Dynamics of Ocean Tides

337 Pages·1989·26.586 MB·English

by  G. I. Marchuk, B. A. Kagan (auth.)

DYNAMICS OF OCEAN TIDES
OCEANOGRAPHIC SCIENCES LIBRARY
VOLUME 3
G.1. MARCHUK and B. A. KAGAN
P. P. Shirshov Institute o/Oceanology, Moscow, U.S.SR.
Dynamics of
Ocean Tides
Translated/rom the Russian by
V. M. DIVID, N. N. PROTSENKO and YU. U. RAJABOV
Kluwer Academic Publishers
Dordrecht I Boston I London
Library of Congress Cataloging in Publication Data
Marchuk, G. 1. (Gur;! 1vanovich), 1925-
[Oinalllika okeanskiKh prilivov. Englishi
Oynalllics of ocean tides I G.1. Marchuk and B.A. Kagan.
p. c~. -- (Oceanographic sciences library)
Translation of: Oinalllika okeanskikh prilivov.
Bibliography: p.
Inc 1u des index.
ISBN 90-277-2552-7
1. Tides. I. Kagan, B. A. (Boris Abralllovich) II. Title.
III. Series: Oceanographic sciences library (Kluwer Acadelllic
Pub 11 shers )
GC301.2.M3713 1989
551.47·08--dc20 89-8049
ISBN-13: 978-94-010-7661-6 e-ISBN-13: 978-94-009-2571-7
DOl: 10.1007/978-94-009-2571-7
Translated from the 1983 Russian edition of
.1lHHAMHKA OKEAHCKHX nPHJlI1BOB
published by Gidrometeoizdat, Leningrad.
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
All Rights Reserved
© 1989 by Kluwer Academic Publishers
Softcover reprint of the hardcover lst edition 1989
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording, or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
FOREWORD TO THE RUSSIAN EDITION ix
FOREWORD TO THE ENGLISH EDmON xi
CHAPTER 1: TIDAL POTENTIAL 1
1.1. The Tide-Generating Forces in the Ocean 1
1.2. Tidal Potential 4
1.3. Harmonie Analysis of the Tidal Potential 7
1.4. Additional Potentials of Deformation 12
CHAPTER 2: METHODS AND RESULTS OF EXPERIMENTAL STUDIES
OF OCEAN TIDES 18
2.1. Direct Measurements of Tidal Elevations 18
2.1.1. Semidiurnal Tides 20
2.1.2. Diurnal Tides 24
2.1.3. Long-Period Tides 26
2.1.4. The Nature of the Ocean's Response to External Action 27
2.2. Satellite Data 29
2.3. Determination of Ocean Tides from Gravimetric Data 39
CHAPTER 3: QUALITATIVE STUDIES OF THE TIDAL DYNAMICS
EQUATIONS 42
3.1. Tidal Dynamics Equations 42
3.2. Simplification of the Tidal Dynamics Equations, Boundary
Conditions 45
3.3. Basic Notions and Definitions 47
3.4. Uniqueness Theorem 49
3.5. A Priori Estimates 51
3.6. Existence Theorem 57
3.7. Solvability of the Three-Dimensional Boundary Value Problem
of Tidal Dynamics: Homogeneous Ocean 62
3.8. Solvability of the Three-Dimensional Boundary Value Problem
of Tidal Dynamics: Stratified Ocean 71
3.9. Asymptotic Behavior of the Solution of Tidal Dynamics Equations
when t ...... 00 84
vi Table of Contents
CHAPTER 4: FREE OSCILLATIONS IN THE WORLD OCEAN 88
4.1. Rayleigh's Ratio 88
4.2. Analytical Solutions 92
4.2.1. Spherical Ocean of Constant Depth 93
4.2.2. Ocean Bounded by Two Meridians 95
4.2.3. A Flat Basin 98
4.3. Numerical Solutions 100
4.3.1. Proudman's Method 102
4.3.2. Finite Difference Method 104
4.3.3. Results 107
4.4. Elementary Modes of Free Oscillations 112
4.4.1. Sverdrup Waves 114
4.4.2. Kelvin Waves 118
4.4.3. Poincare Waves 122
CHAPTER 5: FORCED TIDAL OSCILLATIONS IN THE WORLD OCEAN 128
5.1. Analytical Solutions 128
5.2. Numerical Solutions 135
5.2.1. The Semi-Emperical Approach 135
5.2.2. The Theoretical Approach 136
5.2.3. Semidiurnal Tides 137
5.2.4. Diurnal Tides 141
5.2.5. Long-Period Tides 142
5.3. Numerical Experiments 144
5.3.1. Tides in the World Ocean of Real Configuration 144
5.3.2. Tides in the World Ocean of Idealized Configuration 144
5.3.3. Tides in the Oceans Separated by Barriers 147
5.3.4. Tides in the World Ocean in the Absence of the
Earth's Rotation 148
CHAPTER 6: TIDES IN THE OCEAN-SHELF SYSTEM 151
6.1. Preliminary Remarks 151
6.2. The Existing Methods of Shelf-Effect Parameterization 153
6.2.1. Local Methods of Parameterization 153
6.2.2. Integral Approaches to Parameterization 160
6.3. The Influence of Shelf Effects on the Tides in an Idealized Ocean 164
6.4. The Influence of Shelf Effects on the Tides in the World Ocean 174
CHAPTER 7: GLOBAL INTERACTION OF OCEAN AND TERRESTRIAL
TIDES 1M
7.1. Solvability Conditions of the Problem 184
7.2. Difference Methods of Solution 186
7.3. Results of Numerical Experiments 192
7.3.1. Ocean Tides 192
Table of Contents vii
7.3.2. Terrestrial Tides 194
7.3.3. Tidal Variations of Gravity 198
CHAPTER 8: ENERGETICS OF OCEAN TIDES 205
8.1. Energy Equation 205
8.2. Astronomical, Geophysical, and Satellite Estimations of Tidal
Energy Dissipation 207
8.2.1. Astronomical Estimations 207
8.2.2. Geophysical Estimations 213
8.2.3. Satellite Estimations 219
8.3. The Problem of Tidal Energy Dissipation in the Ocean-Earth
System 223
8.4. Tidal Energy Dissipation in the Paleoocean 229
CHAPTER 9: BOTTOM BOUNDARY LAYER IN TIDAL FLOW:
EXPERIMENTAL DATA 235
9.1. Motion Regime in the Bottom Boundary Layer 235
9.1.1. The Method of Small Perturbations 236
9.1.2. The Energy Method 240
9.2. Hydrodynamic Properties of the Sea Bottom 244
9.3. Mean Velocity Profiles 247
9.4. Statistical Characteristics of Turbulent Fluctuations 253
9.5. Unidimensional and Co-Spectra of Velocity Fluctuations, Reynolds
Stress 259
9.6. Similarity of Turbulence Structures in Boundary Layers of
Different Origin 264
CHAPTER 10: BOTTOM BOUNDARY LAYER IN TIDAL FLOW:
THEORETICAL MODELS 266
10.1. Qualitative Considerations Regarding the Vertical Distribution
of the Tidal Velocity in the Bottom Boundary Layer 266
10.2. Bottom Boundary Layer Models Based on A Priori Assignment
of the Vertical Turbulent Viscosity Coefficient 268
10.3. Bottom Boundary Layer Models Based on the Closure of
Equations with the Help of Semi-Emperical Hypotheses 274
10.3.1. Bottom Boundary Layer with Neutral Stratification 275
10.3.2. Stratified Bottom Boundary Layer 285
10.4. The Resistance Law in Tidal Flow 295
10.4.1. Jonsson's Solution 298
10.4.2. Kajiura's Solution 301
10.4.3. Kagan's Solutions 304
REFERENCES 309
INDEX
325
FOREWORD TO TIlE RUSSIAN EDITION
Applications of ocean-tide information have expanded considerably in recent years.
These data are now being used not only to solve vital problems in oceanography,
but also, in the adjacent fields of geophysics, to study earth tides, elastic properties
of the Earth's crust and tidal gravity variations. They are also used, in space stu
dies to calculate the trajectories of man-made satellites of the Earth and to inter
pret the results of satellite measurements. New prospects have also opened up for
the use of data on tidal currents.
Even now, the shelf zones are intensively utilized, and there are plans to com
mence, in the near future, the construction of major industrial structures such as
nuclear reactors, oil storage facilities and ports. Next in turn is the prospecting for
oil and gas on the floor of the open ocean, which entails performing deep-sea dril
ling and organizing oil and gas production, storage and transportation. The neces
sity of appraising the possible economic, social and ecological consequences of all
these projects imposes increasingly stringent demands on the comprehensiveness
and accuracy of information concerning tides and tidal currents.
These demands were partially taken into account in the program of deep-sea
tidal measurements, prepared in 1966 by Working Group No. 27 of the Scientific
Committee on Ocean Research (SCOR). This program invisaged the recording of
tidal elevations on a network of stations covering the deep part of the open ocean
and the continental slope region. Seventeen years have passed, but the end of the
work is not in Sight. Satellite measurements seem to be the only source of new
empirical ocean-tide data to be realistically contemplated in future.
Observations of the perturbations in satellite orbit elements now make it pos
sible to determine the parameters of the second spherical harmonic of ocean tides.
This is sufficient to estimate the global dissipation of tidal energy but not enough
to reproduce the tidal pattern in the World Ocean. In the latter case, the use of
satellite altimetry may prove to be highly successful; but here, unfortunately,
things are not as simple as one would wish them to be. Firstly, the tidal elevations
usually observed in the open ocean lie within the limits of error in determining the
mean ocean level. Secondly, satellite trajectories vary in space, so that a continuous
series of measurements cannot be obtained for a fixed point but only for a certain
region in its vicinity. Thirdly, the possibilities of satellite altimetry are limited by
the accuracy of determining satellite orbits. Consequently, in the accuracy of
measurements, satellite altimetry is still inferior to deep-sea and standard observa
tions of tidal level oscillations. It is, however, vastly superior for conducting
x Foreword to the Russian Edition
extensive measurements in the open ocean and, therefore, is the method that will
be used in the future.
So far, almost all information on the open ocean tides is obtained from calcu
lations based on more or less trustworthy theoretical models. Quite understand
ably the quality of these calculations is to a considerable extent, determined by the
comprehensiveness and accuracy of the description of the numerous factors taking
part in the formation of the phenomenon in question and, therefore, by the general
level of the development of theory.
Noticeable progress has been made in recent years in elaborating the theory of
ocean tides. In this connection we should like to mention the solution of the prob
lem of global interaction between ocean and terrestrial tides. Important steps have
been made in solving the problem of the parameterization of shelf effects, which
has become the stumbling block of every attempt at a numerical simulation of
ocean tides. The spectral problem of the World Ocean of the real configuration is
now solved, which has at last made it possible to give an exhaustive answer to the
question regarding the nature of ocean tides. Investigations in the benthic boundary
layer of the ocean have been developed further.
The structure of the benthic boundary layer produces a significant effect on
the dynamics of the ocean as a whole: it is through this layer that the thermal and
dynamical interaction between the water mass and the underlying ocean bottom is
realized. In this layer the processes of sedimentation and sediment transport, the
chemical reactions and vital activity of benthal organisms are taking place. These
processes are controlled by the dynamics of the benthic boundary layer and, in
their turn, exert their influence upon it. The simplest example of such an interac
tion is the relationship between the biological activity and the regime of motion in
the benthic layer. The flow structure determines the conditions in the habitat of
living organisms, and they, after dying off, determine the hydrodynamical proper
ties of the ocean floor and its influence of the adjacent flow motion. The interrela
tion between the dynamics and the ecology of the benthic layer is almost com
pletely unknown at present.
In this book we have tried to present systematically the current state of
knowledge on ocean tides and, after a critical interpretation, to focus attention on
questions as yet unsolved, the purposeful and systematic study of which would
faCilitate a speedy progress in ascertaining the dynamics of ocean tides - an impor
tant and intensively developing component part of the science of the ocean.
We are sincerely grateful to S.S. Voit, Ye.N. Dvorkin, A.V. Nekrasov, and A.S.
Sarkisyan for their valuable comments, and to V.B. Zalesny for his participation in
writing Sections 3.7 - 3.9.
January, 1982.
G.I. MARCHUK.
B.A. KAGAN
FOREWORD TO TIlE ENGLISH EDITION
Six years have passed since this book was written. These years have been marked
by rapid progress in theoretical and experimental studies of ocean tides and by the
expansion of their applications, particularly in satellite navigation and space geo
desy. In their turn, improvements in the methods of space navigation and space
geodesy have resulted in the development of new technology to obtain empirical
information and in the ever-increasing use of the data of satellite altimetry.
Apart of this purely experimental aspect, comprehensively discussed in the
publications of recent years, important results have also been obtained in the field
of the theory of ocean tides. These are, firstly, the solution of the problem that has
remained in the background for a long time, regarding the parameterization of soli
tary islands, island chainS, and archipelagoes, secondly, the conclusion of the
investigations into the global interaction of oceanic and atmospheric gravitational
tides and the establishment of the nature of the latter; thirdly, intensive develop
ment of investigations into the vertical structure of tidal flow, based on the high
level models in the hierarchy of the planetary boundary-level models, fourthly,
the appearance of the results produced by modelling the evolution of the spectrum
of natural oscillations and global tides in the Paleoocean and their use in construct
ing the theory of the tidal evolution of the Earth-Moon system; and, finally, the
development of new combined methods to calculate the global ocean tides. It is this
latter point that we should like to dwell on in greater detail.
It is well known that the scarcity of initial empirical data results in the
World Ocean tides being most often appraised by the results of numerical model
ling. This situation will possibly change only in the future, when the traditional
terrestrial measurements of the level are replaced by satellite altimetry. Even this,
however, will be unable to do without the results of the numerical modelling of
ocean tides. This is so because the accuracy of isolating the legitimate signal from
the data of altimetric measurements depends on a priori information on ocean
tides; and, even if the required accuracy is reached, it will still be necessary to
interpolate the altimetric measurement data between the far spaced intersection
points of satellite trajectories. The future progress in describing the ocean tides is
therefore modelling results and the altimetric and terrestrial (coastal, insular,
deep-sea) level measurement data.
The data of terrestrial measurements are not by chance mentioned here. The
matter is that level perturbations, obtained with the help of satellite altimetry,
constitute a small difference between larger quantities (the altitude of the