

The p r i n c i p l e advantage

of t h e f i n i t e element method i s t h e f l e x i b i l i t y of t h e g r i d , making i t more appropri-

a t e f o r s i t u a t i o n s i n v o l v i n g complex geometry,

such a s t h e G r e a t Bay E s t u a r y system.

MODEL A P P L I C A T I O N The g o a l of t h e r e s e a r c h program i s t o model t h e e n t i r e e s t u a r y .

The p r e s e n t

form of CAFE does n o t t r e a t t i d a l f l a t s , t h e r e f o r e Great Bay has been excluded from t h e i n i t i a l model s t u d y .

A model of t h e remainder of t h e e s t u a r y would i n v o l v e a

g r i d composed of approximately 600 elements and 400 nodes, which would s t r a i n t h e l i m i t s of t h e computational s y s t e m a v a i l a b l e t o t h e a u t h o r s .

Therefore, t h e estuary

has been d i v i d e d i n t o s e c t i o n s , which a r e modeled s e p a r a t e l y .

The i n f l u e n c e of top-

ography on t h e hydrodynamics of t h e e s t u a r y make t h e segmentation p o s s i b l e , and determines t h e l o c a t i o n of t h e segment b o u n d a r i e s .

The model of t h e Portsmouth Har-

bor and lower P i s c a t a q u a River s e c t i o n i s p r e s e n t e d h e r e .

The open boundary i n

Portsmouth Harbor was l o c a t e d near t h e mouth of t h e e s t u a r y , where t h e hydrodynamic c h a r a c t e r i s t i c s of t h e e s t u a r y a r e s t i l l p r e s e n t .

The open boundary i n t h e P i s c a t a q u a

River was l o c a t e d below t h e a r e a where L i t t l e Bay j o i n s t h e P i s c a t a q u a River.

The

t i d a l choking which o c c u r s a t Dover P o i n t , and t h e r e l a t i v e i n s i g n i f i c a n c e of t h e upper P i s c a t a q u a R i v e r , a l l o w t h e e s t u a r y t o be d i v i d e d a t t h i s p o i n t . The g r i d was s e l e c t e d t o c o i n c i d e w i t h t h e c o a s t a l and b a t h y m e t r i c f e a t u r e s o f t h e e s t u a r y a s p r e s e n t e d by United S t a t e s Coast and Geodetic Survey c h a r t s N o .

and 2 1 2 .

S e v e r a l f a c t o r s were c o n s i d e r e d when developing t h e g r i d .

211

Features includ-

ed i n t h e g r i d a r e a t l e a s t a s l a r g e a s t h e d i s t a n c e between two nodes ( t h e c h a r a c t e r i s t i c length).

The s i z e and shape of t h e elements may v a r y , b u t t h e most s t a b l e

c o n f i g u r a t i o n i s e q u i l a t e r a l t r i a n g l e s of e q u a l s i z e .

To p r e s e r v e t h e g r i d s t a b i l i t y ,

t h e a r e a of a d j a c e n t elements v a r i e s less t h a n 20 p e r c e n t , and t h e element a n g l e s a r e g r e a t e r t h a n 30 and l e s s t h a n 90 d e g r e s s .

Another f a c t o r c o n s i d e r e d i n s e l e c t i n g

t h e g r i d i s t h a t an element may n o t have more t h a n two of i t s t h r e e nodes on a l a n d boundary.

The f i n a l c o n s i d e r a t i o n i s t h e number of nodes and elements t o be i n c l u d e d .

The l a r g e r t h e number of nodes, t h e g r e a t e r t h e d e t a i l , b u t computer c o r e and time requirements a r e a l s o i n c r e a s e d .

The maximum s i z e of the time s t e p i s r e l a t e d t o t h e

c h a r a c t e r i s t i c l e n g t h , and i n t h i s c a s e t h e s m a l l c h a r a c t e r i s t i c l e n g t h s c o n s t r a i n e d t h e s i z e of t h e time s t e p s .

A s a r e s u l t of t h e s e computational r e s t r i c t i o n s t h e g r i d

was s e l e c t e d w i t h a s l a r g e a c h a r a c t e r i s t i c l e n g t h a s p o s s i b l e .

The g r i d was modified

s e v e r a l times t o o b t a i n an a c c e p t a b l e r e s u l t . Boundary c o n d i t i o n s were s e t up a s s p e c i f i e d f o r t h e model.

Land boundary

c o n d i t i o n s a r e handled i n t e r n a l l y by t h e model, f o r c i n g t h e flow t o be t a n g e n t i a l t o t h e land.

To accomplish t h i s , normal a n g l e s

( w i t h r e s p e c t t o t h e l a n d boundary)

must be s p e c i f i e d f o r each node on a l a n d boundary.

The open boundaries a r e i n -

i t i a l l y t r e a t e d by s p e c i f y i n g t h e t i d a l amplitude, t i d a l frequency and phase l a g . The t i d a l frequency w a s assumed s t a n d a r d , and i s t h e same f o r a l l open b o u n d a r i e s . The t i d a l amplitude was s e l e c t e d f o r each open boundary, and t h e phase f o r each node on each open boundary.

The v a l u e s f o r t h e s e parameters were o b t a i n e d from

t h e Tide Tables of North America.

The bottom f r i c t i o n and eddy v i s c o s i t y

c o e f f i c i e n t s were assumed c o n s t a n t throughout t h e g r i d , and t h e v a l u e s f o r t h e i n i t i a l run were s e l e c t e d based on p r e v i o u s a p p l i c a t i o n s of t h e model. With t h e d a t a s e t up, t h e model was i n i t i a l l y r u n u s i n g a c o n s t a n t depth ( r e p r e s e n t a t i v e of t h e depth of t h e e s t u a r y ) a s i t i s s e n s i t i v e t o d e p t h v a r i a t i o n s . This allowed t h e s t a b i l i t y of t h e g r i d t o be t e s t e d and approximate v a l u e s f o r t h e bottom f r i c t i o n and eddy v i s c o s i t y c o e f f i c i e n t s t o be o b t a i n e d .

A f t e r t h e g r i d has

been e v a l u a t e d and modified, and approximate v a l u e s of t h e c o e f f i c i e n t s determined, t h e depth of t h e e s t u a r y a t each node was e n t e r e d .

The c h a r t d a t a w a s supplemented

f o r t h e P i s c a t a q u a River channel with d a t a from a U n i v e r s i t y of New Hampshire bathym e t r i c survey.

To p r e s e r v e t h e c r o s s s e c t i o n a l a r e a s and volume c h a r a c t e r i s t i c s of

t h e e s t u a r y segment, s l i g h t m o d i f i c a t i o n of t h e d e p t h s was n e c e s s a r y .

The model was

run a g a i n , with v a r y i n g d e p t h , t o t e s t t h e s t a b i l i t y of t h e t h r e e dimensional g r i d . Large changes i n depth w i t h i n an element, and v e r y shallow d e p t h s , cause numerical instabilities,

r e q u i r i n g s l i g h t m o d i f i c a t i o n of t h e d e p t h s .

To determine t h e e f f e c t s of t i d a l a m p l i t u d e , bottom f r i c t i o n c o e f f i c i e n t , and eddy v i s c o s i t y c o e f f i c i e n t on t h e model, a s e r i e s of model r u n s were e x e c u t e d , syst e m a t i c a l l y varying t h e s e parameters

(Reichard and C e l i k k o l 1 9 7 6 ) .

The r e s u l t s of

t h i s experiment i n d i c a t e s t h a t c u r r e n t speed depends on t i d a l amplitude and bottom f r i c t i o n c o e f f i c i e n t , w h i l e c u r r e n t phase i s a f f e c t e d by t i d a l amplitude and eddy viscosity coefficient.

S i n c e t i d a l amplitude i s determined by t h e p h y s i c a l conditions

p r e s e n t a t t h e boundary, c a l i b r a t i o n of t h e model w i t h r e s p e c t t o c u r r e n t speed i s accomplished by changing t h e bottom f r i c t i o n c o e f f i c i e n t , w h i l e c a l i b r a t i o n w i t h r e s p e c t t o phase i s c a r r i e d o u t through m o d i f i c a t i o n of t h e eddy v i s c o s i t y c o e f f i c i ent.

MODEL CALIBRATION

The model uses two parameters which m u s t be e v a l u a t e d f o r t h e e s t u a r y : bottom f r i c t i o n c o e f f i c i e n t and t h e eddy v i s c o s i t y c o e f f i c i e n t .

the

Bottom f r i c t i o n

p l a y s a major r o l e i n e s t u a r y dynamics. This f r i c t i o n a l e f f e c t i s expressed i n t h e 2 model i n t h e form CU /h, where C i s t h e f r i c t i o n c o e f f i c i e n t , U t h e v e r t i c a l l y averagcd v c l o c i t y , and h t h e w a t e r d e p t h .

The eddy v i s c o s i t y term i s a combination

of Reynolds S t r e s s terms, r e s u l t i n g from ensemble a v e r a g i n g , and i n t e r n a l s t r e s s terms, r e s u l t i n g from t h e v e r t i c a l a v e r a g i n g .

a 3~

itxx

au

G) where

current gradient.

Exx

I t i s expressed i n t h e form

i s t h e eddy v i s c o s i t y c o e f f i c i e n t and w a x i s t h e h o r i z o n t a l

Both of t h e c o e f f i c i e n t s a r e assumed c o n s t a n t i n t i m e , and con-

s t a n t f o r an element.

They may be s p e c i f i e d f o r each element i n d i v i d u a l l y , however,

allowing s p a t i a l changes i n t h e c o e f f i c i e n t s from element t o element.

The v a l u e s of

t h e s e parameters a r e s e l e c t e d , i n a p r o c e s s c a l l e d model c a l i b r a t i o n , t o cause model r e s u l t s t o compare f a v o r a b l y w i t h f i e l d d a t a .

A procedure f o r q u a n t i t a t i v e s e l e c t i o n

of t h e bottom f r i c t i o n c o e f f i c i e n t was developed t o a s s i s t i n t h e c a l i b r a t i o n process.

361 The initial step in the development of such a procedure is a review of the physical processes involved. For a one-dimensional, constant depth channel, neglecting the eddy viscosity term, the conservation of momentum equation can be written:

au au 317 cu - + U z = - g z + h at

2

Where U is the vertically averaged current velocity, 17 the surface elevation, h the water depth, and C the friction coefficient.

Evaluating the order of magnitude of

each term in the equation for the Great Bay Estuary system:

au + u a au, = -g a n at

cu 2

+

h

The acceleration terms are small compared to the surface slope and friction terms, and can be neglected.

Therefore, the equation can be written as a balance between

surface slope and bottom friction:

The current velocity U is related to the bottom friction coefficient: U = K/C1’2

K =

an

(gh -)ax

112

This expression is used to select new bottom friction coefficients based o n model results and field data.

The equation is first solved for the constant K using

the current velocity predicted by the model for the specified bottom friction coefficient.

The equation is then solved for a new bottom friction coefficient using

the calculated constant K , and measured current velocity.

This procedure was found

to work quite well, producing acceptable results with only one or two interations.

COMPARISON OF MODEL RESULTS WITH FIELD DATA

The model predicts currents and tidal elevation, which have been measured in the estuary (Brown et a1 1977).

To compare these two types of data, the physical

interpretation of the prediction and measurement must be clearly understood.

The

model predictions are arrived at through a numerical solution of the appropriate differential equations.

The numerical methods used in the solution impose certain

constraints or limitations on the results. The temporal aspect of the model is obtained by solving the equations at a discrete time, and repeating the solutions at fixed intervals in time.

This intro-

duces a minimum temporal scale of resolution approximately the size of the time interval between solutions.

In addition, the equations have been ensemble averaged,

eliminating stochastic variations from the result.

The spatial variation of a

parameter in an element is a linear function of the values of the parameter at the three nodes.

This imposes a minimum spatial scale of resolution approximately the

size of an element.

It also means that the parameter predicted by the model repre-

sents a spatial average over an area approximately the size of an element.

The cumulative e f f e c t of t h e s e r e s t r i c t i o n s on t i d a l e l e v a t i o n p r e d i c t i o n s i s t h a t a p r e d i c t e d e l e v a t i o n r e p r e s e n t s a s p a t i a l average over an a r e a approximately t h e s i z e of an element and a temporal average over a time approximately t h e s i z e of t h e time s t e p , of t h e d e t e r m i n i s t i c

(ensemble averaged) v a l u e of t i d a l e l e v a t i o n .

The d i f f e r e n t i a l equations have been v e r t i c a l l y averaged, so c u r r e n t p r e d i c t i o n s r e p r e s e n t t h e average c u r r e n t i n t h e water column.

Thus, a p r e d i c t e d c u r r e n t repre-

s e n t s a s p a t i a l average over a volume of water approximately t h e s i z e of an element, and a temporal average over a time approximately t h e s i z e of t h e t i m e s t e p , of t h e d e t e r m i n i s t i c value of t h e c u r r e n t . The s t o c h a s t i c v a r i a t i o n s i n water s u r f a c e e l e v a t i o n a r e of l e n g t h and time s c a l e s small enough t h a t they can be e f f e c t i v e l y removed from t h e measurement by mounting t h e t i d e gauge i n a p r o p e r l y designed w e l l .

The s u r f a c e e l e v a t i o n i s mea-

sured a t one p o i n t , while t h e model p r e d i c t i o n s r e p r e s e n t an average over a l a r g e area.

However, t h e d e t e r m i n i s t i c v a r i a t i o n s i n s u r f a c e e l e v a t i o n a r e w e l l behaved

s p a t i a l l y and temporally, and can be a c c u r a t e l y r e p r e s e n t e d by simple mathematical expressions such a s t h o s e used i n t h e model.

A s a r e s u l t , model p r e d i c t i o n s can be

compared d i r e c t l y with t i d e d a t a , and g e n e r a l l y compare q u i t e w e l l . Water i n t h e e s t u a r y e x h i b i t s a v a r i e t y of motion c h a r a c t e r i z e d by s c a l e s ranging from molecular t o some l a r g e r than t h e e s t u a r y .

Any s i n g l e method of mea-

surement can d e a l with only a p o r t i o n of t h e wide spectrum of motion, with t h e remainder appearing a s n o i s e .

An E u l e r i a n measurement with a l e n g t h s c a l e of t h e

o r d e r of 1 0 0 m and a time s c a l e on t h e o r d e r of a minute would be i d e a l l y s u i t e d f o r model comparison. The c u r r e n t d a t a presented i n t h i s paper was obtained from conventional mechanical c u r r e n t meters, which can r e s o l v e l e n g t h and time s c a l e s of motion on t h e o r d e r of a meter and s e v e r a l seconds r e s p e c t i v e l y .

This s c a l e i s two o r d e r s

o f magnitude smaller than t h e r e s o l u t i o n of t h e model, t h e r e f o r e v e l o c i t y f l u c t u a t i o n s and small flow s t r u c t u r e s which appear i n t h e measurement a r e not p r e d i c t e d by t h e model.

A s a result,

there is greater variation present i n the current data

t h a a i n model p r e d i c t i o n s , and i n some c a s e s t h e d i f f e r e n c e s a r e q u i t e l a r g e .

In

a d d i t i o n , t h e model p r e d i c t s t h e v e r t i c a l average v e l o c i t y , which i s g e n e r a l l y l e s s than t h e v e l o c i t y near mid-depth where t h e c u r r e n t d a t a was c o l l e c t e d . These problems can be addressed by making measurements a t s e v e r a l d i f f e r e n t p o i n t s i n an a r e a and processing t h e d a t a a p p r o p r i a t e l y . s e v e r a l c r o s s - s e c t i o n s of t h e e s t u a r y .

t i d a l c y c l e , b u t were not done c o n c u r r e n t l y . and i s not y e t a v a i l a b l e .

This has been done f o r

These t r a n s e c t s were each conducted f o r one This d a t a i s p r e s e n t l y being processed,

We a r e a l s o p r e s e n t l y attempting t o measure t h e flow

through a c r o s s - s e c t i o n of t h e e s t u a r y w i t h a geomagnetic measurement.

The measure-

ment i s marginal and t o d a t e no s i g n i f i c a n t r e s u l t s have been obtained. The model has been run f o r two d i s t i n c t t i d a l c y c l e s , t h e f i r s t o c c u r r i n g

on September 1 9 , 1977, and t h e second on September 2 1 , 1977.

The open boundary

363 conditions were specified directly from tide data collected on these two days, thus eliminating errors .resulting from approximating the boundary condition, and including meterological as well as astronomical effects.

This was accomplished by inter-

polating from hourly tide height data collected at the boundary for the appropriate day.

The model for September 19 was used for calibration, while the model for

September 21 was run with the coefficients developed in the calibration process, changing only the open boundary conditions, for model validation. A surface level isoplot (Figure 6) and current vector plot (Figure 7) at per-

iods of high current velocity are presented as an overall, qualitative view of the model results.

The portions of the estuary having the smallest cross-sectional area

have the largest surface gradients and current velocities as expected.

In addition,

the current vector plots indicate the presence of topographically induced eddies in the flow regime.

These eddies are at the lower limit of the model's spatial resolu-

tion, so the predictions are not quantitatively accurate.

The existence of these

eddies in the estuary has been confirmed by observations. Tidal elevation has been compared with model predictions at the Simplex tide station, and is presented in Fig. 8a and 8b.

Predicted tidal elevation is presented

as a solid line while hourly tide height data is represented by circles.

The agree-

ment is quite good with the difference between predicted and measured values generally being less than the accuracy of the measurement. Current velocities are compared at three locations, stations C-104, C-119, and C-121 in Figs. 9 through 11 respectively, as speed and direction plots.

Pre-

dicted currents are presented as solid lines, while measured currents are represented as circles ( 3 m depth) and squares (6 m depth).

The speed predictions are gen-

erally less than measured values, as expected, since the predicted value is vertically average speed.

The current meter moorings were located to one side of the

channel to avoid the heavy ship traffic, resulting in asymmetries in the data due to irregularities in the shore.

Speed predictions for flood and ebb tide are nearly

symmetrical, and predicted directions align with the main channel, as the grid i s not fine enough to resolve these local topographical features.

An extreme case is

Station C-121, which is in the main flow on the ebb tide, but borders on a reverse flow (eddy) on the flood tide.

CONCLUSION

The surface elevations predicted by the model are as accurate as tidal measurements, and provide better spatial resolution.

Initial comparisons of predicted

and measured currents, within the limitations set forth previously, appear promising. However, the predicted currents should be compared with a data set having a similar scale of resolution.

Currents predicted by the model do not provide enough detail

to be useful for some of the problems in the estuary presently being addressed, but they are satisfactory for problems concerning large scale mass fluxes.

The model is

364

PORTSMOUTH

Figure 6.

Surface level isoplot (5 cm increments) as predicted by the model

one hour after mid-flood.

I M/S

.. PORTSMOUTH

Figure 7 . mid-flood.

Current vector plot as predicted by the model one hour after

365 20c

15C

IOC

-.-

-E

5c

tI

C

'3 W

I

w

D

-50

t-

-100

-150

-200

I

I

I

I

1

I

I

I

I

I

1

TIME ( H R S )

Figure 8a. Comparison of p r e d i c t e d t i d a l e l e v a t i o n ( s o l i d l i n e ) and hourly t i d e h e i g h t d a t a ( c i r c l e s ) a t Simplex on September 19 1975.

200

I5C

100

-E

-

50

0

I-

1

'3 w

0

I

-50 I-

-100

-150

-200

I

I

I

I

I

I

I

I

I

I

I

TIME ( H R S )

Figure 8b. Comparison of p r e d i c t e d t i d a l e l e v a t i o n ( s o l i d l i n e ) and hourly t i d e h e i g h t d a t a ( c i r c l e s ) a t Simplex on September 2 1 1975.

366

t

50 0

0

I

I

I

I

I

I

I

I

I

I

I

2

3

4

5

6

7

8

9

10

I

II

TIME ( H R S )

20

)

01 0

I

I

I

7

4

TIME ( H R S I

Figure 9a.

Comparison of predicted speed and direction (solid lines) with

current data (circles and squares) at station C - 104 for September 19, 1977

12

367

400

3 50

300

2 50

z 0 200

I

‘L

n

0 0

150

100

50 0 0

I

I

I

I

13

14

15

16

I

I

I

I

I

I

I

17

18

19

20

21

22

23

4

TIME ( H R S )

160

140

-

- - loo120

u) \

5

0

> k

0

5 w

80

5

60

-

0

> n w

-

0 O

0

O

n

0

3

40-

1

12

0

20

21

0 0

20 f

O L

0

I

I3

I

I

I

14

15

16

17

18

19

22

23

TIME (HRS)

F i g u r e 9b.

Comparison of p r e d i c t e d speed and d i r e c t i o n ( s o l i d l i n e s ) w i t h

c u r r e n t d a t a ( c i r c l e s and s q u a r e s ) a t s t a t i o n C - 1 0 4 f o r September 2 1 , 1 9 7 7 .

24 24

368

40C

35c

300

I

I

250

z 0

t- 200 W V

LL

n I50

100

0

0

2

3

50

0 0

I

4

5

6

7

8

I

I

7

8

9

9

10

II

10

II

TIME ( H R S )

I60

140

0

120

0

20

0 0

I

0

I

2

I

I

I

I

3

4

5

6

I

TIME ( H R S )

Figure 10a. Comparison of predicted speed and direction (solid lines) with current data (circles and squares) at station C - 119 for September 19, 1977.

12

369 400

p -

350

I

300

250

z

0 I-

g

200

un I50

0

~

0

0 I00

0

0

0

0

I

I

o

o

u

I

I

I

50

0

I

I

I

I

I

I

TIME ( H R S )

I60

140

0 0

0 0

0

0 0

0

0

17

18

0 0

20

0 I

I3

14

15

16

19

20

21

22

23

TIME ( H R S )

Figure l o b .

Comparison of p r e d i c t e d speed and d i r e c t i o n ( s o l i d l i n e s ) w i t h

c u r r e n t d a t a ( c i r c l e s and s q u a r e s ) a t s t a t i o n C

- 119 f o r September 21, 1977.

4

370 40C

0 0

35c

300

250

0

z

2 g

200

n “r I50

0

0

0

0

0

0

100

50

0

0 0

I

2

3

4

5

6

7

8

9

10

II

12

TIME ( H R S )

I60

I40

-c

I20

2

100

I

>

t;

0

80

W

> 60

W

a

a

3 0

40

0

20

0 0

I

I

I

I

I

I

2

3

4

5

6

I

I

7

8

0 9

I

I

10

II

TIME ( H R S )

Figure l l a .

Comparison of p r e d i c t e d speed and d i r e c t i o n ( s o l i d l i n e s ) with

current data (circles) a t station C

- 1 2 1 f o r September 1 9 , 1977.

12

371

0

0

12

I

I

I

13

14

15

I

t

16

17

1

I

I

I

I

I

18

19

20

21

22

23

24

TIME ( H R S )

I60

140

-2

-E

I20

100

>

k

sw 0

80

>

5

60

w

a a

3 0

0 40

20

13

14

15

16

17

18

19

20

21

22

23

TIME ( H R S )

Fiqure I l b .

Comparison of predicted speed and direction (solid lines) with

current data (circles) at station C

-

121 for September 21, 1977.

4

presently being used as an aid to understanding the hydrodynamics of the estuary, and to provide a data base for the advective portion of a dispersion model, for which it appears ideally suited.

ACKNOWLEDGEMENTS We are indebted to the National Ocean Survey and the Public Service Company of New Hampshire for their assistance in collecting field data.

This work is a result

of research sponsored by NOAA Office of Sea Grant, Department of Commerce, under Grant No. 04-6-158-44056.

REFERENCES Brown, W., Swenson, E., Trask, R., 1977, The Great Bay Estuarine Field Program 1975 Data Report Part I Currents and Sea Level, UNH Sea Grant Technical Report UNHSG-157. Connor, J.J., Wang, J.D., 1973, Mathematical Models of the Massachusetts Bay Part I, MIT Sea Grant Technical Report, MITSG 74-4. Connor, J.J., Wang, J.D., 1974, Finite Element Model of Two-Layer Coastal Circulation, 14th Coastal Engineering Conference, Copenhagen. Hansen, W., 1956, Theorie zur Errechnung des Wasserstandes und der Stromungen in Randmeeren Nebst Anwendungen, Tellus, 8-3. Iiess, K.W., White, F.M., 1974, A Numerical Tidal Model of Narrangansett Bay, Univ. of %ode Island, Technical Report MTR No. 20. Leendertse, J.J., 1967, Aspects of a Computational Model for Long-Period Water Wave Propagation, Rand Corp. RM-5294-PR. Leendertse, J.J., 1972, A Water Quality Simulation Model for Well-Mixed Estuaries and Coastal Seas: Vol. IV, Jamaica Bay Tidal Flows, Rand Corp. R-1009 NYC. Leendertse, Jan J., Richard C. Alexander, and Shiao-Kung Liu, 1973, A Three-Dimensiona1 Model for Estuaries and Coastal Seas: Volume I, Principles of Computation, Rand Corp. R-1417-OWRR. Reichard, R.P., Celikkol, B., 1976, Hydrodynamic Model of the Great Bay Estuarine System, UNH Sea Grant Technical Report UNH-SG-153.

373

S P R E A D I N G AND M I X I N G O F THE HUDSON R I V E R EFFLUENT I N T O THE NEW YORK BIGHT

MALCOLM J . BOWMAN

Marine S c i e n c e s R e s e a r c h 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, S t o n y Brook, N e w York 1 1 7 9 4

U.S.A.

ABSTRACT R e s u l t s a r e p r e s e n t e d f r o m t h r e e Hudson R i v e r plume s a m p l i n g c r u i s e s made i n t h e N e w York B i g h t , i n A u g u s t 1 9 7 6 . The d a t a show t h a t t h e s e t and s h a p e o f t h e s p r e a d i n g e f f l u e n t v a r y w i d e l y o v e r t i m e p e r i o d s c 6 d a y s , and a r e c l e a r l y i n f l u e n c e d by l o c a l wind s t r e s s . A p p l i c a t i o n o f T a k a n o ' s model o f a s t e a d y s t a t e plume s p r e a d i n g 8

i n t o a s t a g n a n t o c e a n s u g g e s t s a h o r i z o n t a l e d d y v i s c o s i t y -10

c m 2 sec

-1

,

and a s t r o n g a n t i c y c l o n i c d e f l e c t i o n o f t h e plume.

This

v a l u e i s c o n s i d e r e d t o b e an o v e r e s t i m a t e , s i n c e i n t e r f a c i a l s h e a r

s t r e s s i s n e g l e c t e d i n t h e model. More c a r e f u l m e a s u r e m e n t s a n d c a l c u l a t i o n s a r e n e e d e d t o s e p a r a t e o u t t h e e f f e c t s o f h o r i z o n t a l and v e r t i c a l v i s c o s i t i e s , C o r i o l i s f o r c e , a d v e c t i o n by a p r e v a i l i n g c o a s t a l c u r r e n t and l o c a l wind

s t r e s s , on plume d y n a m i c s . INTRODUCTION

The Hudson R i v e r , a m a j o r s o u r c e o f f r e s h w a t e r t o t h e c o a s t a l o c e a n of t h e n o r t h e a s t e r n U n i t e d S t a t e s , h a s b e e n a n i m p o r t a n t n a v i g a b l e waterway f o r s e v e r a l c e n t u r i e s and h a s p l a y e d an e s s e n t i a l r o l e i n t h e commercial d e v e l o p m e n t o f N e w York C i t y and S t a t e . I n i t s l o w e r r e a c h e s , t h e R i v e r h a s t h e c h a r a c t e r i s t i c s of a c l a s s i c a l p a r t i a l l y mixed drowned r i v e r v a l l e y e s t u a r y .

The e f f l u e n t

from t h e Hudson e s t u a r y f l o w s i n t o t h a t s e c t i o n o f t h e c o a s t a l o c e a n known a s t h e N e w York B i g h t Apex, w h e r e i t s p r e a d s a s a buoya n t plume whose a r e a l e x t e n t , i n p e r i o d s of h i g h r u n o f f , may e x c e e d 2 - 5 0 0 km

.

The p u r p o s e of t h i s p a p e r i s t o p r e s e n t an a n a l y s i s o f d a t a obt a i n e d f r o m t h r e e plume s a m p l i n g c r u i s e s made i n t h e Apex on August 1 3 , 1 6 , and 1 9 , 1 9 7 6 , and t o compare t h i s w i t h t h e p r e d i c t i o n s

374

of T a k a n o ' s

(1954a,b, 1955) s t e a d y s t a t e a n a l y t i c a l t h e o r y of a

b u o y a n t plume s p r e a d i n g i n t o a r o t a t i n g s t a g n a n t o c e a n . PHYSICAL CHARACTERISTICS OF THE LOWER R I V E R The l o n g t e r m mean r u n o f f o f t h e Hudson R i v e r i s -550

m3 sec-I

( G i e s e and B a r r , 1 9 6 7 ) , b u t i t s f l o w i s s u b j e c t t o l a r g e i r r e g u l a r i 2

ties.

The w a t e r s h e d c o v e r s

- 3 . 5 ~ 1 0 km ~

,

and a b o u t one h a l f of

t h e a n n u a l r u n o f f f o r a n o r m a l y e a r o c c u r s d u r i n g March, A p r i l and May, when t h e m o n t h l y mean may e x c e e d t h e l o n g t e r m a v e r a g e by a f a c t o r o f 4 o r more. S a l t i n t r u s i o n up t h e R i v e r may e x c e e d - 1 3 0

km u n d e r low r u n o f f

c o n d i t i o n s , b u t d u r i n g mean d i s c h a r g e , p e n e t r a t e s -70

km u p s t r e a m .

F u r t h e r i n f o r m a t i o n on t h e Hudson R i v e r a n d e s t u a r y may b e f o u n d i n G i e s e and B a r r

( 1 9 6 7 ) , J a y and Bowman ( 1 9 7 5 ) , a n d Bowman and

Wunderlich ( 1 9 7 7 ) . Ketchum e t a 1 ( 1 9 5 1 ) made t h e f i r s t e x t e n s i v e s u r v e y o f t h e Hudson e f f l u e n t d u r i n g s i x c r u i s e s b e t w e e n F e b r u a r y , l 9 4 8 and J a n u a r y , 1950. F u r t h e r s t u d i e s h a v e b e e n p u b l i s h e d by P e a r c e ( 1 9 7 0 ) , N a t i o n a l Marine F i s h e r i e s S e r v i c e ( 1 9 7 2 )

,

Bowman and W u n d e r l i c h

A l a r g e c o l l e c t i o n o f p a p e r s on t h e

(1976).

C h a r n e l 1 and Hansen

( 1 9 7 4 ) and

o c e a n o g r a p h y and w a t e r q u a l i t y ( r a t h e r t h e l a c k o f i t ) o f t h e Apex

i s Gross ( 1 9 7 6 ) . The t o t a l d i s c h a r g e of t h e Hudson d u r i n g a mean y e a r

(-1.7~10

10

3

m ) i s c o m p a r a b l e t o t h e volume of w a t e r c o n t a i n e d i n t h e r e g i o n

surveyed d u r i n g t h e t h r e e 1 9 7 6 c r u i s e s (see F i g . volume of f r e s h w a t e r i n t h e plume

( - 2 . 5 ~ 1 0 ~m3,

1 ) . However, t h e e q u i v a l e n t t o -8

d a y s ' d i s c h a r g e ) d u r i n g t h e s u r v e y r e p r e s e n t e d o n l y -1.5% of t h e t o t a l volume o f s e a w a t e r i n t h e Apex. The Apex w a t e r column i s r e l a t i v e l y u n s t r a t i f i e d i n w i n t e r , b u t s u s t a i n s a strong, almost l i n e a r s t r a t i f i c a t i o n during quiescent summer c o n d i t i o n s d u e t o t h e p r e s e n c e o f a h a l o c l i n e , t h e summer t h e r m o c l i n e and weak s u r f a c e wind and b o t t o m t i d a l s t i r r i n g . Ketchum c a l c u l a t e d t h e r e s i d e n c e t i m e o f w a t e r i n t h e plume t o b e -6

t o 10 days, i n s p i t e of ninefold v a r i a t i o n s i n r i v e r discharge,

and found a r a p i d r e e s t a b l i s h m e n t o f t h e b a s i c c i r c u l a t i o n p a t t e r n a f t e r o n l y 2 d a y s f o l l o w i n g a m a j o r w i n t e r storm d i s r u p t i o n . Under c o n d i t i o n s o f h i g h d i s c h a r g e , t h e e f f l u e n t o f t e n f l o w s southward a s a s l o p e c u r r e n t a l o n g t h e N e w J e r s e y s h o r e , w h i l e under low summer d i s c h a r g e , a much more v a r i a b l e p a t t e r n c a n e x i s t w i t h a weak plume o f t e n s p r e a d i n g o v e r most of t h e Apex.

375

F i g . 1. I s o m e t r i c p e r s p e c t i v e of d e n s i t y ( O T ) o f t h e Hudson e f f l u e n t and t h e N e w York B i g h t Apex. DETAILS OF THE CRUISES T h r e e c r u i s e s (Aug. 1 3 , 1 6 , 1 9 , 1 9 7 6 ) w e r e e x e c u t e d t o map c o n t i n u o u s s u r f a c e t e m p e r a t u r e , s a l i n i t y and c h l o r o p h y l l a , a l o n g c r u i s e t r a c k s c h o s e n i n r e a l t i m e t o d e l i n e a t e t h e plume;

i n addi-

t i o n / during c r u i s e 1 only, p r o f i l e s w e r e taken a t s e l e c t e d s t a t i o n s t o d e t e r m i n e t h e v e r t i c a l s t r u c t u r e o f t h e above v a r i a b l e s p l u s i n o r g a n i c n u t r i e n t s and s u s p e n d e d p a r t i c u l a t e m a t t e r a s w e l l .

Only

t h e p h y s i c a l a s p e c t s a r e d i s c u s s e d i n t h i s p a p e r ; c h e m i c a l and b i o l o g i c a l p r o p e r t i e s w i l l be p u b l i s h e d e l s e w h e r e . T e m p e r a t u r e and s a l i n i t y a t 1 m c o n t i n u o u s l y w e r e m e a s u r e d w i t h a P l e s s e y 6600T Thermosalinograph.

W a t e r column s a m p l e s were g a t h e r e d

w i t h an i n s i t u pumping s y s t e m ( H u l s e , 1 9 7 5 ) a l s o c o n n e c t e d t o t h e Thermosalinograph. Each c r u i s e l a s t e d a b o u t 12-18 h o u r s .

Some e r r o r s i n c o n t o u r i n g

due t o t i d a l a d v e c t i o n a r e u n a v o i d a b l e ; however, away f r o m t h e mouth of t h e e s t u a r y , t h e t i d a l e x c u r s i o n i s s m a l l r e l a t i v e t o t h e s c a l e of t h e plume.

C o n t o u r s n e a r t h e mouth w e r e a d j u s t e d t o r e f l e c t

c o n d i t i o n s a t l o c a l s l a c k a f t e r e s t u a r i n e ebb. An i s o m e t r i c p e r s p e c t i v e of d e n s i t y i s shown i n F i g . 1, viewed

376

from s p a c e t o t h e n o r t h w e s t ( o t h e r d i a g r a m s i l l u s t r a t i n g h y d r o g r a p h i c , c h e m i c a l and b i o l o g i c a l p r o p e r t i e s a r e f o u n d i n Bowman and I v e r s o n , 1977). The r i v e r d i s c h a r g e d u r i n g t h e p e r i o d was s t e a d y a t - 3 8 0 The plume s p r e a d s as a v e r y t h i n

(-5

m3 sec-l.

m ) l e n s o v e r t h e c o a s t a l re-

.

c e i v i n g w a t e r s , e x h i b i t i n g complex i n t e r d i g i t a t i o n s ( F i g . 2 ) The -2 p r e v a i l i n g wind ( s t r e s s - 2 dyne c m ) d u r i n g t h e p r e c e d i n g 3 d a y s was s o u t h w e s t e r l y ; wind s t r e s s c l e a r l y i n f l u e n c e d t h e s e t o f t h e plume and i t d r i f t e d t o t h e n o r t h and t h e e a s t . S t r o n g f r o n t s formed a l o n g t h e 290/00

i s o h a l i n e ( ~ ~ - 2 0 on ) Hudson

e s t u a r y e b b t i d e p a r t i c u l a r l y n e a r t h e b a s e s o f t h e plume l o b e s , s i n c e t h e s u r f a c e s l o p e s o f t h e e f f l u e n t w e r e presumably g r e a t e s t there. S u r f a c e c o n v e r g e n c e c u r r e n t s i n s i d e t h e plume a r e v e r y e f f e c t i v e i n s w e e p i n g f l o t s a m i n t o t h e s e f r o n t s , and v a s t amounts o f . f l o a t i n g g a r b a g e c a n b e t r a c e d a t t i m e s f o r 30 km o r m o r e as f i l a m e n t s s t r e t c h e d along t h e f r o n t .

Such d e b r i s i n c l u d e s

o i l , g r e a s e and

t a r b a l l s , b o t t l e s , p l a s t i c d e v i c e s and c o n t a i n e r s o f a l l d e s c r i p t i o n s , p a p e r , c o n t r a c e p t i v e s , h o s p i t a l r e f u s e , abandoned b o a t s , e n t r a i l s , timber, f r u i t , t o y s , e v e n c a d a v e r s . F i g u r e s 3 and 4 i l l u s t r a t e t h e s u r f a c e s a l i n i t y c h a r a c t e r i s t i c s o f t h e plume on A u g u s t 1 6 t h and 1 9 t h ( c r u i s e s 2 and 3 ) .

Character-

i s t i c s measured d u r i n g c r u i s e 2 r e p r e s e n t e d a t r a n s i t i o n s t a t e :

the

winds d u r i n g t h i s p e r i o d w e r e r o t a t i n g f r o m s o u t h e r l y t h r o u g h w e s t e r l y t o northwesterly,

set.

t h e New J e r s e y Coast. westerly -2

and t h e plume was d r i f t i n g b a c k t o a s o u t h e r l y

By c r u i s e 3 , t h e plume h a d r e v e r t e d t o a l o n g r i b b o n a l o n g

dyne c m

The w i n d s d u r i n g t h i s p e r i o d were n o r t h -

(wind s t r e s s d u r i n g c r u i s e s 2 and 3 r e m a i n e d s t e a d y a t -2 )

.

A v e r t i c a l c r o s s s e c t i o n of t h e plume i s shown i n F i g .

front a t s t a t i o n 4 represented its outer l i m i t .

5.

A sharp

T h i s f r o n t formed

j u s t b e f o r e t h e s e c t i o n was t a k e n and c o i n c i d e d w i t h t h e b e g i n n i n g of e b b t i d e i n t h e mouth o f t h e e s t u a r y .

The i s o h a l i n e s (S=25-29°/oo)

of F i g . 4 c o l l a p s e d i n t o s i n g u l a r f r o n t which w a s s u b s e q u e n t l y followed a f t e r t a k i n g t h e t r a n s e c t , unbroken, northwards i n t o t h e mouth o f t h e e s t u a r y . Discontinuities i n p r o p e r t i e s w e r e dramatic. yellow-brown

Plume w a t e r was

i n c o l o r , w i t h an o i l y smooth a p p e a r a n c e .

water o u t s i d e t h e f r o n t w a s coastal blue-green,

Oceanic

w i t h -1 m s u r f a c e

g r a v i t y waves a p p r o a c h i n g t h e f r o n t , p i l i n g up, and b r e a k i n g . numerous f i l a m e n t s o f f l o t s a m d e l i n e a t e d t h e f r o n t .

Again

It is interesting

%.

hm

10

F i g . 2 . S u r f a c e (1 m) s a l i n i t y o f Hudson e f f l u e n t d u r i n g c r u i s e 1 (Aug. 1 3 , 1 9 7 6 ) .

F i g . 3. S u r f a c e (1 m) s a l i n i t y of Hudson e f f l u e n t d u r i n g c r u i s e 2 (Aug. 1 6 , 1 9 7 6 ) .

F i g . 4 . S u r f a c e (1 m) s a l i n i t y of Hudson e f f l u e n t d u r i n g c r u i s e 3 (Aug. 1 9 , 19761..

378 S A L I NI TY

%o

F i g . 5 . V e r t i c a l s a l i n i t y c r o s s s e c t i o n a c r o s s plume d u r i n g c r u i s e 3 . S t a t i o n p o s i t i o n s a r e shown i n F i g . 4 . t o n o t e t h a t t h e a r e a l e x t e n t o f t h e plume d u r i n g t h e t i m e p e r i o d 2

of t h e s u r v e y s r e m a i n e d c o n s t a n t ( 1 9 0 km ) w i t h i n e x p e r i m e n t a l e r r o r .

FRESH WATER CONTENT O F THE PLUME

I n o r d e r t o compare t h e dynamics o f t h e plume w i t h t h e s t e a d y s t a t e v e r t i c a l l y i n t e g r a t e d model o f Takano ( 1 9 5 4 a , b , 1 9 5 5 ) , s a l i n i t y d a t a from c r u i s e 1 w e r e u s e d t o s e p a r a t e o u t t h e f r e s h w a t e r cont e n t of t h e e f f l u e n t . S(z)

The plume

c o n s i s t s of water of s a l i n i t y

S o , where S o i s some b a c k g r o u n d o c e a n i c s a l i n i t y , and z i s

t h e v e r t i c a l o r d i n a t e ( p o s i t i v e downwards)

.

The e f f l u e n t c a n t h e n b e s e p a r a t e d i n t o a f r e s h w a t e r component o f d e p t h h , o v e r l y i n g a s a l t w a t e r component of s a l i n i t y S o , o f thickness zl.

Thus,

h

=

zo - z 1

i s t h e d e p t h a t which S ( z ) + S o l i . e . , t h e u n d e r s u r f a c e o f 0 t h e plume, and p ( z ) i s d e n s i t y . The r e s u l t s are i l l u s t r a t e d i n

where z

F i g . 6 , where t h e x a x i s i s t a k e n p e r p e n d i c u l a r t o t h e l i n e across t h e mouth of t h e e s t u a r y

(y a x i s ) .

DYNAMICS The c o o r d i n a t e s y s t e m i s i l l u s t r a t e d i n F i g .

7.

The l i n e a r i z e d ,

s t e a d y s t a t e e q u a t i o n s of m o t i o n and c o n t i n u i t y a r e :

379

€QU/ VAL ENT HEIGH T

Fig.

km

6 . F r e s h w a t e r c o n t e n t of t h e plume d u r i n g c r u i s e 1.

MSL

d

Z

Fig.

I . C o o r d i n a t e s y s t e m f o r plume d y n a m i c a l c a l c u l a t i o n s .

380

O = - - 1- -aP+ f u + N v 2 v + - ( Na P aY h az

E) z

az

where p = p r e s s u r e f = C o r i o l i s p a r a m e t e r 210

-4

-1

sec

.

Nh,NZ

= horizontal

u,v

horizontal current velocities

=

and v e r t i c a l eddy v i s c o s i t i e s

The b o u n d a r y c o n d i t i o n s a r e :

aazu = aaz V = o ,

z

=

~

,

1

d

(no stress a t s u r f a c e , bottom) u = v = o

,

2 2 s

2

(no flow i n u n d e r l y i n g f l u i d ) . Takano d e f i n e s l o n g i t u d i n a l and t r a n s v e r s e m a s s t r a n s p o r t s p e r u n i t width as: -

Mx -

Sd 5,

pudz = a j / a y

~d

where J, i s t h e stream f u n c t i o n . The i n t e g r a t e d p r e s s u r e P i s :

[

d

P =

pdz. I

R i v e r f l o w i s c o n s t a n t and u n i f o r m a c r o s s t h e mouth of w i d t h 2 R : < y

M x --M o l

x = O ,

-1

M x = O ,

x = O ,

- R > y > R

x = o ,

jty

M

Y

= o ,

The s o l u t i o n f o r

Note

J,

IJJ

is:

i s independent of f .

The s o l u t i o n f o r P is:

< R

381

(Coriolis contribution)

+

Y+y. x2+ (y+a.)

2Nh

-

Y-Q

x2+ (y-a )

2

2

J j

(eddy v i s c o s i t y c o n t r i b u t i o n ) The t h i c k n e s s o f t h e ( f r e s h w a t e r ) p l u m e , h , i s g i v e n by:

A p = density contrast

across plume e . 0 2 ,

and

g = a c c e l e r a t i o n due t o g r a v i t y . The t h i c k n e s s h ( x , o ) a l o n g t h e x a x i s i s :

The a n t i c y c l o n i c d e f l e c t i o n o f t h e e f f l u e n t a s it s p r e a d s a t sea i s a f u n c t i o n of t h e h o r i z o n t a l Ekman number E:

E = Nh/fR2. COMPARISON W I T H DATA FROM CRUISE 1

Fig.

8 i s a p l o t of h ( x , o ) f r o m t h e above e q u a t i o n and e x p e r i m e n t a l

d a t a p o i n t s t a k e n from F i g .

6.

The c u r v e , v i s u a l l y f i t t e d by ad-

and II, g a v e v a l u e s o f

j u s t i n g Nh

Nh = 1x108 c m 2 sec

-1

,

and

Q = 1 . 9 5 km

T h i s l e a d s t o a v a l u e of E - 2 7 . F i g u r e 9 shows t h e e x p e c t e d d e f l e c t i o n o f t h e plume f o r v a r i o u s v a l u e s of E ,

i n c l u d i n g t h a t d e d u c e d from t h e d a t a .

DISCUSSION The d e d u c e d v a l u e o f Nh i s s o m e two o r d e r s of m a g n i t u d e l a r g e r t h a n v a l u e s o f t e n t a k e n f o r open o c e a n v i s c o s i t i e s .

This is a

d i r e c t c o n s e q u e n c e of t h e l a r g e s u r f a c e s l o p e s f o u n d w i t h i n t h e e f f l u e n t (much l a r g e r t h a n t h o s e f o u n d i n , f o r e x a m p l e , t h e Columbia R i v e r plume

(Barnes e t a l , 1 9 7 2 ) .

The z e r o s h e a r s t r e s s b o u n d a r y c o n d i t i o n s a t t h e s u r f a c e and plume i n t e r f a c e s a r e t a n t a m o u n t t o n e g l e c t i n g v e r t i c a l eddy v i s c o s i t y , and h e n c e a l l f r i c t i o n a l r e t a r d a t i o n i s a s c r i b e d t o l a t e r a l s h e a r

stresses.

Thus t h e computed v a l u e of N h i s o v e r e s t i m a t e d , and u n d e r

a more r e a l i s t i c b o u n d a r y c o n d i t i o n o f f i n i t e i n t e r f a c i a l s t r e s s , would b e a c c o r d i n g l y r e d u c e d .

382

4.00 3.60 3.20

2.00

2.40

h (m)2.00 1.60

1.20 .00 .40 0

5

0

15

10

20

1

I

I

25

30

35

distance from r i v e r mouth, x

tkm) F i g . 8 . P l o t o f t h i c k n e s s o f f r e s h w a t e r c o n t e n t o f plume v e r s u s d i s t a n c e from mouth o b t a i n e d from t h e model. The e x p e r i m e n t a l p o i n t s a r e o b t a i n e d from F i g . 6 . A s seen i n Fig.

1 0 , t h e e s t u a r y s u p p o r t s a w e l l developed upstream

n o n - t i d a l flow a t d e p t h .

This counterflow w i l l increase, perhaps

q u a d r u p l e , t h e i n t e r f a c i a l stress b e t w e e n t h e t w o l a y e r s above t h a t f o r a s t a g n a n t l o w e r l a y e r a s assumed i n t h e model. B e a r d s l e y and Hartman ( 1 9 7 8 ) h a v e e x t e n d e d T a k a n o ' s work by cons i d e r i n g t h e e f f e c t s o f b o t t o m f r i c t i o n and t o p o g r a p h y on a s i n g l e l a y e r d i s c h a r g e , and i n a d d i t i o n , i n t e r f a c i a l f r i c t i o n f o r a d o u b l e l a y e r e d system. However, t h e y go t o t h e o p p o s i t e e x t r e m e and assume t h a t l a t e r a l f r i c t i o n i s dynamically i r r e l e v a n t .

F o r t h e d o u b l e l a y e r model,

t h e s u r f a c e flow d e f l e c t s t o t h e l e f t , whereas t h e i n f l o w i n t o t h e e s t u a r y a t d e p t h a p p r o a c h e s f r o m t h e r i g h t , l o o k i n g o u t t o sea. This opposite d e f l e c t i o n f o r t h e surface l a y e r i s ascribed t o t h e d i r e c t i o n o f t h e i n t e r f a c i a l s h e a r , and t i l t i n g of t h e i n t e r f a c e . I t i s o b v i o u s t h a t f u r t h e r c a r e f u l m e a s u r e m e n t s and c a l c u l a t i o n s

a r e n e e d e d t o a d e q u a t e l y model t h e e f f e c t s o f l c t e r a l and v e r t i c a l v i s c o s i t i e s i n s u c h plumes.

383

Y/ I F i g . 9 . E x p e c t e d d e f l e c t i o n o f t h e plume a s a f u n c t i o n of t h e h o r i The s t r a i g h t d a s h e d l i n e s r e p r e s e n t t h e z o n t a l Ekman number E . e d g e s o f t h e plume f o r f = 0 . The h e a v y c u r v e d l i n e s r e p r e s e n t t h e e x p e c t e d e d g e s o f t h e plume f o r c o n d i t i o n s o b s e r v e d d u r i n g c r u i s e 1 ( n e g l e c t i n g wind s t r e s s and c o a s t a l c u r r e n t s . Adapted from O f f i c e r , 1 9 7 6 ) . a ) E = 5 0 0 ; b ) E = 2 5 0 ; c ) E = 1 2 5 ; d) E = 62.5; e) E = 31.2; f ) E = 15.6. The v a l u e o f 2 1 - 4 e s t u a r y mouth

(-9

km)

km i s a l s o much l e s s t h a n t h e w i d t h o f t h e

.

However, as c a n b e s e e n i n F i g . 1 0 , t h e

s e a w a r d s u r f a c e f l o w i s c o n s t r i c t e d t o a w i d t h of -7.5

km, and i f

t h e plume i s decomposed i n t o f r e s h and o c e a n i c w a t e r , t h e w i d t h r e d u c e s e v e n f u r t h e r , s o t h e computed v a l u e o f R seems r e a s o n a b l e . CONCLUSIONS The a p p l i c a t i o n o f T a k a n o ' s c l a s s i c a l model h a s p r o v i d e d i n s i g h t i n t o t h e dynamic b a l a n c e s o f a s p r e a d i n g e f f l u e n t a t s e a , b u t o b v i o u s l y f u r t h e r c a r e f u l s t u d i e s a r e needed t o s e p a r a t e o u t t h e v a r i o u s i n f l u e n c e s o f h o r i z o n t a l and v e r t i c a l v i s c o s i t i e s , C o r i o l i s e f f e c t , a d v e c t i o n by a p r e v a i l i n g s h e l f c u r r e n t , and l o c a l wind

stress.

S h a l l o w s h e l f w a t e r s i n N e w York B i g h t c a n b e r e v e r s e d

w i t h i n s i x h o u r s by a s u d d e n c h a n g e i n wind s t r e s s ( G . Hann,

384

STATION

POSITIONS

0 I

2 3 4

5 6

- 7 I

1 +

0

h 9 IC II

12 :I

14 I:

AMBROSE CHANNEL

I€

F i g . 1 0 . Averaged n o n - t i d a l v e l o c i t i e s a c r o s s t h e mouth o f t h e Hudson e s t u a r y , J u n e 2 - 7 , 1 9 5 2 ( f r o m Kao, 1 9 7 5 ) . p r i v a t e c o m m u n i c a t i o n ) ; t h e plume i t s e l f would b e e x p e c t e d t o r e s p o n d e v e n more q u i c k l y . And f i n a l l y , w e know l i t t l e a b o u t how t h e a c t u a l m i x i n g o f t h e plume i n t o t h e a m b i e n t s h e l f w a t e r o c c u r s .

I s i t p r i m a r i l y by

upward a n d / o r downward e n t r a i n m e n t b e t w e e n plume and u n d e r l y i n g o c e a n i c w a t e r , o r i s t h e m i x i n g p r i m a r i l y a c o n s e q u e n c e of h o r i z o n t a l exchange?

And what r o l e do t h e u b i q u i t o u s plume f r o n t s p l a y i n

e f f e c t i n g c r o s s plume t r a n s p o r t and m i x i n g ?

The a n s w e r t o t h e s e

q u e s t i o n s i s n o t o n l y i m p o r t a n t t o dynamical i n s i g h t b u t t o environmental concerns a s w e l l . ACKNOWLEDGEMENTS.

T h i s work was s u p p o r t e d by t h e M a r i n e E c o s y s t e m s A n a l y s i s (MESA) Program of NOAA,

and by t h e U n i v e r s i t y Awards Committee o f t h e

R e s e a r c h F o u n d a t i o n and t h e S t a t e U n i v e r s i t y o f N e w York. C o n t r i b u t i o n 2 0 5 of 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) of t h e S t a t e U n i v e r s i t y o f N e w York a t S t o n y Brook.

385 REFERENCES Duxbury, A . C .

Barnes, C.A.,

a n d Morse B . - A . ,

1972.

Circulation

and S e l e c t e d P r o p e r t i e s o f t h e Columbia River E f f l u e n t a t S e a . IN:

The C o l u m b i a

and A d j a c e n t

River Estuary

Bioenvironmental S t u d i e s .

Ocean Waters.

P r u t e r and D.L.

Eds. A.J.

Alversen

U n i v e r s i t y of Washington P r e s s , S e a t t l e , 1972. A s i m p l e t h e o r e t i c a l model

and H a r t , J . , 1978.

Beardsley, R.C.

f o r t h e flow o f an e s t u a r y o n t o a c o n t i n e n t a l s h e l f .

To a p p e a r

i n J . Geophys. R e s . Bowman, M . J .

and Wunderlich, L.D.,

S p e c i a l symposia, V o l . Bowman, M . J .

2.,

1 9 7 6 . Am. SOC. Lim. O c e a n o g r .

pp. 58-68.

and W u n d e r l i c h , L . D . ,

1977.

New York B i g h t A t l a s

monograph #1, New York S t a t e S e a G r a n t I n s t i t u t e , A l b a n y , N . Y . Bowman, M . In:

and I v e r s o n , R.L.,

1 9 7 7 . E s t u a r i n e a n d plume f r o n t s .

Oceanic f r o n t s i n coastal p r o c e s s e s :

w o r k s h o p , May 25-27, C h a r n e l l , R.

and Hansen, D . V . ,

L.

p r o c e e d i n g s o f MSRC

1 9 7 7 . MSRC, S t a t e U n i v . 1974.

of N.Y.,

S t o n y Brook.

Summary a n d a n a l y s i s o f

p h y s i c a l o c e a n o g r a p h y d a t a c o l l e c t e d i n t h e New York B i g h t Apex d u r i n g 1969-1970. Giese,

G.L.

and B a r r , J . W . ,

R e s o u r c e s Comm. Gross, M . G . ,

ed.,

Bull. Am.

74-3.

B o u l d e r , C0:NOAA.

NYS C o n s e r v . Dep. Water

1967.

61, Albany, N.Y.

1976.

N e w York B i g h t . VOl.

MESA Rep.

Middle A t l a n t i c C o n t i n e n t a l S h e l f and SOC. Lim. O c e a n o g r .

S p e c i a l symposia,

2.

H u l s e , G.L.,

1975.

i n g system.

The P l u n k e t :

T e c h . Rep.

a s h i p b o a r d water q u a l i t y monitor-

2 2 , Marine S c i . R e s . C e n t . ,

S t a t e Univ.

o f New York, S t o n y Brook. J a y , D.A.

a n d Bowman, M . J . ,

P h y s i c a l oceanography and w a t e r

1975.

q u a l i t y o f N e w York H a r b o r a n d w e s t e r n Long I s l a n d Sound. Rep.

23.

Marine S c i . R e s .

Cent.,

S t a t e Univ.

Tech.

o f New York,

S t o n y Brook. Kao, A . F . ,

1975.

Hook-Rockaway

A s t u d y o f t h e c u r r e n t s t r u c t u r e i n the Sandy

Point transect.

Brook: M a r i n e S c i . R e s . Ketchum, B . H . ,

Cent.,

R e d f i e l d , A.C.

U n p u b l . MS res. p a p e r . S t a t e Univ.

and A y e r s , J . C . ,

Stony

of New York. 1951.

Papers i n

P h y s . O c e a n o g r . a n d Meterol. 1 2 (1)1-46. N a t i o n a l Marine F i s h e r i e s S e r v i c e , 1972. d i s p o s a l i n t h e N e w York B i g h t , Pearce, J.B., Interim rep. Car.,

1970.

The e f f e c t s o f waste

Summary f i n a l r e p . NTIS AD 743936.

The e f f e c t s o f w a s t e d i s p o s a l i n N e w York B i g h t .

f o r 1 January 1970.

S a n d y Hook, N J :

Nat.

Marine F i s h .

386 O f f i c e r , C.B.,

1976.

P h y s i c a l o c e a n o g r a p h y of e s t u a r i e s ( a n d

a s s o c i a t e d c o a s t a l w a t e r s ) Ch. 4 , J o h n W i l e y , N . Y . Takano, K . ,

J o u r n a l o f t h e O c e a n o g r a p h i c a l S o c i e t y of

1954a.

J a p a n , 1 0 , 60-64. Takano, K . ,

J o u r n a l of t h e O c e a n o g r a p h i c a l S o c i e t y of

195413.

J a p a n , 1 0 , 92-98. Takano, K . ,

1955.

J o u r n a l of t h e O c e a n o g r a p h i c a l S o c i e t y of

J a p a n , 11, 1 4 7 - 1 4 9 .

J. J. Leendertse and S. K. Liu

The Rand Corporation, Santa Jbnica, California (U.S.A.)

ABSTF?AcT

Three-dimensional flms in water bodies with nonhomgeneous density can be cmputed effectively by use of a finite difference model which contains an equation of continuib], equations describing conservation of m n t u m , salinity, temperature, subgridscale energy, and an apation of state. In the model, vertical accelerations are neglected, but not the vertical velocities. The vertical exchange coefficients are ccanputed from the subgridscale energy intensity. The introduction of turbulence closure by ccsrpxlting vertical exchange coefficients in the transport equations eliminates the necessity of evaluating exchange ccefficients in the whole computation field, but rather requires the evaluation of only a few characteristic praters. Fxperhents made with the model produced velocity distributions which typically occur in coastal areas.

BASIC CONCEPTS OF TUFBULEXT EXCHANGE PROCESSES In mathematical models of geophysical problems the approach is taken of using discrete representations of the differential equations which describe the main physical processes. merally IMSS balance equations are used, together with mmentum equations. This approach is taken here also, as we will describe later in more detail. The main difficulty encountered, however, is that the discrete representation forces

us to work with values for the variables which are characteristic for the region

described by the finite representation. If a horizontal grid representation of 1 mile is used, the d e l can only represent the average velocity, pressure, etc. of the grid distance of 1 mile. Naturally, significant variations do occur which, for example, cause m s s and mxra3ntm exchanges which can be considerably larger than the advective transports expressed by average velocities and concentrations. The exchanges are traditionally expressed as functions of the local mass and momentum gradients. In the horizontal mtions it is generally assumed that these functions are linear with the gradients; thus we used horizontal nmwntum and mass exchange coefficients.

388

In the vertical this approach was even far less satisfactory, and generally the exchange coefficient is taken as a function of the square of the vertical velocity gradient. This assumption s e e d justified, as with larger gradients m r e turbulence is generated, thus nmre vertical exchange. In addition to the above-mtioned relationship between the m a n flow field and the vertical exchange coefficient, an additional relationship is generally used between the exchange coefficient and the Richardson number. The Richardson numkr is an expression for the ratio of the density gradient and the local turbulent energy. In the d e l presented here the vertical exchange coefficients are related to the subgridscale turbulent energy. The mvanents of the water on a small scale cannot be described, but we are able to compute the small-scale energy by considering the energy transfer from the larger scales to the smaller scales and the decay of these mall-scale motions. The vertical exchanges are typically induced by the turbulence generated by the flow moving over the bottom, by the turbulence generated by the air mvinq over the water, and by the tugbulence generated by differences in the mean velocities at different locations in the vertical. If the turbulent energy is used as an indicator for the intensity of turbulence, then exchange coefficients can be taken as a function of the subgridscale energy and a length scale. This approach requires that the subgridscale energy be computed. In the model presented here the computed subgridscale energy is transported like a constituent, and is dissipated at a rate dependent on the local energy intensity. The subgridscale energy in our model is prduced as a function of the local mean flow gradients and, as stated above, at the upper and lower boundaries of the

water body. With this concept we generate the subgridscale energy predminantly near the bottom, and if wind is present, also in the fluid near the surface. The vertical moomentun and mass exchange concept presented here also has a length scale which has to be cetermined. Two approaches are available, =ly, use of an algebraic expression or by computing the lenqth scale also frcnn mean flow infomation. Presently the nmst simple approach is used, and we made the length scale a function of the depth. The vertical mass and momentum exchanges are suppressed if vertical density gradients are generated. In the model the vertical exchange rates are also taken as a function of the Richardson number ccmputed fram the local intensities of turbulent energy. The horizontal exchanges are mainly due to horizontal eddies which are considerably larger than the eddies in the turbulence mentioned earlier (Fig. 1). These large eddies are *dimensional, since they will be limited by the water depth in the vertical scale, and also by density differences if these are present. The horizontal exchanges should actually also include an exchange

389

mass and momentum exchange /Vertical by three-dimensional turbulence ,Horizontal mass and moment um exchange by two-dimensional eddies (turbulence)

Fig. 1. Concept of mass and mmentm exchange in the model.

generated by turbulence f r m the bottcgn and the wind-driven w a t e r surface, but i n our d e l we neglected this f o r the present time.

Thus the horizontal mass

and m e n t m exchanges w e r e taken only due t o turbulence w i t h lower frequencies

(smaller wave nmkers) than are qenerated by the h t t m and surface mJmentum transfers. The simplest and mst primitive model f o r the horizontal m t m exchange i s

t o take it as a function of the local qradientwith fixed constant horizontal exchange coefficients f o r the m t m and mass transfer.

This w a s the approach

taken previously by us [1,21. It seems mre appropriate, hmever, t o take this coefficient as a function of the local velocity deformation calculated i n the f i n i t e difference grid. Even though t h e larger subgridscale motions are pred&antly rather than three-dimmsional,

two--sional

upon which this hypothesis is based, t h i s

assumption has prcduced good results i n similar three-dimensional ccorrputations

w i t h predominant two-dimensional horizontal motions.

W e w i l l discuss the cam

putation of the horizontal exchange terms i n mre d e t a i l . "ATICAL

DESCRIPTION OF THE PRCCESSES

Flows i n an estuary o r coastal sea are mainly horizontal and primarily turbulent.

The equations of horizontal m t i o n f o r an incompressible, internally

source-free f l u i d on a rotating earth i n Cartesian coordinates with the z-axis positive upward are:

The v e r t i c a l acceleration of f l u i d m t i o n associated with the p r e d d n a n t hydrdynamic processes such as tidal and wind-induced circulations are

390 extremely small i n ccanparison w i t h the gravitational acceleration.

Therefore we

can neglect the v e r t i c a l acceleration and advection, and the equation of m t i o n kccanes the hydrostatic equation:

The equation of continuity is:

The equations of salt and heat balance are:

ar

+

at

~ ( u T )I ~ ( v T ) ax aY

and the equation of s t a t e

-

p = p

+

(7)

p'(s.T)

Similar balance equations can be written f o r turbulent densities and dissolved pollutant constituents:

and

where u,v,w are respective ccmpnents of velocity; s and T are s a l i n i t y and

temperature; e and P are turbulent (subgridscale) energy density per unit mass and p l l u t a n t concentration, respectively; D i s the dissipation rate f o r turbulent energy; l? S

is the decay rate f o r the pollutant concentration; and Se and

are the source and sink terms f o r the turbulent energy and pollutant concen-

P t r a t i o n , respectively. The &el

described in this paper uses a grid system w i t h equidistant points

i n the horizontal direction.

In the v e r t i c a l direction, it is possible to use

391

an unequal grid distance. time-variable thickness.

The top layer, bounded by the free surface, has a

In the canputation, the origin of the v e r t i c a l c0ord.i-

nates is taken a t the mean sea level, whereas the w a t e r surface z the upper boundary of the system. The f i n i t e differen=

=

r;(x,y,t) i s

approximation of the d i f f e r e n t i a l equations (1)through

( 6 ) , and (7) and (9) are acccanplished i n the f o l l m i n g manner:

F i r s t , the equa-

tions f o r the layers w e r e divided by v e r t i c a l l y integrating the variables over

the layer thickness, and subsequently, f i n i t e difference approxirrations f o r the layer equation w e r e established [ 1 , 2 ] . on the v e r t i c a l grid.

Figure 2 shows the location of variables

A space-staqqered grid (Fig. 3) was selected.

Figure 4

shows the location of velocities and other variables on the horizontal grid.

The

position of the variables in the grid i s indicated by indices i , j , indicating a position iAx and jAy from t h e origin of the coordinates, w i t h i , j = 0 ,

...

i 3/2,

?

1,

The v e r t i c a l p s i t i o n is determined a s t o i t s location in the center

of the layer nmkered f r m the top with integer k = 1 , 2 , 3 interfaces w i t h half-integer values k = 1/2, 3/2, 5/2 cretized i n t o n&er integer value.

... o r a t horizontal

... etc.

tin^ i s dis-

of tirw steps (nAt) from the reference t k , w i t h n an

b7e adopted c o q a c t sum, difference and t h - l e v e l notations f o r

x, y, z and t. I n t h e x direction, -x F

=

'/z I F[(i

I

1 I

+

6 F = - F[(i x Ax I

k ) Ax,jAy,kAz,nAt] +

F[(i

+ 4) Ax,jAy,kAz,nAt] -

F+ = F[iAx,jAy,kAz,(n

+

l)At]

F - = F[iAx,jAy,kAz,(n

-

l)At]

- k)

F[(i

-

Ax,jAy,kAz,nAt]j

1

(10)

I

$) Ax,jAy,kAz,nAt]\

(12) (13)

P J i t h these notations, Eqs. (1)t h o u g h (6), and (8) and (9) f o r the level k

w i t h layer-averaged values take the following form:

392

Fig. 2.

Location of variables on the v e r t i c a l grid.

j+',

I

j

+

j-1:

I

I

I -

Fig. 3. The location of u (-) space-staggered grid.

+

-

I

t

I

1

and v ( ) and other parmters (+) in the

at i, j

a t i , j , k, n

+ 4, k,

n

393

1.03

-

1.02

-

L

1.01

-

1.00

-

-

Fig. 4. Equation of state and graphical representations of several selected ranges of S and T values.

at i, j, k, n

+ S - D h

at i, j, k , n

394 6t(hP)t

=

-

l(3

6x(h UP )

-

6 (?vF) Y

-

hSZ(w?)

at i, j, k, n

The density is conputed using s(i,j,k,t) and T(i,j,k,t) according to the equation of state p =

[5890 + 38T - 0.375T2 + 3s]/[(1779.5 -

(3.8

+

0.OlT)s

+

0.698(5890

+

+ 11.25T -

38T

-

0.375T2

+

0.0745T21 3s)]

(21)

at i, j, k, n + 1 A graphical representation of the density as a function of temperature for water

with different salinity is s b m in Fig. 4. The finite difference equation used to canpute the vertical velocity.ccmpnent W:

6 z =~

-

6,(?u)

-

6y(?v)

at i, j, k, n

This equation is used for the bot-

+1

(22)

layer first, and then for the layer above,

etc. "he horizontal pressure gradients are ccmputed f m the top layer downward with increasing k by use of

Once the horizontal pressure gradients are lmcn\m, the water level and velocities can be cquted again by the sequence of finite difference equations (14) through

(26). Special procedures are required for the canputation of variables on or near the boundaries, particularly the seaward boundaries. They are discussed, together with the specification of the initial conditions in Ref. 2.

395 VEFTICAL EXCHANGE OF M3-I

AND CONSTITUENTS

In the computational model the turbulent exchanges of m n t m and constituents

in the vertical which cannot be represented by the r x q u t a t i o n a l grid are accounted for by means of exchange coefficients.

The mgnitude of these exchange coef-

f i c i e n t s varies not only with space and t k , but it is also a function of local turbulence level.

Their values are also influenced by the local v e r t i c a l sta-

b i l i t y induced by the density s t r a t i f i c a t i o n .

In our computational model, the

vertical exchange ooefficients are taken according t o the basic theoretical considerations of Kolmqorof (1941) and Prandtl (1945) a s a function of local turbulent energy level. E =

(27)

L G

where L i s a lenqth scale, evaluated a s a function of distance to the bottom and

free surface according t o L = k'z(1

-

z/d)'

(28)

is the von Karman constant, z represents the v e r t i c a l distance from the bottom t o the pint considered, and d i s the local water depth.

where

In fluids w i t h v e r t i c a l l y stable density gradients, the turbulent intensity

i s suppressed.

Therefore, lover values of exchange coefficient would result.

The c r i t e r i a for the onset of the turbulence-sumressing process in a nonbmgeneous flm system can be obtained by multiplying by a proportionality coefficient which is a function of the local Richardson number, defined a s

IGmiyav (1958) indicates t h a t the reduction of vertical exchange due t o s t r a t i f i c a t i o n can be expressed in an exponential m e r .

For -tun

exchange

i n the x-direction,

The exponential term i n this equation describes the Richardson n&r

depen-

dency, and m represents a constant. The mass exchange coefficients are cmputed a t a different location, namely, a t the layer interface between the points where the concentrations are computed, a s shown in Fig. 2.

396

Consequently, the expression for the mss-exchange coefficient is s m h a t different than the rmrentm exchange coefficients.

In the model we are using:

L I

K = a

4 LC -

where r

=

exp[r>

a constant.

A factor a4 appears in this formula, as the mass exchange is not the same a s

the momentum exchange. The subgridscale energy i s transported i n a similar manner a s the transport of constituents, thus the energy exchange coefficient can be written in t h e same form as the mass exchange.

In the horizontal direction w e have assumed that the exchange processes are

predominantly governed by two-dimensional turbulence, as the horizontal dimensions a r e much larger than the depth, and thus no isotropic three-dimensional turbulence can be generated.

The two-dimensional turbulence has the property

t h a t the enstrophy (one-half v o r t i c i t y squared) cascades from smaller t o larger scales, while i n three-dimensional turbulence, it cascades from larger scales to

smaller scales.

In the model the local enstrophy has t o be dissipated, a s we

have l i m i t s to the m r t i c i t y which can be expressed on the grid.

This d i s s i -

pation i s acccnnplished by i n t r d u c t i o n of nonlinear horizontal eddy viscosity coefficients :

where

w

i s v o r t i c i t y , y i s a coefficient and A.t i s grid dimension.

GENEXATION AND DISSIPATION OF SUBGRIDSCALE: ENERGY

In the i n t e r i o r of the f l u i d in our &el, it is assumed that the interlayer shear generates the subgridscale energy. This source can be expressed as:

where

= man velocity E

= L&

In the model this source i s determined a t the interface between the layers a t i, j , k

+ 1/2.

391

It w i l l be noted t h a t only the energy is ccouputed a t the lower time level.

Tkis was necessary f o r s t a b i l i t y of the conputation. The energy generated a t this location is assumed to be distributed equally into the adjacent layers. In the httcan layer another source exists.

It i s ass&

that energy which

i s taken out of the man flow through the b o t t m stress inSneaiately enters the subgridscale energy system. The stress term i n the momentum equation i n the direction of the mean flow is

2 P

= gu

2

lc

2

where U = velocity i n bottam layer in the direction of flow.

I f this term i s multiplied by U, we obtain the energy which is taken out of the mean f l m system and the local source ( S ) f o r the subgridscale energy.

s

= gu

3

lc

2

(37)

I n the &el,

the subgridscale energy generation i s computed a t the layer

interfaces and the local f i n i t e difference source term becomes

at i, j, K

+ 4,n

The energy is completely intrcduced i n the layer K. A t the w a t e r surface the generation of the subgridscale energy is different.

H e r e the energy source i s the wind which generates surface waves, and through

these waves, turbulence.

Wave and swell conditions d e p d on wind intensity,

duration of the wind and the fetch.

In the test cases a fully-developed sea under

mderate wind speed was used as input.

Under these conditions the waves are sc-

called deep water waves, and the total wave energy can be found frcnn the PiersonWskowitz spectral sets (Neumann and Pierson [6]). P e r unit area, the total wave energy i s E~ = 5.6

1 0 - ~ ~ ~ W

where uw = wind speed in d s e c a t 19.5 m above man sea surface E

= w a v e energy

(39)

398 Half of this energy i s kinetic energy.

If we assme t h a t a l l this k i n e t i c

energy i s i n the top layer (hl) of the mdel, then the vertically-average subgridscale energy intensity i n this layer i s

at i, j, 1, n

As the wave theory presents an energy intensity f o r a given wind condition, we are not mncerned w i t h influx of the subgridscale energy i n t o the system, but with maintaining this energy level during the duration of the wind condition i n the sirmilation. W e have assumed t h a t a l l the k i n e t i c wave energy i s in the top layer.

mom

deep water wave theory it is knm t h a t the wave-induced w a t e r motions are effectively z e r o a t a depth which i s half the wave length.

This puts an upper

lhit u p n the wind s p e d which we were able to allow i n the simulation.

“his

wind speed can be estimated f r o m the average wave period belonging t o the wind s p e d (Neumann and Pierson [ 6 1 ) .

and from the wave-length-wave-period relation

=

-2 2a gT

The maxinnnn wind that is allowed in a d e l w i t h an u p r layer thickness h f o r use in

Q.

(40) can then be found from Eqs. ( 4 1 ) and ( 4 2 ) :

Higher wind velocities would also involve the computation of subgridscale energy in lower layers.

The &el

a t present does not include inputs other than

i n the surface layer.

For the dissipation of energy, use i s made of the now classical concepts developed by xolmogorov [3] and Prandtl [4] that the dissipation r a t e depends on the transfer process from larger eddies to smaller eddies according t o

To demonstrate

the generation and decay of turbulent energy in a transient

hydrodynamic system, the computational scheme has been tested on a seiche o s c i l l a t i o n w i t h an initial amplitude of 25 cm in a rectangular basin.

This

399

basin was 46 km long, 1 4 km wide and 1 2 reters deep. f i e l d contained 23 layer of 2 meters.

x

7

x

The i n t e r i o r camputationdl

6 grid points and an equal layer thickness f o r each

Constants i n the computation are al = 0.68

a2 = 0 . 1 a3 = 1 . 0

a4 = 1 . 0

!k

= 0.4

The i n i t i a l value of SGS energy density w a s set t o 0.4 ergs per unit mass

throughout the system.

A t the

starting t i n e , a l l the energy resolvable in the

mean flow is stored i n the form of potential energy.

Imneaiately a f t e r the start of the computation, a system of current i s set up. The stresses a t the

bottom intrcduce vertical gradients f o r the horizontal velocity, which causes subgridscale energy generation.

The gradual establishwnt of the vertical

velocity distribution i s i l l u s t r a t e d in Fig. 5.

The velocities obtained from

an analytical solution of the long wave equation in the basin f o r ideal fluid is

also shown in the graph. The subgridscale energy maxim o c m shortly a f t e r the occurrence of the

maximum velocities i n the system.

The subgridscale energy fluctuation f o r the

f i r s t two o s c i l l a t i o n s i s shown in Fig. 6.

Note that k d i a t e l y a f t e r the

start of the simulation the energy decays, as the velocities are s t i l l m a l l , and consequently the generation is small. The simulation was carried out t o 1500 time steps (18.75 hrs r e a l t k ) . A t t h a t time the o s c i l l a t i o n had decayed significantly. minima of the total SGS content are shown.

In Fig. 7 the maxima and

Both the m x k and minima decay

exponentially. Figure 8 s h w s the v e r t i c a l distributions of the u velocity components a t several time intervals.

The distribution appears t o be near l o g a r i t h i c only i f

the flow i s w e l l established and approaches a steady s t a t e condition. In a &el

of Long Island Sound, U.S.A.,

[7],turbulent energy and the total

kinetic energy a t each level appeared t o increase gradually from three and onehalf percent a t the surface to sixteen percent near the bottom, where both local production and dissipation are mst intensive. During ebb tide, the velocity magnitude and the subgridscale energy are plotted in Fig. 9 .

The velocity appears to have a nearly logarithmic profile.

Since i n this simulation the wind speed w a s set to zero, the local production of subgridscale energy is small in the top layer. The inputs of subgridscale energy in that layer are tkrough v e r t i c a l advection

and diffusion and generation a t the interface with the second layer.

This is

400

0.

6.

12.

18.

30.

2%.

'12.

36.

5*

98.

S i m u l a t i o n r i m e Steps

Fig. 5. Comparison of computed mean flaw velocity component u by numerical nethod t o non-viscous fluid by analytical solution.

0 0.

" ' I , 20.

" I

'

90

'

I

60.

'

5

,

I), 80.

I

,

I , '

100.

'

I

120.

'

'

I , , ( I IUO. 160.

I

a

180

SIMULATION T I M E STEPS

Fig. 6. Generation and decay of the total subgridscale k i n e t i c energy i n the o s c i l l a t i n g basin.

401

G e n e r a t i o n and decay of t h e t o t a l s u b q r i d s c a l e e n e r q y i n t h e homogeneous o s c i l l a t i o n b a s i n ( o n l y t h e maxima and minima o f each oscillation are plotted)

32. (n

28.

E

c

S I M U L A T I O N T I M E S T E P S I D E L T A T = 45 SEC)

Fig. 7. The generation and decay of turbulent energy in the honqeneous rectangular o s c i l l a t i n g basin.

-12.

-9.

-6.

-3.

0.

3.

6.

9.

12

V E L O C I T Y COMPONENT U l C M / S E C )

Fig. 8. V e r t i c a l distribution of velocity cconponent u in the middle of hmgeneous o s c i l l a t i n g basin ( l i n e a r v e r t i c a l scale).

402

-10.

0.

10.

20.

30.

90.

50.

V E L O C I T Y l C M / S E C l O R E N E R G Y l E R G S / U N I T MASS1

Fig. 9. Computed vertical distributions of horizontal velocity component u and turbulent energy densities at location m 6 9 , n=ll, during ebb tide.

balanced by the local dissipation. It appears that the m a x b energy is near the bottom, but not in the lowest layer. In Fig. 10 the vertical distribution of the energy in the m a n flow and the energy in the subgridscale range are Shawn. The ccsnputed water surface elevation and the computed horizontal circulation in the surface layer at a particular tim? are shown in Figs. 11 and 12. From the simulation we also abstracted the total subgridscale energy in the system, which varies considerably during the tidal cycles. As shown in Fig. 13, in the beginning of the simulation the totdl subgridscale energy drops, as the tidal f l a w is only effective in the most eastern part of the bay. From the graph it will be noted that after the second tidal cycle the flow is campletely established. The fourth peak is slightly lower than the third peak, as the tideinduced velocities are slightly lmer and less subgridscale energy is generated.

DISCUSSION The introduction of the subgridscale energy in numerical simulations opens m a y possibilities for the investigation of the behavior of time-dependent hydraulic systems and for the investigation of many of the turbulence hypotheses and theories. Introduction of ccsnputations would provide an excellent check on validity of these theories, which otherwise can only be inferred from measuremnts. We do then expect to see many suggestions for improvement, particularly

403

,,/////////

~ 200.

~ 250.

~

"

300.

~ 350.

"l 900.

I

I ~

/

,

950.

~I

I

I

~.

500.

I

"I

I ~I j~ , ~, I ~, I , , I , ' ' '

550.

'

~

600.

"

650.

~

700

ENERGY (ERGS/UNIT MASS1

Fig. 10. Coquted v e r t i c a l distribution of tide-induced t o t a l kinetic energy a t a particular location during ebb t i d e in a model of mng Island Sound (Leendertse and Liu, 1977).

L O N G I S L A N D SOUND

STEP

W A l t H LEVFL

Fig. 11. Cbquted water surface elevation 1 4 hours a f t e r beginning of simulation (tm s t e p 1 7 3 6 ) .

404

Fig. 1 2 . computed horizontal circulation i n the surface layer 1 4 hours a f t e r beginning of simulation (time s t e p 1736).

, , 1 , , 1 , , 1 , , / , , 1 , , ,

13

-

,

I

/

, ,

, ,

TURBULENT ENERGY

12.

Time rtev s i z e = 30 s e c .

I. O

.

0.

"

"

300.

'

~

600.

'

" BOO.

'

~ 1200

~

'

1500.

'

~ ' 1800.

, ~ 2100.

, ~ 2WO.

~ , 2700.

,

~

~

I

I

I

3000.

INTEGRATION TIME STEP

Fig. 13. Computed t k history of t o t a l turbulent subgridscale energy in the Long Island Sound model [ 7 ] .

I

405

since we are publishing the computer program [81.

Ih7ith the prqram l i s t i n g , a

sample case is provided which can be used d i r e c t l y f o r experimmtation.

In the model particular emphasis w a s placed on the v e r t i c a l exchanges. Naturally, a similar approach could have been taken f o r the horizontal exchanges. I f energy is determined in the tm-dimnsional

turbulence, then the influence of

the two-dimensional turbulence on the subgridscale energy i n t e n s i t y can l i k e l y also be computed.

Unfortunately, very limited i n f o m t i o n i s available on energy

i n t e n s i t i e s with the length scales of the horizontal grid.

Mst basic studies

directed toward finding rates of horizontal exchanges are based on the spread of dissolved substances, which are not very useful in determining the basic s t a t i s t i c a l properties of the f l u i d motions we w i l l be needing f o r further developent of n m r i c a l mdels. CONCLUSION Three-dimensional flaws in estuaries, bays and coastal seas w i t h norhmcqenwus density distributions can effectively be computed by use of the model described i n t h i s paper.

The intrcduction of turbulence closure by computing vertical exchang

coefficients i n the t r a n s p r t equations f o r mass, rrcm?ntm

and srrall-scale turbulent

energy f r m the ccarrputed turbulence i n t e n s i t y eliminates the necessity of evaluating exchange coefficients i n the whole ccarrputational f i e l d , but rather requires the evaluation of only a few c h a r a c t e r i s t i c parameters. ACKNCWLElX3EiW

The work presented i n this paper w a s suppOrtea wholly by funds provided by the United S t a t e s Departnent of the I n t e r i o r , as authorized under the Water Resources Research A c t of 1964, Public Law 88-379, as m d e d . REFERENCES

1 Jan J. Leendertse, Richard C. Alexander, and Shiao-Kung Liu, A Three-DimensionaZ Mode?, for Est-uaries and Coastal Seas: VoZwne I , Principles of Computation, The Xmd Corporation, R-1417-CR~E?R, Decenhr 1973. 2 Jan J. Leendertse, and Shiao-Kung Liu, A Three-DimensionaZ Mode2 f o r Estuaries and CoastaZ Seas: VoZwne 11, Aspects of Computation, The Rand Corpration, R-1764-0lF3, June 1975. 3 A. N. Kohmgoroff, Compt. rend. acad. sci. USSR, 30(1941)301 and 32(1941)16. 4 L. Prandtl, “Uber e i n neues Fomelsystem f u r d i e ausgebildete Turbulenz,” flachr. Akad. Wiss. Gottingen, 6(1945)19. 5 0. I. m y e v , “The Influence of S t r a t i f i c a t i o n on Vertical Turbulent Plixing 5n t l e Yea,“ 1.w. Geophys. Ser. , (1958)870-875. Tr. Victor A. Salkind. 6 G. N e u m a n n , and W. ,J. Pierson, PrincipZes of PhysicaZ Oceanography, Prentice-Hall, New Jersey, 1966. 7 Jan J. Leendertse, and Shiao-Kung Liu, A Three-Dimemiom2 Model f o r Estuaries and CoastaZ Seas: VoZwne I V , TurbuZent Energy Computation, The Rand Corporation, R-2187-U*m, Nay 1977. 8 Shiao-Kung Liu, and Alfred B. Nelson, A Three-DimensionuZ Mode2 f o r Estuaries and CoastaZ Seas: VoZwne V , TurbuZent Energy Program, The Rand Corporation, 9-2188-CPJ”, IQy 1977.

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407

A B R I E F REVIEW O F P R E S E N T T H E O R I E S O F F J O R D DYNAMICS

F1. B o P e d e r s e n

I n s t i t u t e of H y d r o d y n a m i c s and H y d r a u l i c E n g i n e e r i n g Technical U n i v e r s i t y of D e n m a r k .

L I S T O F CONTENTS

I.

INTRODUCTION

11.

THE PARAMETERS A F F E C T I N G THE DYNAMICS O F F J O R D S

11.1

The g e o m e t r y o f f j o r d s

11.2

The h y d r o l o g y of t h e adjacent w a t e r s h e d

11.3

T h e oceanographic c o n d i t i o n s o u t s i d e t h e f j o r d

11.4

The w i n d f i e l d

111. THE D E E P B A S I N

IV.

111.1

Fjords without s i l l

111.2

Fjords w i t h s i l l ( s )

THE UPPER LAYERS I N F J O R D S IV.l

V.

Types of f j o r d s

DYNAMICALLY A C T I V E F J O R D S V.l

V.2

C o m m o n f e a t u r e s of p r e s e n t f j o r d t h e o r i e s V.l.l

The f l o w c h a r a c t e r i s t i c a

V.1.2

The d y n a m i c c h a r a c t e r i s t i c a

D i f f e r e n t f e a t u r e s of p r e s e n t f j o r d t h e o r i e s V.2.1

F j o r d s of t y p e 1 - general >> f a c t s <<

V.2.2

F j o r d s of t y p e 1 - p r e s e n t t h e o r i e s

V.2.3

F j o r d s o f t y p e 1 - g e n e r a l remarks

V.2.4

F j o r d s of t y p e 2

- g e n e r a l >> f a c t s <<

V.2.5

F j o r d s of t y p e 2 - p r e s e n t t h e o r i e s

V.2.6

The w i n d e f f e c t on f j o r d s

VI.

CONCLUSIONS AND P O S S I B L E IMPROVEMENTS

VII.

REFERENCES

408 I . INTRODUCTION A s it i s impossible t o t r e a t p e c u l i a r i t i e s i n a b r i e f review, w e s h a l l

c o n f i n e o u r s e l v e s t o t h e dynamics of " g o o d - n a t u r e d " f j o r d s . Even w i t h t h i s res t r i c t i o n a g r e a t v a r i a b i l i t y i s o b s e r v e d i n f j o r d dynamics from one p l a c e t o a n o t h e r and from one t i m e t o a n o t h e r , see C a r s t e n s ( 1 9 7 0 ) , Dyer (1970), Officer Pickard e t a l .

(1976), Pickard

( 1 9 7 3 ) , Gade

(1961, 1967, 1 9 7 1 ) , P i c k a r d e t a l .

(1956), Saelen (1967), Tabata e t a l .

(1959),

(1957), Tully (1949).

I n o r d e r t o d e f i n e t h e frame o f t h e r e v i e w , w e s h a l l f i r s t l o o k a t t h e main p a r a m e t e r s a f f e c t i n g t h e dynamics and t h e n a t t h e i r i n f l u e n c e .

I t then

t u r n s o u t t o b e q u i t e n a t u r a l t o t r e a t t h e lower and t h e u p p e r p a r t o f t h e f j o r d s separately. F i n a l l y , we s h a l l t r y t o o u t l i n e t h e p o s s i b l e improvements i n f u t u r e models of f j o r d s .

11. THE PARAMETERS AFFECTING THE DYNAMICS OF FJORDS

The o b s e r v e d d i f f e r e n c e s i n f j o r d dynamics o r i g i n a t e m a i n l y from t h e variation i n (see f i g .

1):

1. The geometry o f f j o r d s 2 . The h y d r o l o g y o f t h e a d j a c e n t w a t e r s h e d s

3. The o c e a n o g r a p h i c c o n d i t i o n s o u t s i d e t h e f j o r d s 4. The wind f i e l d s .

INLAND WATERSHED

WIND FIELD

OCEAN

QF!JLl

TIDES

U PEST UA RY V

U PWELLl NG

DOWNESTUARY

111

116

QFQ

-_-----__----__

T

111

FIORD

INNER BASIN O

102
+lo< h<102 rn A

P 104<~<105m ___Y

Fig.

1. The main p a r a m e t e r s a f f e c t i n g t h e dynamics o f f j o r d s .

N

409 1 1 . 1 . The geometry of f j o r d s -

r e f l e c t s t h e i r geomorphologic o r i g i n s as g l a c i a l - c u t s t r u c t u r e s . T h e r e

are some

{

common f e a t u r e s and some

steep sides deep b a s i n

c

0,

different features

1, 2 ,

....

sills

s i d e c o n s t r i c t i o n s a t t h e mouth.

The o r d e r of m a g n i t u d e s o f t h e s i l l d e p t h , t h e b a s i n d e p t h , t h e w i d t h and t h e l e n g t h o f f j o r d s a r e shown i n f i g .

1.

11.2. - The h y d r o l o g y o f t h e a d j a c e n t w a t e r s h e d The y e a r l y d i s t r i b u t i o n o f t h e f r e s h w a t e r d i s c h a r g e Q,

t o the fjord

shows a pronounced v a r i a t i o n from one l o c a l i t y t o t h e o t h e r . L e t u s i l l u s t r a t e t h i s f a c t by two e x t r e m e s i t u a t i o n s . A h i g h l e v e l i n l a n d w a t e r s h e d h a s a g r e a t s t o r i n g e f f e c t on t h e p r e c i p i t a t i o n d u r i n g w i n t e r , which r e s u l t s i n a low r u n o f f d u r i n g w i n t e r and a v e r y h i g h r u n o f f d u r i n g summer from g l a c i a l and snow f i e l d melting, see f i g .

1. A low i n l a n d w a t e r s h e d e x h i b i t s n o r m a l l y j u s t t h e

o p p o s i t e v a r i a t i o n d u r i n g t h e y e a r , as t h e r u n o f f i s n e a r l y i n p h a s e w i t h t h e p r e c i p i t a t i o n due t o a weak s t o r i n g e f f e c t o f t h e w a t e r s h e d , see f i g .

1. Between

t h e s e e x t r e m e v a r i a t i o n s w e f i n d r u n o f f v a r i a t i o n s which e x h i b i t a c o m b i n a t i o n of t h e o u t l i n e d behaviour.

II.3.

The o c e a n o g r a p h i c c o n d i t i o n s o u t s i d e t h e f j o r d The s t r a t i f i c a t i o n i n t h e o c e a n i s weak compared w i t h t h e s t r a t i f i c a t i o n

i n e s t u a r i e s , b u t changes i n t h e m e t e o r o l o g i c a l c o n d i t i o n s create v a r i a t i o n s i n t h e d e n s i t y o f t h e s a l t w a t e r which i s i n f r e e c o n n e c t i o n w i t h t h e b o t t o m w a t e r of t h e f j o r d . Another c h a r a c t e r i s t i c f e a t u r e o f t h e o c e a n which i n f l u e n c e s t h e

dynamics o f f j o r d s i s t h e t i d e , see f i g .

.11.4. -

1.

The wind f i e l d The s p e c i a l g e o m e t r i c s h a p e s o f f j o r d s have t h e e f f e c t t h a t t h e wind i s

normally e i t h e r up-estuary

o r down-estuary.

111. THE DEEP BASIN

Oceanographic o b s e r v a t i o n s show c l e a r l y t h a t t h e dynamics o f t h e d e e p b a s i n and of t h e upper s t r a t a a r e p r a c t i c a l l y u n c o r r e l a t e d , and c a n t h u s b e treated separately. As f a r as t h e c o n d i t i o n s i n t h e d e e p b a s i n are c o n c e r n e d , t h e r e i s a r e markable d i f f e r e n c e between f j o r d s w i t h and w i t h o u t s i l l ( s ) .

410 111.1. -

Fjords without s i l l In the f j o r d without sills, t h e density d i s t r i b u t i o n i n the basin r e f l e c t s

t h e d e n s i t y d i s t r i b u t i o n i n t h e a d j a c e n t o c e a n . Hence, c o n t i n u o u s r e p l e n i s h m e n t of t h e b a s i n water t a k e s p l a c e , which means t h a t t h e DO-content

i s n o r m a l l y con-

s t a n t l y high.

111.2. -

Fjords with s i l l ( s ) The c o m b i n a t i o n o f one o r more s i l l s and a s t a b l e d e n s i t y g r a d i e n t c r e a t e

a b l o c k i n g e f f e c t w i t h a n e a r l y s t a g n a n t b a s i n below t h e s i l l l e v e l . The e f f e c t i s a d e c r e a s i n g s a l i n i t y and DO-content

u n t i l t h e boundary c o n d i t i o n s o v e r t h e

s i l l ].eve1 become e x t r a o r d i n a r y s a l t y , see f i g . 2 . Then a n i n f l o w t o t h e b a s i n

t a k e s p l a c e , and p a r t o f t h e bottom w a t e r i s r e p l e n i s h e d . A s t h e i n f l o w i n g w a t e r d e p o s i t s a t a l e v e l c o r r e s p o n d i n g t o i t s d e n s i t y , t h e f r e q u e n c y of r e p l e n i s h m e n t i s d e c r e a s i n g w i t h d e p t h ( a n d d i s t a n c e from t h e mouth i n t h e case when more t h a n

one s i l l i s p r e s e n t ) .

_-

INFLOW CREATED BY UPWELLING

---4-

SILL 1962

1963

1964

1965

i x 3 - 8 0 m ) VESTFJORDEN [ G A D E I F i g . 2. Replenishment o f p a r t of t h e bottom w a t e r i n a f j o r d w i t h s i l l . The obs e r v a t i o n s shown a r e from V e s t f j o r d e n , Norway, s e e Gade ( 1 9 7 1 ) .

The i n f l u e n c e on t h e d e e p water from t h e wind f i e l d i s v e r y week. T h i s c a n be v e r i f i e d by l o o k i n g a t t h e measurements from t h e RGrholt f j o r d i n B a m b l e , Norway, s e e f i g . 3 from C a r s t e n s ( 1 9 7 1 ) . Some 6000 y e a r s ago, it was d i s c o n n e c t e d from t h e o c e a n , b u t t h e lower 1 3

rn s t i l l c o n t a i n t h i s 6000 y e a r s o l d s a l t w a t e r .

411 5

10

Fig. 3. Wind e f f e c t on t h e bottom water. Measurements from t h e Rorholt f j o r d , Carstens (1971).

The mechanism of inflow should c r e a t e a s t e p - l a y e r e d bottom water which becomes more and more s a l t y , because t h e wind e f f e c t i s confined t o t h e upper, say 20 m . B u t t h i s i s not t h e c a s e and a p l a u s i b l e explanation i s t h a t t h e t i d e generates i n t e r n a l waves, which by breaking a g a i n s t t h e boundaries c r e a t e

tur-

bulence and hence mixing. The mixed water spreads t o t h e i n t e r i o r of t h e f j o r d causing an e f f e c t i v e v e r t i c a l mixing, s e e S t i g e b r a n d t (1976) . I n o r d e r t o be a b l e t o c a l c u l a t e t h e bottom water replenishment, t h e boundary c o n d i t i o n s o u t s i d e t h e f j o r d must be known, b u t t h i s knowledge i s seldom a v a i l a b l e . Therefore, i n s t e a d of making dynamic models, s t o c h a s t i c models a r e more f r u i t f u l i n d e s c r i b i n g t h e c o n d i t i o n s i n t h e deep b a s i n , s e e Gade ( 1 9 7 1 ) . A pseudo-dynamic model could be e s t a b l i s h e d by c o r r e l a t i n g t h e inflow with some

r e l e v a n t meteorologic q u a n t i t i e s .

IV.

THE UPPER LAYERS I N FJORDS

IV.1. Types of f j o r d s A s t h e combinations of t h e d i f f e r e n t f a c t o r s a f f e c t i n g f j o r d dynamics

a r e numerous and r a t h e r complex, i t i s more convenient t o look a t some t y p i c a l kinematic and dynamic s i t u a t i o n s found i n t h e upper l a y e r of f j o r d s and then give a g e n e r a l d e s c r i p t i o n of t h e c o n d i t i o n s causing t h i s behaviour. In o r d e r t o cover t h e range of c o n d i t i o n s , t h r e e d i f f e r e n t types a r e suggested, s e e f i g . 4 (Pickard (1961)).

TYPE 1 +------ACTIVE

TYPE 2

Fig. 4 . Types of f j o r d s . A f t e r Pickard ( 1 9 6 1 ) .

TYPE 3

P A S S I V E 4

412 Types 1 and 2 c a n b e c h a r a c t e r i z e d as t h e d y n a m i c a l l y a c t i v e f j o r d s i n t h e s e n s e t h a t f r e s h water s u p p l y i s s u f f i c i e n t t o c r e a t e a c i r c u l a t i o n mode i n t h e upper s t r a t a . Type 3 f j o r d s are i n t h e same s e n s e d y n a m i c a l l y p a s s i v e , a s t h e y j u s t r e f l e c t t h e boundary c o n d i t i o n s i n t h e a d j a c e n t o c e a n

- t h e y are merely a

bay of t h e o c e a n . A l l o t h e r p a r a m e t e r s k e p t unchanged, a f j o r d h a s a t e n d e n c y t o change

from t y p e 1 t o 2 and 3 when: Geometry

:

Length/outlet depth r a t i o increases ( i . e . increases tidal-generated v e l o c i t y and h e n c e m i x i n g ) . Width o f t h e o u t l e t s e c t i o n i n c r e a s e s ( i . e . d e c r e a s e s c r i t i c a l d e p t h and h e n c e d e c r e a s e s t h e l a y e r d e p t h ) . D i s t a n c e from h e a d i n c r e a s e s ( i . e . d e c r e a s e s t h e d e n s i t y d i f f e r e n c e and h e n c e i n c r e a s e s t h e m i x i n g ) .

Hydro l o g y

:

Freshwater d i s c h a r g e d e c r e a s e s ( i . e . d e c r e a s e s t h e c r i t i c a l d e p t h and hence d e c r e a s e s t h e l a y e r d e p t h ) .

Ocean c o n d . : T i d a l a m p l i t u d e i n c r e a s e s ( i . e . i n c r e a s e s t i d a l - g e n e r a t e d and hence i n c r e a s e s m i x i n g ) .

velocities

Wind f i e l d : Wind v e l o c i t y i n c r e a s e s ( i . e . i n c r e a s e s m i x i n g ) . I n t h e f o l l o w i n g , we a r e d e a l i n g o n l y w i t h t h e dynamics o f t h e upper l a y e r s of dynamically a c t i v e f j o r d s .

V.

DYNAMICALLY A C TI VE FJORDS

V.l.

Common f e a t u r e s o f p r e s e n t f j o r d - t h e o r i e s

A l l t h e p r e s e n t t h e o r i e s o f f j o r d dynamics t r e a t t h e s t e a d y s t a t e , i . e .

t h e c o n d i t i o n s averaged o v e r a t i d a l p e r i o d . T h i s i s a r e a s o n a b l e approach t o t h e f j o r d e s t u a r i e s , a s t h e t i d a l i n f l u e n c e i s n o r m a l l y weak, e s p e c i a l l y i n t y p e 1 f j o r d s . For t h e s a k e of s i m p l i c i t y , we f u r t h e r o n c o n f i n e o u r s e l v e s t o c o n s i d e r

a f j o r d with constan t width. Another f e a t u r e common t o t h e p r e s e n t t h e o r i e s o f f j o r d dynamics i s t h a t t h e y a l l assume s i m i l a r i t y i n t h e v e l o c i t y p r o f i l e s as w e l l as i n d e n s i t y ( o r salinity) profiles.

Hence

- with reference t o f i g . 4 - t h e p r e s e n t t h e o r i e s of

f j o r d s cover e i t h e r t y p e 1 o r t y p e 2 f j o r d s . Nevertheless, independently of t h e type of f j o r d , t h e following c o n d i t i o n s apply:

V J . l . The f l o w c h a r a c t e r i s t i c a The main p a r t o f t h e f r e s h w a t e r s u p p l y QF i s n o r m a l l y s u p p l i e d a t t h e head o f t h e f j o r d . During t h e f l o w from t h e head t o t h e mouth, t h e u p p e r downe s t u a r y f l o w i n g w a t e r i s c o n t i n u o u s l y i n c r e a s e d by e n t r a i n m e n t o f s a l t y w a t e r from t h e lower u p - e s t u a r y c o u n t e r f l o w , see f i g . 5.

413

a) MOUTH

HEAD

b)

Fig. 5.

VELOCITY I H EA D)

VELOCITY (MOUTH)

a. Longitudinal variation of the discharge. b. Typical velocity distributions.

By considering volume and salt continuity under steady-state conditions, the following two conditions must be fulfilled: Condition 1:

:1

*)

u dx, = qF

which states that the total cross sectional down-estuary discharge per unit width in any cross section is equal to the fresh water supply upstream to the actual cross section (per unit width) and Condition 2:

:1

A u dx,

=

AFqF

which is the analog mass-deficit flux condition. The massdeficit is defined by

A=-

- P

P,

ps

where p

*)

is a reference density.

Footnote: The first person to apply these conditions to oceanographic data was Martin Knudsen in 1899 when he treated the observations from the Danish estuaries (particularly Kattegat - an enormous saltwater wedge). Known as Knudsen's hydrographical theorem.

414 V.1.2.

The dynamic c h a r a c t e r i s t i c a The e n t r a i n m e n t of s a l t y w a t e r h a s two e f f e c t s on t h e dynamics o f f j o r d s ,

viz. 1. I n f l u e n c e on t h e l o n g i t u d i n a l p r e s s u r e g r a d i e n t s , and h e n c e on t h e s h e a r s t r e s s due t o t h e l o n g i t u d i n a l d e n s i t y g r a d i e n t s , and

2.

i n f l u e n c e on t h e momentum c r e a t e d by t h e upward f l o w a c r o s s t h e main l o n g i t u d i n a l l y d i r e c t e d f l o w , which c r e a t e s s h e a r s t r e s s e s a s w e l l .

A s t y p e 1 f j o r d s a r e a s s o c i a t e d w i t h low d e n s i m e t r i c Froude numbers

and type 2 f j o r d s with high

IF* A

IF2 (but still < l ) , the pressure gradient e f f e c t i s

A

most pronounced f o r t y p e 1 f j o r d s , w h i l e t h e momentum e f f e c t i s most i m p o r t a n t f o r type 2 f j o r d s . I n d e p e n d e n t l y of t h e t y p e of f j o r d , two c o n d i t i o n s must a p p l y i n t h e s t e a d y s t a t e , namely Condition 3: which s t a t e s t h a t t h e p r e s s u r e measured a g a i n s t a c o n s t a n t r e f e r e n c e p r e s s u r e ( t h e p r e s s u r e i n t h e a d j a c e n t o c e a n ) must v a n i s h a t t h e b o t t o m , and Condition 4 :

7

x3=D

- 0

which s t a t e s t h a t no s h e a r s t r e s s i s p r e s e n t a t t h e b o t t o m , as t h e d e e p water b a s i n i s s t a g n a n t . I f we a p p l y c o n d i t i o n 3 , an e x p r e s s i o n f o r t h e s u p e r e l e v a t i o n 11 o f t h e w a t e r s u r f a c e i n t h e f j o r d c a n b e e s t a b l i s h e d . The h y d r o s t a t i c p r e s s u r e i n t h e fjord is pF = 1 i 3 P 9 d x 3

=

x3 Psg(ll + x , ) - Ps91 -rl

I f we s u b t r a c t t h e c o n s t a n t

dX3

r e f e r e n c e p r e s s u r e pR = ps g x 3 , t h e o v e r -

p r e s s u r e becomes

where we have made u s e o f c o n d i t i o n 3:

D

11

=

lnA

dx3

C o n d i t i o n s 1 , 2 , and 3 a r e d i r e c t l y o r i n d i r e c t l y i n c o r p o r a t e d i n a l l t h e present theories of fjords. With r e g a r d t o c o n d i t i o n 4 , t h e v a r i o u s a u t h o r s u s e i t i n d i f f e r e n t ways. The problem i s n o t t o a p p l y t h e c o n d i t i o n , b u t t o make p h y s i c a l l y p l a u s i b l e assumptions t o which t h e c o n d i t i o n c a n b e a p p l i e d . A s a l l t h e p r e s e n t t h e o r i e s assume s i m i l a r i t y i n v e l o c i t y a s w e l l a s i n d e n s i t y ? r o f i l e s ,

it i s n e c e s s a r y t o

415 treat type 1 and 2 fjords separately in the following. This does not mean that it will always be so.

V.2. Different features of present fjord theories

-

With a view to the historic development, type 1 fjords will be treated first.

V.2.1.

Fiords of tvoe 1

-

aeneral >> facts <<

The most characteristic feature of type 1 fjords is the abrupt salinity (and hence the density) jump across the interface, see fig. 6. Above the interface, the density deficit is nearly uniformly distributed, while it decreases gradually below the interface. Accordingly, the velocity is nearly uniform in the upper layer with a weak compensation flow in the lower layer. These features are characteristic for low densimetric Froude numbers, stratified flows, i.e.

MOUTH

Fig. 6. Typical density and velocity distribution in type 1 fjords.

The fact that p 2 is small can be utilized in an illustration of the dyA namics of type 1 fjords, as it means that the shear stress in a first order analysis is balanced by the longitudinal pressure gradients

a (P,/ps9) 3x1

dX3

By taking a plausible density-deficit distribution and making use of conditions 3 and 4, the general shape of the shear stress distribution can be found.

Example : ..- - - - - In order to demonstrate the procedure, we take

A

=

A

=

A 1exp

where y

above the interface, and

{

-

=

F}

below the interface

the upper layer depth, y1

=

a length scale for the lower layer.

416 The hydrostatic overpressure distribution is then

By applying condition 3 we get the surface elevation

rl

=

Auyu + Alyl

By differentiating the pressure with respect to x 1 and integrating the pressure gradient, the shear stress is then obtained. As this example is given only for illustration, no further calculations have been performed, but the general shape of

T

is shown in fig. 7.

Fig. 7. The general shape of the distribution of the 1.ongitudinal pressure gradients and the shear-stress in type 1 fjords.

It is important to realize that the interfacial shear stress is balanced by pressure gradients in the lower layer counterflow. These pressure gradients can be established only by longitudinal density gradients originating from the weak transport of brackish water downwards, combined with the up-estuary convective transport. This gives a recirculation of part of the brackish water, a fact of great importance for the distribution of pollutants. None of the present theories of type 1 fjords describes this mechanism!

V.2.2.

Fjords of type 1

-

present theories

Instead of giving a detailed description of the theories of type 1 fjords, we shall concentrate on the weakness of the theories, see fig. 8:

THE CORRECT ( ? )

417

BO PEDERSEN (1972 a, b )

XI

si balanced by e n t r a i n N.-E OTTESEN HANSEN (1975)

XI

pV2 R.R. LONG (1975)

i balanced by pressure gradients i n t h e lower layer XI

Flg. 8. The distribution of the mass deficit A, the velocity u, the longitudinal pressure gradient a(p,/psg)/ax,, and the shear stress T/p g in the present theories of type 1 fjords.

H. Stommel (1951)

Stommel's theory is consistent in the sense that it includes all the four mentioned conditions, but by putting the interfacial shear stress equal to zero, the theory fails, for instance by predicting

F1. Bo Pedersen (1972a, b)

The main progress in Bo Pedersen's theory was to focus on the problem

of balancing the interfacial shear stress Ti/P

=

f v2 7

in order to fulfil condition 4

(7

x3=D

N

0 ) .

By using the work energy equation for the mean flow, a relation between the friction factor f and the entrainment velocity V E was established

418

The physical meaning of this result is that the interfacial shear stress is balanced by the entrainment. This has later been shown by Ottesen Hansen and hence the (1975) to be contradictory to the assumption of a small IFz A theory failed. N.-E. Ottesen Hansen (1975)

Ottesen Hansen's theory is based on the assumption that the interfacial shear stress is mainly balanced by pressure gradients in the compensation flow although the mechanism for transferring brackish water downwards was not incorporated in the theory. Even by introducing probable assumptions concerning the interfacial shear stress and the entrainment, it turned out to be difficult to solve the equations. As the introduced simplifications do not seem convincing, it is sufficient to conclude that the basic ideas in Ottesen Hansen's theory are the physically most acceptable presented so far (March, 1977) concerning type 1 fjords.

R.R. Long (1975, 1976) The fjord theory presented by Long contains an entrainment function and the interfacial shear stress. Nevertheless, the theory is inconsistent, as the interfacial shear stress is not balanced (condition 4 not fulfilled)

V.2.3. - Fjords of type 1

-

general remarks

It is interesting to notice that the interfacial shear stress, when incorporated in the theories, is represented by

and, accordingly, the mass flux through the interface as

But this is not sufficient for a type 1 fjord theory, because the downward mass flux is necessary for creating the pressure gradients in the counterflow. This can be facilitated by using an approach which incorporates an eddy diffusivity KAv,

defined by

and accordingly an eddy viscosity K .-

V;V;

= T/p

=

KT

defined by

aU 3

It is evident that a K-theory with known values is superior to the presented theories - and needed. So far, lack of knowledge of the K-values has stopped to this approach. We shall revert to this problem later.

419 By chance, we can demonstrate the K-theory approach in this review as the fjords of type 2 have been treated in this way.

-

V.2.4. Fjords of type 2 - general >> facts << In fjords of type 2, the two-layer characteristic is absent. The velocity as well as the mass-deficit decrease continuously with the distance from the water surface, see fig. 9. These features are characteristic of stratified flows with higher densimetric Froude numbers and with a more intense vertical mixing. For the dynamics of the fjord, this implies that the shear stress is not balanced solely by the pressure gradients as the momentum is important as well. Hence, a simple illustration as that presented for type 1 fjords is not possible.

Fig. 9. The general shape of the distribution of the mass deficit, the velocity and the shear stress in type 2 fjords. W.M. Cameron (1951)

Unfortunately, the doctoral dissertation by Cameron has not been at the author's disposal. M. Rattray, jr. (1967)

By applying the well-known procedure used by Rattray et al. (1965) to coastal plain estuaries, Rattray transformed the conservation equations for mass, volume, and momentum to a set of two coupled, non-linear, ordinary differential equations*) and a definite integral. These equations were then solved to the lowest order of approximation

-

i.e. with focus

on the upper part of the circulation. In order to obtain similarity, Rattray was forced to restrict the theory to fjords, in which the fresh-water discharge is a small fraction of the total circulation. This condition is likely to occur in the down-estuary part of type 2 fjords, as demonstrated by the measurements of various authors, see the introduction.

*)

Footnote: There are two rather misleading misprints in Rattray's eq. 26: V@$" should be y$@"' + V @ ' @ ' ' .

y$$" +

420 In order to compare the theory with actual measurements, Rattray estimated the K-values as exponentially damped linear functions, but any attempt to correlate the absolute values with the dynamics was not made. Hence the theory can be used in a descriptive model but not in a predictive model. D.F. Winther (1972, 1973)

The approach used by Winther is quite similar to that of Rattray. The main progress in Winther's model was the introduction of a distorted scale, in which the characteristic horizontal and vertical length scales were related to the physical behaviour of the flow. This improvement makes it easier to ascribe values to the involved parameters, when the theory is compared with actual field measurements. As an example can be mentioned that the K-values are proportional to the characteristic vertical length and velocity scale, and hence probably of a more general nature than the K-values used by Rattray. As shown by examples in Winther's papers the theory reflects the behaviour of type 2 fjords reasonably well.

V.2.6. - The wind effect on fjords The mixing of the upper layer in a fjord due to wind is normally very strong, but as the problems concerning the wind effect are not specific for fjords they will not be treated in this review. The reader is referred to the

papers by Farmer (19721, Ottesen Hansen (19751, and Bo Pedersen (1977b), where further reference lists are given.

VI. CONCLUSIONS AND POSSIBLE IMPROVEMENTS Measurements in fjords (see the introduction) and in laboratories (see Bo Pedersen et al. (1974)) as well as theories (see Rattray (1967), Winther (1972,

1973), and Ottesen Hansen (1975)) show clearly that the basic assumption in all the present theories concerning similarity is not fulfilled. Furtheron, it has been demonstrated that an approach using an interfacial shear stress and an entrainment function is insufficient. This gain in knowledge could not have been obtained without the present theories, and in that respect, they are all of great value. The next step in developing a fjord theory is therefore to cancel the similarity assumptions concerning the velocity and density profiles. This demands that quite new ideas are introduced in the fjord theories. One practicable way is to extend the set of equations by the conservation equations for the turbulent

kinetic energy, the Reynolds' shear stress, the mass deficit flux, etc. In this way it may be possible to set up a general fjord theory applicable to all types of fjords. The only problem is how to simulate the conservation equations for the turbulent properties.

421

A first step in this direction has been made by the author (1916, 1977a). If we confine ourselves to the interfacial equilibrium layer for type 1 fjords, the extended sets of equations can be solved analytically. In this way the f o l lowing quantities have been evaluated: The eddy viscosity for momentum and salt, the velocity and the density gradient, the Monin-Ubokhov length scale, the dissipation length scale, the Brunt-Vaiasala frequency, the gradient and the flux Richardson number. It is probably impossible to go much further analytically, but by means of computers the way is prepared for further progress in fjord models.

VII. REFERENCES Cameron, W.M., 1951. On the dynamics of inlet circulations. Doctoral dissertation. Univ. of California, Los Angeles, California. Carstens, T., 1970. Turbulent diffusion and entrainment in two-layer flow. Proc. ASCE, W W I , Feb. 1970. Carstens, T , 1971. Transport processes in water courses and fjords (in Norwegian). Institutt for Vassbygging. Dyer, K.R., 1973. Estuaries. Wiley and Sons, 140 pp. Farmer, D.M., 1972. The influence of wind on the surface waters of ALBERNI INLET. Pacific Marine Science Rep. No. 12-16. July 1972. Gade, H.G., 1910. Hydrographic investigations in the Oslo fjord, a study of water circulation and exchange processes. Geofysisk Institutt, Rep. 24. Gade, H.G., 1971. Deep water renewals in a sill fjord. Geofysisk Institutt, Rep. 25, 1971/72. Long, R.R., 1975. Circulations and density distributions in a deep, strongly stratified, two-layer estuary. J. Fluid Mech., 71. Long, R.R., 1976. Estuarine circulations and mass distributions. Tech. Rep. No. 9 (Series C). John Hopkins University. Dep. of Earth and Planetary Sciences and Mechanics and Materials Science. Baltimore, Maryland, Dec. 1976. Officer, C.B., 1976. Physical oceanography of estuaries (and associated coastal waters). New York, Wiley, 465 pp. Ottesen Hansen, N.-E., 1975. Entrainment in two-layered flows. Series Paper 7. Institute of Hydrodynamics and Hydraulic Engineering, Tech. Univ. of Denmark, 99 pp. Pedersen, F1. Bo, 1972a. Gradually varying two-layer stratified flow. ASCE, Journal of Hydraulics Division. Vol. 98, No. HY1, Jan. 1972. Pedersen, F1. Bo, 191213. Gradually varying two-layer stratified flow in fjords. International Symposium on Stratified Flows, Paper - 19. Novosibirsk, pp 413-429. Pedersen, F1. Bo and Ottesen Hansen, N.-E., 1914. Entrainment in tvo-layered flows, part 1. Prog. Rep. 33, Aug. 1974. Institute of Hydrodynamics and Hydraulic Engineering, Tech. Univ. of Denmark. Pedcrsen, F1. Bo, 1976. The flux Richardson number and the entrainment function. Prog. Rep. 39, Aug. 1976. Institute of Hydrodynamics and Hydraulic Engineering, Tech. Univ. of Denmark.

422 Pedersen, F1. Bo, 1977a. Part 1: The entrainment function for gradually varying two-layered stratified flow. Part 2: A turbulence model and its application to fjord estuaries. Copies of preprint available from the Institute of Hydrodynamics and Hydraulic Engineering, Tech. Univ. of Denmark, March 1977. Pedersen, F1. Bo, 197733. The hulk flux Richardson number applied on turbulent entrainment at a stable density interface. Prog. Rep. 43, in print. Institute of Hydrodynamics and Hydraulic Engineering, Tech. Univ. of Denmark. Pickard, G.L., 1961. Oceanographic features of inlets in the British Columbia Mainland Coast. J. Fish. Res. Bd. Canada, 1816): 907-999. Pickard, G.L., 1967. Some oceanographic characteristics of the larger inlets of southeast Alaska. J. Fish. Res. Bd. Canada, 24(7): 1475-1506. Pickard, G.L., 1971. Some physical oceanographic features of inlets of Chile. J. Fish. Res. Bod. Canada, 28: 1077-1106. Pickard, G.L. and Rodgers, K., 1959. Current measurements in Knight Inlet, British Columbia. J. Fish. Res. Bd. Canada, 16(5): 635-678. Pickard, G.L. and Trites, R.W., 1956. Fresh water transport determination from heat budget with applications to British Columbia Inlets. J. Fish. Res. Bd. Canada, 14(4): 605-616. Rattray, M., jr., 1967. Some aspects of the dynamics of circulation in fjords. Estuaries, edit. G.H. Lauff, pp 52-62. AAAC, Washington, D.C. Rattray, M., jr. and Hansen, D.V., 1965. Gravitational circulation in straits and estuaries. J. Marine Res. 23(2): 104-122. Saelen, O.H., 1967. Some features of the hydrography of Norwegian fjords. Estuaries, edit. G.H. Lauff, pp 63-70. AAAC, Washinton, D.C. Stigebrandt, A., 1976. Vertical diffusion driven by internal waves in a sill fjord. Jour. Phys. Ocean. 6(4): 486-495. Stommel, H., 1951. Recent developments in the study of tidal estuaries, Ref. N o . 51-33, Woods Hole Oceanographic Institution, Woods Hole, Mass. Tabata, S. and Pickard, G.L., 1957. The physical oceanography of Bute Inlet, British Columbia. J. Fish. Res. Bd. Canada, 14(4): 487-520. Tully, J.P., 1949. Oceanography and prediction of pulp mill pollution in Alberni Inlet. Bull. Fish. Res. Bd. Canada, 83, 169 pp. Winther, D.F., 1972. A similarity solution for circulation in stratified fjords. Intern. Symp. on Stratified Flows, Novosibirsk, pp 715-724. Winther, D.F., 1973. A similarity solution for steady state gravitational circulation in fjords. Estuary and coastal marine science, 1: 387-400.

423

P R O P E R T I E S OF THE ROBERT R. LONG MODEL OF ESTUARINE C I R C U L A T I O N IN

FJORDS H.G.

GADE a n d E .

SVENDSEN

G e o p h y s i c a l I n s t i t u t e , U n i v e r s i t y o f B e r g e n (Norway)

ABSTRACT I t i s shown t h a t t h e c i r c u l a t i o n model p r o p o s e d by Long ( 1 9 7 5 ) i m p l i e s t h a t t h e b r a c k i s h water t h i c k n e s s i s completely determined by t h e i n t e r f a c i a l F r o u d e n u m b e r , t h e d r a g r a t i o a n d t h e d e n s i t y d i f f e r e n c e a c r o s s t h e i n t e r f a c e . Near t h e h e a d t h i s r e l a t i o n s h i p approaches h*xAp=constant. Numerical s o l u t i o n s t o t h e s e t o f d i f f e r e n t i a l e q u a t i o n s a r e e s t a b l i s h e d t o show t h e b e h a v i o u r o f t h e f l o w o f t h e b r a c k i s h l a y e r . The s o l u t i o n s a r e i l l u s t r a t e d i n t e r m s of i n t e g r a l curves r e f e r r i n g t o s e l e c t e d values o f l a y e r t h i c k n e s s , i n i t i a l i n t e r f a c i a l Froude number, d r a g r a t i o and growth f a c t o r of t h e width. INTRODUCTION

The p r o b l e m o f e s t u a r i n e c i r c u l a t i o n i n f j o r d s h a s b e e n d e a l t w i t h i n t h e l i t e r a t u r e a l o n g two l i n e s , o n e p e r t a i n i n g t o c o n t i n u o u s density distributions, the other requiring layers.

v e r t i c a l l y homogeneous

Both a p p r o a c h e s h a v e i n p r i n c i p l e i n v o l v e d i n d u c e d h o r i z o n t a l

a d v e c t i o n as a c o n s e q u e n c e o f m i x i n g , m a i n l y v e r t i c a l , a n d t h e r e s u l t s h a v e d e p e n d e d h e a v i l y upon t h e z h o i c e o f a s s u m p t i o n s made w i t h r e g a r d t o t h e m a g n i t u d e of t h e m i x i n g i n v o l v e d . The model p r e s e n t e d by Long ( 1 9 7 5 ) i s o f t h e l a y e r e d k i n d .

It

i n v o l v e s two v e r t i c a l l y homogeneous l a y e r s c o r r e s p o n d i n g t o a b r a c k i s k t o p l a y e r a n d a s e a w a t e r body b e l o w .

The d y n a m i c s f o r t h e two-

d i m e n s i o n a l c a s e h a s b e e n s t u d i e d by Stommel ( 1 9 5 1 ) a n d Stommel a n d

F a r m e r ( l 9 5 2 ) , b u t t h e i r r e s u l t s a r e i n c o m p l e t e as l i t t l e was known a t t h a t t i m e o f t h e e x c h a n g e mechanisms t o b e e n c o u n t e r e d i n t h e system. I n h i s model L,ong makes u s e o f r e s u l t s o b t a i n e d from l a b o r a t o r y e x p e r i m e n t s oE m i x i n g a c r o s s a n i n t e r f a c e ( K a t o a n d P h i l l i p s 1 9 6 9 , Moore and Long 1 9 7 1 ) . From t h e s e e x p e r i m e n t s it w a s p o s s i b l e t o s p e c i f y an entrainment v e l o c i t y p r o p o r t i o n a l t o t h e cube o f t h e r m s t u r b u l e n t v e l o c i t y a n d i n v e r s e l y p r o p o r t i o n a l t o t h e t h i c k n e s s of

424 t h e u p p e r l a y e r a n d t h e d e n s i t y jump a c r o s s t h e i n t e r f a c e . I n t h e integrations t h e r m s turDulent velocity applicable t o t h e brackish l a y e r i s assumed c o n s t a n t i n t i m e a n d s p a c e . D e p a r t i n g f r o m e s s e n t i a l l y t h e same c o n d i t i o n s a s u s e d b y Stommel a n d F a r m e r (1952),

and w i t h t h e i n c l u s i o n o f t h e e n t r a i n m e n t

c o n d i t i o n Long p r o c e e d s t o e s t a b l i s h a n a l y t i c a l s o l u t i o n s for t h e c a s e o f a f j o r d of c o n s t a n t w i d t h .

S i m i l a r l y , for t h e case o f

v a r y i n g w i d t h , t h e s y s t e m i s r e d u c e d to a s e t o f t w o p a r t i a l d i f f e r e n -

t i a l e q u a t i o n s f r o m which t h e p r o p e r t i e s o f t h e f l o w c a n be o b t a i n e d by n u m e r i c a l m e t h o d s . The s o l u t i o n s d e p e n d o n t h e c h o i c e o f a n o u t e r b o u n d a r y c o n d i t i o n w h i c h i s t a k e n t o b e a t t h e mouth o f t h e f j o r d . H e r e , t h e f l o w i s a s s u m e d t o become c r i t i c d l , a c o n d i t i o n c o n s i s t e n t w i t h t h e r e s u l t s f r o m t h e t a n k e x p e r i m e n t s by Stommel a n d F a r m e r ( 1 9 5 3 ) .

By t h i s

condition t h e s o l u t i o n i n t h e i n t e r i o r i s completely determined. I n d e r i v i n g t h e a n a l y t i c a l s o l u t i o n s Long i s f o r c e d t o make r a t h e r c o m p l i c a t e d s u b s t i t u t i o n s which r e n d e r t h e d i s c u s s i o n d i f f i c u l t t o f o l l o w . A l s o , some o f t h e v e r y s i m p l e p r o p e r t i e s o f t h e model a r e

not i m m e d i a t e l y a p p a r e n t . I n t h e p r e s e n t p a p e r a n e f f o r t i s b e i n g made t o e l u c i d a t e some o f t h e v e r y b a s i c p r o p e r t i e s o f t h e m o d e l a n d f u r t h e r m o r e g i v e e x s n p l e s o f s o l u t i o n s w h i c h h a v e b e e n o b t a i n e d by n u m e r i c a l i n t e g r a t i o n . I n d o i n g s o i t i s h o p e d t o make t h e model more r e a d i l y a v a i l a b l e as a means t o t h e s t u d y o f t h e b e h a v i o u r o f two-layered f j o r d s .

THEORY

In t h e f o l l o w i n g a b r i e f o u t l i n e o f t h e t h e o r y w i l l b e g i v e n . T o r d e t a i l s o f t h e d e r i v a t i o n s t h e r e a d e r is r e f e r r e d t o t h e p a p e r s

by Stommel dnd F a r m e r ( 1 9 5 2 1 , Long ( 1 9 7 5 ) a n d S t i g e b r a n d t ( 1 9 7 6 ) . However, a f e w a d d i t i o n a l r e m a r k s o f t h e m o d e l a r e i n c l u d e d . ‘The f j o r d i s t r e a t e d a s a d e e p c h a n n e l in w h i c h t h e c i r c u l a t i o n

i s e s s e n t i a l l y two-dimensional.

The c h a n n e l i s , h o w e v e r , a l l o w e d

t o var’y i n w i d t h . The f l o w i n t h e u p p e r l a y e r is a s s u m e d t o b e s u f f i c i e n t t u r b u l e n t t o e r o d e t h e i n t e r f a c e w i t h t h e sea w a t e r b e l o w . The s e a w a t e r t h u s e n t r a i n e d i n t o t h e b r a c k i s h l a y e r is assumed

t o be i n s t a n t l y mixed v e r t i c a l l y s o t h a t t h e u p p e r l a y e r

r e m a i n s v e r t i c a l l y homogeneous. C o n t i n u i t y i s m a i n t a i n e d by a r e t u r n flow i n t h e s e a w a t e r l a y e r . A s t h e m i x i n g i s assumed t o b e u n i d i r e c t i o n a l , t h e l o w e r l a y e r i s u n a f f e c t e d by t h e b r a c k i s h f l o w a b o v e a n d r e m a i n s c o m p l e t e l y hornogeneous.

The l o w e r l a y e r i s f u r t h e r m o r e a s s u m e d t o b e i n f i n i t e l y

425 d e e p s o t h a t t h e b a r o t r o p i c f l o w becomes i n f i n i t e s i m a l .

Velocities

and a c c e l e r a t i o n s i n t h e lower l a y e r a r e a c c o r d i n g l y n e g l e c t e d . The d y n a m i c a l e q u a t i o n s d e r i v e f r o m v e r t i c a l i n t e g r a t i o n of t h e momentum e q u a t i o n s f o r e a c h l a y e r . e n c o u n t e r e d i n t h e er:trainment,

In addition t o the resistance

i n t e r c a c i a l boundary f r i c t i o n i s

e x p e c t e d t o b e o f some i m p o r t a n c e a n d i s r e t a i n e d i n t h e m o d e l .

Shear

s t r e s s from wind a c t i o n a t t h e s u r f a c e i s n o t a s p e c i f i c f e a t u r e o f t h e m o d e l , a l t h o u g h wind i s c o n s i d e r e d o f p r i m e i m p o r t a n c e i n maint a i n i n g t h e turbulence of t h e upper l a y e r . Tides and e f f e c t s from r o t a t i o n o f t h e e a r t h are n e g l e c t e d .

/ b

Fig.

-t

X

1. S c h e m a t i c d i a g r a m o f model c o n d i t i o n s a n d n o t a t i o n e m p l o y e d .

_In t h e model t h e l o w e r l a y e r i s a s s u m e d t o b e i n f i n i t e l y d e e p . With r e f e r e n c e t o t h e n o t a t i o n i n F i g .

1 t h e v e r t i c a l l y and

l a t e r a l l y i n t e g r a t e d momentum e q u a t i o n s f o r s t a t i o n a r y f l o w a r e

w h e r e u i s t h e mean v e l o c i t y o f t h e u p p e r l a y e r a n d W t h e w i d t h o f t h e c h a n n e l . Whenever p o s s i b l e ,

indexes r e f e r r i n g t o t h e upper

426

l a y e r h a v e b e e n l e f t o u t . H e r e , a c o e f f i c i e n t y a p p e a r s as a r e s u l t o f r e p l a c i n g t h e v e r t i c a l a v e r a g e e n e r g y by t h e k i n e t i c e n e r g y o f t h e mean f l o w . The q u a n t i t y y d e p e n d s upon t h e d i s t r i b u t i o n o f v e l o c i t y i n t h e l a y e r . A c c o r d i n g t o Long t h e v a l u e o f y l i e s i n t h e range of 1 t o 2 .

I n t h e subsequent computations t h e value 1 . 3 has

been u s e d c o n s i s t e n t l y . Similarly, the vertically integrated continuity

equation

becomes

Furthermore, an e n t r a i n m e n t r e l a t i o n o f t h e form

i s u s e d , where on i s t h e r m s t u r b u l e n t v e l o c i t y o f t h e u p p e r l a y e r , a n d k a c o n s t a n t . The p r o b l e m i s t h u s d e s c r i b e d by t h e f o u r p a r t i a l d i f f e r e n t i a l e q u a t i o n s above. The s l o p e o f t h e f r e e s u r f a c e c a n e a s i l y b e e l i m i n a t e d f r o m E q n . ( l ) by means o f Eqn.

Eqn.

(2), l e a d i n g t o t h e f o l l o w i n g f o r m

( 3 ) , ( 4 ) and ( 5 ) c o n s t i t u t e t h e r e d u c e d s e t of e q u a t i o n s w e

s h a l l use. P R O P E R T I E S OF THE SOLUTIONS A s Long (1975) h a s shown E q n s .

( 3 ) , (4) a n d (5) c a n b e i n t e g r a t e d

by a s u i t a b l e c h a n g e o f v a r i a b l e s . I t i s , h o w e v e r , t e m p t i n g t o d e r i v e some o f t h e p r o p e r t i e s o f t h e s o l u t i o n s d i r e c t l y f r o m t h e basic equations. Consider f i r s t t h e c a s e of a f j o r d w i t h c o n s t a n t width and w i t h no s h e a r s t r e s s b e i n g t r a n s m i t t e d t o t h e l o w e r l a y e r . Near t h e h e a d o f t h e f j o r d t h e v e l o c i t i e s a r e o f t e n q u i t e s m a l l , making i t p o s s i b l e t o n e g l e c t t h e l e f t h a n d s i d e o f Eqn.

(5).

In t h i s case

t h e e q u a t i o n can be i n t e g r a t e d d i r e c t l y t o y i e l d t h e fundamenta relationship ; i 2 - ~ p=

c

= constant,

g o v e r n i n g t h e i n i t i a l , o r i n n e r form o f t h e s o l u t i o n s . Thus, as

6)

427

t h e d e n s i t y d i f f e r e n c e g r a d u a l l y d i m i n i s h e s as t h e b r a c k i s h w a t e r becomes more s a l t y , t h e t h i c k n e s s o f t h e b r a c k i s h l a y e r grows accordingly. The c o r r e s p o n d i n g f o r m o f t h e i n t e r f a c i a l F r o u d e number

i n d i c a t e s t h a t a t l e a s t i n i t i a l l y t h e Froude

number w i l l grow as

u 2 h . I f , h o w e v e r , a t some l o c a t i o n i n t h e f j o r d dh/dx=O Eqn.

(5)

w i l l reduce t o

which t o g e t h e r w i t h t h e e q u a t i o n o f c o n t i n u i t y

permit t h e e l i m i n a t i o n o f d uh/dx, l e a v i n g

which i s e q u i v a l e n t t o t h e c o n d i t i o n

t h a t t h e i n t e r f a c i a l Froude

number b e l / 4 .

I t i s c o n s i s t e n t w i t h t h i s r e s u l t t h a t f o r v a l u e s o f F2 s m a l l e r than 1 / 4 ,

dh/dx i s p o s i t i v e ,

a n d for v a l u e s o f F'

larger than l / 4 ,

d h / d x becomes n e g a t i v e . A c t u a l c o m p u t a t i o n s o f t h e c o m p l e t e s o l u t i o n s

show t h a t f o r a g i v e n i n n e r b o u n d a r y c o n d i t i o n t h e i n t e r f a c i a l F r o u d e number w i l l grow

m o n o t o n i c a l l y u n t i l t h e v a l u e 1/4 i s

r e a c h e d a f t e r w h i c h i t v e r y r a p i d l y w i l l r e a c h t h e v a l u e 1. A t t h i s p o i n t t h e f l o w becomes c r i t i c a l . I n h i s d i s c u s s i o n o f t h e s o l u t i o n s Long assumed t h e c r i t i c a l flow c o n d i t i o n t o be a s s o c i a t e d w i t h t h e opening o f t h e f j o r d towards t h e s e a . I n t h i s way t h e s u d d e n w i d e n i n g a t t h e mouth would s e r v e

a s a h y d r a u l i c c o n t r o l . By a p p l y i n g t h i s c o n d i t i o n t h e s o l u t i o n s would b e c o m p l e t e l y d e t e r m i n e d t h r o u g h o u t t h e fjord.

It i s n o t always p o s s i b l e t o i d e n t i f y such a sudden widening of a f j o r d as i t o p e n s up t o w a r d s t h e s e a . N e i t h e r i s i t a l w a y s

p o s s i b l e t o r e c o g n i z e two w e l l d e f i n e d l a y e r s t h r o u g h o u t t h e e n t i r e l e n g t h o f t h e fjord. Most o f t e n t h e t r a n s i t i o n b e t w e e n t h e l a y e r s widens toward t h e s e a , g r a d u a l l y e r o d i n g i n t o t h e lower l a y e r and

also e v e n t u a l l y e x t e n d i n g a l l t h e way t o t h e s u r f a c e . I n t h i s c a s e

428

t h e concept of a two-layer

s y s t e m may w e l l b e a p p l i c a b l e t o t h e

i n n e r p a r t s o f t h e f j o r d , b u t w i l l f a i l in t h e o u t e r r e a c h e s . For t h i s r e a s o n i t may b e more r e w a r d i n g t o d i s r e g a r d t h e o u t e r b o u n d a r y condi-tion and r a t h e r d e v e l o p e t h e s o l u t i o n s from t h e i n n e r end. Long f o u n d a n a l y t i c a l s o l u t i o n s t o t h e p r o b l e m f o r t h e c a s e of uniform width.

I n h i s a n a l y s i s i t i s shown t h a t t h e s e s o l u t i o n s

depend upon t w o n o n - d i m e n s i o n a l p a r a m e t e r s m a n d s , b o t h i n v o l v i n g t h e r m s t u r b u l e n t v e l o c i t y . I t i s p o s s i b l e t o show t h a t a l t h o u g h i n p l i c i t l y , t h e s o l u t i o n s a r e more r e a d i l y d i s c u s s e d i n t e r m s o f two o t h e r r e l a t e d p a r a m e t e r s , Fo a n d C 2 / C 1 ,

where t h e f o r m e r s t a n d s

for t h e i n t e r f a c i a l F r o u d e number a t t h e h e a d o f t h e f j o r d , a n d t h e l a t t e r being t h e r a t i o

0:-

the drag coefficient

(C2)

t o the

r e l a t i v e e n t r a i n m e n t v e l o c i t y Cl=2we/uo a t t h e head o f t h e f j o r d . The s o l u t t o n s do dlso d e p e n d upcn t h e v a l u e o f t h e e n t r a i n m e n t p a r a m e t e r O,, b u t i n s u c h a way t h a t v a r i a t i o n o f t h i s p a r a m e t e r c a u s e s t h e s o l u t i o n s t o c o n t r a c t or e x p a n d a l o n g t h e x - a x i s .

Thus,

i f C,x i s u s e d a s h o r i z o n t a l a x i s , a l l s o l u t i o n s c a n b e e x p r e s s e d i n t e r m s of t h e two p a r a m e t e r s Fo a n d C 2 / C 1 . The e n t r a i n m e n t p a r a m e t e r C 1 d o e s n o t e n t e r a t a l l i n t o r e l a t i o n s h i p s s u c h as t h a t between t h e t h i c k n e s s and t h e d e n s i t y

o f t h e u p p e r l a y e r . F o r t h e case o f c o n s t a n t w i d t h t h e s o l u t i o n s a r e p a r t i c u l a r l y s i m p l e . By a s u i t a b l e s u b s t i t u t i o n L o n g ‘ s E q n . ( 2 1 ) (Long 1975) t r a n s f o r m s t o

s+l

where s = l + ( 1 + 2 C 2 / C 1 shown i n F i g s .

)-’.

2 and 3 .

The g e n e r a l b e h a v i o u r o f t h e s e s o l u t i o n s i s In Fig. 2 t h e density versus thickness relation

s h i p tor t h e c d s e o f n o b o u n d a r y f r i c t i o n (C2.0.

s.2)

i s shown f o r

s e l e c t e d F r o u d e numbers a t t h e h e a d o f t h e f j o r d (Fo), a l l w i t h i n t h e range o f naturdlly occurring values.

The r e s u l t s a r e drawn i n

a double l o g a r i t h m i c diagram which i s p a r t i c u l a r l y s u i t e d f o r comparison w i t h observed f e a t u r e s

i n fjords.

The d i a g r a m e x h i b i t s two s t r i k i n g f e a t u r e s . F i r s t l y , i t i s s e e n t h a t a l l l i n e s d e p a r t from o r i g i n w i t h t h e s l o p e w i t h t h e r e s u l t o f Eqn.

-4

i n accordance

6 . The o t h e r f e a t u r e i s t h a t t h e t h i c k n e s s

r e a c h e s a maximum a t s o m e p o i n t b e y o n d w h i c h t h e t h i c k n e s s o f t h e l a y e r d r o p s d r a s t i c a l l y . T h e r e i s a l s o a t e n d e n c y for t h e l i n e s t o c u r v e b a c k w a r d i m p l y i n g l o s s o f d e n s i t y . T h i s c a n n o t b e a t t a i n e d by t h e p h y s i c a l mechanisms i n v o l v e d s o t h a t t h e m a t h e m a t i c a l s o l u t i o n c a n n o t b e c a r r i e d behond t h e p o i n t o f maximum d e n s i t y w h i c h c o r r e s p o n d : t o t h e p o i n t w h e r e t h e i n t e r f a c i a l F r o u d e number r e a c h e s o n e .

429

In h/h.

Fig.

2.

Logarithmic p r e s e n t a t i o n o f t h i c k n e s s o f upper l a y e r

r e l a t i v e t o head c o n d i t i o n s versus r e l a t i v e d e n s i t y d i f f e r e n c e between t h e l a y e r s .

I n t e g r a l c u r v e s f o r t h e c o n s t a n t w i d t h , non-

f r i c t i o n a l case f o r i n d i c a t e d i n i t i a l ( i n n e r e n d ) i n t e r f a c i a l F r o u d e number s q u a r e d .

In

F:: 0.0017

1.5 C>/CI

.o in

0.5 C>/C, = 1

- 30

Fig.

3.

- 2.5

- 2.0

-1.5

- 1.0

- 0.5

..

Logarithmic p r e s e n t a t i o n o f t h i c k n e s s of upper l a y e r r e l a t i v e

t o head c o n d i t i o n s v e r s u s r e l a t i v e d e n s i t y d i f f e r e n c e between t h e l a y e r s . C o n s t a n t w i d t h i n t e g r a l c u r v e s for i n d i c a t e d i n t e r f a c i a l F r o u d e number s q u a r e d a n d d r a g r a t i o s .

430 The i n f l u e n c e o f boundary f r i c t i o n on t h e t h i c k n e s s v e r s u s

d e n s i t y r e l a t i o n s h i p is i l l u s t r a t e d in F i g .

3 . F o r t h i s purpose an

i n i t i d 1 F r o u d e number i s c h o s e n t o b e t y p i c a l o f o n e o f t h e m a j o r Norwegian w e s t z o a s t f j o r d s , t h e S o g n e f j o r d , w h e r e F i w a s e s t i m a t e d at

The f a m i l y o f c u r v e s c o r r e s p o n d s t o t h e v a r i o u s v a l u e s

.0011.

of the d r a g r a t i o C 2 / C 1 ,

t h e uppermost curve r e p r e s e n t i n g t h e

irictionless c a s e . i t f o l l o w s from F i g .

tend:

3 that i n t e r f a c i a l b o u n d a r y f r i c t i o n

t o l i m i t t h e v a r i a t i o n of b o t h t h i c k n e s s a n d d e n s i t y g r o w t h

of t h e u p p e r l d y e r . For e x t r e m e l a r g e v a l u e s o f b o u n d a r y f r i c t i o n t h e u p p e r l a y e r may n o t grow a t a l l , b u t r e d u c e s t e a d i l y . NIJMEllICAL I N T E G R A T I O N

The a c t u a l f o r m o f t h e s o l u t i o n s t h r o u g h o u t t h e f j o r d i s .;pec.i.fied

by a c o u p l e o f

partial^ f i r s t o r d e r e q u a t i o n s w h i c h c a n n o t

r e a d . i l y b e i n t e g r a t e d e x c e p t by n u m e r i c a l m e t h o d s . T h i s r e n d e r s t h e

d i : : c u s s j o n somewhat cumbersome a n d o n e i s o b l i g e d t o go t h r o u g h a s e r i e s oi c a s e s w i t h s e l e c t e d p a r a m e t e r v a l u e s .

'The c o m p l e t e e q u a t i o n s a r e g i v e n b y Long (1975, E q n s .

( 3 5 ) and

(36)).In c o n c o r d a n c e w i t h t h e n o t a t i o n u s e d h e r e t h e s e e q u a t i o n s car1 be w r i t t e n

where W is t h e w i d t h o f t h e fjord,Wo b e i n g t h e v a l u e a t t h e h e a d .

lor the c a s e o f c o n s t a n t w i d t h C l x c a n b e e l i m i n a t e d f r o m L,qri:i.

( 9 ) arid ( 1 0 ) m a k i n g it p o s s i b l e t o i n t e g r a t e t h e s e t d i r e c t l y

to yield I

F' h=K. ( i + 1 t 2 - -CZ ; ) F 2 ) "

(11)

Cl

wliere

c* It-4 a=- 3 c ,

C C, and K is a c o n s t a n t o f i n t e g r a t i o n . l t 2 2

Eqns.

(9) a n d ( 1 0 ) h a v e b e e n i n t e g r a t e d by n u m e r i c a l m e t h o d s

431 f o r a range of parameter values.

Beginning from t h e i n n e r end t h e

t h i c k n e s s , d e n s i t y 2nd mean v e l o c i t y o f t h e u p p e r l a y e r a r e d e t e r m i n e d ds f u n c t i o n s o f l o c a t i o n i n t h e f j o r d , i . e . t h e h o r i z o n t a l c o o r d i n a t e .

The s o l u t i o n s a r e d e v e l o p e d f o r s e l e c t e d v a l u e s o f i n i t i a l F r o u d e number, i n n e r e n d b r a c k i s h w a t e r t h i c k n e s s , d r a g . r a t i o C 2 / C 1 a n d t h e r a t e o f c h a n g e o f w i d t h of t h e f j o r d . Corresponding t o a r e f e r e n c e v a l u e o f t h e i n n e r end Froude number s q u a r e d F 2 = .002 a n d a d r a g r a t i o C 2 / C l = 1 0 ,t h e s o l u t i o n s f o r c o n s t a n t w i d t h a r e p r e s e n t e d i n F i g s . 4a, b a n d c f o r i n d i c a t e d i n n e r e n d t h i c k n e s s e s . A s a l s o i n d i c a t e d by Eqn. c u r v e s shown i n F i g .

(9) t h e f a m i l y o f

a l l b e g i n w i t h t h e s l o p e (4y)-'.

4a

S i m i l a r l y , f o r t h e case of g i v e n i n n e r end t h i c k n e s s o f t h e b r a c k i s h l a y e r h,=4m,

t h e s o l u t i o n s f o r i n d i c a t e d r a n g e of F2 a r e 0

shown i n F i g s . S a , b a n d c , t h e o t h e r p a r a m e t e r s b e i n g t h e s a m e .

In F i g s . 6 a , b a n d c i s shown t h e i n f l u e n c e o f v a r y i n g t h e driag r a t i o C 2 / C 1 . The o t h e r p a r a m e t e r s a r e t h e same as i n t h e previous computations. F i n a l l y , t h e case of a f j o r d o f varying width is i l l u s t r a t e d i n Figs.

7 a , b and c .

S o l u t i o n s are shown f o r s e l e c t e d v a l u e s o f

the c o n s t a n t g r a d i e n t o f w i d t h b=dW/dC,x. A P P L I C A T I O N TO OBSERVATIONS

R e l i a b l e o b s e r v a t i o n s from f j o r d s s u i t a b l e f o r t e s t i n g t h e model a r e r a t h e r s c a r c e . A s a g e n e r a l r u l e , t h e f j o r d s h o u l d h a v e a " n i c e " g e o m e t r y , t h e w i d t h b e i n g n e a r c o n s t a n t or c h a n g i n g o n l y

g r a d u a l l y w i t h d i s t a n c e , and t h e l o n g i t u d i n a l a x i s € r e e o f s t r o n g curvatures. Immediately p r i o r t o and a t t h e t i m e o f observation t h e w i n d s i n t h e a r e a s h o u l d b e w e a k t o a l l o w f o r a dynamic b a l a n c e f r e e of s u r f a c e s t r e s s e s . Furthermore, a long term n e a r steady s t a t e

condition should prevail. A s i n g l e s e t of d a t a i s a v a i l a b l e from t h e S o g n e f j o r d , one

oi t h e l a r g e r Norwegian w e s t c o a s t f j o r d s . The main f j o r d , some 2 0 0 km l o n g , i s f a i r l y u n i f o r m i n w i d t h ( a = .02, b = 1 . 6 x 1 0 4 ) , b u t

has a number o f m i n o r b r a n c h e s . A l m o s t a l l o f t h e f r e s h w a t e r d i s c h a r g e i n t o t h e f j o r d occurs i n t h e i n n e r reaches of t h e system, o u t s i d e which t h e f j o r d c a n b e t r e a t e d as a s i n g l e c h a n n e l . C o r r e s p o n d i n g t o a n e s t i m a t e d v a l u e o f F 2 = .0017 t h e o b s e r v e d 0

a v e r a g e d e n s i t i e s a n d l a y e r t h i c k n e s s e s f r o m 20-21 J u n e 1971 a r e p l o t t e d on F i g .

3 . A l t h o u g h t h e f i t is n o t p a r t i c u l a r l y g o o d ,

g e n e r a l form o f t h e s o l u t i o n s i s a c c e p t a b l e w i t h t h e b e s t f i t being obtained with a drag r a t i o near 3 .

the

432

4

ho

1

12

8 I

1

16

12 I

I

1

I

20 I

I

I

32 m

28

24 I

I

1

I

1

- cl

-

8

4

"0 m/s

0.24

I

I

I

I

I

16

12 I

I

I

1

20 1

1

'

24 I

'

y r nC,X

28 1

h:, 5m

-

0.16 -

4 b

24

28

-

32 rn

Cl

-

4 c

Fig.

4.

~

C o n s t a n t w i d t h i n t e g r a l c u r v e s for C2/C1=10 a n d F 02 = .002

S o l u t i o n s o f t h i c k n e s s ( F i g . 4a), mean v e l o c i t y ( F i g . 4b), a n d d e n s i t y ( F i g . 4c) v e r s u s l o n g i t u d i n a l d i s t a n c e .

c

Fig.

F, =0.05 2,

I

" I

I

5 . C o n s t a n t w i d t h i n t e g r a l c u r v e s for Cp/C1=10 a n d ho=4m.

S o l u t i o n s of t h i c k n e s s ( F i g . 5 a ) , mean v e l o c i t y ( F i g . 5 b ) , a n d d e n s i t y ( F i g . 5 c ) v e r s u s l o n g i t u d i n a l d i s t a n c e for i n d i c a t e d i n n e r e n d i n t e r f a c i a l F r o u d e number s q u a r e d .

434

4

Aeo 3

kg/m

1

1

8 1

1

12 1

1

16 1

1

24

20 1

1

1

28

32m

ClX

1

-

-

-

6 c

-

Fig.

6.

C o n s t a n t w i d t h i n t e g r a l curves f o r ho=4m a n d F 2 = .002. S o l u t i o n s

of t h i c k n e s s

0

( F i g . 6 a ) , mean v e l o c i t y ( F i g . 6 b ) , a n d d e n s i t y ( F i g .

6c) v e r s u s l o n g i t u d i n a l d i s t a n c e for i n d i c a t e d d r a g r a t i o s .

435

i

0

2

-

70

b=10

,

Fig.

-

,

I

1

I

I

,

I

-

7. E f f e c t o f a c o n s t a n t w i d t h g r a d i e n t h o n t h e v a r i a t i o n o f

t h i c k n e s s ( F i g . 7 a ) , m e a n v e l o c i t y ( F i g . 7b), a n d d e n s i t y ( F i g . 7c) versus l o n g i t u d i n a l distance. a n d F 2 = ,002. 0

I n t e g r a l c u r v e s for C,/C,=lO, h o = 4 m ,

436 Similarly, with t h e observations plotted i n t h e thickness distance r e l a t i o n s h i p i n Fig.

6a,

it i s p o s s i b l e t o o b t a i n a

r e a s o n a b l e f i t w i t h a n e n t r a i n m e n t p a r a m e t e r C 1 = 8 ~ 1 0 ,- ~ again i n d i c a t i n g a d r a g r a t i o i n t h e r a n g e of 1 - 1 0 ,

the higher values

p e r t a i n i n g t o t h e middle and i n n e r p a r t s o f t h e f j o r d .

DISCUSSION A t t h e o u t s e t o f t h i s s t u d y it w a s n o t obvious t o t h e a u t h o r

t h a t i n t e r f a c i a l boundary f r i c t i o n w a s e s s e n t i a l t o t h e problem.

In h i s d i s c u s s i o n Long (1975) h a d s u g g e s t e d a d r a g c o e f f i c i e n t of a m a g n i t u d e of t h e o r d e r of o n e , a v a l u e w h i c h would l e a d t o a n

o v e r w h e l m i n g i n f l u e n c e o f t h e b o u n d a r y f r i c t i o n . Such a h i g h v a l u e would a l s o b e i n c o n s i s t e n t w i t h t h a t no a c c o u n t w a s t a k e n o f e f f e c t s o€ s h e a r s t r e s s e s being t r a n s m i t t e d t o t h e lower l a y e r

(Bo P e d e r s e n , p e r s o n a l c o m m u n i c a t i o n ) . The d a t a f r o m S o g n e f j o r d p r e s e n t e d i n t h i s p a p e r i n d i c a t e a n e n t r a i n m e n t p a r a m e t e r of t h e o r d e r o f magnitude o f

where-

a s t h e i n t e r f a c i a l b o u n d a r y f r i c t i o n a p p e a r s t o b e d e t e r m i n e d by a drag coefficient of t h e order

T h e s e v a l u e s may, h o w e v e r ,

be m a r g i n a l a n d t h u s n o t t y p i c a l f o r t h e m a j o r i t y o f f j o r d s . Although q u i t e simple t o o b t a i n , and u s u a l l y w i t h a f a i r d e g r e e o f a c c u r a c y , measurements o f u p p e r l a y e r t h i c k n e s s e s a r e n e v e r t h e l e s s p o o r l y s u i t e d for t e s t i n g t h e v a l i d i t y o f a c i r c u l a t i o n model. A s t y p i c a l f o r h o r w e g i a n w e s t c o a s t f j o r d s i t a p p e a r s t h a t t h e e n t r a i n m e n t o f sea w a t e r i n t o t h e b r a c k i s h l a y e r d e p e n d s l a r g e l y on r h e a c t i o n o f w i n d . F o r t h i s r e a s o n i t would s e e m r a t h e r m e a n i n g l e s s t o s e e k o u t p e r i o d s o f c a l m for c a r r y i n g o u t o b s e r v a t i o n a l p r o g r a m s . On t h e o t h e r h a n d , w i n d s t e n d t o c o n v e r g e or d i v e r g e t h e w a t e r masses, t h e r e b y s e r i o u s l y a f f e c t i n g t h e d e p t h o f t h e pycnocline.

The c o n f l i c t c a n p o s s i b l y b e a v o i d e d by f o c u s s i n g t h e

a t t e n t i o n on t h e c h a n g e o f d e n s i t y o f t h e u p p e r l a y e r a l o n g t h e fjord. A n e f f o r t i n t h i s d i r e c t i o n i s shown i n F i g .

observed d e n s i t i e s

6c, w h e r e t h e

from t h e S o g n e f j o r d a r e p l o t t e d a g a i n s t t h e

l o n g i t u d i n a l a x i s C , x . H e r e , C 1 i s m a i n t a i n e d 8 ~ 1 0 - ~a ,v a l u e which a p p e a r s t o be t o o h i g h i n t h e i n n e r r e a c h e s , a n d t o o s m a l l f u r t h e r o u t i n t h e f j o r d . T h i s i s a c t u a l l y no s u r p r i s e , as t h e i n n e r r e a c h e s of t h e f j o r d are c h a r a c t e r i z e d

by

g e n e r a l l y much

l i g t h e r w i n d s t h a n f u r t h e r o u t . The same a r g u m e n t h o l d s f o r t i d a l a n d o t h e r e x t e r n a l l y i n d u c e d d i s t u r b a n c e s . The e f f e c t s o f t h e

437 1.atter i n t h i s r e s p e c t a r e , however, c a t i o n i n Fig. ratio

not ascertained.

The a p p l i -

6 c g i v e s n o i n d i c a t i o n of t h e v a l u e of t h e C 2 / C 1

.

REFERENCES

Kato, €3. a n d P h i l l i p s , O . M . ,

1 9 6 9 . On t h e p e n e t r a t i o n o f a t u r b u l e n t

l a y e r i n t o a s t r a t i f i e d f l u i d . J . F l u i d Mech., Long, R . R . ,

1975.

37:643-655.

C i r c u l a t i o n s and d e n s i t y d i s t r i b u t i o n s i n a deep,

strongly s t r a t i f i e d , two-layer

estuary.

J. F l u i d M e c h . , 7 1 :

529-540. Moore, M . J .

a n d Long, R . R . ,

1 9 7 1 . An e x p e r i m e n t a l i n v e s t i g a t i o n of

turbulen-i- s t r a t i f i e d s h e a r i n g f l o w .

J . F l u i d Mech.,

49:635-

655. Stigebrandt, A.,

1976. Stasjonar tvzlagerstr@mning i e s t u a r i e r .

( I n S w e d i s h ) . R i v e r and Harbour L a b o r a t o r y , Trondheim. Stommel, H . ,

1 9 5 1 . Recent developments i n t h e study o f t i d a l

e s t u a r i e s . WHOI Rep. No. Stommel, H .

and Farmer,

H.G.,

51-33. 1 9 5 2 . On t h e n a t u r e o f e s t u a r i n e

c i r c u l a t i o n . WHOI R e p . No. 5 2 - 8 8 . S t o m e l , I L a n d F a r m e r , H.G., 1 9 5 2 . A b r u p t c h a n g e i n w i d t h i n twol a y e r o p e n f l o w . J. M a r . Stommel, H .

Res.,

11(2):205-214.

a n d F a r m e r , H.G., 1 9 5 3 . C o n t r o l of s a l i n i t y i n a n

e s t u a r y by a t r a n s i t i o n .

J.Mar.Res.,

12(1):13-20.

This page intentionally left blank This Page Intentionally Left Blank

439

WIND-DRIVEN CIRCULATION IN A FJORD 13. Svendsen

Geophysical Institute, Univerqsity of Bergen (Norway) and

9.O.R.Y. Thompson CSIRO Divison of Atmospheric Physics, Aspendale (Australia)

AH STRAC 7' Currents, temperature, salinity, wind, runoff and water level

were observedfora month in the J d s e n f j o r d of Southern

Norway. Tide gauges and currents show little semi-diurnal tide. There is a s t r o n g diurnal signal in the upper 20 m, which a linear There is a week-long event in which the entire water-mass above the sill is flushed out; this is interpreted to be caused by down-welling outside the fjord. The strong stratification near the surface of the fjord greatly modifies the diurnal response of the fjord, but any density-driven mean circulation is at least an order of magnitude smaller than model shows to be caused by the wind.

the wind-driven currents.

(Manuscript submitted to Journal of Physical Oceanograohy. October 1977.1

This page intentionally left blank This Page Intentionally Left Blank

441

SUMMER REPLACEMENT OF DEEP WATER IN BYFJORD, WESTERN NORWAY: MASS EXCHANGE ACROSS THE SILL INDUCED BY COASTAL UPWELLING. Hans B. Helle Geophysical Institute, University of Bergen (Norway)

ABSTRACT From previous published data on replacement 3f deep water in Byfjord (Linde, 1970) and obvious relation i s found to exist between northerly winds along the west coast of Norway during summer and the volume of deep water being replaced. Northerly summer winds induce upwelling of denser water to shallower depths, which then can spill over the sill and intrude into the fjord basin. The magnitude Of replacement appears to be independent of variations in freshwater discharge and climate. Recent observations of current and water properties reveal that the exchange flow across the sill is basicly two-layered: a landward flow in the lower and a seaward flow in the upper layer, with the level of no-net motion located at mid-depth. Short period (2-7 days) variations in the long-shore wind component are well reflected in the exchange flow. Wind speed extremes are followed one to two days later by corresponding extremes in the exchange flow, indicating the approximate response time for the upwelling process. In periods of sustained northerly winds along the coast the inflow attains a saturation speed of about 30 cm/s, which i s slightly less than the critical speed obtained from the mean stratification by assuming an ideal two -layer flow across the sill. INTRODUCTION The fjord under consideration is a larger estuarine system contiguous to the town of Bergen in the western Norway (Fig. 1). Major subdivisions are Byfjord, Herdlafjord, Osterfjord and Sdrfjord with basin depths ranging from 350 m in Byfjord to a maximum of 650 m in Osterfjord. The interconnections between the different basins are quite open, except for the inner part of S@rfjord where a 400 m deep basin is secluded behind a 160 m deep sill.

442

Figure

1 .

Map o f t h e a r e a w i t h l o c a t i o n s o f c u r r e n t meter s t a t i o n (HI and hydrographic s t a t i o n s ( €

- 5)

443

The estuarine system, denoted by Byfjord for abbrevation in the following text, is connected to the adjacent coastal water through a complex network of narrow sounds formed by an archipelago to the west and the north. The sounds are all constricted by sills. The deepest passage is through Hjeltesund at the southwestern entrance to the estuarine system through Hjeltefjord. This narrow passage coincides with the location of a 170 m deep sill. No other sound provides a free passage deeper than 40 m. Hjeltefjord is relative wide, deep (300-500 m) and openly connected to the coastal water at its northmost end where a 300-350 m deep submarine canyon provides a free connection with the deepest portions of the Norwegian Trench. Despite its rather complex topography Byfjord is considered to be a typical Norwegian sill or treshold fjord. The volume of the f j o r d is 47 km3 and its surface covers an area of 55 km2. The drainage basin for the fjord is an area of 3700 km with extraordinarily high precipitation. Annual fresh-water run-off amounts to about 10 km3, which is mainly discharged during the summer months by rivers located in the innermost parts of the fjord. Mean tidal range in Bergen is approximately 0.9 m, predominately of semi-diurnal (M2) type. Owing to it proximity to oceanographers from Bergen (the Bergen Museum and later the University of Bergen) the fjord has been subject to investigations, for various purposes, since the end of last century. There is no doubt the Byfjord is the place where the first "Nansen cast" was taken. Since that the general hydrographic features of Byfjord have been well known, mainly from the long-term serial observations of water properties as treated by Helland-Hansen (1935) and by Linde (1970). The first paper deals with the general hydrography of Byfjord, with special attention to seasonal and long-term variations of the water temperatures related to climatic changes. Linde's report deals with the deep water exchanges, and using a specially developed method he was able to determine the amount of deep water being replaced over a long series of years. Both papers reveal that the bottom and deep water masses of the fjord are almost completely replaced during an extended period every year, with very few exceptions, some time in the period March-August. The replacement events are characterized by increasing oxygen conceiitration in the deep water, usually followed by rising salinity and falling temperature. Hence the new water,

444

which originates from) water masses occurring outside the fjord mouth, has a greater density than the resident water of the fjord. The inflow itself thus constitutes a gravity flow following the appearance of denser water outside the mouth, at sill level or above, than that existing within the fjord basin. As pointed out by Gade (1973) and by Salen (19761, the principal mechanisms governing these exchanges in a sill fjord are linked with the sustained density reduction of the resident water thriugh vertical eddy diffusion and the natural occurring density variations in the adjacent coastal water. Along the west coast of Norway these density variations are clearly linked with the seasonal oscillation of the Norwegian Coastal Current assumably due to the seasonal changes in wind pattern, fresh water discharge from land and possibly influenced by climatic conditions. Which one of these is the governing factor, however, has been subjected to some discussion in the litterature but their relative importance has previously not been properly documented. The main goal of this paper, by using Byfjord as an example of a typical west coast fjord, is to demonstrate the relative importance of the different factors to the deep water exchange, and hence to the seasonal density variation in the near-shore coastal water. Because deep water replacements previously only have been documented at monthly and bimonthly intervals with water properties, little knowledge exists on the inflow process itself. Particular questions being raised are whether the inflow occurs as a slow and steady flow or as a sudden discharge.

In most cases for which data is available, the deep water renewal of the fjord basin is known to be a relatively rapid event, which often is complpted within the course of a few weeks. This relatively short time scale of the process, the large quantities of water involved, as apparent from Linde's results, and the small density difference between the new and original basin water points to the possibility that the exchange flow may be subjected to hydraulic control at the sill.

Some unique observations of deep water exhange

made in Byfjord in 1973 and 1974 will be presented. The result shed some light upon some principal questions related to the exchange process and improve our understanding of the oceanography of the Byfjord and other fjords with similar characteristics.

445

MEAN SEASONAL VARIATIONS I N THE COASTAL AREA The n o r m a l mean s e a s o n a l c h a n g e s i n t h e wind p a t t e r n a l o n g t h e west c o a s t a r e d i s p l a y e d i n t h e v e c t o r p l o t

F i g u r e 2 showing

mean m o n t h l y wind v e c t o r s a t t h r e e d i f f e r e n t l o c a t i o n s a l o n g t h e The m e t e o r o l o g i c a l s t a t i o n s ( l i g h t h o u s e s ) a r e s i t u a t e d

coast.

on i s l a n d s some d i s t a n c e from t h e m a i n l a n d ; U t s i r a a n d Kinn t o t h e s o u t h a n d t h e n o r t h r e s p e c t i v e l y a n d w e l l w i t h i n t h e e d g e s of t h e s t r a i g h t n o r t h - s o u t h r u n n i n g p a r t of t h e c o a s t l i n e .

The m e t e -

o r o l o g i c a l s t a t i o n H e l l e s d y l i e s a p p r o x i m a t e l y midway between

U t s i r a and Kinn a n d j u s t o u t s i d e t h e f j o r d a r e a u n d e r c o n s i d e r a t i o n . The wind v e c t o r s a r e c o n s t r u c t e d on t h e b a s e of ? h e s t a n d a r d n o r m a l s 1931-1960 o f m o n t h l y wind € r e q u e n c i e s a n d mean wind f o r c e ( J o h a n n e s s e n and Hdland,

,

J

'

F

'

M

.

A

'

M

'

J

KlNN

\

:

J

'

A

1969).

. S .O .N .D ,

\ \

'5 HELL I S0Y

0

Fig.

2

1

2.

3 mfsec

3 0 y e a r mean m o n t h l y wind v e c t o r s

of t h r e e d i f f e r e n t c o a s t a l s t a t i o n s .

( 1 9 3 1 -60).

G e n e r a l l y , t h e wind a l o n g t h e w e s t c o a s t i s d o m i n a t e d by winds f r o m s o u t h i n t h e w i n t e r a n d f r o m n o r t h d u r i n g t h e summer months.

Mean s t r e n g t h a n d f r e q u e n c y a r e h i g h e r i n t h e w i n t e r a n d

t h e e x t r e m e wind s i t u a t i o n s u s u a l l y o c c u r i n t h e p e r i o d DecemberFebruary.

The Norwegian h i g h l a n d h a s a t e n d e n c y t o i n f l u e n c e t h e

446

wind in a 30-40 km wide belt out from the coast, and the dominant wind directions are parallel to the coast. There is a marked long-shore gradient in the mean wind field, and also a cross-shore gradient due to friction from the rugged landscape. The water masses occuring outside the coast are characterized by an upper layer of low-salinity water, forming the Norwegian Coastal Current with a mean northward flow, flowing upon a heavier Usually the 3 5 o/oo isohaline is taken as the boundary between coastal and oceanic water, but as it will appear from the 35 year mean isopleths in figure 3 from a near-by coastal station the 34.5 o/oo isohaline coincides with the base of the main pycnocline and hence has more physical significance as far as the near-shore zone is concerned. This water of Atlantic origin.

isohaline follows an almost harmonic varying path, from approximately 100 m depth at the turn of the year to approximately 50 m depth in June through July. The variation of the pycnocline depth quite closely follows the normal mean long-shore wind component. J F M A M J J A S O N D 0

50

150

200

250

300

Fig. 3. 3 5 year mean salinity isopleths from the coastal station Sognesj@en (1935-70).

The typical changes in the distribution of water masses outside the west coast may be best illustrated by the two consecutive salinity sections shown in Figure 10 taken between Feie Island (see Figure 1) and Shetland Islands (U.K.).

The first

441 s e c t i o n , o b t a i n e d d u r i n g t h e f i r s t d a y s o f March 1 9 7 4 , d i s p l a y s a t y p i c a l w i n t e r s i t u a t i o n with t h e h i g h l y s a l i n e and v e r t i c a l l y

w e l l - m i x e d A t l a n t i c s u r f a c e w a t e r o c c u r r i n g on t h e s h e l f a n d t h e shelf-slope.

There i s c l e a r evidence f o r convection o f t h e s h e l f

w a t e r down i n t o t h e d e e p e r p o r t i o n s o f t h e Norwegian T r e n c h , a phenomenon w h i c h i s known t o t a k e p l a c e i n t h e some d e e p e r p a r t o f t h e T r e n c h i n S k a g e r r a k d u r i n g c o l d w i n t e r s , a s d e s c r i b e d by Ljrpen a n d S v a n s s o n ( 1 9 7 2 ) , b u t t h i s more l i k e l y o c c u r s e v e r y w i n t e r i n t h e shallower p a r t o f t h e Trench o f f t h e w e s t c o a s t .

This convection

i s o f c o n s i d e r a b l e i m p o r t a n c e t o t h e f j o r d s b e c a u s e it a s s u r e s h i g h oxygen c o n c e n t r a t i o n i n t h e w a t e r a v a i l a b l e for i n f l o w a t s p r i n g a n d summer t i m e . I n t h e w i n t e r s i t u a t i o n shown, t h e c o a s t a l w a t e r i t s e l f i s confined t o a narrow s t r i p of width 6 0 - 7 0

km a t t h e c o a s t a n d

r e a c h i n g i t s g r e a t e s t d e p t h o f a b o u t 1 8 0 m some d i s t a n c e f r o m t h e coast.

The n e x t s e c t i o n w a s t a k e n a b o u t two months l a t e r f o l l o w i n g

a p e r i o d o g 3 - 4 weeks w i t h s u s t a i n e d n o r t h e r l y w i n d s ( s e e F i g u r e 11).

The d i s t r i b u t i o n o f w a t e r masses h a s e n t i r a l y c h a n g e d ; a t h i n

surface layer (20-30

m t h i c k ) o f c o a s t a l w a t e r i s d i s p l a c e d westward

t o a b o u t 1 8 0 km f r o m t h e c o a s t a n d u p w e l l i n g i s c l e a r l y s e e n t o occur i n a 20-25

km w i d e s t r i p a t t h e c o a s t .

The c h a n g e s a p p a r e n t

f r o m t h e two c o n s e c u t i v e s e c t i o n s a r e b o t h q u a l i t a t i v e l y a n d q u a n t i t a t i v e l y w e l l i n a g r e e m e n t w i t h t h e c l a s s i c a l t h e o r i e s of c o a s t a l upwelling (see e . g . Yoshida, 1 9 5 5 ) . of t h e near-shore

The marked d i l u t i o n

coastal w a t e r i s not necessarily related t o

increased r i v e r run-off,

which a t t h e end o f A p r i l s t i l l w a s

r e l a t i v e weak i n t h e w e s t e r n r e g i o n s o f Norway, b u t i s m e r e l y a n e f f e c t of t h e d e e p w a t e r i n f l o w t o t h e w e s t c o a s t f j o r d s d u r i n g t h i s period.

Even i n w i n t e r t h e r e i s a c o n s i d e r a b l e f r e s h w a t e r

e x c e s s s t o r e d i n t h e u p p e r l a y e r f j o r d w a t e r and as a consequence t o deep

w a t e r i n f l o w t h i s low s a l i n i t y u p p e r f j o r d w a t e r w i l l b e

poured o u t i n t o t h e c o a s t a l water.

A TIME SERIES OF DEEP WATER REPLACEMENTS. The d e e p w a t e r i s e x c h a n g e d o r r e p l a c e d by more d e n s e w a t e r which s p i l l s o r i n t r u d e s o v e r t h e s i l l .

If t h e i n t r u d i n g w a t e r

spreads o u t along t h e b a s i n bottom, f i l l i n g t h e deeper p o r t i o n s of t h e b a s i n a n d g e n t l y e l e v a t i n g t h e o r i g i n a l w a t e r , t h e amount of w a t e r b e i n g r e p l a c e d c a n b e c e t e r m i n e d from t h e o b s e r v e d e l e v a t i o n of t h e o r i g i n a l water.

A c o n v e n i e n t method f o r making

s u c h e s t i m a t e s w a s d e v e l o p e d by L i n d e ( 1 9 7 0 ) . method on t h e

deep

water replacements i n

He applied h i s

Byfjord

448 a n d q u a n t i f i e d a l t o g e t h e r 1 8 i n d i v i d u a l r e p l a c e m e n t s i n t h e two p e r i o d s of i n v e s t i g a t i o n 1 9 2 7 - 3 9 and 1 9 4 7 - 5 6 . period constitutes t h e best

of replacement

The l a s t 1 0 y e a r

d a t a s e t from which a n unbroken s e r i e s

estimates can be c o n s t r u c t e d .

H i s r e s u l t s a r e a t f i r s t u t i l i z e d f o r simple s t a t i s t i c a l t r e a t m e n t as shown i n T a b l e 1 a n d i n F i g u r e 4 TABLE 1 Replacement ( km3 ) Me a n S t . dev. Min Max

Initial

Final

U

27.277 0.019 27.253 27.309

13.2

3.9 0.0 21.0

Incremental

Ao

U

27.337 0.026

0.060 0.026

-

0.127

27.394

T h e s e s t a t i s t i c s a r e b a s e d on a l l t h e 1 8 r e p l a c e m e n t e v e n t s . The

at values refer t o 300 m depth a t a f i x e d s t a t i o n i n Byfjord,

but because of t h e high degree of u n i f o r m i t y of t h e deep water values are f a i r l y representative

masses i n t h e f j o r d t h e g i v e n crt throughout t h e basin.

(Although t h e u p p e r 1 0 0 m of t h e f j o r d

usually is strongly stratified,

local density variations i n the

d e e p w a t e r masses b e l o w may b e w i t h i n t h e a c c u r a c y o f common determinations).

The i n i t i a l ut

t o the inflow event, t h e f i n a l ment i s c o m p l e t e d a n d not

ist

refers t o the situation prior i s t h e value after t h e replace-

i s t h e increment.

I n T a b l e 1 i s g i v e n t h e r a n g e of v a r i a t i o n o f t h e r e p l a c e d volumes dnd t h e c o r r e s p o n d i n g c h a n g e s i n t h e d e e p water d e n s i t y . Then t h e mean a n d t h e maximum r e p l a c e d volume c o r r r e s p o n d s t o 2 8 % a n d 4 5 % r e s p e c t i v e l y o f t h e t o t a l f j o r d volume.

a p p a r e n t from t h e t a b l e i s t h e r e l a t i v e l y v a r i a t i o r , o f t h e deep water d e n s i t y .

As also

r e s t r i c t e d r a n g e of

A remarkable p o i n t i s t h e

very s m a l l standard d e v i a t i o n o f t h e i n i t i a l at

-

v a l u e s , which

i s almost l e s s t h a n t h e expected s t a n d a r d d e v i a t i o n due t o e r r o r s i n r o u t i n e d e t e r m i n a t i o n o f s a l i n i t y by t h e t i t r a t i o n method u s e d . I n F i g . 4 a r e p l o t t e d t h e volumes r e p l a c e d v e r s u s t h e incremental d e n s i t i e s .

The d o t t e d l i n e i s a s e c o n d o r d e r l e a s t

square f i t t o t h e data, constrained t o pass through t h e o r i g i n . I n vipw o f t h e f a c t t h a t t h e b a s i c h y d r o g r a p h i c d a t a a r e c o l l e c t e d over

d

l o n g p e r i o d o f t i m e a n d t h e r e l a t i v e l y s i m p l e method u s e d

t o make r e p l a c e m e n t e s t i m a t e s , t h e r e i s a s u r p r i s i n g l y s m a l l s c a t t e r around t h e f i t t e d l i n e .

T h i s r e s u l t l e a v e s a f e e l i n g of

c o n f i d e n c e t o L i n d e ' s and c l e a r l y p r o v e s t h e e x i s t a n c e o f a c l o s e

449

r e l a t i o n s h i p b e t w e e n t h e volume b e i n g r e p l a c e d a n d t h e c o r r e s p o n d i n g d e n s i t y change.

The l e a s t s q u a r e f i t may i n f a c t b e u t i l i z e d

with reasonable accuracy

t o d e t e x n i n e t h e r e p l a c e d volum d i r e c t l y

from d e n s i t y r e c o r d s .

rm3

25

. / /

/

4.

.

/ / 0

Incremental sigma-t "

Fig. 4.

0.02

"

0.04

"

Replacement

0.06

-

"

0.08

"

0-10

"

0.12

"

0.14

0.16

d e n s i t y d i a g r a m b a s e d on 1 8 i n d i v i d u a l

replacement e v e n t s i n 1 9 2 7 - 3 9 and 1 9 4 7 - 5 6

according t o Linde ( 1 9 7 0 )

From t h e s t a t i s t i c a l t r e a t m e n t it a p p e a r s tha':

t h e deep

w a t e r replacement i s a f u n c t i o n o f t h e f i n a l d e n s i t y r a t h e r t h a n

of t h e i n i t i a l d e n s i t y , and t h a t t h e i n i t i a l d e n s i t y a t t a i n s p r a c t i c a l l y t h e s a m e contant value p r i o r t o each inflow event. T h i s means t h a t t h e d e n s i t y r e d u c t i o n f o l l o w i n g a l a r g e r e p l a c e m e n t w i t h h i g h d e n s i t y w a t e r must b e more r a p i d t h a n t h a t f o l l o w i n g

a smaller replacement with u s u a l l y lower d e n s i t y water.

This

r e s u l t i n d i c a t e s t h a t t h e e n e r g y w h i c h i s r e l e a s e d , when t h e a b o v e e s t i m a t e d q u a n t i t i e s of w a t e r c h a n g e l e v e l , i s a n i m p o r t a n t c o n t r i b u t i o n t o t h e t o t a l energy a v a i l a b l e f o r i n t e r n a l mixing and d i f f u s i o n p r o c e s s e s i n t h e s h e l t e r e d b a s i n . The a s s u m p t i o n z o f a s l o w i n f l o w a l o n g t h e b a s i n b o t t o m a n d a g e n t l y e l e v a t i o n o f t h e o r i g i n a l w a t e r are n o t r e a l i s t i c i n t h e

g e n e r a l case.

Some m i x i n g b e t w e e n new a n d o r i g i n a l w a t e r w i l l

a l w a y s o c c u r , p a r t i c u l a r l y a t t h e s i l l by s h e a r i n s t a b i l i t y o f t h e e x c h a n g e f l o w a n d by s t i r r i n g i n d u c e d i n t h e b a s i n .

As the

450

inflow proceeds fractions of the new water will escape with the outflowing admixture. When quantifying deep water exchanges it is therefore of some importance to distinguish between the amount which in fact spills over the sill and the replaced volume. From the two basic methods available; Linde's indirect method which determine net replacement and the direct measurement of the exchange flow from which the total input can be determined, the two quantities obtained will probably differ by an appreciable amount and a factor of 2 to 3 seems to be a realistic magnitude (Helle, 1975). Thus, although Linde's estimates represent reasonable values for the replacements, they clearly are underestimates of the total amounts of water which are involved in these exchanges.

+

0

1948

1950

;

:

:

:

1952 :

:

1954 :

:

1956 :

I

c

1948

Fig. 5.

1950

1952

1954

1956

10 year series of deep water replacement (a),

Northerly wind-flux anomaly (b), local ( c ) and revional (d: run-off and mean winter air-temDerature (e).

X

0.1)

451

The next step in utilizing Linde's result is to contruct a time series o f the total annual replacements (which all are

confined to the period March-August) which enables us to compare these variations with time series of the different quantities assumed to effect the replacement process. Such an intercomparison is set up in Fig. 5 where the different time series, all covering the 10 year period 1947-56 corresponding to the most comprehensive data set used by Linde, are as foolows: a) The annual replacement in Byfjord b) The net northerly wind-flux anomaly y s i n ~Utsira as reference station. The anomaly r is defined by:

r =

"-

"SN

"SN where V is the weighted sum of the monthly mean longshore wind component given the weight one if southward directed and the weight zero if northward directed, and summed over the six months March-August. The subscript c)

d)

e)

SN refers to the standard normal 1 9 3 1 - 6 0 . The summer run-off (March-August) to Byfjord based on discharge records from the main rivers draining to the fjord. The summer run-off (March-August) from the southern Norway is estimated by a procedure equivalent to that used by Tolland (1976) for the normal monthly run-off f r o m the different regions of Norway. The estimates are based on all available discharge records from the rivers in the regions (region I and I1 according to Tollan). The mean air-temperature in winter (November-April) at Utsira is assumed to reflect the average winter climate in the coastal region.

As stated above, the governing factor for the deep water exchange is the density of the water occurring outside the west coast at spring and summer time, which clearly is affected by the climatic conditions during the previous winter. Cold winters are usually associated with little winter run-off and cooling of the coastal water and the Atlantic surface water occurring on the North S e d shelf. Both the upper coastal water regime and the lower ocpanic regime at the coast will hence be more dense than after a mild winter.

452

Th- vertical displacement of the isohalines at the coast of the magnitude indicated by the mean isopleth (Fig. 3 1 , however,

introduce local density variations which noramlly far exceed the effect from climatic variation.

The absence of covariation

between air-temperature and replacement in Fig. 5 is thus not unexpected, and it is reasonable to seek for covariations with the factors which are assumed to be major responsible for these vertical oscillations, namely the run-off from land and the coastal wind.

A such covariation appears surprisingly clear to exist

between replacement and wind-flux, may be with an exception f o r 1952 when an appreciable amount of deep water was replaced despite

weak northerly wind-flux at the coast during the summer. fn the first three years (1947-49) there is a l s o a similar tendency in the run-off diagram, but thereafter the replacement and run-off are completely uncorrelated. No replacement were detected at a l l in 1950 despite a very cold wintei. and that mdximum run-off

for this 10 year period

was recorded.

The results of this intercomparison are very convincing and it seems to be no doubt left that the deep water replacements are more or less completely governed by the coastal wind. The most surprising point, however, is that this process seems to be completely independent of the magnitude of fresh water discharge,

both locally and reionally.

Though longer time

series would be needed in order to make more reliable conclusions about the effect of run-off, this point is of considerable interest in view of the extensive use of hydroelectric resources, with resulting changes in the natural run-off pattern.

It appears

at least, that although artificial run-off regualtion may cause considerable changes in the upper layer circulation pattern and l o c ? a l ice problems in winter, it will not alter the natural regu1 , i r i t y and extpnt of deep water exchanges. SOME OBSERVATIONS OF REPLACEMENT EVENTS

An observational program, which included current meassurements at the sill and hydrographic observation at monthly and sometimes

bimonthly intervals at selected locations in Byfjord (see Fig. l), w a s carried out during 1973 and 1974.

The main purpose of this

investigation was to determine the capability of the fjord to absorb sewage from the surrounding population centers.

453 A s t r i n g of t h r e e Aanderaa c u r r e n t meters, equipped with

t e m p e r a t u r e s e n s o r s , w a s moored a t t h e s i l l ( s t . H ) s u r f a c e f l o a t i n 2 5 m d e p t h f o r buoyancy.

u s i n g a sub-

The c u r r e n t meters

were p o s i t i o n e d a t d e p t h s o f 3 0 , 8 0 and 1 6 0 m and o p e r a t i l l g a t a

Two t y p i c a l r e p l a c e m e n t e v e n t s

sampling i n t e r v a l o f 1 0 minutes.

w e r e d e t e c t e d d u r i n g t h i s p e r i o d , o n e i n summer 1 9 7 3 a n d t h e o t h e r

early i n spring 1974.

Background h y d r o g r a p h i c d a t a f r o m t h e

a d j a c e n t c o a s t a l area w e r e o b t a i n e d from t h e I n s t i t u t e o f Marine R e a s e a r c h i n Bergen.

Their ships q u i t e regularly occupies s t a t i o n s

a l o n g t h e s e c t i o n b e t w e e n F e i e a n d S h e t l a n d a n d f o r t u n a t e l y two sections are coincident wiht t h e 1 9 7 4 inflow event.

Wind d a t a

f r o m U t s i r a l i g h t h o u s e , s a m p l e d e v e r y t h i r d h o u r , a r e assumed r e p r e s e n t a t i v e for t h e c o a s t a l a r e a .

Water l e v e l r e c o r d s w e r e t a k e n

from t h e t i d e gauge i n Bergen Harbour. The t i m e s e r i e s w e r e t r e a t e d a s f o l l o w s .

Current v e l o c i t i e s

w e r e r e s o l v e d i n t o c o m p o n e n t s a l o n g t h e a x e s of maximum a n d minimum Wind v e l o c i t i e s w e r e r e s o l v e d i n t o

v a i r i a n c e of t o t a l r e c o r d s .

componenis a l o n g a n d p e r p e n d i c u l a r t o t h e c o a s t l i n e .

A l l time

s e r i = s were l o w - p a s s E i l t e r e d by l i t i l i z i n g a 2 5 h o u r r u n n i n g mean. The s e a - l e v e l

v a r i a t i o n a r e g i v e n r e l a t i v e t h e mean o f t o t a l

records i n each series.

Run-off

and s a l i n i t y diagrams from Byfjord

a r e shown f o r r e f e r e n c e i n F i g u r e 6 a n d F i g u r e 7. The 1 9 7 3 r e p l a c e m e n t ,

This season t h e replacement took

p l a c e i n two s e p a r a t e e v e n t s as s e e n f r o m t h e two s u c c e s s i v e peaks o c c u r r i n g i n t h e s a l i n i t y i s o p l e t h s Fig.

7.

The m o s t

p r o n o u n c e d e v e n t o c c u r r e d i n March t h r o u g h A p r i l when t h e new a n d s l i g h t l y more s a l i n e w a t e r e n t e r e d t h e b a s i n .

The o r i g i n a l

w a t e r w a s l i f t e d t o t h e l e v e l s a b o v e 1 0 0 m w h e r e it c a u s e d a marked s a l i n i t y i n c r e a s e .

This comparatively high s a l i n i t y and

low oxygen c o n t e n t w a t e r o c c u r r i n g i n t h e u p p e r l a y e r s g r a d u a l l y e s c a p e d f r o m t h e fjord i n t h e c o u r s e o f May.

The i n f l o w i t s e l f

a p p e a r s l i k e a n i n c l i n e d plume i n t h e h y d r o g r a p h i c s e c t i o n t a k e n a l o n g t h e f j o r d a x i s o n A p r i l 2 8 ( F i g . 8). plume c o r e s l i g h t l y e x c e e d 3 5

O/oo

The s a l i n i t y i n t h e

at s i l l l e v e l a t t h e entrance

a n d g r a d u a l l y d e c r e a s e d t o w a r d s t h e f r o n t n o s e a s i n d i c a t e d by t h e 34.95

O/oo i s o h a l i n e .

D i s o l v e d oxygen r e v e a l s a s i m i l a r

p i c t u r e w i t h t h e h i g h e s t c o n c e n t r a t i o n c e n t r e d a l o n g t h e c o r e and decreasing concentration towards t h e f r o n t .

A t t h i s stage the

new w a t e r h a d n o t y e t p e n e t r a t e d down i n t o t h e d e e p e s t t r o u g h

as s e e n by t h e c o m p a r a t i v e l y low o x y g e n c o n s e n t r a t i o n n e a r t h e b o t t o m . Oxygen minimums a b o v e t h e plume i n d i c a t e s t h e l o c a t i o n o f d i s p l a y c e d o r i g i n a l waters.

A RF

- 900 mSs

- 800

- 700 - 600

- 500

- 400 - 300

- 200 - 100 -0

0'

J

'

F

'

M

I

A

'

M

'

J

'

J

'

A

'

S

'

O

1973

Figure 6.

'

N

'

D

~

J

~

F

~

1976

Monthly mean run-off rate RF and

equivalent thickness

HF of the frrsh-water layer from monthly observations of salinty

proiiles.

Figure 7. Byfjord

Salinity isopleths from

St. 4.

455 St H 0

St 5

m

6

St 3

StL

0

3L 0 3L 5

5

8.

St 1

33

10km

500

Fig.

St 2

\

S e c t i o n a l d i a g r a m s o f s a l i n i t y a n d oxygen c o n t e n t

showing t h e i n f l o w plume ( B y f j o r d , A p r i l 2 8 , 1 9 7 3 ) .

U n f o r t u n a t e l y no c u r r e n t r e c o r d s e x i s t f r o m t h e e a r l y i n f l o w e v e n t . T h e c u r r e n t m e t e r p r o g r a m f i r s t s t a r t e d up J u n e 1 8 a n d t h e f i r s t

r e c o r d s e r i e s o b t a i n e d , l a s t i n g u n t i l August 3 , w a s c o i n c i d e n t with t h e second 1 9 7 3 inflow.

The l o w - p a s s e d wind v e c t o r , c u r r e n t

a n d s e a - l e v e l r e c o r d s a r e p l o t t e d a g a i n s t a common t i m e a x i s i n Fig. 9 . a t 1 6 0 m,

Net c u r r e n t t h r o u g h o u t t h e r e c o r d s e r i e s i s l a n d w a r d seaward a t 3 0 m and n e g l i g e i b l y s m a l l a t 8 0 m.

The

456 v e r t i c a l s t r u c t u r e e v i d e n t l y composes a t w o - l a y e r i n t h e lower and outflow i n t h e upper p o r t i o n .

flow; inflow

There i s clear

e v i d e n c e f o r a l e v e l o f no n e t m o t i o n l o c a t e d a t a p p r o x i m a t e l y The e x c h a n g e f l o w i s f a r f r o m s t e a d y .

mid-depth.

The i n f l o w a s

d e t e c t e d by t h e c u r r e n t m e t e r i n 1 6 0 m forms smooth v a r y i n g p u l s e s o f o n e d a y t o a week d u r a t i o n a r r i v i n g a t a p p r o x i m a t e l y w e e k l y i n t e r v a l s , with speed amplitudes varying between 1 0 cm/s and 3 3 cm/s.

The v a r i a t i o n s i n t h e l o w e r f l o w a r e w e l l r e f l e c t e d i n

t h e u p p e r f l o w and t h e s p e e d a m p l i t u d e s i n 30 m and 1 6 0 m a r e

of s i m i l a r magnitude.

JUNE 20

JULY

25

30

5

10

15

25

20

r

30

I

20-

CURRENT

80"

- 10 CURRENT

0-

- 10

160m

AS E A L E V E L I N B E R G E N HARBOUR

-10

Fig.

9.

Low-passed t i m e s e r i e s o f c o a s t a l w i n d , c u r r e n t s a t t h e

s i l l and s e a - l e v e l

i n Bergen.

J u n e 1 8 - August 3 1 9 7 3 .

A d i r e c t c o m p a r i s i o n o f t h e c u r r e n t a n d wind v e c t o r p l o t

r e v e a l s a s i m i l a r i t y where e v e n m i n o r d e t a i l s a r e r e p r o d u c e d .

Nor7therly w i n d s a l o n g t h e c o a s t a r e f o l l o w e d by i n c r e a s i n g c u r r e n t s

457

I

400

200

250 km

150

100

50

0

0

m

100

200 SHIP

:

JOHAN HJORT 1974

-

DATE : 30/4 115

300

Figure 1 0 .

Two d i s t r i b u t i o n s o f s a l i n i t y w i t h i n a v e r t i c a l

s e c t i o n a c r o s s t h e Norwegian T r e n c h a n d t h e s h e l f ( a l o n g t h e l a t i t u d e N 60'

45'

between F e i e and S h e t l a n d ) .

458

a c r o s s t h e s i l l , s o u t h e r l y w i n d s a r e f o l l o w e d by d e c r e a s i n g c u r r e n t s a n d some t i m e s by t h e r e v e r s e f l o w , o r p o s s i b l y a more complex flow s t r u c t u r e

which c a n n o t b e d e t e r m i n e d by t h e o n l y t h r e e

c u r r e n t meters used.

C u r r e n t s a n d wind a r e l a g g e d by 1 t o 2 d a y s

which i n d i c a t e s t h e a p p r o x i m a t e r e s n o n s e t i m e f o r t h e u n w e l l i n v Drocess involved.

Only t h e s e a - l e v e l d r o p o c c u r r i n g a t t h e e n d

o f t h e r e c o r d series, f o l l o w i n g a r e l a t i v e long p e r i o d w i t h n o r t h e r l y winds a l o n g t h e coast, i s c o n s i s t e n t w i t h t h e e x p e c t e d behaviour of t h e sea-level

during c o a s t a l upwelling.

The 1 9 7 4 r e p l a c e m e n t .

Following t h e s t a g n a n t p e r i o d from

December t o March, d u r i n g which s a l t e x t r a c t i o n p r o c e s s e s d o m i n a t e d i n t h e d e e p w a t e r masses o f t h e F j o r d ( F i g . 7 ) ,

a new c o l d e r a n d

more s a l i n e t y p e o f w a t e r w a s i n t r o d u c e d i n t h e b a s i n f r o m t h e m i d d l e o f March.

The i n f l o w s t a r t e d w i t h a s e r i e s o f marked

i n f l o w p u l s e s s i m i l a r t o what w a s o b s e r v e d d u r i n g summer 1 9 7 3 a s described above, b u t t h e n t u r n e d o u t l i k e a s t e a d y flow w i t h c u r r e n t s p e e d w i t h i n t h e r a n g e 3 0 - 3 5 c m / s a t 1 6 0 m ( F i g . 11).

The f l o w

maintained i t s steady c h a r a c t e r throughout t h e rest of t h e record s e r i e s , i n a p e r i o d o f a l m o s t a month. Mean s e a w a r d c u r r e n t a t 3 0 m d u r i n g t h i s e a r l y e v e n t w a s c o n s i d e r a b l y l e s s p r o n o u n c e d (15 c m / s )

t h a n d u r i n g summer 1 9 7 3 ,

a n d o n l y a b o u t h a l f t h e mean l a n d w a r d c u r r e n t a t 1 6 0 m .

The two

s e t s o f r e c o r d s , however, were o b t a i n e d u n d e r e n t i r e l y d i f f e r e n t run-off

conditions.

seasonal run-off

Whereas t h e 1 9 7 3 e v e n t c o i n c i d e d w i t h t h e

maximum t h e e a r l y 1 9 7 4 i n f l o w o c c u r r e d

i n extra-

o r d i n a r i l y d r y and r e l a t i v e l y c o l d p e r i o d w i t h i n s i g n i f i c a n t d i s charge from t h e r i v e r s . Two s e c t i o n s o f t h e r e c o r d s e r i e s p r e s e n t e d i n F i g . 11 r e v e a l new f e a t u r e s w h i c h i n s e v e r a l r e s p e c t s d i f f e r f r o m what w a s previously observed. S p e c i a l a t t e n t i o n i s a t t r a c t e d towards t h e f i r - s t inflow pulse occurring i n the records, following a period w i t h r e l a t i v e weak e a s t e r l y a n d n o r t h e r l y w i n d s a t t h e c o a s t a n d a s s o c i a t e d with a remarkable of t h e s e a - l e v e l

high surface elevation.

The r i s e

o f more t h a n 0 . 5 m i n t h e c o u r s e o f 8 d a y s i s

r e l a t e d t o a n atmospherj-c f r o n t a r r i v i n g a t t h e North Sea from w e s t o v e r S c o t l a n d w h i c h c a u s e d a 6 0 mbar d r o p i n t h e a i r - p r e s s u r e

over t h e c e n t r a l p a r t s o f t h e N o r t h S e a .

Whether t h i s i n f l o w ,

o r t h e associated baroclinic response i n t h e adjacent s t r a t i f i e d

c o a s t a l water (see Fig. wave"

( G i l l and Clarke,

10) w a s i n d u c e d by some " c o a s t a l t r a p p e d 1974)

g e n e r a t e d by t h e a t m o s p h e r i c

459

disturbance, or solely induced by local winds at the coast cannot, however, be clearly verified from the present data. It seems at least that both the barometric surface elevation and the change to southerly winds were responsible for the following decrease and reversing of the flow across the s i l l .

L

S E A L E V E L IN BERGEN HARBOUR

Fig. 11.

Low-passed time series of coastal wind, currents at the

s i l l and sea-level in Bergen.

March 7 - May 4 1974.

The other section in the records which reveals special features is from about April 10 until the end of the record series May 4.

This is a

unique example of a major seasonal

replacement event with it5 characteristically marked change of water properties (Fig. 7).

Northerly winds dominate at the

coast during this period, but despite highly variable strength d n d even shorter intervals with appreciable southerly wind components

460

the flow across the sill maintains its steady features.

In the

salinity section taken across the Norwegian Trench (Fig. 10) upwelling in a narrow strip at the coastline is clearly seen and the temperature records at 160 m (Fig. 12) indicates that the intruding watery with temperatures decreasing from 7.6 to 7.2OC, originates from gradually deeper layers in the Trench as the upwelling proceeds.

I

I

1

I

I

8-

7-2

1

I

1

I

I

1974

I

I

I

1

1

Fig. 12. Low-passed time series Mdrch 7 - May 4 1974.

I

1

I

I

of temperature at the sill.

Vertical profiles of water properties and current at the are shown in Fig. 13 as observed May 3. Ebb and flood current profiles indicates a range of variations which is sill itself

between 20 and 40 cm/s landward at 160 m y between 7 and 22 cm/s seaward at 30 m and between 5 and 7 cm/s landward and seaward respectivaly at 80 m. A two layer flow structure is also evident

from the sigma-t and the oxygen profiles showing the dense and oxygen rich water which intrudes with the deep flow and the original water escaping with the upper flow. A fairly homogenous 20-30 m thick layer contered at mid-depth indicates strong mixing due to shear instability or due to breaking of internal waves.

461 26.9

27.0

,

27.1

27.2

27.3 6, Seaward

t

0-

Landward

:

rn

25-

Current

crn/s

5 0..

75100.-

125-

150. 7nnnyr

Fig. 1 3 .

V e r t i c a l profiles o f at, O 2 a n d c u r r e n t a t t h e s i l l

May 3 1974.

E V I D E N C E FOR H Y D R A U L I C CONTROL OF THE EXCHANGE FLOW. The p a s s a g e t h r o u g h H j e l t e s u n d c o n s t i t u t e s a c o n s t r i c t i o n of t h e f j o r d both i n depth and width and hence a c t s as a c o n t r o l s e c t i o n f o r t h e exchange f l o w . I n o r d e r t o o b t a i n a n esti-mate o f t h e c u r r e n t v e l o c i t i e s neccessary t o e s t a b l i s h a c r i t i c a l two-layer

flow, w e adopt t h e

c r i t i c a l c o n d i t i o n s o f Stommel a n d F a r m e r ( 1 9 5 3 ) :

1.

2

Fl +

F; = 1

where Fl a n d F2 are t h e j n t e r n a l F r o u d e numbers d e f i n e d b y :

S u b s c r i p t 1 and 2 r e f e r t o upper and lower l a y e r r e s p e c t i v e l y , U

1,2

i s t h e c u r r e n t v e l o c i t y of t h e l a y e r averaged over t h e c r o s s

s e c t i o n a r e a A 1,2.

H1,2

i s t h e mean t h i c k n e s s , p; i s t h e a c c e l l e -

r a t i o n d u e t o g r a v i t y , p i s sea w a t e r d e n s i t y a n d A p i s t h e d e n s i t y jump a c r o s s t h e i n t e r f a c e . The c o n d i t i o n o f n o - n e t

is 2.

UIAl

+ U2A2 = 0 ,

t r a n s p o r t through t h e c o n t r o l secti-on

462

or f o r small fresh water discharges as outlined by Long (1976), 4.

111

H2 = ;D

where D is nimimum channel depth (sill depth). The combination of equation (I), ( 3 ) and ( 4 ) yields

lhe quantities determined by the geometry of the control section 3 2 ar’e: A1 = 128x10 m , A2 61x103 m2 and D = 160 m. The cross section area of the shallower passage ( 5 5 ~ 1 0 ~is m ~included in Al.

Representative values of the remaining quantity

Ap/p can be An obtained f r o m Table 1 using the mean and the maximum Ao t’ appropriate value of A p l p is also obtainable by depth averaging thP observed u t profile (Fip. 13) over each of the two layer above and below the level of no-net motion. The resulting current

vrlocities are piven in Table 2 below. TABLE 2

0.060 0.126 0. 5

20.4 29. 5 59

9.8 14.2 28

mean not from Table 1. max Ao, from Table 1. Ao, f r o m actual profile.

In thc two first cases there is very good agreement with range of current velocities observed. In the latter case, using the ob:;erved density profile, the computed velocities are some too high hut still of an acceptable magnitude.

The relative great

density j u m p in this case, and hence the high critical velocities arc due the contribution from the upper bracki.sh water.

This

dilution of the outflowing water is not accounted f o r in the first cases where the actual density measurements are made at 300 m depth in the basin. A more appropriate value of

P

which also is in accordance

with the critical flow conditions as the existance of a long nonpropagating interfacial wave, may be obteined by taking the differpnce between the a v e r a o e densities o f a s a y , 10-20 m thick

463

layer on b o t h s i d e s o f t h e m i x e d l a y e r i n d i c a t e d i n F i g . 1 3 . whdtever' d e f i n i t i o n o f t h e d e n s i t y jump i s a d o p t e d , h o w e v e r , t h e i d e a l a s s u m p t i - o n i n t h e h y d r a u l i c t h e o r y ; homogenous l a y e r s , s t e a d y f r i c t i o n l e s s f l o w s w i t h no i n t e r f a c i a l mixing, a r e c l e a r l y n o t ?Jevertheless, t h e e s t i m a t e d v e l o c i t i e s i n Table2 i n d i c a t e

satisfied.

thdt c r i t i c a l flow conditions represent q u i t e a r e a l i s t i c s i t u a t i o n tiur>ing d e e p w a t e r r e p l a c e m e n t i n B y f j o r d .

From t h e c u r r e n t r e c o r d s

it i s a l s o e v i d e n t t h a t c e r t a i n u p p e r l i m i t s e x i s t t o t h e e x c h a n g e rdte.

RE FE RE N CE S Br,;iatcn, U a n d Satre, R . ,

1973.

Oppdrett av l a k s e f i s k i norske

k y s t f a r v a n n ; M i l j g og a n l e g g s t y p e r .

Fisker. og h a v e t , s e r i e

B , 9: 1-95. Gade, H . G . ,

Deep w a t e r e x c h a n g e i n a s i l l f j o r d : A s t o c h -

1973.

as-tic p r o c e s s . J.Phys.Ocean.,

Gill, A . E .

213-219.

3:

a n d C l a r k e , A.J., 1974.

Wind-induced u p w e l l i n g ,

c o d s t a l c u r r e n t s and s e a - l e v e l changes.

Deep-sea Res., 2 1 :

325-345. Iielland-Haiisen,

B.

In:

fjord.

,

1935.

T e m p e r a t u r e v a r i a t i o n s i n a Norwegian

S. Hedin ( E d i t o r ) ,

Geografiska Annaler 1 9 3 5 ,

S v e n s k d S B l l s k a p e t f6r. a n t r o p o l o g i o c h o c e a n o g r a f i, Stockholm, pp.

Helle, H . B . ,

1975.

563-570. Oseanografisk resepientundersdkelse av

f j o r d e n e r u n d t Bergen.

Geophysical

Bergen, 7 2 pp.

Institute, Johdrinessen,

Unpublished r e p o r t .

T.W.

and H S l a n d , L., 1 9 6 9 .

Standard normals 1931-60

o f m o n t h l y wind s u m m a r i e s f o r Norway C l i m a t o l o g i c a l s u m m a r i e s

o f Norway. Linde, E . ,

1970.

D e t n o r s k e m e t e o r o l o g i s k e I n s t i t u t t , Oslo. H y d r o g r a p h y of t h e B y f j o r d .

Geophysical i n s t i t u t e , Bergen, Lj$en,

K. a n d S v a n s s o n , A . ,

1972.

Rept.

20,

38 p p .

Long-term

s u r f a c e t e m p e r a t u r e s i n t h e Skagerak.

v a r i a t i o n s of s u b Deep-sea R e s . ,

19:

277-288. ILong,

R., 1 9 7 6 . estuary.

Stommel. H .

Mass arid s a l t t r a n s f e r a n d h a l o c l i n e d e p t h i n a n Tellus,

28,s: 460-472.

and Farmer, H . G . ,

e s t u a r y by t r a n s i t i o n .

1953.

Control of s a l i n i t y i n an

J.Har.Res.,

ll: 2 0 5 - 2 1 4 .

464

Szlen, O . H . , 1976. General hydrography of f j o r d s . In: S.Skreslet et al. (Editors), Fresh water on the sea. The Association of Norwegian Oceanographers, Oslo, pp. 43-49. TOlldn,

A., 1976.

River run-off in Norway.

In: S.Skreslet

et al.

(Editors), Fresh water on the sea. The Association of Norwegian Oceanographers, Oslo, p p . 11-13. Yoshida, K., 1955. Coastal upwelling off the Californian coast. Records of 0cean.Works in Japan, 2,2: 8-20.

465

NONLINEAR INTERNAL WAVES I N A FJORD

DAVID FARMER

I n s t i t u t e of Ocean S c i e n c e s , Sidney, B.C.,

Canada

and J . DUNGAN SMITH

Department o f Oceanography, University o f Washington, S e a t t l e , Washington, U.S .A.

ABSTRACT A p i l o t study i n Knight I n l e t , B r i t i s h Columbia has y i e l d e d measurements of l a r g e amplitude, n o n l i n e a r i n t e r n a l waves i n t h e s t r o n g l y s t r a t i f i e d s u r f a c e l a y e r . A t r a i n of s e v e r a l waves w i t h amplitudes exceeding 1 0 meters and p e r i o d s o f a few minutes, passed t h e v e s s e l on each f l o o d t i d e . Observations w i t h p r o f i l i n g and f i x e d depth instruments showed t h a t the waves t r a v e l l e d i n a s t r o n g l y sheared flow. The r e s u l t s are i n t e r p r e t e d using Benjamin's (1967) approximate model f o r n o n l i n e a r waves i n deep f l u i d s , s u i t a b l y modified t o allow f o r shear. This model then provides t h e b a s i s f o r an examination o f c o n d i t i o n s l e a d i n g t o s h e a r flow i n s t a b i l i t y . A s o l u t i o n t o t h e eigenvalue problem f o r exponential s t r a t i f i c a t i o n and s h e a r i s used t o examine t h e i n t e r a c t i o n o f the waves w i t h t h e background flow; c r i t i c a l amplitudes corresponding t o c o n d i t i o n s o f marginal s t a b i l i t y are found t o b e comparable t o t h o s e observed i n Knight I n l e t . The waves a r e d i s c u s s e d i n t e r m s o f their l i k e l y c o n t r i b u t i o n t o v e r t i c a l exchange p r o c e s s e s i n f j o r d s .

1.

INTRODUCTION

This paper is concerned w i t h one o f the many unsteady a s p e c t s o f flow i n

fjords.

H i s t o r i c a l l y , a t t e m p t s t o study c i r c u l a t i o n i n deep, s t r o n g l y

s t r a t i f i e d e s t u a r i e s have c o n c e n t r a t e d on a s t e a d y s t a t e o r t i d a l l y averaged description.

This s i m p l i f i c a t i o n i s understandable and may indeed be j u s t i f i e d

f o r t h e d e s c r i p t i o n o f c i r c u l a t i o n i n p l a c e s t h a t have a very s m a l l t i d a l range; i t a l s o makes s e n s e as a f i r s t s t e p towards understanding what i s c l e a r l y a most complicated system.

N e v e r t h e l e s s , time-dependent

effects are

o f t e n dominant and t h e assumption o f a s t e a d y flow i n t h e t i d a l l y averaged s e n s e masks much of t h e underlying p h y s i c s t h a t must be understood i f w e are t o g a i n a deeper knowledge o f e s t u a r i n e c i r c u l a t i o n .

As

an example t h a t i s d i r e c t l y

r e l a t e d t o t h e o b s e r v a t i o n s d e s c r i b e d h e r e i n , t h e i n t e r a c t i o n of t i d a l flow w i t h topographic f e a t u r e s may i n f l u e n c e mixing i n a very important b u t n o n l i n e a r way. Y e t w i t h o u t an understanding o f t h e p h y s i c s involved, it i s impossible t o s e l e c t

a p p r o p r i a t e entrainment models t o use w i t h t i d a l l y averaged c a l c u l a t i o n s of t h e circulation.

466 Of course t i d a l flow i s n o t t h e only t i m e dependent s i g n a l w i t h which one must contend.

Wind-stress

i s o f t e n more v a r i a b l e and always less p r e d i c t a b l e

than t i d a l f o r c i n g and can p l a y a major r o l e i n a l t e r i n g the c i r c u l a t i o n of deep estuaries.

S i m i l a r l y f l u c t u a t i o n s i n fresh-water

during p e r i o d s of high run-off,

discharge, which c a n be l a r g e

w i l l i n t r o d u c e a t i m e dependence i n t o t h e

c i r c u l a t i o n as w i l l v a r i a t i o n s i n t h e d e n s i t y s t r u c t u r e o u t s i d e t h e f j o r d . However, one of oscillations.

our primary concerns i s w i t h t h e response of f j o r d s t o t i d a l This i n t e r e s t i s motivated p a r t l y by t h e c o n s i d e r a t i o n t h a t

t i d a l e f f e c t s a r e ubiquitous and t i d a l ranges a r e q u i t e l a r g e (Q4m) i n t h e i n l e t s of t h e North E a s t P a c i f i c , and p a r t l y by t h e s t r o n g t i d a l l y generated e f f e c t s w e have observed i n Knight I n l e t , t h e e s t u a r y t h a t w e have chosen t o examine i n d e t a i l . Knight I n l e t has been t h e s i t e of s e v e r a l previous oceanographic s u r v e y s , so a s u b s t a n t i a l body of h i s t o r i c a l d a t a i s a v a i l a b l e ; perhaps f o r t h i s reason i t has been a t e s t i n g ground f o r r e c e n t t h e o r e t i c a l models o f f j o r d c i r c u l a t i o n (Winter, 1973, Long, 1975, Pearson and Winter, 1978).

I n particular a detailed

s e r i e s of c u r r e n t measurements were taken by Pickard and Rodgers (1959).

I t is

of i n t e r e s t t h a t t h e i r o b s e r v a t i o n s taken over 20 y e a r s ago, a r e n o t i n c o n s i s t e n t with our own measurements made i n 1976 w i t h a modern p r o f i l i n g instrument and i n t e r n a l l y r e c o r d i n g c u r r e n t meters. Although somewhat longer than most ( 1 0 2 km), Knight I n l e t (Figure 1) i s f a i r l y t y p i c a l of B r i t i s h Columbia f j o r d s . a maximum depth of 550 m.

I t has an average width of 3 km and

The i n l e t has two s i l l s , t h e innermost of which i s

74 km from t h e head and i s l o c a t e d near t h e middle of a s t r a i g h t reach; this

s i l l has a maximum depth of 63 m.

Most o f t h e f r e s h water e n t e r s a t t h e i n l e t

head from t h e K l i n a k l i n i and F r a n k l i n Rivers. maximum c l o s e t o 800 m3s-’

Mean monthly run-off

i n June (Pickard and Rodgers, 1959) b u t t h e mean

annual discharge i s e s t i m a t e d t o be only 410 m3s-’. t h e discharge as type A l .

reaches a

Pickard (1961) c l a s s i f i e s

A s m a l l secondary maximum occurs i n t h e autumn due

t o r a i n f a l l r a t h e r than snowmelt b u t it i s followed by r e l a t i v e l y low d i s c h a r g e s during t h e w i n t e r months. An a i r reconnaissance of t h e i n l e t w a s made i n August 1976.

Flying t h e

l e n g t h of t h e channel, s e v e r a l f e a t u r e s w e r e observed i n d i c a t i n g t h e g r e a t v a r i a b i l i t y t h a t must be expected.

Lenses o f c l e a r f r e s h water from streams

along the s i d e s of the i n l e t w e r e f l o a t i n g on t h e s u r f a c e of t h e b r a c k i s h zone, coloured grey by i t s suspended load of rock f l o u r , and showed t h e e x i s t e n c e of complex eddy motions.

Even more s t r i k i n g w e r e the s u r f a c e e f f e c t s of i n t e r n a l

waves; t h e s e were c l e a r l y v i s i b l e over the s i l l a s a p a t t e r n of overlapping curves and f u r t h e r u p - i n l e t as a sequence of s t r a i g h t l i n e s s e p a r a t e d by 50-

150 m.

(See Figure 2 ) .

Pickard (1961) has made s i m i l a r o b s e r v a t i o n s and has

described t r a i n s of i n t e r n a l waves c e n t e r e d on t h e pycnocline i n Knight, as w e l l

461 a s o t h e r B r i t i s h Columbia i n l e t s .

I n calm weather the presence o f t h e waves i s

made apparent a t t h e s u r f a c e by s u r f a c e r u f f l e s thought t o be caused by convergence e f f e c t s .

KNIGHT I N L E T BRITISH COLUMBIA

STATION

I

5

10

15

20

Figure 1. Map of Knight I n l e t i n d i c a t i n g CTD < s t a t i o n l o c a t i o n s , numbered from 1 t o 2 0 , and a l o n g i t u d i n a l s e c t i o n showing t h e approximate depth a t each s t a t i o n . Most o f the f r e s h water e n t e r s t h e f j o r d from t h e K l i n a k l i n i and F r a n k l i n Rivers a t i t s head. The time s e r i e s d e s c r i b e d i n t h e t e x t was taken a t s t a t i o n 5.

During t h e week of November 9-14,

1476 w e undertook a p i l o t study i n Knight

I n l e t using t h e U n i v e r s i t y of Washington Research Vessel ONAR, i n o r d e r t o determine t h e temporal v a r i a b i l i t y t h a t would have t o be considered i n planning a more d e t a i l e d experiment.

This study included CTD measurements along t h e

channel w i t h some t i m e s e r i e s o b s e r v a t i o n s a t a s t a t i o n approximately 6 km u p - i n l e t of t h e s i l l ( F i g u r e 1).

468

Figure 2 . Knight I n l e t looking seaward. The i n n e r s i l l extends from Hoeya Head, about 2 / 3 of t h e way from the bottom o f t h e photograph on t h e r i g h t hand s i d e , t o Prominent P o i n t which i s t h e low f e a t u r e extending i n t o t h e i n l e t 3/4 of t h e way up on t h e l e f t hand s i d e . The l e a d i n g edge of a t r a i n of i n t e r n a l waves t r a v e l l i n g up i n l e t , i s v i s i b l e i n t h e c e n t e r of t h e photograph. The wave p a t t e r n becomes q u i t e complicated i n t h e neighbourhood o f t h e s i l l .

2.

INSTRUMENTATION Observation o f t h e s a l i e n t f e a t u r e s o f t i d a l , e s t u a r i n e and wind d r i v e n

c i r c u l a t i o n s from a s i n g l e r e s e a r c h v e s s e l concurrent w i t h t h e study of r e l a t i v e l y s h o r t p e r i o d i n t e r n a l waves r e q u i r e s the use o f s e v e r a l instrument systems, each based on a d i f f e r e n t sampling s t r a t e g y .

I n t h e November 1976

study t h e primary p h y s i c a l v a r i a b l e s o f i n t e r e s t were temperature, s a l i n i t y and flow v e l o c i t y and t h e s e had t o be r e s o l v e d over the upper 100 meters i n a temporal band ranging from a f r a c t i o n o f a minute t o s e v e r a l t i d a l days. Low frequency information over t h e e n t i r e p e r i o d and depth range was o b t a i n e d with a v e l o c i t y and d e n s i t y p r o f i l i n g system designed f o r use i n t h e A r c t i c . This u n i t cycled continuously between t h e s u r f a c e and 140 meters approximately once every f i v e minutes.

The A r c t i c P r o f i l i n g System (APS) could n o t be h e l d

a t one depth t o provide a n a c c u r a t e time series, because v e r t i c a l motion w a s r e q u i r e d t o f l u s h the c o n d u c t i v i t y c e l l .

To s a t i s f y a need f o r t h e l a t t e r type

of information, a Surface Layer Frame (SLF) , s u p p o r t i n g a h o r i z o n t a l l y o r i e n t e d

469 c o n d u c t i v i t y c e l l , a temperature s e n s o r and s e v e r a l types of c u r r e n t meters was used.

However, even w i t h both of t h e s e systems t h e c o n d u c t i v i t y and v e l o c i t y

f i e l d s due t o p a s s i n g i n t e r n a l waves could n o t be r e s o l v e d w i t h s u f f i c i e n t s p a t i a l and temporal r e s o l u t i o n and a t h i r d approach was r e q u i r e d .

This

comprised a s t r i n g of i n t e r n a l l y r e c o r d i n g (Aanderaa) c u r r e n t m e t e r s each f i t t e d with c o n d u c t i v i t y cells and f a s t response t h e r m i s t o r s and s e t t o sample The three systems a r e sketched i n Figure 3 i n t h e manner

every 30 seconds.

t h a t they w e r e suspended from R/V ONAR.

-,NAVIGATION TRANSDUCER

AANDERAA ARRAY

c

#

COMPASS-

EMCM

\

MCM'S

S E E C,T SENSOR!

ACM.

SLF

Q Figure 3. Cartoon showing the c o n f i g u r a t i o n o f t h e r e s e a r c h v e s s e l and t h e oceanographic s e n s o r s t h a t w e r e suspended from i t while anchored a t s t a t i o n 5. Three b a s i c systems were employed: a) t h e A r c t i c P r o f i l i n g System (APS) from which v e l o c i t y and d e n s i t y p r o f i l e d a t a w e r e obtained; b) t h e Aanderaa c u r r e n t m e t e r a r r a y from which time s e r i e s of temperature, s a l i n i t y , and c u r r e n t speed were procured and c ) t h e Surface Layer Frame (SLF) from which t i m e s e r i e s of near s u r f a c e s a l i n i t y and c u r r e n t v e l o c i t y were measured.

470 The A r c t i c P r o f i l i n g System i s c o n s t r u c t e d around a G u i l d l i n e Model 8101 CTD. On t h e nose of the device t h r e e p r e c i s i o n mechanical c u r r e n t meters are mounted orthogonally and symmetrically, and behind t h e G u i l d l i n e p r e s s u r e c a s e a second s t a i n l e s s s t e e l tube e n c l o s e s an i n e r t i a l r e f e r e n c e u n i t capable of r e s o l v i n g bearing to 2'

and p r o v i d i n g accelerometer and r a t e gyro d a t a from which t h e

a c c e l e r a t i o n s and t i l t s experienced by t h e device can be c a l c u l a t e d .

The

l a t t e r i s r e q u i r e d f o r removing any high frequency h o r i z o n t a l motion of t h e p r o f i l i n g f i s h t h a t a r e induced by t h e s h i p p l u s a l l v e r t i c a l flow components. The c u r r e n t meters deployed w i t h t h e APS w e r e developed f o r measuring t u r b u l e n t f l u c t u a t i o n s i n r i v e r s , e s t u a r i e s and ocean boundary l a y e r s , and a r e d e s c r i b e d by Smith ( 1 9 7 4 ) .

When used i n t h e p r o f i l i n g mode, t h e i r 1 cm/sec t h r e s h o l d

v e l o c i t y i s n o t a problem even i n very slow flows and their c a l i b r a t e d accuracy of b e t t e r than 0 . 1 cm/sec ensures t h a t t h e measurements w i l l be obtained w i t h more p r e c i s i o n than the s t a b i l i t y of t h e r e f e r e n c e frame can be guaranteed. Pickard and Rodgers (1959) have emphasized t h e n e c e s s i t y o f c o r r e c t i n g v e l o c i t y measurements f o r s h i p motion; t o this end we used a Mini-Ranger

I1

microwave n a v i g a t i o n system w i t h t h e o u t p u t recorded on magnetic t a p e alongside t h e APS d a t a .

A f t e r analyzing t h e APS d a t a , i t was.discovered

t h a t when t h e

v e l o c i t y p r o f i l e s w e r e averaged over a p e r i o d of approximately one hour (10 t o 1 2 p r o f i l e s ) the r e s u l t i n g v e l o c i t y component p r o f i l e s were w i t h i n a f r a c t i o n

of a centimeter per second of each o t h e r a t a l l l e v e l s whether o r n o t t h e navigation c o r r e c t i o n was made.

Since s h i p motion c o r r e c t i o n c e r t a i n l y should

be made t o each i n d i v i d u a l p r o f i l e when t h e i n s t r u m e n t i s near t h e s u r f a c e and a s such a c o r r e c t i o n probably should n o t be made when t h e APS i s a t g r e a t depth, t h i s important r e s u l t g r e a t l y s i m p l i f i e s t h e d a t a a n a l y s i s procedure.

It

implies also t h a t the accuracy o b t a i n e d w i t h s h i p mounted p r o f i l i n g systems may be s u b s t a n t i a l l y i n c r e a s e d by averaging p r o f i l e s over a s u i t a b l e p e r i o d . The Surface Layer Frame w a s designed t o head i n t o t h e mean flow even under r e l a t i v e l y rough c o n d i t i o n s and i t has proven s u c c e s s f u l i n t h i s t a s k on a

number o f occasions.

I n the Knight I n l e t experiment the frame supported a

Seabird E l e c t r o n i c s (SBE) c o n d u c t i v i t y c e l l , a S e a b i r d E l e c t r o n i c s temperature s e n s o r , a Marsh-McBirney two a x i s electromagnetic c u r r e n t meter, two p a i r s of p r e c i s i o n mechanical c u r r e n t meters, a two a x i s t r a v e l t i m e a c o u s t i c c u r r e n t meter made by t h e C h r i s t i a n Michaelson I n s t i t u t e i n Bergen, Norway, a Paros c i e n t i f i c p r e s s u r e c e l l and an Aanderaa compass. t h e instrument l a y o u t on the frame. oriented t o flush horizontally,

The i n s e t i n Figure 3 shows

A s the c o n d u c t i v i t y c e l l on t h e SLF was

t h i s system could be l e f t suspended a t a f i x e d

depth o r r a i s e d and lowered slowly r e l a t i v e t o the h o r i z o n t a l c u r r e n t speed. S e v e r a l of t h e s e n s o r s on this frame i n c l u d i n g the SBE temperature and c o n d u c t i v i t y s e n s o r s were q u i t e new and t h e i r s p e c i f i c a t i o n s had n o t y e t been checked under f i e l d c o n d i t i o n s .

R e s u l t s from t h e l a t t e r were compared t o

471 p r o f i l e s from t h e G u i l d l i n e System on the APS during q u i e t p e r i o d s a s was done

w i t h the Aanderaa c o n d u c t i v i t y c e l l s d e s c r i b e d subsequently.

I n both c a s e s this

approach w a s s u f f i c i e n t f o r t h e s c i e n t i f i c g o a l s of t h e study b u t i t d i d n o t comprise a very s t r i n g e n t t e s t o f t h e s t a b i l i t y of t h e SBE s e n s o r s .

Similarly,

t h e electromagnetic flow s e n s o r o u t p u t s were compared t o r e s u l t s from the p r e c i s i o n mechanical c u r r e n t meters and were found t o be r e l i a b l e t o w i t h i n 2 cm/sec.

The Aanderaa c u r r e n t meters w e r e suspended w i t h o u t vanes from two p a r a l l e l cables.

The a r r a y w a s deployed twice; on t h e f i r s t occasion it remained i n t h e

water f o r about 8 hours a f t e r which t h e t a p e s w e r e removed from t h e instrument and processed i n the f i e l d .

Once i t w a s c l e a r t h a t t h e i n s t r u m e n t s w e r e s e t

a t reasonable l e v e l s and running s a t i s f a c t o r i l y , they w e r e redeployed w i t h new tapes f o r the r e s t of t h e experiment.

For t h e most p a r t t h e i n t e r n a l l y

recording c u r r e n t meter d a t a d i s c u s s e d i n this paper comes from t h e second deployment during which t h e instruments were a t 7 , 1 2 , 13.5, below t h e s u r f a c e .

1 5 and 20 meters

Use of such a s h o r t sampling p e r i o d s e v e r e l y l i m i t s t h e

r e s o l u t i o n of speed d a t a ; i n t h e p r e s e n t case r e s o l u t i o n i s k.041 m s - l .

3.

OBSERVATIONS

3.1

Background saZinity, temperature and v e Z o c i 9 f i e Z d .

S a l i n i t y p r o f i l e s ( F i g u r e 4 ) taken along t h e channel show the t r a n s i t i o n from the highly s t r a t i f i e d c o n d i t i o n s between Cascade P o i n t and the I n l e t head, and the weaker s t r a t i f i c a t i o n between Cascade P o i n t and the s i l l ( S t a t i o n 3 ) .

In

Figure 5 this a s p e c t i s p r e s e n t e d a g a i n i n t h e p l o t s of r e p r e s e n t a t i v e s a l i n i t i e s and pycnocline depths.

This f i g u r e a l s o emphasizes t h e v a r i a b i l i t y

of the s t r a t i f i c a t i o n between s t a t i o n s 3 and 1 0 .

A fundamental d i f f i c u l t y i n

t h e i n t e r p r e t a t i o n o f p r o f i l e sequences such as t h e s e a r i s e s from a l i a s i n g due t o i n t e r n a l waves.

The s t r u c t u r e o f t h e s e waves may o n l y be r e s o l v e d by

r a p i d time s e r i e s taken a t one l o c a t i o n . Throughout t h e time series a t S t a t i o n 5 , s u r f a c e c u r r e n t s o f 0.4 t o 0 . 7 ms-’ were observed, s e t t i n g predominately down i n l e t ; t h i s s u r f a c e flow appeared t o be due p r i m a r i l y t o the wind which blew down i n l e t a t moderate speed.

Beneath

t h e s u r f a c e the flow v e l o c i t y f e l l o f f rather s h a r p l y , e s p e c i a l l y i n the upper 8 meters reaching zero near t h i s l e v e l d u r i n g the f l o o d .

water w a s i n t h i s s u r f a c e l a y e r .

Most of t h e r i v e r

A s t h e wind d r i v e n and e s t u a r i n e c i r c u l a t i o n s

were enhancing one a n o t h e r throughout t h e e n t i r e experimental p e r i o d i t w a s n o t p o s s i b l e t o s e p a r a t e them w i t h any degree of c e r t a i n t y .

Furthermore, the

s i t u a t i o n was complicated by t h e t i d e s which a t t h e experimental s i t e caused

the flow to r e v e r s e below about 8 m, b u t w i t h a phase l a g o f two and one h a l f hours

r e l a t i v e t o t h e t i d a l e l e v a t i o n and t h e t i d a l c u r r e n t over t h e s i l l a t

Hoeya Head.

472

STATIONS

5%0

Figure 4 . S a l i n i t y p r o f i l e s a t 20 s t a t i o n s along the a x i s o f Knight I n l e t . From s t a t i o n 1 7 t o the f j o r d head t h e s u r f a c e w a t e r w a s f r e s h and t h e s u r f a c e l a y e r was less than o r e q u a l t o 1 m; t o s i m p l i f y t h e diagram the upper p a r t s of these p r o f i l e s a r e n o t shown.

.-.-.-.-.\.A-

0

5

15

20

DISTANCE

25

30

35

40

45

UPSTREAM I N KILOMETERS

Figure 5. Typical s u r f a c e l a y e r c h a r a c t e r i s t i c s between Hoeya Head and the K l i n a k l i n i River. The upper graph shows a r e f e r e n c e s a l i n i t y a t 35 meters below the s u r f a c e , t h e s a l i n i t y a t t h e base o f the pycnocline and the s u r f a c e s a l i n i t y a s a f u n c t i o n o f p o s i t i o n along t h e a x i s o f t h e i n l e t ; t h e lower one shows t h e depth o f t h e pycnocline and t h a t o f the mixed l a y e r base.

50

473 Figure 6 shows approximately one hour average s a l i n i t y , temperature and velocity profiles.

Both t h e temperature and s a l i n i t y p r o f i l e s i n d i c a t e a

r e l a t i v e l y poorly mixed s u r f a c e l a y e r and a c l e a r t i d a l v a r i a t i o n i n i t s thickness.

The temperature v a r i a t i o n i s s m a l l and t h e d e n s i t y p r o f i l e i s

dominated by s a l i n i t y .

Although t h e s t r a t i f i c a t i o n i s q u i t e s t r o n g , s a l i n i t y

decreases a l l t h e way t o t h e s u r f a c e a t a l l s t a g e s o f t h e t i d a l c y c l e and t h e pycnocline has no w e l l d e f i n e d base because t h e r a t e o f s a l i n i t y i n c r e a s e i s more o r l e s s uniform i n i t s lower p a r t .

A s can be seen from Figure 7 , this

g e n e r a l s t r u c t u r e i s n o t an a r t i f a c t of averaging, b u t i s d i s p l a y e d i n each of t h e i n d i v i d u a l p r o f i l e s as w e l l .

I n f a c t , t h e only major d i f f e r e n c e between

t h e averaged p r o f i l e s o f Figure 6 and t h e ones p r e s e n t e d i n t h e subsequent f i g u r e i s t h e removal o f high frequency and high wave number s t r u c t u r e i n t h e former.

TEMPERATURE

7" L - L d 9°C

L

UPSTREAM VELOCITY

CROSS-STREAM VELOCITY

1800

I I NOV 76

0000

0600

1200

1600

12 NOV 76

Figure 6 . Sequence o f one hour average p r o f i l e s o f s a l i n i t y , temperature and upstream and cross-stream v e l o c i t y components f o r t h e p e r i o d between 1000 on 11 November and 1600 on 12 November. These p r o f i l e s w e r e obtained by averaging d a t a from approximately 1 0 i n d i v i d u a l lowerings o f t h e APS.

474

w

a

12 N G V 76

I / NGV 76

Y

t w

I%r

a

I2

NGV 7 6

Figure 7. I n d i v i d u a l s a l i n i t y p r o f i l e s from which t h e averaged s a l i n i t y data displayed i n Figure 6 were obtained. Note the e f f e c t s o f l a r g e i n t e r n a l waves around 0100 and a g a i n around 1300 on 1 2 November. I n a d d i t i o n note t h e l a y e r e d n a t u r e o f t h e pycnocline d u r i n g t h e ebb t i d e e s p e c i a l l y i n t h e e a r l y p a r t of t h e experiment.

The v e l o c i t y f i e l d is shown i n Figure 6c and d; Figure 6 c g i v e s the down i n l e t and Figure 6d t h e cross-stream component.

These i n d i c a t e a s u b s t a n t i a l

c r o s s - i n l e t component i n the observed v e l o c i t y f i e l d .

The p r o f i l e s i n Figure 6c

f o r t h e p e r i o d j u s t p r i o r t o s l a c k water are r e p r e s e n t a t i v e o f t h e measured flow during passage o f t h e waves.

Here the s h e a r i s l a r g e s t i n the upper 1 0

meters o f the flow and i s f a i r l y s m a l l below 30 meters.

Moreover the downstream

v e l o c i t y component i s c l o s e t o zero below 20.meters a t t h e t i m e t h e i n t e r n a l waves a r r i v e a t t h e experimental l o c a t i o n .

I n c o n t r a s t , t h e down i n l e t component

o f t h e s u r f a c e v e l o c i t y i s q u i t e high a t these t i m e s , p a r t i c u l a r l y a t mid-day. A t the v e s s e l l o c a t i o n the c u r r e n t i n the upper 100 meters i n c r e a s e s n o t i c e a b l y

with t h e passage o f the f i r s t n o n l i n e a r i n t e r n a l wave. Figure 8 shows the depth change o f c h a r a c t e r i s t i c isopycnals a s a f u n c t i o n of t i m e .

Here two main depth zones can be recognized.

Below about 30 m t h e

v a r i a t i o n i s approximately s i n u s o i d a l ; whereas, i n t h e pycnocline t h e r e i s a s h a r p plunge with passage o f t h e l a r g e i n t e r n a l waves followed by a gradual

rise over t h e rest o f t h e t i d a l c y c l e .

These f e a t u r e s are c l e a r l y t i d a l i n

c h a r a c t e r and appear t o be r e l a t i v e l y f r e e o f wind e f f e c t s .

The d a t a

comprising Figure 8 show the o v e r a l l temporal and v e r t i c a l s t r u c t u r e a t t h e experimental s i t e b u t a l s o have been averaged over a one hour p e r i o d t o remove high frequency f e a t u r e s , such a s i n t e r n a l waves.

475

I I NOV 76

12 NOV 76

Figure 8. Isopycnal depths as a f u n c t i o n of t i m e obtained from t h e averaged d a t a d i s p l a y e d i n Figure 6. Note t h e g r a d u a l r i s e then r a p i d d e s c e n t of the pycnocline depth i n c o n c e r t w i t h the semi-diurnal t i d e .

3.2

Internal Waves

Unfortunately, even i n t h e s a l i n i t y p r o f i l e s from i n s t a n t a n e o u s APS lowerings, t h e i n t e r n a l waves are incompletely sampled.

Nevertheless,

t h e i r presence i s

v i s i b l e and t h e n a t u r e of t h e changes i n v e r t i c a l s t r u c t u r e caused by their passage i s e v i d e n t .

I n g e n e r a l , the pycnocline i s d r a s t i o a l l y d i s t o r t e d and

deepened w i t h the passage of each wave, b u t r e t u r n s almost t o i t s o r i g i n a l shape a f t e r t h e wave goes by.

Figure 7 shows t h a t t h e s a l i n i t y i s s u b s t a n t i a l l y

decreased t o a depth of a t l e a s t 30 meters w i t h the passage of each wave and that under t h e trough t h e r e i s an almost l i n e a r decrease i n s a l i n i t y between

t h i s c h a r a c t e r i s t i c depth and t h e bottom of t h e s u r f a c e l a y e r . Figure 9 p r e s e n t s t h r e e t i m e s e r i e s , t h e deeper two of which c l e a r l y i n d i c a t e the r e l a t i o n s h i p between t h e l a r g e i n t e r n a l waves and t h e r a p i d deepening of

t h e s u r f a c e l a y e r , followed by a slow r i s e i n pycnocline depth.

I n Figure 1 0

the t i m e series from seven i n s t r u m e n t s are shown on a g r e a t l y expanded scale. A s each wave p a s s e s , the s a l i n i t y decreases corresponding to a deepening o f the

476 isohalines.

A t no p o i n t does the s a l i n i t y rise above t h e e q u i l i b r i u m l e v e l :

thus t h e waves may be viewed a s i s o h a l i n e d e p r e s s i o n s t r a v e l l i n g p a s t t h e p o i n t of o b s e r v a t i o n .

7 m

135

20 ,

,

,

,

~

1200 Hrs

Nov 12. 76

~

I

,,

,

,

,

l ,

,

, ~

,

i , , -~ - r - T

~

~

~

1200

Nov 13

Detailed t i m e s e r i e s from t h r e e Aanderaa instruments f o r 1 2 and 1 3 Figure 9. November. Note t h e high frequency o s c i l l a t i o n s a s s o c i a t e d w i t h t h e r i s e i n s a l i n i t y a t t h e two deeper i n s t r u m e n t s .

--

7 m

27 38 27 40

u

105 12

I35 15 165

20 *

Nov 12, 76

,

.

;

:

:

:

1200 Hrs

:

:

.

:

,

:

:

:

*

:

;

:

1300

Figure 1 0 . Seven d e t a i l e d s a l i n i t y time s e r i e s f o r t h e p e r i o d of r a p i d pycnocline depth i n c r e a s e and high frequency s a l i n i t y o s c i l l a t i o n s . Here a t r a i n of f i v e s o l i t a r y - l i k e i n t e r n a l waves i s e v i d e n t .

The instrument from which t h e t o p r e c o r d i n g w a s o b t a i n e d i n F i g u r e s 9 and 10 was s i t u a t e d r e l a t i v e l y c l o s e t o t h e s u r f a c e i n a l a y e r w i t h a s u b s t a n t i a l l y lower s a l i n i t y g r a d i e n t and i t d i s p l a y s somewhat d i f f e r e n t f e a t u r e s .

For

~

~

477 example, d u r i n g t h e e v e n t j u s t a f t e r 1200 hours on 1 2 November, the i n t e r n a l waves i n t h e s u r f a c e l a y e r produce a s i g n a l t h a t i s n o t of s i g n i f i c a n t l y l a r g e r amplitude than a number o f o t h e r u n i d e n t i f i e d e v e n t s a t this l e v e l and the former can be found only by i t s coherence w i t h t h e c l e a r l y defined i n t e r n a l wave induced d i s t u r b a n c e s

i n the o t h e r two t r a c e s .

F u r t h e r , d u r i n g both the

e a r l y morning and e a r l y afternoon f l o o d t i d e s on 1 3 November, no evidence o f t h e l a r g e i n t e r n a l waves i s s e e n i n the s u r f a c e s a l i n i t y t i m e s e r i e s , whereas they c e r t a i n l y are v i s i b l e i n t h e o t h e r two r e c o r d s .

A l l a v a i l a b l e d a t a show

that t h e a b r u p t deepening was most pronounced on the a f t e r n o o n of 1 2 November, b u t that i t occurred t o some degree on a l l ebb t i d e s d u r i n g which measurements

w e r e being made.

From Figure 9 and even more c l e a r l y from Figure 10 it can be

seen that the change i n s a l i n i t y due t o t h e presence o f i n t e r n a l waves i s comparable t o t h a t due t o t h e a b r u p t change i n pycnocline depth. Although i n t e r n a l waves were observed d u r i n g s e v e r a l t i d a l c y c l e s , our b e s t d a t a are f o r the mid-day ebb on 1 2 November.

The f i r s t wave, which appears t o

be l o n g e r than t h o s e following it, a r r i v e d j u s t a f t e r 1220 P a c i f i c Standard Time.

I t l e d a well-defined

t r a i n of seven waves, the s i g n a t u r e s o f which a r e

c l e a r l y e v i d e n t i n the s a l i n i t y t i m e series o f Figure 10.

Data from t h e

Aanderaa m e t e r s (Figure 11) i n d i c a t e the change i n v e l o c i t y due t o passage of the f i r s t wave w a s c o n s i d e r a b l y less than that due t o subsequent e v e n t s .

Speeds from t h e Aanderaa i n s t r u m e n t s w e r e somewhat h i g h e r than those o b t a i n e d from the SLF, probably as a r e s u l t o f s h i p motion.

During most of t h e

experimental p e r i o d , s h i p motion w a s r e l a t i v e l y s m a l l b u t d u r i n g passage o f the f i r s t wave the s h i p d r i f t e d s i g n i f i c a n t l y i n t h e upstream d i r e c t i o n causing s u b s t a n t i a l d i s t o r t i o n of the s i g n a l s . n e g l i g i b l y a f f e c t e d by s h i p d r i f t .

S i g n a t u r e s of the subsequent waves were

I t i s e v i d e n t from Figure 1 0 t h a t t h e p e r i o d

between each of these succeeding waves d e c r e a s e s as they p a s s . Conductivity and v e l o c i t y r e c o r d s from the Surface Layer Frame during passage of the f i r s t two i n t e r n a l waves a l s o show t h e v e l o c i t y d i s t u r b a n c e a s s o c i a t e d w i t h t h e f i r s t wave t o be s u b s t a n t i a l l y s m a l l e r i n magnitude and t o l a g w e l l behind t h e s a l i n i t y response.

A f i r s t c o r r e c t i o n using the navigation d a t a

and t h e flow measurements from this frame y i e l d s an a c t u a l wave p e r i o d of approximately h a l f t h a t observed.

A s the s h i p motion followed

the wave motion

the wave shape a l s o was d i s t o r t e d r e s u l t i n g i n a broader a p p a r e n t trough.

4.

THEORETICAL CONSIDERATIONS

4.1

Theoretica 2 mode Zs f o r internal waves

E s t i m a t e s based on observed s a l i n i t y excursions d u r i n g passage of the i n t e r n a l waves shown i n Figures 9 and 1 0 s u g g e s t amplitudes i n excess o f 10 m.

As the

s t r o n g l y s t r a t i f i e d p a r t o f the water column i s only 15-20 meters deep i t i s e v i d e n t t h a t n o n l i n e a r e f f e c t s w i l l be important.

478

7 m -

0

20 -0

1200

Nov 12. 76

Hrs

I300

Figure 11. Aanderaa c u r r e n t meter r e c o r d s from f o u r depths f o r t h e time p e r i o d shown i n Figure 1 0 . I n o r d e r t o r e s o l v e t h e n o n l i n e a r i n t e r n a l waves, a 30 second sampling i n t e r v a l had t o be used i n t h e Aanderaa instruments causing the v e l o c i t y r e s o l u t i o n t o be reduced t o .041 cms-l. During passage o f t h e f i r s t wave t h e s h i p moved w i t h t h e wave producing a n e g l i g i b l e r e l a t i v e v e l o c i t y The v e l o c i t y f i e l d whereas this d i d n o t occur w i t h t h e subsequent two waves. a s s o c i a t e d w i t h t h e s m a l l e r i n t e r n a l waves a t t h e end o f t h e t r a i n was n o t w e l l resolved by Aanderaa instruments.

These l a r g e amplitude waves may be compared t o those photographed during t h e summer.

Figure 2 shows a view looking seaward from approximately 1 0 km up-

i n l e t of t h e s i l l .

The waves show up as colour v a r i a t i o n s presumably a s s o c i a t e d

with t h e varying t h i c k n e s s o f t h e s u r f a c e l a y e r , which c a r r i e s a suspension of rock f l o u r , although t h e s u r f a c e r u f f l e s d e s c r i b e d by Pickard (1961) a l s o h e l p t o make t h e waves v i s i b l e . flood t i d e .

In other

This photograph i s one o f t h e sequence taken d u r i n g

photographs i n t e r n a l waves can be seen r i g h t over t h e

s i l l , b u t i n t h e s e t h e p a t t e r n i s h i g h l y complex; t h e simple banded s t r u c t u r e of Figure 2 w a s found some d i s t a n c e from t h e s i l l and appeared t o be advancing up-inlet. From t h e observed d e n s i t y v a r i a t i o n i n the s u r f a c e l a y e r a phase-speed o f

0.5 ms-’

may be e s t i m a t e d u s i n g long wave theory.

much h i g h e r speed (1 ms-’)

This may be compared w i t h t h e

r e p o r t e d by Pickard (19611, presumably based mainly

on summer measurements when t h e s t r a t i f i c a t i o n i s more i n t e n s e .

Since t h e

479 i s o h a l i n e depressions p a s s t h e p o i n t o f o b s e r v a t i o n i n t h r e e o r f o u r minutes,

the h o r i z o n t a l s c a l e of the wave must be of o r d e r 70-100 m.

Thus, a t l e a s t

on t h e b a s i s of this c a l c u l a t i o n t h e waves appear t o be long w i t h r e s p e c t t o t h e depth o f t h e s u r f a c e l a y e r .

On t h e o t h e r hand t h e t o t a l depth of t h e

f j o r d i n t h i s reach exceeds 400 m, s o t h a t t h e shallow w a t e r approximation i s n o t s t r i c t l y v a l i d beneath t h e s t r a t i f i e d l a y e r . Trains o f l a r g e amplitude waves which b e a r a q u a l i t a t i v e resemblance

to those

shown i n Figure 10 have been observed both i n l a k e s (Hunkins and F l i e g e l , 1973, Thorpe e t a l , 1972, Farmer 1978) and i n t h e ocean (Halpern, 1971 and Ziegenbein, 1969).

Hunkins and F l i e g e l i n t e r p r e t e d the Seneca Lake waves i n terms of

s o l u t i o n s t o t h e Korteweg-de Vries equation which w a s d e r i v e d f o r long nonlinear s u r f a c e g r a v i t y waves and subsequently found t o be a p p l i c a b l e t o c e r t a i n c l a s s e s of long i n t e r n a l waves.

The same equation was l a t e r extended

t o second o r d e r by Lee and Beardsley (1974) and a numerical i n t e g r a t i o n o f t h e i n i t i a l value problem w a s s u c c e s s f u l l y a p p l i e d t o Halpern's Massachusetts Bay observations

.

The g r e a t depth of Knight I n l e t p r e c l u d e s a n a n a l y s i s based on t h e long wave approximation throughout t h e f l u i d .

However t h e e x i s t e n c e o f i n t e r n a l waves of

permanent form i n f l u i d s o f g r e a t depth h a s been demonstrated by Benjamin (1967) and by Davis and Acrivos (1967).

Benjamin considered a d e n s i t y s t r u c t u r e

composed of two r e g i o n s : a t h i n heterogenous l a y e r (Region I ) and a n i n f i n i t e l y deep l a y e r (Region 11) o f uniform d e n s i t y .

We s h a l l apply a modified v e r s i o n

of h i s model t o Knight I n l e t and then compare the t h e o r e t i c a l r e s u l t s w i t h our observations. A f e a t u r e of t h e waves t o be expected i n t h i s c a s e i s t h a t t h e r e l e v a n t

d i s p e r s i o n r e l a t i o n has l e a d i n g terms of t h e form w = k c , ( l where c, i s t h e speed o f an i n f i n i t e s i m a l wave. s t e a d y wave s o l u t i o n s o f the Korteweg-de

-

y Ik/),y>O

A s i s a l s o t h e case f o r the

V r i e s e q u a t i o n , the d e p a r t u r e o f the

phase speed from t h a t o f a n i n f i n i t e s i m a l wave i n c r e a s e s with amplitude

a.

However the r e l a t i o n s h i p between t h e h o r i z o n t a l s c a l e h and t h e v e r t i c a l s c a l e of s t r a t i f i c a t i o n h, is

aX = O(h2) i n c o n t r a s t t o t h e classical s o l i t a r y wave

r e l a t i o n s h i p ah2 = O ( h 3 ) . Benjamin p o s t u l a t e d t h e equation:

and C i s a c o n s t a n t , and i n e f f e c t , he examined s o l i t a r y and p e r i o d i c s o l u t i o n s of

(1) depending only on t h e wave co-ordinate x ' = x-ct.

Although e q u a t i o n (1)

480 was p r e s e n t e d by Benjamin, it was n o t formally d e r i v e d by him and he d i d n o t use i t i n h i s subsequent a n a l y s i s .

I t i s o f i n t e r e s t however, i n t h a t i t

c o n s t i t u t e s a c o u n t e r p a r t t o t h e Korteweg-de

Vries e q u a t i o n f o r t h e case of an

i n f i n i t e l y deep f l u i d ; it was f i r s t d e r i v e d by G a r g e t t (1976) who was a b l e t o show by a comparison theorem t h a t a t l e a s t f o r harmonic i n i t i a l c o n d i t i o n s , a t r a i n o f s o l i t a r y waves evolves, s i m i l a r t o t h e known s o l u t i o n t o t h e Kortewegde Vries equation.

What has n o t y e t been e s t a b l i s h e d i s t h a t equation (1)

y i e l d s a t r a i n o f s o l i t a r y waves f o r an a r b i t r a r y i n i t i a l c o n d i t i o n , a s does t h e KdV equation b u t t h e evidence i s s u g g e s t i v e .

The p r e s e n t o b s e r v a t i o n s do

not p e r m i t a d e t a i l e d a n a l y s i s o f t h e g e n e r a t i o n mechanism; however w e have a c l e a r p i c t u r e o f t h e waves themselves as they p a s s t h e p o i n t of o b s e r v a t i o n , and t h e s e o b s e r v a t i o n s permit a comparison of t h e wave p r o p e r t i e s w i t h Benjamin's theoretical predictions. While r e f e r e n c e may be made t o the o r i g i n a l paper (Benjamin, 1967) f o r mathematical d e t a i l s , we n o t e h e r e those a s p e c t s d i r e c t l y a p p l i c a b l e t o an a n a l y s i s of the Knight I n l e t d a t a .

Figure 1 2 shows p r o f i l e s o f d e n s i t y and

v e l o c i t y averaged over approximately one hour p r i o r t o t h e a r r i v a l of t h e waves presented i n Figure 1 0 ; i t a l s o i n d i c a t e s how t h e f l u i d was d i v i d e d i n t o two regions as r e q u i r e d by the theory.

The s t r o n g down-inlet s u r f a c e flow i s l i k e l y

t o be important i n the p r e s e n t c a s e and t h e a n a l y s i s i s modified as suggested i n S e c t i o n 6 of Benjamin's

(1967) paper.

Application of t h e theory f o r s p e c i f i c d e n s i t y and v e l o c i t y d i s t r i b u t i o n s must begin w i t h t h e s o l u t i o n of the eigenvalue problem f o r i n f i n i t e s i m a l waves.

5 be t h e v e r t i c a l displacement p e r t u r b a t i o n and S ( x , n )

Let

= +,(q)eikx w i t h k r e a l ,

and q t h e depth below t h e e q u i l i b r i u m f r e e s u r f a c e ; t h e r e s u l t i n g SturmL i o u v i l l e problem i n Region I t h e n becomes

with t h e f r e e s u r f a c e c o n d i t i o n :

and

(3)

A l t e r n a t i v e l y , with l i t t l e loss of accuracy f o r most p r a c t i c a l a p p l i c a t i o n s we may t a k e t h e f i x e d upper boundary c o n d i t i o n , + , ( O )

= 0.

The e q u i v a l e n t problem

f o r Region 11, i n which t h e r e i s no s h e a r , remains unchanged.

Solutions

s a t i s f y i n g t h e s e boundary c o n d i t i o n s must be found by choosing t h e a p p r o p r i a t e

481 wave speeds c, 1

,

c, 2

,...

f o r s u c c e s s i v e modes.

However, i n what follows w e

s h a l l drop t h e s u b s c r i p t and t r e a t only the lowest i n t e r n a l mode, a s only this mode appears to be a p p l i c a b l e t o t h e Knight I n l e t d a t a .

I t i s a l s o convenient

t o normalize t h e e i g e n f u n c t i o n by t a k i n g $,(h) = 1.

0,2

,

t

' 0 '

U

I

Figure 1 2 . Sigma-t and down-inlet v e l o c i t y component p r i o r t o passage of t h e I n t h e numerical c a l c u l a t i o n s described i n f i r s t s o l i t a r y - l i k e i n t e r n a l wave. s e c t i o n 5.1, t h e measured a t and v e l o c i t y component d a t a o f Region I w e r e used. I n t h e s i m p l i f i e d a n a l y t i c a l model d e s c r i b e d i n s e c t i o n 4.3, t h e dashed curve was used. For both models v e l o c i t y and u t a r e assumed independent of depth i n Region I1 as i n d i c a t e d by s o l i d l i n e s . I n t h e c a l c u l a t i o n s the v e l o c i t y a x i s i s s h i f t e d so that t h e v e l o c i t y i n Region I1 can be taken as zero. I n e q u a t i o n ( 2 ) W, r e f e r s t o the q u a n t i t y U*c, of t h e undisturbed f l u i d i n t h e x - d i r e c t i o n ,

where U ( q ) denotes t h e v e l o c i t y

c a l c u l a t e d with r e s p e c t t o t h e

v e l o c i t y i n Region I1 ( w e t a k e U = 0 i n Region 11). S i m i l a r l y , W = U + c where c is t h e t r u e wave speed, so t h a t W r e p r e s e n t s t h e primary v e l o c i t y as observed i n a frame o f r e f e r e n c e t r a v e l l i n g w i t h t h e wave; W, f u n c t i o n f o r i n f i n i t e s i m a l waves. propagation w i t h ( - )

i s t h e corresponding

The ambiguous s i g n covers t h e c a s e of

o r a g a i n s t (+) t h e c u r r e n t .

482 The e x i s t e n c e of a non-uniform b a s i c flow introduces some s p e c i a l p h y s i c a l considerations.

F i r s t , t h e a n a l y s i s i s based upon the c o n d i t i o n o f dynamic

s t a b i l i t y t o small disturbances; w e may assume s t a b i l i t y i n t h i s sense provided 4g p

-' p,>.U; .

The mean flow and d e n s i t y d i s t r i b u t i o n s shown i n Figure 1 2

s a t i s f y this condition, however these p r o f i l e s are based on an average o f s e v e r a l p r o f i l e s each of which i s s e p a r a t e d by s e v e r a l minutes.

Even i f t h e

i n d i v i d u a l c a s t s showed dynamically s t a b l e c o n d i t i o n s , we cannot be s u r e that t h e c o n d i t i o n i s instantaneously s a t i s f i e d ; moreover the p e r t u r b a t i o n induced by the passage of t h e wave i t s e l f c o n t r i b u t e s t o t h e s h e a r .

This i n t e r a c t i o n

of the wave with t h e b a s i c v e l o c i t y f i e l d m u s t be considered a primary candidate f o r s h e a r flow i n s t a b i l i t y and r e s u l t a n t generation of turbulence and mixing. Notwithstanding this complication and its oceanographic i m p l i c a t i o n s Benjamin's r e s u l t s appear t o o f f e r a reasonable d e s c r i p t i o n o f the waves, and a l s o permit an examination of t h e c o n d i t i o n s l i k e l y t o lead t o such i n s t a b i l i t i e s . Secondly, t h e e x i s t e n c e o f t h e b a s i c flow has a d d i t i o n a l i m p l i c a t i o n s f o r t h e wave p r o p e r t i e s as discussed by Benjamin (1962).

I n p a r t i c u l a r , both t h e phase-

speed and t h e h o r i z o n t a l s c a l e of wave motion a r e modified although t h e f u n c t i o n a l form o f the wave shape remains unchanged.

However, a d i f f i c u l t y

a r i s e s i f t h e b a s i c flow i s such that t h e primary v e l o c i t y W ( n ) becomes very small o r negative a t some p o i n t of t h e p r o f i l e .

I n t h i s case s e p a r a t i o n of t h e

flow must be expected, t h a t is, a s t a g n a n t region of f l u i d i s c a r r i e d along by t h e wave, so t h a t although a steady wave might s t i l l be p o s s i b l e , t h e a n a l y s i s used here w o u l d not be a p p r o p r i a t e .

Recent work by Redekopp (1977) has shown

how such closed s t r e a m l i n e problems may be handled and has i n d i c a t e d t h e complex eddy s t r u c t u r e s that r e s u l t . I n t h e p r e s e n t c o n t e x t , and f o r a b a s i c flow U ( q ) of t h e shape shown i n Figure 1 2 , s e p a r a t i o n of flow would be expected a t t h e f o o t of Region I i f t h e flow opposing a wave was s u f f i c i e n t t o cause W ( n ) become negative.

= U(q)

+

c t o vanish o r

For a wave t r a v e l l i n g W i t h t h e s u r f a c e flow W ( q )

= U(q)

-

c

would be l e a s t near t h e s u r f a c e so t h a t s e p a r a t i o n would be expected a t this point.

Observations taken a t one l o c a t i o n cannot unambiguously determine the

d i r e c t i o n of propagation.

However, our observations during the summer a s w e l l

a s p h y s i c a l c o n s i d e r a t i o n s , i n d i c a t e a t r a i n of waves t r a v e l l i n g up-inlet,

i s to say a g a i n s t the s u r f a c e flow.

that

Moreover our photographs show t h e waves

t o be q u i t e s t r a i g h t s o t h a t it seems j u s t i f i e d t o t r e a t them as plane waves t r a v e l l i n g along t h e a x i s o f t h e i n l e t .

On t h i s assumption, c a l c u l a t i o n of

W ( n ) shows t h a t t h i s q u a n t i t y i s everywhere g r e a t e r than zero f o r lowest mode

i n t e r n a l waves, and the problem of flow s e p a r a t i o n w i l l n o t be considered further

.

Having found t h e phase speed c o of i n f i n i t e s i m a l waves, a s w e l l a s t h e function

+n and

-

-

+ r l q , we n e x t c a l c u l a t e t h e i n t e g r a l c o e f f i c i e n t s U and V f o r t h e nonlinear

483 equation

-

U f

-

v f2

+P

of t h e i n t e r f a c e 17 = h.

[f]

= 0

where f (x) i s now t h e v e r t i c a l displacement

Again allowing f o r a b a s i c flow U ( n ) , Benjamin's

formulae f o r t h e s e c o e f f i c i e n t s ( e q u a t i o n 3.52,

3.53) become

For s o l i t a r y waves t h e r e l a t i o n s h i p between t h e h o r i z o n t a l s c a l e A and amplitude a i s then given by:

Solving f o r t h e ( n o n l i n e a r ) wave speed c w e f i n d

and where q = .75 1 3 / 1 2 .

where p = - j I l / I 2 ,

The i n t e g r a l s a r e o b t a i n e d by w r i t i n g o u t t h e c o e f f i c i e n t s

and

i n full:

where j = +1 o r -1 f o r waves t r a v e l l i n g a g a i n s t o r w i t h t h e flow r e s p e c t i v e l y . The s i g n o f t h e r o o t i n ( 7 ) i s e s t a b l i s h e d by t h e requirement t h a t c = c o f o r

a

= 0.

Knowing t h e phase-speed,

the horizontal s c a l e A .

is

f ( x ) = aA2(x2

4.2

+

the coefficient

may be evaluated and by ( 6 )

F i n a l l y , t h e v e r t i c a l displacement o f a s t r e a m l i n e

~21-l.

D y m i c a l s t a b i l i t y f o r a wave i n sheared f l o w

The dynamical s t a b i l i t y of t h e f l u i d i n t h e neighbourhood of t h e wave can be examined i n terms of changes to t h e g r a d i e n t Richardson N u m b e r , R i .

I n most

484 p r a c t i c a l c a s e s h o r i z o n t a l g r a d i e n t s are much less than v e r t i c a l g r a d i e n t s , s o we may take t h e customary d e f i n i t i o n

The d i s t o r t i o n of t h e flow f i e l d due t o passage of an i n t e r n a l wave can be found from t h e s t r e a m l i n e displacement r e l a t i o n z =

u

R.

= (c ?

U)/z

I-

= g p-l

03

where O = 1

I-

+ af(x) +(I-).Since

( 9 ) becomes

[uI- -

+ af(x)$

(c

,-11-2

c U) af

.

The dominant term i n b r a c k e t s

[ ] i n (10) w i l l determine which of two e f f e c t s

a l t e r the Richardson N u m b e r as t h e wave p a s s e s .

Suppose f i r s t t h a t

IU,[>>I ( c i U) f~$ O-’[ corresponding t o t h e s i t u a t i o n i n which t h e s h e a r of

nn

t h e unperturbed flow was f a r g r e a t e r than s h e a r induced by the wave i t s e l f . This s i t u a t i o n might be expected c l o s e t o t h e s u r f a c e as i n d i c a t e d by t h e Knight I n l e t p r o f i l e s shown i n Figure (12): then fi

P-l 03u --2

pn

(11)

n

and t h e Richardson Number i s seen t o be modulated by t h e term (1 + a f Thus s t a b i l i t y

increases d u r i n g passage o f t h e wave.

+n ) 3.

P h y s i c a l l y we may

i n t e r p r e t t h i s as t h e e f f e c t of s t r e t c h i n g of t h e s t r e a m l i n e s l e a d i n g t o r e d u c t i o n of t h e b a s i c s h e a r .

Of course i s o p y c n a l s are also s t r e t c h e d , b u t

t h e Richardson N u m b e r depends upon t h e square of t h e v e l o c i t y g r a d i e n t s o t h e d e s t a b i l i z i n g e f f e c t o f isopycnal s t r e t c h i n g i s r e l a t i v e l y l e s s important. I f t h e b a s i c s h e a r i s s m a l l and [ U

il

1<<1

(c

+_

U) f $I-IV 1 [ ,wave induced s h e a r

dominates and w e f i n d

with the x-dependence now governed by t h e term

I f we n e g l e c t f o r t h e p r e s e n t t h e i n f l u e n c e of amplitude on wave speed, d i f f e r e n t i a t i o n of

(13) with respect t o

a shows t h a t t h e Richardson Number w i l l

485 decrease w i t h i n c r e a s i n g amplitude u n l e s s t h e amplitude exceeds a value o f

2/(3f$q).

However t h e r e w i l l always be a depth a t which $

example a t

TI

= h i n Benjamin's

r7

vanishes, f o r

(1967) model, so t h i s c o n s t r a i n t on the i n f l u e n c e

of amplitude on s h e a r flow i n s t a b i l i t y d i s a p p e a r s i n t h a t p a r t of t h e water column where $

r7

i s s u f f i c i e n t l y s m a l l , and the Richardson Number w i l l always

drop d u r i n g passage o f t h e wave.

I n p a r t i c u l a r , f o r $q<
I t is i n t h e lower p a r t o f t h e s t r a t i f i e d l a y e r , where e q u a t i o n ( 1 4 ) i s

a p p l i c a b l e , t h a t w e t h e r e f o r e e x p e c t minimum Richardson numbers and thus a g r e a t e r o p p o r t u n i t y f o r s h e a r flow i n s t a b i l i t y d u r i n g passage of t h e wave. Taking t h e c r i t e r i o n Ri = 1/4 f o r marginal s t a b i l i t y , w e can compute a c r i t i c a l amplitude such t h a t t h i s v a l u e i s j u s t achieved a t the wave c e n t e r x = 0.

Thus,

These r e s u l t s are q u i t e g e n e r a l ; a s p e c i f i c a p p l i c a t i o n using Benjamin's model allows comparison w i t h t h e waves observed i n Knight I n l e t .

(1967)

I n t h i s case

we combine e q u a t i o n s (15) and ( 7 ) t o o b t a i n :

where

r2

= (c,

- P I ' + acq

and

For given c o e f f i c i e n t s c,,

k,

p , q , $ r 7 q , t h e c r i t i c a l amplitude a

may be

recovered using Cardan's s o l u t i o n t o t h e c u b i c (16) o r by numerical methods.

4.3

S o l u t i o n for exponential s t r a t i f i c a t i o n ahd shear'

While t h e eigenvalue problem ( 2 ) i n Region I may be solved numerically f o r observed v e l o c i t y and d e n s i t y d i s t r i b u t i o n s , it i s o f i n t e r e s t f i r s t t o c o n s i d e r a simple a n a l y t i c a l r e s u l t .

Benjamin (1967) p r e s e n t e d a n example i n

which t h e d e n s i t y changed e x p o n e n t i a l l y ; w e s h a l l keep t h e same d e n s i t y d i s t r i b u t i o n b u t now i n c l u d e a v e l o c i t y p r o f i l e that d e c r e a s e s e x p o n e n t i a l l y with

486 depth t o zero a t t h e i n t e r f a c e h.

W e s h a l l r e s t r i c t t h e a n a l y s i s t o waves

t r a v e l l i n g upstream, as observed i n Knight I n l e t and d e s c r i b e t h e d e n s i t y and v e l o c i t y s t r u c t u r e with three independent parameters

5, 6 and h.

Let

P(Q) =

P

e

5 (r7-h)

The c o n s t a n t a i s n o t known

a pfio.i,

b u t i s determined as a s o l u t i o n t o t h e

eigenvalue problem f o r a p a r t i c u l a r mode and f o r given values of 6 , 6 and h. I n p a r t i c u l a r we r e l a t e u t o t h e phase speed:

c, = u - U ( 0 ) .

The v e l o c i t y

and d e n s i t y f i e l d of (17) may be r e p r e s e n t e d by the dimensionless parameter and t h e eigenvalue problem now c o n s i s t s o f determining 6h a s . a

R, =

f u n c t i o n of R,

and t h u s u and c, a s a f u n c t i o n o f

5, 6 and h .

From ( 1 7 ) we have

which when s u b s t i t u t e d i n t o ( 2 ) g i v e s

The g e n e r a l s o l u t i o n t o (18) i s

where j i =

i

= 1

+

KOexp(6n)

6/26.

and where t h e o r d e r o f t h e Bessel f u n c t i o n s i s

Even f o r weak s h e a r s 5<<26 so we may s e t

p r o f i l e s i n Figure 1 2 , 5/26

2

i

= 1.

For t h e

0.5 x

Application o f t h e boundary c o n d i t i o n s t h e n l e a d s t o the e q u a t i o n

where $,

=

KO,$ J ~=

ioexp(6h).

Zeroes o f ( 1 9 ) a r e t a b u l a t e d ( c . f . Abramowitz

and Stegun, 1968, page 4 1 5 ) ; t h e s e zeroes now d e f i n e t h e r e l a t i o n s h i p between R,

and 6h from which u and c, may be determined a s a f u n c t i o n of 5, 6 and h .

487 The i n t e g r a l s ( 8 ) must be numerically e v a l u a t e d t o f i n d t h e c o e f f i c i e n t s

-

V.

Table I g i v e s t h e r e l a t i o n s h i p between 6h, R,,

p/a,

q / a 2 , aX/h2,

and

acfi f o r

t a b u l a t e d f i r s t mode zeroes of ( 1 9 ) .

TABLE I

Dimensionless wave parameters f o r e x p o n e n t i a l s h e a r and d e n s i t y model 6h

RO 43.00 6.78 2.15 1.54 1.17 -617

-~

0.223 0.511 0.811 0.916 1.012 1.253 ~~

P/a

s/a

- .128 - -217

.0201

-685

.0161

-555

-0125 .0114 .0105 .0083

-447 .413 .385 .321

- .245 - - 243

-.238 215

-.

aX/h2

ac/h -796 -667 .560 .528 .501 -441

~~

A s a numerical example, c o n s i d e r t h e v a l u e s

6

= 1.37 x

m-l

6 = 2.56 x

m-l

h = 2 0 m

a = 10m which a r e a n approximate r e p r e s e n t a t i o n o f t h e c o n d i t i o n s shown i n Figure 12. From T a b l e I w e f i n d R,

= 6 . 7 8 which i m p l i e s a = .550 ms-’,

I n Table I1 w e compare t h e values f o r c o , c and

X with

U(0) = - 2 2 0 ms.-l.

t h o s e o b t a i n e d from

Benjamin’s model w i t h the same d e n s i t y d i s t r i b u t i o n b u t w i t h o u t s h e a r .

TABLE I1

Comparison of wave p r o p e r t i e s w i t h and w i t h o u t opposing s h e a r Wave amplitude 1 0 m With s h e a r co

.330 ms-l

.467 ms-’

*

C

.420 m s

.572 ms..’

*

A

*

Without s h e a r

22.2

-’

m

m

32.4

I n a f i x e d r e f e r e n c e frame t h e s e speeds are i n c r e a s e d by the mean Region I1 v e l o c i t y of - 0 6 2 m s --l.

The presence o f an opposing s h e a r h a s t h e e f f e c t of slowing down the waves and decreasing t h e i r h o r i z o n t a l l e n g t h s c a l e .

The s h e a r i n f l u e n c e s t h e

dynamical s t a b i l i t y by reducing t h e c r i t i c a l amplitude a marginal s t a b i l i t y a t t h e f o o t o f Region I .

Solving (16) f o r t h e numerical

values given above w e f i n d t h a t i n the sheared flow a unsheared case a

= 17.5

m.

r e q u i r e d t o produce

= 13.34

m, while i n t h e

C a l c u l a t i o n o f t h e g r a d i e n t Richardson Number from

(10) throughout t h e water column, shows t h a t f o r a wave o f 1 3 . 3 4 m t h e s t a b i l i t y

488 a t x = 0 i s i n c r e a s e d above 1 7 I? b u t s h a r p l y decreased below this depth. (13) gives c r i t i c a l amplitudes f o r o t h e r values of R,,,

Figure

together with the

r e l a t i o n s h i p f o r 6h.

Figure 1 3 . Curves r e l a t i n g R, t o 6h and ac/h t o 6h. The f i r s t curve g i v e s t h e r e l a t i o n s h i p between t h e flow parameters a , 8 , 6 and h s a t i s f y i n g t h e f i r s t mode s o l u t i o n t o equation (19). The second permits e v a l u a t i o n of t h e c r i t i c a l amplitude of a wave corresponding t o a given v a l u e of 6h.

Our a n a l y s i s has been based upon a model which assumes Region I1 t o be u n s t r a t i f i e d and this has n e c e s s a r i l y r e s t r i c t e d t h e d i s c u s s i o n of Richardson Numbers t o the s u r f a c e l a y e r .

I n p r a c t i c e however, t h e r e i s weak s t r a t i f i c a t i o n

i n Region 11, as s e e n i n Figure 1 2 and wave induced s h e a r s may be expected t o lead t o i n s t a b i l i t i e s and mixing on b o t h s i d e s of t h e i n t e r f a c e . To summarize t h z s e r e s u l t s we may say t h a t the waves t e n d t o i n c r e a s e the

s t a b i l i t y of sheared flow a g a i n s t which they a r e t r a v e l l i n g i n t h e upper p a r t of the s t r a t i f i e d l a y e r , while d e c r e a s i n g t h e s t a b i l i t y near i t s b a s e .

For t h e

c o n d i t i o n s p r e v a i l i n g i n Knight I n l e t d u r i n g November q u i t e l a r g e waves are r e q u i r e d t o reduce Ri below 1/4.

O f course t h e background wave f i e l d needs t o

be considered a s w e l l and i n t h e presence o f o t h e r i n t e r n a l waves t u r b u l e n c e may be generated by smaller s o l i t a r y waves.

I t i s of i n t e r e s t t h a t the waves

observed i n o u r p i l o t s t u d y w e r e comparable i n s i z e t o t h o s e r e q u i r e d t o produce marginal i n s t a b i l i t y , s u g g e s t i n g that waves of t h i s amplitude c o n s t i t u t e an upper l i m i t f o r t h e s e c o n d i t i o n s .

Larger waves would l o s e energy r a p i d l y u n t i l

they approached t h i s c r i t i c a l amplitude.

489

5.

DISCUSSION AND CONCLUSIONS

5.1

Comparison w i t h nwnerically evaluated wave p r o p e r t i e s

A c l o s e r comparison between o b s e r v a t i o n s and t h e t h e o r e t i c a l p r e d i c t i o n s can

be made by s o l v i n g ( 2 ) numerically f o r t h e measured d e n s i t y and v e l o c i t y p r o f i l e of Figure 1 2 .

The eigenvalues were o b t a i n e d by a shooting method.

phase speed c o , i n t e g r a t i o n by t h e f o u r t h o r d e r Runge-Kutta a t Q = h and t h e f r e e s u r f a c e c o n d i t i o n checked a t Q =

0.

For a t r i a l

scheme was begun Convergence was t h e n

sought by the method of s u c c e s s i v e b i s e c t i o n and i n v e r s e p a r a b o l i c i n t e r p o l a t i o n . Observations a t a s i n g l e p o i n t cannot be used t o determine t h e wave speed c directly.

However comparison o f the measured p e r t u r b a t i o n s w i t h Benjamin's

theory may be made using t h e c a l c u l a t e d wave speed t o transform t h e observed temporal v a r i a t i o n s i n t o s p a t i a l v a r i a t i o n s .

For example t h e t i m e A t

f o r the displacement of an i s o h a l i n e a t depth h , t o change from 1 / 2

required

a t o a, i s

r e l a t e d t o t h e h o r i z o n t a l s c a l e as cAt = A . The a c t u a l v e r t i c a l displacements may be i n f e r r e d from t h e observed s a l i n i t y excursion and t h e measured s a l i n i t y p r o f i l e i n t o which t h e waves a r e t r a v e l l i n g . This seemed a more s a t i s f a c t o r y procedure than computing contours o f c o n s t a n t s a l i n i t y based on an i n t e r p o l a t i o n of o b s e r v a t i o n s a t d i s c r e t e d e p t h s , p a r t i c u l a r l y s i n c e t h e time s e r i e s measurements o n l y covered t h e depth range 7 whereas the p r o f i l e s began a t the s u r f a c e .

-

20 m

Figure 1 4 shows t h e e s t i m a t e d

v e r t i c a l excursion a t 20 m f o r t h e p e r i o d d u r i n g which t h e waves shown i n Figure

1 0 w e r e observed.

I t w i l l be noted t h a t the waves r e p r e s e n t a depression of

s t r e a m l i n e s a s expected f o r a d e n s i t y s t r u c t u r e o f t h i s type.

The f i r s t o r d e r

theory i n d i c a t e s t h a t l a r g e r amplitude waves would t r a v e l f a s t e r and a l s o have a smaller horizontal s cal e. The amplitude o f each wave i s such t h a t t h e v a l i d i t y o f an a p p l i c a t i o n o f Benjamin's a n a l y s i s i s q u e s t i o n a b l e .

I n p a r t i c u l a r we a r e s c a r c e l y j u s t i f i e d

i n t h e assumption made f o r t h e expansion i n Regi.on I , that a / h < < l .

This a l s o

becomes e v i d e n t upon c a l c u l a t i o n o f the h o r i z o n t a l s c a l e l e n g t h A , which f o r l a r g e waves may be comparable to the v e r t i c a l s c a l e o f s t r a t i f i c a t i o n h. Nevertheless, t h e r e s u l t s a r e of some i n t e r e s t and a r e p r e s e n t e d i n Figure.14. C a l c u l a t i o n s were c a r r i e d o u t b o t h with and w i t h o u t i n c l u s i o n o f t h e v e l o c i t y profile. Figure 15.

The corresponding e i g e n f u n c t i o n s a r e compared w i t h observed values i n A wave of amplitude 1 2 m i s p r e d i c t e d t o have a speed o f 0.555 ms-'

n e g l e c t i n g t h e s u r f a c e flow and 0.467 allowing f o r t h e s u r f a c e flow.

Note t h a t

t h e term amplitude a s used h e r e r e f e r s t o t h e m a x i m u m v e r t i c a l excursion o f t h e s t r e a m l i n e a t Q = h r a t h e r than one h a l f t h a t v a l u e a s would be a p p r o p r i a t e f o r a s i n u s o i d a l wave. We have a l r e a d y mentioned t h a t t h e f i r s t wave was d i s t o r t e d by s h i p motion producing a much broader a p p a r e n t trough.

The t h e o r e t i c a l p r e d i c t i o n f o r

490 subsequent waves shows that t h e shape i s s i m i l a r although t h e observed wave width i s somewhat g r e a t e r t h a n p r e d i c t e d .

~.

t t15-

5 '

4 10

Figure 1 4 . Comparison o f measured and c a l c u l a t e d pycnocline depths. The s o l i d l i n e i n t h i s f i g u r e i n d i c a t e s t h e pycnocline base displacement measured w i t h t h e Aanderaa i n s t r u m e n t s t r i n g ; t h e d o t t e d l i n e i n d i c a t e s a displacement p r e d i c t e d by the modified theory of Benjamin ( 1 9 6 7 ) . The a p p a r e n t broadening of the crest i n t h e f i r s t wave r e s u l t s from s h i p motion d u r i n g passage of t h e i n t e r n a l wave.

.2

Ot e-

-

c (m.s-1) .3

.4

.5

I

2

'-

-xI0I.. LL E

v

w n

._

15.-

20 y

Figure 1 5 . Comparison o f c a l c u l a t e d i n t e r n a l wave p r o p e r t i e s w i t h and w i t h o u t s h e a r . The right-hand graph shows the v a r i a t i o n o f phase speed w i t h wave amplitude and l e n g t h , and t h e v a r i a t i o n of wave l e n g t h w i t h wave amplitude and phase speed; the l e f t - h a n d one i n d i c a t e s the s t r u c t u r e of t h e e i g e n f u n c t i o n . Heavy l i n e s correspond t o c a l c u l a t i o n s i n c l u d i n g s h e a r e f f e c t s , l i g h t ones are f o r zero s h e a r .

.6 I

491 A f u r t h e r t e s t o f the theory could be made i f t h e

generation were known.

t i m e and p o s i t i o n o f wave

I d e n t i f i c a t i o n o f t h e s e w a s t h e s u b j e c t of a r e c e n t l y

completed experiment, most of the d a t a from which have n o t y e t been analyzed. Nevertheless,

from t h e a v a i l a b l e r e s u l t s we now know t h a t t h e waves a r e generated

during t h e breakdown of an i n t e r n a l h y d r a u l i c jump over t h e down i n l e t s i d e o f t h e s i l l an hour o r so b e f o r e low w a t e r .

Taking t h e speed c a l c u l a t e d f o r t h e

n o n l i n e a r waves i n t h e absence of s h e a r t o be a p p l i c a b l e throughout t h e e n t i r e d i s t a n c e over which they have propagated and assuming a z e r o b a s i c flow i n t h e i n l e t , y i e l d s a t r a v e l t i m e o f 3.9 hours; use of a phase speed lowered by t h e presence of t h e opposing s h e a r observed a t t h e experimental s i t e y i e l d s a t r a v e l time of 5 . 2 hours.

The former i s n o t unreasonable and g i v e s a g e n e r a t i o n time

about 1 1 / 2 hours b e f o r e low water.

The l a t t e r i s somewhat t o o l a r g e .

Unfortunately we cannot assume t h a t t h e waves t r a v e l a t a l l t i m e s through a flow s i m i l a r t o t h a t observed a t t h e experimental s i t e .

Both t h e v e r t i c a l

s h e a r and t i d a l advection change w i t h t i d a l phase and l o c a t i o n .

Pickard and

Rodgers (1959) found r e l a t i v e l y less s h e a r near t h e s i l l than a t o t h e r l o c a t i o n s ; moreover t h e t i d a l flow i s r e l a t i v e l y s m a l l d u r i n g t h e f i r s t two t o t h r e e hours a f t e r t h e waves a r e formed.

A t this s t a g e w e might t h e r e f o r e e x p e c t t h e waves

t o t r a v e l a t a speed c l o s e r t o .55 ms-' of s h e a r .

which i s t h a t p r e d i c t e d i n t h e absence

Without simultaneous o b s e r v a t i o n s taken a t s e v e r a l l o c a t i o n s i t i s

hard t o e s t i m a t e t h e expected wave speed, b u t i t i s reasonable to suppose t h a t t h e i r average v e l o c i t y w a s between these two extremes.

I f t h e c o r r e c t value

averages t o 0.50 ms-'l it would t a k e 4.2 hours f o r t h e waves t o t r a v e l t h e 7.6 km from t h e i r source t o t h e experimental s i t e .

This i s c o n s i s t e n t w i t h a

g e n e r a t i o n time approximately 1 . 7 hours p r i o r to low w a t e r .

While t h e s e r e s u l t s

do not e x a c t l y coincide w i t h t h e summer o b s e r v a t i o n s they a r e c l o s e enough t o suggest t h a t t h e g e n e r a t i o n mechanism i s s i m i l a r , although p o s s i b l y modified by t h e weaker s t r a t i f i c a t i o n d u r i n g w i n t e r .

5.2

I n t e r n 2 uaves and fjord circu2ation

W e have d i s c u s s e d o b s e r v a t i o n s of l a r g e amplitude i n t e r n a l waves i n Knight

I n l e t accompanied by a deepening of the s u r f a c e l a y e r .

This deepening i s

c o n s i s t e n t w i t h t h e concept of an undular b o r e , which w e b e l i e v e t o be generated by t h e breakdown o f an i n t e r n a l h y d r a u l i c jump o v e r t h e s i l l .

Our

exponential s h e a r and d e n s i t y d i s t r i b u t i o n model s u g g e s t s t h a t c r i t i c a l c o n d i t i o n s were approached a t t h e base of t h e s t r a t i f i e d l a y e r f o r waves of t h e amplitude observed i n Knight I n l e t .

F u r t h e r numerical c a l c u l a t i o n s based on

t h e observed p r o f i l e s and numerically e v a l u a t e d e i g e n f u n c t i o n , y i e l d FLichardson Numbers i n t h e wave trough o f o r d e r u n i t y .

This may be compared w i t h v a l u e s

1 0 t o 100 times g r e a t e r f o r t h e undisturbed flow.

Whether t h e former value i s

s u f f i c i e n t l y low t o p e r m i t t u r b u l e n c e production w i t h t h e passage of each wave

492 i s an unresolved q u e s t i o n .

However, t h e r e probably are s u f f i c i e n t s m a l l s c a l e

d i s t u r b a n c e s a v a i l a b l e t o make t h i s a d i s t i n c t p o s s i b i l i t y and we s u g g e s t t h a t the passage of t h e n o n l i n e a r i n t e r n a l wave t r a i n causes t u r b u l e n t mixing i n t h e lower p a r t o f t h e pycnocline s e r v i n g t o keep i t broad a t l e a s t under w i n t e r conditions.

Of course a p e r t u r b a t i o n model based on s m a l l values o f t h e

parameter a/h could be i n a c c u r a t e when a p p l i e d t o o b s e r v a t i o n s of l a r g e amplitude waves.

However, it seems l i k e l y t h a t s u c c e s s f u l comparison of t h e o r e t i c a l

models with o b s e r v a t i o n s w i l l be as dependent on a c c u r a t e e s t i m a t e s of t h e t h r e e dimensional flow f i e l d as on any f u r t h e r refinements o f t h e model. I t appears t h a t l a r g e amplitude i n t e r n a l waves o f this s o r t a r e an e f f e c t i v e

mechanism f o r the e x t r a c t i o n o f energy from t i d a l flows over s i l l s .

I f the

waves are l a r g e enough s h e a r flow i n s t a b i l i t i e s c o n v e r t p a r t of t h i s energy i n t o t u r b u l e n c e , l e a d i n g t o mixing near t h e base of the s t r a t i f i e d l a y e r .

Once

below t h i s c r i t i c a l amplitude t h e waves may proceed r e l a t i v e l y unattenuated, probably e v e n t u a l l y being destroyed through i n t e r a c t i o n s with t h e i n l e t s i d e s ,

as discussed by S t i g e b r a n d t (1976) i n connection w i t h i n t e r n a l t i d e s .

During

t h e summer o f 1977 we have observed waves t r a v e l l i n g as f a r as Kwalate P o i n t , about 1 2 km beyond t h e s t r a i g h t reach, b u t no f u r t h e r . The wave p r o p e r t i e s a r e s t r o n g l y i n f l u e n c e d by s h e a r .

Taking Benjamin's

(1967) model as a s t a r t i n g p o i n t w e have shown t h a t when t r a v e l l i n g upstream through t h e type of v e l o c i t y p r o f i l e observed i n Knight I n l e t , t h e waves are slowed down and their h o r i z o n t a l s c a l e i s diminished.

However, t h e s h e a r

i n t h e lower p a r t of the s t r a t i f i e d zone i s i n c r e a s e d under t h e s e f e a t u r e s , thus enhancing t h e o p p o r t u n i t y f o r s h e a r flow i n s t a b i l i t y and mixing.

Numerical

c a l c u l a t i o n s f o r waves t r a v e l l i n g w i t h t h e flow show t h a t i n t h i s c a s e t h e wave speed i s g r e a t e r and t h e h o r i z o n t a l scale

X is

increased.

Thus t h e

d i r e c t i o n i n which t h e waves t r a v e l w i l l have a s i g n i f i c a n t e f f e c t on t h e c r i t i c a l amplitude and hence t h e o p p o r t u n i t y f o r s h e a r flow i n s t a b i l i t i e s l e a d i n g t o turbulence production and mixing.

I t i s t h e waves t r a v e l l i n g

upstream t h a t w i l l have t h e s m a l l e s t c r i t i c a l amplitudes and w i l l thus be most s u s c e p t i b l e t o i n s t a b i l i t y . Our p i l o t study was n o t designed t o p r o v i d e d i r e c t evidence of mixing due t o i n t e r n a l waves.

However i t i s c l e a r from e a r l i e r hydrographic s u r v e y s , as w e l l

a s our own CTD d a t a , t h a t t h e smoothly changing d e n s i t y s t r u c t u r e i n t h e sinuous region of Knight I n l e t has a much s m a l l e r h o r i z o n t a l g r a d i e n t than does t h e s t r a i g h t reach up i n l e t o f t h e s i l l .

Undoubtedly t h e waves have caused some

a l i a s i n g o f p r e v i o u s l y c o l l e c t e d hydrographic d a t a , b u t a l l t h e evidence supports t h e hypothesis of enhanced v e r t i c a l exchange i n t h i s s t r a i g h t reach, and t h e i n t e r n a l waves t h a t we have d e s c r i b e d appear t o be a most l i k e l y source

of energy f o r t h i s p r o c e s s .

493

References Abramowitz, M. and Stegun, I . A . Functions, Dover.

( E d i t o r s ) , 1968.

Handbook of Mathematical

Benjamin, T.B., 1962. The s o l i t a r y wave on a stream w i t h an a r b i t r a r y d i s t r i b u t i o n o f v o r t i c i t y , J . F l u i d Mech., 1 2 : 97. Benjamin, T.B., 1967. I n t e r n a l waves of permanent form i n f l u i d s o f g r e a t depth, J. F l u i d Mech., 29: 559-592. Davis, R.E. and Acrivos, A . , 1967. J. F l u i d Mech., 29: 593-607. Farmer, D.M., 1978. J . Phys. Oceanog.,

S o l i t a r y i n t e r n a l waves i n deep w a t e r ,

Observations o f long n o n l i n e a r i n t e r n a l waves i n a l a k e , 8 , in press

1976. Generation of i n t e r n a l waves i n t h e S t r a i t o f Georgia, Gargett, A.E., B r i t i s h Columbia, Deep Sea Res., 23: 17-32. Halpern, D . , 1971. Observations o f s h o r t p e r i o d i n t e r n a l waves i n Massachusetts Bay, J. Mar. R e s . , 24: 116. Hunkins, K. and F l i e g e l , M . , 1973. I n t e r n a l undular s u r g e s i n Seneca Lake: n a t u r a l occurence of s o l i t o n s , J. Geophys. Res., 7 8 ( 3 ) : 539-548.

a

Lee, C . and Beardsley, P . , 1974. The g e n e r a t i o n o f long n o n l i n e a r i n t e r n a l waves i n a weakly s t r a t i f i e d s h e a r flow, J. Geophys. R e s . , 7 9 ( 3 ) : 453-462. 1975. C i r c u l a t i o n s and d e n s i t y d i s t r i b u t i o n s i n a deep, s t r o n g l y Long, R.R., s t r a t i f i e d , two-layer e s t u a r y , J. F l u i d Mech. 71(3) : 529-540. Pearson, C.E. and Winter, D.F., 1978. Two-layer a n a l y s i s o f s t e a d y c i r c u l a t i o n i n s t r a t i f i e d fjords. In: J. Nihoul ( E d i t o r ) , Hydrodynamics of E s t u a r i e s and F j o r d s , E l s e v i e r , Amsterdam. 1961. Oceanographic f e a t u r e s of i n l e t s i n the B r i t i s h Columbia Pickard, G . L . , c o a s t , J. Fish. R e s . Board Can., 1 8 ( 6 ) : 907-999. Pickard, G.L. and Rodgers, K . , 1959. C u r r e n t measurements i n Knight I n l e t , B r i t i s h C o l u m b i a , J. F i s h . R e s . Board Can., 1 6 ( 5 ) : 635-678. Redekopp, L.G., 82: 725-746.

1977.

On the theory of s o l i t a r y Rossby Waves, J. F l u i d Mech.,

1974. Turbulent s t r u c t u r e of t h e s u r f a c e boundary l a y e r i n a n i c e covered ocean, Rapp.P. -v Reun. Cons. I n t . Explor. M e r .

S m i t h , J.D.,

S t i g e b r a n d t , A . , 1976. V e r t i c a l d i f f u s i o n d r i v e n by i n t e r n a l waves i n a s i l l f j o r d . J . Phys. Oc. 6 ( 4 ) : 486-495. Thorpe, S . , H a l l , A . and C r o f t s , I . , 1972. Nature, 237: 96-98.

The i n t e r n a l s u r g e on Loch N e s s

Winter, D.F., 1973. A s i m i l a r i t y s o l u t i o n f o r s t e a d y s t a t e g r a v i t a t i o n a l c i r c u l a t i o n s i n f j o r d s , E s t u a r i n e and C o a s t a l M e r . S c i . , 1: 387-400. Ziegenbein, Sea Res.,

J .,

1969- S h o r t i n t e r n a l waves i n the S t r a i t o f G i b r a l t a r , Deep 16: 479.

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495

TWO-LAYER ANALYSIS OF STEADY CIRCULATION IN STRATIFIED FJORDS C.E. PEARSON and D.F. WINTER Department of Aeronautics and Astronautics and Department of Oceanography, University of Washington, Seattle, WA 98195

ABSTRACT In deep, narrow, fjord-type inlets with copious runoff near the head, freshwater inflow produces a surface slope and a pressure gradient which drives a brackish, near-surface layer seaward; at greater depths a denser, saline layer derived from oceanic water moves landward. We describe in this paper a self-consistent twolayer representation of this mode of inlet flow, generally referred to as "estuarine circulation." Our approach is different from other layered analyses in that the present model includes important effects of variations in mass density, channel width and depth, and a l s o allows for turbulent and advective exchange between the deep and near-surface layers. The starting point of the analysis is a set of equations expressing incompressibility and conservation of mass and horizontal momentum in each zone. Transfer of mass and momentum across the interface between the layers is parameterized by two interzonal exchange flux rates, FU and Fd, representing the upward and downward rate of fluid flow per square meter of interfacial area. When the time-averaged mass density variations can be estimated from field data, then the flux rates, FU and Fa, can be inferred entirely from known or measurable quantities. Two integrals of the motion are immediately available, and the mathematical problem is reduced to solving a pair of nonlinear equations for the layer cross-sectional areas. By way of illustration, the procedure is applied to Knight Inlet, a deep, stratified fjord on the Pacific Northwest coastline. TWO-LAYER ANALYSIS Inlets with appreciable fresh water runoff frequently exhibit a two-layer circulation pattern in which a brackish near-surface layer moves seaward while a deeper saline layer moves landward. The near-surface layer salinity generally increases in the seaward

496

-- - LAYER 2

-- - -

LAYER

I

. . . : ..., ......:

Figure 1.

'; :

.

' '

Sketch of an inlet cross-section (a) and longitudinal section (b), depicting the two major flow layers and illustrating geometrical quantities. The horizontal scale is compressed.

't

I I

I

I

I

I

+.-aX+l

I

...... ...

Figure 2.

Sketch of fluid sections used to derive equations of conservation of volume, mass, and horizontal momentum.

497

direction because of entrainment of water from the lower layer. The water in the lower layer is derived largely from sea water external to the inlet, although it may be freshened somewhat during its progress by brackish water from above due to turbulent mixing between the layers.

We consider here a simplified model of the

estuarine circulation mode, constructed from an hydraulic analysis similar to that developed by Stoker (1957) for river flow and later adapted to flow in stratified estuaries by Vreugdenhil (1970), Long ( 1 9 7 5 ) , and others. The present analysis extends previous layer models by including bathymetric effects and by allowing for interlayer exchange; the discussion is restricted to time-averaged conditions (over a tidal cycle, say). It is assumed that the inlet is sufficiently narrow that a one-dimensional treatment is appropriate, in the sense that all quantities in each layer depend only on the distance x fro= the seaward end (Fig. 1). The z-axis is directed vertically upward from a horizontal datum level. Subscripts 1 and 2 are used to denote variables in the lower and upper layers, respectively. It is assumed that the fresh water in the inlet is introduced exclusively into the upper layer with zero horizontal velocity (except at x =L) at a cumulative volumetric rate given by R(x); (thus, R(x) denotes the total influx between x and L). Because of the typically small depth of the near-surface layer, its breadth b(x) may be taken as constant over its depth. Let d 2 be an appropriate fixed reference depth, such as the layer thickness at the inlet mouth, and denote by hl, h2 the displacements of the lower and upper surfaces, respectively (Fig.1). The cross-sectional area of the near-surface layer, denoted by A2(x), can be written as

Next, suppose that the deep layer has cross-sectional area A(x) relative to the datum level: the breadth of its top surface will be b(x). The actual cross-section of the deep layer can then be represented as

(It will be convenient subsequently to refer to a maximum depth dl (x) of the deep layer when the fluid is at rest.)

Finally, the

498

horizontal velocities of the deep and near-surface layers will be denoted by u,(x) and u2(x), respectively; each is positive when directed landward. Equations expressing incompressibility (or conservation of volume) and conservation of mass and horizontal momentum in each layer can now be written down. In so doing, it is necessary to allow for (upward) convecti-Je motion and for turbulent mixing between the layers. In models where flow parameters are continuous functions of depth, vertical transport is described by vertical flow velocities and turbulent fluxes, the latter involving eddy coefficients of viscosity and diffusion. Since turbulent exchange mechanisms in estuarine flow are not at all well understood, the eddy coefficients are usually assigned values or functional forms which are adjusted until calculated flow and hydrographic patterns are similar to those observed in practice. In layered models of coastal plain and salt wedge estuaries, the approach has not been much different; the traditional procedure is to assign adjustable values to friction coefficients at a layer interface until some correspondence is achieved between computation and observation. In the present work, convective and turbulent transfer between the layers is represented by two interzonal exchange flux rates denoted by FU and Fd. The symbol FU represents the upward volume rate of flow of fluid from the deep layer to the near-surface layer per square meter of interfacial area. Likewise, Fd denotes the downward volumetric flux rate from the upper to the lower layer. Equations expressing incompressibility or volume conservation for each layer are derived from considerations of volume flow rates in and out of sectional "slices" of thickness Ax, illustrated in Fig. 2.

For the deep and near-surface layers respectively, one

obtains (UIA1)'

=

b(Fd

-

Fu)

(3)

and

where a prime denotes differentiation with respect to x. Similarly, the equations expressing conservation of mass in the lower and upper layers are (PIUIA1)' =

b(P2Fd

- PIFU)

(5)

499

and

where p o denotes the mass density of fresh water.

In writing down

(5) and ( 6 ) , it has been assumed that horizontal diffusion is negligible compared with advective transport. This is a common assumption in deep inlet studies and is based upon the findings of Eqs.

certain oceanographic field studies of fjord waters (e.g., McAlister et al., 1959; Dyer, 1973). On the other hand, it may not be justified in the vicinity of a long shallow sill where tidally induced longitudinal dispersion of mass can become competitive with advection. To derive the momentum equations, we use the fact that the net force on each slice of fluid, in Fig. 2 , must equal the net rate of efflux of momentum from that slice. Consider first the lower slice. The net force on it is made up of a pressure imbalance across its two faces, the pressure force on the sides due to changes in breadth, the horizontal component of the pressure force acting on the interface, and the frictional stress T ~ ( x )acting on the wetted perimeter Cw:

where p is the pressure, g the acceleration of gravity, and B(x,z) the breadth of the lower layer at vertical position z . The usual convention of shallow water theory is adopted to the effect that p results from hydrostatic forces only, so that

In carrying out the differentiation indicated for the first term of the momentum conservation equation above, there is no contribution from the x-dependence of the lower limit, since B(x,-[d2+dl]) = 0. Furthermore, there is no contribution from this term if channel has

500

a flat bottom since, in that case, the momentum equation contains an additional term of equal magnitude and opposite sign. The final result can be written as

where y is defined by the relation

y(-d2

-

1 -2d 1) A

=

r2

zB(x,z) dz

-d2 - d1

Clearly, in the case of a rectangular channel, y =l. A similar line of argument leads to the equation of horizontal momentum applicable to the near-surface layer:

R'

+ - TWb 2A2

+

POU2

'

where ~ ~ ( is x )the stress exerted on the upper surface by wind. Equations (3) - ( 6 ) , ( 9 ) , and (10) can be regarded as a system of equations for the six unknown dependent variables p l , p 2 , A1, Two integrals can be obtained immediately by A 2 , ul, and u 2 . adding Eqs. (3) and ( 4 ) ,

and

and Eqs. (5) and ( 6 ) , respectively:

501

These two equations will be used subsequently to eliminate a pair of dependent variables from the system. We suppose now that p1 and p 2 (and therefore also p l ' and p 2 ' ) are known from field measurements as functions of x; R(x) is also

supposed known. The quantities (u A ) and (u2A2) can then be deter1 1 mined by solving the linear equation system (12) and (13): R(P2 - Po) u,A, I

I

u A

2 2

=

p1

-

02

=

5), we obtain expressions for the

Using Eq. (14) in Eqs. ( 3 ) and exchange flux rates FU and Fd:

I

L

From Eqs. (14) and ( 1 5 ) , u1 and u2 can be expressed in terms of A1 and A2. Since Eqs. (16) and (17) determine Fd and F U , it now follows that Eqs. (9) and (11) constitute a coupled pair of differential equations for A1 and A

2

of the form

allAll

+

a12A2'

=

bl

a21A1'

+ a22A2'

=

b2

where the a. and b . coefficients are certain functions of A 1' A2' +j and axial distance x. (The coefficients are readily derived from information provided above, but inasmuch as they are quite lengthy, the expressions are not presented here.) The system (18) and (19) can be readily integrated by, say, a conventional fourth order Runge-Kutta method to obtain Al and A2 as functions of x. Of course, initial conditions at x = O must be specified; because the choice of the datum level is arbitrary, we can set h2 = O at x = O , and the value of hl at x = O can be obtained from the measured

502

salinity (or, preferably,. horizontal current) profiles. In turn, these two values determine A1 and A2 at x = O . Once A1(x) and A2(x) have been calculated, ul(x) and u2(x) may be obtained Trom Eqs. (14) and (15), and the solution is complete.

An example is provided in

the last section. STABILITY The numerical values of the coefficients in Eqs. (18) and (19) are frequently such that the two equations are very similar to one another, and consequently it is advantageous to solve the modified system allAl' + a12A2'

=

bl

(20)

where the second of these equations is obtained by analytically subtracting Eq. (19) from Eq. (18). Even so, the solution of the modified system is fairly sensitive to the quantity ( p l - p 2 ) , as might be expected from its appearance in denominators in Eqs. (14) through (17), and from the fact that pl- p 2 is generally small compared with either p1 or p 2 . It turns out that for sufficiently small values of ( p 1 - p 2 ) the two equations (20) and (21) can cease to have a solution. This can occur in the neighborhood of a sill, for example, where intense turbulent mixing may take place by means of physical processes whose effects are not adequately represented in the present model. In order to gain further insight into such a situation, we confine our attention to the simple case in which R , B, and A are constant. Then Eqs. (20) and (21) become

+ - P2bFd PIAl

(u2

-

ul)

-

bul

- (Fd A1

- Fu)

503

U

2

A1,

+

I

- u2) +

(ul

[s +1 -

-

[q+ bul 1 1

(Fu - Fa) bu2

PIAl

P2A2

Typical numerical values are such that, to a first approximation, Eq. (22) states that A 1 + A 2 is approximately constant and equal to (say) K. Then in Eq. ( 2 3 ) , A1' can be replaced by - A 2 ' . Next we write p1 = p 2 + A p and assume that Ap and p 2 ' are approximately constant over some range of x-values of interest. In most examples we have considered, A1 has been much greater than A 2 , and the dominant terms on the right-hand side of Eq. (23) have been the first three.

It then turns out that Eq. (23) can be approximated by

If p l - p o is treated as approximately constant, integration of Eq. (24) yields

where

- E X , where Write p1 - p10 approximate a by

-E

is the (constant) value of p l l

and

504

where we have inserted representative values for ( p l - po) and for (in metric units). Then Eq. (25) may be written as PO

where A

is close to the value of A2 at x = O . The left-hand side 20 of Eq. (28) has a minimum value when A2 = and, consequently, from Eq. (28) the maximum value of x must be given by

3%

If the average depth of the channel is D , then from Eq. (27) it follows that the maximum length over which the quasi-laminar flow model can be valid is given by

where h is the thickness of the upper layer at x = O . This equa0 tion yields results compatible with numerical solutions of Eqs. (20) and (21) in appropriate cases, in the sense that the thickness of the upper layer, as computed from Eqs. (20) and (21) vanishes near this value of x. APPLICATION TO KNIGHT INLET, BRITISH COLUMBIA The method is illustrated by application to Knight Inlet in the southern portion of the mainland British Columbia coastline. Knight Inlet is a deep fjord located about 320 km northwest of Vancouver, British Columbia. It is approximately 110 km long and has an average width of about 3 km (see map in Fig. 3). Knight Inlet is a positive, high-runoff fjord estuary which is in communication with the Pacific Ocean by way of Johnstone Strait and Queen Charlotte Strait. The inlet is divided into two basins by a threshold of approximately 65 m depth, located at 75 km from the head. The outer basin, which is the shallower of the two, has a second 65 m deep threshold at the confluence of the inlet with marine waters in Johnstone Strait. The outer basin is characterized by depths in the range 150 to 200 m, and has an irregular

505

KNIGHT INLET BRITISH COLUMBIA

STATION

I

2

3

3 ! ’ 2 4 5

6

7

8

9

1011

IMEASURED)

E 200 c I a

600 MAXIMUM CHANNEL DEPTH (MODEL SIMULATION)

F i g u r e 3.

Map of K n i g h t I n l e t , B r i t i s h Columbia, t o g e t h e r w i t h o b s e r v e d l o n g i t u d i n a l s e c t i o n ( a f t e r P i c k a r d and R o d g e r s , 1959), and maximum d e p t h v a r i a t i o n u s e d i n model c a l c u l a t i o n s .

506

shoreline indented by passes leading to adjacent bodies of water. The inner basin is deep over most of its extent, with the maximum depth in excess of 550 m. A longitudinal section of Knight Inlet is shown in Fig. 3; in the same figure is shown an idealized representation of the inner basin and inner sill depth profile, based on elementary functions (trigonometric and quadratic forms), which was used in the model calculations below. Most of the fresh water in Knight Inlet is derived from the Klinaklini and Franklin Rivers which discharge into the head. Although runoff is significant throughout most of the year, its intensity exhibits a seasonal variation characterized by a JanuaryMarch minimum and a May-July maximum, the latter being due to summertime melting of snow which fell on the adjacent mountains during the winter. A secondary runoff maximum may occur during the rainy season in October-November. The Klinaklini and Franklin Rivers are not routinely gauged, but estimates of fresh water discharge for Knight Inlet are available (Trites, 1955; Pickard, 1961), which indicate an annual mean of 410 m3 sec-l, and high runoff rates in the neighborhood of 600 and 750 m3 sec-l during the months of June and July, respectively. Inlets with the yearly discharge pattern of Knight Inlet have been termed Type A.l (stored runoff) fjords by Pickard (1961). Typically, the surface salinity is well below 2 o/oo near the head when runoff is intense and it remains at low values over substantial portions of the inlet. Figures 4a and 4b show continuous profiles of salinity obtained during 3-4 June 1951 and 15 July 1953, as presented by Trites (1955). Pronounced stratification over the major portion of the inlet is evident in these traces; the data from Stations 3-5 near the sill can only be regarded as qualitative since it is now known that aliassing is a serious problem throughout the region which includes these stations, (D.M. Farmer and J.D. Smith, personal communication, 1977). Model calculations were performed for comparison with the Trites' data from June and July, and the computed results for the salinity distributions are exhibited in Figs. 4c and 4d. The depth profile in Fig. 3 was used, together with an assumed constant main channel width of 2 km. The mean monthly cumulative fresh water discharges R for June and July were taken to be independent of x and equal to 600 and 750 m3 sec-1 , respectively. Average upper and lower layer salinities were estimated from the data. The thickness of the near surface layer near the head was assigned a value of I m

507

SALINITY

(o/oo)

L

I

I

20

I

KNIGHT INLET, JUNE 3-4,1951

Figure 4.

I

KNIGHT INLET, JULY 15, 1953

Depth profiles of salinity measured on (a) 3-4 June 1951 (1955); and (b) 15 July 1953, as presented by Trites model salinity profiles for conditions in (c) June 1951 and ( d ) July 1953.

508

for June and 6 m for July, as suggested by the measurements, and the integration of Eqs. (20) and (21) was carried out in the direction of decreasing x. It can be seen from Fig. 4 that the general features of the salinity and zonal thickness data are adequately represented over much of the inlet length, except near the sill where aliassing is known to occur. Figure 5 shows the calculated axial variation of upper layer thickness, zonal velocities, and the interzonal exchange rate FU for the month of June (Fd was equal to zero in this calculation since p1 was taken to be independent of x). Unfortunately, we lack appropriate field data for making meaningful comparisons between time- and depth-averaged horizontal current profiles and calculated values of u in the upper and lower zones. Field data described by Pickard and Rodgers (1959) suggests that, in the absence of wind stress, near the Knight Inlet sill the average horizontal current speed may be as high as a few decimeters per second at the surface, but it decreases rapidly in magnitude toward zero below the halocline. A consideration by Winter (1973) of distance and velocity scales appropriate to near-surface circulation in stratified fjords suggests that the characteristic transport speed uo in the upper zone is u0

=

Ro/bo

z0 IS 0

(31)

'

where Ro is a representative cumulative runoff value, b 0 is an effective channel width, zo is a representative thickness of the upper zone, and u0 is the fractional salinity excursion from great depths to the surface in the inlet segment under consideration. we assign representative mid-inlet values of R = 6 0 0 m3 sec-1,

If

0

bo = Z O O 0 m, z o = 8 m, and o0=2/3, then the characteristic transport speed for Knight Inlet during the month of June turns out to be u = 0.056 m sec-l, which is in agreement with the calculated value 0

of u (x) in the vicinity of Stations 5 and 7, but lower than might 2 be inferred from Pickard and Rodgers' measurements. If the characteristic horizontal scale xo is determined by a balance between mixing and advective salt transport, as assumed by Winter (1973), then

509

K N I G H T INLET (JUNE)

-

-E

10

0

01

I

30

20 1

I

1

1

1

50

40 1

I

1

1

60 1

I

8 0 (km)

70 I

I

1

'

"

I

a , In

E

1 6 ~ -

F i g u r e 5.

C a l c u l a t e d a x i a l v a r i a t i o n s of u p p e r l a y e r t h i c k n e s s , a v e r a g e u p p e r and l o w e r l a y e r s p e e d s , and upward f l u x

rate F

U

i n Knight I n l e t , B . C . ,

f o r June c o n d i t i o n s .

C o o r d i n a t e x i s measured p o s i t i v e landward from S t a t i o n 3 , seaward of t h e s i l l .

510

where K is a representative value of the vertical eddy diffusivity. 0 From the equation of continuity it follows that the characteristic vertical velocity w 0 is of the order of

w0 =

uo zo/xo

.

(33)

Trites (1955) has estimated that KO in Knight Inlet for early summer conditions is of the order of 2 x 1 0-4 m2 sec-’; it follows from -1 m sec . the equations above that x0 = 18000 m and wo = 2 . 5 x The latter quantity is of the same order of magnitude as the computed values of the upward interzonal flux rate FU. In this same connection, it is also interesting to note that from physical model experiments, Keulegan (1949) established an empirical relationship between the entrainment velocity and the longitudinal velocity of the upper layer: F~

=

3.5 x 1 0 - ,~ ~ ~

(34)

in the present notation. The value 0 . 0 5 6 m sec-’ for u 0 leads to an estimate of 2 x lob5 m sec-I for FU, which is of the same order as calculated mid-inlet values (Fig. 5). It would appear from the figure that in Knight Inlet FU attains its maximum value at x = 0 , i.e., at the seaward end of the main inlet segment.

However, this

feature is an artifact of the functional representation of the upper layer mass density variation p 2 (x). Field data indicate that in reaches of the inlet seaward of x = O , p 2 ’ eventually decreases faster than ( p - P , ) ~ , and it follows from Eq. (17) that FU ultimately declines to smaller values. Figure 6 presents longitudinal variations of surface salinity and salinity at depth as functions of x, where x is measured positive landward from Station 3 , located a few kilometers seaward of the inner sill.

The points in the figure represent the average of

field measurements acquired during the month of June in the years 1951, 1973, and 1974. It would appear from the figure (and from other field data) that mixing processes taking place near the inner threshold cause the salinity of the surface layer to increase substantially from about 30 km upstream from the sill. Figure 7 shows averages of salinity measurements at selected depths at fixed stations in the inlet during the months of March and June in the three years 1951, 1973, 1974.

The runoff during

511

I

DEEP ZONE

3

Figure 6.

4

5

6

7

9 STATION NUMBER

Longitudinal variations of surface salinity and salinity at depth for March and June conditions in Knight Inlet. The data points are averages of field measurements acquired in the years 1951, 1973, and 1974.

March is estimated to be of the order of 250 m3 sec-l.

It may be

noted that the scatter in the measurements in the pycnocline leads to an apparent distortion of the variation of salinity with depth, as exemplified by the traces in Fig. 4 . Nevertheless, calculations based on the averaged measurements seem to give reasonable representations of the upper layer thickness variations for both runoff regimes in Knight Inlet. It can be seen from Fig. 5 that, during high runoff, the upper layer thickness increases monotonically in the seaward direction from 6 or 7 m near the head to about 20 meters on the seaward side of the sill. Throughout the greater part of the inlet, however, the calculated interface depth is between 6 and 10 m, in accord with observation. On the basis of a somewhat different analysis (lower layer at rest, critical flow at the mouth, specified entrainment rate), Long (1975) has calculated a halocline depth of 21 m for Knight Inlet, but this appears to be somewhat of an overestimate. Only on the seaward side of the sill does the time-averaged upper layer thickness attain a value of the order of 20 meters. It is an interesting fact that when integrations of Eqs. (20) and (21) are performed with the same longitudinal density profiles p l and p 2 ,

512

KNIGHT INLET (MARCH 1 ST4.3 ( O h m i

ST4.5iZOkmi

STA.7 (40krr.l

ST4.9!60kml

ST4.11(75km1

KNIGHT INLET (JUNE) STA 3 ( 0 km)

STA 5 ( 2 0 k m )

STA 7 (40 km)

STA 9 ( 6 0 k m )

I

401

60

Figure 7.

n

I

I

j

I

!

!

I

I I

I

Measured and calculated salinity profiles at f i v e stations along Knight Inlet for average conditions during March and June.

513

but with somewhat different starting values of upper layer thickness, the calculated upper layer thicknesses from mid-inlet toward the head are still of the order of a decameter or less. We conclude from the results for Knight Inlet, and from calculations performed for several other fjords along the British Columbia coastline, that the present model can provide reasonable, self-consistent representations of the main features of time-averaged estuarine circulation in stratified inlets. ACKNOWLEDGMENTS We are grateful to Dr. David M. Farmer for several useful discussions of the physical oceanography of British Columbia inlets. The research described in this paper was partially supported by the National Coastal Pollution Research Program of the Environmental Protection Agency under Grant No. R-801320, and by the National Science Foundation, Oceanography Section, under Grant No. OCE7680720. A travel grant from the National Science Foundation made it possible to present this work at the Ninth International Lisge Colloquium on Ocean Hydrodynamics. Salinity data for Knight Inlet were obtained from Data Reports prepared by the Institute of Oceanography, University of British Columbia, Vancouver, B.C. REFERENCES 1.

2. 3.

4.

5. 6. 7. 8.

Dyer, K. R., 1973. Estuaries: A Physical Introduction. John Wiley, London, 140 pp. Keulegan, G. H., 1949. Interfacial instability and mixing in stratified flows. Journal of Research, National Bureau of Standards, 43: 487-500. Long, R. R., 1975. Circulations and density distributions in a deep, strongly stratified, two-layer estuary. Journal of Fluid Mechanics, 71: 529-540. McAlister, W. B., Rattray, M., Jr., and Barnes, C. A., 1959. The dynamics of a fjord estuary: Silver Bay, Alaska. Department of Oceanography Technical Report No. 62, University of Washington, Seattle, Washington, 70 pp. Pickard, G. L., 1961. Oceanographic features of inlets in the British Columbia mainland coast. Journal of the Fisheries Research Board of Canada, 18: 907-999. Pickard, G. L. and Rodgers, K., 1959. Current measurements in Knight Inlet, British Columbia. Journal of the Fisheries Research Board of Canada, 16: 635-678. Stokes, J. J., 1957. FJater Waves. Interscience, New York, 567 pp. Trites, R. W., 1955. A study of the oceanographic structure in British Columbia inlets and some of the determining factors. Ph.D. Thesis, Institute of Oceanography, University of British Columbia, Vancouver, B.C., 125 pp.

514 9. 10.

V r e u g d e n h i l , C . B . , 1 9 7 0 . C o m p u t a t i o n of g r a v i t y c u r r e n t s i n D e l f t Hydraulics Laboratory P u b l i c a t i o n N o . 86, estuaries. 1 0 8 pp. Winter, D. F . , 1973. A s i m i l a r i t y s o l u t i o n f o r s t e a d y s t a t e 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 f j o r d s . E s t u a r i e s and C o a s t a l M a r i n e S c i e n c e , 1: 387-400.

C o n t r i b u t i o n N o . 1 0 0 1 , D e p a r t m e n t of O c e a n o g r a p h y , U n i v e r s i t y of W a s h i n g t o n , S e a t t l e , WA.

515

V A R I A B I L I T Y OF CURRENTS AND WATER PROPERTIES FROM YEAR-LONG

OBSERVATIONS

I N A FJORD ESTUARY

G . A.

CANNON and N .

P.

LAIRD

P a c i f i c Marine Environmental Laboratory, Environmental Research L a b o r a t o r i e s , National Oceanic and Atmospheric Administration, S e a t t l e , Washington 98105, U . S . A .

ABSTRACT Observations were made of v e l o c i t y , temperature, and c o n d u c t i v i t y from a v e r t i c a l a r r a y of s e n s o r s a t a l o c a t i o n i n Puget Sound f o r 1 year commencing i n September 1975.

Winds w e r e measured nearby on l a n d , and w a t e r p r o p e r t i e s were

measured along t h e Sound a t 2- t o 3-month i n t e r v a l s .

Most dense bottom water

e n t e r e d t h e Sound below s i l l depth a t about f o r t n i g h t l y i n t e r v a l s during e a r l y f a l l near t h e end of t h e c o a s t a l upwelling season. a s s o c i a t e d w i t h l a r g e t i d e s over t h e e n t r a n c e s i l l .

The f o r t n i g h t l y events were Density decreased about

l i n e a r l y u n t i l e a r l y w i n t e r when c o l d water e n t e r e d t h e Sound f i r s t a t t h e bottom i n a series of s t e p d e c r e a s e s , a l s o a t about f o r t n i g h t l y i n t e r v a l s .

w a s 1.6OC i n 3-4 days.

The l a r g e s t

Density i n c r e a s e d d u r i n g each inflow i n t e r v a l l a s t i n g

about 5 days, b u t t h e o v e r a l l d e n s i t y continued t o decrease u n t i l e a r l y February when t h e water column became isothermal b u t w a s s t i l l s t r a t i f i e d .

During l a t e

w i n t e r and e a r l y s p r i n g , d e n s i t y a g a i n i n c r e a s e d through a s e r i e s of f o r t n i g h t l y s a l i n i t y i n t r u s i o n s , and t h e i n i t i a l one was more than 0 . 5 d a i l y near-bottom speeds w e r e up t o about 2 0 cm/sec, exceeding t h e l e n g t h o f t h e Sound.

o/oo

i n 5 days.

Mean

implying a p o s s i b l e excursion

Winds were observed t o a l t e r mean d a i l y

c u r r e n t p r o f i l e s t o more than h a l f t h e w a t e r depth.

These and o t h e r w i n t e r

o b s e r v a t i o n s i n d i c a t e d deep-water renewal was much quicker than e a r l i e r e s t i m a t e s .

INTRODUCTION

puget Sound i s a f j o r d - l i k e e s t u a r y connecting through Admiralty I n l e t and Deception Pass t o t h e S t r a i t o f Juan de Fuca and then t o t h e P a c i f i c Ocean ( F i g . 1 ) . I t i s e n t i r e l y w i t h i n t h e S t a t e of Washington i n t h e United S t a t e s and i s p a r t

of a l a r g e r e s t u a r i n e system contiguous t o s e v e r a l major p o p u l a t i o n c e n t e r s i n t h e United S t a t e s and Canada.

About 98% of t h e t i d a l prism flows through

Admiralty I n l e t over a s i l l of about 64 m depth near P o r t Townsend. i s about 150 km from t h e ocean and extends s o u t h about 30 km.

This s i l l

The main b a s i n

has depths exceeding 200 m and extends south about 60 km from t h e major j u n c t i o n with Admiralty I n l e t near Possession P o i n t t o The Narrows, a c o n s t r i c t i o n of

516 about 44 m s i l l depth s e p a r a t i n g a southern b a s i n .

Within t h e main b a s i n t h e

183-m contour d e l i n e a t e s a deeper s e c t i o n approximately 50 km long and 3-5 km wide.

The o t h e r main s u b d i v i s i o n s i n c l u d e Hood Canal and t h e Whidbey b a s i n , t h e

l a t t e r extending from Possession P o i n t t o Deception Pass through Saratoga Passage. Numerous r i v e r s e n t e r t h e Puget Sound system.

However, t h e S k a g i t e n t e r i n g i n

t h e n o r t h s u p p l i e s more than 60% o f t h e f r e s h w a t e r , about h a l f o f which flows southward toward t h e main b a s i n (Barnes and Ebbesmeyer, 1 9 7 7 ) .

5'

18"

15'

17' 30

15'

F i g . 1.

Puget Sound r e g i o n showing s t a t i o n l o c a t i o n s .

517 Replacement of Puget Sound bottom w a t e r w i t h more s a l i n e and d e n s e r water h a s been thought t o o c c u r i n July-October

i n response t o n o r t h e r l y summer winds

N o r t h e r l y winds induce upwelling of d e n s e r o c e a n i c

along t h e P a c i f i c c o a s t .

w a t e r t o shallow d e p t h s which t h e n can be t r a n s p o r t e d a l o n g t h e bottom of t h e S t r a i t of Juan de Fuca i n t o Puget Sound (Barnes and C o l l i a s , 1 9 5 8 ) . long-term hydrographic o b s e r v a t i o n s show t h a t bottom-water a l s o i n c r e a s e s d u r i n g April-May ( C o l l i a s et a l . ,

1974).

Additionally,

d e n s i t y i n t h e Sound

and d e c r e a s e s d u r i n g May-June and October-February

During t h e w i n t e r d e c r e a s e i n d e n s i t y i n November-March,

mixing and/or replacement r e s u l t s i n t h e d e e p e r w a t e r i n t h e main and Whidbey b a s i n s becoming r e l a t i v e l y i s o t h e r m a l , l e s s s a l i n e , and h i g h e r i n d i s s o l v e d oxygen (Barnes and C o l l i a s , 1958; Cannon, 1 9 7 5 ) .

Because deep-water

replacement

p r e v i o u s l y only had been documented a t a b o u t monthly i n t e r v a l s w i t h water p r o p e r t i e o b s e r v a t i o n s of c u r r e n t s and w a t e r p r o p e r t i e s were made a t s e v e r a l l e v e l s a t a c e n t r a l l o c a t i o n f o r 1 month d u r i n g w i n t e r 1973 (Cannon and Ebbesmeyer, 1 9 7 7 ) . Our p r e v i o u s work d u r i n g t h e w i n t e r month showed t h a t w a t e r n e a r t h e bottom was cooled i n a s e r i e s of s t e p d e c r e a s e s of t e m p e r a t u r e of up t o 0.6OC a t about 2-week i n t e r v a l s c o i n c i d e n t with t h e o n s e t of about 5-day i n t e r v a l s of n e t landward flow n e a r t h e bottom. with a s m a l l seaward component.

Between t h e s e i n t e r v a l s , bottom flow was t i d a l Temperature s e c t i o n s a l o n g t h e a x i s of t h e Sound

showed f o l l o w i n g t h e f i r s t s t e p a v e r t i c a l temperature f r o n t c l o s e t o t h e mooring, whereas a t t h e beginning and end of t h e month t h e r e was no evidence of t h i s f r o n t . Density s e c t i o n s showed t h a t , f o l l o w i n g t h e s t e p , l e s s dense w a t e r ( c o l d e r b u t l e s s s a l i n e ) was i n t r u d i n g t o near t h e bottom j u s t n o r t h of t h e mooring and t h e p a s s i n g temperature f r o n t , and t h a t an o v e r a l l d e c r e a s e i n d e n s i t y o c c u r r e d d u r i n g t h e month ( F i g . 2 ) . Deep water was shown t o be r e p l a c e d when f l o o d - t i d e ranges a t S e a t t l e exceeded 3.5 m .

A t this range S t r a i t of Juan d e Fuca w a t e r had been shown by h y d r a u l i c

model s t u d i e s t o completely t r a n s i t t h e f i n i t e - l e n g t h s i l l i n Admiralty I n l e t i n one f l o o d p e r i o d and t h u s be l e a s t mixed (Farmer and R a t t r a y , 1 9 6 3 ) .

I n our

study t h e l a r g e s t temperature decrease occurred following flood t i d e s concurrent with below-freezing a i r t e m p e r a t u r e s over Admiralty I n l e t .

Density d i f f e r e n c e s

d u r i n g t h e temperature s t e p s were s m a l l , and it was n o t p o s s i b l e t o determine whether t h e new w a t e r i n i t i a l l y was more dense.

A l s o , an 8-10 day t r a n s i t time

was r e q u i r e d f o r t h i s w a t e r t o flow from t h e s i l l t o t h e mooring s i t e . F i n a l l y , i t was s p e c u l a t e d t h a t a s i m i l a r replacement p r o c e s s might o c c u r i n l a t e summer when t h e most dense bottom w a t e r can e n t e r t h e Sound.

Confirming

c u r r e n t o b s e r v a t i o n s on t h e s i l l of P o r t Susan, a secondary b a s i n i n Puget Sound, showed flow i n t o t h e b a s i n a t a b o u t f o r t n i g h t l y i n t e r v a l s l a s t i n g a b o u t 5 days each time (Cannon, 1 9 7 5 ) .

I t a l s o was s p e c u l a t e d t h a t deep water might b e

r e p l a c e d q u a s i - c o n t i n u o u s l y a t about 2-week

intervals.

of o b s e r v a t i o n s was planned and made d u r i n g 1975-76.

Thus, a year-long s e r i e s This paper d e s c r i b e s t h o s e

518 observations and presents additional ideas concerning deep water replacement which might be applicable to other fjords.

I

I

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8.81

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10

0

a 0 Y,

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SEAWARD FLOWS v

7.2

4 h

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3

v

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2 4

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-4 I

5

1973

15

10

20

25

30 I

FEB

JAN

Fig. 2 . Daily average longitudinal bottom currents and temperature in Puget Sound; range of greatest daily Seattle flood tide; and daily average Port Townsend air temperature (from winter 1973). EXPERIMENT DESCRIPTION The single current-meter mooring with seven Aanderaa meters with temperature and conductivity sensors was first deployed in mid-September 1975 in the main basin of Puget Sound in about 200 m of water (Fig. 1).

It was recovered and

redeployed in November, February, April, and June, and it was finally recovered in early October 1 9 7 6 .

The NOAA ship M c A r t h u r was used on a l l but the June

cruise which was carried out from the University of Washington's Onar.

Nominal

depths of the instruments were 15, 30, 50, 105, 145, and 1 9 5 m (5 m above bottom). The meters recorded every 20 min samples of average speed and instantaneous

values of pressure, direction, temperature, and conductivity.

The meters had

relatively large vanes, making them fairly insensitive to high-frequency direction fluctuations. Buoyancy of about 450 kg was provided by an ORE sphere; the mooring cable was torque-balanced 3 wheels weighing about 1000 kg.

x

19 cable; and the anchor was three railroad

The mooring was deployed anchor first from the

McArthur and buoy first from the O n a r .

The buoy or anchor was then lowered with

519 The mooring w a s r e t r i e v e d each time by AMF a c o u s t i c

a t e n s i o n - r e l e a s e t r i p hook. r e l e a s e o f t h e anchor.

Unfortunately,

t h e mooring was snagged, a p p a r e n t l y by a

tug and barge, on 20 J u l y ( a s shown i n t h e d a t a ) and moved about 1 4 km s o u t h . Thus t h e year-long s e r i e s a t t h e s i n g l e l o c a t i o n was n o t completed.

However,

d a t a from a s p a t i a l a r r a y deployed i n t h e same v i c i n i t y on 20-21 J u l y (by chance) and from t h e r e l o c a t e d mooring showed s i m i l a r c h a r a c t e r i s t i c s f o r t h e remainder o f t h e year. A survey of water p r o p e r t i e s along Puget Sound was made f i v e times during t h e

y e a r , p r i m a r i l y c o i n c i d e n t with t h e deployment c r u i s e s (Fig. 1 ) . Observations were made a t each s t a t i o n t o w i t h i n a few meters of t h e bottom using a pinger attached t o a salinity-temperature-depth 9006).

(STD) r e c o r d e r

(Bissett-Berman, model

Nansen b o t t l e samples were o b t a i n e d t o c a l i b r a t e t h e STD.

The occupied

s t a t i o n s ( t a k i n g about 1 2 h r ) were chosen because of t h e r e l a t i v e l y l a r g e h i s t o r i c a l d a t a base a t t h e s e s i t e s , and t h e northern two s t a t i o n s were seaward of th e entrance s i l l .

A 25-hr

time s e r i e s made i n February a t t h r e e s t a t i o n s

near t h e mooring showed no t i d a l s i g n a l i n t h e water p r o p e r t i e s . Wind speed and d i r e c t i o n were measured u s i n g a Meteorology Research, I n c . mechanical weather s t a t i o n on a nearby low p o i n t of l a n d , where winds seem t o be r e p r e s e n t a t i v e of t h e main b a s i n .

The analog r e c o r d s were averaged hourly

by hand, r e s o l v e d i n t o north and e a s t components, and then d a i l y averaged.

Gaps

i n t h i s d a t a s e t were f i l l e d from U.S. Coast Guard o b s e r v a t i o n s a t t h e same location. Aanderaa c u r r e n t d a t a were f i r s t reduced t o north and e a s t components, 2.86-hr f i l t e r e d f o r t i m e series p l o t s and p r o g r e s s i v e v e c t o r diagrams, and 35-hr f i l t e r e d for s t i c k plots.

Instantaneous v a l u e s of temperature and s a l i n i t y were p l o t t e d ,

and simple d a i l y averages c a l c u l a t e d because of t h e absence of any t i d a l s i g n a l . V e l o c i t i e s then were resolved i n t o components along axes of maximum and minimum variance of t h e t o t a l r e c o r d (Cannon, 1 9 7 1 ) . t h e s e axes change with depth. upper water and about 40'-45' f i l t e r e d from t h e along-axis

P r i m a r i l y due t o changing topography,

The along-channel axes were about 20'-25' i n t h e deeper water.

Tidal p e r i o d i c i t i e s w e r e

r e c o r d s u t i l i z i n g fast-Fourier-transform

Fourier c o e f f i c i e n t s were c a l c u l a t e d f o r 30-day

i n the

intervals.

techniques.

Coefficients for

periods s h o r t e r than 30 h r were s e t t o z e r o , t h e r e s u l t i n g s e r i e s retransformed, and d a i l y averages c a l c u l a t e d f o r conceptual convenience.

This technique has

been compared t o v a r i o u s o t h e r f i l t e r s , and no s i g n i f i c a n t d i f f e r e n c e s have been found .

HYDROGRAPHY

The d a i l y averages of d e n s i t y a s sigma-t

(where

lo3

ot

= p

- 1),salinity,

and temperature from t h e year-long mooring have been p l o t t e d t o show t h e C h a r a c t e r i s t i c s of t h e v a r i a t i o n s ( F i g s . 3-5).

S i m i l a r c h a r a c t e r i s t i c s were

520 observed a t s p a t i a l a r r a y moorings following r e l o c a t i o n of t h e year-long mooring. Density changes predominately were r e f l e c t e d i n s a l i n i t y changes with some important exceptions noted below near t h e bottom. s t r a t i f i e d , and t h e s m a l l e s t top-to-bottom

Note t h e e s t u a r y was always

differences i n density (or s a l i n i t y )

probably occurred i n l a t e October.

1975 Fig. 3 .

28

Daily average sigma-t i n Puget Sound main b a s i n .

-,?

+

S T D CRUISES I

Fig. 4.

1976

I

I

I

I

e I

I

e I

I

Daily average s a l i n i t y i n Puget Sound main b a s i n .

I

e I

I

I

521

OCT

NOV

DEC

JAN

FEB

MAR

APR

1975 5.

MAY

JUNE

JULY

AUG

SEPT

1976

Daily average temperature i n Puget Sound main b a s i n .

The change i n bottom sigma-t d u r i n g t h e year was r e l a t i v e l y l a r g e , more than a 1.1 g/R decrease from t h e maximum i n October t o t h e minimum i n February. maximum d e n s i t y appeared t o occur l a t e r n e a r e r t h e s u r f a c e t h a n a t depth.

The The

r e l a t i v e i n c r e a s e s i n d e n s i t y a t t h e deeper l e v e l s l a s t i n g f o r a few days t o a week a r e i n t e r p r e t e d a s new water e n t e r i n g t h e Sound a c r o s s Admiralty I n l e t s i l l . Current-meter d a t a shown below v e r i f y t h i s concept. i n c r e a s e s occurred i n September-October.

Two prominent d e n s i t y

These were followed by a r e l a t i v e l y

steady decrease i n d e n s i t y u n t i l l a t e December when two more r e l a t i v e i n c r e a s e s occurred.

The o v e r a l l d e n s i t y , however, continued t o decrease u n t i l t h e minimum

was reached i n e a r l y February.

The s i n g l e l a r g e s t sigma-t i n c r e a s e of about

0.6 g/l, more than h a l f t h e y e a r l y change, then occurred i n about 1 week, followed by a s e r i e s of i n c r e a s e s r a i s i n g t h e o v e r a l l bottom d e n s i t y d u r i n g t h e n e x t 2 months.

During one o f t h e s e i n c r e a s e s i n l a t e February, i t appeared t h a t t h e

i n c r e a s e occurqgd a t shallower depths b e f o r e it occurred a t t h e bottom.

If this

was s o , t h e lower p a r t of t h e water column may have been u n s t a b l e t e m p o r a r i l y . STD o b s e r v a t i o n s j u s t b e f o r e t h i s e v e n t gave no i n d i c a t i o n s of an i n s t a b i l i t y , hence i t may have been i n s t r u m e n t a l e r r o r . I n c r e a s e s i n d e n s i t y a t t h e bottom almost always were e v i d e n t a t shallower depths up t o about 50 m .

Exceptions occurred i n e a r l y August and i n e a r l y

September when i n c r e a s e s observed a t 70 and 105 m b u t not a t 149 and 192 m

522 indicated the only intermediate l e v e l intrusions.

The o v e r a l l m a g n i t u d e o f

d e n s i t y w a s l o w e r a t t h e end of t h e y e a r l y o b s e r v a t i o n s . p a r t l y t h e r e s u l t o f u n u s u a l summer c o n d i t i o n s i n 1976.

T h i s may h a v e b e e n There w e r e m o r e s t o r m s

t h a n u s u a l , h e n c e more f r e s h w a t e r w a s a v a i l a b l e t o mix w i t h incoming s a l t w a t e r . A l s o , t h e i n c r e a s e d abundance of s t o r m s would r e s u l t i n lesser d e v e l o p e d c o a s t a l upwelling,

t h u s less d e n s e w a t e r would p r o g r e s s landward t o t h e e n t r a n c e s i l l .

The t e m p e r a t u r e changed from a maximum o f 11°-120C i n l a t e summer and e a r l y f a l l t o a minimum of l e s s t h a n 7.5OC i n March.

The top-to-bottom

temperature

r a n g e w a s g r e a t e s t d u r i n g h i g h e s t t e m p e r a t u r e s and l e a s t d u r i n g l o w e s t t e m p e r a t u r e s . I n f a c t , t h e top-to-bottom

r a n g e w a s o n l y 0.1°-0.20C

d u r i n g March.

However, t h e

d e n s i t y r a n g e w a s r e l a t i v e l y l a r g e , and t h e w a t e r column remained s t a b l y s t r a t i f i e d . The f a l l 1976 v a l u e s were a b o u t t h e same a s t h o s e o f 1975 f o r similar d e p t h s . There w a s a r e m a r k a b l e l a r g e d e c r e a s e i n b o t t o m t e m p e r a t u r e of a b o u t 1.6OC which o c c u r r e d i n 3-4 d a y s commencing on 25 December.

T h i s d e c r e a s e was f o l l o w e d

by a n o t h e r of a b o u t 0 . 9 O C commencing a b o u t 8 J a n u a r y .

These c a n b e e x p l a i n e d by

t h e p r o c e s s o u t l i n e d above u s i n g t h e w i n t e r d a t a from 1 9 7 3 (Cannon and Ebbesmeyer, 1977).

However, t h e s e t e m p e r a t u r e s t e p d e c r e a s e s w e r e l a r g e r t h a n a n y o b s e r v e d

i n 1973, and, u n l i k e those i n 1973, they w e r e evi dent as a d e f i n i t e i ncr ease i n bottom d e n s i t y shown by t h e d e n s i t y p e a k s i n l a t e December and e a r l y J a n u a r y . Note, t h e r e w a s no c o r r e s p o n d i n g s a l i n i t y i n c r e a s e d u r i n g t h e s e t w o e v e n t s .

All

o t h e r d e n s i t y i n c r e a s e s w e r e accompanied by s a l i n i t y i n c r e a s e s . There w e r e s e v e n t e m p e r a t u r e s t e p d e c r e a s e s o b s e r v e d a t t h e b o t t o m from e a r l y O c t o b e r u n t i l March when t h e water column became i s o t h e r m a l .

A l l b u t t h e one

d e c r e a s e i n e a r l y November (more s i m i l a r t o t h o s e i n 1973) w a s accompanied b y a p r o m i n e n t d e n s i t y i n c r e a s e , and even i t showed a s l i g h t i n c r e a s e .

Density

i n c r e a s e s f o l l o w i n g t h e w a t e r column becoming i s o t h e r m a l i n l a t e March must b e due t o s a l i n i t y i n c r e a s e s .

However, t h e i n t r u s i o n s t h e n w e r e c h a r a c t e r i z e d by

r e l a t i v e l y warmer water as e v i d e n c e d i n A p r i l and t w i c e i n J u l y .

Commencing i n

August, t h e i n t r u s i o n s a g a i n w e r e c h a r a c t e r i z e d by c o l d e r w a t e r a t a l l o b s e r v e d d e p t h s , and t h o s e a t 73 m ( a b o u t s i l l d e p t h ) o c c u r r e d b e f o r e those a t d e e p e r d e p t h s by s e v e r a l d a y s . STD p r o f i l e s show t h e v e r t i c a l c h a r a c t e r i s t i c s o f w a t e r p r o p e r t i e s a t t h e moorinq d u r i n q t h e two e x t r e m e s e a s o n s ( F i g . 6 ) .

S a l i n i t y and sigma-t c u r v e s

werc p a r a l l e l , and o n l y a s l i g h t p y c n o c l i n e was o b s e r v e d i n F e b r u a r y b e c a u s e freshwater i n p u t w a s n o t nearby.

Temperatures w e r e w a r m e r a t t h g s u r f a c e i n

September and a b o u t i s o t h e r m a l i n F e b r u a r y .

V a l u e s o b s e r v e d w i t h t h e moored

i n s t r u m e n t s a g r e e d w e l l w i t h t h e STD o b s e r v a t i o n s . S e c t i o n s a l o n g t h e Sound show t h e h o r i z o n t a l d i s t r i b u t i o n s of p r o p e r t i e s f o r t h e same two t i m e p e r i o d s ( F i g s . 7 - 8 ) .

The September sec-iion w a s o c c u p i e d d u r i n g

a n i n t r u s i o n a s shown by t h e t e m p e r a t u r e and s a l i n i t y c o n t o u r s .

In fact, a t the

moor.ing a l l w a t e r below a b o u t s i l l d e p t h (64 m ) w a s f l o w i n g s o u t h w a r d .

The

523 February s e c t i o n w a s occupied a b o u t 10 days a f t e r t h e l a r g e s t s a l i n i t y i n t r u s i o n and a t t h e o n s e t o f a smaller i n t r u s i o n , b u t t h e o n l y e v i d e n c e o f i t i s t h e c o l d e r t e m p e r a t u r e s a n d h i g h e r s a l i n i t i e s of t h e d e e p w a t e r

(see F i g s . 4 - 5 ) .

P r i o r t o t h e l a r g e s t i n t r u s i o n , s a l i n i t y w a s l e s s t h a n a b o u t 29.0

o/oo

a t the

mooring, a n d , a f t e r w a r d s , w a t e r d e e p e r t h a n 100 m i s a l l . g r e a t e r t h a n 29.2 Temperatures changed from a b o u t 8.0°C t o a b o u t 7.5OC.

o/oo.

The w a r m e s t d e e p w a t e r

o c c u r r e d a t t h e s o u t h e r n end of t h e main b a s i n and i s c h a r a c t e r i s t i c t h r o u g h o u t the cooling cycle.

SA L I N I T Y 28 I

1

I

I

(%o)

30

32

I

I

I

I

TEMPERATURE ("C) 10

8 I

I

s FEE

I

I

20

I

I

I2

3

I

I

I

I

S SEPT 75

s'

76

I

22

I

24

I

1 26

SIGMA-T (g11) F i g . 6. S a l i n i t y , t e m p e r a t u r e , and s i g m a - t v e r t i c a l p r o f i l e s n e a r t h e mooring i n summer and w i n t e r .

524 MOORING

0

100

200

TEMPERATURE

PC)

0

-E -

I@@

I t a w 200 n

SALINITY

(%o)

xnn

0

I00

200

300

F i g . 7. T e m p e r a t u r e , s a l i n i t y , and s i g m a - t s e c t i o n s a l o n g P u g e t Sound i n l a t e summer 1975. S a l i n i t y s e c t i o n s o n l y show t h e p r o g r e s s i o n t h r o u g h t h e r e m a i n d e r of t h e y e a r (Fig. 9 ) .

The A p r i l s e c t i o n w a s i m m e d i a t e l y f o l l o w i n g a n i n t r u s i o n ; J u l y w a s

d u r i n g an i n t r u s i o n ; and September w a s w e l l a f t e r a n i n t r u s i o n .

Spring-summer

c o n d i t i o n s u s u a l l y showed l o w e r s u r f a c e s a l i n i t i e s n e a r t h e s o u t h end o f Whidbey I s l a n d (two s t a t i o n s n o r t h o f t h e mooring) which w e r e t h e r e s u l t o f t h e major S k a g i t R i v e r d i s c h a r g e i n t o P u g e t Sound. the sections.

T h e r e w e r e s o m e common f e a t u r e s between

I s o p l e t h s t h a t w e r e c o n t i n u o u s a t d e p t h t h r o u g h t h e main b a s i n

and were more o r l e s s h o r i z o n t a l , s l o p e d upward i n A d m i r a l t y I n l e t and i n t e r s e c t e d the surface.

I s o p l e t h s from t h e S t r a i t o f J u a n d e Fuca which c r o s s e d t h e A d m i r a l t y

I n l e t s i l l s l o p e d downward and u s u a l l y i n t e r s e c t e d t h e b o t t o m i n t h e I n l e t . These f e a t u r e s w e r e c a u s e d by t h e i n t e n s e t i d a l m i x i n g t h r o u g h t h e I n l e t .

Although

t h e c o n t o u r s a r e h o r i z o n t a l , i n d i v i d u a l STD's i n d i c a t e d s l i g h t h o r i z o n t a l g r a d i e n t s i n t h e deep w a t e r i n t h e main b a s i n .

S a l t i e r ( d e n s e r ) w a t e r t h a n i n t h e bottom

o f t h e main b a s i n w a s a l w a y s p r e s e n t a t a b o u t s i l l d e p t h j u s t s e a w a r d o f t h e Admiralty I n l e t s i l l .

525

0

100

zoo TEMPERATURE ("C)

0

-E

I

100

I F

200

n SALINITY

(%o)

0

I00

200

SIGMA-t

25 FEB 1976

300

Fig. 8 . Temperature, salinity, and sigma-t sections along Puget Sound in winter 1976.

FLOW CHARACTERISTICS Inflow occurred at the bottom during each of the density intrusions (Fig. 3), and the details of the flow were similar throughout the year.

This section

describes some of the flow characteristics during the intrusions observed in the November-February deployment.

The flow was tidal at all depths and was

predominately semidiurnal (Fig. 10).

Maximum currents exceeded 30 cm/sec at

the surface and were about 20 cm/sec at the bottom.

Tidal amplitudes were similar

at all depths except right near the bottom which was slightly reduced and was probably the result of frictional influence. same at all depths.

Phases a l s o were approximately the

Net seaward flow is obvious at the upper three levels, and

net landward flow at 113 m.

Onset of the largest salinity intrusion occurred

midday on 5 February both at the bottom and at 113 m.

Tidal signals were still

evident in both flows, but t h e increased average landward flow prevented any reversals at the bottom.

Vector plots of the tidally filtered records more

clearly depicted the average flows, the influence of topography on the direction o € flow, and the onset of the intrusion (Fig. 11).

526

0

-E

I00

v

I t 200

a ALINITY (%J

22 JULY 76

0

I00

zoo

300

Fig. 9.

S a l i n i t y s e c t i o n s a l o n g P u g e t Sound f o r s p r i n g , summer, and f a l l 1976.

The c y c l i c n a t u r e and magnitude o f t h e t h r e e b o t t o m i n f l o w s d u r i n g t h i s deployment c a n b e r e p r e s e n t e d u s i n g a p r o g r e s s i v e v e c t o r d i a g r a m ( F i g . 1 2 ) .

If

t h e s e v e c t o r s r e p r e s e n t e d c o n t i n u o u s f l o w a l o n g t h e Sound, i n s t e a d of t h e v e c t o r summation of f l o w a t a p o i n t , t h e n t h e e x c u r s i o n s would h a v e b e e n 35, 35, and 140 km, r e s p e c t i v e l y .

These w e r e a l l r e l a t i v e l y l a r g e , and t h e l a s t o n e ,

c o i n c i d i n g w i t h t h e l a r g e s t s a l i n i t y i n t r u s i o n , w a s remarkably l a r g e c o n s i d e r i n g t h e l e n g t h of t h e Sound a t t h e s e d e p t h s i s o n l y a b o u t 50 km.

The i m p l i c a t i o n s

a l s o a r e l a r g e i n t h a t new b o t t o m w a t e r c o u l d f i l l t h e Sound i n l e s s t h a n 1 week. Average d a i l y a l o n g - c h a n n e l c u r r e n t s , c a l c u l a t e d a s i n d i c a t e d a b o v e , a r e shown

as v e r t i c a l p r o f i l e s f o r p a r t s of December and F e b r u a r y ( F i g . 1 3 ) .

These p a r t i c u l a r

examples a r e shown b e c a u s e t h e y i n d i c a t e d i n f l o w a t t h e b o t t o m c o r r e s p o n d i n g w i t h two major b o t t o m d e n s i t y i n c r e a s e s , o n e r e s u l t i n g p r i m a r i l y from t e m p e r a t u r e change and o n e by s a l i n i t y change.

The i n c r e a s e i n December c o r r e s p o n d e d w i t h

t h e l a r g e t e m p e r a t u r e d e c r e a s e , and i n f l o w o c c u r r e d from 26-30 December.

The

i n c r e a s e i n F e b r u a r y c o r r e s p o n d e d w i t h t h e l a r g e s t s a l i n i t y i n c r e a s e which f o l l o w e d t h e minimum b o t t o m s a l i n i t y ( a n d d e n s i t y ) o b s e r v e d d u r i n g t h e y e a r ,

527 and inflow occurred from 5-14 February. d a i l y c u r r e n t s a t t h e bottom i n February. consecutive days.

Note t h e e x t r a o r d i n a r i l y l a r g e average Speeds were about 20 cm/sec f o r 5

This would correspond t o a h o r i z o n t a l excursion of 86 km,

which i s s l i g h t l y longer than t h e main b a s i n .

Otherwise, t h e p r o f i l e s a r e

c h a r a c t e r i s t i c of e s t u a r i n e flow w i t h landward flow a t depth and seaward flow i n t h e upper water.

There a r e l a r g e v a r i a t i o n s , however.

The average depth of

change from average landward t o average seaward flow h a s been shown t o occur i n winter a t t h i s l o c a t i o n a t about 50 m and t o have s i g n i f i c a n t p e r t u r b a t i o n s r e l a t e d t o changing winds (Cannon and L a i r d , 1 9 7 6 ) .

The average depth-of-change

i n flow d i r e c t i o n i s about 25% of the t o t a l depth, and t h i s i s i n t e r m e d i a t e between t h e shallower changeover f o r m o s t f j o r d s (10-15%) and t h e deeper changeover

f o r p a r t i a l l y mixed e s t u a r i e s ( 4 0 - 5 0 % ) .

T

30

T

30

Fig. 1 0 . Along-channel c u r r e n t s f o r p a r t of t h e r e c o r d a t t h e mooring which have been 2.8-hr f i l t e r e d t o remove high-frequency n o i s e . P o s i t i v e c u r r e n t s a r e seaward.

528

1'-

-

-

/ , /

-

I

/ / , , .

.

/

/

W

/-/

27 w

a

78 rn

15 , , , , , J

/'

\

n

,\ , /

15

113rn

1'5 /

/

/

/

I

/// 190 rn

15

Fig. 11. Vector p l o t s o f c u r r e n t s a t t h e mooring c o n s t r u c t e d from 35-hr d a t a and p l o t t e d every 6 h r f o r t h e same d a t a a s i n F i g . 1 0 .

filtered

Winds along t h e Sound f o r December-February were p r i m a r i l y t o t h e n o r t h a s s o c i a t e d with w i n t e r storms, b u t t h e r e were a few r e v e r s a l s a s s o c i a t e d with c e s s a t i o n o f storms (Fig. 1 4 ) .

The 3-month north-south

average wind was 3 . 8 m/sec,

while t h e f a l l and s p r i n g were only about h a l f a s l a r g e a t 1 . 8 and 1 . 3 m/sec, respectively.

Only i n t h e summer 3 months was t h e average t o t h e s o u t h , and then

i t was r e l a t i v e l y small a t 0 . 2 m/sec.

Variance about t h e mean was comparable i n

fall, w i n t e r , and s p r i n g , b u t i t was about h a l f i n summer.

The wind r e v e r s a l

commencing a t t h e end o f January a p p a r e n t l y w a s s u f f i c i e n t t o change t h e h o r i z o n t a l p r e s s u r e g r a d i e n t and t o s i g n i f i c a n t l y a l t e r the c u r r e n t p r o f i l e on 4 February, t h e peak of t h e wind r e v e r s a l . t h e o r e t i c a l study of f j o r d s .

These s h i f t s were c o n s i s t e n t with R a t t r a y ' s

(1967)

Out-estuary winds should r a i s e t h e inflowing l a y e r s

t o shallow depths and i n c r e a s e t h e magnitude of t h e maximum inflow. e v i d e n t h e r e around 2 3 December, 5 January, and 18 February.

This was

Relaxation of t h e

529 o u t - e s t u a r y winds o r r e v e r s a l t o i n - e s t u a r y winds should r e s u l t i n deepening t h e outflowing l a y e r and sometimes t o development o f a t h i r d l a y e r flowing i n a t the This was e v i d e n t h e r e following t h e wind r e v e r s a l s around 1 January

surface.

and 4 February.

Cannon and L a i r d (1976) have d i s c u s s e d o t h e r examples of t h e s e

reversals.

0 0

0 cr)

10 N O V 7 5 0

E

c!

+ b

00

Fig. 1 2 . P r o g r e s s i v e v e c t o r diagram from t h e near-bottom c u r r e n t meter f o r t h e November 1975-February 1976 deployment. A 25-hr

time s e r i e s o f t h r e e STD s t a t i o n s a c r o s s Puget Sound a t t h e mooring

showed t h a t v a r i a t i o n s i n temperature and s a l i n i t y caused by t i d a l advection were small.

F i g u r e 1 5 shows t h e average temperature, s a l i n i t y , and sigma-t i n t h e

s e c t i o n f o r 24 February and t h e average d a i l y c u r r e n t p r o f i l e s f o r 23 and 25 February.

A l e v e l o f no-net-motion

i s i n d i c a t e d a t about 50 m.

The isopycnals

sloped down from west t o e a s t , and t h e sigma-t s e c t i o n showed t h e 22.6 g/R isopycnal a t about t h e bottom of t h e mixed l a y e r and t h e depth of no-net-motion. If t h e isopycnals were a d j u s t e d f o r 22.6 g/R t o be f l a t , then t h e isopycnal s l o p e

would be roughly c o n s i s t e n t with outflow i n t h e s u r f a c e l a y e r and inflow i n t h e deeper l a y e r , perhaps i n quasi-geostrophic

balance.

The temperature s e c t i o n

showed a well-mixed l a y e r a t middepths ( s t i p p l e d ) which extended down t o about t h e depth o f no-net-motion,

which might be i n d i c a t i v e of t h e water coming from

530 f a r t h e r south.

In f a c t , i t probably i s water which has been mixed i n The Narrows

and i s then r e f l u x e d i n t o t h e main b a s i n somewhat s o u t h of t h e mooring (Barnes and Ebbesmeyer, 1977).

-20

o

20

crn/sec

h

E

-r 2oo DEC 23

2oo

FEB

3

24

25

26

27

28

4

5

6

7

8

29

30

31JAN I

2

U, PUGET

2oo FEB 17

18

19

20

21

22

3

4

5 JAN

SOUND, 1975-76

23

Fig. 1 3 . V e r t i c a l p r o f i l e s of d a i l y average along-channel c u r r e n t s f o r p a r t s of December 1975-February 1976. S t i p p l e d a r e a s d e s i g n a t e landward flow. V e r t i c a l l i n e s d e s i g n a t e zero flow f o r each p r o f i l e , and a d j a c e n t v e r t i c a l l i n e s d e s i g n a t e i 2 0 cm/sec. REPLACEMENT PROCESS The replacement of deep water i n Puget Sound can be summarized u t i l i z i n g Figure 1 6 showing deep-water range p r e d i c t e d f o r S e a t t l e .

d e n s i t y , along-channel flow, and t h e f l o o d - t i d e A s i n our e a r l i e r work a t t h e same l o c a t i o n

(Cannon

531 and Ebbesmeyer,

1 9 7 7 ) , we observed t h e w i n t e r i n t r u s i o n s (December-February) t o This

occur 8-10 days following p e r i o d s when t h e f l o o d - t i d e range exceeded 3.5 m.

t i m e l a g i n d i c a t e d t h e time r e q u i r e d f o r t h e i n t r u s i o n s t o flow w i t h t h e t i d a l c u r r e n t s from t h e e n t r a n c e s i l l t o t h e mooring; t h e t i d a l range p r e d i c t e d when t h e flooding t i d e s would be g r e a t e s t i n Admiralty I n l e t over t h e s i l l .

Similar

t i m e l a g s have been observed f o r t h e approximate f o r t n i g h t l y renewal during summer-fall f o r t h e P o r t Susan e s t u a r y , a secondary b a s i n w i t h i n Puget Sound w i t h

about a 100-m s i l l depth ( F i g . 1 ) . About a week was r e q u i r e d f o r water t o flow from Admiralty I n l e t s i l l t o t h e s i l l of P o r t Susan (Cannon, 1975).

s -20 DEC

J A N 76

FEB

Fig. 1 4 . Hourly average winds measured a t W e s t P o i n t f o r December 1975February 1976.

TEMPERATURE [ " C ) WEST

-

SALINITY

EAST

WEST

SIGMA-t

("Lo)

EAST

WEST

EAST

50

E

I

I 100

c

a

w n 150

200

-20

0

20

E I

I 100

23 FEE

I-

n w 150

IN

OUT

200

Fig. 15. Daily average water p r o p e r t i e s i n a s e c t i o n a c r o s s Puget Sound a t t h e mooring f o r 24 February, and d a i l y average along-channel c u r r e n t s f o r 23 and 25 February.

532

P i g . 1 6 . Daily average sigma-t a t 5 and 50 m above t h e bottom i n Puget Sound; range (R) and h e i g h t ( H ) of g r e a t e s t d a i l y S e a t t l e f l o o d t i d e ; and d a i l y average along-channel c u r r e n t s a t 5 and 50 m above t h e bottom f o r 1975-76. There a r e major v a r i a t i o n s , however, from our w i n t e r concepts. were occurrences of l a r g e t i d a l ranges d u r i n g November-January

F i r s t , there

w i t h almost no o r

very small inflow and p a r t i c u l a r l y with no apparent changes i n water p r o p e r t i e s . Second, t h e r e were occurrences of i n f l o w during low t i d a l ranges during AugustOctober and d u r i n g March-April.

These l a t t e r two p e r i o d s a l s o had d i f f e r e n c e s

i n t h a t l a t e summer-fall was when t h e most dense water could e n t e r t h e Sound and t h a t winter-spring was about when t h e l e a s t dense water w a s i n t h e Sound. Thus, o t h e r f a c t o r s must be important i n determining when t h e r e w i l l be i n t r u s i o n s o f deep water.

One apparent f a c t o r i n d i s c u s s i o n s concerning t h e

problem (Barnes, p e r s o n a l communication) was t o c o n s i d e r t h e degree of mixing which t a k e s p l a c e when c r o s s i n g t h e f i n i t e - l e n g t h e n t r a n c e s i l l .

During summer-

f a l l t h e water a p p a r e n t l y i s s u f f i c i e n t l y s a l t (dense) t h a t i t does n o t m a t t e r t h a t more than one t i d e i s r e q u i r e d t o t r a n s p o r t i t a c r o s s t h e s i l l .

Also,

t h e r e i s minimum freshwater e n t e r i n g t h e Sound t o be a v a i l a b l e f o r mixing over the s i l l .

On t h e c o n t r a r y , t h e l a c k of s i g n i f i c a n t inflow d u r i n g f a l l when

t h e r e a r e l a r g e t i d e s probably i s i n f l u e n c e d by having an abundance of freshwater a v a i l a b l e f o r mixing over t h e s i l l .

The l a r g e s t temperature s t e p decrease i n

December occurred following a r e l a t i v e extr-eme c o l d s p e l l c o i n c i d e n t with t h e l a r g e t i d a l range.

This c o l d s p e l l a l s o was c h a r a c t e r i z e d by c l e a r weather and

a l a c k o f f r e s h w a t e r , a s was t h e c a s e w i t h t h e 1973 d a t a . F i n a l l y , t h e two l a r g e i n t r u s i o n s i n March occurred following r e l a t i v e l y small tides.

They r e p r e s e n t e d excursions a s l a r g e a s t h e one i n February.

This c a s e

is least well understood.

Apparently, the combination of greater density outside

and minimum inside is sufficient to cause the intrusion, even though more than one tidal cycle would be required to cross the sill.

This period of renewal was

observed to be most important in other years and needs additional study.

Our

earlier work showed it to be the dominant time of renewal of oxygen in the secondary basin of Port Susan (Cannon, 1975). The overall rate of replacement of intermediate and deep water was approximated for the 1973 data using the net landward transport and assuming negligible effects of entrainment and diffusion (Cannon and Ebbesmeyer, 1977).

Computations were

made using average daily currents, assumed uniform across channel.

The landward

transport was divided into the volume beneath the average depth of no-net-motion (about 35 km3 beneath 52 m), yielding an effective replacement time of about 9 days.

This value is applicable here for all the major intrusions.

There has been a variety of shorter duration studies of renewal of deep water in other fjord estuaries in western North America (Anderson and Devol, 1973; Muench and Heggie, 1977; Pickard, 1976; Stucchi and Farmer, 1976). characteristics described here.

None show the

Perhaps one difference is that the head of the

main basin of Puget Sound is really a relatively narrow, shallow passage (The Narrows in Fig. 1) entering another basin.

There are extremely high tidal

currents through this passage, which appear to be capable of pumping deep water up from about twice the depth of the sill.

Physically, the process would be

explained by application of Bernoulli's principle, similar to that proposed by Stommel et al. (1973) for the Mediterranean deep water passi:ig over its shallow sill into the Atlantic Ocean.

Helseth et al. (1976) have shown this effect in

Puget Sound using a hydraulic model.

It is not yet known whether the process

occurs continuously on all flood tides, or whether it occurs only on the largest, fortnightly tides.

In either case, the effect of The Narrows is to greatly assist

the movement of deep water through Puget Sound. This information is of interest to city managers of wastewater discharges in the Puget Sound system.

The renewal rates of 1-2 weeks indicated by these studies

is significantly quicker than earlier estimates of 2-10 months based on water property observations. Additional field experiments are planned to provide better descriptions of the processes at the head of the main basin (The Narrows) and on the entrance sill (Admiralty Inlet).

SUMMARY

A year-long series of observations from moored instruments has provided significant new information concerning the replacement of bottom water below sill depth in Puget Sound, a fjord estuary.

Deep water was shown to be replaced

by intrusions at about fortnightly intervals (but not every fortnight) when flood-tide ranges exceeded 3.5 m at Seattle.

At this range, Strait of Juan de Fuca

534 water could completely t r a n s i t t h e Admiralty I n l e t e n t r a n c e s i l l on one flood t i d e and t h u s be l e a s t mixed.

An 8- t o 10-day time l a g w a s r e q u i r e d f o r an

i n t r u s i o n t o t r a n s i t from. t h e s i l l t o t h e mooring. Most dense water e n t e r e d i n l a t e summer and e a r l y f a l l followed by an approximate steady decrease i n d e n s i t y u n t i l e a r l y w i n t e r .

I n t r u s i o n s of c o l d e r and denser

water were then observed l a s t i n g about 5 days each.

The c o l d e s t w a s 1.6'C

and

followed flood t i d e s concurrent w i t h below-freezing a i r temperatures over Admiralty I n l e t .

However, following each i n t r u s i o n , mixing r e s u l t e d i n t h e

o v e r a l l d e n s i t y of deep water c o n t i n u i n g t o decrease u n t i l February when t h e water column became l e a s t dense.

Density then i n c r e a s e d through a s e r i e s of

i n t r u s i o n s of h i g h e r s a l i n i t y , t h e i n i t i a l one being g r e a t e r than 0 . 5 Some of t h e i n t r u s i o n s i n l a t e summer and i n midwinter were n o t c o i n c i d e n t with l a r g e f l o o d - t i d e ranges, and t h e s e r e q u i r e a d d i t i o n a l study. Mean d a i l y near-bottom speeds a t t h e mooring were up t o 2 0 cm/sec d u r i n g some of the intrusions.

This speed implied a p o s s i b l e excursion exceeding t h e l e n g t h

of t h e Sound i n 5 days.

Computations of replacement t i m e a l s o were made using

average d a i l y c u r r e n t s , assumed uniform a c r o s s t h e channel a t t h e mooring.

The

landward t r a n s p o r t was d i v i d e d i n t o t h e volume of water below t h e average depth o f no-net-motion,

y i e l d i n g an e f f e c t i v e replacement time of about 9 days.

Thus,

i t seems p o s s i b l e t h a t t h e deep water could move through Puget Sound i n less

than 2 weeks. Water of g r e a t e r d e n s i t y than t h e r e s i d e n t water i n t h e Sound was always a v a i l a b l e i n t h e S t r a i t of Juan de Fuca a t s i l l depth.

Additional study i s

needed of t h e mixing p r o c e s s e s a c r o s s t h e f i n i t e - l e n g t h s i l l i n Admiralty I n l e t t o determine t h e c o n d i t i o n s which w i l l r e s u l t i n an i n t r u s i o n when t h e r e a r e la rge flood t i des.

F u r t h e r s t u d y a l s o i s r e q u i r e d a t The Narrows, t h e r e l a t i v e l y

shallow c o n s t r i c t i o n a t t h e head o f t h e main b a s i n which provides a c c e s s t o t h e There i s evidence t h a t t h e l a r g e f l o o d c u r r e n t s i n The Narrows

southern b a s i n .

a c t a s a h y d r a u l i c pump t o remove deep water from t h e Puget Sound main b a s i n .

ACKNOWLEDGMENTS Supporting s e r v i c e s by many i n our C o a s t a l Physics Group a r e g r a t e f u l l y acknowledged.

S p e c i a l thanks f o r t h e a s s i s t a n c e i n t h e f i e l d a r e due t h e crew

and o f f i c e r s of t h e NOAA s h i p McArthur.

One of u s (GAC) i s g r a t e f u l t o

Professor Emeritus C l i f f o r d A . Barnes a t t h e U n i v e r s i t y of Washington f o r d i s c u s s i o n s concerning Puget Sound c i r c u l a t i o n .

F i n a n c i a l support was provided

by t h e Marine Ecosystems Analysis Program o f t h e National Oceanic and Atmospheric Administration.

535 REFERENCES Anderson, J. J. and Devol, A. H., 1973. Deep water renewal and Saanich Inlet, an intermittently anoxic basin. Estuarine and Coastal Marine Sci., 1:l-10. Barnes, C. A. and Collias, E. E., 1958. Some consideration of oxygen utilization rates in Puget Sound. J. Marine Res., 17:68-80. Barnes, C. A. and Ebbesmeyer, C. C., 1977. Some aspects of Puget Sound's circulation and water properties. In: B. Kjerfve (Editor), Transport Processes in Estuarine Environments. Proc. 7th Belle W. Baruch Symp., Univ. South Carolina Press, Columbia, South Carolina (in press). Cannon, G. A., 1971. Statistical characteristics of velocity fluctuations at intermediate scales in a coastal plane estuary. J. Geophysical Res., 76:5852-5858. Cannon, G. A., 1975. Observations of bottom-water flushing in a fjord-like estuary. Estuarine and Coastal Marine Sci., 3:95-102. Cannon, G. A., and Ebbesmeyer, C. C., 1977. Winter replacement of bottom water in Puget Sound. In: B. Kjerfve (Editor), Transport Processes in Estuarine Environments. Proc. 7th Belle W. Baruch Symp., Univ. South Carolina Press, Columbia, South Carolina (in press). Cannon, G. A. and Laird, N. P., 1976. Wind effects on tidally averaged current profiles in a fjord estuary. Trans. Am. Geophys. Union, 57:933 (Abstract). Collias, E. E., McGary, N. and Barnes, C. A . , 1974. Atlas of physical and chemical properties of Puget Sound and its approaches. Washington Sea Grant Publ., Univ. of Washington Press, Seattle, washington, 235 pp. Farmer, H. G. and Rattray, M., 1963. A model of the steady-state salinity distribution in Puget Sound. Technical Report No. 85, Dept. of Oceanography, Univ. of Washington, Seattle, Washington, 33 pp. Helseth, J. M., Ebbesmeyer, C. C., Barnes, C. A. and Lincoln, 3. H., 1976. Bathymetrically driven transport in a fjord: A simple demonstration using a physical model of Puget Sound. Trans. Am. Geophys. Union, 57:934 (Abstract) Muench, R. C., and Heggie, D. T., 1977. Deep water exchange in Alaskan subarctic fjords. In: B. Kjerfve (Editor), Transport Processes in Estuarine Environments. Proc. 7th Belle W. Baruch Symp., Univ. South Carolina Press, Columbia, South Carolina (in press). Pickard, G. L., 1975. Annual and longer term variations of deepwater properties in the coastal waters of southern British Columbia. J. Fish. Res. Bd. Canada, 32:1561-1587. Rattray, M., 1967. Some aspects of the dynamics of circulation in fjords. In: G. H. Lauff (Editor), Estuaries. American Association Advancement Sci., Washington, D. C., pp. 52-62. Stommel, H., Bryden, H. and Mangelsdorf, P., 1973. Does some of the Mediterranean outflow come from great depth? Basel, Birkhauser Verlag, Pageoph, 105:887-889. Stucchi, D. and Farmer, D. M., 1976. Deep water exchange in Rupert-Holberg Inlet. Pacific Marine Science Report No. 76-10, Marine Sciences Directorate, Pacific Region, Victoria, B. C., Canada, 32 pp. Winter, D. F., Banse, K. and Anderson, G. C., 1975. The dynamics of phytoplankton blooms in Puget Sound, a fjord in the northwestern United States. Marine Biology, 29:139-176.

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537

SUBJECT I N D E X Aberdeen, Abidjan.

165. Ivory Coast,

Adeane p o i n t ,

472.

Admiralty I n l e t , Advective Algonac,

515, 517, 521, 524, 531, 533, 534.

transport,

29, 55.

Axe point,

472.

Baltimore Harbour,

Bath,

387.

254, 256, 257.

Antwerpen,

Bangor,

198.

142.

267.

68.

Blaue Balje,

346.

Blocking e f f e c t s ,

4

Boundary P a s s a g e ,

288.

BrGnt V a i s d l a

frequency,

15.

Buoyancy, see a l s o s t r a t i f i c a t i o n , s a l i n i t y , d e n s i t y . - Buoyancy e f f e c t , 2 1 9 , 222, 225, 226, 230. - Buoyancy e f f e c t upon d i s p e r s i o n , 217. Buoyant

spreading,

Byfjord

(Western Norway),

CAFE

373, 374, 383. 441, 443, 444, 447, 4 4 8 , 452

( C i r c u l a t i o n A n a l y s i s by F i n i t e E l e m e n t s ) ,

Camp p e a k ,

T

455, 463.

357, 359.

472.

cascade point, chalk point,

471.

2.

C h e s a p e a k e Bay,

2, 3, 6 , 127 - 129, 132, 134, 135, 137, 138, 143, 144.

-

s e e a l s o c u r r e n t , v e l o c i t y , 8 2 , 221, 224 - 227, 325, 419, 424, 439, 465, 466, 471, 476, 491. A n o m a l i e s o f c i r c u l a t i o n , 201, 202. B r a n c h i n g of c i r c u l a t i o n , 199.

-

E s t u a r i n e c i r c u l a t i o n : see e s t u a r y . R e s i d u a l c i r c u l a t i o n , 5 0 , 209. T i d a l c i r c u l a t i o n : see t i d e s , t i d a l

Circulation,

-

Coastal w a t e r ,

current.

127, 389, 504.

10 - 1 2 , 57, 61, 6 3 - 6 5 , 6 9 , 73, 7 4 , 108, 109, 115, 1 2 3 , 2 1 3 , 218, 224, 225, 233, 234, 238, 241, 453.

Concentration,

Conservation equation, Continuity

equation,

419, 420.

357, 387, 400.

Coriol is - C o r i o l i s a c c e l e r a t i o n . 262, 263. - C o r i o l i s e f f e c t , 107, 217, 219 - 222.

538 -

C o r i o l i s e f f e c t u p o n d i s p e r s i o n , 217. C o r i o l i s f o r c e , 40. C o r i o l i s p a r a m e t e r , 170, 221 - 223, 291, 348, 358, 380.

C o r p u s C h r i s t i B a y , T e x a s , 147 - 150, 154, 155, 157. C u r r e n t , see a l s o v e l o c i t y , circulation. C u r r e n t p r o f i l e , see v e l o c i t y profile. Residual c u r r e n t , see r e s i d u a l velocity. T i d a l c u r r e n t , see tides. C u r r e n t m e t e r , 1 , 3 , 4 , 82, 8 4 , 127, 128, 142, 147, 149, 150, 155, 156, 285, 286, 298 - 300, 307, 363, 442, 453, 455, 456, 458, 466, 469, 470, 5 1 8 , 521, 529. - Aanderaa c u r r e n t m e t e r , 453, 469, 471, 477, 478, 490, 518, 519. Mechanical c u r r e n t m e t e r , 470, 471. C u r v a t u r e , 330. - C u r v a t u r e e f f e c t : 219, 222, 225, 230. C u r v a t u r e e f f e c t upon d i s p e r s i o n , 217. D a b o b R a y , 267. Decay - of p o l l u a n t c o n c e n t r a t i o n , 390. o f e n e r g y d e n s i t y , 390. Deep water e x c h a n g e , 441, 444, 447. D e e p water r e p l e n i s h m e n t , 410, 411. D e l f t , 107, 110, 114, 120. D e n s i t y , see a l s o b u o y a n c y , s t r a t i f i c a t i o n , salinity. - D e n s i t y d i s t r i b u t i o n , 115, 119, 239, 415, 423, 482, 485, 491. - Density g r a d i e n t s , 226, 388, 410, 414, 416, 421. D e n s i t y p r o f i l , 8 2 , 420, 412, 414, 462, 468, 469, 480, 489, 511. - D e n s i t y s t r a t i f i c a t i o n , 395. D i f f u s i o n , see a l s o t u r b u l e n c e , m i x i n g , eddy. Diffusing c l o u d , 235 - 237. D i f f u s i o n c o e f f i c i e n t , 4 4 , 58, 217. E f f e c t i v e l o n g i t u d i n a l d i f f u s i o n c o e f f i c i e n t , 207, 212, 213. Molecular d i f f u s i o n , 236, 237. - T u r b u l e n t d i f f u s i o n , s e e turbulence. D i s p e r s i o n , see t u r b u l e n c e , m i x i n g , e d d y , d i f f u s i o n , 1 3 , 14. Dispersi.on c o e f f i c i e n t , 9 , 15, 5 2 , 217, 226, 227, 230, 231 D i s s i p a t i o n r a t e o f t u r b u l e n t e n e r g y , 390. D o v e r , 177, 178. Down

-

w e l l i n g , 439.

D r a g , see a l s o f r i c t i o n , shear. Bottom d r a g , 264. D r a g c o e f f i c i e n t , 428. - D r a g r a t i o , 4 2 3 , 429 - 431, 434, 436. T r a n s v e r s e d r a g , 220, 221. Dutchman h e a d , 472. Dutch Wadden S e a , 215. Duwarnish R i v e r , 79

-

8 2 , 84 - 8 7 , 9 6 , 9 7 , 100

Ebb

-

-

-

Ebb Ebb Ebb Ebb

c h a n n e l , 27, 3 1 , 46. c u r r e n t , 199, 200, 460. t i d e , 27, 79, 399, 402, 403, 474. v e l o c i t y , 14, 302, 304.

539 Ebrie

Lagoon. Ivory C o a s t , 7 9 , 197, 198.

Eddy , s e e a l s o turbulence, mixing E d d y d i f f u s ivity, 3 6 , 3 7 , 3 9 , 2 0 3 , 204, 2 1 7 , 2 1 9 , 2 2 1 , 2 2 2 , 2 2 6 , 2 3 0 , 4 1 8 , 4 9 6 , 510. - Eddy m o t i o n , 466. E d d y v i s c o s i ty, 3 7 , 3 9 , 2 0 3 , 2 0 4 , 2 9 1 , 3 2 4 , 3 4 9 , 3 5 7 , 3 5 8 , 3 6 1 , 3 7 3 , 3 8 0 , 3 8 1 , 3 8 8 , 3 9 9 , 4 1 8 , 4 2 1 , 498. - Eddy viscosity coefficient, 3 5 8 , 360. T u r b u l e n t e d d y , 108, 234. Effl u e n t , 3 7 3 , 3 7 5 , 3 7 7 , 3 7 8 , 3 8 1 , 383. Ekman - Ekman d y n a m i c s , 127. - E k m a n f l u x , 135. - E k m a n n u m b e r , 3 8 1 , 383. Energy - k i n e t i c , 398. - Subgridscal'e, 3 8 7 , 388. - g e n e r a t i o n o f , 388. - d i s s i p a t i o n o f , 398. T r a n s f e r , 388. - W a v e , 3 9 7 , 398. Enst r o p h y , 396. Entr a i n m e n t , 3 0 8 , 3 8 4 , 4 1 4 , 4 1 8 , 4 2 4 - 4 2 6 , 496. - E n t r a i n m e n t function, 420. - E n t r a i n m e n t parameter, 4 2 8 , 436. - En t r a i n m e n t v e l o c i t y , 4 1 6 , 4 2 3 , 4 2 8 , 510. EOF A n a l y s i s , 1 31, 1 3 4 , 138. Escaut E s t u a r y , see also Scheldt Estuary, 15, 55. Estuary Estuarine c i rculation, 1 3 1 , 2 6 7 , 2 8 3 , 2 8 6 , 2 8 7 , 2 9 0 , 307. Estuarine d i s c h a r g e , 3 7 3 , 376. - Ho m o g e n e o u s e s t u a r y , 2 6 2 , 2 6 7 , 272. - La g o o n , 79. - Pa r t i a l l y mixed estuary, 131, 142, 1 4 3 , 193. Pa r t i a l l y stratified estuary, 3 4 , 109. Sa l t w e d g e e stuary, 7 9 , 8 0 , 8 2 , 87. Well-mixed e stuary, 4 6 , 6 4 , 109. Exchange c o e f f i cients - hor i z o n t a l , 390. ver t i c a l , 3 8 7 , 388. Feie I s l a n d , 4 4 6 , 4 5 3 , 457. Fernda l e , 298. Finite d i f f e r e n ce - Eq u a t i o n , 391 - 396. G r i d , 392. Mod e l , 387. Finit e d i f f e r e n ce m e t h o d , 2 4 6 , 2 4 7 , 3 4 9 , 358. Finit e e l e m e n t m e t h o d , 2 6 1 , 2 6 2 , 2 6 6 , 2 7 2 , 3 5 0 , 3 5 7 , 358. Finite e l e m e n t m o d e l , 3 4 9 , 350. Flood , 8 8 , 90. Flood c h a n n e l , 2 7 , 3 1 , 46. Flood c u r r e n t , 199, 200, 4 6 0 , 534.

540

-

F l o o d p e r i o d , 30. F l o o d t i d e , 27, 28, 211, 305, 465, 477, 517, 531 - 534. F l o o d v e l o c i t y , 117, 302, 304, 305.

Fort Gratiot, Franklin

254, 257.

River,

Fresh water

466, 467, 505, 506. 9 , 12, 13, 16, 34, 156, 158, 198, 199, 204, 495,

inflow,

521, 531. F r i c t i o n , s e e a l s o d r a g , s h e a r , 108, 246, 269, 446. - B o u n d a r y f r i c t i o n , 425, 426, 430, 436. - B o t t o m f r i c t i o n , 208, 210, 262, 263, 269, 317 - 324, 331, 348, 350,

360, 382.

-

B o t t o m f r i c t i o n c o e f f i c i e n t , 349, 350, 358, 360, 361. F r i c t i o n c o e f f i c i e n t , 210, 251, 263, 269, 290, 2 9 1 , 296, 3 6 1 , 498,

-

F r i c t i o n f a c t o r , 247, 255, 257, 416. F r i c t i o n v e l o c i t y , 240. F r i c t i o n n a l s t r e s s , 499. Q u a d r a t i c f r i c t i o n t e r m , 163, 176, 178, 180, 184, 317.

580.

F r o u d e N u m b e r , 243, 414, 415, 419, 431. I n t e r f a c i a l F r o u d e N u m b e r , 423, 427 I n t e r n a l F r o u d e N u m b e r , 8 2 , 9 2 , 461.

-

-

Fuljord Harbour, Fundy

298.

Ossenisse Channel,

307

-

( S t r a i t of), 283

187, 191, 193.

Graph method,

252.

Graph t h e o r y ,

259.

G r e a t Bay E s t u a r y , Grid s t r u c t u r e , Island,

349 - 351, 355, 359, 361.

392.

283, 287, 289, 295, 296, 300, 305, 306.

Guinea,

o f Mexico,

Hansweert,

Haro

291, 295, 296, 298, 300, 302, 303, 305,

323.

Gironde Estuary,

Harle,

-

309.

German B i g h t ,

Gulf of

27.

27, 29, 30, 31.

Gentbrugge, Georgia

Gulf

429, 433.

( B a y o f ) , 290.

G a t van

Gulf

-

197. 147

-

150, 152, 153, 156 - 158.

30

334, 335, 346, 347.

Strait,

283, 286, 288, 291, 298, 302, 305.

Hatchet Point, Hazel P o i n t ,

472.

269.

Heat balance e q u a t i o n , Hellesoy

390.

( M e t e o r o l o g i c a l S t a t i o n ) , 445.

Herdlafjord, Hermiksen,

441.

435.

Herries Point,

472.

541 443.

Hjelte Fjord, Hjeltesund,

443, 461.

Hood C a n a l ,

Washington,

Hoeya

468, 471, 472.

Head,

Hudson

River,

Huron,

Lake,

373

-

261, 262, 267 - 269, 516

375, 377, 384.

254, 256, 257. 496.

Hydraulic analysis,

H y d r a u l i c J u m p , 79, 102. I n t e r n a l h y d r a u l i c jump, Hydrostatic

equation,

Inlet

471.

head,

I n t e r n a l waves Breaking of

i n t e r n a l waves,

82.

138.

James R i v e r ,

283, 285, 286, 290, 308, 504.

Johnstone S t r a i t , Josenf jord

8 2 , 9 0 , 9 3 , 9 9 , 105.

390.

( S o u t h e r n Norway)

, 439.

d e F u c a S t r a i t , 267, 283, 286 515, 516, 533, 534.

Juan

445.

Kinn,

Klinaklini Knight

-

466, 467, 472, 505, 506.

River,

465 - 467, 470, 471, 479 - 481, 486, 488, 491, 492, 495, 504 - 513.

Inlet,

484

B r i t i s h Columbia,

Korteweg d e V r i e s Kwalate

equation,

estuary,

198. 134.

Lerwick,

165.

Lewisetta,

133.

turbulence

level,

I s l a n d Sound,

Lowestoft, Margate,

479, 480.

472, 490.

Point,

Lanezos f i l t e r ,

Lagoon

Long

288, 290, 300, 3 0 1 , 302, 307,

129.

Kelvin wave,

Local

-

388.

290, 399, 403, 404.

1 6 3 , 165, 167, 168, 173, 174, 176 - 178, 184.

167, 169.

M a s s a c h u s e t t s Bay,

349, 350.

Mass b a l a n c e

equation,

Mathematical

model

Mersey R i v e r ,

387.

three-dimensional,

387.

241.

M e t e o r o l o g i c a l f o r c i n g , 127, 131, 143, 147 - M e t e o r o l o g i c a l f o r c i n g m e c h a n i s m , 157. - M e t e o r o l o g i c a l l y f o r c e d e x c h a n g e s , 156. Middelgat Channel, Minsener Oog,

346.

27.

-

150, 157, 158

542 80.

Mississipi,

t u r b u l e n c e , e d d y , 3 1 , 34, 3 6 , 55, 63 - 6 5 , 108, 109, 177, 218, 239, 286, 290, 302, 309, 335, 373, 384, 411, 420, 423, 424, 465, 481, 488, 510, 517, 531, 534. I n t e n s e m i x i n g e v e n t , 79, 8 0 , 87, 8 8 , 9 0 , 95 - 9 7 , 105. I n t e r f a c i a l m i x i n g , 463. L o n g i t u d i n a l m i x i n g , 10. V e r t i c a l m i x i n g , 79, 108, 198, 202, 204, 218, 419.

Mixing,

-

-

Mixing

see

also

187 - 189, 191, 192, 195.

coefficient,

Monin-Ubokhov

length

Momentum e q u a t i o n , Narragansett

Bay,

421.

scale,

387, 389.

1 3 2 , 349.

Narrows,

515, 530, 533, 534.

National

Ocean

Navier-Stokes

Shore,

374.

New J e r s e y C o a s t ,

376.

New Y o r k

Bight

373

Apex,

(Atlantic coast),

hydrodynamic

Norwegian Trench, Oostgat,

Plateau,

Piscataqua

373

River,

-

3. 350.

P o r t Aransas,

-

384.

373, 380. 390.

concentration,

Portsmouth,

3 , 4 , 17 - 22, 134, 136.

376, 378, 379, 3 8 1

Plume d y n a m i c s , Polluant

1, 2 ,

300.

Bay,

Piedmont

Texas,

148, 149, 158.

350, 352, 353, 355, 356, 359.

P o r t S u s a n , 517, 533. P o r t Susan Estuary, Townsend,

531.

515, 518.

Possession Point, Pototnac

349, 357

443

PatuxentRiver Estuary,

Port

model,

447, 457, 460.

27.

Osterfjord,

Plume,

134, 136

177, 178, 184.

Ostend,

Pedder

375.

-

165, 167.

North Shields, Numerical

249, 252, 253, 259, 267, 283, 443.

373, 383.

Bight,

New Y o r k

-

349, 351.

Hampshire,

New J e r s e y

Norfolk

350, 357.

equation,

96, 244, 245, 247

Network, New

372.

Survey,

Estuary,

515. 127, 129

-

135, 1 3 8 - 140, 143.

543 Pres s u r e - Pr e s s u r e g r a dient, 157, 158, 2 2 4 , 3 0 8 , 3 5 0 , 3 9 4 , 4 1 4 - 4 1 9 , 4 9 5 , 528. 7-

Providence R i v e r, Rhode I s l a n d , 156. Puget S o u n d , 7 9 , 2 6 1 , 2 6 2 , 2 6 7 , 2 6 8 , 2 9 7 , 3 0 5 , 515 - 5 1 9 , 5 2 1 , 5 2 4 5 2 6 , 5 3 0 , 5 3 2 , 533. Pycno c l i n e , 9 0 , 9 5 , 9 7 , 9 8 , 100 - 1 0 3 , 105, 4 3 6 , 4 4 6 , 4 6 6 , 471 - 4 7 5 , 4 7 7 , 4 9 0 , 4 9 1 , 5 1 1 , 522. Quantico Marine B a s e , 134. Queen C h a r l o t t e S t r a i t , 504. Reynolds n u m b e r , 2 3 8 , 239. Rhine, 6 4 , 6 9 , 178, 184. Richardson N u m b e r , 3 9 , 8 8 , 104, 105, 109, 111, 123, 2 0 0 , 202 - 2 0 4 , 20 6 , 3 8 8 , 3 9 5 - 3 9 7 , 4 2 1 , 483 - 4 8 5 , 4 8 7 , 4 8 8 , 491. Rorholt Fjord

( Bamble - N o r w a y ) , 4 1 0 , 41 1.

Rosario S t r a i t , 2 8 3 , 286. Rotterdam W a t e r way, 107, 110. Rough P o i n t , 4 7 2. Runge-Kutta m e t h o d , 4 8 9 , 501. Runof f , 2 3 , 80 - 8 2 , 9 8 , 153, 4 0 9 , 4 3 9 , 4 4 7 , 4 5 0 - 4 5 4 , 4 5 8 , 4 6 6 , 4 9 5 , 5 0 4 , 5 0 6 , 5 0 8 , 5 1 0 , 511. Rupel, 2 8 , 3 0 , 3 4 , 51 St. C l a i r , L a k e , 2 5 4 , 256.

St. Clair R i v e r , 2 5 4

-

259.

St. Venant e q u a tions, 2 4 9 , 253.

Salin i t y , 1, 2 , 7 , 9 , 14, 3 1 , 5 1 , 5 5 , 6 4 , 7 1 , 7 3 , 7 4 , 7 7 , 187, 3 0 9 , 375 - 3 7 8 , 3 8 1 , 3 8 7 , 3 9 0 , 4 3 9 , 4 6 8 , 4 7 5 , 4 7 7 , 4 9 5 , 5 0 6 - 5 0 8 , 5 1 0 , 5 1 1 , 5 1 8 , 5 1 9 , 5 2 1 , 530. Sa l i n i t y p r o f i l e , 3 1 , 3 2 , 6 3 , 6 5 , 6 6 , 9 5 , 9 7 9 9 , 1 0 1 , 1 0 5 , 107 11 0 , 1 9 0 , 2 0 7 , 2 8 6 , 2 9 7 , 4 5 3 , 4 7 1 , 4 7 4 , 4 7 5 , 4 8 9 , 5 0 2 , 512. Residual s a l inity, 189, 194, 201.

-

Salinity g r a d i e n t, 3 1 , 3 4 , 3 6 , 5 1 , 143, 190, 191, 2 0 4 , 2 2 5 , 283. Salinity f l u c t u ation, 2 0 1 , 203. Salt flux, 7 9 , 9 3 , 9 5 . Sa l t balance equation, 390. Sa l t c i r c u l a tion, 28. - Salt d i s t r i b u tion, 108, 109, 120. - S a l t i n t r u s i o n, 107, 309. Tu r b u l e n t s a l t f l u x , 8 0 , 94. San J u a n I s l a n d , 283 - 2 8 7 , 2 8 9 - 2 9 1 , 2 9 6 , 2 9 8 , 305 - 307. Saratoga P a s s a g e , 516. Scheldt Estuary. Belgium. Holland, 2 7 - 32, 3 4 , 3 8 , 3 9 , 4 1 , 4 2 , 4 5 , 4 7 , 4 8 , 5 1 , 5 5 , 5 7 , 6 4 - 7 2 , 7 4 , 76. Schel l e , 28. Scheu r , 27. Sea-b r e e z e . 134.

544 Sea l e v e l , 135, 349. Secondary f l o w , 103. S e c o n d a r y c i r c u l a t i o n , 104. Seiche m o t i o n , 127. S e i c h e , 129. Seiche o s c i l l a t i o n , 135, 138. S i w a s h B a y , 472. S h e a r , see a l s o d r a g , friction. - I n t e r f a c i a l shear s t r e s s , 382, 416 - 418, 420. - Reynolds shear s t r e s s , 358, 420. - S h e a r e f f e c t , 50. - S h e a r s t r e s s , 373, 381, 414, 416, 417, 420, 425, 426, 436 - S h e a r s t r e s s d i s t r i b u t i o n , 415. Shetland I s l a n d , 446, 453, 457. S i l l , 267, 283, 286, 408 - 410, 441, 444, 447, 450 - 453, 456, 458 460, 466 - 468, 471, 478, 490 - 492, 502, 506, 508 - 510, 515 517, 519, 521, 522, 524, 531 - 534. S k a g e r r a k , 447. S k a g i t R i v e r , 524. Sognefjord

:

N o r w e g i a n W e s t C o a s t F j o r d , 430, 431, 436.

S o r f j o r d , 441. S o u t h e n d , 165, 167, 168, 173, 175 - 177. S o u t h P o i n t , 267. Sparse m a t r i x t e c h n i q u e s , 252, 259. Spectral a n a l y s i s , 134, 234, 237, 238. S p i e k e r o o g , 346, 347. Stability a n a l y s i s , 495, 497. Stag I s l a n d , 254. S t a t e ( e q u a t i o n of), 392, 393. S t r a t i f i c a t i o n , see a l s o b u o y a n c y , d e n s i t y , s a l i n i t y , 31, 3 6 , 39, 45, 108, 202, 225, 374, 409, 439, 441, 453, 471, 473, 478, 488, 489, 491, 506, 518. - S t r a t i f i e d f l o w , 6 4 , 415, 419. T w o - l a y e r e d f j o r d , 424. - T w o - l a y e r e d s t r a t i f i e d flows, 142. Three-layered f l o w , 142. Susquehanna R i v e r , N o r f o l k , V i r g i n i a , 129. T a k a n o m o d e l , 373, 374, 378, 382, 383. T a y R i v e r , 241 T e l e g r a p h e q u a t i o n , 209. Temperature

-

s a l i n i t y d i a g r a m , 201, 202.

T h a m e s R i v e r , 161, 163, 165, 167, 169, 170, 182, 184. T h a m e s R i v e r E s t u a r y , 161, 178, 184. T h r e e - d i m e n s i o n a l m o d e l , 387. Tides T i d a l a m p l i t u d e , 29, 30, 6 3 , 120, 123, 172, 184, 199, 201, 359, 412, 525.

545 -

T i d a l c i r c u l a t i o n , see t i d a l current. T i d a l c u r r e n t , 34, 188, 207, 208, 212, 215, 241, 261, 267, 286, 307, 3 1 5 , 334, 346, 389, 471, 531, 533. T i d a l c u r v e , 332 - 335. T i d a l e l e v a t i o n , 349, 350, 352, 353, 355, 356, 361 - 363, 365. T i d a l m i x i n g , see a l s o mixing. T i d a l m o d e l , 243, 290, 293, 308, 315, 318, 324. T i d a l p h a s e , 9 0 , 166, 359, 491. T i d a l p r o p a g a t i o n , 161, 163, 178, 184, 259.

T i l b u r y , 165. T o m a k s t u m , 412. Tower P i e r , 165, 167, 169. T r a n s i t h e a d , 472. Transport - A d v e c t i v e t r a n s p o r t , 495, 498, 499, 508. T u r b u l e n t t r a n s p o r t , 495, 498, 508. T s u k o l a P o i n t , 472. Tumbo

C h a n n e l , 298.

T u r b u l e n c e , see a l s o m i x i n g , e d d y , 34, 9 3 , 100, 108, 109, 1 1 1 , 187, 188, 203, 204, 208, 210, 212, 214, 215, 218, 225 - 227, 230, 233, 236, 239 - 241, 243, 263, 324, 352, 388, 389, 396, 397, 402, 407, 411, 423 - 426, 428, 444, 480, 488, 491, 492, 499. Relative t u r b u l e n t d i f f u s i o n , 233. T u r b u l e n t d i f f u s i o n in n a t u r a l f l o w , 238 - 2 4 1 . T u r b u l e n t d i f f u s i o n , 40, 1 1 1 , 115, 215. T u r b u l e n t d i s p e r s i o n , 37, 44, 50, 115. T u r b u l e n t e x c h a n g e , 9 5 , 495. T u r b u l e n t f l u c t u a t i o n , 36, 470. T u r b u l e n t f l u x , 498. T u r b u l e n t k i n e t i c e n e r g y , 80, 420. T u r b u l e n t m i x i n g , 50, 8 0 , 492, 497, 498, 501. T u r b u l e n t e n e r g y , 388. Underway m e a s u r e m e n t s y s t e m , 82. Unequal grid d i s t a n c e , 391. U p w e l l i n g , 143, 204, 286, 408, 441, 447, 458, 460, 515, 517, 521. U t s i r a , 445, 451, 453. Vancouver I s l a n d , 283 309.

-

237, 289, 290, 291, 296, 293, 302, 304, 305,

Velocity, see a l s o c u r r e n t , c i r c u l a t i o n . Velocity d i s t r i b u t i o n , 110, 178, 184, 415, 485. - Velocity g r a d i e n t , 3 6 3 , 422, 484. Velocity p r o f i l e , 9 8 , 103, 105, 109, 112, 114, 115, 117 - 119, 1 2 3 , 201, 308, 407, 410, 412, 414, 468 - 470, 473, 480, 485, 488, 491. Vertical v e l o c i t y g r a d i e n t , 388. Vertical v e l o c i t y , 387. - Residual v e l o c i t y , 189, 190, 194, 201, 207, 208, 210 - 214, 307, 324. V e s t f j o r d e n , N o r w a y , 410. V l i s s i n q e n , 30, 3 1 , 68. Von Karman c o n s t a n t , 395.

546 Vorticity transfer, Wahkash P o i n t ,

207.

472.

Walsoorden,

27, 30, 31, 34.

Wangerooge,

327, 331, 336, 337, 3 4 1 , 344, 346.

Walton,

163, 165, 167, 169.

Wave n u m b e r , Western

389.

branch,

West P o i n t ,

16, 23.

531.

Whidbey B a s s i n ,

516, 517.

Whidbey I s l a n d ,

524.

Wick,

165, 167.

W i n d , 204, 205, 388, 397 - 399. - Wind e f f e c t , 420, 474. - Wind f i e l d , 132, 408 - 410. - Wind f o r c i n g , 173. - Wind i n f l u e n c e ( s e e w i n d e f f e c t ) , 107. - Wind s p e c t r a , 135. - Wind s t r e s s , 127, 1 3 2 , 134, 136, 142, 147, 157, 158, 251, 253,

262, 263, 309, 373, 376, 383, 466. Work e n e r g y e q u a t i o n ,

417.