Advances in Applied Mechanics, Volume 16

Advances in Applied Mechanics Volume 16

Editorial Board T. BROOKE BENJAMIN Y.

c. FUNG

PAULGERMAIN

L. HOWARTH WILLIAM PRAGER T. Y . Wu HANSZIEGLER

Contributors to Volume 16 G. K. BATCHELOR ROBIN D. HILL

H. KOLSKY H. K. MOFFATT D. H. PEREGRINE GEORGE I. N. ROZVANY CHIA-SHUN YIH

ADVANCES IN

APPLIED MECHANICS Edited by Chia-Shun Yih DEPARTMENT OF APPLIED MECHANICS AND ENGINEERING SCIENCE THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN

V O L U M E 16

1976

A C A D E M I C PRESS

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A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT 0 1976, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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Contents

vii viii

LISTOF CONTRIBUTORS PREFACE

1

G. I. Taylor as I Knew Him

G.K.Batchelor Interaction of Water Waves and Currents

D. H.Peregrine 10 17 70 76 103 106 111

I. Introduction 11. Large-Scale Currents 111. Small-Scale Currents IV. Currents Varying with Depth V. Turbulence VI. Ship Waves References

Generation of Magnetic Fields by Fluid Motion

H.K. Moflatt I. Introduction 11. 111. IV. V.

Magnetokinematic Preliminaries Convection, Distortion, and Diffusion of 5Lines Some Basic Results The Mean Electromotive Force Generated by a Random Velocity Field VI. Braginskii's Theory of Nearly +xisymmetric Fields VII. Analytical and Numerical Solutions of the Dynamo Equations VIII. Dynamic Effects and Self-Equilibration References V

120 125 130 135 139 154 163 168 176

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Contents

The Theory of Optimal Load Transmission by Flexure George I . N . Romany and Robin D . Hill I. 11. 111. IV. V. VI. VII. VIII.

Introduction Static-Kinematic Optimelity Criteria-Plastic Design General Optimality Criteria-Elastic Design Optimal Flexure Fields-Basic Geometrical Properties Optimal Flexure Fields-Clamped Boundaries Optimal Flexure Fields-Mixed Boundary Conditions Optimal Flexure Fields-Further Developments Optimal Flexure Fields of Constrained Geometry List of Symbols References

i84 187 20 I 206 226 2 46 292 299 303 304

The Role of Experiment in the Development of Solid Mechanics-Some Examples H . Kolsky I. Introduction 11. Viscoelastic Behavior 111. Experimental Determination of Viscoelastic Response IV. Tensile Shock Waves in Rubber V. Plastic Wave Propagation VI. Experimental Studies in Dynamic Fracture VII. Conclusion References

309 3 16 322 345 349 354 364 365

Instability of Surface and Internal Waves Chiu-Shun Y ih I. General Introduction 11. Instability of Surface Waves 111. Instability of Internal Waves References AUTHOR INDEX SUBJECT INDEX

369 37 1 393 419 42 1 42 7

List of Contributors

Numbers in parentheses indicate the pngs on which the authors' contributions begin.

G. K. BATCHELOR, Department of Applied Mathematics and Theoretical Physics, University of Cambtidge, Cambridge, England (1)

ROBIN D. HILL, Faculty of Engineering, Monash University, Clayton, Victoria, Australia (183) H. KOLSKY,Division of Applied Mathematics, Brown University, Providence, Rhode Island (309)

H. K. MOFFAIT, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England (1 19) Department of Mathematics, University of Bristol, Bristol, D. H. PEREGRINE, England (9) GEORGEI. N. ROZVANY,Faculty of Engineering, Monash University, Clayton, Victoria, Australia (183) CHIA-SHUNYIH, Department of Applied Mechanics and Engineering Science, University of Michigan, Ann Arbor, Michigan (369)

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Preface

This volume is dedicated to the memory of Sir Geoffrey Ingram Taylor (March 7, 1886-June 27, 1975), a great scientist and a wonderful man who was held in esteem and admiration by all who knew his work, and in affection by all who had the good fortune to enjoy his friendship. Much has been written about him and his work since his death, and immediately following these few pages is an intimate sketch of G. I. (for it is thus that many of us refer to him) by Professor G. K. Batchelor, G. I.’s associate for 30 years, who is also writing a more complete biographical article of him for publication by the Royal Society of London. Here I shall only say a few things in remembrance of him. His work was always marked by an originality of thought and a freshness of approach that continue to delight his readers, and a characteristic welding of analysis to experiment that is rarely attempted, let alone attained, by others. Since his collected works fill four large volumes, it is natural to wonder which of his works gave him the most satisfaction. In 1967, when he came to Ann Arbor to receive an honorary degree, I asked him that question, and he answered by singling out his work on the stability of Couette flows. At this, Professor A. M. Kuethe, who was walking with us, expressed some surprise, for he had thought G. 1,’s statistical theory of turbulence occupied that position of honor. I suppose G. I.’s answer might well have been different if I had asked him which of his works he considered the most important. But I do not think that question would have been as congenial to him. He considered himself an amateur in science. It is therefore not surprising that the spirit of adventure is often evident in his work, for one who works out ofcuriosity and love of the subject is likely to explore where others dare not tread or, more often, do not think of treading. But if he called himself an amateur, the term was appropriate in the etymological sense only. The spirit of adventure was evident in his life as well. He learned to fly an airplane and to parachute when aviation was young, he rode viii

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ix

balloons,t he sailed all over Europe, he skied when bindings were primitive and skis were without steel edges, and he tried (unsuccessfully) to waterski when he was eighty years old! On his holidays he was often accompanied by his wife (nee Grace Stephanie F. Ravenhill), who shared his interests in travels and was an ideal companion. In 1929 they visited Borneo and then Japan, where they slept in picturesque ryokans (wayside inns). In 1933 and 1934 they visited Canada. The trip in 1934 was one on horseback through the Canadian Rockies, taken with two friends and a cook. What he enjoyed most of his holidays was the beauty of nature and the feeling of solitude that it imparted. Once I showed him three volumes of old prints by Hiroshige, and the silent figures, the swaying willows, the blooming cherry trees, the temples, the bridges, and the waters gave him so much pleasure that he told Brooke Benjamin about them when he returned to England. The Japan G. I. knew was relatively unspoiled compared to the Japan of today, but the Japan Hiroshige depicted would have suited him even better. His love of beauty might well be inherited from his father, who was an artist. In 1967, while we were looking at a book of Camille Pissaro’s paintings, G. I. recalled a visit of the Pissaros (Camille and his son Lucien, who was also an artist) at his father’s home in St. John’s Wood in London. The Pissaros were in London in 1892 and 1897. Thus he was able to recall things that happened when he was six years or eleven years old. On Sunday afternoons at St. John’s Wood, G. 1,’s father would make pencil drawings of roses without using an eraser. G. 1. gave me one of these drawings, of two sprigs of roses. The roses look soft and moist, as if the dew had just evaporated from them. In G. I.’s living room was an oil painting by his father, a purplish landscape of Wales with wet sand on the beach: not so wet as to give perfect reflection,just wet enough to give the feeling of wetness. G. I. seemed to have inherited his father’s love of the beauty of Wales, for he had a cottage in Llanfair, Wales, where he kept his sailboat. G. 1.3excellent health and adventurous spirit were only part of his gift for happiness; his capacity for enjoying the simple things in life and a natural

t He once described the hazards in landing balloons. One pulled the cord (to let gas escape) in order to descend, discarded sand bags to let it rise again when it descended too fast or toward the wrong place, and repeated the procedure over and over until one succeeded in landing. In one such maneuver, after some sand was discarded, “the sand rapidly dispersed and fell more slowly than the balloon so, relative to us, it went upwards. Then as our downward velocity stopped and reversed the sand caught up and filled the air all round us. It was like a sand storm.”

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detachment from the worldly aspirations that are a burden to less fortunate people were, I think, what gave his life the happiness that mere health and adventures could not have bestowed. His detachment arose not only from a natural simplicity but also from a desire to conserve time and energy to do what he wanted most to do. Indeed he received his numerous honors with a simple pleasure. But he never let these honors change him, and it is easy to believe that he could have been as happy without them. I shall now quote two letters from him, because these are interesting, and because they will make the points I have made, only more eloquently. The first was written on July 28, 1969, and the second on October 30, 1969. Many thanks for your letter and congratulations. I had just come back from Cape Breton Island when my 0. M.t was announced. I had a beautifully quiet time there at my cousin’s place which is on a dirt road 6 miles from the nearest paved highway: n o noise from cars or planes-only the wind in the trees and the surf on the shore. One could safely drink out of any of the streams and I used t o bathe in one close to where it ran into the sea. It has been so dry since I got back that I have been trying to keep the plants alive by watering them. I had a very pleasant time (contrary to my expectations) when I was received by the Queen and given the 0. M.insignia. I had expected officials all round, but I was introduced by an equerry who retired and shut the door, leaving me to talk with the Queen for a quarter of an hour. She is very easy to talk to and to listen to, and though nothing of any particular importance was said by either of us I thoroughly enjoyed the interview. I formed the impression that she is a very nice person whom one would much like to have as a friend under other circumstances. I also have been looking at problems of interest, in particular what is the mechanism which drives sap up a tree and sugar down from the leaves? It is a very interesting problem but I doubt whether it has much hydrodynamic content. I think that individualists like you and me should not feel the unimportance of what we d o compared with the work of the thousands in the space programmes. Of course our works are minute by comparison; but as long as we enjoy doing them that is the thing.

Your writing of Gaspe reminds me of a holiday Stephanie and I took in 1933, sleeping in a barn which was, I think, then the nearest building to the lighthouse at Gaspe Point. I tried bathing but it was terribly cold. . . .

Although G. I. was never in China he did have one connection with it. Mary Boole, his mother’s elder sister, married Howard Hinton. The Hintons’ grandson William$ and his wife were in China during the Sino-Japanese f “This isa very illustrious order, confined to 24 people who make outstanding contributions of a non-political kind, and it is certainly the highest distinction which has been conferred on him. His close friend Adrian (whom you may remember as a former master of Trinity) is also a member of the Order of Merit. He will be able to wear a very fine medal and ribbon around his neck on dress-up occasions in the future.” So wrote George Batchelor in a letter to me. William’s sister Joan has been in China for many years and, like him, is very sympathetic to the Chinese people.

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War, and stayed until many years after the Revolution of 1949. Indeed, at least one daughter was born to them in China. I once asked G. I. why the Hintons stayed so long in China, and he answered that they did so because they were idealistic. The mechanics community has lost a leading light and a living example of excellence, goodness, and the possibility of happiness. His life and work will forever be an inspiration, but those of us who knew him shall always miss him.

YIH CHIA-SHUN

Sir Geoffrey I. Taylor, 1886-1975

G. I. Taylor as I Knew Him

I first met G. I. Taylor in April 1945 when I arrived in Cambridge to work for a Ph.D. under his supervision. Our meeting place was his rather crowded room in the old Cavendish Laboratory in which one maneuvered around a wind tunnel. My immediate impression was of a kindly and unpretentious man with an informal manner who did not think it necessary to remove the bicycle clips from his trouser legs. He showed me the wind tunnel that D. C . MacPhail had designed and built in 1940 to have a low level of turbulence in the working section. No use had been made of the wind tunnel during the war years, and G. I. (as I later learned to call him) hoped that a new research student would be interested in an investigation ofgrid-generated turbulence to follow up some of the work he did in the late thirties. My own taste was for theoretical studies, but A. A. Townsend joined me as a research student working under G. I. in June 1945 and he began what proved to be a long and fruitful series of experiments on turbulence in this little wind tunnel with a working section about 160 cm long and 50 cm square in cross section. G. I. had no specific plans for our research and, while always interested in our results, was content to let us do what we wished. This suited both of us well, since we had done a little research before coming to Cambridge and were prepared to be independent. Some supervisors like their research students to be junior collaborators on some definite problem, but G. I. seldom needed or wanted a collaborator, junior or senior, and assumed that everyone was as happy to go his own way in research as he was. Three years later (in 1948) I did in fact collaborate with him on the calculation of the effect of wire gauze on small disturbances in the velocity of a uniform stream. We had each done a little previous work on the problem, and some new measurements of the effect of wire gauze on turbulence by Dryden and Schubauer stimulated us into looking for a complete mathematical solution. I saw then the difficulty ofcollaboration with G. I. His perception was incredibly quick, and I felt like a tortoise trying to play with a hare. 1

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G. K . Batchelor

Thinking back to that first meeting with Taylor 31 years ago, it is difficult for me to appreciate that he was older then than I am now. He was presumably well past his research peak, but at the age of 59 he seemed tremendously fertile and inventive. He wrote and published over 60 papers in a period of 27 years after the end of the second world war, and this is the period in which I knew him and could observe him at work. In duration and achievement it is equivalent to a normal and successful scientific career. But by 1945 he had already published over 140 papers and had made outstanding contributions in meteorology, oceanography, theory of metal crystals, turbulent flow, gas dynamics, and a host ofpractical problems thrown up by the two wars. Whenever I am tempted to claim that I knew him well, I need to remember that his creative working life lasted from 1909 until he suffered a severe stroke in 1972 and that I saw nothing of his early and middle years in which the bulk of his work was done. G. I. was a member of a talented and interesting family. His mother’s father was George Boole, the mathematician who pioneered the study of symbolic logic and what we now call Booleian algebra. Boole was a selfeducated schoolmaster at Lincoln when he did his best-known work, and he was later appointed to the professorship of mathematics at Queen’s College, Cork. This great and good man, who was said to be “as innocent as a child,” apparently inspired feelings of love and reverence among his pupils, colleagues, and acquaintances. G. I. has written with affection about George Boole in an article prepared for the Dublin centenary celebrations of the publication of Boole’s “ Laws of Thought ” [published in Proceedings ofthe Royal Irish Academy (1954, pp. 6 6 7 3 ) and, in slightly different form, in Notes and Records ofthe Royal Society (Vol. 12, 1956, pp. 44-52)]. Boole married Mary Everest, a niece of Sir George Everest, who held the post of Surveyor-General of India and was one of the pioneers of geodesy; and they had five daughters: Mary, Margaret (G. I.’s mother), Alice, Lucy, and Ethel. George died in 1864, the year in which Ethel was born, and his wife and children moved to London where Mrs. Boole supported the family by working as a matron at a hospital. The third daughter, Alice, also possessed a mathematical mind and, despite having no more training than school Euclid, became interested in four-dimensional geometry and discovered for herself the six regular figures that can exist in four-dimensional space and their three-dimensional sections. Alice Boole dropped her mathematical hobby when she married, whereas George Boole was able to take up his professionally on his appointment to Cork; both were amateurs, in the sense of being independent and self-reliant and free to do what they wished, and G. I. liked, I think, to see himself as a scientific amateur in something of the same way. The fourth daughter, Lucy, became a professor of chemistryprobably the first woman to do so in England-at the Royal Free Hospital in London.

G . I . Taylor as I Knew Him

3

The youngest daughter, Ethel Lilian, had literary and musical ability. She married a Polish-born Russian revolutionary, W. Voynich, who later became a rare-book dealer in London and New York. Ethel Voynich’s first novel, “The Gadfly,” is a powerful and tragic story of young lovers involved in revolutionary struggles in Italy during the mid-nineteenth century, and, perhaps for reasons to do with the subject matter, is immensely popular in the socialist countries, especially in USSR where over 5 million copies have been sold and the author has been ranked with Dickens. Ethel Voynich moved to New York in 1920 and remained there until her death in 1960. There is a fascinating article by Anne Fremantle about her life and the remarkable success of “The Gadfly” in History Today (Sept. 1975). Margaret Boole married Edward Taylor, who was an artist, and the house in St. John’s Wood (in London) in which G. I. and his younger brother grew up was part studio, part home. G. I.%father designed the decoration of the public rooms of big ships and painted landscapes, but will perhaps be remembered most for his fine pencil drawings of flowers, many of which are on display in museums and galleries in Britain. G. I. may have acquired from his father his love of plants and gardens and an interest in rare specimens. In 1925 G. I. married Stephanie Ravenhill, who had been a school mistress; they had no children. The Taylors were keen sailing and traveling partners in the ten years after their marriage and made many enterprising journeys, including a voyage in their own sailing boat up the coast of Norway to the Lofoten Islands and overland expeditions to the interior of Borneo and some of the more remote parts of Japan. In the postwar period when I first knew her, Stephanie’s role was mistress of “Farmfield,” their home in Cambridge. She was very hospitable, and especially kind to newcomers like my wife and me. I recall their regular Christmas parties at Farmfield with hot punch and mince pies, very English in atmosphere and diverting for me as a raw and impressionable research student, since any one of the people with whom I was playing games might turn out to be a knight or an FRS or even both. I should say that I have learned much more about G. I.’s family in the course of tidying up the papers left after his death than from him directly. He was a shy and reserved man, and seldom spoke about such personal matters. In conversation he seemed to me to be not very articulate, although on paper he had a charming style and in his later years wrote a number of general articles about his life and work which make delightful reading. He enjoyed these reminiscences, especially those with a scientific content, and from them we get some insight into a happy, uncomplicated man with mainly conservative attitudes and a taste for thinking about concrete matters rather than abstractions. One thing is made abundantly clear from these articles written in later life, if it was not already evident from his scientific papers, and that is his love of

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experiments, apparatus, and gadgets. He was, ofcourse, more than competent with mathematical manipulations and was supremely good at constructing theories, but it was a well-designed experiment that gave him the most pleasure. A visitor to his room in the Cavendish Laboratory would usually be shown the experiment in progress or some photographic results, and, although it was not always easy to understand what the idea was, the twinkling eyes and enthusiastic description made G. 1,’s feelings clear. G. I. did sometimes refer to himself as an applied mathematician (and he was, of course, using those words in the sense that is conventional in Britain), but his interest was very much more in the “application” than in the “mathematics,” as I think is evident from the following remark made in 1952 in what was probably the first of his reminiscing lectures (called “A Scientist Remembers”): “I think if I were to start again I should still try to be an applied mathematician, because the number of amusing activities to which mathematics can lead one is so great.” And elsewhere he says that he got his satisfaction chiefly “ from the interplay between applied mathematics and experiment.” Inventiveness in devising a simple experiment which reveals the physical process relevant to some phenomenon or which gives accurately a desired quantity was one of the hallmarks of his genius. I recall an example of this originality which is easily explained and which appears almost trivial-once you have “seen” it. In the early fifties G. I. was exploring various cases of diffusion and convection of salt dissolved in water in a tube and was led to think about the conditions for stability of the stationary liquid in a vertical circular tube when the salt concentration increases upward. It is not too difficult to solve the eigenvalue problem that gives the maximum concentration gradient compatible with stability, and, as was his custom, G. I. wished also to measure the critical value of the concentration gradient. There arises then the problem of setting up stratified fluid in a vertical tube and of varying the concentration gradient, presumably increasing it from zero until overturning first occurs. G. I. saw, and I do not think any period of thought was required, that the right way-the “obvious” way-to do the experiment is to connect the top of a long vertical tube of pure water to a reservoir of dyed salt solution of known concentration and then to measure the length of the column of dye when it has stopped being carried down the tube by overturning. I feel sure I could have thought of that myself; why didn’t I ? Most of G. I.’s experiments were done to test some idea or calculation after he felt he understood what was going on rather than to acquire data from which the understanding would come. This is probably a reflection of the quantitative and analytical character of most problems in mechanics. At all events, he usually knew in advance what to expect and was seeking a measurement for comparison with some theoretical prediction. Because his

G.I . Taylor as I Knew Him

5

insight was so acute, it seldom happened that the experiment failed to yield the expected result; and if there was some numerical disagreement, it could often be accounted for by some extraneous factor. The form of experimental check he liked best was the observation of a single number of which a theoretical value was already available, and I think he sometimes stopped making measurements as soon as he had obtained agreement with the theoretical value, with blithe unconcern for the rules about number of significant figures, theory of errors, etc. Sometimes his conscience seems to have pricked him, although not to the point of compelling stern action, for in several of his papers there is a gleeful admission that the agreement obtained is closer than is justified by the accuracy of the measurements or the theory (see, for example, the end of his second paper in 1950 on the blast wave from an intense explosion, where he estimates the initial velocity of rise of the luminous hemisphere left by the first atomic bomb explosion by regarding it as a vacuous bubble of radius equal to that where the density of the heated air is half that of the surrounding cold air-why a half?-and ends up with a value which differs by only 2% from the observed value, which itself is uncertain to within a much bigger range). As I have said, G. I. was seldom in doubt about the general validity of his theory, and I am sure that any apparently cavalier treatment of observations never led him to incorrect conclusions; but it gave his close associates some innocent fun. G. I.’s interest in science seems to have been aroused first by physical experiments. He records that he knew he wanted to be a scientist from the time when as a schoolboy he heard Oliver Lodge give the traditional Christmas lectures for children at the Royal Institution in London. The lectures and demonstrations were on wireless telegraphy, and they stimulated G. I. and his friends to construct a Wimshurst machine with which they generated weak X rays and photographed the bones in his mother’s hand. Boat building, sailing, and navigation were other early hobbies that developed his practical skills and the adventurous curiosity that I think was the driving force of his life. When he did these things he thought for himself, and the independence and originality which came to him as easily as breathing led to some improvement, some technical development, or some new concept. His love of small-boat sailing showed him that the conventional anchors were inconveniently heavy to handle, and in about 1933 he designed a new one which is a triumph of geometrical imagination and which has a holding power three times that of the best anchor of traditional type and the same weight. The merits of this so-called CQR anchor are now generally recognized, and it is widely used for small boats. An even more impressive example of his capacity for original thinking is his work as meteorologist on the “Scotia” expedition which was sent in 1913 to report on icebergs in the North Atlantic following the loss of the

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“Titanic” due to collision with an iceberg. He was just 27; it was less than five years since the completion of his mathematics and physics course at Cambridge; and he had only recently taken up the study of meteorology. Yet this tyro foresaw that he would have an opportunity of measuring the vertical distributions of velocity, temperature, and humidity in the friction layer of the atmosphere and so of finding the rates of turbulent transfer. Singlehanded he organized and executed experiments involving the launching of kites and balloons carrying instruments from the deck of a sailing ship and flying them up to 6000 feet, and he produced a classic paper on turbulent transfer in the atmosphere which contained as incidentals the vorticitytransfer theory and an independent derivation of the “Ekman spiral distribution of mean velocity. It is easy to see how G. I. was drawn to science, but what made him choose the mechanics of fluids and solids for his life’s work? His third undergraduate year at Cambridge was spent in studying physics, and, being given a further scholarship by Trinity College, he stayed on at the Cavendish Laboratory in 1908. J. J. Thomson suggested he might like to test the recent suggestion that the energy of light waves is quantized by observing whether interference fringes continue to be formed by light when the intensity is exceedingly small and the incident quanta would be widely separated. G. I. did this, at his parents’ home in London with home-made equipment costing about two dollars, and described the result, namely, that the fringes were unaffected by the feebleness of the incident light, in his first paper, a onepage note published in 1909. That seems to have been the end of his association with “pure” physics, and his next research was the well-known investigation of the structure of shock waves, published in 1910. I believe a paper by Rayleigh drew his attention to this problem. The work evidently impressed people in Cambridge, for he was. awarded a Smith’s Prize for it and was elected to a Fellowship at Trinity College in the same year. In 191 1 a temporary readership in dynamical meteorology was set up at Cambridge with money provided by a wealthy professor at Manchester named Schuster who wanted to encourage a mathematician to study the subject, and G. I. was appointed, so far as I know without any previous experience or specialist knowledge (which, of course, would be inconceivable nowadays). This appears to have been the beginning of a major commitment to applied and classical physics which was maintained for the rest of his life. His experiences during the early part of the first world war, when he joined a group at Farnborough helping to put the new field of aeronautics on a scientific basis, may have consolidated his interest in mechanical subjects. In practical aerodynamics he certainly found plenty of scope for the theoretical analysis of quite new problems, some with the spice of adventure about them, as when he made what were probably the first measurements of the pressure distribution over a wing in full-scale flight. ”

G.I . Taylor as I Knew Him G. I. knew how to conserve his time and energy for the things that he was interested in. He tended not to get involved in affairs going on around him and stayed out of movements, administration, and all activities which men undertake collectively. He could not avoid being put on committees, but I think the only ones on which he played more than a listening part were technical committees such as those connected with the Aeronautical Research Council. He steered clear of teaching and was not tempted to write books or pedagogical articles; and when he did agree to write something with an educational purpose, such as his chapter on turbulence in “ Modern Developments in Fluid Dynamics,” it was usually done as a straightforward adaptation of his research papers. His research aids were simple and unpretentious, and were typified by those projection slides that he took to conferences; G. I. used to make these by writing or drawing with his fountain pen on one piece of glass 34 in. square and binding it with adhesive tape to a second similar blank piece of glass. Correspondence is an unavoidable part of research work, but G. I. kept the time spent on that to a minimum by carrying around letters in his pockets until they had either been lost or answered and by writing all replies briefly and by hand. He did not try to “ keep up with the literature” (and did not need to do so),and his knowledge of past work on a topic of interest to him was often patchy. I have the impression that, in the postwar years at least, he tended to avoid problems that other people were working on and to take up ones that were entirely new or had been mistakenly passed by. This was perhaps in keeping with his view of himself as an amateur in science. He had no wish to be a leader or to influence or enlighten people, and wanted only to continue to indulge his scientific curiosity in a way that would make minimum demands on other people. It is perhaps a significant comment on his way of life that, so far as I know, he never took leave of absence and never visited some other institution for the purpose of pursuing his research there (except when he went to Los Alamos for periods of several weeks for highly confidential work under wartime conditions). He would not have felt any need to get away from Cambridge, since he had no duties there; and it is unlikely that he felt any need for the stimulation of new associates and a new environment. He had ideas in plenty to investigate, and no need of facilities other than his oneroom laboratory and his one technical assistant, both located in the Cavendish Laboratory. He liked to travel, yes, but the notion of going away from one’s own institution to work would have seemed strange to him. If I had to sum up G. I.’s character and life in a few words I should describe him as a scientific “natural.” He had the gift of being able to see clearly how things worked-what in more impressive language we call physical insight-and he had a remarkable capacity for quantitative and analytical thought. Physical science was the obvious subject for his enquiring

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mind, and he was drawn to the more applicable aspects linked with engineering, and to mechanics in particular, by the scope they offered for more adventurous activities. Meteorology and aeronautics were the first fields to attract him, and the state of their development 60 years ago made them just right for him, particularly aeronautics which was then in its pioneering phase. He was given the best possible scientific education and opportunities for research as a consequence of wise decisions first by Trinity College, which gave him an undergraduate scholarship and a research fellowship; second, by Cambridge University, which appointed him to the Schuster readership in meteorology at an early age; and third, by the Royal Society, which supported him as a Yarrow Research Professor from 1923 until his (official) retirement in 1952 and enabled him to devote all his time to research during those 29 years. All these factors and circumstances together provided the possibility of an unusually fruitful scientific career, and the additional ingredient of an uncomplicated, modest, contented, purposeful character, with supreme self-confidence which came early in life as a consequence of being able to exercise his innate skills, turned this possibility into a magnificent actuality. Opportunities were made available to him, and he took them all and used them to the fullest. I have never met a man who fitted so naturally into the pattern of his life. His character and his activities were perfectly matched, and this enabled him to live and work with freedom from the tensions, the maladjustments, and the pretentions that limit and handicap ordinary men. The result: a lovable man and a contribution to science of the highest quality which will be an inspiration for many generations. G . K. BATCHELOR

Interaction of Water Waves and Currents D. H. PEREGRINE Department of Mathematics University of Bristol. Bvistol. England

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 A . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 B. SeaWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 C. Coastal Waves . . . . . . . . . . . . . . . . . . . . . . . . . 12 D . Waves in Rivers and Channels . . . . . . . . . . . . . . . . . . 13 E . Hydraulic Breakwaters . . . . . . . . . . . . . . . . . . . . . 14 F . ShipWaves . . . . . . . . . . . . . . . . . . . . . . . . . . 15 G. Generation of Currents . . . . . . . . . . . . . . . . . . . . . 16 H . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 I1. Large-Scale Currents . . . . . . . . . . . . . . . . . . . . . . . . 17 A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 17 B. Waves on Uniform Currents . . . . . . . . . . . . . . . . . . . 18 C. Waves on Slowly Varying Currents . . . . . . . . . . . . . . . . 26 D . Steady Current, Varying with Distance along the Stream . . . . . . . 39 E . Steady Current, Varying across the Stream . . . . . . . . . . . . . 53 F. Flows with Significant Vertical Accelerations . . . . . . . . . . . . 63 I11 . Small-Scale Currents . . . . . . . . . . . . . . . . . . . . . . . . 70 IV . Currents Varying with Depth . . . . . . . . . . . . . . . . . . . . 76 A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 76 B. Infinitesimal Waves . . . . . . . . . . . . . . . . . . . . . . . 77 C . Finite-Amplitude Waves . . . . . . . . . . . . . . . . . . . . . 91 D . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 E . Waves on Flow in Channels . . . . . . . . . . . . . . . . . . . 102 V. Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 VI . ShipWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Notes Added in Proof . . . . . . . . . . . . . . . . . . . . 117

9

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D . H.Peregrine

I. Introduction

A. SUMMARY The varied physical circumstances in which interactions between water waves and currents occur are described in this introduction. Different mathematical approaches, relevant observations, and experiments that are applicable to all or some of these physical Circumstances are described in the other sections. The paper has been written with gravity waves and currents such as those in seas or rivers in mind: thus there are only incidental references to the effects of surface tension and viscosity, which are of greatest significance for length scales of the order of centimeters and smaller. The emphasis is on waves and their interaction with preexisting currents rather than on wavegenerated currents, although these are mentioned where they are relevant. In all water wave problems approximations must be made to find mathematical solutions in order to gain physical understanding. Almost always the water is supposed to be inviscid and the flow irrotational. Here the first of these approximations is made but in few cases can the second approximation hold. Another common simplifying assumption is that the waves are of sufficiently small amplitude for the free surface boundary conditions to be linearized and evaluated at, or close to, the mean free surface. Most progress can be made in this subject with such a constraint, but wherever possible finite-amplitude effects are discussed. In order to get a reasonably wide class of solutions further approximations are necessary, the most important being for short waves and long waves, that is, for waves short (or long) compared with the length scale in which significant current variations occur. Sections I1 and 111 are on large- and small-scale currents, respectively. Much of the theory of water waves on large-scale currents only differs in detail from theory for any short waves in a moving medium (e.g., see Bretherton, 1971, for a review of linear theory). An adequate theoretical description was first given by Longuet-Higgins and Stewart (1960, 1961), who introduced the idea of radiation stress. Further improvement in our understanding has come from the use of Whitham’s method of averaging a Lagrangian and the concept of wave action. Since this is probably the most important field of wave-current interaction a number of simple situations are examined in detail in Section 11. Relatively little work has been done on small-scale currents, and Section I11 is mainly about waves in the presence of thin shear layers. Unlike some other common forms of wave motion, water waves involve water motion varying with direction perpendicular to the space in which

Interaction of Water Waves and Currents

11

they propagate. That is, water waves propagate on the surface of the water, but their motion also varies with the depth. Thus if there is current variation with depth it may affect the waves on the water surface. This is the topic of Section IV, which includes appreciable detail because of the possible applications to waves on streams and to the effects of a wind-driven surface current. In most applications the currents are turbulent and are approximated by a corresponding mean flow. However, there is interaction between waves and turbulence, so the few results are discussed in Section V. The paper concludes with another short section on the interaction of waves generated by a ship with the flow around it. A number of new results are incorporated in the text at various points. Particular examples are the errors involved in neglecting currents (Section II,B), the behavior of small-amplitude waves at a stopping point (Section I1,D) and of finite-amplitude waves approaching a caustic (Section II,E), and the surface layer solution for waves above a critical layer Section IV,B). B. SEA WAVES Over a large part of the world’s oceans and seas the spatial distribution of surface currents due to the tides and ocean circulation is on such a large scale that even the largest ocean waves are on an effectivelyuniform current, unless global propagation is being considered. It is mainly near continental margins and in shallow seas that currents influence waves significantly and this is discussed further in Section 1,C. However, the strong western boundary currents of oceans can and do strongly affect ocean waves. This is especially true of waves propagating onto and against such currents. They are shortened and steepened by the adverse current and refracted into caustics and foci by the shear at the currents’ boundaries. Examples of damage to ships by waves on the Agulhas current are mentioned in Section I1,E. Even on a uniform current difficult problems of practical importance arise. This is particularly so with nonlinear properties of waves such as the forces they exert on structures. With the increasing number of offshore structures the prediction of forces due to combinations of currents and waves is of growing importance. An appreciation of the problem may be gained from Hogben (1974). The surface drift caused by wind stress plays a role in wave dynamics as Banner and Phillips (1974) show (Section IV,C). It is probably also important in the generation of waves by wind; the tangential stress at the interface is influenced by its velocity and it in turn will influence the flow of air over

12

D. H. Peregrine

the wave. Phillips and Banner (1974) have studied this boundary layer in the water, but its implications for the wind-wave system need further study. Another effect, whose importance is debatable but not negligible, is the interaction between water waves. The interaction of short waves with much longer waves can be treated in the same manner as interaction with a current (Section II,F) and such effects are easily observed when a long swell meets short wind waves. Less commonly seen is the interaction of short waves with the velocity field of long internal waves (Section II,D) but Gargett and Hughes (1972) show photographs of wave patterns that are best explained in this manner. The shortest waves are capillary waves, and their ubiquity among wind waves is in part due to the steep gravity waves. The small radius of curvature at the crest of the steepest gravity waves and the surface tension there act rather like a moving pressure distribution and generate capillary waves. However, in Section II,F another mechanism for forming such waves is pointed out. Short gravity-capillary waves being overtaken by a larger gravity wave can be reflected near its crest and propagate away from it as capillary-gravity waves. The wavelength of this second class of waves is longer than those generated by the first mechanism.

C. COASTAL WAVES Water waves have their greatest economic importance when they arrive at coastlines. This is due to their ability to erode and build up land, their power to damage man-made structures, and the difficulties they cause in the handling of ships. Practically all the work to date on predicting and assessing coastal wave problems has neglected their interaction with currents, even though currents are often strong in coastal regions. There are two main reason for this neglect. One is that the transformation of waves due to lessening depth and their refraction and diffraction by underwater and coastal topography are often more important. The other reason is that the currents are often inadequately known. This is because of their complexity. Tidal currents vary in cycles so that in many parts of the world a good measurement of them requires recording for at least a lunar month. Furthermore, local winds, especially in storms, can produce currents comparable in magnitude with those due to tides in many places. Man-made structures are often associated with river mouths (which may be the reason for a harbor’s existence) and the river flow introduces further variability. To complicate matters still further, the mass transport associated with an irrotational wave train usually sets up mean currents in the region of the

Interaction of Water Waves and Currents

13

shore. For more detail, Longuet-Higgins (1972) gives an account of the longshore currents generated by waves and James (1974) gives further detailed calculations. The waves also generate rip currents and other return flows, which complete a cycle of interaction by influencing the incoming waves (Section 11,D). These effects are observed in hydraulic models, but sometimes efforts are made to minimize rather than measure them, especially if the wave-generated currents are unsteady. It is not clear whether attempts should be made to incorporate current effects into most coastal wave studies. The substantial variations in typical current fields mean that getting the basic data is very expensive unless there is a strong, welldefined current system that clearly cannot be ignored. On the other hand, it is advisable to estimate the “worst” conditions together with their probability of occurrence. As is indicated in Section II,F, the common impression that waves are higher at high tide than at low tide receives support from theory, but focusing effects are probably more important. For example, tide races, in which steep waves occur on strong currents around headlands and in channels, are due to waves, propagating partially against the current, being concentrated. Fortunately such concentrations of wave energy are usually in the strongest current, which is not usually a point where structures are erected. Indeed, wave action at the shoreline may be substantially overestimated in some circumstances if currents are ignored, since they can have a sheltering effect. Right on a beach, in the surf zone, there are appreciable problems in actually describing the waves. One promising model is to use the finiteamplitude shallow-water equations with bores fitted. In such a model a separation into waves and currents does not assist analysis. The relatively small scale of the currents also accentuates the difficulties. For example, rip currents might be expected to accentuate waves incident upon them; but observations indicate that waves on rip currents tend to be lower and break less. This is probably a case of diffraction being more important than refract ion. Two interesting minor points are a proposal by Dagan (1975) that short wave-long wave interaction may contribute to wave breaking (Section II,F), and an occasional wave feature in the backwash from surf that Peregrine (1974) interprets as being due to current shear in the vertical (Section IV,C). D. WAVES IN RIVERSAND CHANNELS Rivers normally have a nonuniform current distribution and this directly affects all the waves that occur on rivers (with the possible exception of flood and tide waves). If the currents are sufficiently swift then variations in the

14

D. H. Peregrine

river’s bed and banks (e.g., a projecting obstacle) may cause stationary waves on the surface, like ship waves. If such waves are caused by a constriction in the channel they are usually confined to the region of maximum current velocity and a train of several crests may be seen (Section 11,D). Stationary waves caused by a sluice or weir may have a very large amplitude (surface shear waves, Section IV,C). Shorter waves are commonly generated by the wind and by boats. The most usual interaction of these waves with a typical current profile (i.e., maximum velocity away from the shore) is for waves traveling upstream to be refracted toward the maximum current, while waves traveling downstream are refracted toward the bank (Section 11,E). Thus, in midstream the former waves may persist for a considerable time since they are not dissipated by interaction with the banks, while the latter waves soon meet the bank and decay rapidly. This also means that river banks may suffer more wave action from the upstream direction. The variation of current along a river also affects waves (Section 11,C). For example, a boat traveling upstream at a constant speed relative to the water produces waves of constant amplitude, but if they propagate upstream into a region of stronger current they may be considerably amplified. Thus, it is possible in such a region to see a boat pass traveling upstream and for the waves following it to increase continually in amplitude for a considerable time after the boat has passed. The author has experienced this while sculling and written a note about it (Peregrine, 1972). Similarly, the boat may generate waves that are “stopped (that is, propagating upstream but with a group velocity equal to the stream velocity). These waves, which appear to be moving since their phase velocity is upstream, persist until dissipated, which may take a surprisingly long time. ”

E. HYDRAULIC BREAKWATERS A hydraulic breakwater is simply an extensive current of water directed toward waves in order to stop them. It works; but for long waves very substantial currents are needed. The power needed to pump the water is such that it is rarely an economically feasible proposition. A considerable number of experiments have been performed-most designed to assess the power requirements of specfic designs. Evans (1955) gives an historical perspective as well as some experimental results. More recent reports of experiments are Nece ef al. (1968), Bulson (1963), and Williams and Wiegel (1962). A closely related device is the bubble or pneumatic, breakwater. The

Interaction of Water Waves and Currents

15

original idea was that a stream of air flowing up through the water would form a region with a lower effective density than water, thus reflecting some of the wave energy. In fact, the entrainment of water by the air results in an outward surface current from a line of bubble generators, which is effective in stopping waves. Evans (1955) also includes and compares results from a pneumatic breakwater. A more recent paper is that by Green (1961), which also has a summary of previous work. Naturally there is no need for a breakwater current to extend down to the full depth of the water if the incident waves are in “deep water.” Experiments such as Evans’ (1955) show that a surface current need only extend to a depth that is only a small fraction of a wavelength in order to stop waves, if it is strong enough. While there is linear and nonlinear theory available for waves on a slowly varying current that is uniform with depth (Section II,D), when the vertical structure of the current is also important results of linear theory are only sufficient to find the local wavelength, although this gives a first approximation to the stopping velocity (Taylor, 1955). Witham’s method of using an averaged Lagrangian may provide a way of finding the variation of wave amplitude in such cases.

F. SHIPWAVES The greatest interest in ship waves is in their contribution to resisting a ship’s motion. This may be assessed by measuring the waves radiated by a ship. However, some of the energy and momentum that may be assigned to the wave field close to the ship is lost from the wave field, for example, by wave breaking. On the other hand, the waves generated by a ship depend on the flow of water around it, and if this is altered (for example, by boundary layer suction) then so is the radiated wave pattern. The interactions between flow and waves are discussed in more detail in Section VI. Theoretical methods of estimating wave resistance are complementary to the measurements since these are usually made on ship models and need to be scaled for use in ship design. Most mathematical models assume inviscid irrotational flow, but it has been shown by experiment and theory that the wake and boundary layer lead to significant effects on the waves. These are not adequately described by increasing the size of the ship to account for the displacement thickness of the boundary layer. Many theoretical models involve several different approximations and considerable care needs to be taken to ensure consistency when proceeding beyond the first approximation. The further approximations usually involve wave-current interaction terms.

16

D.H.Peregrine G. GENERATION OF CURRENTS

The mass transport associated with water waves is of second order in the amplitude but still makes an appreciable contribution to currents in the vicinity of coasts. Details of the mass transport in a uniform wave train are calculated in Longuet-Higgins (1953) and some more recent work is in Sleath (1973, 1974). Mass transport is transformed into a current that is not directly coupled with waves whenever there is wave dissipation. Similarly, when waves gain or lose momentum because of interaction with currents there is a corresponding change in the current. This must be taken into account in any theory dealing with wave-current interactions in water of finite depth unless the waves have infinitesimal amplitude. Equations governing such an interaction are given in Section II,C, but there have been few direct applications to current generation, mostly to longshore currents. Another mechanism for generating a current from waves is given by Craik (1970). He describes a nonlinear interaction between two wave trains propagating over a depth-dependent shear flow. The idea is that the shear flow models the shear due to wind stress. The current generated has streamlines that superficially resemble those of vortices aligned with the wind. It is suggested that this mechanism may be partly responsible for Langmuir vortices, which are a similar feature observed in the sea. H. NOTATION

Notation is generally explained as it is introduced, except for some conventions that are uniform throughout the article. Coordinate axes are chosen with Oz vertically upward and Ox usually in the direction of the current if it is unidirectional. The plane z = 0 is a horizontal surface at or near the mean free surface. The current field in the absence of waves is U(r, t) with components ( U , V, W), and it is U + u with components (V + u, V + u, W + w ) in the presence of waves. Similarly the free surface is

although Z is zero sufficiently often that the symbol is used with other meanings. The wave frequency relative to a fixed reference frame is w and relative to the water is 0. The phase velocity relative to the water is denoted by c in Section 11, but elsewhere c is the phase velocity relative to the frame of reference. The wave number vector k is taken to be (k, 0,O) or (I, rn, 0)

Interaction of Water Waves and Currents

17

depending on circumstances, and 8 denotes the angle between k and U.The amplitude of the wave motion of the water surface is denoted by a. When tensor notation is used, Greek suffixes have the values 1, 2 and Roman suffixes 1, 2, 3, where ( X I , x2

9

x3)

= ( x , y, z ) .

The two-dimensional vector operator @/ax, a/ay, 0) is denoted by V,. In various places one symbol is used with different meanings to avoid using unusual letters or substantial numbers of suffixes. However, when results are cited from other works the notation is changed to agree with that in use in this paper.

11. Large-Scale Currents

A. INTRODUCTION In many instances of waves riding upon currents, the time and length scales determined by the current are many times larger than the period or wavelength of the waves. The natural assumption is to suppose that at any particular point the waves may have the same properties as a plane wave train on a uniform current, and further that the parameters describing the wave train, such as amplitude and wavelength, may vary slowly with the current. It is intuitively clear that such an approximation is likely to be effective, but the problem can be approached more formally, by requiring

where k and w are the wave number (= 2n/wavelength) and frequency (= 2n/period) of the waves. From ratios of such wave and current scales one or more small parameters may be constructed and formal expansions of variables in powers of a small parameter can be used to obtain solutions. It is expected, but not proven, that such solutions are asymptotic to exact solutions. This short-wave, or large-scale current, approximation has to be used in conjunction with solutions for uniform plane waves on water moving with uniform velocity. Such a flow is irrotational, and hence a velocity potential may be introduced to simplify the analysis, but it is still necessary to approximate to obtain water wave solutions. Some of these approximate solutions are briefly reviewed in Section II,B.

18

D. H.Peregrine

It is commonly the case in short-wave approximations that the solutions are singular on certain lines or points. In water wave examples it is not always clear which approximation is responsible for the singularity. It may be the water wave approximation, e.g., if a small-amplitude assumption is made, or it may be the short-wave approximation of a locally plane wave. For small-amplitude waves the plane-wave approximation can usually be improved in those cases where two possible solutions converge on the same singularity, one representing waves propagating toward it and the other representing waves propagating away from it. The resulting solution will describe waves being reflected. The maximum steepness of such a solution will then indicate whether the water wave approximation is sufficient or not. Specific examples are described in Sections II,D and II,E, but no such solutions have been produced for finite-amplitude waves, although a singularity, which requires such a description, is noted in Section II,E. The subject of this section is only one aspect of the problem of wave propagation in a slowly varying medium and most work on the subject is not specifically confined to water waves or to moving media. In particular, work published in the last decade on nonlinear short-wave problems, all originating from Whitham’s (1965a,b) method of averaging, has shed considerable light on the propagation of linear and nonlinear waves in nonuniform, slowly varying media. The concept of wave action, crystallized by Bretherton and Garrett (1968), is particularly valuable for moving media. An extensive and up-to-date account of the subject is Whitham (1974). Water waves are also influenced by the depth of water and where this too varies slowly its variation can usually be included. This is done in the rest of this section wherever it is convenient to do so. B. WAVESON UNIFORM CURRENTS In many applications of short-wave approximations, either the equations are linear, as in vacuum electromagnetic theory, or a linear approximation is an excellent first approximation, as in acoustics. In water wave problems a linear approximation can be quite sufficient, but it is more usually only a rough guide when waves have an appreciable amplitude and hence greater importance. Hence a few parameters, which are most often used for sinusoidal linear waves, are defined for nonlinear plane waves. If a wave is periodic in both space and time, then the physical variables describing it will all be functions of a phase

-

x = k r - ot + 6,

(2.2)

in which k, the wave number vector, is perpendicular to planes of constant

Interaction of Water Waves and Currents

19

phase (e.g., wave crests for water waves). Water waves are an example of modal waves, that is, waves that have structure in a dimension in which they do not propagate, in this case down into the water. Thus k is essentially parallel to the mean water surface and has no component perpendicular to it. The wave number k and frequency w are made unique by choosing x so that its period is 27r; they then correspond to the usual definitions of wave number and radian frequency for sinusoidal waves. The phase velocity c defined by

c=w/k, k = lkl, (2.3) is also defined for nonlinear waves. It is relevant to note that phase velocity is not a vector, e.g., the phase velocity along a line in the direction of a unit vector e is w/(k * e). If r is a position vector in a frame of reference in which water is moving with uniform velocity U, then the corresponding position vector r’ in a frame of reference moving with the water is given by r’ = r - Ut.

(2.4)

Thus a wave on moving water described by

f(k

- r - wt)

(2.5)

is also described by f(k * r’ + k * Ut - wt) =f(k

- r’ - at).

(2.6) Thus if any wave property (e.g., a dispersion relation) is given for still water for a wave of frequency a, the corresponding property for a wave on water in uniform motion is given by the relation 0

=w

- k * U.

(2.7)

A uniform plane wave train of infinitesimal amplitude, propagating over still water of uniform depth h, with vertical surface displacement

C = a exp[i(k * r’ - at + S)]

(2.8)

above the mean level, has velocity potential

’=

+

iaa cosh k(z h ) exp[i(k r’ - at k sinh kh

+ S)]

and dispersion relation rsz = g k tanh kh.

(2.10) In these expressions the physical quantity is the real part of a complex expression, and z is measured upward from the mean free surface. If surface

20

D. H.Peregrine

tension T is to be included, then g should be replaced by g + Tk2/p in the dispersion relation (2.10). The group velocity, the velocity of energy propagation, is

:$( 1 + sin2hkhZkh)

c =--

(2.11)

for gravity waves. This linear approximation is a good approximation when all three of the parameters ak, alh, and a/k2h3

(2.12)

are much less than one. For finite amplitude waves there are various different approximations, of which two are most relevant. A straightforward perturbation expansion in powers of ak gives the Stokes’ wave approximation, which is appropriate for water of moderate or great depth, specifically when a/k2h3is small. For shallow-water waves a more subtle expansion, balancing the effects of a/h against ak, is needed to produce the cnoidal wave solution (e.g., see Whitham, 1974). Such expansion procedures are cumbersome for dealing with the highest waves and usually separate approaches have been made to that problem. However, recently, computer-assisted calculations have enabled expansions to be carried out to high orders giving results for most of the range of possible periodic waves (Schwartz, 1974) and for the limiting case of the solitary wave (Longuet-Higgins and Fenton, 1974). For purely capillary waves Crapper’s (1957) exact solution covers the whole range of amplitudes. Stokes (1847) noted that there is ambiguity in defining “still water” for a finite-amplitude wave train on water of finite depth. The two natural definitions, (i) the average velocity is zero at any point that is always submerged, and (ii) the average flow of water through any vertical plane is zero, are not equivalent. This is clearly shown by considering the most general form of the velocity potential for a periodic wave. Since only the physical variables need be periodic (2.13) 4 = B * r - Yt + @(x), where y, is the phase (2.2). The constant fi corresponds to a uniform velocity.

The physical interpretation of y is less clear, but it contributes to the mean pressure and is thus related to the mean level of the water. If definition (i) for still water is chosen, then B = 0. On the other hand, with definition (ii), 2nio

Bh=

-j [j‘VO dr] dt. 0

-h

(2.14)

Interaction of Water Waves and Currents

21

This value of b is often called the mass transport velocity of the waves. The contribution to the integral comes mainly from the region above the lowest value of ( in the troughs of the waves. Thus for small amplitude waves it is of order a’ and can often be neglected. For deep water this ambiguity disappears since the still water at great depth provides a reference frame. However the mass transport is still nonzero. As an example of a finite-amplitude solution the first terms in a Stokes’ wave expansion are ak(3 - Ti) 4Ti cos 2% O(a’k’)],

+

C#J

=b

-r-

yt

+-aak

cosh k(z + h) sin sinh kh

I

+ h, sin 2% + 3ak 8‘Oshsinh2k(z 4kh

+

(2.15)

x

I.

O(a2k2)

(2.16)

and the dispersion relation

.

(a - b k)’ = gk tanh k(h

+ 9Ti) a’k’ + O(a4k4)], + b) 1 + (9 - 10Ti 8 T:

I

(2.17)

where To = tanh kh. The parameters k, w, a and fl, y, b define a specific wave train within a phase shift 6, and the dispersion relation (2.17) provides one equation between them. The choice of a frame of reference determines fl, e.g., definition (ii) for still water gives = tka’c coth kh

+ O(a3k2c).

(2.18)

Either y or b may be chosen arbitrarily but they must satisfy the relation y = $pz

+ gb + [(l - Ti)aZa2/4Ti]+ O(a3kaz)

(2.19)

obtained from the constant terms in Bernoulli’s equation. More details of finite-amplitude waves are given in Section II,C, which shows the advantage of leaving fl, y, b in these expressions. Now, consider infinitesimal waves on the surface of a uniform stream U . The dispersion relation (2.10) becomes (w - k U)’ = gk tanh kh

(2.20)

after using (2.7). This may be conveniently rewritten W =

+a(k)+k*U,

(2.21)

D . H . Peregrine

22 where

a(k) = +(gk tanh kh)’”.

(2.22)

A common and direct use of dispersion relations is to find the value of k once w is known (or vice versa) in order to calculate other wave properties. For example: (i) Measurements of [ ( t )at one point may be available and a measure of velocity fluctuations on the bottom may be required; or (ii) waves may be generated at a fixed frequency w, as in many experiments. In the absence of a current, k is determined uniquely but the direction of k is undetermined. In the k plane (i.e., a plane where k is a position vector) the locus of possible solutions is a circle. There is a greater lack of uniqueness in solving Eq. (2.21) for k when there is a current, even if U is known. The easiest way of appreciating the solution of the dispersion relation for k is to consider the intersection of the plane

m=w-k.U

(2.23)

m = ka(k)

(2.24)

with the surface of revolution in (k,m) space. [This is a development of the graphical method of solution given by Jonsson et al. (1970) for k parallel to U.] The general form of the locus of solutions for U # 0 is seen by noting that if k is perpendicular to U then the current does not affect the solution, while for k in any other direction, specified by a unit vector e, a diametral section of (2.24) yields a curve as shown in Fig. 1. The trace of a typical plane (2.23) is also shown, and four solution points A, B, C, and D in that diametral plane are labeled. The solution point A corresponds to waves with a component of k in the direction of the current, being swept along by it so that the measured frequency w is greater than the frequency a relative to the water. Similarly, B represents waves with a component of k opposed to the current direction traveling more slowly relative to a fixed observer so that w is less than a. These solutions effectively exhibit the Doppler effect, with appropriate corrections for dispersion. The solution represented by point C does not occur without the current, or for nondispersive waves. It corresponds to waves propagating against the current, in the sense that their crests move upstream, but their energy is being swept downstream. That is, - c < u c o s e < -c*,

(2.25)

Interaction of Water Waves and Currents

23

FIG. 1. Solution of the dispersion relation showing multiple values of k for given w, h, and U.

where 6 is the angle between k and U.These waves have to be generated on the current. The point D corresponds to waves with a M > 1,

-&I
0 if the center of curvature is within the fluid. The equations of motion for the basic flow give

-

-ap/an = p v p t ) - g ii,

(2.174)

where g is the gravitational field and ii the unit vector normal to the free surface. Thus, taking (2.169) into account the boundary condition (2.173) may be rewritten p =(-g

a - K u2)q.

(2.175)

Flow along the surface streamline will satisfy Bernoulli's equation so that U is of the order (gL)1'2,where L represents the length scale of the basic flow. This is the case for deep-water long waves since U is then approximately equal to the phase velocity, which is ( g L / 2 ~ ) 'where / ~ , L is the wavelength. If the radius of curvature K - ' is also of order L, then the terms uU2 and g ii are of the same order of magnitude and should both be included in the first approximation. For a small-amplitude deep-water long wave K - is of order AIL2, where A is the wave amplitude, and KU' is of order (A/L)g ii. For small AIL it might be assumed negligible. However, this is nat the case since the variations in U that are of interest are also of order (A/L)U.Thus the term K U ~ should again be included. Typical terms in the equations of motion are

-

-

For these terms, the primary assumption that the waves are short compared with L clearly indicates that velocity gradients and the curvature of streamlines need not be included in a first approximation. Thus the dispersion relation for short waves is a2 = (o- kU)2 z

(-g

*

ii

+ DV/Dt)k = g*k,

(2.176)

Interaction of Water Waves and Currents

65

where g* is the “effective gravity” in a frame of reference moving with the free surface of the large-scale flow. For a steady situation it is reasonable to suppose that wave action is conserved and that the wave action flux

B = E( U + cg)/c

(2.177)

E = 4pg*aZ.

(2.178)

is constant, where However, as Bretherton and Garrett (1968) point out, there is ambiguity possible in defining E, the “perturbation energy density” in moving reference frames. This is made apparent by Longuet-Higgins and Stewart (1960) where the form

E = apg*az + $pgaz

(2.179)

is chosen for waves on a basic flow with small surface slopes, the second term representing potential energy in the gravitational field only. The choice (2.178) seems more appropriate since it corresponds to equipartition of kinetic and potential energy densities. This choice is also supported by consideration of the Lagrangian for a perturbed flow derived by Bretherton and Garrett, Eq. (4.19), although it should be noted that this involves energy per unit horizontal area rather than per unit area of mean free surface. The solution of these equations differs from that for the simpler flows of Section II,D only in the substitution of g* for g, so that most results carry over directly once g*(s) and U(s) are prescribed. However, the example of short waves riding on long waves traveling in the same direction involves a different solution from that discussed in Section II,D. In a frame of reference in which the long waves appear stationary, the current is U ( s )=

-c

cos II/

+ U*(s),

(2.180)

where C is the phase velocity of the long waves, $(s) their surface slope, and U*(s) the water velocity due to the waves. That is, the short waves in this frame of reference are meeting an adverse stream of magnitude C . The phase velocity of the short waves is much less than C, so that 0 = k(c

+ U)

(2.181)

is negative, showing that c is always less than - U ( s ) . Similarly the wave action flux B is also negative, and the solution illustrated in Fig. 4 does not represent this situation. For the case g* = g, the variation of a, k, and ak relative to their values at U = -2c are shown in Fig. 11. A large range of velocities are shown in the

66

D . H . Peregrine

/

0.1

4

FIG.11. The variation ofamplitude, wave number, and wave steepness ak on a current U ( S ) for negative values of w and E. Note both ordinates are logarithmic. The suffix 2 refers to values where U = -2c.

figure, since for a large amplitude long wave, - U ( s )varies from a value greater than C in the trough to almost zero near the crest. The relatively rapid variation of steepness a k of the short wave with velocity U(s)is clear. The solutions shown in Figs. 4 and 11 can also be used when g* differs from g. Let

Y = g*/go

(2.182)

7

where go is the value of g* at a point where values of the wave parameters are known. Introduce new variables c

k* = k/y,

c* = yc,

a* = a/y,

U* = yU.

(2.183)

Then the equations to be solved for the starred variables are those for constant g* = g o . The simplest example with significant vertical acceleration is where the long waves are of infinitesimal amplitude with surface displacement A sin(Kx - Of).

Interaction of Water Waves and Currents

67

Then

U ( S )1: - C

+ AR Wth Kh sin Ks,

g*(s) 1: g - ARZ sin Ks.

(2.184) (2.185)

Since A is small the analysis is simplified by assuming U = -C

+ dU

and

g*(s) = g

+ dg*,

(2.186)

and using the differentials of Eq. (2.177) and (2.181) and the dispersion relation. The results of this analysis, after identifying dU and dg* with the appropriate terms in (2.184) and (2.185), and after simplifying even further by neglecting c compared with C, are

da/o = iAK(coth K h - tanh Kh) sin Ks,

(2.187)

dklk = AK coth Kh sin Ks,

(2.188)

da/a = $AK(3 coth Kh + tanh Kh) sin Ks,

(2.189)

d(ak)/ak = tAK(7 coth Kh + tanh Kh) sin Ks.

(2.190)

Longuet-Higgins and Stewart (1960) derive results corresponding to and agreeing with (2.188) and (2.189) in their perturbation analysis. Vincent (1975) points out that although c c C,it is often not small enough to be neglected. There is only a simplification of algebra gained by its neglect. It is clear that short waves steepen as the crest of a long wave overtakes them. If the long wave has appreciable amplitude they may steepen sufficiently to break at some point on the forward facing slope of the long wave. If short waves travel in the opposite direction to the long waves, then U ( s ) is positive (supposing that the short-wave direction is again taken to be positive). However, Eqs. (2.187)-(2.190) still hold for infinitesimal long waves since dU/U still has the same value. It is suggested by Longuet-Higgins (1969) that the variation of steepness of short waves on longer waves may contribute to the growth or decay of the longer waves. Two mechanisms are proposed, one weak and one strong interaction. The weak effect is the viscous decay of the short waves, which is proportional to wave steepness. One result of the decay is to transfer momentum from the short waves to the “current.” Since the short waves are steeper at crests than in troughs, more momentum is transferred at the crest, leading to a growth or decay according to whether the short waves are traveling with or against the longer waves. Longuet-Higginsdiscusses details of the momentum transfer, introducing a virtual tangential stress at the surface, and makes an estimate of its value for a typical wind-driven system.

68

D. H. Peregrine

It may account for more than 10%of the stress due to the wind at low wind speeds, and less for high winds. It is also shown that

_1dA - - - - 4v (ak)2fJf23, A dt

(2.191)

g2

where v is the kinematic viscosity; the strong dependence on the frequency of the long waves is evident, but it is possible that the effect may be significant in their amplification. A stronger effect can be expected when the short waves steepen sufficiently to break. This gives a very direct transfer of momentum, and also, since much of the wave energy may be dissipated, more of the wave momentum may be transferred. If the wind can regenerate the short waves between the long-wave crests, this could be an efficient mechanism for generating waves with phase velocities greater than the wind speed. Longuet-Higgins’ (1969) estimates for the rate of growth of the waves are consistent with this mechanism being important for certain sea states. A contradictory result is obtained by Hasselmann (1971). The inviscid equations of motion are averaged and the interactions are considered from an Eulerian viewpoint. This represents the mass flow associated with the short waves as a surface flow occurring between the wave troughs and their crests. This results in a new kinematic boundary condition for the mean flow. At the mean free surface, z = Z ,

+

(dZ/dt) (U * V1)Z - W = -V1

. M/p,

(2.192)

where M is the mass flow associated with the short waves. The term on the right-hand side gives the rate at which the mean surface Z must be lowered to supply water to feed increases in the mass flow of the short waves. This mass flow is greatest down the front face of a crest of the longer wave, effectively transferring water from the crest to the trough of the wave, and hence reducing its potential energy. Hasselmann (1971) deduces that this term is as effective in damping a wave, as the momentum transfer is at amplifying it. Using infinitesimal wave theory he analyzes the residual terms in an expression for the rate of change of the long wave and deduces that, whichever way the short waves travel, the long wave is damped. In a discussion of radar measurements of short-wave spectra he concludes that the magnitude of this damping is “ of marginal significance.” The approaches in these two papers are difficult to reconcile. Clearly Hasselmann introduces an important interaction, which Longuet-Higgins was unaware of, but the momentum transfer in his inviscid model may be a poor representation. This is especially so for breaking short waves, where the vorticity generated may spread below the surface layer. Similarly, the re-

Interaction of Water Waves and Currents

69

liance on linear theory for order of magnitude estimates may not be adequate, especially as for short waves of small amplitude there is no significant energy transfer. More work is needed on this problem, but it is reasonable to conclude that Longuet-Higgins overestimates any amplification of long waves. A closely related subject is the generation of capillary waves near the crest of a steep gravity wave. These are interpreted as waves of the same phase velocity generated by the excess pressure at the sharply curved crest due to surface tension. Longuet-Higgins (1963) analyzes their generation by a perturbation scheme and discusses the results in the context of short waves on a large-scale flow. Crapper (1970) starts from this point of view and uses his solution for finite-amplitude capillary waves (Crapper, 1957) to calculate the variation in their steepness along the gravity wave profile. In this problem, as in most problems where surface tension is important, it is necessary to take account of the dissipation of the short-wave motion by viscosity. For this reason wave action is not conserved and it is more appropriate to use an energy equation, with radiation stress, dissipation, and an input function. For short waves of this type, if surface tension is dominant then both gravity and surface accelerations are unimportant and may be neglected. It is clear from the last few paragraphs that the dissipative effects of these capillary waves may be important and should be investigated further. Another mechanism for generating capillary waves on the front of a gravity wave becomes evident once the properties of capillary-gravity waves are considered. These waves have a minimum value of their group velocity. We have noted that U c is always negative for short waves traveling with longer waves, in the frame of reference with the long waves stationary. For gravity waves, this implies U c, is never zero, but this is not so for gravitycapillary waves. U cg may be zero and there is then a stopping point. The magnitude of the velocity U at the stopping point, and hence at the crest of the long wave, must be less than the wave velocity at which c = c,, which equals the minimum phase velocity. For normal gravity and clean water, this is 0.23 m sec- These reflected capillary-gravity waves are longer than those generated at the crest. This is easily seen from graphical consideration of the equations

+

+

+

’.

- kU,

(T

=

(T

= (gk

+ Tk3/p)’”.

The reflected waves have w < 0, while those generated at the crest have o = 0 (note that U < 0). A further factor influencing short waves riding on long waves on the sea is the vorticity distribution in the water due to direct wind stress and the

70

D . H. Peregrine

dissipation and breaking of waves. Shemdin (1972) gives experimental measurements of velocity profiles and estimates the effect on the dispersion relation, and Banner and Phillips (1974) discuss its effect on the maximum amplitude of the shorter waves (see Section IV). As already described, when the crest of a steep long wave catches up with short waves they increase rapidly in steepness. Dagan (1975) makes the interesting suggestion that for the highest waves, where U(s)is near zero at the crests, the rapid increase of amplitude of the short waves may be interpreted as an instability of the basic flow. That is, the initiation of breaking might be described in these terms. Two properties of the breaking process are described by this hypothesis: (i) its initiation is rapid, (ii) breaking occurs on the front face of a wave. On the other hand, breaking does not resemble an oscillatory short wave. If U ( s ) is always greater than the minimum phase velocity, then sufficiently smallamplitude wave disturbances may pass over the wave crest, while if U ( s )has a lower value, there is a stopping point near the crest where waves may be reflected, and for sufficiently small initial waves, infinitesimal wave theory may be used to find the amplification. Thus the usual requirement for instability of indefinite amplification of an arbitrarily small disturbance is not met. Dagan’s analysis is for a steady flow and makes no assumption about the rate of change of the short waves, in which respect it is valuable. It certainly indicates that short waves may sometimes precipitate or influence breaking. In confirmation of this the author has a 16-mm film showing a small wave disturbance meeting a wave on the point of breaking on a beach. The larger wave breaks with two sheets of water projected forward. By running the film backward frame by frame it is clearly seen that one of the sheets of water is directly connected with the incident disturbance. 111. Small-Scale Currents

A common way of finding solutions to difficult mathematical problems is to look for parameters that may be either very large or very small and then to solve the problem for those cases by making appropriate approximations. This approach is successfully used in Section I1 to deal with large-scale currents. At the other extreme are current distributions with a scale much smaller than a wavelength. Some important examples are best discussed in a context covering all length scales; thus flows that vary with depth are treated in the next section, and the interaction of the flow around a ship with the waves it generates is considered in Section V. This leaves few situations of interest where much analysis may be done, and most of this section is devoted to thin shear layers.

Interaction of Water Waves and Currents

71

There has been very little work on this problem, so it is of interest to note an analogous problem that has been studied more intensively.The propagation of sound through moving fluids is one such case; indeed, twodimensional sound waves in a uniform atmosphere satisfy exactly the same equations and boundary conditions as infinitesimal shallow-water waves in water of constant depth. This subject is reviewed in a paper by Lighthill (1972) and the book by Goldstein (1974) gives mathematical details of some of the topics. Naturally many of the problems that arise in acoustics have little relevance to water waves and vice versa, although the mathematical methods are applicable to both fields, or at least provide a useful starting point. When all dimensions of a current system are very small compared with those of the waves, the wave may simply be taken as giving the local mean water level and current as slowly varying functions of time. For example, this is often done in relation to the tides for small-scale coastal problems. The effect on the waves is negligible, unless there are many such small current systems. In any case, the author does not know of important or interesting examples. More interesting currents are those which have one long length scale but which are otherwise short compared with the waves. A thin shear layer between two different, nearly uniform flows is the simplest example. A thin jet is another example. In real flows regions of strong velocity gradients only remain thin if there is some factor opposing their usual turbulent spread. However, in many cases a portion of the flow field might be well described as a thin shear layer, and any solution for that case can be of value in interpreting or predicting behavior in the problem where the current scale is of the same order as that of incident waves. In searching for mathematical solutions to problems involving thin shear layers, it is a natural step to look at the limit as the thickness of the shear layer goes to zero. That is, to consider a flow with a vortex sheet across which the velocity is discontinuous. In practice both vortex sheets and thin shear layers are unstable flows, so that steady solutions to such problems cannot be expected to give more than a crude approximation to real situations. This is better than nothing and may be quite adequate in some circumstances. If a vortex sheet at y = q(x, z, t ) separates two regions of flow, denoted by subscripts 1 and 2, then the boundary conditions on the vortex sheet are (i) that the pressure is continuous, Pl

=P2

9

and (ii) the fluid particles each side of the vortex sheet move with the sheet,

72

D . H. Peregrine

that is,

art

- + u.-aY

at

+ w . -art = u i ,

lax

for i = 1, 2.

laz

This latter boundary condition has been incorrectly formulated in a number of papers, both for compressible flows and for water wave problems, by the omission of the last two terms on the left-hand side of Eq. (3.2). The simplest problem is for an undisturbed flow that consists of a plane vortex sheet with uniform flows ( U1, 0,O)and (V,, 0,O)on its two sides. The incident wave is simplest if it is plane periodic, making an angle 8, with the flow direction. Solutions for the acoustic problem were given independently by Miles (1957) and Ribner (1957). The corresponding linear shallow-water problem is quite straightforward. Matching phases on both sides of the vortex sheet gives

kl cos 8, sec 8,

= k,

cos 0 2 ,

(3.3)

+ F , = sec O2 + F, ,

(3.4) where tI2 is the angle the transmitted wave makes with the flow and Fi is the Froude number Ui/c= Ui(gh)-’”. These show that there are two critical angles, and incident waves are totally reflected if

- 1 I sec 8,

+ F , - F , I 1.

(3.5) The amplitudes of waves are easily found from the linearized boundary conditions. If A iand Biare the amplitudes of waves propagating in the + y and -y directions, respectively, on the appropriate sides of the sheet, then A1

+ B1 = A , + B 2 ,

k , sin $ , ( A , - B , ) = k , sin $ , ( A , - B2).

(3.6) (3.7)

If the signs of cos 8, and cos 8, differ, then the reflected and transmitted waves may be many times larger than the incident wave. This solution only occurs for lF,-F,l>2 (3.8) and appears to be a mathematical curiosity since the amplification is much diminished if a finite shear layer is incorporated into the mathematical model (Graham and Graham, 1969). Much further work has been done on the acoustic problem: in particular, Jones and Morgan (1972) solve for an instantaneous line source situated off the vortex sheet. After the wave produced has interacted with the vortex sheet for a finite time, an “instability wave” arises, which has an exponentially growing amplitude. Jones and Morgan (1974) use a very simple

Interaction of Water Waves and Currents

73

method of modeling the turbulence that must arise, and this leads to a more complete discussion of the scattered sound. The papers are also of interest for the mathematical techniques used to ensure that the solutions satisfy causality. For waves in deep or moderately deep water, the acoustic analogy is not available. Even the simplest problem of linear plane waves on a vortex sheet has not been solved. The difficulty arises in satisfying the boundary conditions on the vortex sheet at all depths. Matching the phase of the wave in x gives

k, cos 8, = k, cos 0,

(3.9)

again, but matching the frequencies leads now to (3.10) for deep water instead of (3.4), because of the different dispersion relation. Given k l and el, this is sufficient to determine k, and B,, and thus the range of total reflection, which again lies between two critical angles, and is given by However, kl # k,, except in one isolated case. Thus the variation of the wave motion with depth, that is, exp(k,z), is different on the two sides of the vortex sheet. A solution that includes terms whose influence is confined to the neighborhood of the vortex sheet is needed. If the velocity potentials are assumed to vary like exp{i(lx - wt)}, where 1 = ki cos Bi ,

(3.12)

then the remaining problem looks deceptively simple and symmetrical. The equations to be solved are (a24i/ay2)

+ (a2+i/az2)

-124= ~

o

(3.13)

in the region y I0, z I0, if i = 1, and in y 2 0, z I 0, if i = 2. The boundary conditions on z = 0 are

a4i/az = a ; 4 i ,

(3.14)

and those on y = 0 are (3.15)

74

D. H. Peregrine

where ai =

10

- lUi lg-1’2.

(3.16)

Those who are mathematically inclined may like to prove the existence, or nonexistence, of solutions, or find some. Evans (1975) has succeeded in reducing the problem to that of solving a pair of singular integral equations. Conservation of wave action is proven and approximate solutions are found. Figures 12 and 13 show the results for four angles of incidence. The reflection and transmission coefficients given are simply the ratio of the surface amplitudes in the relevant waves. It may be noted that unless 8 is small the reflection is low except in the vicinity of critical angles, or when there is total reflection. When there is little reflection the transmission coefficient differs little from that for a wide shear layer for which results are given in Section II,E and shown on Fig. 12. Evans used two different approximations; both are shown in Fig. 13 and thus give an idea of the accuracy that may be expected. For

I u, - u2 1 s c1,

(3.17)

it appears to be quite adequate for any application. Another solution involving vortex sheets is given by Peregrine and Smith (1975). The solution is for stationary waves on a “ top-hat ” jet. This type of solution is relatively simple since, on introducing a velocity potential within the jet and noting that there can be no motion outside it, the boundary condition (3.1) on the bounding vortex sheet reduces to

a4lax = 0,

(3.18)

while (3.2) becomes an equation for finding its displacement. For a rectangular jet of width b and depth h, surface waves of the form

5 = a sin(nxy/b) cos lx

(3.19)

gk = U212tanh kh,

(3.20)

have a dispersion relation where

k2 = 1’

+ n2n2/b2.

(3.21)

If the currents are taken to be as weak as the water velocities in the wave motion, then it is appropriate to make a perturbation expansion with the first approximation being a simple superposition of the two (as mentioned in Section II,C,3).

Interaction of Water Waves and Currents

75

3.0 5"

'\

2.5

2 .o

0.5

-3

-2

0

-1

1

2

[u, - u,)/c,

FIG. 12. The modulus 17'1 of the transmission coefficient for waves of unit amplitude incident on a vortex sheet (solid lines) compared with the transmission coefficient for a slowly varying change of velocity (dashed lines). The angle O1 is the angle between the crests of the incident waves and the current and differs from the angle O1 used in the text. (From Evans, 1975, Fig. 1.)

75"

0.6

(u, - u, )/c, FIG. 13. The modulus of the reflection coefficient for waves of unit amplitude incident on a vortex sheet, two different approximations, solid lines and dashed lines. The angle O1 is the angle between the crests of the incident waves and the current and differs from the angle O1 used in the text. (From Evans, 1975, Fig. 2.)

76

D . H.Peregrine

IV. Currents Varying with Depth

A. INTRODUCTION There are two major causes of steady currents that vary with depth. Wind stresses at the surface and frictional stresses acting on the bottom. Viscous stresses and turbulent Reynold’s stresses transmit these to the body of the flow, setting up a mean velocity profile. One class of such flows are those where viscosity and surface tension are important. These may be described as thin-film flows and are of great importance in chemical engineering and hence have a substantial literature, both theoretical and experimental. No more than occasional reference is made to those flows here since they are outside the scope of this paper. For two-dimensional high-Reynolds-number flow in a stream with a free surface, the velocity profile is often taken to have the form U ( Z )= A Z ” ~ , (44 although measurements indicate a maximum velocity below the free surface (this may be due to surface stress from still air or to three-dimensional effects). Near a rigid bottom, z = - h, the velocity may be better represented by the “law of the wall” logarithmic profile U ( z ) = ( U * / K ) log[(z

+Wol.

(4.2) Similarly, near the free surface, z = 0, the wind-induced current may be described by

Shemdin (1972) reports measurements from a wind/wave flume and shows that they are in reasonable agreement with this formula. Wu (1975) also reports measurements from a flume that indicate a linear variation of velocity in the top few millimeters, that is, a laminar sublayer. In many cases the major part of the velocity variation is confined to boundary layers at the surface and on the bottom. Then the wind drift will only directly affect the shortest waves and only long waves, influenced by the whole velocity profile, will be affected by the bottom boundary layer. Thus a whole range of “ intermediate” length waves will be only slightly influenced by the velocity variations. Two other causes of flows varying with depth are (i) density stratification, which can lead to internal waves and to “selective withdrawal” from a stratified reservoir, and (ii) sudden increases in depth of river beds or

Interaction of Water Waves and Currents

77

artificial channels with a resulting separation of the flow such that in extreme cases it may form a surface jet. Hydraulic breakwaters also take the form of a fast surface current. In the rest of this section it is assumed that the current is in one direction only. Waves may be at an angle to the current. However, this threedimensional problem is not discussed in most examples since if a twodimensional solution is known then a corresponding solution at an angle 8 to the current is readily found, as indicated below, following Benney (1966). The x axis is chosen in the direction of wave propagation so that the basic flow is

( V ( Z )cos 8, U ( Z )sin 8, 0),

(4.4)

and the wave motion depends on x and z only. The momentum equations in the x and z directions, the continuity equation, and the boundary conditions are then exactly the same as for two-dimensional waves on the flow V ( z )cos 8 in the direction of their propagation. (That is, as long as the pressure boundary condition is not rewritten by using a form of Bernoulli’s equation.) This is because the motion is independent of y, and the velocity components in the y direction only occur in the y momentum equation, which thus becomes an equation for finding the y component of velocity once the rest of the problem is solved. This is true for finite-amplitude waves, but in that case can only apply for a single wave train or for two wave trains traveling in exactly opposite directions. B. INFINITESIMAL WAVES

1. Equations for Variation of Wave Motion with Depth Taking a basic velocity field

u = (V(z),0, 0)

(4.5)

and the inviscid equations of motion, linearized equations for a perturbation may be written. If the depth is assumed constant with the bottom of the flow at z = - h, the perturbation quantities may be taken to vary like

f (z) exp{i(kx - w t ) }

(4.6)

without loss of generality. It is then straightforward to eliminate all but one variable, giving a second-order equation for its z variation. If pressure is chosen, the equation is P”(z) - [2U’/(U- c ) ] ~ ’ ( z-) k 2 p ( z ) = 0,

(4.7)

where a prime denotes a derivative with respect to z, and c = o / k . Another

D. H . Peregrine

78

convenient variable is the vertical velocity, which gives the alternative equation W”

(

L)

- k2 + _ _ w = 0.

This equation is the “ inviscid Orr-Sommerfeld equation” or “ Rayleigh equation ” of hydrodynamic stability theory. Clearly, if the velocity profile is such that it is unstable, the unstable perturbations are possible solutions as well as solutions corresponding to periodic surface waves. Stability is discussed in Section IV,D. A full discussion of this equation, in the context of the stability of flows with rigid boundaries, is given by Drazin and Howard (1966) and a shorter account may be found in Yih (1969, Ch. 9, Sect. 6). At a rigid bottom z = -h, the boundary conditions for these equations are p’/( U

- c) = 0,

(4-9)

w = 0.

(4.10)

At the mean free surface z = 0, the linearized boundary conditions are gp’ = k2( U - c ) ~ P ,

(U - C y w ’ = [ g

+ ( U - c)U’]w.

(4.1 1) (4.12)

Equations (4.7) and (4.8) can only be solved explicitly for a few simple functions U(z),so that it is often useful to consider composite profiles. The matching conditions at a discontinuity of velocity and/or velocity gradient are that either p

and p’/(U - c)’

(4.13)

or w/(U - c) and ( U - c)w’ - U’w

(4.14)

be continuous. Occasionally continuity of w’ has been wrongly used at discontinuities of ve1ocity.gradient instead of the second of (4.14). For general wave numbers and frequencies, analytic solutions are only available for uniform currents and for currents depending linearly on z. Solutions for w are easily found and p is obtained from the relation k 2 p = U’W- ( U - c)w‘.

(4.15)

Crude approximations to most velocity profiles may be made with two or more linear regions. Although analytic dispersion relations are found, even a bilinear profile leads to complicated relations that take some effort to interpret. A number of authors have used linear and bilinear velocity profiles in

Interaction of Water Waves and Currents

79

different circumstances. For example, Taylor (1955)finds stopping velocities for a hydraulic breakwater, and Betts (1970) studies instabilities in a flume where the flow’emerges from a closed section. For stationary waves, c = 0, analytic solutions may be found for a wider range of velocity profiles. Peregrine and Smith (1975) give a short table of solutions for various jetlike flows over still water, and corresponding solutions for finite depth are possible. Lighthill (1953) gives the solution for U ( z )= Uo(h

+ z):

(4.16)

and Freds~e(1974) uses the velocity profile U ( z )= U1 cos p(z - z o )

(4.17)

very effectively to model stream flow over an obstacle. There is no intrinsic difficulty in numerical integration. Fenton (1973) gives a method that is appropriate when an analytic expression is available for U(z),but it may need modification if tabulated values of V ( z )are used since it involves v”(z). Fenton presents full dispersion diagrams showing c as a function of stream velocity for a wide range of wave numbers for the simple linear profile and the one-seventh power profile (4.1). Numerical integration is also used by Shemdin (1972). For wind waves, he takes the profile (4.3) together with the corresponding velocity profile in the air. The results for the phase velocity of short waves are in agreement with his experiments.

2. A Particular Class of Velocity Profiles One can give a.picture of the effect of different velocity profiles on waves by considering the solutions for stationary waves on flows satisfying U f f= aU,

(4.18)

where a is a constant. That is, U ( z )= U o cosh al/’z

+ Voa-‘I’ sinh al/’z,

(4.19)

where a may be positive, zero, or negative. There are three disposable constants a, U o, and Vo, so that this form can be used as a rough approximation to a variety of flows, and, although only stationary waves are considered, traveling waves can be included if c is supposed known. The solution for w ( z )for the profile (4.19),with a rigid bottom at z = - h, is W ( Z ) = sinh{(k’

+ LY)~/’(z+ h)},

(4.20)

D. H.Peregrine

80

and the surface boundary conditions give the dispersion relation

+

+

(k2 + a)’’’h coth{(k2 a)’/’h} = ( g h / U i ) ( Uoh/Uo).

(4.21)

If k2 + a is negative, this relation still holds, noting that coth ix = - i cot x.

(4.22)

The corresponding dispersion relation for a uniform flow kh coth kh = gh/Ug

(4.23)

Z = gh/Ug,

(4.24)

is included. Introducing the inverse of a Froude number squared, we consider the two relations (4.21) and (4.23) as functions of (2, k2h2).In the (2, k2h2)plane the curve (4.21)is identical to (4.23) if the latter is displaced (- Ub h / U , , -ah2), or more conveniently, if a graph of (4.23) is given, then a new origin is chosen at the point (Uoh / U o , ah2) to give the curve (4.21). Figure 14 shows Eq. (4.23) on the (Z, k2h2)plane. One advantage of this plot is that local, exponentially decaying surface disturbances are also included, in the region k2 c 0. The curve has other branches, for k2h2 < -n2, which do not appear in the diagram. Such disturbances are needed to describe fully flow near an obstacle or wave-generating object. Figure 15 is a complementary diagram showing velocity profiles corresponding to different choices of origin. The effects of velocity gradient and curvature in determining the position of the curve are clear. The origin is on the curve if V (- h) is zero, and if the velocity profile has a zero above the bottom the corresponding origin is on the right-hand side of the dispersion curve. These cases are discussed below. An advantage of these velocity profiles is that Eq. (4.8) has constant coefficients and it is thus straightforward, in principle, to use Fourier transforms to find solutions. Fredsere (1974) makes use of this. He chooses an appropriate “ truncated cosine ” profile to model the stream flow and proceeds to calculate details of the stationary waves formed by a small oblique ridge on the bottom of a wide stream. The theory agrees well with experimental results, which Fredsere presents for the variation of wave number with Froude number. Another example is calculated corresponding to a rounded bulge in the middle of a channel of finite width. The results of these two calculations are shown in various ways. When compared with a uniform flow with the same surface velocity they exhibit a reduction in surface amplitudes. However, the paths of fluid particles at the bottom of the flow show that their transverse displacement is considerably larger (e.g., twice as much) in the shear flow than it is in irrotational flow. A

Interaction of Water Waves and Currents

81

-8

FIG.14. The dispersion relation for stationary waves on a uniform stream. Z = gh/Vi

f 1,-11

f 0. -11

fl,-fl

FIG.15. The velocity profiles corresponding to the dispersion relation given by choosing the point indicated as a new origin in Fig. 14.

D. H. Peregrine

82

rough physical interpretation is given. To first order the fluid moves with the velocity appropriate to its level in the flow. Thus if its path has curvature K, a horizontal pressure gradient of magnitude pxU2 is needed. However, the perturbation pressures are unlikely to vary in the same way as V’(z), so the curvature must vary. For the cases where perturbation pressures vary slowly compared with U 2 ( z )the curvature must increase relatively rapidly to balance any marked reduction in V ( z ) .(Compare with boundary layer theory, where pressure is taken as constant through the layer.) A more mathematical way of looking at this particular phenomenon is to note that Eqs. (4.7) and (4.8) have a singular point just below the bottom, in the cases studied by Fredsere (1974). 3. A Critical Layer in the Flow

A singular point of Eqs. (4.7) and (4.8) occurs at z = z1 if V(zl) - c = 0.

(4.25)

If z = z l is in the fluid, this means that there is a critical layer at that level. The solutions in the neighborhood of a singular point, for sufficientlydifferentiable U(z),may be found by expanding V ( z )in a Taylor series in

z = z - 21. As is well known from the theory of second-order differential equations, one solution is always regular and the other may be singular. In Fredsere’s example, both solutions for p and w are regular but u and u have a singular solution. If part of the singular solution is needed to satisfy the boundary condition at the bottom, u and u could be large without there being a critical layer actually in the flow. It is instructive to look at the general solution for an oblique wave exp{i(lx + my - ot))near a critical layer. In this case IU(Zl) - 0 = 0,

(4.26) (4.27)

The results are = B[ 1

- + k 2 z 2- 2k2(v; / v ; ) zlog ~1 z 11 + A Z +~ 0(z4log I z I ), (4.28)

Interaction of Water Waves and Currents

+-1’(u;)2(31’ + m’,

83

ml, -ik2Ul/U‘;) (4.29)

where A, B are constants multiplying the regular and singular solutions, respectively, and kZ = 1’ + m’. The most striking aspect of this result is that the most singular behavior is in the perturbation velocity in a direction perpendicular to the wave number vector of the wave train. This Z - ’ term goes to zero for waves traveling against the current since its variation with 8, the angle between k and U,is sin 8 sec’ 8.This becomes very large for 8 near 4 2 , but the possibility of a critical layer then is remote because condition (4.27) becomes dificult to satisfy at a depth where the effects of the wave motion are significant. The singular solution cannot be used directly as a description of wave motion. At a critical layer other properties of the flow, neglected in the present analysis, must be introduced in order to find a physically sensible description. This aspect of critical layers is extensively studied in the theory of hydrodynamic stability. The usual method of proceeding is to include the effects of viscosity, which are used to get a solution valid in the neighborhood of the critical layer. This may be matched with an inviscid solution each side of a layer. This is relatively straightforward for unstable, growing modes of which there are usually only a finite number. The inclusion of viscosity also introduces a set of damped modes. For the water wave problem, Craik (1968) presents an analysis for resonant interactions among a triad of waves on flow with a uniform shear, and Velthuizen and van Wijngaarden (1969) consider long waves in a channel and attempt to find their rate of decay. Velthuizen and van Wijngaarden are concerned about the problem of upstream propagation against fast flows (see Section IV,B,S) so they assume a critical layer for very long waves even though there is a solution without a critical layer. However, a full discussion of solutions with critical layers should take into account how the waves may be generated, and a new proposal is presented below. For high-Reynolds-number flows it may be more appropriate to include nonlinear effects or effects due to the turbulence in the flow to find a local solution for the critical layer. No such applications have been made to this field.

D. H. Peregrine

84

For a realistic class of flows satisfying V(0)< 0,

v”(z) 2 0,

V ( z ) finite,

(4.30)

in - h 5 z I 0, Yih (1972) shows that if there is a critical layer at z = zlr then c must be real and the boundary conditions are not consistent with a solution for which w ( z l ) is nonzero. He incorrectly excludes the case w ( z l ) = 0. By multiplying Eq. (4.8) by the complex conjugate function w*(z) and integrating from - h to zl, he finds

after putting w ( z l ) = w( - h) = 0. Since conditions (4.30)imply U”/(U - c) is positive in - h I z -= zl, the only possible value for W ( Z ) in that interval is zero. There may be a discontinuity in w‘(z) at a singular point, and thus a solution regular for z 2 z1 and zero for z I zi is possible.? Direct examination of the equations of motion shows that this type of solution satisfies them, and it will be called a “surface layer solution.” Such a solution is also possible for other flows; they do not need to satisfy conditions (4.30),so that other solutions may also be possible in some cases. Where more than one solution is possible, the relevant one in any circumstance might be determined by solving an initial value problem. It seems reasonable thdtoif the waves are generated by surface disturbances or by disturbances above the critical layer then the surface layer solution is appropriate; but if the wave generation is by a disturbance extending below the critical layer, other possible solutions may be expected to be relevant. It is desirable to ascertain when the surface layer dispersion relation differs significantly from the “conventional” solution. As usual, the only case that is simple to investigate analytically is the linear profile, for example, U(Z) = U,(1

+ z/h).

” .

(4.32)

t When this result was communicated to Professor Yih, he agreed that it is possible for the differential equation (4.8) to have a solution with a discontinuous w’ at z = zl, a possibility that had simply escaped his attention. He notes, however, that for k = 0 the solution regular at z=z,is w=u-c,

and this cannot possibly satisfy the Gee-surface condition. Thus for long waves Yih’s conclusion still stands. How large k2 has to be in order to have a solution with discontinuous w’ can be decided by following the development in Yih’s paper ‘(1972, pp. 214-216) for the case U”(zl)= 0. The condition U”(z,)= 0 can now be removed since we admit a solution with a discontinuous w‘, and an estimate of k for the longest possible waves may be made using the long-wave approximation given in Section IV,B,4.

Interaction of Water Waves and Currents

85

This has the “conventional ” dispersion relation

+

k coth kh = g/( Uo - c ) ~ Uo/h(Uo - c).

(4.33)

If there is a critical layer at a depth hl, hl

= (UO- C)h/UO

1

(4.34)

and the surface layer dispersion relation is

+

k coth khl = g/(Uo - c ) ~ Uo/h(Uo - c).

(4.35)

The two dispersion relations can only differ appreciably if khl is sufficiently small for tanh kh, to be noticeably less than one, say kh, < 2. The dispersion relation (4.35) can be rewritten in this particular case as khl coth khl = 1 + gh/(Uo - c ) U o ,

(4.36)

the right-hand side of which has a minimum value 1 + gh/Ui for 0 I c IU o. For khl to be less than 2, the left-hand side of (4.36) must also be less than 2. Thus the surface layer dispersion relation differs significantly from the conventional one only if gh/Ui < 1.

(4.37)

But this condition is appropriate for the case c = O , where in fact they agree, so in practice the condition is U i %. gh,

(4.38)

which implies that the shear must be quite large. The two dispersion relations are plotted, in two different ways in Fig. 16 for the case

U i = 4gh0 .

(4.39)

These results are of academic interest only since such high-speed flows develop finite-amplitude waves as instabilities of greater practical importance (see Section IV,D). For short waves, for which the depth of the flow is not significant, it is straightforward to show that the surface layer dispersion relation is only significantly different if uo

> 30,

(4.40)

where o is the frequency of the waves relative to the surface water, that is, o - k U o . Again, this is a strong shear.

D. H . Peregrine

86

1.2

-

1.0

t

wh/U(O)

A/h

I

-

-

0

4-=+= 1

3

wh/UlO)

FIG.16. Dispersion curves for the velocity profile U ( z )= 2(gh)'"( 1 is for the surface layer solution.

+ z/h).The dashed line

4. Approximate Solutions

Approximations may be made for long waves and for short waves. For long waves, kh 6 1 and an appropriate way to write Eq. (4.7) is [p'/( U

- c)']'

= k2p/(U - c),,

(4.41)

which may be integrated twice to give the integral equation p ( z )= A

+ B j' ( U , - c)' -h

dz,

+ k2

z2

(" j-h j-h

(u1

- c ) : p ( z l ) dz, dz, ,

- c)

(4.42) in which the abbreviation

is used, and A and B are constants to be determined by boundary conditions. It is now easy to find successive approximations to p ( z ) as a power series in k.

Interaction of Water Waves and Currents

87

For flow over a rigid bottom, B = 0, and, setting A = 1 without loss of generality,

+ O(k6h6),

(4.44)

with a dispersion relation

(4.45)

This result is given by Thompson (1949), but the first approximation 0

g

j

dzl/(Ul - c)’ = 1

(4.46)

-h

is better known from Burns’ (1953) paper. The same approach can be used for problems where the flow is uniform except for a thin layer, e.g., a boundary layer at the bottom or at the free surface, or to the surface layer solution when that layer is thin. For example, consider waves on still, deep water with a thin wind-driven boundary layer of thickness h. The pressure perturbation p(z)must vary as exp(kz) below the layer, and if U is effectively zero at z = - h, matching p and p‘ with solution (4.42) leads to c Z A= kB,

(4.47)

and to the approximate dispersion relation

Not one of the dispersion relations (4.45),(4.46),and (4.48) is easy to use or interpret. Perhaps the simplest is the first approximation to stationary

88

D. H.Peregrine

waves on a surface jet. This is obtained by putting o = 0 in (4.48) and leads to the result (4.49)

given by Peregrine and Smith (1975). The integral is proportional to the momentum flow in the surface jet. A thin sheet of momentum flow at the surface of a fluid acts rather like a negative surface tension. Compare Eq. (4.49) with g

+ ( T k * / p ) = kc’,

(4.50)

the deep-water dispersion relation, when c = 0 (but also see “surface shear waves” in Section IV,C). At the other extreme, when waves are short compared with the current variations, wave properties are determined by the flow close to the surface. One can either use a WKB approximation for W ( Z ) as Dalrymple (1973, Appendix 1) does, or expand systematically in inverse powers of ko = duo - c)’,

(4.51)

as is done by Peregrine and Smith [1975, Eq. (48)]. The first three terms of the dispersion relation are (4.52)

where a zero subscript indicates that the function is evaluated at the surface. Further approximations involve higher derivatives of U ( z ) that would be difficult to evaluate from measurements of a real flow. Some idea of the accuracy of these approximations may be obtained from Fig. 17 which shows the dispersion relation for stationary waves on a deep flow of the form U ( z )= Uoea‘,

with k plotted against U o, using appropriate dimensionless variables. A few bounds for c are available. Thompson (1949) proves Umin- (gh)”’ Ic I U,,,

+ (gh)”’,

(4.53)

and that when U is a monotonic and nondecreasing function of height above the bed, c I U,,,

+ (g/k)’’’.

(4.54)

Interaction of Water Waves and Currents

1

-

2

89

3

s/u:

FIG. 17. Dispersion relation for two-dimensional stationary waves on the velocity profile U ( z )= LI, exp az, together with approximations. S,, S,, and S, are successive short-wave approximationsand L,is the first long-wave approximation. (From Peregrine and Smith, 1975, Fig. 4.)

With similar conditions,

U' 2 0 and finite,

U 5 0,

(4.55)

Yih (1972) extends a result of Burns (1953) to prove that there is one solution with c IU ( - h), (4.56) and another with c 2 U(0). (4.57) It is worth noting that there is no simple equivalent of the linear longwave equations for irrotational flow:

au ay -++-=o, at ax

ay

au ax

-+hh-=OO. at

There is only the result (4.46) for the long-wave velocity.

(4.58)

90

D . H . Peregrine

5. Upstream Propagation Conditions (4.55)are such that many profiles V ( z )that may be chosen to represent stream flow would satisfy them. If U ( - h) is zero, condition (4.56) indicates that that solution corresponds to upstream propagation of waves, regardless of how large the surface or mean velocities may be. Benjamin (1962,p. 108)suggests that the solution (4.56)may not be physically realizable when the mean velocity is much greater than (gh)’/’. He argues that the relatively high velocities near the bed due to the wave in such a solution severely limit the amplitude of the wave if separation of the boundary layer is not to occur. Yih (1972)discusses this further, confirming the high perturbation velocities near the bed, and arguing against a conjecture af Benjamin’s that the maximum velocity of propagation upstream should be of the order -(gh)’/’ + 0. There are several facets to this problem. There is no doubt of the existence of the mathematical solution corresponding to (4.56)with conditions (4.55) for any value of 0. The conditions (4.55)certainly apply to a real flow if the Reynolds number is low enough for it to be laminar. However, in that case it is less realistic to omit viscosity in the analysis. If viscosity is included these waves do not occur, as is shown by a stability analysis [see Benjamin (1957) or Yih (1969,Sect. 9.9)for further results]. For a turbulent high-Reynolds-number flow, many model profiles U ( z ) would give either a nonzero velocity at the bottom or an infinite velocity gradient U’(- h) as in the frequently used one-seventh power profile (4.1).In the latter case, Lighthill (1953)shows there is no upstream propagation for

e

U > 1.0353(gh)’’’.

(4.59)

There is also the problem of generation and detection of such waves. For example, using the linear velocity profile (4.34)a rough calculation shows that if a low-frequency oscillating surface pressure is applied over an appropriate length of the surface, the two long-wave modes are generated with amplitudes inversely proportional to their phase velocities. This means that for a high-Froude-number flow, the controversial upstream propagating mode would have a substantially smaller amplitude than the waves propagating downstream. Its group velocity upstream would also be small, so that it would suffer appreciable damping, by neglected effects, before it got clear of the generating area. In summary, these particular upstream propagating waves appear to be a mathematical solution with little physical relevance. There is an exception, when the flow separates from the bed. This is discussed further in Sections IV,C and D.

Interaction of Water Waves and Currents

91

6. Group Velocity

In many applications, the most important wave parameter is the group velocity; for example, it is needed to find the stopping velocity in a hydraulic breakwater. Once again, the only simple case is the linear velocity profile U(Z) = u,

+ ZU, .

(4.60)

The group velocity cu is given by

in which B(kh) is the ratio cU/c for waves of the same wave number in still water of depth h. The denominator of the right-hand side of (4.61) cannot be less than f since the maximum value of (c - U,)Ub/g is 1, which it attains at k = 0. From Eq. (4.61) one may see that (c,, - U,)/(c - U , ) behaves rather like B(kh) but “skewed” in the direction one would expect from the underlying shear. Inspection of Fig. 16 shows that the surface layer solution has similar properties, except at c = 0, a point that may merit further attention. For most applications, numerical solutions need to be found; even when analytic solutions are found it may be more convenient to determine c, graphically or numerically, e.g., Taylor (1955) finds the stopping velocities of a sectionally linear profile graphically.

C. FINITE-AMPLITUDE WAVES One familiar method of finding finite-amplitude wave solutions becomes relatively inappropriate when the basic flow varies with depth. This is the method of expanding the free-surface boundary condition in a Taylor series about the mean level. If this approach is adopted, the mean flow V(z) must also be expanded in a Taylor series. While this may be sensible for a flow chosen for its mathematical convenience, such as a linear profile with constant vorticity, it is quite inappropriate if actual velocity measurements are used, since even second derivatives may be quite uncertain. A number of transformations of the equations of motion for steady flow enable this problem to be avoided, at least for steady periodic waves. A few transformations are now given, followed by some solutions.

D. H. Peregrim

92

1. Transformations of the Equations

For steady flow in two dimensions, the introduction of a stream function $ leads to the expression -V2$ for vorticity and to the equation

v’* =f (*)

(4.62)

to express the fact that for inviscid flows vorticity is constant along streamlines (Batchelor, 1967, Sect. 7.4). Transformations of coordinates that effectively replace z with II/ thus have two desirable properties. The free surface becomes a fixed boundary, I) = const, and the basic distribution of vorticity is explicitly stated. The most obvious transformation is the direct one, (4.63)

u(x, z ) = u(x, *),

a von Mises transformation. The continuity equation and Eq. (4.62)become

au + w a u - u-a w = 0, a+ a* aw aw a u -+ w-+u-=f(*), ax a* a* -

ax

(4.64)

-

(4.65)

respectively. These are two equations for the two components of the total velocity in a reference frame moving with the wave. Gouyon (1958) and Moiseev (1960) have used these equations for existence proofs in the case where the variation in the basic flow is small compared with the wave velocity, so that to a first approximation they are additive. A less direct approach is to use the height z of a streamline as an independent variable, that is, z = z(x, *). (4.66) The total velocity components are then u = l/z$

and

w = z,/z*,

(4.67)

where subscripts are used to denote partial derivatives. The vorticity equation (4.62) becomes z,,z$ - 224ZZ,Z$+ Z$$(l

+ z:)

= Z;f(*).

(4.68)

Dubreil-Jacotin (1934) uses this equation for an existence proof. Dalrymple (1973) gives a finite-difference approximation to Eq. (4.68) and presents sample results of finite-amplitude waves on a linear shear and on a oneseventh power profile. Benjamin (1962) takes this approach a step further in his derivation of a solitary-wave solution. He introduces a new height variable s, equal to the

Interaction of Water Waves and Currents

93

value of z in the undisturbed flow. Thus the undisturbed flow is given by

*

=

w,

(4.69)

where

U ( S )- c = dY/ds.

(4.70)

The vorticity equation now becomes

22,,2,2,

+ Z,,(l + zf)} + vl(s){zj - z,( 1 +

Zf)}

= 0.

(4.71)

An advantage of this equation is that U ( s )appears explicitly. Note that it is a nontrivial matter to find f ($) to substitute in Eqs. (4.62), (4.65), and (4.68) for most velocity profiles, since it is given by

f(*) = yT”(s),

*

=

W).

(4.72)

2. Solutions The earliest finite-amplitude wave solution is Gerstner’s (1802, in Lamb, 1932, Sect. 251) and it has a vorticity distribution. As Lamb shows, following Stokes, the uniform flow corresponding to that vorticity distribution is

U ( z )= -ce2kb,

(4.73)

where k(z - z,) = kb - ie2kb,

c2 = g / k ,

kz, = 4a2k2 - In ak.

The flow is in the opposite direction to the waves’ propagation and for the highest wave the vorticity is singular at the free surface. This solution is unlikely to be relevant to waves on real flows. As might be expected from the difficulty of finding solutions for infinitesimal waves for most velocity profiles, the only analytic solution corresponding to the Stokes wave for irrotational flow is for flow with uniform vorticity. Tsao (1959) gives a third-order approximation for arbitrary depth. The algebraic complexity of the solution is somewhat daunting, despite the fact that for uniform vorticity it is possible to introduce a velocity potential for the wave motion. For the linear profile U = bz,

(4.74)

it is relatively simple to show that for deep-water waves, the wave motion is given by a velocity potential acekz sin k(x - c t ) + $a2b(3+ S)e2k’ sin 2k(x - ct) + O(a3k3),

(4.75)

D. H. Peregrine

94

with a surface elevation [ = u cos k ( -~ C t )

+ $i2k(l + 2 s + is’) cos 2k(x - ct) + O(a2), (4.76)

in which kc2 + bc - = 0

(4.77)

S = b/ck = b/a.

(4.78)

The constant in Bernoulli’s equation is increased by

+

C2 = fa2b(ck tb),

(4.79)

and this may be interpreted as either a change of level of the free surface by C2/(g - be) or an additional uniform velocity - C2/c. Tsao (1959), Eq. (2.19), is not in agreement with the result (4.79). From the wave elevation (4.76) it is easily seen that the usual asymmetry between crest and trough increases as the shear increases for waves traveling in the + x direction. That is, if the maximum current is at the surface, waves traveling in that direction have sharper crests than corresponding irrotational waves. Conversely, waves traveling in the opposite direction are more nearly symmetrical about the mean level. The wave profile is sinusoidal to second order when

s=-2+Jz.

(4.80)

s 2 -1

(4.81)

We may see that by rewriting the dispersion relation (4.77) in terms of S, c2(1

+ S) = g/k.

(4.82)

However, for c negative a surface layer solution should be found, which may modify the results. Numerical techniques to solve the finite-amplitude problem for steady waves have been developed by Dalrymple (1973, 1974). He extends Dean’s stream function method (Dean, 1965), which essentially is a double Fourier expansion of the stream function. A considerable number of results are presented for waves of flows with linear velocity profiles. The results include large-amplitude waves. Dalrymple (1973) also presents some results from a finite-difference approximation to Eq. (4.68) for other velocity profiles. These methods appear to be an effective approach to solving specific problems, and by comparison with irrotational waves, the effects of vorticity may be better understood.

Interaction of Water Waves and Currents

95

Solitary-wave solutions may be found for any velocity profile without a critical layer. The solution is given by Benjamin (1962) and has been derived by several other authors since then. The corresponding Korteweg de Vries equation is given by Benney (1966) [which appears to have an error in Eq. (54)] and by Freeman and Johnson (1970). The equation is (4.83) where (4.84) (4.85) and co is the linear long-wave velocity given by g l , = 1.

(4.86)

Note that the I, are negative for n odd. The solitary-wave solution is = a sech’ P(x

- c1 t),

(4.87)

in which c1 = co - aI,g/213,

(4.88)

8’

(4.89)

= a14g/4J.

These reduce to the usual irrotational results for water at rest far from the wave, c1 = (gh)”’( 1

+ $a/h),

P’

(4.90) Benjamin (1962) discusses the results for waves propagating on a stream and deduces that the effect of the vorticity is small unless the Froude number of the flow is near one. In that case the waves most commonly met are stationary waves. Strictly in such a case there is a critical layer at the bottom of the flow, but if the one-seventh power velocity distribution (4.1) is used to model the flow the analysis is not affected since all the integrals converge. A comparison, of stationary waves on such a flow, with the corresponding irrotational flow, is given in Fig. 18. The example calculated corresponds to the waves generated by a small obstacle, stationary in a stream with Froude number

F = U(gh)-”’.

= 3a/4h2.

(4.91)

96

D. H. Peregrine

FIG. 18. Stationary waves on a stream caused by a reduction pU2hD2in the momentum flow.(a) Uniform stream of velocity 0.(b) Stream with velocity profile. (8/7)0(z + h)'"h-"'.

It is supposed that the momentum flow (4.92)

is reduced by a small amount p02hD2,

(4.93)

Interaction of Water Waves and Currents

97

which corresponds to the force on the obstacle. The energy of the flow is assumed to be unchanged. [See Benjamin and Lighthill (1954) for a full discussion in the case of uniform flows, and Fenton (1973) for an application to a linear profile.] The right-hand boundary in the figures dorresponds to the solitary-wave solution. At first sight the slope of this curve is surprising, since the larger the wave the faster it travels. However, one result of the loss of momentum flow is to reduce the mean level of the stream, so there is no inconsistency in a larger wave being on a slower stream. The most noticeable difference between Figs. 18a and b is the increase with vorticity of the area of the (a,F 2 ) plane in which waves may occur. Essentially this is due to the increased speed of the solitary wave, which is noted by Benjamin (1962). Other differences are quantitative rather than qualitative. The biggest of these is an increase in wavelength for the stream with vorticity when F 2 c 0.85, but the theory is less appropriate there. For finite-amplitude shallow-water waves, the position is similar to that for linear long waves. Benney (1974) shows that there is no pair of “simple shallow water equations to characterize long waves in a general flow.” On the other hand, if attention is focused on waves propagating in one direction only, some progress has been made by Blythe et al. (1972). By looking for an equation of the form

( a r p t ) + c aypx = 0,

(4.94)

r,

where c is a function of equations leading to a simple-wave solution are found for flows without a critical layer. An example is given for the case where the flow has uniform vorticity, but no guidance is given for other less simple flows. A rather specialized class of finite-amplitude waves has been described by Peregrine (1974) and named “surface shear waves.” The basic flow configuration is a sheet of rapidly moving water traveling over water at rest or nearly so. Stationary waves may form on such a flow. Below weirs or sluices they may have an amplitude much greater than the initial thickness of the surface sheet. The wide range of conditions in which this form of wave occurs is shown by Moore and Morgan (1959), who call it a “wave hydraulic jump.” A very simple theory is possible when the Froude number

(Mhh2)”*

(4.95)

is large. Here h is the thickness of the jet and M its momentum flow,

5

0

p U 2 ( z ) dz. -h

(4.96)

98

D. H. Peregrine

To a first approximation the surface flow is deflected only by the pressure difference across it. That is the difference between atmospheric pressure and the approximately hydrostatic pressure in the slow-moving water beneath. The appropriate equation for the surface elevation is Mlc = pgr,

(4.97)

where IC is the curvature of the surface. This is also the “elastica” equation for the bending of a thin sheet of elastic material. Thus the shape of the waves is easily reproduced by bending a sheet of paper. The steepness of the waves is not limited in theory, and in practice it is easy to produce them with slopes greater than 30”, the maximum for Stokes waves. The crests and troughs are rounded and symmetrical. Peregrine (1974) also draws attention to superficially similar waves that occur on beaches in the backwash from surf, described in more detail in the next section. If their structure is also similar, then they are formed by the high-velocity backwash separating from the beach and riding up over a separation bubble. This hypothesis is supported by experiments with the wave hydraulic jump, where by raising backwater levels, it can be formed by separation of flow from the plane spillway of a weir. 3. Highest Waves For irrotational waves, Stokes (Lamb, 1932, Sect. 250) showed that the highest waves in steady motion have a 120” corner at their crest. The same method of local analysis can be applied to waves on a rotational flow, and Miche (1944, pp. 386-406) shows that the result is unchanged. Miche proceeds further and shows that vorticity affects the curvature each side of the crest. In particular, vorticity b gives a free surface

!(y2

6 = & !!+ 3 3 9

+ ...,

(4.98)

where 6 is measured from the downward vertical. Grant (1973) shows that for irrotational waves the next term in an expansion like (4.98) has a (probably) transcendental power of r approximately r 1.2. Delachenal(l973) also considers this problem but assumes that the vorticity has the form Ar- l‘zf (e),

(4.99)

which is singular at the crest. Thus his solution is in the same unrealistic class as Gerstner’s highest wave. A different aspect of highest waves is the maximum amplitude that may be attained. If the phase velocity of the wave in question is known, then the use

Interaction of Water Waves and Currents

99

of Bernoulli's theorem in a frame of reference moving with the wave gives the maximum amplitude corresponding to a stagnation point at the crest. Banner and Phillips (1974) draw attention to the effect a surface drift due to wind has in this context. If the surface drift velocity has magnitude q, then the maximum amplitude of a wave of phase velocity c relative to deep water in the same direction as q is (4.100) Note that for an irrotational wave of given wavelength c only varies by 20 % as the amplitude increases to its maximum. It is unlikely that a thin shear layer can cause a much larger variation. Thus if q/c is near 1, the maximum wave can be expected to have quite a small amplitude, and any small amplitude approximation is likely to be very limited in its applicability. As Banner and Phillips point out, this is likely to be the case for the shorter waves in a wind-driven sea. Phillips and Banner (1974) investigate the effect of large waves on a surface drift layer. The water motion in the large wave causes the surface drift to vary, with its maximum velocity at the crests of the long wave. As discussed in Section II,F, short waves become shorter near long-wave crests, and thus their phase velocity relative to the water below the surface layer decreases, just when the velocity of that layer increases. When q = c they cannot propagate, and however small their amplitude, it seems that they must break. Phillips and Banner estimate these effects using linear wave theory. If the long wave has appreciable steepness, then the amplification of the surface drift at the crest of the wave is substantial. Thus the interaction of long surface waves with a surface drift layer may have appreciable effects in suppressing shorter waves. The proportionate reduction in wave energy is estimated for the shorter waves and is found to be in surprisingly good agreement with experimental measurements. For irrotational waves, Longuet-Higginsand Fenton (1974) and LonguetHiggins (1975) show that the highest wave is not the wave with most mass, momentum, or energy. This is very relevant to both wave breaking and to actually producing a highest wave. Presumably, similar results are likely to hold for rotational waves.

D. STABILITY Except for flows in thin films, which are not considered in this work, the currents of interest are turbulent flows, so discussion of stability may seem inappropriate. This is not so. If an instability transfers energy to surface

100

D . H . Peregrine

wave motion, then it is of direct interest in the present context. Also it has been suggested, with respect to other turbulent flows, that “instabilities of the mean velocity may determine the large-scale structure of the flow (Landahl, 1967). The stability of inviscid flows (U(z),0, 0) over a rigid bottom to infinitesimal disturbances is considered by Yih (1972). He extends several theorems for flow between two rigid walls to this case and shows that the requirements for stability are very similar in that unstable modes are associated with inflection points in the velocity profile. Silcock (1975) examines the surface jet flows ”

U = sech z

and

U = exp( - j z 2 )

(4.101)

in detail and computes the growth rates of infinitesimal disturbances. The stationary-wave solutions form part of the stability boundary, but for small Froude numbers (based on the jet “thickness”) the growth rates of the associated instabilities are very small. There are also instabilities that have little effect on the surface. Unlike many topics in this paper, there are some experimental results. Sarpkaya (1957) reports on a substantial experimental work in which waves were propagated against a stream flowing under gravity. Measurements were made of wave phase velocity, amplitude, wavelength, and shape, for those particular waves that propagated unchanged in amplitude. Higher and shorter waves were amplified, and smaller and longer waves were damped. Figures 19 and 20 are taken from Sarpkaya (1957) and summarize some of the results. It may be noted that only waves of finite amplitude are amplified. It is odd that a set of neutral waves, varying in frequency, was not found for each flow.

0

1.o

0.5

1.5

value of R X ~ O - ~

FIG. 19. Stability boundaries of stream flows of diNerent Froude numbers with waves propagating upstream. Amplitude/wavelength is plotted against Reynolds number. (From Sarpkaya, 1957, Fig. 2, p. 575.)

Interaction of Water Waves and Currents 0.4(

I

I

I

101

I

F e 0.10 0.15 B 0.20 e 0.25 e 0.30 o 0.35

-

0.30 G \

10

1 c

0

-0.15

E

p0.2

0.20

L

-%

5

0,

e -(

0.10

I

I

I

!5

0.35 1.5

v m of ~

~ 1 0 - 5

FIG.20. Stability boundaries of stream flows of different Froude numbers with waves propagating upstream. Depth/wavelength is plotted against Reynolds number. (From Sarpkaya, 1957, Fig. 3, p. 575.)

It is of particular interest that there is no critical layer for any of the waves measured in these experiments; thus an explanation may need to include the interaction of the water waves and the turbulence in the flow, or interaction with the boundary layer on the bed of the channel. The latter seems most likely. By using the results shown and linear irrotational theory it is possible to work out u( - h)/O (that is, the particle velocity due to the waves at the bed divided by the mean velocity). For a high proportion of the experimental results, this ratio lies in the range 0.6-0.7with no systematic variation apparent. This may we11 be sufficient to cause separation or substantial thickening of the bottom boundary layer. A more marked instability occurs in uniform streams at high Froude numbers. Large-amplitude waves form and develop bores at their fronts. These progress downstream with variable frequency and amplitude. They are called roll waves. This instability may be demonstrated theoretically (Jeffreys, 1925) by using the linear long-wave equations for irrotational flow and adding a Chezy friction term, that is, a quadratic resistance term that also varies inversely with depth of water. Dressler (1949) gives more details of solutions and Dressler and Pohle (1953) consider more general friction

D. H . Peregrine

102

laws. Experimental measurements of the development of roll waves are given by Brock (1969), and numerical examples are calculated and discussed by Jolly and Yevjevich (1974). Friction laws for streams are one way of representing the turbulence that also gives rise to the mean velocity profile. Laminar flow down a plane is also unstable if the Reynolds number is greater than

2 cot p,

(4.102)

where is the inclination of the plane to the horizontal, and Benjamin (1957) points out the analogy with roll waves. More details may be found in Yih (1969, Ch. 9, Sect. 9) and experimental results in Benjamin (1961). Corresponding calculations for turbulent stream flow would require an eddy viscosity or other hypothesis to represent the turbulent Reynolds stresses. Another type of instability gives rise to the surface shear wave in backwash on a beach, mentioned at the end of Section IV,C,3 (Peregrine, 1974). When the backwash is not affected by a following wave it usually forms a nearly stationary turbulent bore where it meets the still water. After a time a long smooth wave may emerge in front of the bore and travel upstream, gaining height, usually until it dwarfs the bore it sprang from. If a hypothesis of a separation of the flow from the bed is correct, the wave may start in the following way. According to linear theory, a small disturbance, exponentially decaying upstream, may precede the bore. If this disturbance is sufficient to cause flow separation on the bed, the wave may appear. It will grow by entrainment of water into the separation “bubble.” It seems possible that such an instability may only occur for certain velocity profiles, e.g., flow that starts from rest on a slope, or may depend on the Reynolds number of the flow. E. WAVESON FLOWIN

CHANNELS

This pertains to waves on a flow

u = ( q y , z), 0, O),

(4.103)

confined in a channel. Peters (1966) treats the case of long waves, deriving the equation

jjdY d z / ( W , z ) -

C l 2 = h/g

(4.104)

S

for the long-wave velocity, in which S is the cross-sectional area of the channel and b its surface breadth.

Interaction of Water Waves and Currents

103

The linearized equation corresponding to Eq. (4.7) is (4.105)

with

dppn =o

(4.106)

on the walls of the channel and (4.107)

on the mean free surface. No solutions have been found with any y variation. Peters (1966) also finds the equation for a solitary-wave solution, but a subsidiary partial differential equation, similar to (4.107), must be solved in S . However long the waves are, such solutions are only likely to be of value for channels that do not have a large aspect ratio (width/depth). In the irrotational case, Peregrine (1968) shows that a second long-wave approximation (which is needed for the solitary-wave solution) has a term that increases with the square of the aspect ratio for nonrectangular channels. V. Turbulence

In considering the interaction of waves and turbulence, a useful way to develop ideas is to take a simple view of the turbulence. That is, characterize the turbulence by a length scale and a typical maximum fluctuation of velocity. Although turbulence has important small-scale properties, it seems likely that interactions are dominated by the most prominent turbulent motions, and the ratio of their lengths and velocities to those of the waves. Perhaps the simplest case to understand, though a difficult one to analyze, is when the turbulence has a scale much greater than the waves and velocities comparable with the wave group velocity. The waves are then refracted in accord with the equations derived in Section I1,C. However, consideration of the solutions in Sections II,D and E shows that unless the waves are reflected, by refraction, out of the region of turbulence, they are likely eventually to encounter currents causing their wavelength to diminish considerably so that a high proportion of their energy is lost by breaking. Thus large-scale turbulence acts as a wave absorber. Such behavior is easy to see on rivers, where wind waves get little chance to grow if the turbulence is strong enough. Usually this seems to coincide with a level of turbulence, which noticeably deforms the free surface so that its dominant features are small ripples and dips above vortex cores.

104

D . H.Peregrine

Much stronger turbulence leads to relatively violent surface motions that can generate some propagating waves, or in extremes as in turbulent hydraulic jumps, it leads to the surface breaking up into drops and irregular masses of water. The whole range of behavior may be observed, on a relatively small scale, in the boundary layer of ships. An example that is commonly observed is the effect of a ship’s wake on short wind waves that are incident upon it. The turbulence is relatively strong and often of larger scale and thus acts to absorb or reflect the waves by refracting and steepening them. However, the mean motions associated with a wake are also likely to be important. There is the flow along the wake in the direction of motion of the generating vessel and also the transverse motions due to trailing vortices (from bilges or propellors or both). That these latter may be dominant is indicated by the relatively stronger effect that a curved wake has on waves. When the turbulent velocity fluctuations of large-scale turbulence are weak compared with the wave velocity, one may think in terms of waves being scattered by the turbulence. Phillips (1959) attempts to analyze this scattering for very weak turbulence, using a Fourier decomposition of the velocity field. It is difficult to make use of such an approach since the components of a spatial Fourier decomposition of turbulence are virtually unknown except for very special cases. Phillips uses estimates based on the inertial subrange of turbulence, but the author does not think this is likely to be an important part of the interaction. It seems somewhat more likely that Howe’s (1973) method for treating scattering may be applicable. Turning to the other extreme, we have small-scale turbulence. This may actually be the same turbulent flow but viewed with respect to different, longer waves. Small-scale turbulence is more amenable to study in laboratory experiments and to the extension of empirical methods established in other fields. For example, the “ friction laws ” established for steady flows in channels are often extended to unsteady flows such as long waves (see the case of roll waves mentioned in Section IV,D). On the other hand, Sarpkaya’s (1957) experiments (also mentioned in Section IV,D) show that some waves on a turbulent flow are amplified. Taking another viewpoint, the rate of strain tensor for a plane irrotational wave train is eij = a24/axi axj .

(5.1) This is oscillatory, but as Phillips (1959) points out, the Stokes’drift may be more important. More precisely, the finite-strain tensor has a component of second order in ak increasing linearly with time. It acts to stretch vortex lines and thus may lead to stronger interactions than the oscillatory part. Experiments by Green et al. (1972), described below, appear to give some support

Interaction of Water Waves and Currents

105

to this idea. If it is important, then the common eddy viscosity hypothesis may be of limited value. A few experiments have been performed to measure the scattering and dissipation of water waves by turbulence. In none of these experiments has scattering attributable to the turbulence been detected. The most interesting, from the viewpoint of the rest of this paper, are two experiments Savitsky (1970) reports. In both experiments turbulence was generated by towing a grid in a ship model testing tank. Waves were sent along the tank to overtake the grid and its turbulence. In the first experiment the grid spanned a 12-ft (3.7 m) tank. In the second experiment, the grid was only 3 ft (0.9 m) wide in a 75-ft (23 m) tank. Savitsky was unable to detect any scattering or dissipation by the turbulence, since in both cases the effects of the mean currents set up by the moving grid dominated the wave behavior. In the first experiment, there was a velocity defect at each side of the tank and the waves became unsteady with curving crests. In the second experiment the waketype flow refracted and diffracted the waves. The maximum mean velocities in these experiments were more than 10%of the group velocity of the waves. A more successful experiment is reported by Green et al. (1972). In a laboratory tank, turbulence was generated by a grid oscillating vertically. A false bottom was inserted, for all but the longest waves, to protect the surface layer of water from mean currents. The turbulent eddies had a scale of around 1 cm and the shortest waves had a wavelength of about 5 cm. Damping of the waves was observed. Measurements were made on various waves before and after propagating through the turbulent region, the turbulence being approximately the same in all cases. The results are presented in two different ways, one assuming an exponential decay with distance, the other assuming a quadratic decay law daldx = - ya2 (5.2) for the turbulent damping. The coefficients obtained from the measurements show appreciable scatter, but assumption (5.2) gives the smaller scatter. The coefficient y is found to depend on frequency in such a way that y = Cb5,

(5.3)

where C is a dimensional constant. Green et al. (1972) note Phillips’ (1959) suggestion that the most effectiveinteraction with the turbulence may be of second order in ak. Also, one may note that the time that any wave group spends in the turbulent region is inversely proportional to its group velocity. For this purpose it is adequate to assume the linear deep-water gravity wave dispersion relation, in which case Eqs. (5.2) and (5.3) may be rewritten daldx = - Cg3a2k2/2c,, (5.4) which supports the hypothesis that second-order effects are relevant.

106

D. H . Peregrine

A rather different experiment is reported by Green and Kang (1976). In this case, a single long wave, the fundamental resonant mode of a wave tank, was allowed to decay in the presence of different intensities of turbulence. The turbulence was due to thermal convection generated by heating the bottom of the tank. The intensity of this turbulence depends on the Rayleigh number. A major problem in interpreting the results of this experiment is that less than 20% of the observed damping could be ascribed to turbulence. However, after careful analysis and estimation of the other dissipative effects, Green and Kang present results for the turbulent damping that show an appreciable dependence on Rayleigh number. They also provide an interpretation of the interaction. Convective turbulence takes the form of intermittent thermals arising from the bottom boundary layer. When there is a horizontal flow, such as that due to long waves, a rising thermal carries relatively stationary fluid from the bottom boundary layer into the main moving mass of fluid. This can be interpreted as giving a Reynolds stress approximately equal to npuw, where n is the fraction of the horizontal area over which thermals occur at any instant, u the horizontal velocity of the main mass of fluid, and w a typical vertical velocity in a thermal. Green and Kang’s (1976) results are consistent with the estimate of this Reynolds stress that they make.

VI. Ship Waves A major aim in the study of ship hydrodynamics is the prediction of the total hydrodynamic resistance of a ship. This is a difficult and complex problem, so that another more practical topic is also studied: how to relate measurements on ship models to the behavior of the prototype. The traditional approach has been to divide the resistance into two parts, “viscous resistance” and “wave resistance,” and to scale the first according to the Reynolds numbers of the ship and its model and scale the second with their Froude numbers. A similar approach is used for studying the primary problem of predicting ship resistance theoretically. Independent methods of measuring the viscous and wave components of resistance, by measuring the water velocities and wave amplitudes behind ship models, have shown that this simple view is inadequate. Explanation of the actual resistance involves consideration of the interaction between waves generated by a ship and the flow around and in the wake of the ship. Important papers illustrating this point are those by Lackenby (1965) and Shearer and Cross (1965). The wave resistance of ships is the subject of a recent substantial survey by Wehausen (1973), which gives more details on many of the topics mentioned here.

Interaction of Water Waves and Currents

107

There are different ways of looking at this subject. One can look at physical quantities and interpret them directly, or, from a theoretical viewpoint, start with some approximation and interpret higher-order terms as interactions. For example, a direct physical approach to a finite-amplitude water wave does not lead to the concepts of linear and nonlinear waves that arises from mathematical approximations. However, interpreting and estimating the physical quantities depends on adequate theoretical backing. A “physical” analysis of ship resistance is illustrated in Fig. 21, which is an adaptation and extension of a diagram in Brard (1972, Fig. 2). Some of the subdivisions in the diagram are not easy to define, especially with respect to form drag, but it does help to understand the exchanges between the viscous and wave components. Two different ways these may be defined are indicated by the dotted lines. However, in neither case can the viscous component be expected to be entirely independent of the Froude number or the wave component independent of the Reynolds number. The direct viscous drag is approximately dependent on the wetted surface area, but this depends both on the shape of the water surface around the ship and on the trim of the ship. Both of these depend on the waves generated by the ship in its own vicinity. In the same way the form drag, which may be largely due to regions of flow separation and to trailing vortices, depends on the same two “ wave ” variables, waterline and trim. When waves break, as is very often the case near the bows of ships, momentum is transferred from the wave motion into the water. Even for steady ship motion, breaking can be an unsteady process in which case there can be momentum transfer into other wave components. For full-size ships, breaking is probably the most important form of wave dissipation, but there is also some due to turbulence in the boundary layer and wake and due to viscosity. At the lower Reynolds number of ship models, viscosity may be more important and surface tension effects near wave crests can also be relevant. The current field associated with flow around a ship may also be considered a wave generator. It is propagating into still water at the same speed as the ship. This notion is given approximate quantitative form in the papers by Beck (1971), Brard (1972), and Tatinclaux (1970). There is some imprecision here since this effect could also be termed a wave-current interaction. Most wave-current interactions involve a transfer of momentum between the two components. The mean flow and waves are steady in a frame of reference moving with the ship, and as the work in Section I1 indicates, in a steady situation wave action flux is conserved in those cases where it can be defined; but conservation of wave action does not normally imply conservation of momentum in one part of the system. The waves generated by the ship interact with the flow around it, the approximately inviscid flow, as well as the boundary layer and wake.

D. H.Peregrine

108

G Total resistance

I/ 1

..

J

of s t r e s s over ship's surface

tangential s t r e s s

Direct viscous drag

/normal

Form drag

stress

Direct wave drag

Momentum flow through a control

a Total resistance

FIG.21. An analysis of the hydrodynamic resistance to a ship's motion. The quantities within ovals are often measured for ship models. The dotted lines indicate two different divisions into viscous and wave resistance.

Interaction of Water Waves and Currents

109

While the mathematical problem to be solved in predicting ship resistance can be stated, its solution requires a number of substantial approximations and simplifications. The usual approximations are to assume inviscid irrotational flow, a linearized free-surface boundary condition, and some geometrical ratio, such as breadth/length, to be small. Such approximations may be thought of as a first term in a perturbation expansion, but with so many parameters in the problem it needs careful analysis when looking for higher approximations to ensure a consistent approach. This is discussed in Wehamen (1973), so only the current-wave interactions are mentioned here. There are two particularly important aspects in the mathematical problem of the interaction of the waves generated by the ship and the flow around it. One is the interaction with the potential flow, and the other is the interaction with the boundary layer and wake. If the inviscid problem were solved without any approximation involving the speed and shape of the ship, then the first-named interaction would be automatically satisfied. However, this is not usually the case. (Note that unless the ship is very slender or deeply submerged, linearizing the freesurface boundary condition is a poor approximation in the vicinity of the ship.) The ship is usually taken to be “slender ” or “thin,” or else the Froude number is supposed to be small or large. The thin-ship approximation does take some wave interaction into account at first approximation; the sinkage and trim of the vessel may be calculated. However, the flow around (or perhaps one should say along) the ship and the wave motion are both small, so that interactions come in at the second approximation. On the other hand, if the ship is assumed to have finite bulk and a low Froude number, the first approximation has no waves, so the second approximation necessarily includes solving the wave pattern on the flow field of the first approximation. Dagan (1972) gives an interesting account of this last type of problem, using as a simple example a two-dimensional submerged body. Submerged bodies introduce some further considerations (e.g., see also Farell and Guven, 1973)but two-dimensional potential flow is much simpler than flow in three dimensions. The interaction of waves with the turbulent flow in the boundary layer and wake is also a difficult problem, even when only mean flows are considered. The approach of representing both the boundary layer and wake by a corresponding displacement thickness, that is, taking a semiinfinite body that is an appropriate amount larger than the ship, has been tried several times with only a relatively small improvement in the results. However, other approximations made at the same time may be more important. Another approach is to assume that the flow is inviscid but has a vorticity distribution that is chosen to model the actual flow. This seems better than introducing a simple eddy viscosity since the effect of the eddy viscosity on

110

D. H.Peregrine

the waves may not be representative of the effect of the effect of the turbulence (see Section V). The simplest example is to consider a wave-making source at the center of a “wake ” that is uniform in the direction of motion of the source. Peregrine (1971) uses this model and simplifies the analysis by supposing the waves are so short that the “ray theory” of Section I1 applies. The resulting wave pattern differs from that for a point source in motion through water at rest. The envelope of wave cusps is inside the Kelvin angle of 1%”once the maximum wake velocity is greater than 0.1 times the velocity of the wave-making source; the transverse waves behind the source are strongly distorted, although since these are the longest waves the approximation is less likely to be accurate. No information is given about wave amplitudes and with any ray theory approach to the problem initial values on a ray for the wave amplitude are difficult to ascertain. However, the approach is easy to understand. Rather more detailed descriptions of wake flow are used by Tatinclaux (1970) and Beck (1971). The former has a distribution of vorticity to represent the wake behind a thin vertical two-dimensionalcylinder of ogival cross section. Beck uses vortex sheets to model the wake behind a thin ship. Both papers assume that fluid velocities are small and use linearized boundary conditions at the free surface. The contribution of the wake to the wave resistance is calculated for one ship in each paper. Tatinclaux (1970) chooses a particular vorticity distribution, which decays relatively rapidly behind the cylinder, and calculates solutions for a range of Froude numbers. The wake has most effect for Froude numbers less than 0.5. It increases the wave resistance by an amount that varies considerably with Froude number, in an oscillatory manner, from over + 10 to - 35 %. Beck (1971) considers variation of the dimensions of the wake in his model. The effect of his wake is around f 10% of the irrotational wave resistance. All these papers indicate the importance of wave-wake interaction. A very direct experiment on the interaction of waves and a wake has been performed by Gadd (1975). Two identical ship models were towed in a catamaran arrangement. Where the bow waves of the models intersected, a steep pyramidal wave formed at high enough speeds. A vertical flat plate was introduced along the centerline between the two hulls, so that its trailing edge was just ahead of the steep pyramidal wave. This meant that the bow waves of the twin hulls met the wake of the plate. The introduction of the plate considerably modified the steep wave. It flattened and moved forward the wave peak and caused extensive turbulent flow. A large superficially similar wave often occurs with its crest at a ship’s stern. This experiment is expected to give an insight into the interaction between that wave and the ship’s boundary layer and wake. The behavior of this flow brings to mind Banner and Phillips’ (1974)

Interaction of Water Waves and Currents

111

paper, which is discussed in Section IV,C. The effect of the reduced velocity in the boundary layer relative to the waves is a decrease in the maximum height attainable. For example, if the velocity of the flow is one-half the ship’s speed, use of Eq. (4.100) shows that the maximum elevation of the water is one-quarter its maximum for irrotational flow. Gadd’s (1975, Fig. 3) photograph shows that the wave is breaking. One consequence of this observation is that one must expect nonlinear effects to become important at much lower amplitudes in this type of problem than in cases where irrotational flow is a good approximation. Further details, including wake traverses and wave measurements, are included in Gadd’s (1975) paper. They show more details of the interaction that occurs in this experiment. Interesting points are the appreciable changes in the wake behind the hulls when the plate is introduced and the associated differences in the waterline near the stern of the models and in the waves radiated. These appear to be largely due to the bow wave of one hull influencing the stern of the other and its modification when the plate is introduced. ACKNOWLEDGMENTS I wish to thank the many people who have assisted by sending me preprints, reprints, and copies of reports and inaccessible papers. The report by Dalrymple (1973) was particularly useful for Section IV,C. REFERENCES

ARTHUR,R. S. (1950). Refraction of shallow water waves: the combined effect of currents and underwater topography. Trans. Amer. Geophys. Union 31, 549-552. BANNER,M. L, and PHILLIPS 0. M. (1974).On the incipient breaking ofsmall scale waves. J . Fluid Mech. 65, 647-656. BARBER, N. F. (1949).The behavior of waves on tidal streams. Proc. Roy. SOC.,Ser. A 198,81-93. BATCHELOR, G . K. (1967). “An Introduction to Fluid Mechanics.” Cambridge Univ. Press, London and New York. BECK,R. F. (1971).The wave resistance of a thin ship with a rotational wake. J. Ship Res. 15, 196-216. BENJAMIN, T. B. (1957).Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554-574. BENJAMIN, T. B. (1961).The development of threz-dimensional disturbances in an unstable film of liquid flowing down an inclined plane. J . Fluid Mech. 10, 401-419. BENJAMIN, T. B. (1962).The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12, 97-116. BENJAMIN, T. B. (1967).Instability of periodic wavetrains in nonlinear dispersive systems. Proc. Roy. SOC., Ser. A 299, 59-75. BENJAMIN, T. B., and LIGHTHILL, M. J. (1954).On cnoidal waves and bores. Proc. Roy. SOC.,Ser. A 144, 425-451.

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BENNEY, D. J. (1966). Long nonlinear waves in fluid flows. J. Math. Phys. (Cambridge, Mass.) 45, 52-63.

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NOTES ADDED IN PROOF BIRKEMEIER, W. A., and DALRYMPLE, R. A. (1975). Nearshore water circulation induced by wind and waves. Proc. Symp. Modeling Techniques, Amer. SOC.Civil Eng. pp. 1062-1081. Extends the work of Noda er al. (1974).

PURI,K. K. (1974). Waves on a shear flow. Bull. Aust. Math. SOC.11, 263-277. Unsteady linear waves on a flow U,,+ bz are calculated; the solutions may be of help in a more detailed study of critical layers (e.g., see p. 84). RUPERT,V. C. (1976). Wave current interaction in a.random wave field: a practical approach. J . Geophys. Res. 81, 363-367. This paper differs appreciably from those discussed on p. 53. SMITH,R. (1975). The reflection of short gravity waves on a non-uniform current. Proc. Cambridge Phil. Sac. 78, 517-525. An account of caustics which is complementary to that given in Sections 1I.D and 1I.E.

THOMSON, J. A,, and WEST,B. J. (1975). Interaction of small-amplitude surface gravity waves with surface currents. J. Phys. Oceanog. 5, 736-749. An interesting treatment of the effect of currents of the type described in Section I1,D.

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Generation of Magnetic Fields by Fluid Motion H. K . MOFFATT Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge. England

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 I1 . Magnetokinematic Preliminaries . . . . . . . . . . . . . . . . . . 125 A . Idealization of the Kinematic Dynamo Problem . . . . . . . . . . 125 B. Magnetic Field Representations . . . . . . . . . . . . . . . . . 127 C. Alfven's Theorem and Woltjer's Invariant . . . . . . . . . . . . . 128 D. Natural Decay Modes and Force-Free Fields . . . . . . . . . . . 129 111. Convection, Distortion. and Diffusion of B Lines . . . . . . . . . . . 130 A . Balance of Stretching and Diffusion in a Magnetic Flux Rope . . . . 131 B. Flux Expulsion by Flows with Closed Streamlines . . . . . . . . . 131 C. Topological Pumping of Magnetic Flux . . . . . . . . . . . . . . 133 D. Generation of Toroidal Field by Differential Rotation . . . . . . . 134 IV. Some Basic Results . . . . . . . . . . . . . . . . . . . . . . . . 135 A . Cowling's Theorem and Related Results . . . . . . . . . . . . . 135 B. Rotor Dynamos . . . . . . . . . . . . . . . . . . . . . . . . 137 V . The Mean Electromotive Force Generated by a Random Velocity Field . 139 A . The Two-Scale Approach . . . . . . . . . . . . . . . . . . . . 139 B. The Strong Diffusion Limit . . . . . . . . . . . . . . . . . . . 141 C. Evaluation of all and But for a Random Wave Field . . . . . . . . 145 D. Effect of Turbulence in the Weak Diffusion Limit, 1-0 . . . . . . 147 E. The Forms of ail and /Iuh in Axisymmetric Turbulence . . . . . . . 150 F. Dynamo Equations for Axisymmetric Mean Fields Including Mean 152 Flow Effects . . . . . . . . . . . . . . . . . . . . . . . . . VI . Braginskii's Theory of Nearly Axisymmetric Fields . . . . . . . . . . 154 A . Lagrangian Transformation of the Induction Equation . . . . . . . 154 B. Nearly Axisymmetric Systems . . . . . . . . . . . . . . . . . . 155 C. Nearly Rectilinear Flows; Effective Fields . . . . . . . . . . . . . 157 D. Dynamo Equations for Nearly Rectilinear Flows . . . . . . . . . 158 E. Comments on the General Approach of Soward . . . . . . . . . . 160 F. Comparison between the Two-Scale and Nearly Axisymmetric 161 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . VII . Analytical and Numerical Solutions of the Dynamo Equations . . . . . 163 A . The a2 Dynamo with a Constant . . . . . . . . . . . . . . . . 163

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B. a’ Dynamos with Antisymmetric a . . . . . . . . . . . . . . . . C. Local Behavior of am Dynamos . . . . . . . . . . . . . . . . . D. Global Behavior of am Dynamos . . . . . . . . . . . . . . . . VIII. Dynamic Effects and Self-Equilibration . . . . . . . . . . . . . . . A. Waves Influenced by Coriolis Forces, and Associated Dynamo Action B. Magnetostrophic Flow and the Taylor Constraint . . . . . . . . . C. Excitation of Magnetostrophic (MAC) Waves by Unstable Stratification D. Mean Flow Equilibration . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164 165 166 168 168 170 171 174 176

I. Introduction The existence of the magnetic field of the Earth, and its variation with time, presents a profound challenge to geophysics. This field, though influenced slightly by electric currents in the ionosphere, is predominantly of internal origin and is associated with a large-scale azimuthal current distribution in the liquid core of the Earth. It is well known (see, e.g., Hide and Roberts, 1961) that the temperature of the core is far above the critical value (the “Curie point”) at which permanent magnetization can persist. Moreover, in the absence of any regenerative action, the electric currents in the core would decay through ordinary resistive (“ohmic”) dissipation in a time of order 104-105 years. Geomagnetic studies indicate, however, that the Earth’s field has existed in one form or another for at least lo8 years and is probably as old as the Earth itself, and further that, although the main dipole field exhibits random rapid reversals, a phenomenon reviewed by Bullard (1968), it remains at least quasi-steady between reversals for periods up to order lo6 years, i.e, one or two orders of magnitude greater than the natural decay time. It is now generally agreed that this persistence of the Earth’s field can only be explained in terms of electromagnetic induction, whereby the electric currents that provide the field are generated by motion of the fluid in the core across the self-same field, which permeates the core region as we11 as the nonconducting exterior. The characteristic feature of such dynamo action is that the field is maintained exclusively by the action of the fluid velocity and without the help of any external source of field. This type of behavior is most simply illustrated in terms of the simple disk dynamo illustrated in Fig. 1. The electrically conducting disk rotates about its axis with angular velocity Q and a conducting wire makes sliding contact with the rim of the disk and with its axis, as shown; the wire is twisted into a circle in its passage from the rim to the

Generation of Magnetic Fields by Fluid Motion

12 1

U FIG.1. The self-exciting disk dynamo; note particularly the concentrated shear at the sliding contact and the lack of reflectional symmetry of the system.

axis in such a way that any current flowing in the wire gives rise to a magnetic field with nonzero flux across the disk. The rotation of the disk in the presence of this flux generates a radial electromotive force, and since a closed circuit is available, a net current I ( t ) flows along the wire. The flux then equals MI, where M is the mutual inductance between the wire and the rim of the disk, and I is given simply by

L dlldt + RI

= MRI,

(1.1)

where L and R are, respectively, the self-inductance and resistance of the complete current circuit. Clearly, if R < MQ the current grows, and if R is maintained at a constant value, this growth is exponential. The state with I = 0 is then unstable to the growth of small electromagnetic disturbances. Such growth cannot, of course, continue indefinitely. The Lorentz force j A B (where j is the current distribution in the disk and B the magnetic field) provides a net resisting torque MI’, and (1.1) must be coupled with the equation of angular motion of the disk C dRldt = G - MI’,

(1.2)

where C is the moment of inertia of the disk about its axis, and G the applied torque. If G (rather than R) is kept constant, then as I increases, R decreases until an equilibrium is reached in which G = MI’,

R = RIM.

(1.3)

Note that the angular velocity of the disk in this equilibrium state does not depend on the appiied torque ! This simple dynamo relies for its success on the carefully contrived path that the current is forced to follow. The conductor (disk + wire) occupies a region of space that is not simply connected, a feature that is of course not shared by the conducting core of the Earth. There are, however, two features of the disk dynamo configuration that deserve particular emphasis, because

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122

these features do recur in the fluid context and are both intimately associated with successful dynamo action. First, there is a region of concentrated shear at the sliding contact on the rim of the disk; the counterpart of this in the fluid context is differential rotation, which plays an important part in generating toroidal magnetic field from poloidal magnetic field (see Section 111,D). Second, the configuration lacks reflectional symmetry in that the sense of twist of the wire in Fig. 1 bears a very definite relation to the sense of the angular velocity of the disk: if the twist is reversed, then Mi21 in (1.1) is replaced by - MRZ, and rather than dynamo action we have accelerated decay of any transient current in the wire. This lack of reflectional symmetry also has its counterpart in homogeneous fluid systems. The simplest measure of the lack of reflectional symmetry of a localized fluid motion u(x) is its helicity

a quantity that admits interpretation in terms of the degree of linkage (or " knottedness") of its constituent vortex lines (Moffatt, 1969). We shall describe in the following sections (particularly Section V) how the presence of helicity is of vital importance for the process of regeneration of poloidal field from toroidal field, i.e., for the closing of the dynamo cycle that makes field regeneration a reality. The Earth is, of course, not the only celestial body that exhibits a significant large-scale magnetic field. Among the planets, Jupiter, Mars, and Mercury are now known to have this property also; the radii, rotation rates, and dipole moments of these planets in comparison with the Earth are displayed in Table 1 (from Dolginov, 1975). A theory that successfully explains the Earth's field may clearly have relevance in the context of these TABLE 1 COMPARATIVE FIGURES FOR THE EARTH, JUPITER,' MARS:

AND

MERCURY~

~~

Radius R (km) Earth Jupiter Mars Mercury a

6371 7 1,351

3386 2439

Dipole moment p (G km3)

P/R3 (GI

Rotation period (days)

8.05 x 10" 1.31 x 1015 2.47 x 107 4.8 x 107

3.11 x lo-' 3.61 6.36 x 3.31 x 10-3

1 0.415

Warwick (1963);Smith et a/. (1974).

* Dolginov et a/. (1973). Ness et al. (1974).

1

58

Generation of Magnetic Fields by Fluid Motion

123

other planetary fields. The internal constitution of Mercury, Mars, and Jupiter is, of course, largely a matter of speculation at present; it may be that detailed observation of the surface magnetic fields of these planets will in the long run provide (via theoretical argument) information about their interiors. In the case of Jupiter, it has been argued (see, e.g., Hide, 1974) that the core consists of a liquid alloy of metallic hydrogen and helium under high pressure and that this core provides the seat of magnetohydrodynamic dynamo action. The magnetic field of the Sun (which is believed to be typical of “cool” stars with convective outer envelopes) is much more complex in its structure and behavior than that of the Earth, and it too is widely (though not universally) believed to be the result of dynamo action involving the two features, differential rotation and motions lacking reflectional symmetry, described above. Paradoxically, although the Sun is certainly remote as compared WKi-the Earth’s liquid core, we have much more detailed information about its magnetic field, simply because it may be detected and measured at its visible surface, i.e., at the surface of the convective region where, from a magnetohydrodynamic point of view, all the interesting action takes place. In the case of the Earth, we have no prospect or hope of any direct measurement of the magnetic field, or indeed of any other quantity, either in the liquid core or on its surface, and we must make do with what we know of the field on the surface of the solid Earth, a pale shadow of the field of the deep interior, and a dim and distant reflection of the fluid turmoil in that deep interior which is at the heart of our problem. The Sun’s field is characterized by active regions and by a weak general field near the north and south poles of its axis of rotation. Active regions are associated with strong upwelling from the convective envelope with an associated vertical stretching of any magnetic field lines that are convected upward. When this stretching is particularly localized and intense, the strong vertical field that is created (of the order of thousands of gauss) can locally suppress thermal convection; the resulting decrease in heat transport leads to a local cooling of the surface; radiation from this local region (of the order of hundreds of kilometers in horizontal extent) is largely suppressed, and in consequence it is seen from the Earth as a dark spot on the surface of the Sun. Such sunspots occur in pairs, roughly along a line of latitude, but with the leading spot (i.e., that to the East) slightly nearer the equatorial plane. Sunspot activity has been followed for more than 300 years and is known to follow a roughly periodic cycle with half-period of about 11 years. At the beginning of a sunspot cycle, pairs of spots appear within the band of latitudes about & 30” from the equatorial plane, with statistical symmetry about this plane, first at the higher latitudes only, then gradually over a wider band of latitudes that drifts, as the cycle proceeds, toward the equatorial plane. In

124

H. K. Mofatt

any pair of sunspots, the vertical magnetic field is positive in one and negative in the other; if positive in the westerly spot, the pair has positive polarity, otherwise negative. In any half-cycle of 11 years, all sunspot pairs in the northern hemisphere have the same polarity, and all in the southern hemisphere have the opposite polarity. In the following half-cycle, these polarities are reversed. The weak polar field of the Sun also follows a somewhat irregular periodic evolution with approximately the same period as that of the sunspot cycle. The field was first measured by direct magnetograph measurements in 1952 (Babcock and Babcock, 1955) and it has been followed closely since that date. The field around the north pole reversed in 1958 and again in 1971;the field around the south pole reversed in 1957 and again in 1972! At each reversal, for about one year, the fields at north and south poles therefore had quadrupole rather than dipole symmetry about the equatorial plane. The reversals apparently occur during that part of the sunspot cycle when sunspot activity is at its maximum. These observations are compatible with the following qualitative picture (essentially conceived by Parker, 1955a): the global magnetic field of the Sun includes poloidal and toroidal ingredients that are not steady but vary periodically in time, with period approximately 22 years. The poloidal field can be observed and has the polar reversal behavior described above; the toroidal field is contained in some way beneath the visible surface of the sun and cannot be directly detected. This toroidal field is coupled with the poloidal field and in a typical half-period drifts like a wave from polar regions toward equatorial regions, intensifying as it progresses. When this field reaches a certain critical level of intensity, local upwelling instabilities may develop in which ropes of toroidal flux are stretched vertically upward, breaking through the visible surface of the sun and forming sunspots as described above. As the buoyant fluid rises through several scale-heights, it expands due to the decreasing ambient pressure; as a result of the tendency to conserve angular momentum, the rising blob develops a rotation (the sense of rotation being such that it has negative helicity in the northern hemisphere, positive in the southern): hence the twist of the sunspot pair from the original line of latitude of the underlying toroidal field. As the periodic evolution proceeds, the toroidal fields of opposite signs from the two hemispheres interpenetrate and annihilate each other in the equatorial zone and the sunspot activity in consequence dies away. The process then repeats itself, the toroidal field again growing from polar to equatorial regions (but with a complete change of polarity from one half-cycle to the next). What part does the weak poloidal field play in this process? It just must be present, for otherwise the dynamo cycle cannot proceed. On the one hand, all the little eruptions over the solar surface generate a field with a

Generation of Magnetic Fields by Fluid Motion

125

radial component (i.e., poloidal field); since the surface eruptions are most intense in equatorial latitudes, one might expect the poloidal field to be most evident in these regions also. However, poloidal field can be redistributed by large-scale meridional circulation in the convective zone and this must presumably play a part in sweeping poloidal field back to the polar regions. Meridional circulation also tends to generate differential rotation (conservation of angular momentum again) and this differential rotation is the means by which the toroidal field is regenerated from the poloidal. These complicated interactions may seem far removed from the simplicity of the disk dynamo described at the outset; yet the two features-differential rotation and lack of reflexional symmetry-appear in the solar context as vital ingredients in its periodic behavior; the lack of reflexional symmetry appears in the rising, twisting blobs, described by Parker (1955a) as “cyclonic events,” and directly responsible for sunspot formation. The above description is, of course, purely qualitative and suggestive. In the sections that follow, we shall endeavor to show how the various physical ideas implicit in the description may be placed on a secure mathematical foundation, and to relate the various approaches to dynamo theory that have made progress in this direction over the last 20 years. The reader who wishes further background material in the terrestrial and solar contexts may consult a number of review articles that have appeared in recent years (Parker, 1970a; Roberts, 1971; Weiss, 1971, 1974; Roberts and Soward, 1972; Vainshtein and Zel’dovich, 1972; Moffatt, 1973; Gubbins, 1974) and the very extensive detailed references that these articles contain.

11. Magnetokinematic Preliminaries A. IDEALIZATION OF THE KINEMATIC DYNAMO PROBLEM Suppose that fluid of uniform electrical conductivity 0 is confined to a simply connected region of space V inside a closed surface S, and suppose that the region p exterior to S (extending to infinity) is nonconducting. Let u(x, t ) be the velocity field in V, satisfying

and let p(x, t ) be the density field satisfying the equation of mass conservation appt

+v

(pu) = 0.

(2.2)

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H. K. Mofatt

For many purposes it will be sufficient to restrict attention to incompressible fluids of uniform density for which p=po, V.u=O. (2.31 Let j(x, t), B(x, t), and E(x, t) denote electric current, magnetic field, and electric field, respectively. Neglecting displacement current (certainly valid for phenomena on the long time scales considered), the equations relating j, B, and E in V are

V*B=O, poj = V A B =poa(E + UAB), aB/at = - V where p o = 411 x

(2.4) (2.5)

E,

(2.6) S.I. units. In p, where j = 0, B is determined by A

V*B=O, VAB=O. (2.7) Moreover, both normal and tangential components of B must be continuous across S, i.e., [n*B]: =0, [ ~ A B ] != O on S, (2.8) where n is the unit outward normal on S. Finally, we require that B be without singularities in V and in p, and that there should be no sources of magnetic field at infinity; this means that B must be at most dipole, i.e., 0 ( ~ - 3 ) as I = 1x1+ Elimination of j and E from (2.4)-(2.6) gives the well-known induction equation (which holds in V), aB/at = v A (u A B) + WB,

(2.9) where A = (p0a)-' is the magnetic diffusiuity of the fluid. For given u, we wish to explore the evolution of the field B as determined by (2.7)-(2.9) and the subsidiary conditions mentioned. A simple measure of the field level is the total magnetic energy M ( t ) = (2p0)-

j

B2 d 3 x.

(2.10)

V+P

If, for given u, M ( t )+ 0 as t -,co, then the motion u does not act as a dynamo. If M ( t ) f ,0 as t + 00, then the motion does act as a dynamo, there being ultimately sufficient rate of generation of magnetic energy by fluid motion to counteract the natural decay of magnetic energy due to ohmic dissipation associated with the finite conductivity of the fluid. In a complete theory that takes account of the dynamics of the fluid motion, u is, of course, constrained to satisfy the Navier-Stokes equation

127

Generation of Magnetic Fields by Fluid Motion

(with Coriolis forces, Lorentz forces, buoyancy forces, etc., included if the context so requires). In a purely kinematic approach at the outset, it proves useful to widen the scope of the investigation and to imagine that u is any kinematically possible velocity field (without dynamical restriction); the influence of dynamic constraints, which will be considered in Section VIII, is more easily comprehended after close investigation of the kinematic problem. B. MAGNETIC FIELDREPRESENTATIONS

1. Poloidal and Toroidal Decomposition Any solenoidal field B may be expressed as the sum of a poloidal ingredient B, and a toroidal ingredient B,, where

B, = v h v h ( X ~ ( X ) ) ,

B, = v A (xT(x)); S and T are the defining scalars for these fields. Note that

(2.11)

VAB, = VAVA(XT)

(2.12)

is a poloidal field with defining scalar T, while

V A B, = - V2(V A xS)= - V A (xVZS)

(2.13)

.

is a toroidal field with defining scalar -VzS. Note further that x & = 0, i.e., the lines of force of the % field lie on spheres r = const, as do the Hnes of force of V A %, a property that makes the representation particularly useful when problems with spherical boundaries are considered.

2. Axisymmetric Fields A field B is axisymmetric about an axis Oz when its defining scalars Sand T are independent of the azimuth angle 4 about 02. If S = S(r, 8), T = T(r, 8),where 8 is measured from 02, then

where

x

= - r sin 8 asps,

B6 = -a T@l.

(2.15)

x, theflux function, is the analog of the Stokes stream function for solenoidal velocity fields, and the lines of force of the B, field are given by x = const. The B, field may also be expressed in the form B, = V A = X/r sin 8, and i4 is a unit vector in the 9 direction.

A

(A&), where

H. K. Moffatt

128

3. Two-Dimensional Fields

It is frequently illuminating to consider configurations in which B depends only on two artesian coordinates, say x and y. In this case, the representation analogous to the above is B = B, B,, where now

+

and the BP lines are given by A = const. C. ALFVBN'STHEOREM AND WOLTJER'S INVARIANT It is an immediate consequent of (2.4)-(2.6) that, if @(t) is the flux of B across any surface spanning a closed curve C(t) that moves with the fluid, then

(E + u A B) dx = -

d@/dt = 'cw

a-'j

*

dx,

(2.17)

'cw

so that, in the perfect conductivity limit (a = co), @ is constant for any material curve C(t).It follows that in this limit B lines are frozen in the fluid, and in an incompressible flow, stretching of the B lines leads to proportionate intensification of the B field. Closely associated with the " frozen-field " concept is the invariance of the integral

4"

H - A .Bd3x,

(2.18)

(Woltjer, 1958). Here A is the vector potential of B, i.e., B = V A A, and it is supposed that B is a localized field so that the integral exists. The volume V in (2.18) may be any volume bounded by a material surface S on which B II = 0. The interpretation of H, is identical with that for the helicity integral (1.4) (which is likewise constant whenever circumstances are such that vortex lines move with the fluid), i.e., HM is a measure of the degree of topological complexity of the field B within the surface S-and this measure cannot, of course, change when the B lines are frozen in the fluid. We may note in passing that the solution of (2.9) in the limit a = co (i.e., A = 0) may be expressed in Lagrangian variables in the form (due to Cauchy) Bi(x, t ) = Bj(a, 0) 8 x i / a a j ,

(2.19)

where x(a, t ) is the position at time t of the fluid particle that was at position a at time t = 0.

Generation of Magnetic Fields by Fluid Motion

129

D. NATURAL DECAYMODESAND FORCE-FREE FIELDS If u is steady, i.e., u = u(x), then the problem (2.7)-(2.9) admits solutions proportional to exp( -pt), where possible values of p are determinate as eigenvalues of the problem. If, for all these values, R e p > 0, then B inevitably decays with time, while if for any eigenvalue Re p < 0, the corresponding field structure (the eigenfunction) grows exponentially in time until Lorentz forces modify the velocity field (cf. the rotating-disk situation discussed in the introduction). If p = p , + ipi then the condition pr = 0 is critical in that it determines the onset of dynamo action for the corresponding field structure Bp(x).If, when pr = 0, it also happens that p i = 0, then the resulting mode is steady under critical conditions; this is the sort of behavior that we look for in the context of the Earth's magnetic field, which is steady (with weak fluctuations) over very long periods. If, on the other hand, p i # 0 when pr = 0, then the resulting mode is oscillating under critical conditions; this type of behavior would be relevant in the solar context. Of course, when u = 0, all the p's are real and positive; in the important case when the volume V is spherical with radius R, the eigenvalues are given by pnq = l R - ' x i q , (2.20) where xnqis the qth zero of the Bessel function J , , 1 / 2 ( x )The . structure of the corresponding fields for r < R are given, in the notation of Bullard and Gellman (1954), by

SF + iSy = V A V A [xr- '/'J*+ 112(A-'pn4r)eid],

(2.21) (2.22)

The field S0,e matches to a dipole field in the exterior region r > R, the field Sy matches to an axisymmetric quadrupole field, and so on. From (2.20) the time scale of decay of these modes of simple structure is O(R'A- ') (as can, of course, be anticipated from dimensional analysis). The natural decay modes are closely related to field structures that are force-free, i.e., for which the Lorentz force j A B everywhere vanishes. Such fields arise naturally in the context of the kinematic dynamo problem, and it will be useful to set out some of their properties here. First, for such fields there exists a scalar field K(x) such that (2.23) p; 'j = V A B = K(x)B, and, since V * j = V * B = 0, it follows that B.VK=O and j-VK=O, i.e., B lines and j lines lie on a surface K = const.

(2.24)

130

H. K. Mofatt

The simplest example of a force-free field, with K constant everywhere, is (in Cartesians) B = Bo(sin Kz, cos Kz, 0);

(2.25)

the property V A B = KB is trivially verified. The B lines lie in planes z = const and rotate in a left-handed sense with increasing z. The vector potential of B is just A = K- ‘B, so that A B = K-’B2 = K - ’ B i .

(2.26)

The magnetic helicity density A * B is thus uniform; the field structure (2.25) has “ maximal helicity ” (Kraichnan, 1973). This and other similar examples have the property that the j field extends to infinity. There are in fact no force-free fields for which j is confined to a finite volume and B is everywhere continuous and O(r- 3, at infinity (see, e.g., Roberts, 1967, p. 109). It is, however, possible (Chandrasekhar, 1956) to construct force-free fields in a sphere I/ that match smoothly onto currentfree fields in the exterior region P that do not vanish at infinity: let ~ ( re), = Ar-”’J3/2(~r) cos e,

(2.27)

and B = VA(XS) + K - ’ V A V A ( X S )

for r

-= R.

(2.28)

Then it may be readily verified that, since S satisfies the Helmholtz equation (V’ + K’)S = 0, the field (2.28) does satisfy V A B = KB. However, since j n = 0 on r = R, and since B is parallel to j, we must also satisfy B n = 0 on r = R (a condition that the decay modes do not satisfy) and this requires that J3,2(KR) = 0. The exterior field in P is purely poloidal and is given by B = K-’VAVA(XS), where

-

S = -Bo(r - R3/r2)cos 8,

Bo = iAR-”2 dJ,,,(KR)/dR, (2.29)

the latter condition ensuring smoothness across r = R.

111. Convection, Distortion, and Diffusion of B Lines

In this section, we consider certain particular solutions of the induction equation (2.9) when u(x) is prescribed and steady. The behavior is strongly dependent on the order of magnitude of the magnetic Reynolds number R, = uo l/A [where uo and 1 are, respectively, velocity and length scales characteristic of u(x)]. Moreover, great care must be exercised when the two

Generation of Magnetic Fields by Fluid Motion

131

limiting processes R, --t co and t -+ 00 are considered-the solution may depend in a most sensitive manner on the order in which these limits are taken. A. BALANCE OF STRETCHING AND DIFFUSION IN A MAGNETIC FLUXROPE

Equation (2.9) represents the evolution of B under the joint action of the stretching of magnetic field lines and of their diffusion relative to the fluid. A well-known steady solution of the equation in which these effects exactly balance exists when u is the uniform extensional straining motion given by u = (-ax, -ay, 2az),

a > 0.

(3.1)

+ y’)),

(3.2)

The steady solution of (2.9) is then

B = (0, 0, Bo exp( -a/1)(x2

representing a flux rope of gaussian structure aligned with the z axis. The field (3.2), in fact, provides the asymptotic solution of (2.9) as t --+ co (with 1 fixed and nonzero). The flux in the rope is nBo1/a, so that for given flux, Bo becomes very large when 1 is very small, under this type of persistent stretching. This type of motion may be expected (locally) to generate the strong vertical magnetic fields observed in sunspots, as described in the introduction.

B. FLUXEXPULSION BY FLOWS WITH CLOSED STREAMLINES The phenomenon of flux expulsion was first explicitly considered by Parker (1963) and Weiss (1966). Suppose we have a steady two-dimensional incompressible flow with stream function $(x, y) and suppose that the fluid is permeated at time t = 0 by a magnetic field that is uniform and in the plane of the flow. For t > 0 the flow distorts the field and diffusion, of course, also influences its behavior. If R, 4 1, diffusion dominates and the field perturbations remain small [in fact, O(R,) relative to the initial field]. If R , $- 1, the picture is very much more complicated. During an initial phase whose duration is O(Ri/2)l/uo(1 and uo being the scales introduced above), diffusion is negligible and the field perturbations grow in intensity to O(Ri/’) times the initial field. There is then an intermediate stage, which has been studied in a particular case by Parker (1966), and whose duration is O ( R ~ ~ 2 ) l / uduring ,, which diffusion causes the breaking off of closed flux loops in regions of closed streamlines; these flux loops slowly decay and disappear and there is an associated net reduction in the total flux threading

H . K . Moffatt

132

the region of closed streamlines. Finally, the field settles down to a steady state, in which the flux across any region of closed streamlines is exponentially small. The difference between limtdmlimA-.o and limAdolimtdm is quite striking in this context: the former procedure gives a field whose energy density increases without limit (as t’); the latter procedure gives a steady field whose energy density is generally less than that of the uniform field that we started with! The following simple proof that the flux across any region of closed streamlines must vanish (under the second limiting procedure) appears to be new. Using the representation (2.16a) for the magnetic field, (2.5) may be expressed in the simple form DA/Dt E aA/at + u V A = IV’A. (3.3) It is supposed here that there is no applied electric field, so that E, = -aA/at. Under steady conditions then, V * (uA) = u * V A = IV’A,

(3.4) and in the limit 1+ 0, u * V A = 0, so that A is constant on streamlines, or equivalently A = A($). We now adapt the argument of Batchelor (1956) (as applied to the vorticity equation) to the present context: let C be any closed streamline, and integrate (3.4) over the area enclosed by C. Since n * u = 0 on C (where n is normal to C), the left-hand side integrates to zero. This focuses attention on the effects of diffusion when I is small but not quite zero. The right-hand side, on integration, gives

-

I n V A ds = I dA/d$

0= $C

$ a$/an ds = I K c dA/d$,

(3.5)

C

where Kc is the circulation around C. It follows that in the steady state (which may, of course, take a long time to attain) dA/d$ = 0, i.e., A = const, i.e., B = 0 throughout the region of closed streamlines. A net flux of field across, say, a periodic array of eddies with closed streamlines cannot, of course, be simply eliminated by this mechanism. What happens, as is clearly demonstrated in the numerical solutions of Weiss (1966), is that the flux is concentrated into sheets of thickness O(R; ‘ I 2 ) at the boundaries between adjacent eddies. A horizontal row of eddies will concentrate vertical flux in this way at the vertical cell boundaries, whereas any horizontal flux will be expelled to the regions above and below the eddies. The above proof can be simply adapted to cover the corresponding axisymmetric situation when steady meridional circulation acts on an axisymmetric poloidal field. Again, if the relevant magnetic Reynolds number is large, the field is ultimately excluded from any region of closed

Generation of Magnetic Fields by Fluid Motion

133

streamlines. This result has important implications for dynamo theory: meridional circulation that is weak can be conducive to efficient dynamo action, but meridional circulation that is too strong merely expels poloidal field from regions of closed streamlines (i.e., from the whole fluid region for an enclosed flow) and this effect is bound to be counterproductive, as indeed demonstrated in the numerical studies of P. H. Roberts (1972).

C. TOFQLOGICAL PUMPING OF MAGNETIC FLUX A rather fundamental variant of the flux expulsion mechanism has recently been discovered by Drobyshevski and Yuferev (1974). This study was motivated by the observation that in steady Benard convection between horizontal planes, fluid rises at the center of the convection cells and falls at the periphery; the regions of rising fluid are therefore separated from each other, whereas the regions of falling fluid are all connected. If the fluid is permeated by a horizontal magnetic field, then a field line near the upper p:ane will be distorted to lie entirely in a region of falling fluid and will therefore tend to be convected downward, a tendency that may be resisted to some extent by diffusion. A field line near the lower plate cannot be distorted to lie everywhere in the disconnected regions of rising fluid and cannot therefore be convected upward. One would therefore expect a net tendency for flux to be transported toward the lower plate; the Benard layer should act as a valve, allowing horizontal flux to pass downward but not upward. The particular velocity field chosen by Drobyshevski and Yuferev to demonstrate. this effect is (in dimensionless form) u=(-sinx(l+$cosy)cosz,

- (1 + f cos x) sin y cos z, (cos x + cos y + cos x cos y) sin z), (3.6) for which the cell boundaries are square; the more realistic choice of hexagonal cell boundaries would complicate the analysis without greatly adding to the insight provided. The magnetic field in the absence of fluid motion (or equivalently at zero magnetic Reynolds number) is taken to be ( B o , 0, 0), i.e., uniform in the x direction, and it is supposed that the boundaries z = 0, n are perfectly conducting so that the flux nBo is trapped in the gap between them. The steady solution of (2.9) may be obtained as a power series in R, when R, + 1; what is of interest is the average of this field over the horizontal plane, which turns out to have the expansion 7Ri cos 22 + 7 R: (28 cos z - 3 cos 32) + O ( R i ) B(z) = Bo 1 + __ 48n2 240x

(

(3.7)

134

H . K . Moflatt

The asymmetry about z = 71/2 appears at the O(R:) level, and this term indeed shows the expected increase in flux in the lower half of the gap. This phenomenon is of potential interest in the solar context. Convection in the Sun’s outer layers is turbulent, but there are also fairly stable and persistent large-scale structures, reminiscent of Benard cells, that survive even in the presence of this turbulence. These cells do show a preferred tendency for fluid to rise in discrete regions and to fall in connected regions, and any toroidal flux permeating this region will in consequence tend to be transported downward. The relevant magnetic Reynolds number is R,, = us1JIe where us and 1, are scales characteristic of the large-scale structures and I, is an effective eddy diffusivity associated with the smallscale turbulence (see Section V); R,, will probably be of order unity (even though the magnetic Reynolds number based on molecular diffusivity is very large). The phenomenon of flux pumping has been further investigated by Proctor (1975), who has demonstrated that when R, G 1, pumping can occur even when the topological distinction between upward- and downwardmoving fluid is absent. Proctor analyzes the effect of two-dimensional motions in detail and shows that a lack of geometrical symmetry about the midplane is sufficient to lead to a net transport of flux either up or down; e.g., if (w3)# 0, where w is the vertical velocity at the midplane and the angular brackets denote a horizontal average, then there will be a net transport, which Proctor describes as geometrical (as opposed to topological) pumping. When R, % 1, however, he demonstrates that this type of twodimensional geometrical pumping is almost nonexistent, whereas in this limit the Drobyshevski and Yuferev mechanism may be expected to be most effective (although no detailed analysis has yet been carried out).

D. GENERATION OF TOROIDAL FIELD BY DIFFERENTIAL ROTATION We conclude this section with a brief discussion of the process by which toroidal field is generated from poloidal field by differential rotation. Physically it is clear that differential rotation about an axis will tend to distort the field lines of an initially poloidal axisymmetric field B, if the angular velocity w varies along the field lines. In fact, if u = w(s, z)k A x, where (s, 4, z ) are cylindrical polar coordinates and k a unit vector along Oz, and if B = B,(s, z ) + BJs, z, t)i4, then the 4 component of (2.9) is For the moment we shall suppose that B, is maintained steadily by some unspecified mechanism. As expected, it is the gradient of w along B, lines that gives rise to generation of toroidal field. If I is small, and if initially

Generation of Magnetic Fields by Fluid Motion

135

B4 = 0, then B4 grows linearly with time until = O(R,)lB,I, at which stage a steady state is established. There is no question of flux expulsion in this case since B, is axisymmetric (if B, included any nonaxisymmetric ingredients then these would be expelled). The ultimate steady solution of (3.8) may be easily obtained if B,,and w are prescribed. For example, if B, = B, k where Bo is uniform, and if w = w(r)where r2 = s2 + z2, then the steady solution of (3.8) is r

B 4 = - -:B, (Lr3)-’ sin 8 cos 8 r;lw(rl) d r l .

jo

(3.9)

Note that B4 as given by this solution is antisymmetric about the “equatorial” plane 8 = 4 2 , and that if o ( r l ) decreases sufficiently rapidly as rl -+ co for the integral in (3.9) to converge, then B4 =.O ( r - 3 )as r + a. Contrast the situation when B, is an irrotational field with uniform gradient, i.e.,

B,

+ zk),

= C(-2si,

(3.10)

where is is a unit vector in the s direction. The steady solution for B6 then has two ingredients proportional to sin 0 and ~ P ~ ( c0)/88, o s but the former ingredient dominates for large I , and again provided o ( r l ) decreases sufficiently rapidly as rl -+ co, the asymptotic behavior of B4 for large r is

B4

-

(2C sin 8/3Lr2)

1

03

r:w(rl) dr,.

(3.11)

JO

This field is symmetric about 8 = 4 2 , and O(r-’) at infinity. In general, therefore, if a poloidal field B, is weakly varying in a region of differential rotation, then it is the local gradient of the field (rather than the local field itself) that determines the toroidal field generated at remote points when conditions are steady.

IV. Some Basic Results

A. COWLING’S THEOREM AND RELATEDRESULTS The impossibility of steady maintenance of an axisymmetric magnetic field by motions axisyrnmetric about the same axis (Cowling, 1934) is so well known as hardly to require comment here. The modifications and extensions of the theorem are numerous, but all reflect the basic fact that axisymmetric meridional circulation can redistribute poloidal flux but cannot systematically regenerate it. The equation for the flux function ~ ( r8), under

H. K. Moflatt

136

axisymmetric conditions [analogous to (3.35)] is

axlat + 4 - vx = A D ~ X ,

(4.1)

where 4 is the meridional velocity (assumed axisymmetric) and D2 the Stokes operator. The structure of this parabolic equation essentially ensures (Braginskii, 1964a)that V Xeverywhere tends to zero. Even if 1is nonuniform but satisfies merely the natural condition up V 1 = 0, standard manipulation of (4.1) and the relevant boundary conditions leads to this same conclusion (yet another minor extension of Cowling’s celebrated result !). Of course, when ( V x ( and so lBPl have reached a negligibly weak level, the toroidal field B+ must likewise decay, again essentially because of the parabolic structure of (3.8). An interesting variation on Cowling’s theorem has been claimed by Pichakhchi (1966). This is that steady dynamo action is impossible if the electric field E vanishes everywhere. (In the axisymmetric case with B, = 0, this is just a rewording of Cowling’s theorem since E I 0 in this situation under steady conditions.) If a steady dynamo with E = 0 were possible, then the field B would [from (2.5)] satisfy B * (V A B)= 0, i.e., it would have zero helicity everywhere and a correspondingly simple topological structure. The fact that this is apparently not possible is of course significant. There is also a counterpart of Cowling’s theorem for two-dimensional velocity and magnetic fields (Zel’dovich, 1957; Lortz, 1968). In this case, (3.3) is the governing equation, and by standard manipulations, provided A = o(r- I ) at infinity,

.

(dldt)

55

‘A2dx d y = -21

55 (VA)’

d x dy.

(4.2)

It follows that a steady state is possible only if VA = 0, i.e., only if B, = By= 0. Again, this removes the source of any possible regeneration of B,, which must also therefore vanish in a steady state. If u and B are stationary random functions (with zero mean) of x and y, then (4.2) must be replaced by d ( A 2 ) / d t = -2A((VA)’),

(4.3)

where ( ) indicates averaging over the x-y plane. No matter how small 1 may be, this again implies ultimate decay of the magnetic energy density. In the limit 1 + 0, ( A z ) apparently remains constant; however, this holds only so long as ((VA)’) remains finite; in fact, ( ( V A ) * ) increases (just as in the particular situation described in Section III,B) until it is O(A-’); at this stage and the inexorable decay of the field the length scale of the field is 0(1’/2) then sets in.

Generation of Magnetic. Fields by Fluid Motion

137

The impossibility of dynamo action under purely toroidal motion (Bullard and Gellman, 1954; Backus, 1958) is also closely related to Cowling’s theorem, in that it follows from the equation

D(x B)/Dt = (B * V)(X

*

U)

+ rZVZ(x

B),

(4.4)

.

which may be derived from (2.9) together with V u = 0. If u is purely toroidal, then x u = 0 and so x B inevitably decays everywhere to zero. The decay of Bp and is almost an immediate consequence. Busse (1975) has on the basis of Eq. (4.4) obtained a necessary condition (in terms of a lower bound on the poloidal velocity) that must be satisfied for successful dynamo action. B. ROTOR DYNAMOS In view of the various antidynamo theorems described above, it was of course of crucial importance that the possibility of steady dynamo action in a fluid of uniform conductivity occupying a simply connected domain be unambiguously established for at least one kinematically possible velocity field (no matter how artificial from a dynamical point of view). Until this was done (Herzenberg, 1958) it was by no means clear that a master theorem proving the absolute impossibility of such dynamo action might not at some stage be proved. Herzenberg’s dynamo consisted of two spherical rotors (i.e., quasi-eddies) embedded in a fluid sphere, the conductivity being uniform throughout. In fact, it is easier to comprehend the three-rotor problem (Fig. 2) considered? by Gibson (1968). Suppose that three spheres S,, S, , S, each of radius a, with centers located at the points (d, 0, 0), (0, d, 0), and (0, 0, d), where d 9 a, rotate with angular velocities (0, 0, -a),( -0, 0, 0), and (0,-o,0), respectively, where o > 0; the conductivity c throughout the whole space is assumed uniform. Note immediately the lack of reflectional symmetry in this configuration. The principle of the dynamo is roughly as follows: suppose that there are nearly uniform fields of the form B, x (0, 0, B), B, x (B, 0, 0), and B, x (0, B, 0) in the neighborhoods of S , , S 2 , and S , ,respectively. Then we have the possibility of a cyclic generation in which the toroidal field generated by rotation of S, (a = 1, 2) acts as the “applied” field Be+ in the neighborhood of S,+ and the toroidal field generated by rotation of S , acts as the applied field B, in the neighborhood of S,. The subtlety of the problem derives from the fact that, as mentioned at the end of Section III,D, the fields generated by differential rotation in each case are determined to an important degree by the local field gradient as well as by the local field itself, and this has to be taken into account in the

,

,,

t The particular configuration considered here was also discussed by Venezian (1967).

138

H. K . MofSatt

FIG.2. The three-rotor dynamo of Gibson (1968). The spheres rotate as indicated and generate toroidal fields B,, which act as the applied poloidal fields for S , (a = 1, 2, 3).

detailed calculation. The condition obtained by Gibson for steady dynamo action (adapted to this particular configuration) is R , = wa2/A= l O f i ( d / ~ ) ~ ,

(4.5)

correct to leading order in the small parameter a/d. A working dynamo based on the interaction of two rotors has been constructed in the laboratory by Lowes and Wilkinson (1963, 1968).The rotors are cylinders inclined to each other and embedded in a block of material of the same conductivity, electrical communication between the rotors and the block being provided by a lubricating film of mercury. Not only was dynamo action demonstrated with this model [through observation of a sudden large increase in the local magnetic field when the angular velocities of the cylinders were increased beyond a certain critical value-analogous to that given by (4.5)], but also reversals of the field were observed when the dynamo was functioning in the fully nonlinear regime-an observation of the greatest interest in view of the known random reversals of the Earth’s magnetic field mentioned in the introduction. A further ingenious example of dynamo action associated with a pair of rotors has been analyzed by Gailitis (1970). In this case, the rotors are toroidal rather than spherical, and the velocity field is axisymmetric about the common axis of the two toruses. The magnetic field that is maintained is, however, nonaxisymmetric, and there is no conflict with Cowling’s theorem.

Generation of Magnetic Fields by Fluid Motion

139

V. The Mean Electromotive Force Generated by a Random Velocity Field

A. THETWO-SCALE APPROACH Motion in the convective zone of the Sun is certainly turbulent, and any dynamo theory that fails to take account of this fact is hardly realistic. Likewise, motion in the core of the Earth almost certainly consists of a mean and a random ingredient. It is not clear whether the random ingredient is turbulence in the normal sense of the word, or rather a random field of waves influenced by Lorentz, Coriolis, and buoyancy forces; from a purely kinematic point of view, this distinction is not crucial, and we merely assume throughout this section that u(x, t) is a stationary random function of both x and t with zero mean-effects of nonzero mean velocity will be considered in Section VI. We are primarily concerned with the evolution of the mean magnetic field, which may be supposed to have a characteristic length scale L large compared with the scale l that characterizes the “background” velocity fiel’d u; in the case of turbulence, 1 will be the scale of the energy-containingeddies (Batchelor, 1953), while in the case of random waves, 1 will be, say, the wavelength of the most energetic modes represented in the spectrum of u. Either way, we can average the induction equation (2.9) over a spatial scale intermediate between 1 and L to obtain

dB,/dt = V A B

+ IV’B,

,

(54

where B, = (B), B = B, + b, and d = (u A b). This two-scale approach was introduced by Steenbeck et al. (1966) and has since had a revolutionary impact on the subject?. A series of papers by these authors, developing their approach to “mean field electrodynamics” has been collected together in English translation by Roberts and Stix (1971). The main problem, of course, analogous to the closure problem of turbulence dynamics, is to find a means of expressing d in terms of B, so that (5.1) may be integrated. The task is, however, easier because the basic equation for B is linear, albeit with a random coefficient. The equation for b, obtained by subtracting (5.1) from (2.9), is

db/dt = VA(UAB,)+ VA(uhb - (uhb))

+ IV’b,

(5.2)

and if we suppose that b = 0 at same initial instant t = 0, this establishes a linear relationship between b and B, , and so between 6” and B,; since the t The concepts of a mean electromotive force and of an associated eddy conductivity were already present in earlier work (e.g, Kovasznay, 1960).

H . K. Moflatt

140

scale L of Bo is very large, such a relationship may presumably be developed as a series di = q j & j

+ p i j k dBOj/dxk +

d2BOj/dxk 8x1

+

(5.3) the coefficients a i j , pis, . .., being pseudotensors determined in principle by the statistical properties of the u field and the parameter I (which, of course, plays an important part in the solution of (5.2) (pseudo because 8 is a polar vector, whereas Bo is an axial vector). It is important to note that these pseudotensors do not depend on B,; hence aij may be evaluated on the simplifying assumption that B, is uniform; then B i l k may be evaluated on the assumption that dBo,/dx, is uniform, and so on. Attention in most investigations has been focused on the first two terms of (5.3) on the grounds that it is essentially a series in ascending powers of l/L, and subsequent terms are likely to have negligible effect when L is large. There are, however, considerable subtleties here, particularly when I is very small, and the possible influence of subsequent terms perhaps deserves investigation. For the present, however, we truncate the series (5.3) after the second term and investigate some of the consequences. First, suppose that the u field exhibits no preferred direction in its statistical properties; since ail, flijk, are essentially statistical properties of the turbulence, they must then be invariant under rotations of the frame of reference and must therefore take the form a.. = ma.ij IJ

9

YijkI

pijk

=b i j k

9

“ ‘ 9

(5.4)

where a is a pseudoscalar and fl a pure scalar. Here the crucial role played by “lack of reflectional symmetry” comes into evidence. If the u field is reflectionally (as well as rotationally) symmetric in its statistical properties, then a (being a pseudoscalar that is not invariant under change from a rightto a left-handed frame of reference) must vanish. No such conclusion applies to the fl term. If the turbulence lacks reflectional symmetry, then the a term will in general be nonzero, and the relation between B and B, becomes

d = aB, - P A B , .

(5.5)

In view of the assumed statistical homogeneity of the u field, a and p are constants, and (5.1) becomes

aB,/at = aV A Bo ’+ ( I

+ p)V2Bo.

(54 Hence fl plays the role of a turbulent diffusivity; it is to be expected that 1> 0, although no general proof of this appears yet to be available. The a term, on the other hand, has quite a novel structure (from the point of view of conventional electrodynamics) and is in fact of crucial importance for dynamo theory.

Generation of Magnetic Fields by Fluid Motion The average of Ohm's law (2.5), incorporating (5.5), becomes J, = a(E, aB, - BV A B,),

+

141

(5-7)

where Jo = (j), E, = (E), or equivalently, ae = cr( 1 /?upo)-'. Jo = a,(E, aB,), (5.8) The a term therefore tends to drive mean current along the lines of mean magnetic field. In a spherical geometry, this effect in the presence of a toroidal field will generate a toroidal current, which acts as the source of a poloidal field. This therefore is the key to the means by which poloidal field may be regenerated from toroidal field by nonaxisymmetric random motions. The explosive character of Eq. (5.6)may be recognized very quickly if we suppose for the moment that our fluid fills all space and if we consider a magnetic field having an initial structure satisfying V A B, = KB, (i.e., one of the force-free fields of Section I1,D). For such a field, V'B, = -K2BQ, and so according to (5.6) the field will retain its spatial structure and develop exponen tially like exp cot, where o = aK - ( A + B)K2. (5.9)

+

+

Clearly we have exponential growth (i.e., dynamo action) if aK > ( A + B)K2

(5.10)

(and clearly we must choose K to have the same sign as a in order to ensure this possibility). Provided 1 K I is sufficiently small (i.e., provided the scale of Bo is sufficiently large), condition (5.10) is satisfied. Dynamo growth is then assured for force-free modes of sufficiently large length scale. It is, of course, important to obtain an explicit representation of aij and Bijk in terms of the statistical properties of the u field; this stage of the problem is analogous to the statistical mechanics problem of obtaining expressions for the various transport coefficients in terms of the statistical substructure of the medium considered. In the present context the substructure is provided by the background random velocity field. Unfortunately, determination of aij and Bijk is possible only in certain limiting situations; however, these limiting situations are in themselves of particular interest and will be considered in the following sections.

B. THESTRONG DIFFUSION LIMIT If R , = uol/A ao, on the other hand, then differential rotation is responsible for the generation of toroidal field from poloidal, whereas the a effect, through the term aB in (5.52) regenerates poloidal from toroidal field; a dynamo operating in this way is described as an aw dynamo (avo would perhaps be more accurate, since it is only the gradient of w that is relevant here). The equations for the 010dynamo in the form

+ up VB = - v w + - s - z ) ~ , aA/at + up VA = aB + l ( v z - s - ~ ) A , aB/at

S B ~

rz(v2

(5.55) (5.56)

were obtained originally by Parker (1955a). In this form, the sole effect of the

154

H. K. Moffatt

background random motions is the appearance of the term aB in (5.56), and Parker derived this on the basis of his “random cyclonic events” model (see also Parker, 1970b, 197la,b,c,d,e,f, 1975; Lerche, 197la,b). Equations having the same structure were derived by systematic perturbation procedures by Braginskii (1964a,b), whose approach will be described in the following section ;this latter approach leads, moreover, to the appropriate determination of a in terms of properties of the background fluctuating (nonaxisymmetric) motions. The approach that we have adopted in this section (following Steenbeck, Krause, and Radler) seems the most general and the most easily comprehended-and ease of comprehension is of crucial importance when attention is turned to the far more difficult dynamic aspects of the problem. It is, of course, reassuring that the various approaches, from rather different standpoints, do converge at the same destination [in the form of Eqs. (5.55) and (5.56)], and this lends confidence to the extensive analytical and numerical studies of these equations that have been carried out, some of which will be described in Section VII. VI. Braginskii’s Theory of Nearly Axisymmetric Fields

A. LAGRANGIAN TRANSFORMATION OF THE INDUCTION EQUATION An approach to the dynamo problem that is in some respects complementary to that described in Section V was developed by Braginskii (196hb) and has since been elucidated and extended by Soward (1972).The approach is based on the idea that, although an axisymmetric field cannot be maintained by axisymmetric motions, weak departures from axisymmetry in both velocity and magnetic fields may be sufficient when 1 is small to provide the means of regeneration of the field against ohmic decay. In describing the essence of this theory we follow the approach advocated by Soward. The starting point is a property of invariance of the induction equation in the frozen field (A = 0) limit. For reasons that emerge, it is helpful to modify the notation slightly: let s(%,t ) , a(ji t) represent field and velocity at (2,t), and consider a 1-1 continuous mapping 2 + x(ji t ) satisfying the incom11 be unity. Then if we pressibility condition that the determinant J(dx,/d%, define new fields U , ( q t ) = dx,/df Bk(& t ) = 8, dxk/dzi, the invariance property is that the equation

&/dt = 6 A (ii A 8)

+

&i

dxk/d%i,

(6.1)

Generation of Magnetic Fields by Fluid Motion

155

transforms into aB/at = V A (U A B).

(6.3) B(x, t) is (physically) the field that would result from fs(ji t) under the frozen-field distortion 3 + x. Suppose now that we subject the full induction equation

aB/at - P A(ii A fs) = nP28

(6.4) to the same transformation. We obtain, after some elaborate manipulation of the right-hand side, aB/dt - VA(UAB)= V h 8 + nv2B,

(6.5)

where

8,= a,Bj

+ cikpppjaBk/aXj,

(6.6)

and

Note that aij (a pseudotensor) and Bpj (a tensor) are both quadratic in the displacement x - f. The similarity between (6.5), (6.6) and (5.1), (5.3) is immediately striking; note, however, that in the present context, the term V A 8 is wholly diffusive in origin. Paradoxically, although diffusion is responsible for the natural tendency of a field to decay, it can also be of crucial importance in creating the electromotive force that can counteract this decay.

B. NEARLY AXISYMMETRICSYSTEMS Throughout this section, we shall use angular brackets ( ) to denote an average over the azimuth angle 6 in cylindrical polar coordinates (s, 4, z). Thus for any scalar $(s, 4, z),

With this convention, we may talk of the mean toroidal field (f4)i4 and the mean poloidal field (f,)i, + (f,)i,, which are by definition axisymmetric.

H. K. Moffatt

156

If the velocity field ii(x, t) is nearly axisymmetric, then we may express it in the form

a = (a)

+ EU’,

E

< 1.

(6.10)

A magnetic field P(x, t ) convected and distorted by such a velocity field must exhibit at least a similar degree ofasymmetry, and it is consistent to suppose that

P = (P) + E b .

(6.11)

When E = 0, the field (P)cannot survive, by Cowling’s theorem. Braginskii’s model is based on the expectation that the mean electromotive force E’(u’ A b) may compensate the erosive effects of the term AV’(S) in the averaged induction equation. For this to be possible it is necessary that A be not greater than O(E’);we therefore put A = A. E’ and keep lofixed as E -+ 0. Expansion in powers of E may then be expressed equivalently as expansion in powers of R , ‘I’ where R, = U oL/A,where U o and L are scales chracterThis was the procedure adopted by istic of the mean velocity field (i). Braginskii. The mean velocity may be expressed as the sum of toroidal and poloidal parts:

(ii)

=

U(s, z)i,

+ Up(s,z).

(6.12)

The toroidal part generates toroidal field from poloidal field by differential rotation (Section II1,D) and this is certainly conducive to dynamo action (although, of course, not sufficient in itself). The poloidal part U&, z) tends to redistribute the mean toroidal field; it also tends to wind up the mean poloidal field and to expel it (Section II1,B) from regions of closed streamlines of Up (i.e., from the whole fluid region for an enclosed flow) if a magnetic Reynolds number based on a typical value of I UpI is large. This latter process is certainly not conducive to dynamo action, and the only way to control it is to suppose that I Up / U o is at most O(Rk1), so that

I

UoL/l= R , ,

]Up

= O(1).

(6.13)

The dominant ingredient of the mean velocity field is then the toroidal ingredient Ui, whose typical order of magnitude is U o. To emphasize this scaling, we rewrite (6.12) in the form

(a)

= ~ ( s z)i, ,

+ &’uP(s,

z).

(6.14)

Note that, in proceeding in this way, attention is automatically restricted to velocity fields that have some a priori chance of success as potential dynamos.

Generation of Magnetic Fields by Fluid Motion

157

The mean field (P) may likewise be expressed as the sum of toroidal and poloidal parts :

(P) = B(s, z)i4 + B,(s, z).

(6.15)

The dominant differential rotation Ui4 generates Bi4 from Bp and it may be anticipated (Section II1,D) that B = O(R,) I Bp I, or equivalently I Bp I = O(E’)B.Hence we may also rewrite (6.15) in the form

(P) = ~ ( s z)i4 , + E’L+(s,

z).

(6.16)

It is, of course, implicit in the notation that B, Ib 1, and I b, I are all of the same order of magnitude as E --* 0; similarly for U, Iu’ 1, and Iup I. C. NEARLY RECTILINEAR FLOWS; EFFECTIVE FIELDS It is mathematically simpler to focus attention on the Cartesian analog of the situation described in Section VI,B, in which (s, 4, z) are replaced by ( x , y, z), and ( ) in consequence now indicates an average with respect to the y variable. The additional difficulties of dealing with the cylindrical coordinate system are purely geometrical and need not concern us here. The fact that the fluctuating fields of (6.10) and (6.11) are O ( E )suggests that it may be possible to “accommodate” them through the use ofa transformation function

I=x

+ q(x, t),

(q) = 0,

v - 7 = 0.

(6.17)

If this is possible, then the related fields u and B will have the simple form (the time dependence being for the moment understood)

+ &’ueP(x,z) + O(c3), B = B(x, z)i, + EZbcp(x,z) + O(c3), u = ~ ( xz)i, ,

(6.18) (6.19)

where the effective fields uep and beP are related in some way (to be determined) to the fields up and bp . Let us first obtain the relationship between the fields bepand b, . First, by expanding B,(x) in Taylor series about the point 3 and using Bi(W) = (dik E a1i/dxk)Bk(x),we obtain

+

+

o(E3).

(6.20)

158

H. K. Moflatr

Comparison with (6.11) shows that

b = V A ( ~ A B=) B aq/ay - (q * V)Bi,

(6.21)

(to leading order), and the mean poloidal ingredient of (6.20) gives

(6.22)

-

where q, = q - (q iy)iy is the meridional projection of q. In terms of the vector potentials a, a, defined by b, = V

A

(ai,),

be, = V

(a, i,,),

(6.23)

dqu/ay)y.

(6.24)

A

this result takes the simpler form a = a,

- wB,

w = +(q,

A

The relationship between u,, and up is similarly obtained with the sole difference that it is uk - ax,/at (rather than uk) that appears in the transformation relationship (6.1). This leads to the expression [analogous to (6.21)] = (a/at

+ u a/ayh- (11 - V)ui,,

(6.25)

and the result [analogous to 6.22)]

up = ucp

+ +v A (4,

A

(slat + u aiayh,).

(6.26)

D. DYNAMO EQUATIONS FOR NEARLY RECTILINEAR FLOWS Equations for the evolution of B(x, z, t) and beP(x,z, t) may now be obtained by averaging (6.5) with respect to y. The y component (i.e., the “ toroidal” ingredient) of the resulting equation is aB/at

+ A,,

VB = E2bcp

- vu + (v

A

+

+ o(E41,

B,)~ IV~B

(6.27)

where

(6.28) Since I = O(E’)and aij and /Ipjare both O(E’),A, is evidently O(c4),so that to leading order (6.27) becomes aB/at

+ E2Uep .VB =

&2b,,

- vu + I ,

E ~ v ~ B .

(6.29)

In this equation the effect of departures from exact rectilinearity are wholly absorbed through the introduction of the effective fields uePand be, . Note that the time scale of evolution of the B field is O ( E - ~ L / U ~ ) .

Generation of Magnetic Fields by Fluid Motion

159

The x and z components of the averaged equation (6.5) may be “uncurled ” to give c2 aa,/dt

+ e4uep - Va, = &oy + E2AV2ae+ O(E’),

(6.30)

and here it is clear that the term &‘oy is of the same order as the other terms in the equation. Moreover, the dominant contribution to goy[from (6.2811 is evidently given by g o y = A 0,

(2.18)

the minimum cost may be calculated from any one of the following three expressions (Rozvany, 1973~): (2.19) (2.20) (2.21)

192

George 1. N . Rozvany and Robin D . Hill

in which Q, q, P,and p are statically and kinematically admissible and satisfy (2.11). Naturally, (2.19) is valid for any convex specific cost function. For homogeneous specific cost functions, the optimality criterion (2.11) also implies the condition q Q/$ = d'/$ = const

on D,

(2.22)

which is termed the uniform energy dissipation principle and was obtained originally by Drucker and Shield (1956, 1957). Homogeneous specific cost functions of order p with (0 < p < co)may also be defined by making use of the operator Y:

Q . S[$(Q)I= MQ).

(2.23)

For homogeneous specific cost functions of order one, the cost gradient Y[$(Q)] depends only on the direction of the stress vector Q but not on its magnitude :

Y[$(kQ)l= S[lCI(Q)],

for k > 0.

(2.24)

The locus of Y[$(Q)] for such functions is termed the cost gradient surface. Upper and lower bounds, respectively, on nmin may be obtained by evaluating the following quantities (Rozvany and Adidam, 1972d): =

jD *(Q? dx,

(2.25)

1

(2.26)

fi = P * pk dx, D

in which Q" is statically admissible, and pk is kinematically admissible and is associated with strain rates qk, which represent points on or inside the cost gradient surface. The validity of (2.26) is restricted to convex homogeneous specific cost functions of order one. For the same class of functions, the Prager-Shield (Prager and Shield, 1967) optimality condition (2.11) may be given another physical interpretation, namely, the optimal stress and strain rate fields (Q,,,, q,,,) are identical with stress and strain fields for a structure of ideal locking material (Prager, 1957), having the same static and kinematic continuity conditions and constraintst as the plastic system to be optimized and a locking surface that is identical with the cost gradient surface of the latter. This means that the Prager-Shield condition (2.11) transforms the problem of optimal design of perfectly plastic systems to the problem of analysis of structures made out of a locking material.

t In earlier terminology: the same boundary conditions, loading, equilibrium, and compatibility equations.

Optimal Load Transmission by Flexure

193

If absolute limits [t,hmin(x), $max(x)]are imposed on the specific cost, then (2.11) must be supplemented by the following conditions (Prager, 1974a): q=

,I*[$(a)], " I *[$(Q)]

for $(Q)< $min for $(Q)= $mi"

$(a)=

for

$ma,

(2.27)

3

9

3

0 I v I 1,

(2.28)

1 I v I a*

(2.29)

C. UNSPECIFIED COSTDISTRIBUTION-SEVERAL ALTERNATE LOADS Let a structure be subject to alternative loads Ph(X) (h = 1, .. ., t) and let the corresponding velocities, stresses, and strain rates be ph(x), Qh(x),and qh(x), respectively. At a point x E D,each load condition Ph(X)gives a different cost requirement $(Qh).The design value $ of the cost is then given by

(2.30) For alternate loads, the Prager-Shield optimality condition (2.11) changes to (Rozvany and Adidam, 1972d; Charrett and Rozvany, 1972a)

(2.31)

qh = &y[$(Qh)l, Ah

> 0,

only if

$(Oh) =

$, h = 1, ..., t,

(2.32) (2.33)

where qh and Qh( h = 1, .. ., t), respectively, are statically and kinematically is given by the following two admissible. Then the minimum cost amin expressions :

(2.34) (2.35) Upper and lower bounds on the minimum cost are furnished by max

=

"=s where

q h

D

$(a;)dx,

(2.36)

h

iPh'PkdX,

(2.37)

D h=l

is statically admissible and pk is kinematically admissible and is

George I . N . Rozvany and Robin D . Hill

194

associated with strain rates

(2.38) in which Y* may define any point on or inside the cost gradient surface. Equation (2.37) is restricted to convex homogeneous specific cost functions of order one. Minimum-weight plastic design of beams and frames for alternate loads are discussed by Mayeda and Prager (1967) and Prager (1967),respectively. If a structure having the domain D is subjected to an infinite number of loading conditions such that each load system is associated with a vector 6 E B (where B is the set of all admissible values of t),then (2.33) and (2.32) change to

A(% 6)d6 = 1,

(2.39)

for all x E D,

1.4

A(x,

S) > 0,

only if $(x) = max c

$[Qk 6)l = $[Q(x,513,

(2.40)

where $(x) is the design value of the specific cost at x and Q(x, 6)is the state of stress at x equilibrating the load associated with vector 6. It is important to note that 6 may define not only the location but any other property of a load system. For example, the alternate load systems may consist of a line load of length Land intensity P with LP = const, and 6 may then take the form

(2.41)

6 = ( y , z, 8,L),

in which y and z are the coordinates of the centroid of the line load, 8 defines its orientation, and L its length. Problems involving moving loads were discussed by Save and Prager (1963), Save and Shield (1966), and Lamblin and Save (1971).

D. UNSPECIFIED COSTDISTRIBUTION-MULTICOMPONENT SYSTEMS The design value $ of the specific cost may consist of several components $k ( k = 1, . . ., v ) such that v

(2.42) (2.43) where

Qh

equilibrates the hth load condition and

I(/k(Qh)

are termed cost

Optimal Load Transmission by Flexure

195

component functions. The foregoing problem differs from the one discussed in Section II,C, because the cost components $k are independent from each other at any point x and thus they may be governed by different load conditions. An example of a multicomponent system is an axisymmetric fiber-reinforced plate having top and bottom reinforcing fibers in radial and circumferential directions. Extension of the Prager-Shield optimality condition (2.11) to this problem gives (Charrett and Rozvany, 1972a) U

qh

=

h = 1,

&g[$k(Qh)]t k= 1

t,

(2.4) (2.45)

qk= +k(Qh),k = 1, ..., u,

A,, 2 0 and Ak,, > 0 only if

h = 1, ..., t,

(2.46)

where qh and Qhare, respectively, kinematically and statically admissible. An upper bound on the minimum cost SZmin is given by (2.47)

where Q; (h = 1, . . ., t ) are statically admissible. A lower bound on the minimum cost for convex homogeneous cost component functions of order one is furnished by (2.37),in which pf:is kinematically admiskble and is associated with a strain rate field v

(2.48)

with

c &, t

= 1,

k

= 1,

. . ., 0, &,2 0,

(2.49)

h= 1

where 9; is a vector representing a point on or inside the cost gradient surface for t+kk(Q). E. PARTIALLY PRESCRIBED COST DISTRIBUTION-FIXED SEGMENTATION Let the domain D be divided into segments D, (g = 1, . . ., w ) and let the design cost value $ ( x ) be prescribed in the form

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George I . N . Rozvany and Robin D . Hill

where Ag ( g = 1, . . ., w ) are unknown constants and [,(x) are given functions termed shape functions. Then the Prager-Shield optimality condition takes on the following modified form (Rozvany, 1973~): q h = &Y[d'(Q,)],

h = 1,

.

.) t,

(2.51) (2.52)

1, > 0

only if $ ( x ) = IC/(Qh).

(2.53)

Special cases of the foregoing problem were discussed by Foulkes (1954), Prager (1971% 1974a), and Sheu and Prager (1969). If absolute limits (Agmin,Agma.Jare imposed on the multiplier A,, then for Ag = Agminand Ag = Agmsxthe equality sign in (2.52) changes to 2 and ,I respectively. An upper bound on Rminmay be obtained by finding statically admissible stress fields Q; (h = 1, . . ., t ) that satisfy (2.50) and then integrating $ over D. For homogeneous specific cost functions of order one, a lower bound on Rminmay be established by evaluating (2.37) such that pk are kinematically admissible and are associated with strain rates satisfying (2.51) and (2.52), in which Y[$(Qh)] is replaced by a vector Y* representing any arbitrary point on or inside the cost gradient surface for $(Oh).

F. GENERALIZED LOADSAND REACTIONSOF UNSPECIFIED MAGNITUDE AND NONZERO COSTIN GIVENLOCATION Considering a perfectly plastic structure having the domain D and the specific cost function $(Q),let the generalized reactions (consisting of forces and couples) or loads R acting on a given subdomain S E D be associated with the cost function $(R). Then the optimization problem becomes

(2.54) For an unspecified distribution of $ ( x ) , the extended version of the Prager-Shield optimality condition for the above problem becomes (Rozvany, 1974) 4 x 1 = %[$(Q)I, on D. (2.55) P(X)

= 9[4(R)],

on

s.

(2.56)

In the case of several alternative loads, (2.31)-(2.33) replace (2.55), and (2.56) must also be changed in a similar manner. For a partially prescribed be distribution of $, (2.51)-(2.53) replace (2.55). An upper bound on Rmincan

Optimal Load Transmission by Flexure

197

obtained from any statically admissible system of stresses and reactions on the basis of (2.54). For homogeneous cost functions ($ and 4 ) of order one, a lower bound on Qmi, is given by (2.57) if pk is kinematically admissible with

qk =8*($), on D,

(2.58)

pk = Y*(4),

(2.59)

on S,

where Q*($) and %*(4),respectively, denote any vector representing a point on or inside the cost gradient surfaces for $ and 4. For convex homogeneous specific cost functions of order p, Omi,,may also be calculated from

where p satisfies (2.55) and (2.56). G. JOINTS (CONNECTIONS) OF NONZERO COST

The optimality criteria presented in this section are discussed in the particular context of one-dimensional continua (e.g., beams, frames, rings) but they can be extended readily to multidimensional systems. Assuming that connections of nonzero cost occur at x = x i (i = 1, . . ., z ) and have an infinitesimal length but nonzero cost, the total cost for the system becomes (2.61) where qi( ) is the cost function for the joint at x = x i . For the foregoing problem, the Prager-Shield condition (2.1I) changes to (Rozvany and Mroz, 1975) Q = Q[$(Q)]+

Q { ~ i [ Q ( x i ) ]S(X > - Xi),

(2.62)

i= 1

where 6 is the Dirac delta function (impulse function) and Q and q, respectively, are statically and kinematically admissible. The idea of including the cost of joints in the optimization procedure is due to Prager (1974b) and has also been used by Parkes (1975).

198

George I . N . Romany and Robin D. Hill

(b)

(a)

FIG.4. Optimal velocity fields for joints of nonzero cost.

Considering the connection of a beam, the joint cost function qi( ) may take on the following form: Vi(M

V ) = c l M I +c,

pq,

(2.63)

where c and c1 are known constants, M the bending moment, and V the shear force. Then at x = x i slope discontinuities (relative angular velocities) Oi and velocity discontinuities Ai (Fig. 4) take on the following values (Rozvany and Mroz, 1975):

Oi = c sgn M ( x i ) , lei

1 5 C,

Ai = c1 sgn V(x,),

IAil

IC,,

for M ( x i )# 0,

(2.64)

for M ( x i )= 0,

(2.65)

for

V ( x , )# 0,

(2.66)

for

V ( x , )= 0.

(2.67)

Remark. All optimality criteria discussed in Sections I1,B-G represent necessary and sufficient conditions for a (global) minimum of the cost if the cost functions involved are convex. The criteria discussed in Sections I1,H-L hereinafter are only necessary, even in the case of convex cost functions. Further, the latter criteria will be discussed only in the context of beams under flexure, although their extension to other systems is readily available.

H. DISCONTIN uous COSTFUNCTIONS

+,

Let the specific cost function +(M)for a beam have a discontinuity at M = M i(Fig. 5a). Then the Prager-Shield criterion (2.11) can still be used

M, (a)

DO&O+!

M

(b)

(C)

FIG.5. Problems involving discontinuities in the specific cost.

Optimal Load Transmission by Flexure

199

as a necessary condition if an angular velocity 8 is inserted at the cross sections (x = X) with M = M i . The magnitude of 8 is given by (Rozvany, 1974b)

6=

$i//

V(x)l,

(2.68)

where V is the shear force (Fig. 5b). Modified conditions have been derived for cases when V is discontinuous or M has a local extremum at x = X. I.

PARTIALLY PRESCRIBED

COST DISTRIBUTION-OPTIMAL SEGMENTATION

Considering again a beam with a momentdependent specific cost function $(M), let the cost distribution be prescribed in the form given in (2.50) and let the boundaries of the segments D, (g = 1, . . ., w )be unspecified and to be optimized. Denoting the cost discontinuity at a segment boundary again by A$i (Fig. 5c), (2.51)-(2.53) together with (2.68) will constitute a necessary condition for optimality. This means that in the optimal velocity field u(x), the angular velocities 8 furnished by (2.68) will appear at plastic hinges at the segment boundaries. In relation to optimal segmentation, a proof of (2.68) based on principles of mechanics was obtained by Prager (Prager and Rozvany, 1975) and a variational proof by Rozvany (19744.

J. GENERALIZED REACTIONSAND LOADS OF UNSPECIFIED AND LOCATION MAGNITUDE If the specific cost $(M)is moment dependent and a load or reaction consists of a normal force R1 in an unspecified location x B and has a cost function 4 ( R , , xB),then a necessary condition for the optimal location and magnitude of R , becomes (Rozvany, 1975a)

*;

- $I; + 0,v,

- 0;v; + +,,

= (P,RI~

= 0,

(2.69) (2.70)

where $; and I); are specific costs (Fig. 6), 0;and 0: absolute angular velocities, and V ; and V : shear forces at infinitesimal distances to the left and right of B. If the cost q5( ) does not depend on the location x B of the reaction and there is no cost discontinuity at B, then (2.69) reduces to (Fig. 6)

v,

0;

v;,

= 0;

(2.7 1)

which has also been obtained by Prager (Prager and Rozvany, 1975), who used principles of mechanics.

200

George I . N . Rozvany and Robin D . Hill

(b) FIG.6. Condition for optimal support location.

If, in addition, the cost distribution is unspecified and $(M)is continuous, then there is no slope discontinuity (relative angular velocity) at Band hence (2.71) reduces to WB

= 0,

for R , = 1;'

- V , # 0,

(2.72)

where oB= u ,is ~the absolute angular velocity at B. Equation (2.72) has also been obtained by Mroz (Mroz and Rozvany, 1975). If the reaction R , has a zero cost, then (2.70) changes to u s = 0.

(2.73)

I

If the specific cost is shear dependent in the form $( V ) = k V I and the cost of the load or reaction R , is independent of its location x B , then (2.71) is replaced by (Rozvany, 1975a)

-v,

=

v;.

(2.74)

Considering now an unspecified reaction or load consisting ofa couple R , having a cost function 4 ( R 2 , xB), a necessary condition for the optimal location and magnitude of R , becomes (Rozvany, 1975a)

ll/i - $; + 4.u = = (b.R2'

0 9

(2.75) (2.76)

= 0 and thus (2.75) becomes For location-independent reaction cost, 4,x8

*,

= ?;$

(2.77)

which has also been derived by Mroz (Mroz and Rozvany, 1975). Conditions (2.69)-(2.77) can be extended readily to several alternate load

Optimal Load Transmission by Flexure

20 1

conditions. For example, the generalized form of (2.71) is t

t

(2.78) where the subscript h denotes quantities associated with the hth load condition.

K. A SUPERPOSITION PRINCIPLE

A very useful principle has been introduced by Nagtegaal and Prager (1973), who have shown that an optimal plastic design for two alternative loads P, and P, may be obtained by first deriving optimal solutions for the loads P+ = +(PI + P2) and P- = i(P, - P,) and then superimposing the values of the strength parameters in the solutions for P+ and P-. The same principle was stated in the context of linear programming and trusses by Hemp (1973). 111. General Optimality Criteria-Elastic

Design

A. BASICCONCEPTS In minimizing the cost of an elastic structure, one of the following constraints may be introduced: (i) Strength constraints, where the maximum permitted value of stresses is specified. (ii) Compliance constraints, where the work of generalized loads on the actual elastic displacements is prescribed :

where C is a given constant. (iii) Dejection constraints, where the generalized displacements at a given point or some linear combination of the displacements at a finite or infinite number of points is prescribed. This constraint may be expressed in the form

Jb*P p dx = C, where D* G D, P(x) is a given vector function, and p(x) is the elastic displacement field associated with the loads P(x). P(x) can be regarded as a

202

George 1. N . Rozvany and Robin D . Hill

virtual load system that may contain point loads and distributed loads. For example, a constraint on the deflection or rotation, respectively, at a single point corresponds to a virtual load P(x) consisting of a single point load or couple. Clearly (3.1) is a special case of (3.2). B. STRENGTH DESIGN The field of application of the Prager-Shield condition (2.11) can be extended to optimal elastic strength design if it gives a generalized stress field Q(x) for which the corresponding elastic strains q,,(x) are kinematically admissible. Considering beams obeying Bernoulli’s hypothesis, the foregoing cond ition is, in general, satisfied if the specific cost function JI( ) and stiffness S = E l are related to the moment capacity M in the following form: $ ( M ) = k l IM(”,

S = k , IM12-”.

(3.3)

Then (2.1 1) gives the following optimal plastic curvature rates:

(3.4)

K=klalMP-’sgn M . By (2.3), the corresponding elastic curvatures are K , ~=

MIS = M / k , I M

I P-’

= M

sgn M l k ,

(3.5)

Since (2.1 1) requires K(X) to be kinematically admissible, K , , ( x ) also satisfies the kinematic constraints if the latter consist of prescribed zero displacements at rigid supports, giving 4x1

= 4l(X)7

(3.7)

after integrating (3.6) twice. However, KJX) satisfying (3.6) may be kinematically inadmissible if a given deflection is assigned to a point as in the case of the optimality condition in (2.70). Several problems of practical importance fall into the category defined by (3.3). For various types of rectangular beams, for example, cc in (3.3) takes on the following values:

(i) constant depth but variable width: a = 1; (ii) constant width but variable depth: a = 4; (iii) constant depth/width ratio: a = 4. The first case above is of particular interest to us, because a set of beams of

Optimal Load Transmission b y Flexure

203

(b) M, =PLI8

-Mo=-PL/8

(el

K=-1

K=-1

A

dX)

(0

I

-+

K

FIG.7. Example in which optimal elastic and plastic designs coincide.

prescribed depth constitutes the grillages in the main problem discussed in this study. The foregoing conclusions imply that optimal plastic design of grillages of prescribed constant depth also minimizes their structural weight in elastic strength design. Optimal plastic and elastic strength design may yield the same results even in cases when (3.6) is violated. To demonstrate this point, we shall consider a clamped beam subjected to a central point load P (Fig. 7a). Assuming that the beam has a prescribed constant depth but variable width, which must exceed a specified minimum value, let the specific cost function

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George I . N . Rozvany and Robin D . Hill

for the beam take the form (Fig. 7b) $(M)=

kJM(,

for M 2 M o = P L / 8 ,

(3.8)

klMoI,

for M s M , ,

(3.9)

where k is a constant and M o is the yield moment of the cross section having the prescribed minimum width. The curvature requirements furnished by (2.11) are shown in Fig. 7c and the optimal statically admissible moment diagram ( M ) together with the corresponding velocity field ( u ) and curvaturerates (K)for k = 1 in Fig. 7d-f. The elastic curvature rates induced by the same moments take on the values shown in Fig. 7g if all cross sections with M 2 M o develop the same permissible stress. In Fig. 7 g ko = M O P 0

9

(3.10)

where So is the elastic stiffness of the cross section having the prescribed minimum width. It will be seen that the curvatures in Fig. 7g are also kinematically admissible, and thus the solution in Fig. 7d is optimal for both plastic design and elastic strength design in spite of the fact that (3.6) is violated. However, a simple calculation shows that the two solutions do not coincide if the point load on the beam in Fig. 7a is replaced by a uniformly distributed load. If (2.11) gives a moment field for which the elastic curvatures would be kinematically inadmissible without discontinuities in the slope, then the solution can still be rendered optimal in elastic strength design if it is stipulated that, in order to restore kinematic admissibility, hinges may be inserted at cross sections where the optimal moment field takes on a zero value.

c. DESIGN FOR COMPLIANCE OR DEFLECTION CONSTRAINTS Let the design constraints for a beam be of the type shown in (3.1) or (3.2) and let the specific cost $ depend on the elastic beam stiffness S of unspecijied distribution in the form $ = $(S), where S is required to exceed a given value S o . Then the optimality condition becomes (Rozvany, 1975a) $,s = c M M / S 2 ,

for S > S o ,

(3.11)

2 cMM/S2,

for S = S o ,

(3.12)

$,s

where c is a constant, M and A, respectively, equilibrate with the loads P and virtual loads P [see (3.2)], and both curvature fields MIS and M / S are

Optimal Load Transmission b y Flexure

205

kinematically admissible.? For the case of $ ( S ) = S, (3.11) and (3.12) were obtained by Prager (1971b). If the stiffnessdistribution is prescribed on segments D1,. . .,D, in the form S = A8&,(x),

on D,, g = 1, ..., w,

(3.13)

then the optimality condition changes to (Rozvany, 1975a) (3.14) For the particular case of $ ( S ) = S and LJx) = const, (3.14) reduces to 1 = (c/A,ZL,)

.6,M A dx,

(3.15)

which was first obtained by Prager (1971b). Equations (3.11) and (3.12) are of particular interest to the main grillage problem discussed herein because for beams of given depth $ ( S ) = kS and for the compliance constraint in (3.1) P = F, M = A,and hence by (3.11) and (3.12) k = M 2 / S 2 = K:, ,

for S 2 0,

(3.16)

k 2 M 2 / S 2 = K:, ,

for S

(3.17)

= 0,

which gives the same solution as the static-kinematic criteria (2.13)and (2.14) for optimal plastic design. Since, in a sense, the compliance constraint prescribes the average stiffness of a structure for its design load, the PragerShield condition (2.11) minimizes the weight of elastic grillages of given depth both for a prescribed strength and for a prescribed average stiffness.

D. OPTIMAL LDCATION

OF

SUPPORTS, HINGES,AND SEGMENT BOUNDARIES

Considering a deflection constraint given in (3.2) and introducing the notation

5 = $(S) + c M A / S ,

(3.18)

the condition for the optimum location X~ of a support with Q hinge becomes (Rozvany, 1975a)

I);

+ we Pi +

V , - I); - w B+ PB - QB+ VB+ = 0,

in which w; and w i are slopes and V ; and

(3.19)

V i are shear forces associated

t If the slope of $ ( S ) is discontinuous for some values of S, then +,s is replaced by 9 [ $ ( S ) ] in (3.11), (3.12). and (3.14).

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George I . N . Romany and Robin D . Hill

with the real loads P, and the corresponding quantities with overbars are associated with the virtual loads P. The superscripts + and - denote the fact that the quantity is taken at an infinitesimal distance to the right or left of the cross section B. A specific cost discontinuity may be due to a segment boundary at B or to a discontinuity of the specific cost function owing to the availability of only a finite number of cross sections. If there is a hinge but no support, point load, or cost discontinuity at B, then 3, V, and P are continuous across B and (3.19) reduces to (VO

+ vqB= 0,

(3.20)

where 8 and 8 are relative rotations associated with P and P, respectively. If there is a support but neither hinge nor cost discontinuity at B then (3.19) changes to (cBR + wR), = 0,

(3.21)

where and w are slopes and R and R reactions. If there is a discontinuity in the specific cost but no support, point load, or hinge at B, then (3.19) gives

3;

- I&

= 0.

(3.22)

Equations (3.20) and (3.22)were derived originally by Masur (1974, 1975). The optimization of elastic beams with various design constraints was considered also by Barnett (1961), Haug and Kirmser (1967), Prager and Taylor (1968), Sheu and Prager (1968), Prager (1968, 1970, 1971b), Shield and Prager (1970), Chern and Prager (1970),Chern (1971),and Dafalias and Dupuis (1972). General optimality conditions for elastic systems were proposed also by Masur (1970) and Mroz (1972). A number of papers have dealt with the optimization of beams with dynamic constraints but the latter class of problems is outside the scope of this article. IV. Optimal Flexure Fields-Basic

Geometrical Properties

A. DERIVATION OF OPTIMAL MOMENT-CURVATURE RATERELATIONS We have already established that (2.13) and (2.14) constitute necessary and sufficient conditions for the minimum weight of perfectly plastic beams of prescribed strength and elastic beams of prescribed strength or compliance if the cross sections are rectangular and their depth is constrained to

Optimal Load Transmission by Ffexure

207

a given value.? Considering an arbitrary point P of the middle surface of a grillage of such beams, we may start from a basic layout (Prager, 1974a.b) of potential beams in all directions. Then, (2.14) requires the rates of curvature to take on an absolute value not exceeding k in any direction. It follows that condition (2.13) for a nonzero beam moment may only be satisfied at P in directions of maximum or minimum curvature rates, which are termed principal directions (1, 2) and are known to be orthogonal. Therefore, the moment-curvature rate relations for point P can be characterized fully by xi = k sgn M i , for M i# 0, (4.11 1 K i 1 5 k, for M i= 0, (i = 1, 2) in which xi are kinematically admissible principal curvature rates having the same directions as the statically admissible principal moments M i . The relations in (4.1)are represented graphically in Fig. 8% in which the

(+> R+

o + o o s+

R-

s-

T

(b) FIG.8. (a) Optimal moment-curvature rate relation and (b) symbols for optimal regions.

principal moments M , and M , are the Cartesian coordinates, contour lines represent a constant value of the specific cost $, and vectors normal to the contour lines give the same rates of curvatures as (4.1). If MI 2 M , , then the t See Sections II,B, 111,B.and II1,C.

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George I . N . Rozvany and Robin D . Hill

foregoing relation admits the following type of optimal regions in the so1ution:t

(A) R +

~1

(B) R-

(K1

=k,

1 Ik,

KZ=

= K, = k,

(AC) S+

ic1

(BD) S -

I C ~= K Z =

(AB) T

~1

=

-K,

(K11 I

(E) 0

I k,

12 .1

Ml 20,

-k,

MI 2 0,

M2=0,

MI =O,

M, 2 0,

-k,

M1 I0,

M , 5 0,

= k,

M1 2 0,

M, 5 0,

k,

Ik,

1 ~ 2 1

M , 50, (44

M,=M,=O,

where the letters in parentheses refer to points and line segments in Fig. 8a. If all points of a domain are subjected to a nonzero loading, then‘O-type regions of finite area cannot occur in the solution. In this article, R-type regions in figures are marked with arrows in one direction (Fig. 8b), S-type regions with small circles, and T-type regions with arrows in two directions. Arrows signify the directions of nonzero principal moments if they are determinate and the sign of principal moments is also indicated (Fig. 8b). Moment and curvature rate fields satisfying (4.1) are termed optimalJewre Jields. It is a point of interest that apart from the optimization problems considered, the same fields represent the solution to the problem of analysis of plates made out of ideal locking material (Prager, 1957) having the locking surface K~ = k ( i = 1, 2). In addition to the Prager-Shield condition, (4.1) may also be derived from theories of Masur (1970), Mroz (1972) and Save (1972), or by a direct variational method (Rozvany, 1974f).The same condition gives a minimum reinforcement volume for perfectly plastic fiber-reinforced plates of prescribed depth and strength (Morley, 1966). Since beam moments may only occur in principal directions and the specific cost function for all beams is t,b = k I M 1, where k is a known constant and M the beam moment, the optimal “cost” representing the minimum weight of the grillage becomes

*

Qmin=j(IM,I D

+

I M , I ) ~ ~ ~ Y ,

(4.3)

where M, and M, satisfy (4.1) and x and y are Cartesian coordinates. By (2.20) and (2.21) the minimum cost of the grillage may also be calculated t Symbols denoting the regions were proposed by Prager (1974a).

Optimal Load Transmission by Flexure

209

from either of the following expressions:

Qmin

=

I,

PU dx dy,

(4.5)

where M i and x i(i = 1, 2) satisfy (4.1), u(x, y) is the velocity field associated with the foregoing curvature rates, and P(x, y ) is the load. Obviously (4.3) gives an upper bound on the minimum cost for any statically admissible moment field and by (2.26), (4.5) gives a lower bound if the rates of curvature associated with u(x, y ) satisfy the condition

I K ~ I < k,

i = 1, 2.

(4.6)

B. PROPERTIES OF S- AND T-TYPEREGIONS In S-type regions both principal curvature rates take on the same value (k or -k) and therefore the rate of curvature is the same in all directions. Then, by (2.13), the directions of nonzero moments are indeterminate and only their sign is prescribed. If we replace K~ and I C with ~ - u ,and~ ~ in (4.2), then the curvature conditions for S-type regions furnish, after integration, the following general equations for the velocity fields u(x, y):

+ y 2 ) + a + bx + cy, u = &k(x2+ y2) + a + bx + cy,

S + : u = -&k(x2

(4-7)

S- :

(4.8)

where a, b, and c are constants. Next, it is shown that in R- and T-type regions, the centroidal axis of beams having a nonzero moment capacity is always straight. Shield (1960) has obtained the following relation for principal rates of curvature in plates: dKl/dS2 = (l/Pl)(K2

- Kl),

(4.9)

where s2 is a curvilinear coordinate measured in the direction of a line of principal curvature rate x2, and p1 is the in-plane curvature of the line of principal curvature rate K ~ . If M # 0, then by (4.1), icl = const (ie., k or - k). The derivative in (4.9) therefore takes on a zero value, and hence p1 = co for K~ # K ~ Q.E.D. .

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George I. N. Rozvany and Robin D. Hill

This means that T-type regions consist of principal lines in constant directions at right angles, and the general equation for their velocity field is T: u = fk(y2 - x 2 ) + u T: u = kxy + a

+ bx + CY,

+ bx + cy,

(4.10) (4.11)

respectively, for principal directions parallel and at 45" to the coordinate axes x and y. The symbols a, b, and c denote constants. In T-type regions, the direction and signs of the principal moments are prescribed. However, for both T- and S-type regions, the foregoing conditions, in general, admit an infinite number of moment fields, which all yield the same optimal cost.

c. PROPERTIES OF R-TYPEREGIONS R-Type regions are less restricted kinematically than other types and therefore require a longer treatment. It is convenient to consider first a more general class of velocity fields that may be used for constructing R + regions. Defnition I . A velocity field u(x,y ) defined on D t E 2 is of Rt-type if (i) u(x, y ) is differentiable on the interior of the domain 6, and (ii) for each point P E D there exists at least one straight line 1 termed basic line with P E 1, such that along 1 n d and in the direction of 1 the curvature rate K of u(x, y ) takes on a constant value k. If at all points contained in 1 K = k is a principal curvature in the direction of 1, then 1 is termed a principal line. Defnition 2. For a given constant C, the contour lines K1 and K 2 contained in the domain D of an R+-type velocity field are defined by the relations y =f;(x) (i = 1, 2) such that u[x,f;.(x)] = C,

i = 1, 2.

(4.12)

Remarks. All R+-type regions contain an 8'-type velocity field, but the latter may form an R+-type region only if all its basic lines are principal Iines and the absolute value of the curvatures in a direction normal to basic lines nowhere exceeds k. In the propositions that follow, the terms basic line, principal line, and contour lines will always refer to lines in the domain of an R+-type velocity field. A trivial case of R+-type regions consists of parallel principal lines. If the latter are in the y direction, then the velocity field is given by u =f(x) + a + by - ky2/2 with 1 f x x I 5 k, where a and b are constants. R-type regions with nonparallel principal lines are termed fans.

Optimal Load Transmission by Flexure

211

It is known that at all points of a principal line the angular velocity (slope) in the direction normal to such a line is the same. Proposition 1 . A basic line 1 is a principal line if and only if it intersects K1 and K , at the same angle (a in Fig. 9a). Proposition 2. If 1 is a basic line and if at both PI E 1 and P , E 1 with PI # P , the angular velocity o in a direction normal to 1 takes on the same value, then 1 is a principal line. Proposition 3. The angular velocity co, along and in the direction normal is given by to a principal line W,

(4.13)

= k(FW),

't

1

(bf

~

N

P W

(Cf

(dl

FIG.9. Properties of R-type velocity fields.

George I . N . Rozvany and Robin D . Hill

212

in which the distance F W is furnished by the construction in Fig. 9a, where AW I K,, BW I K , , and AW = BW.? Proposition 4 . The curvature K, normal to a principal line AB (Fig. 9a) is given by

(4.14) with 4 = (f;'x+ 1 -lr,.xx) L o

Y

P =W

X

+ 1 +ffi.xx)

lx=o

9

where ( i = 1, 2 ) are the functions in (4.12), the coordinate Fig. 9a,f(O) =f1(0)= -f2(0), andXx(0) =f,,x(O) = -f,,x(O).

b =f(O),

(4.15)

is defined in

Proposition 5 . Considering a principal line D, the point N for which - k and the point Q for which IC, = - co can be found on the basis of the construction in Fig. 9c, in which p and q are furnished by (4.15).Another meaning of p and q is explained in Fig. 9d, in which AB and 71'8; are principal lines, A 4 -+ 0, and BPIAO = p/q.

K, =

Proposition 6. Let F and F' be points of maximum velocity along the and A", respectively, and let A 4 be the angle between AB and A B . Then for A 4 -0,

principal__ lines

z = o2A4/k,

(4.16)

where the distance z is defined in Fig. 9b and 0,is the angular velocity along Equation (4.16) implies that tan y = and in a direction normal to m2/kr(4) in Fig. 9b.

a.

Dejnition 3. The point W (Fig. 9a) with AW = BW, A E K , , B E K , , AW I K , , BW I K , , on K , and K, u = const, is called an intercept point of the principal line 3. The locus of all intercept points of an R'-type field is termed the intercept line. By Proposition 3, the intercept line is the same for any pair of contour lines of an R'-type field. Proposition 7. FW having the properties FW I tangent to the intercept line at W (Fig. 9a).

a, max,,

u = u ( F ) is a

Dejnition 4 . The curve that is contained in the domain of an R'-type field and to which all principal lines are tangent is termed the envelope of the 17'-type field. Point Q in Fig. 9c is contained in the envelope. t The symbol I indicates that two lines or curves are orthogonal at their intersection.

Optimal Load Transmission by Flexure

213

Instead of expressing velocity fields u( ) in R-type regions in Cartesian coordinates (x,y), it is often convenient to adopt coordinates (4, O, M,SO,

M, S O .

M,=O, Mo=O.

(4.59) (4.60) (4.61) (4.62)

As was indicated earlier, the optimal moment field is usually nonunique in S-type regions. Optimal velocity fields satisfying (4.1) for nonnegative (downward) loads are shown in Fig. 13 and the corresponding solutions are given in Fig. 14. Here, and throughout this article, thick lines in plan view denote clamped edges, double lines simple supports, and single lines free edges. Solutions in Fig. 13% b, e, f, h, i, j, k, 1, m, and n are similar to the corresponding solutions for optimal beams obeying Heyman’s (1959) condition (2.13) and therefore require no further explanation. The value of p in Fig. 13c may be calculated by considering the extension (broken line) of u(r) to the axis of symmetry,

~ ( 0=) - kpi/2,

(4.63)

and thus the curvature rates indicated give the following condition for u(p) = 0 :

-Pi/2

+ Pl(P - P

d ) - (P - P1I2/2 = 0,

(4.64)

or p

- p1 = (p2/2 - p6/2)”2.

(4.65)

When po = p/3, (4.65) gives p1 = p/3 and thus no R, region occurs in the solution. Hence for po p/3, the solution becomes the one shown in Figs. 13d and 1 4 . The validity of the latter solution may be checked by the construction shown in Fig. 13d, in which the point P must lie inside the inner support Q for the solution indicated. It can also be checked on the basis of (4.14) or the construction in Proposition 5 (Fig. 9c) that u2 = -k along the inner support if po = p/3. FIG. 14. Optimal solutions for axially symmetric supports and loading. In S + regions, in S- regions K, = = - k ; in R C regions K, = k and in R - regions K, = - k .

K, = K~ = k ;

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George I . N . Rozvany and Robin D . Hill

The position of the optimal region boundary in Fig. 14g may be derived from the boundary conditions ~ ( 0= ) -p:/2,

u,,(O) = 0,

~ ( p=) 0,

u,,(p) = 0.

(4.66)

The region boundaries in Fig. 14a-n do not depend on the distribution of a nonnegative load P ( r ) 2 0. However, due to simple supports on the interior of the domain D, some region boundaries are dependent on P ( r ) in Fig. 140-r. It can be seen in ~ the simple support is kinematFig. 130, for example, that the slope u , at ically indeterminate, but it becomes unique if equilibrium together with conditions (4.1) are also taken into consideration. For further details and applications of the solutions listed herein, the reader is referred to earlier papers (Rozvany, 1968; Rozvany and Melchers, 1970). It was pointed out correctly by Melchers (1975a) that region boundaries for axisymmetric domains without internal supports (Fig. 14a-n) are valid for nonaxisymmetric loads also. In fact, all optimal velocity fields discussed in Sections V and VI are known to be (Rozvany, 1972,1973g)independent of the choice of a nonnegative load P( ) 2 0. V. Optimal Flexure Fields-Clamped

Boundaries

A. REVIEWOF LITERATURE ON NONAXISYMMETRIC OPTIMAL FLEXURE FIELDS

Early theories for fiber-reinforced plates of minimum reinforcement content were proposed by Mroz (1958) and Kaliszky (1965). The optimal moment-curvature rate relations in (4.1) were obtained independently by Morley (1966) and Mroz (1967), in the context of reinforced concrete slabs. The optimal flexure field for square simply supported domains was derived by Morley (1966). The same solution was presented earlier for slightly different design constraints by Rozvany (1966a,b). Morley (1966) also obtained the correct solution for simply supported rectangular domains. Reitman (1967) and Clyde (1967) confirmed by linear programming the analytical optimal solution (Rozvany, 1966a) for square simply supported domains. Sacchi and Save (1969) derived an optimal solution for elliptic simply supported plates. From static (upper bound) considerations and without a proof of global optimality, Melchers (Lowe and Melchers, 1972) obtained the correct solutions for square and polygonal clamped boundaries and for regular simply supported triangular boundaries. The validity of Melchers’ solutions was soon proved on the basis of purely kinematic considerations arising from (4.1) (Rozvany and Adidam, 1972b).

Optimal Load Transmission by Flexure

227

The optimal moment-curvature rate relations in (4.1) yielded solutions and methods for rectangular boundaries with various support conditions (Romany and Adidam, 1972c), clamped boundaries of arbitrary shape (Romany, 1972, 1973e,f), simple supports (Romany e! al., 1973), mixed boundary conditions (Romany, 19738, 1975b), corners (Romany, 1974g), interior supports and alternate loading conditions (Romany, 1974d, 1973g), and edge beam supported domains (Lowe and Melchers, 1974a).

B. PROPERTIES OF SOLUTIONS FOR CLAMPED BOUNDARIES Topographical properties of optimal flexure fields were discussed in Section IV,E. The particular case of clamped boundaries (bp branches) will be considered first because of the comparative simplicity of their geometry. pfl Branches are bounded by two clamped edges ( d , and d , in Fig. 15) and have the following properties.

FIG. 15. Property 1.

Property 1 . A BB branch consists of one inner and two outer regions. In general, the outer regions are of R- type and the inner region of R+ type (which may degenerate into S - and T - or S+-type regions, respectively; see Properties 4 and 5). The region boundaries (b, and b, in Fig. 15) between the inner and outer regions and the centerline h may be determined on the basis of the construction in Fig. 15 in which C A = AW = EB = BW;

228

George I . N . Rozvany and Robin D. Hill

AF = BF; CW I d , ; EW I d , ; A E b , ;B E b,; F E h ; K , = - k on C A and BE; and K~ = k on AB. W is termed the intercept point. The construction in Fig. 15 is subject to the restrictions CW < r ( d l ) c , EW < r(d&,

for r(dl)c > 0, for ~ ( d , > ) ~0,

(54

where r( ) denotes the radius of curvature of the domain boundary and the sign convention for r( ) is given in Fig. 16. d

d

6

(b)

(a)

FIG. 16. Sign convention for radius of curvature of domain boundaries.

Property 2. A branch centerline ( h in Fig. 17) and the region boundary (b in Fig. 17) may only have their intersection G at a distance r(&/2 from the point H of locally maximum boundary curvature, measured at right angles to the domain boundary d. d

FIG. 17. Property 2.

Property 3. If the intercept point (W in Figs. 15 and 18) is at an equal distance from n boundary points with n > 2, then the inner region changes into an n-sided S+-type region in the vicinity of such intercept point, and its boundaries can be constructed on the basis of Fig. 18 (which shows the specific case n = 3 ) .

Optimal Load Transmission b y Flexure

229

FIG. 18. Property 3.

The foregoing S+-type region is termed a junction of n /?/?-typebranches and the point W is called the center of the junction. If W is contained in the junction, then it represents a local maximum of the velocity field u( ). Remark. Property 2 can be used for locating the outer end (points A, B, C, D, and E in Fig. 12) of centerlines of branches, which can be used as a starting point for the construction of branches on the basis of Property 1. The inner end of branches (points I, J, K , L in Fig. 12) is found by employing Property 3 and further branches may initiate from such inner ends (branches ZJ, JK, and K L in Fig. 12). Branches with at least one outer end (on the domain boundary) are termed exterior brunches (AZ, BZ, CJ, DL, and EL in Fig. 12) and branches with two inner ends interior brunches. A pair of junction centers (e.g., K and L in Fig. 12) may be connected by two interior branches. The boundary curvature (l/r) takes on a value - co at F and G and CQ at A, B, and D. A local maximum of the curvature with r = r/2 = 0 occurs only at the latter three, and hence by Property 2, the region boundary is at a zero distance from the domain boundary at A, B, and D.At C and E in Fig. 12, r/2 takes on a finite value.

+

Property 4 . The outer regions degenerate into S--type regions around reentrant corners and clamped-point supports, and the inner region degenerates into a T-type region between two such S--type regions. The foregoing regions can be constructed on the basis of Fig. 19. Boundaries between S - and R--type regions are normal to the domain boundary segments adjacent to the reentrant corners.

230

George I . N . Romany and Robin D . Hill

FIG.

19. Property 4.

Property 5 . When an intercept point ( W in Fig. 20) is at a constant distance r from a circular domain boundary segment, then the inner region degenerates into an S+-type region (Fig. 20) and the centerline intersects the circular domain boundary segment at its midpoint (E in Fig. 12).

\ FIG.20. Property 5.

Property 6. If a fifi branch is associated with a clamped point or reentrant corner and a straight boundary segment at a distance a (Fig. 21) from each other, then the region boundaries are parabolas:

and the centerline (dash-dot line), intercept line (broken line), and envelope

Optimal Load Transmission by Flexure

23 1

-x

FIG.21. Property 6.

(dash-double dot line), respectively, are given by y , = a/2 + 4x:/9a,

yw = a/2 + x&/2a,

7 = 1.25a - 3(Js/4)2f3a1f3,

(5.4) where x and y are the Cartesian coordinates shown in Fig. 21. The principal lines in the inner (R') regions, are normal to the parabola given by (5.3). [Properties 1-6 were obtained by Rozvany (1972).]

c. PROOF OF PROPERTIES 1-6 1. Proposition 8 Let the velocity field u( ) be defined on D such that u( )is continuous on D and is differentiable on the interiors of R, c D, R, c D, and R , c D. Further, let the points A and B be contained in the common boundaries of R , / R 2 and R , f R 3 ,respectively, and let (Fig. 22a)

(i) the curvature rate on AB c R, be K = k; (ii) the angular velocities at A and B in R , and R J , respectively, be

o , / k = AW,

o , / k = BW,

aA= 0, O B = 0, where F E AB with FW I a (Fig. 22a);

(5.5)

(5.6)

U A - u s = (w: - Wi)/2. (5.7) (iii) Then the slope is continuous across the region boundaries at A and B.

232

George I . N . Rozvany and Robin D . Hill

(b) FIG.22. Propositions 8 and 9.

2. Proof of Proposition 8 Considering the velocities at A, M, and N with ( A M )-P 0 in Fig. 22,

+ o A ( A M ) + O(AM)’, UN = u y + O(MN)’,

UM = U A

(5.8) (5.9)

and hence u,? I A

cos 4,

= (UN - u&AN = wAAM/AN =

on R , .

(5.10)

Similarly, u , IB~ = oBcos rl,

on R3 ,

(5.11)

Next, it is shown that max u = u(F).

(5.12)

AB

If (5.12) holds, then by condition (i) and Fig. 22b, UA

- U S = +k[(AW)’ COS’ 4 - (BW)’ COS’ q].

(5.13)

However, by the sine rule (AW)sin (b = ( B W )sin q,

(5.14)

Optimal Load Transmission by Flexure

233

which implies (BW), cos2 q = (BW)Z - (AW)2(1 - cos2 4).

(5.15)

Substituting (5.15) into (5.13) and making use of (5.5), we get

(5.16)

Since the value of (uA- us) uniquely determines the location of u,,, along AB with K = k, the identity of (5.7) and (5.16) proves (5.12). Then it follows from Fig. 22b and (5.5) that u , IA~ = k(AW) cos

4 = O ~ ' C O S 4,

on R , ,

(5.17)

u , lg~ = k(BW) cos

= wg cos q,

on R2 .

(5.18)

Equations (5.10), (5.11), (5.17), and (5.18) establish slope continuity at A and B in direction AB. At A and B, the slope is also continuous in the direction of the region boundaries and then, due to slope continuity on the interior of R1, R, ,and R , , the slope is also continuous in any arbitrary direction.

3. Proposition 9 If AB in Fig. 22 is a basic line (see Definition 1) then it is a principal line. 4. Proof of Proposition 9

In the direction normal to AB, the angular velocities at A and B, respectively, are 0: = w A sin

4,

(5.19)

wz = w g sin q.

Then by (5.5) and Fig. 22, (5.20)

w: = k(FW) = o;,

and thus Proposition 2 implies that AB is a principal line.

5. Proof of Property 1 The construction in Fig. 15 satisfies all requirements of Proposition 8 with = k(AW) = m g = k(BW),

UA

-

= 0.

(5.21)

The restriction a,= a,= 0 follows from the fact that at C and E the angular velocity w is zero in all directions, and CA and EB are principal lines along which the angular velocity (slope) is constant in the normal direction. The principality of CA and EB can be shown by replacing each clamped boundary (e.g., dl in Fig. 15) with two simply supported lines (d;and d'; in

234

George I . N . Rozvany and Robin D. Hill

FIG.23. Proof of Property 1.

Fig. 23) at a constant distance 6 and then applying Proposition 1 to the simple supports. The condition of principality in Proposition 1 does not depend on 6 and thus it still holds when 6 -+ 0, giving the same kinematic constraint as a clamped edge. The limitation (5.1) follows I'rom (4.14) and (4.15) and Fig. 23, in which points C' and C" correspond to A and B in Fig. 9a. The quantities in (4.15) take on the following values:

Then (4.14) gives for S

0 (5.23)

which gives K , < - k for 5 > r(dl)c/2 and hence at A (4.1) is violated when the requirements in (5.1) are not met. A different proof of (5.1) was given elsewhere (Rozvany, 1973f). It follows from Proposition 9 that AB in Fig. 15 is also a principal line. This concludes the proof that the velocity field u( ) generated by the construction in Fig. 15 ensures kinematic admissibility and satisfies (4.1) if the corresponding moment field (Ml, M,) is statically admissible. The latter condition can always be satisfied for a nonnegative (downward) load P( ) 2 0, as can be seen from Fig. 24, in which s is the beam spacing and F A

Optimal Load Transmission by Flexure

235

(b)

(a)

FIG.24. Beam moments in (a) outer and (b) inner regions.

and F , the forces transmitted from the “suspended” beam AB to the “cantilever” beams C A and EB. 6. Proof of Property 2

Considering the boundary configuration in Fig. 25a, it follows from (5.1) that a clamped circular domain boundary segment cannot be adjoined by another circular segment of greater curvature. If a -+ 0, fl --t 0, in Fig. 25a

(a)

FIG.25. Proof of Property 2.

then the same diagram indicates that the curvature cannot increase as we move away from the optimal centerline. Hence the centerline must intersect the domain boundary at a point of locally maximum curvature. However, the converse of the foregoing statement is not always true, that is, not all boundary points of maximum curvature represent an outer end of a centerline. In Fig. 25b, for example, the boundary segment CT is adjoined by boundary segments of smaller curvature, and yet no centerline intersects the boundary segment because the distance C W is smaller than the radius of curvature r. The same would apply if CT had a nonuniform curvature but the radius of curvature at the local maximum curvature exceeded the corresponding distance CW. The distance of points H and G in Fig. 17 can be determined by applying the construction in Fig. 15 to domain boundary points on both sides of H

236

George I. N. Rozvany and Robin D. Hill

H

r(d),+/2 . I

FIG.26. Proof of Property 2.

and then taking the limiting case for A 4 -+ 0 (see Fig. 26), giving

+ H F = H G + GF = $WC + O(A@)

WC = ~ ( d ) O(A@), ~

=fr(d)H

+ O(A@).

7. Proof of Property 3

Using the construction in Fig. 18 for determining the region boundaries of a junction of fifi branches, it follows from Proposition 3 that in the R+-type regions around the junction the angular velocities along and in a direction normal to the sides AB, AZ, and BZ in Fig. 27 take on the values k(FW),

% FIG.27. Proof of Property 3.

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237

k(F‘W), and k(F”W), respectively. If the junction consists of an S+-type region with the maximum velocity at Wand we adopt Cartesian coordinates as shown, then (4.7) gives 0

= -u,xx

IxqW

= k(FW).

(5.24)

This establishes slope continuity along AB and similar conclusions may be obtained for A Z and BZ by an appropriate rotation of the coordinate axes.

8. Proof of Property 4 Considering the two reentrant corners in Fig. 28, along the line AB the slope in the normal direction is (Proposition 3) u,x I A 8 = k ( W .

(5.25)

FIG.28. Proof of Property 4.

Since xF = (TF) + r sin A+ = (TF)+ O ( r ) = ( F W ) + O(r), for r 40, u,x JAB

= kx \ A 8

7

(5.26)

and thus K~ = - u , , ~= - k . The region boundaries follow from the construction in Fig. 15.

9. Proof of Property 5 For a clamped circular boundary segment, the construction in Fig. 15 gives a constant velocity u = C along the region boundary b (Fig. 29) and hence the latter is a contour line. Then the quantities in (4.15) become (Fig. 29) J,(O) = cot a, f(0) = ( r sin a)/2, (5.27) f,,,,= -f2,xx = -2(1 + cot2 ~ t ) ~ ’ /= ” / -2/r r sin3 a,

238

George I. N.Rozvany and Robin D.Hill

FIG.29. Proof of Property 5.

and hence p = q = 1 + cot’ a - l/sin’ a = 0.

(5.28)

Then by (4.14) ~2

= k,

(5.29)

giving an S+-type region along AB. 10. Proposition 10 Let an S-type region and an R+-type region have a common boundary b. Then the principal lines in the foregoing R+-type region are normal to b. 11. Proof of Proposition 10

Considering a point P E b, the curvature in a direction that is tangent to b takes on a value - k. Then by (4.1), any tangent to b is a principal direction for the velocity field in the adjacent R+-type region. Due to the orthogonality of the two principal directions, the principal line containing P is normal to b. 12. Proof of Property 6 In Fig. 21, by the Pythagorean relation,

- a)’ ( 2 ~=~(2yB ) ~

+ xi ,

(5.30)

Optimal Load Transmission by Flexure

239

which gives (5.2). Further, EW = 2yA- a,

(WC)’ = 2’(y, - a)’

+ 2’x2,,

(5.31)

and hence the condition E W = C W reduces to (5.3). Equation (5.4) follows readily from (5.2), (5.3), and the relations (see Fig. 21) y~ = ( y ,

+ yB)/2,

XF

= 1.sxA = 0.75xB.

(5.32)

The second equation (5.4) can be derived from the relations (Fig. 21) yW=2yB,

xW=xB.

(5.33)

The orthogonality of the principal lines in the inner region (R’) to the region boundary given by (5.3) follows from Proposition 10. The equation of the envelope can be obtained by adding to the x and y values in (5.2) the coordinates of the radius of curvature for the curve represented by (5.2). D. EXAMPLES 1. Convex Polygonal Domains Using the foregoing properties, solutions may be derived readily for triangular (Fig. 30a), quadrilateral (Fig. 30b-e), and other polygonal (Fig. 30f-g) domains. It can be seen from Fig. 30b-e that quadrilateral domains may contain one or two S+-type regions (junctions). The optimal solution for regular n-sided polygonal domains (Fig. 30f) contains n R--type regions, n R+-type regions, and one S+-type region. In the case of irregular polygonal boundaries, the S+-type region splits into several parts separated by R+-type regions (Fig. 30g). 2. Domains Consisting of the Complement of Bounded Sets In Fig. 31, solutions are given for point supports (Fig. 31a-b), rectangular supports (Fig. 31c), a combination of octogonal and square supports (Fig. 31d), cross-shaped supports (Fig. 31e), circular supports (Fig. 31f,g), and square supports having diagonals parallel to the grid lines (Fig. 31h). In Fig. 314 the region boundaries in polar (8, p ) and Cartesian (x, y ) coordinates are given by

+ b], x = 3(a tan 8 + b sin 8),

(5.34)

p = $[(a/cos 0)

y = f(a + b cos 0).

(5.35)

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George I . N . Rozvany and Robin D. Hill

(el (Cl

FIG.30. Solutions for clamped convex polygonal domains.

c

I

b

(a)

X

I,

-3b

,

*I

(g) FIG.31. Solutions for the complement of bounded sets.

242

George I . N . Rozvany and Robin D. Hill 3. Combination of Straight-Line and Point or Circular Supports

Figure 32 gives solutions for a rectangular domain with a point support (Fig. 32a), combination of point and line supports (Fig. 32b), and circular and line support (Fig. 32c). Curved lines in Fig. 32a,b are given by (5.2) and (5.3). The value of b in Fig. 32b can also be calculated on the basis of (5.2), in which x B = a12 giving b = 5a/16. (5.36) .12

,

./2

,

.

I

,

./2

, r

.I2

,

FIG. 32. Combination of line and point or circular supports.

In Fig. 32c, the region boundaries may be derived in polar coordinates (0, p ) from the relation z = (a - R cos 0)/(1

+ cos 0).

(5.37)

4. Polygonal Domains with Reentrant Corners

Figure 33 shows the solution for L-shaped (Fig. 33a), T-shaped (Fig. 33b), and cross-shaped domains (Fig. 33c).

Optimal Load Transmission by Flexure

,"

243

(b) (a)

(C)

FIG.33. Solutions for domains with reentrant corners.

In Fig. 33a,

b = (a - c)/2.

(5.38)

Substituting xe = c, y e = b, (5.2) and (5.38) yield c = ($

- l)a,

b = (2 - $)a/2.

(5.39)

The value of b in Fig. 33b is furnished by (5.37). 5 . Domains Consisting of the Union of a Half-Space and Bounded Set

Figure 34 shows solutions for a combination of half-space and semicircle (Fig. 34a), triangle (Fig. 34b), rectangles (Fig. 34c-d), rectangle and two quarter-circles (Fig. 24e), half-ellipse (Fig. 34f), triangle and trapezium (Fig. 34g), and a solution obtained by superposition (Fig. 34h).

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George I . N . Romany and Robin D. Hill

(h)

FIG.34. Solutions for the union of a half-space and a bounded set.

6. Domains Having Curved Boundaries

For a symmetric curved boundary y = k f ( x )(Fig. 35a), the region boundaries on the basis of Property 1, Fig. 15, are given by

7 = y/2,

R =x

+ y,,y/2,

Y,x = Y,,/(2 + Y Y , x x + Y

3.

For a parabolic boundary (Fig. 35b) with y = _+xl/’ (5.40) yields j j = (x - 1 4) 11212.

(5.40)

(5.41) (5.42)

Optimal Load Transmission by Flexure

&

$

Q

+

x

YY

(a)

FIG. 35. Solutions for domains with curved boundaries.

245

246

George I. N. Romany and Robin D.Hill

For an elliptic boundary (Fig. 5.352) given by y = + b ( l - x2/a2)1’2,

(5.43)

(5.40) furnishes the elliptic boundary jj

= b[ 1

- js2/a2(1 - b2/2a2)2]”2/2.

(5.44)

For semicircular and diamond-shaped domains the solutions are given in Figs. 35d and 35e-g, respectively. In the former, y, = R sin O/2( 1 + sin O),

X,

cos 8/2( 1 + sin O),

=R

pc = R[ 1 - sin O/2( 1

+ sin O)],

(5.45)

(5.46)

where R is the radius of the domain boundary. In Fig. 35f polar (6, r ) and Cartesian (x, y) coordinates of a general region boundary point and Cartesian coordinates (xH, yH)of the point H,respectively, are

”(

P =2 1 y = (R/2)(1

- cos O),

+

A),

x = (R/2)(sin 0 + tan O),

x = 2(R - y)(Ry - y2)”’/(R

YH = R - XH = R($

VI. Optimal Flexure Fields-Mixed

- 2y),

- I)/@.

(5.47) (5.48) (5.49) (5.50)

Boundary Conditions

A. BRANCHES CONTAINING a

AND

p FIELDS ONLY

Along simply supported boundary segments, three classes of optimal fields termed a, y, and 6 types may occur in the solution. Because of their relative geometrical simplicity, aa- and ap-type branches will be discussed first and branches containing y- or &type fields will be dealt with in Section VI,D. Basic properties of a fields are (i) The field consists of an R+-type region?. t In limiting cases, a fields may degenerate into an S + - or T-type region. The latter permits nonzero moments along the boundary.

Optimal Load Transmission b y Flexure

247

(ii) Both principal moments take on a zero value along the domain boundary. (iii) The curvature is IC, = k along principal lines. It will be seen later that y fields consist of an S--type region and in 6 fields the two principal curvatures take on values of K 1 = k and I C ~= - k along the ~ - k on the interior of the field. domain boundary with I C = uu Branches are bounded by two simply supported edges and afl branches by one simply supported edge and one clamped edge. Their properties are as follows: Property 7. The principal lines of an aa branch may be constructed on the basis of Fig. 36, in which AW = BW, AW I d , , BW 1 d , , AF = BF,

FIG.36. Property 7.

F E h, and ic, = k on AB, where d , and dz are simply supported domain boundaries and h is the branch centerline. The foregoing construction is admissible only if

I f t x(o)

I5

i = 1, 2,

(6.1) where y = f , ( x ) and y = f z ( x ) represent the boundaries d , and d 2 in the coordinates shown in Fig. 36. +fi(O)fi,XX(O)

1 3

Property 8. If (6.1) is satisfied as an inequality then the corresponding part of an ua branch is an R+-type region. If fi(x) = f 2 ( x ) = f ( x ) and (f:x IX=,, = 1, - 1, respectively, then the uu branch degenerates into an S + - and a T-type region. The second principal curvature at domain boundaries is furnished by

+xxx)

K

~

-=k ( f : x + l ; . f i , x x ) I x = O ,

i = 42.

(6.2)

248

George I . N . Romany arid Robin D. Hill

Property 9. The centerline h of a branch may only intersect the domain boundary d at a local maximum of the curvature of d. Positive boundary curvature is defined in Fig. 16. Property 10. If the intercept point W in Fig. 36 is at an equal distance from n domain boundary points with n > 2, then an n-sided S+-typeregion occurs in the solution, which can be constructed on the basis of Fig. 37 (in which n = 3).

FIG.37. Property 10.

Property 1 1 . The principal lines of a/3 branches may be constructed on BW = CB, A W I d,, BC I d , , the basis of Fig. 38, in which A W =

&2

&2

FIG. 38. Property 11.

FW I AB, F E h, and B E b, where d , is a simply supported and d , a clamped domain boundary, h the branch centerline, and b a region boundary. AB and BC, respectively, are principal lines in R + - and R--type regions. The foregoing construction is admissible only if

I

f,’x + f x x

1,

(6.3)

for rc(d2)> 0,

(6.4)

lx=o

and C W c rc(d,),

where y = f ( x ) represents d,, axes x and y are given by FW and FA (Fig. 38), and rc(d2)is the radius of curvature of d, at C .

Optimal Load Transmission b y Flexure

249

Property 12. The centerline of an a/? branch may only intersect the domain boundary at a point where a line of simple support changes to a line of clamped support.

For the construction of junctions of aa, ap, and Property 27.

pp branches, refer to

Property 13. An S--type region occurs at a corner of a domain if the angle between two adjacent sides is

(i) > n/2 for two simply supported edges (Fig. 39a), (ii) > 3 4 4 for one simply supported and one clamped edge (Fig. 39b), and (iii) > n for two clamped edges (Fig. 39c).

(b)

(C)

FIG.39. Optimal flexure fields for corners.

Property 14. Let an ap branch be associated with a clamped-point support or reentrant corner C and a straight simply supported line support AA'

FIG.40. Property 14.

George I. N . Rozvany and Robin D. Hill

250

at a distance a from each other. Then the region boundary is given by (Fig. 40) y i = x i + a2/2.

(6.5)

B. PROOF OF PROPERTIES 7-14 1. Proof of Properties 7 and 8

It follows from Proposition 1 that the line AB in Fig. 36 is a principal line. By substituting = b and = - b into (4.14), we get

r


r 2 , w < - 1 and hence (6.1) is violated. If we extend the foregoing argument to the case a -+ 0, -+ 0, then it follows that the curvature cannot decrease as we move away from the intersection of the centerline h and the domain boundary d. 3. Proof of Property 10 For this property the proof is the same as for Property 3 (see Section V,C,7). 4. Proof of Property 11

The construction in Fig. 38 satisfies all requirements of Proposition 8 with = kJZ(BW),

COB

= k(BW),

UA

- U S = k(BW)’/2.

(6.9)

It follows from Proposition 9 that AB is a principal line and the proof of Property 1 implies that CB is a principal line. 5. Proof of Property 12 Since the construction in Fig. 38 is only possible for a point A of a simple supported line and a point C of a clamped line, an intersection of the domain boundary d and the centerline h is at a point where AW -+ 0 and CW + 0, and hence AC 40. 6. Proof of Property 13 This property follows directly from the constructions in Properties 1 and 11, and from the fact that at a corner between two simply supported edges the angular velocity must be zero in all directions. Thus such a corner is kinematically equivalent to a clamped point. 7. Proof of Property 14

In Fig. 40, (BC)2 = X’B

+ (y’B- a)2,

2(yB - a ) = ~ ( B c -) a,

which imply (6.5).

(6.10) (6.11)

252

George I . N . Rozvany and Robin D. Hill

c. c1 AND B FIELDS-EXAMPLES 1. Rectangular Domains with Various Support Conditions Optimal solutions for square and rectangular domains with various support conditions are given in Figs. 42 and 43. For the total "cost '' (volume) of various optimal solutions, the reader is referred to an earlier paper (Rozvany and Adidam, 1972~). Considering a rectangular domain having changes in support conditions at points other than corners, the solution takes the form shown in Fig. 44.

FIG.42. Optimal flexure fields for square domains.

25 3

Optimal Load Transmission by Flexure

'I

L/4 ' L/4

I

L/4

I

L/4

q(td

a L

a L

la,Al

4 (e)

FIG.43(a)-(f). Optimal flexure fields for rectangular domains.

254

George I . N . Rozvany and Robin D. Hill

Optimal Load Transmission by Flexure

255

FIG.44. Rectangular domain with changes in support conditions along the edges.

2. Simply Supported Triangular Domains Figure 45 shows the solution for simply supported triangular boundaries. The solution for three acute corners (Fig. 45a) is due to Lowe and Melchers (1972). If one of the corners forms an obtuse angle then the S+-typeregion splits into two parts separated by an R+-type fan (Fig. 45b). 3. Simply Supported Quadrilateral Domains Figures 46-50 (on pp. 257-265) show the 32 different topographies that the optimal solution may take on for simply supported quadrilateral domains. It is rather surprising that for such a simple class of boundary conditions of constant topography (four sides, four corners), the solution may have so many different configurations. Properties of various topographies are listed in Table 2 (on p. 266). Under the classification “type”, 0 refer to obtuse and N to nonobtuse corners (including corners enclosing n/2),and the corners are listed in a cyclic sequence. S S Connections refer to an R+-type region having principal lines in a constant direction and connecting two opposite sides of the domain (see Fig. 46d, for example). Sides are denoted by stating the type of corners at their ends, and subscripts are used in identifying corners only if it is necessary for distinguishing between two topographies.

256

George I. N. Romany and Robin D. Hill

(b) FIG.45. Triangular domains with simple supports.

CS Connections take place between a corner and a side in the form of an S--type region at the corner and an R+-type fan at the side. Two such connections may be observed in Fig. 46c. CC Connections link two corners via two S--type regions and a T-type region, as at the top of Fig. 46h. Lowe and Melchers (1973% Fig. 2) proposed a solution for quadrilateral domains that fails to satisfy the inequality condition in (4.1) at obtuse corners. The above solution would lead to the erroneous conclusion that the two topographies given in Fig. 46a,b can be used for all convex quadrilateral domains. Although the region configurations given in Figs. 46-50 are often valid for some more general cases, many of the actual examples given.belong to a specific class of boundary shapes. For example, Figs. 46c,d and f-h show

Optimal Load Transmission b y Flexure

(a)

(b)

a =c

FIG.&(a)-(d).

Quadrilateral domains with simple supports.

257

258

George I . N . Romany and Robin D. Hill

FIG.46(e)-(h)

Optimal Load Transmission by Flexure

0 -

FIG.47(a)-(d). Quadrilateral domains with simple supports.

259

260

George I . N . Romany and Robin D . Hill

(i)

FIG.47(e)-(i)

Optimal Load Transmission by Flexure

(C)

FIG.48(ab(c). Quadrilateral domains with simple supports.

26 1

262

George I . N . Romany and Robin D . Hill

t 5 FIG.48(d)-(f)

Optimal Load Transmission by Flexure

263

(C) /

FIG.49(a)-(c).

x

Quadrilateraldomains with simple supports.

264

George I . N . Romany and Robin D. Hill

v

FIG.49(d)-(f)

Optimal Load Transmission by Flexure

FIG.50. Quadrilateral domains with simple supports.

265

TABLE 2 CLASSIFICATIONOF THE TOPOGRAPHIES OF SIMPLY SUPPORTED QUADRILATERAL DOMAINS

Type

1

SS Connection

NNNN

NN-NN

I

I

ONNN

ON-NN

1

ON-ON

CC Connection

00-NN

0-0

0-0

CS Con-

O-ON

nection

Ol-NN O,-NN

Ol-NN 02-NN

0-NN

22;

02-01N

Ol-NN 02-NN

47b

47a

47c

46c

Figure 46b

47i

47h

48c

I

number

46d

46h

46f

I

ON-ON

1

47d

I

ONON S S Connection

46g

47g

OOON

OO-ON

146el

47f

1

48b

I

48a

I

47e

49a

49c

48e

49b

50a

50b

48f

49

49e

49f

48d

5Oc

Optimal Load Transmission by Flexure

267

solutions for symmetric trapezoidal domains. The four parameters that govern the choice of optimal topography are a, c, 1; and g, where

f = c - (c2/2 + a2/16)'/2, g = a tan(n/4

+ a)/4.

(6.12) (6.13)

The value off is based on the construction given in Fig. 38 with f i C B =

f i BW = AW and the additional requirement that Wand A fall on the axis of symmetry (Fig. 46h). Point B in Fig. 46h represents the corner of S - - and S+-type regions and f is the distance of B from the boundary segment E. g represents the distance of point L in Fig. 46g from E. Figures 46f, g, and h give solutions for domains with c > a and for f = g, f> g, and f< g, respectively. Figures 46c and d correspond to cases with a = c and a > c. 4. Domain Bounded by Two Simply Supported Circular Arcs (Fig. 51a)

Before considering the above problem, it is convenient to determine first the optimal solution for a simply supported circular domain that is clamped at one boundary point only (point C in Fig. 51b). Using the construction given in Fig. 38, the three side lengths of the triangle CWZ in Fig. 51b take on the values 2a, R - $a, and R, respectively. Then the length a can be easily expressed using the cosine law: a = R(2 cos a -

fi),

(6.14)

which gives the region boundary in Fig. 51b in polar coordinates. Parts of the R + / S region boundary and of the displacement fields shown in Fig. 51a will be used in the optimal solutions to be considered next. Returning to the problem in Fig. 51% the optimal region topography depends on the relative proportions of the domain, which can be defined uniquely in terms of the angle 1in Fig. 51a. (i) 1 2 45". It follows directly from Property 13 that the simplest optimal solution shown in Fig. 52a is valid only if 12 45". However, it is still necessary to show that (6.1)is satisfied for all points of the boundary. Adopting the coordinates shown in Fig. 52a, the equation of the boundary becomes & y = & (R2 - x')''~ T A.

(6.15)

It can then be checked easily that (6.15) satisfies (6.1)even for the limiting case 1 = 4 4 , A = R/$, Q Ix I (ii) 19.47"< 1 < 45". For this range of values, the optimal solution is shown in Fig. 52b, where the curved boundary is given by (6.14) and the

RIG.

268

George I. N. Romany and Robin D. Hill P

R

4 (b)

FIG.51. Solution for simple supports along two simply supported circular arcs.

adjacent R+-type region contains the same displacement field as the corresponding portion of the R+-type region in Fig. 51b. (iii) 1= 19.47'. This is a special case when the solution contains only S + - and S--type regions and R+-type fans. On the basis of Fig. 38, R cos 1 =

Jz ~

( -1sin A),

(6.16)

which gives 1 = arc sin(+) = 19.47'. (iv) 0 < 1 19.47' and -90" I 1< 0. For these ranges of values, the optimal solutions are shown in Fig. 52d,e, respectively. 1= 0 is excluded from the range of values considered because at that value the boundary reduces to a simply supported circle and hence the fixity at the corners is temporarily removed.

-=

Optimal Load Transmission by Flexure

t'

n

A>- 4

FIG

9

(a) 19.47" 1, the material behaves like a Maxwell solid and tan 6 is inversely proportional to w . In covering the frequency spectrum, we also find that at low frequencies (COT m), whose difference is the wave number k of the primary wave motion. These two wave motions interact with the primary wave motion and can thus grow in amplitude as a result of resonance. The major part of this paper is devoted to finding the condition of instability of stationary gravity waves in a liquid flowing over a wavy bottom, and the rate of growth. The effects of surface tension are then investigated. In all cases, when the flow is unstable, the rate of growth of the disturbances is proportional to the amplitude of the primary waves. When the analysis is applied to progressive gravity waves by adopting a moving frame to render them stationary, the Froude number is the ratio of the wave velocity to long-wave velocity, and therefore depends on the wave number. It turns out that progressive gravity waves are unstable for all nonzero wave numbers. This result differs from the result of Benjamin (1967) and Whitham (1966)that progressive gravity waves are stable if the dimensionless wave number is less than 1.363. The theories of Benjamin and Whitham agree, and Benjamin’s, in particular, has been abundantly verified by experiments conducted by Feir (see Benjamin, 1967), so that these theories can be considered well established. The present linear theory does not contradict them; rather it presents a new mechanism of resonance,

372

Chia-Shun Yih

which renders all progressive gravity waves unstable, although for deepwater waves the instabilities found by Benjamin and Whitham are often stronger and always more detectable. Detailed comparison with Benjamin’s results will be given, with explanations for the differences. However, it should be noted here that since the Froude number is no longer arbitrary, the growth rate of disturbances in a progressive wave train can be of the order of 8’ (j3 being the amplitude of the primary waves), as shown by (2.59) in Section 1,H. It can also be of the order of j3.Whether it is of the order of j3 or P2 or of some other order of j3 depends on the wavelength of the waves under study and on the value of j3 itself. As will be seen in Section II,D, the method of approach used here is similar to that used by Davis and Acrivos (1967) to study the stability of progressive internal waves. Their method, like the present one, also gives a growth rate proportional to the amplitude of the waves when they are unstable, in contrast with the results of Benjamin, as a result of the difference in the mechanism of resonance. The difference between the present work and the work of Davis and Acrivos is twofold. First, stationary waves over a wavy boundary are considered, with the Froude number arbitrary. This is unlike the progressive waves, which when made stationary correspond to a definite Froude number. Second, when the stability of progressive waves is considered, instability is found in this paper by considering the increase of the velocity of progressive waves with amplitude, whereas Davis and Acrivos did not consider this increase [or decrease, since they considered internal waves, for which the wave velocity may increase or decrease with amplitude, as shown by Yih (1974)], and found instability only by considering the interaction of various internal modes. B. THEPRIMARY FLOW The primary flow is a steady irrotational flow over a wavy bottom and with a free surface where the pressure is constant. Since it is the flow at the free surface that is the most important, the shape of the bottom being of only secondary importance, we shall use, for the primary flow, a flow that satisfies the nonlinear free-surface condition exactly, with a wavy streamline at the bottom. For this purpose we use one of the two formulas given by Richardson (1920). [These two formulas are reducible to each other, as shown by Yih (1957).] The formula we use is

Instability of Surface and Internal Waves

373

in which H ( w ) and the radical are real for real w,

z = x + iy

w = q5 + i$,

and

where x and y are Cartesian coordinates measured in units of depth d, and q5 and $ are, respectively, the velocity potential and the stream function of the primary flow, measured in units of Ud, with U denoting a mean velocity of the flow. The quantity a in (2.1) is defined by a = dF2, F 2 = U2/gd, (2.2) where g is the gravitational acceleration, acting in the direction of decreasing y . Thus (2.1) is in dimensionless terms. The Bernoulli equation for the free surface is, if all quantities are made dimensionlessand q denotes y on the free surface,

42+ a-'q = const,

(293)

where q2 = u2

+'U

I

= dw/dz

12,

with velocity components u and v in the directions of increasing x and y, respectively. If we take $ = 0 on the free surface, and make sure that both H and the radical in (2.1) are real on the free surface (on which w is real), then for the free surface, (2.1) gives

I dwldz l2 = H(w),

(2.4)

and, by integrating the imaginary part of (2.1), q = -aH(w)

+ const.

(2.5)

It is evident then that (2.3) is exactly satisfied. For our purpose, we shall take U ( W= ) 1 - b cos kw,

(2.6)

where

f l < 1. The free surface and the wavy bed are shown schematicallyin Fig. 1, and the disturbed free surface is shown in Fig. 2. It is immediately clear that if /I= 0, Eqs. (2.1) and (2.6)give a uniform flow in the x direction, with dimensionless velocity 1 or dimensional velocity U,which has been used as the velocity scale. For small B, (2.1) and (2.5) give a flow with a wavy free surface at $ = 0 and a wavy bottom at $ = - 1. From (2.5) one sees that the amplitude of the free surface is a/3 and its wave number is k, which is also the wave number of the wavy bed. Were /? zero, d would be the depth of the flow

374

Chia-Shun Y i h

FIG.1. Free-surface flow over a wavy bed.

everywhere. Since B is not zero, we can, for instance, identify d as the depth at a particular section (such as a section at a crest of the wavy bed), and U as the mean velocity at that section. It is important that the region of flow between and

$=O

$=-1

be free from singularities. If B is sufficiently small this can always be achieved, as one can see from (2.1) and (2.4), which never give dz/dw = 0

in the region of flow, for sufficiently small B. From (2.1) we obtain dw/dz = u - iu = H [ ( H - ’

- a2H’2)’’2+ iaH’],

(2.7)

in which H is written for H ( w ) for brevity. For the H given by (2.5), (2.4) becomes 42

= 1 - /3 cos kw = 1 -

cos kf$

(2.8) on the free surface. Another quantity needed later is u-’q2 on the free surface. This can be evaluated from (2.7) and (2.8), and after some straightforward calculation and much simplification, is given by u-’q2 = (H-’- a2H”)”2 = 1

- &(l - 4a2k2)f12 - fj?

cos k+

+ A(7 - 4a2k2)B2 cos 2k4 + 0 ( p 3 ) . (2.9)

FIG.2. Disturbed free surface of a liquid flowing over a wavy bed. The undisturbed free surface is not shown.

Instability of Surface and Internal Waves

375

Higher-order terms can be calculated, but we shall need only terms of the zeroth and first order in 8, so that we shall use u-'q2 = 1 - f / 3 cos k4.

(2.9a)

Series (2.9) is absolutely convergent if 1 ~-

1-8

1 + a2k2f12< 1,

(2.10)

as can be seen from the square-root term in (2.9). Condition (2.10) can always be satisfied with a sufficiently small 8. Finally we note that, with subscripts indicating differentiations, the kinematic condition at the free surface is uqx = v = f#ly.

(2.11)

The flow is shown schematically in Fig. 1. We emphasize that (2.1) has been used only for the convenience of constructing an exact solution for stationary waves. For any other wavy bed and any other solution, there will be a term containing cos k$ in q2 and u-'q2-quantities appearing as coefficients in the crucial free-surface condition (2.21) to be presented laterand the analysis will follow exactly the same line to reach exactly the same results. OF THE STABILITY PROBLEM C. FORMULATION

For the disturbance we assume a complex potential

w' = 4' + iv, where 4' is the velocity potential and $' the stream function of the irrotational flow caused by the disturbance, both regarded as functions of 4 and $. It is well known that since w and w' are analytic functions of z, w' is an analytic function of w,except at the singularities, which are excluded from the fluid domain. Using 4 and $ as independent variables then, we enjoy the advantage of applying the boundary condition at $ = - 1 and the freesurface condition at $ = 0. If s denotes the speed of the combined flow, sz=l

+

d(w dz w')

[=I

d(w + w') dw - ( I = q ' J l + & Jdw' , dz dw

so that on the free surface, with Idw'/dw 1' neglected, (2.12)

376

Chia-Shun Yih

The Bernoulli equation at the free surface, where the pressure is zero, is

4; + 4s’

t F-2(q

+ 11’) = const,

(2.13)

where is the free-surface displacement in the vertical direction, due to the disturbance. Taking the differencebetween (2.13) and (2.3),and using (2.12), we have (2.14)

where we have assigned the constant in (2.13) the value zero because we expect # and q’ to be sinusoidal in 4. If u’ and d are the perturbations to u and u, corresponding to the potential &‘, the kinematic condition on the free surface is,

+

(‘I ‘I’)= 0

+ u’,

(2.15)

where the time t is measured in units of U- ‘d. The difference between this equation and (2.11) is

(2.16) However,

since

u=&x=*y,

u=&=

-I+bx.

As in (2.17) U‘IX

= q2‘14

(2.18)

9

so that the right-hand side of (2.16) is #y

- #:u- Iq2q+ = u#* = u- 1

+ u#* (u + uq2q4)#;.

U-1q2q4(u#@

2

since, by (2.18), q2‘Io = u‘Ix = u.

- u#*)

= u- ‘q2&;. ,

(2.19)

Instability of Surface and Internal Waves

377

Thus (2.16) can be written

(2.20) Combining (2.14) and (2.20), we have

This is the free-surface condition. As to the condition at the solid boundary constituting the wavy bed, it is simply

@*=O

(2.22)

at $ = - 1 .

The velocity potential satisfies the Laplace equation

4;s + 4;*

(2.23)

= 0.

Equations (2.21)-(2.23) are the differential system governing the stability of the disturbance or of the flow.

D. DEMONSTRATION OF INSTABILITY Before presenting the formal demonstration of instability, we shall explain in intuitive terms the motivation for concentrating on disturbances of certain wave numbers. If /3 were zero and a wavy disturbance of wave number y were imposed on the uniformly flowing stream, the solution of (2.23)satisfying (2.22) would simply be

# = C exp i(yq5 - c o t ) cosh y($

+ l),

and (2.21) would give the secular equation (oo - y)2 = F-’y

tanh y.

(2.24)

We shall denote by R and L the right- and left-hand sides of (2.24),respectively. In Fig. 3, R and L are plotted against y, and where the two curves intersect the wave numbers are denoted by m and m‘. We claim that if

m’ - m = k,

(2.25)

the waviness of the bed may cause instability by a sort of resonance. That this is indeed the case will be shown in this section. Since the Lcurve is a parabola touching the y-axis at y = co, and the R-curve rises monotonically as y increases, it is evident that both m and

378

Chia-Shun Yih

Y FIG.3. The existence of wave numbers of resonating modes.

m‘ - m increases with ( i o , from zero onwards. The variations of m and oo with k for various values of F 2 will be given in the next section. Knowing that m and m’ are the crucial wave numbers, and anticipating the repeated interactions of the m-waves and m’-waves with the primary waves with wave number k, we assume a perturbation velocity potential & to have the form

4‘ = e-iut[acosh m(@ + l)ei“

+ b cosh m’(@+ l)eim‘$ m + n = cnPcosh{(m - nk)($ + 1)) exp i(m - nk)+ 1

+

m

dn/3” cosh{(m’ n= 1

+ nk)(@ + 1))exp i(m’ + nk)4],

(2.26)

in which m and m’ satisfy (2.25).Note that the expansion (2.26) implies that the harmonics with wave numbers m - nk and m’ nk are results of interactions of the harmonics with wave numbers m and m’ (which we shall, in anticipation, call the resonant harmonics) with the primary waves with wave number k. Thus the terms under the summation signs in (2.26) are, so to speak, the “entourage of the resonant harmonics. The question may immediately be raised: Why must harmonics of wave numbers m - nk and m‘ + nk be merely the results of interactions of the resonant harmonics with the primary waves? Why can they not exist on their own? The answer to this question is simply that of course they can exist on their own, but when they do their circular frequency (i will not be the same as that for the resonant harmonics, since their (i would be determined from (2.24) with y equal to m - nk or m’ + nk, whatever positive integer n is.

+

”

Instability of Surface and Internal Waves

379

(Indeed, y does not even have to assume these values, but in that case the harmonic with wave number y will not, on its own, have the same u as the resonant harmonics, and the harmonics with wave numbers y and y L- nk cannot belong to the “entourage ” of the resonant harmonics either.) For this reason the harmonics with wave numbers m - nk and m‘ + nk arise only from interactions of the resonant harmonics with the primary waves, and are calculated from nonhomogeneous equations, the stability question being exclusively and once and for all settled by the calculation for the resonant pair alone.? This is an extremely important point for the understanding of what follows. From the following analysis one can see, even if the terms involving wave numbers m - nk and m‘ nk are shown in (2.27) only for n = 1 and not shown in (2.28) at all, how the coefficients c, and d, in (2.26)are determined by (2.21), step by step. Indeed, it can be shown not only for the flow being considered here but also for any flow with stationary waves that expansion (2.26),with c, and d, determined by (2.21),is convergent up to the value of /3 for which the primary flow exists. There is then nothing arbitrary in the form of & given by (2.26) and nothing indefinite or obscure in using (2.21) and (2.26) to determine u and to make conclusions on the stability or instability of the primary flow. As we have said, in this determination the calculation need be done for the resonant pair only. [We note here also that in the application of the theory for stationary waves to progressive waves made stationary, F 2 will depend on /3 and have a term of 0(B2). Inclusion of this amplitude-dependent term in F2,using (2.24) to determine, Once and fur all, rn and m’ for a given k, and then using (2.26)for the analysis, as in the following analysis for stationary waves, does not involve any unallowable truncation and should not lead to any confusion concerning the collection of terms of various orders in 8.1 The first term in (2.21), after a straightforward calculation is then

+

+ terms containing c, and d,, t Even if for higher-order terms in u feedbacks from overtones are needed.

(2.27)

Chia-Shun Yih

380 in which

+ @bm’ cosh m‘, B = b(c - m‘) cosh m‘ + @am cosh m,

A = a(o - m) cosh m

a, = am cosh m,

b, = bm’ cosh m‘.

By virtue of (2.9a), the second term in (2.21) is

F- ’u- ‘q2c#$ = e- iufF-’{(am sinh m - @bm’ sinh m’)ei‘@

+ (bm’ sinh m‘ - @am sinh m)eimU} + terms of O(B’) or involving c, and d, .

(2.28)

Equating terms of the same wave number (m,m‘, etc.) in (2.21), and retaining terms of O(1) and O(B) only for the first approximation, we obtain then a(. - m)’ cosh m

+ ifibm‘(2o - m - m’) cosh m’

= F-’(am sinh m

- &%m’ sinh m’),

(2.29)

and

b(o - m’)’ cosh m’ + fBam(2o - m = F-’(bm‘

- m’) cosh m

sinh m‘ - #am sinh m).

(2.30)

Let (o - m)’ = (oo - m)’

+ A, + O(B’),

(2.31)

so that o - m = (oo- m)[l

+ &(o0

- m)-’

+ O(B’)],

(2.32)

in which A, is of order O(8). Then (2.29) and (2.30) become, upon neglect of terms of 0(p2), d, cosh m

bA, cosh m‘-

Bb [2m‘(200 - m - m‘) cosh m’ + F-’m‘ +4

oo - m’

sinh m’] = 0, (2.33)

+ Ba [2m(200 - m - m’)cosh m + F-’m

sinh m] = 0. go-m 4 (2.34) It is evident from Fig. 3 that oo - m‘ is negative and oo - m is positive, so that their ratio in the first term of (2.34) is always negative. The bracketed terms in (2.33) and (2.34) are both negative. This can be shown in the following way. We know that

(oo - m)’ = F-’m (go

- m’)’

= F-’m’

tanh m

(2.35)

tanh m‘.

(2.36)

Instability of Surface and Internal Waves

38 1

Writing (2.35) as

-

(o0 m)’

+ F-‘(a0 - m) tanh m - F-’a0

tanh m = 0

we have no - m = $[ -F-’

tanh m

+ (F-4tanh’ m + 4F-’a0

the positive sign before the radical being taken because cr0 Similarly, from (2.36) we obtain o0 - m’ = $ [ - F 2 tanh m’

- (F4 tanh’

tanh m)l/’], (2.37)

-m

is positive.

m’ + 4F-’ao tanh n ~ ’ ) ~ ’ ~ ] , (2.38)

the negative sign before the radical being taken because go - m’ is negative. Thus 20, - m - m’ = f[ -F-’(tanh

m

+ tanh m’)-MI,

(2.39)

where

M

=f(m’)-f(m),

f(m) = [F-’ tanh m(F-’ tanh m

+ 400)]’/’. (2.40)

Thus M > 0.

Substituting (2.40) into (2.33) and (2.34), we have

a l l cosh m =

4

(F-’ tanh m

as

go - m’ bAl cosh m’- (F-’ tanh in‘ oO-m 4

+ M)m’ cosh m‘

(2.33a)

+ M ) m cosh m.

(2.34a)

Multiplying (2.33a) by (2.34a), we have

A+--.--

c0 - m B’mm’ (F-‘ tanh m -t- M)(F-’ tanh m’ o0 - m’ 16

+ M).

(2.41)

Thus A! is negative and A1 purely imaginary. Returning to (2.32), we see that Q

= o0

+ $A1(ao- m)-’ + O(B’),

A1/2(ao - m) = kilpIA,

(2.42)

2 > 0.

The choice of the positive sign then gives - iar =

exp (- iao +

la p)t,

and the disturbance (2.26) is unstable, for whatever F2, whatever k, and whatever B, provided that m is not zero. (See Section 11,E.)

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Chia-Shun Yih

It is important to bear in mind that, given /3, F2, and k, rn and 1 are determined uniquely. The m and rn’ for the unstable disturbance may be quite different from k, and thus not in the neighborhood of k. Furthermore, the present theory is linear, and the growth rate I /3 11 is proportional to I fi I rather than /I2.These facts distinguish the present theory from the theories of Benjamin (1967) and Whitham (1966) for progressive gravity waves. We shall treat the stability of progressive gravity waves in Section II,H, to make a direct comparison. We note that the instability of stationary waves is caused by the resonant interaction of the primary waves with the rn-waves and the rn’-waves. The rn-waves propagate with the flowing stream, but the rn’-waves propagate against it. E. WAVELENGTHS AND GROWTHRATESOF UNSTABLE MODES Given m, we can determine go from (2.24) upon substituting m for y. The other root rn‘ of (2.24) is found by numerical computation and then k is found from (2.25).In this way we obtain corresponding values of go,m, and k. In Fig. 4.m is plotted against k for various values of F2,and in Fig. 5 o0 is plotted against k for the same set of values of F2. We see immediately from Figs. 4 and 5 that the curves go through the origin if F 2 2 1, but intersect the k axis at positive values (kc)of k if F 2 c 1. We have worked with positive k. For negative k, rn and rn’ are negative, but the conclusions remain. By virtue of (2.39), or directly from (2.33) and (2.34), Eq. (2.41) giving 1: canbe written ’

I

84 F 2 =I

k

FIG.4. Variation of m with k.

Instability of Surface and Internal Waves

38 3

k

FIG.5. Variation of u,, with k.

where

G=

1 mm' uo - m 2a0 - m - m' + - F-' tanh m 2 4 o0 -m'

(2t~,-m-m'+-F-'tanhm' 1 2

.

1

(2.43a)

The quantity G is plotted against k in Fig. 6 for various values of F2.It is seen that G is positive for F 2 2 1 and for any positive k, whereas, for F 2 c 1,

k

FIG.6 . Variation of G with k .

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Chia-Shun Yih

k has to be greater than a cut-off value k, for G to be positive. Thus for F 2 2 1 the flow is always unstable for any nonzero k, whereas for F 2 1 the flow is unstable only if k is greater than a cut-off value k, (or, if k is negative,

-=

smaller than - k,). From Fig. 4 we see that m (and hence m')increases with F 2 for any given k. From Fig. 6 we see that G (hence A,) increases with F 2 for any given k. Thus for any given k, the unstable modes have shorter and shorter wavelengths and become more and more unstable as F 2 increases. Furthermore, it is seen from Fig. 6 that G changes but little between F 2 = 4 and F 2 = 100-a fact of some practical usefulness when estimating G at high Froude numbers. Thus the rather short wavelengths of the surface disturbances and their very strong instability when F 2 is high, as observed in pulp conveyed on Fourdrinier wires (or screens), find their explanation here. It is important to note that if F 2 is sufficiently greater than 1, the growth rate 11, 1 is of the order O(p), whereas the growth rate of progressive waves for the instability mechanism of Benjamin (1967) is of the order? O@') only, in the present notation, and is therefore weaker than that for the present mechanism. However, when progressive waves are made stationary by using a moving frame of coordinates, the F 2 is not arbitrary, and the investigationof instability due to the present mechanism will be presented in Section I1,H.

F. EXTENSION TO OTHER W A V Y FLOWS So far we have only considered the primary flow given by (2.1) and (2.6), but Eq. (2.21) governing the stability of the disturbances is quite general. It is both important and useful to see whether the instability that has just been found will exist for other wavy flows. To reach a decision on this question, we need only to see what forms q2 and u-'q2 assume for other flows, and what effect these forms will have, through (2.21), on the stability. For the flow defined by (2.1) and (2.6), q2 is given by (2.8) exactly and u - l q 2 by (2.9a) if terms of order O(8') or higher are neglected, and the stability analysis shows that only terms of orders O(1) and O(p), with the terms of order O(p)being multiples of p cos k4, are needed for q2 and u- lq2. Now there are infinitely many flows over a wavy bottom for which, in dimensionless terms,

u = 1 - @ cos k 4

+ 0(p2),

u = O(p), (2.44) on the free surface, so that the q2 and u- lq2 on the free surface are given by (2.8) and (2.9a), if terms of order 0 ( p 2 )and higher are neglected. In (2.44),p is an amplitude of wave motion, and the form for u can always be achieved t B is proportional to Benjamin's ka. See the two lines after (2.51).

Instability of Surface and Internal Waves

385

since the origin of 4 is arbitrary. Any potential flow for which (2.44) holds on the free surface has wavy streamlines, any one of which below the free surface can be taken as the solid, wavy bed. For such flows, the stability analysis given in Section 11,Dholds exactly. Hence the conclusions reached in that section are not restricted to the flow given by (2.1) and (2.6), but are much more general. For instance, for gravity waves in a liquid of uniform depth d, the x component of the velocity at the free surface is, when the flow is made steady by a set of moving coordinates and velocity components are measured in terms of the wave velocity, ~ = l - + j 3 ~ 0 ~ k 4 + ~b 02 ~ 2 k 4 + b~ ,0 ~ 3 k 4 + * . * , in which b3 = O(P3), etc. If terms of order higher than O(P) are neglected, u has the form given in (2.44),and the stability analysis in Section II,D applies. If, in this flow, we take a (wavy) streamline between the flat bottom and the wavy free surface as a solid lower boundary, the value of F 2 can be made as large as we please by making the mean depth as small as we please. For F 2 greater than 1 then, that flow is unstable for all positive values of the wave number k. We note, incidentally, that the position of the wavy bed determines F2, and is therefore very important in deciding whether or not the flow is stable. One might say that making any internal wavy streamline rigid destabilizes the flow. b2

= O(j3’),

G. EFFECTSOF SURFACE TENSION Surface tension affects the solution for the primary flow as well as the perturbation flow. Crapper (1957) gave an elegant, exact solution for water waves of arbitrary amplitude and for infinite depth of the liquid, with surface tension fully taken into account. But gravity is entirely neglected in his solution. Fortunately, as the analysis in Section II,C shows, only terms of the first order in the amplitude of the waves are important in the determination of stability, and these terms are easily obtainable by the linear theory, even when surface tension is taken into account. Thus, if the wavy bottom is given in dimensionless terms by y = -1

+ a’ cos kx,

and the mean depth (dimensionless)is 1, the solution for stationary waves is, again in dimensionless terms, q’~ = x

- (a cosh ky

+ b sinh ky) sin kx,

(2.45)

Chia-Shun Yih

386 in which

+ Sk2)b = 0, b = a’[cosh k - (F-’k- + Sk) sinh k ] - l ,

( 2 . 41 (2.47)

S = T/pU’d’,

(2.48)

ka - (F-’

where T is the surface tension, d the (dimensional) mean depth, and U the (dimensional) mean velocity. The free-surface displacement is q = b cos kx

and

- ak cos kx + O(a2k2)= 1 - ib cos k 4 + O(j3’), in which 2ak = B. If terms of order O(b2)and higher are neglected, again 4’ = 1 - j? cos k 4 . u=1

We need, however, to modify (2.14) to

(i+

4’

+ F-’q + S[q(qq&)&]= 0

$)cf

at $ = 0,

(2.49)

in which certain terms definitely of order O(b2)have been omitted. Then the demonstration of instability can be given as in Section II,D, with m‘ and m being now the roots of (go

- y)2 = ( F -

+ S y z ) y tanh y,

(2.50)

and with no determined by the condition m’ - m = k. We shall not pursue the details at this time.

H. INSTABILITY OF PROGRESSIVE GRAVITY WAVES IN A LIQUIDOF CONSTANT DEPTH Benjamin (1967) and Whitham (1966) have studied the stability of gravity waves using two different approaches. They agree that when k (kh or K in their notation) is less than 1.363 the waves are stable. Benjamin uses as the disturbance a side-band of wave numbers and frequencies centered around, and therefore in the neighborhood of, the wave number and frequency of the basic waves. When k > 1.363 the rate of growth of the unstable disturbances is of the order O(f12)in the present notation, O(k2a2)in Benjamin’s notation. We shall show that by the present mechanism progressive gravity waves are

Instability of Surface and Internal Waves

387

always unstable, but that the instability is significant only when Benjamin’s theory predicts stability. If we use a frame of reference moving with the waves, and use our dimensionless notation (the velocity scale U now being the wave velocity c), then for Stokes wavest l p ’ X - 2

/? cosh k(y + 1) sin kx + O(/?’), cosh k

(2.51)

where our /3 corresponds to Benjamin’s (1967, p. 65)

-2kBa coth K,

K = k,d

= k,

d = depth, a = wave amplitude,

where kB is used for the dimensional wave number, to distinguish it from the dimensionless k. We use the /? in this way in order to have agreement with (2.8) and (2.44). The dimensional wave velocity c, which is used as the velocity scale, is given in the present notation by (Benjamin, 1967, p. 66), (2.52)

where

Equation (2.52) is valid only if /3’ < k3 if k is small (see Benjamin, 1967, p. 66). Equation (2.21) still governs stability, and we can repeat the analysis of Section I1,D. Thus we have again (2.41) or its equivalent, (2.43),but we need to investigate the values of m and m’.For this purpose we return to (2.35) and (2.36). If we ignore the term of order O(1’) in (2.52), upon taking the proper root as before, these become

- m = (km tanh m/tanh k)l/’, m’ - go = (km’tanh rn‘ltanh k)’]’.

(2.53)

o,,

(2.54)

Taking the sum of (2.53) and (2.54) and using (2.25) we have km tanh m k = ( tanh k

)”’+(

k(k

+ m) tanh(k + m) ‘ I 2 tanhk

It is then obvious that rn = 0,

1.

(2.55)

m’ = k,

t It is important to note that the (p given by (2.51) and the corresponding stream function now serve as the independent variables in (2.21).

+

388

Chia-Shun Yih

and according to (2.41),A, = 0 and there is no instability.? Any instability then must be due to the term fl’f(k) in (2.52).With (2.52) substituted into (2.35) and (2.36), we obtain the equation

[l

+ ;/?2f(k)]k = right-hand side of (2.55).

(2.56)

From (2.55) and (2.56) it follows that

Thus, for small k, m = *fl’kf(k),

m‘ = k + m.

If terms of order O(fl’) are not neglected, the left-hand sides of (2.53) and (2.54) should be multiplied by

1 + -tPf(k), and from the resulting equations one finds that

m‘ - uo = k

kfl’ + m - __ + O(fl”). 8

For small k, we have, with higher-order terms (in fl) neglected,

F-’ tanh m’ = k,

M = k,

no- m = m

+ O(P4),

so that

A:

= -im2k2f12.

and the growth rate is

- 41 = t k I f l I .

I ~ l P ( ~ 0

(2.57)

Recalling that our fl is Benjamin’s - 2ka coth K and our k Benjamin’s K , we see that the right-hand side of (2.57)is Benjamin’s ka, where k is the dimensional wave number and a the dimensional wave amplitude.

7 Professor 0.M. Phillips kindly showed the author the manuscript of the revised edition of his book (1966).In it he mentions the nonexistence ofresonant wave triads, in agreement with our statement here.

Instability of Surface and Internal Waves

389

For large k (deep-water waves), if we assume k fixed and let B be as small as we please, m will be small. In that case from (2.56) we have

m = lkl/ZB2, 8 m‘ = k + m, go - m = k’l2m = ‘k /3 2, m‘-oo=k+O(B2), F-’=k,

M=k,

so that (2.41) becomes

Jt = -4k2.5m2f12, and the growth rate is

1 R1/2(00 - m ) 1 = ko-’5/3/4fi. We have used c/d as the scale for g o and hence the growth rate also. When d is large this scale is inadequate. If we change the scale for the growth rate to c/L, where L is the wavelength, for large k we have (2.58) Since m is supposed small, /3 4 k - 0 . 2 5 ,hence the growth rate based on c/Lis then very weak for very large k, being of the order k-’/’. If, for large k we assume 1 4 m 4 k, from (2.56) we have m = &kp4, go

- m = (km)’”

= ik/3’,

m‘ = k + m,

m‘ - go = k + O(p2),

F-’ = k = M ,

so that (2.41) becomes 2: = - (&k2/?4)2,

and 12, /2(a0 - m) I = &kfi2.

If again c/L is used as the scale of the growth rate rather than c/d, we have growth rate = in/?’.

(2.59)

Equations (2.57)-(2.59) give the rate of growth of unstable disturbances (the rn-waves and the m’-waves) for long or short waves. But since, whatever the wavelength of the primary waves, m is positive and G is positive in (2.43a), progressive gravity waves of all wavelengths are unstable. The result that long waves are unstable, so long as the wavelength is finite, can be compared with the result of Benjamin (1967) and Whitham (1966) that progressive gravity waves are stable for k < 1.363. The explanation for the difference in results lies in the difference in the mechanism of resonance, theirs being the resonance of disturbances with side-band frequencies with

390

Chia-Shun Yih

the second harmonic of the primary waves, ours being the resonance of two wave trains with the basic harmonic of the primary waves. Benjamin’s theory as applied to deep-water waves has been abundantly confirmed by Feir’s experiments and by observations at the National Physical Laboratory (NPL), reported in Benjamin‘s paper (1967). It is therefore necessary to compare the present results for deep water with Benjamin’s and with available experimental data. We shall first compare the exponential growth rates given by (2.58) and (2.59) with the exponential growth rate of the side-band disturbances given by Benjamin, which is, for deep water,

+kia2w = ip’o,

(2.60) where o is the (dimensional) circular frequency of the primary waves. Since

c/d = o / k g d = w/k, the dimensional growth rate corresponding to (2.58) is z

1 /3 10/4$

k1.25,

(2.61)

and that according to (2.59) is a/3’o/8 k.

(2.62)

In the NPL observations k = 50~17.2,1 1 = 0.34, so that both (2.61) and (2.62) are less than (2.60), but (2.61) is relevant because m is small compared with 1. Hence Benjamin’s mechanism dominates the mechanism treated here, under the experimental conditions at NPL. For the Cambridge experiments o=~R/s,

so that

kB = gn’ = 7/ft. The depth of water was not given. Assuming that it is 3 ft, we have k = 21, and in the range of /Icovered by the experiments, m is small and (2.61) should be used. In this case (2.61)is greater than (2.60)for the smaller values I , but the orders of the of p and less than (2.60) for the larger values / magnitudes are quite the same. One must nevertheless pose the question of why the growth rate (2.61)was not observed, especially for very small /3. The answer is that the dimensionless number 6 representing the fractional deviation from o in the side-band frequencies (1 5 6)w was fixed at 0.1 in the Cambridge experiments, but is far from that for the m-waves and m‘-waves treated here. Thus naturally they cannot be observed.

Instability of Surface and Internal Waves

391

Now the m-waves travel with the stream if the primary waves are stationary, and hence against the primary progressive waves in the fixed frame of reference. The opposite is true for the m‘-waves. Since m is small or at least much smaller than k, the 6 in (1 - 6)ois slightly greater than 1, in any case very near 1. Since m’ is very near k, the 6 in (1 + 6)w is very near zero. Thus the Cambridge experiments, with 6 fixed at 0.1, could not have revealed the m-waves or m‘-waves. Since the frequency of the m-waves is small and the deviation of the frequency of the m’-waves from o of the primary waves is small, to detect these small quantities time records such as shown on p. 63 of Benjamin’s paper must be very long. Analysis with records covering a short period of time can only reveal Benjamin’s instability and would necessarily fail to reveal the instability treated here, even when it happens to be stronger than Benjamin’s (which it is not if the amplitude is not too small). The main conclusion of this section is that all progressive gravity waves are unstable. (See note added in proof on p. 419.)

I. CONCLUSIONS (i) Stationary gravity waves in water (or any liquid) flowing over a wavy bottom are unstable for any nonzero wave number of these waves if F 2 2 1, and for any wave number greater than a critical value k, (depending on F 2 ) if F 2 < 1. (ii) Gravity waves of all nonzero wavelengths are unstable. J. DISCUSSION

As mentioned in the introduction, Hasselmann (1967) gave the theorem that for dispersive waves propagating in one dimension, for which the frequency 0 is a function of the wave number k, if there are three wave numbers satisfying the relation k3

= kl

+ k2,

with their corresponding frequencies satisfying 03

= 01

+ 02,

then, on the understanding that the k,-waves are finite and the other two wave trains infinitesimal, one has bl = 0 and a2 = iho2alaf,

(2.63)

a3 = -iho3a:a2,

(2.64)

392

Chiu-Shun Yih

where u l , uz ,and u3 are the amplitudes, however defined, of the three wave trains, the dot indicates the time derivative, the asterisk denotes the complex conjugate, i is, as usual, the square root of - 1, and h is real. From (2.63) and (2.64) it is easy to show that

so that the k2-waves are unstable. The same is true of the k,-waves. In this paper the k, m,and m' correspond to k,, kz , and k , , respectively, and since the frame of reference is stationary with respect to the k-waves, the quantities corresponding to ol,uz , and 0, are, respectively, k,

go

- m, and

go

- m'.

With this in mind, and remembering that u and b in (2.26), (2.33a), and (2.34a) correspond to u2 and u3 in (2.63) and (2.64), we can identify [see (2.4211

with the u2 and i, in (2.63) and (2.64). Upon multiplying (2.33a) by go - m and (2.34a) by go - m', using (2.35) and (2.36), and multiplying b by an appropriate factor, (2.33a) and (2.34a) can be reduced to (2.63) and (2.64). Thus there seems to be a similarity between the present results and the result of Hasselmann embodied in (2.63) and (2.64). Yet the similarity is only superficial. First of all the fact that the bracketed terms in (2.33) and (2.34) are both negative, a fact crucial for the proof of instability and ensuring the existence of a real h in (2.63) and (2.64), has been established in such a special and unobvious way that one can justifiably doubt that a general formula established without regard to the particularity of the problems can apply to the problems considered here u priori. It may be useful to point out further the differences of the present results and the result of Hasselmann. As far as we can see, Hasselmann treated the stability of waves freely propagating in water of uniform depth (i.e., where his theory touches upon the present results). If these are made stationary, the general velocity of flow is not arbitrary but equal to the speed of propagation in quiet water. In this paper, where stationary waves are treated, the general velocity of flow embodied in the Froude number is entirely arbitrary. Thus the conditions on which Hasselmann's theory stands do not seem to be met here, and consequently the instability described here cannot be included in that theorem. Furthermore, whether the waves are due to the wavy bottom or are free

Instability of Surface and Internal Waves

393

waves, the finiteness of their amplitude demands application of the freesurface condition on the free surface and not at a constant elevation. This is taken care of in this article by using d, and II/ as independent variables instead of x and y . There is nothing in Hasselmann’s paper that indicates how this situation is dealt with.

111. Instability of Internal Waves

A. INTRODUCTION Davis and Acrivos (1967) found that Hasselmann’s conditions (mentioned in our Section I) can be satisfied for internal waves in a stratified fluid by resorting to interaction between trains of internal waves not only of different wavelengths but also of different modes. At least one of the wave trains must be of a different mode from the other two. The discovery of Davis and Acrivos is a very interesting one, for the instability they found is truly characteristic of internal waves. It has no counterpart for surface waves in a homogeneous fluid. Our investigations in the instability of internal waves supplement the work of Davis and Acrivos. In the following sections y e will show that Hasselmann’s conditions can be met if (i) the primary wave train is due to a flow of a stratilied fluid over a boundary, or if (ii) it is a wave train freely progressing over a flat bottom, provided.the wave velocity increases with the amplitude. When Hasselmann’s conditions are met, the primary wave train is indeed unstable. In category (i) conditions of stability or instability are found for two cases: (1) waves in two superposed fluid layers, with the upper fluid infinite in extent, and (2) waves in a continuously stratified fluid. The growth rate of the disturbances is given whenever they are unstable. In Case 2 Boussinesq’s approximation is used. We emphasize that the mechanism of instability is, as for surface waves, the resonance of a pair of disturbances with the basic waves produced by flows over wavy surfaces. For interfacial waves the results are nearly the same as for surface waves, but for internal waves in a continuously stratilied fluid the results are somewhat different, the main difference being that these waves are stable for sufficiently high wave numbers and unstable for sufficiently low wave numbers, whatever the internal Froude number may be. The stability of progressive internal waves in an otherwise quiet fluid is briefly discussed after stationary waves over a wavy boundary have been treated.

Chia-Shun Yih

394

B. STATIONARY INTERFACIAL WAVES The primary flow, the stability of which will be studied in the next section, is the flow of two superposed incompressible fluids with a “general velocity U over a wavy bed, which is described by ”

=

-d

+ 7,

cos kx,

(3.1) where x and y are Cartesian coordinates, with y measured in the direction opposite to that of the gravitational acceleration, d and y, are constants, and k is the wave number of the corrugation of the bed. The general velocity U would be the actual velocity of the fluids if y, were zero. The lower fluid has density p, ,and extends from the bed to the interface JJ

Y = 45

q being the (sinusoidal) elevation of the interface above its mean position y = 0. That is to say, if y1 were zero the lower fluid would extend from y = - d to y = 0. The upper fluid has density p1 and extends from the interface to positive infinity. Neglecting viscous effects, we assume the flow in each layer to be irrotational. Velocity potentials 4, and 4, then exist for the upper and lower layers, respectively, both of which satisfy the Laplace equation. Since wave motion must die out as y approaches infinity, we have =

4,

U x + A l e - k y sin kx,

= Ux

(3.2)

+ ( A , cosh k y + B ,

sinh k y ) sin kx.

(3.3) and $, , which are harmonic conju-

The corresponding stream functions gates of q51 and 4,, are

l(li = U y - A , e - k y cos kx, $, = U y + ( A , sinh k y

+ B,

(3.4) cosh k y ) cos kx.

(3.5) We shall give a linear theory for the primary flow, because we need only the first harmonic of the wave motion for our stability study, and a linear theory gives that. At the wavy bed, $, = L U d , so that y = -d

+ U-’(A,

sinh kd

- B,

cosh kd) cos kx.

Uy, = A , sinh kd

- B,

cosh kd.

Hence On the interface,

Instability of Surface and Internal Waves

395

B2 = - A , .

(3.7)

so that

Furthermore, at the interface the Bernoulli equation gives, after the mean quantities have been filtered out, P1

U(41 - w

P2 w

2

x

=

-P - PlSrl,

w,= -P - PZSrl,

-

(3.8) (3.9)

where the subscript x denotes partial differentiation and p denotes the pressure due to waves. The interface displacement '1 is obtained by setting 11/, equal to zero in (3.4), and is q = A, U-' cos kx.

(3.10)

When (3.2), (3.3), and (3.10) are used in (3.8) and (3.9), and p is eliminated, we have (3.11)

A2 = j A l ,

where (3.12) Fi being an internal Froude number. From (3.6) and (3.7), A2 = 7 UYl - A , coth kd, sinh kd

(3.13)

which, together with (3.1 l), gives p z cosh kd

+p,

sinh kd -

kU2

For the convenience of the subsequent development, we shall make the results given above in dimensionless terms. If 4 and 11/ are measured in units of U d and linear dimensions are measured in units of d, after using (3.7), we have

41 = x + a l e - k y sin kx, Ic/l = y cos kx, where

+ (az cosh k y - a , sinh k y ) sin kx, 11/2 = y + (a2 sinh k y - a , cosh k y ) cos ky,

42 = x

(3.15)

Chiu-Shun Yih

396

and k is now and henceforth dimensionless (equal to the original kd). We have a2 = j u l ,

a, = y(cosh k

+j sinh k)-’,

(3.16)

where (3.17) If we write, in accordance with the usage in Section 11,

B = -2a,k, the horizontal dimensionless velocity components u1 and u2 at the interface, to the first order in B, are

so that, to the first order,

u;’q; = 1 - - cos kcp2 .

(3.18)

These formulas will be useful in the next section.

C. INSTABILITY OF STATIONARY INTERFACIAL WAVES It can be readily verified that for waves with the exponential factor exp i(ax - at), where a is the dimensionless wave number and a the dimensionless circular frequency (i.e., with a measured in units of U/d),a is equal to a,, (positive), with a. given by (ao- a)’ = F;’(a

tanh a)(l

+ r tanh a)-’,

with r = p1/p2, (3.19)

provided that y is zero, that is, provided the bottom is flat. Since y is not zero a will differ from a,,, and the purpose of this section is to show that the difference may be a positive imaginary number, signifying instability, if the

Instability o$ Sur$ace and Internal Waves

397

disturbance consists of two wave trains of wave numbers m and m’ satisfying (3.19) and the condition

m’- m = k,

(3.20)

where k is the dimensionless wave number of the primary waves treated in the preceding section. We shall first discuss (3.19)and (3.20)before going on to the study of stability. For a given r and a given Ff, Eq. (3.19)has two roots, one greater than go, which we shall denote by m’,and one less than g o ,which we shall denote by m. That there are two such real roots can be seen in the following way. We shall denote the right-hand side of (3.19) by R and its left-hand side by L. If we plot L against a the curve is a parabola touching the a axis at a = go. If we plot R against a the curve touches the a axis at the origin, rises monotonically (the monotonicity can be easily established) as a increases, but is asymptotically linear for large a. Hence the two curves must intersect at two on its points, one (m)on the descending branch of the L a curve and one (m’) ascending branch. For given k, r, and Ff, we wish to find m and m’ satisfying (3.19)and (3.20). To find m (and therefore m’)and go we adopt the following procedure. Given r and F!, we assign various values to m. Substituting m for a in (3.19),we find go (greater than m). Then with this value of go we solve (3.19) for the other root m‘.The value k is then given by (3.20).In this way we obtain values of m and go corresponding to k, and m-k and uo-k curves can be so constructed. In Figs. 7 and 8 are such curves for various values of FZ and for I = 1. The value 1 is of course never reached by r, and assigning the value 1 to I amounts to adopting the Boussinesq approximation, valid when 1 - r 6 1.

k

FIG.7. Variation of m with k for r = 1 and various values of F f .

398

Chia-Shun Yih

k

FIG. 8. Variation of uo with k for r = 1 and various values of F:.

From Fig. 7 it is seen that for Ff 2 1 we have a nonzero m for a nonzero k, whereas for F: < 1 m is not zero only if k > k,, k , depending on F?.Figures 9 and 10 give m-k and ao-k curves for r = 0.5, and Figs. 11 and 12 give these curves for r = 0.25. It is easy to determine k,. From (3.19) we have F,(a, - m)= J(m),

(3.21)

Fi(o, - m‘) = -J(m’),

(3.22)

k

FIG.9. Variation of rn with k for r = 0.5 and various values of Ff

Instability of Surface and Internal Waves

399

k

FIG.10. Variation of c0 with k for r = 0.5 and various values of F:.

in which m tanh m ‘I2 J(m) = (1 + r tanh m) By virtue of (3.20), the difference between (3.21) and (3.22) is

F ik = J ( m ) + J(m’).

(3. 3

(3.24)

Upon equating m to zero in this equation, we have (3.25)

F i= k - ’ J ( k ) , .I1003616

4

I

3

4

‘/7

5

6

7

8

k

FIG. 11. Variation of m with k for r = 0.25 and various values of Ft.

400

Chia-Shun Yih

k

FIG.12. Variation of o,, with k for r = 0.25 and various values of F ? .

which states that the general velocity U , on which F iis based, is the velocity of internal waves of wave number k in the fluid layers under consideration, if they were at rest and fi = 0. This fact can be readily demonstrated independently in the same way as (3.19).The solution of (3.25) for k is k,. But (3.25) has a real solution only if Fi < 1, for Fi = 1 for k = 0 (very long waves), and the wave velocity decreases with k. We note, however, that the case k = k, must be excluded,? for if (3.25) is satisfied A, is infinite according to (3.14) (where k is still dimensional). This discussion is intended to show that if Fi < 1 but k, < k, Eq. (3.24), in which in' = m k, has a nonzero solution for in. We now give the primary waves a perturbation. The perturbation flow is still irrotational except, of course, at the interface, which is a vortex sheet, as is well known. The velocity potential for the perturbation flow is, for the upper layer,

+

t#~'~

= e-'"'[(o!, exp(

+ irn&l) + p1 exp( -in'$l + iin'&,)],

(3.26)

and, for the lower layer,

t#J'z = e-'"'[a2 cosh

+ l)ei"@*+ fi2 cosh in'($, + l)eim'bz]. (3.27)

It is important to note that at the interface both and $ 2 are equal to zero, is not equal to dZ. but By a procedure similar to that used in Section I1 [see Eqs. (2.12)-(2.14)], the Bernoulli equation at the interface is, after the part corresponding to the ? This exclusion is not necessary if nonlinearity is taken into account.

Instability of Surface and Internal Waves

401

primary flow is filtered out and quadratic terms in the perturbation quantities are neglected, pi Ll C#J'~

+ p' + p1 F2q' = 0

(3.28)

p2 L2#2

+ p' + p 2 F2q' = 0

(3.29)

for the upper layer, and

for the lower layer. In (3.28) and (3.29), p' is the pressure perturbation at the interface, q' the perturbation of q, and

The difference between (3.28) and (3.29) is, after division by p 2 ,

L 2 q 2- rLl#l = - F r 2 q .

(3.31)

As in Section I1 [see Eqs. (2.15)-(2.20)], the kinematic condition at the interface is Ll q' = u; '4: a4% IWi

(3.32)

L2V' = u; l4;

(3.33)

for the upper fluid, and aq21alCI2

for the lower fluid. We recall that, to the first order in u1 = q l

and

fl,

u2 = q 2 .

Applying the operator L2 to (3.31) and using (3.33), we have

(3.34)

since 42

a1842 = 41

a l w l = alas,

s being the distance measured along the interface. The next step is to express

& in terms of 42at the interface. For this purpose we use (3.32) and (3.33), and express q' in the form q' = exp( - iut)[l exp(irn4,)

+ r' exp(irn'~$~)].

(3.35)

402

Chia-Shun Yih

From (3.32) and (3.33) we have, if terms of O(PZ) are neglected, (3.36) which shows that we need not evaluate q’ to the order 41

- 92 = -2ha

cos k 4 2 ,

q2/q1 = 1

fi, since

+ 2hB cos k 4 2 ,

where h = (1 -j)/4J.

(3.37)

Using (3.33), we obtain, upon separating the terms containing exp(im4,) from those containing exp(im’&),

t=

iu,m sinh m + O(B), no- m

r‘ =

iB2mtsinh m’ oo - m’

+ O(B).

(3.38)

Substituting (3.35) and (3.38) into (3.36), we can evaluate a1 and fll in CP; in terms of a2 and fi2, but to do this we have to convert the exponentials exp(im4,) and exp(im’41) into sums of exponentials exp(im4,) and exp(im’4,). With higher-order terms omitted, we have

41- 42= (a, - a2)sin k 4 2 = - 2hB sin k + 2 . k Hence

= cos

m42 + hBm [cos(k - m)& - cos m’&] + O(fi2). k

(3.39)

The term containing cos(k - m)@zcan be omitted, for the components with wave numbers m - nk (n = 1,2,3, . ..) and m’ nk are governed by nonhomogeneous equations with the nonhomogeneous parts produced by the basic components with wave numbers m and m’,which are being considered here. Thus the other components are expressible in terms of the basic components, which determine the stability of the flow. Returning to (3.39), then, we see that, as far as the basic components are concerned,

+

hmB exp(im4,) = exp(im42)- -exp(im’&). k

(3.40)

Instability of Surface and Internal Waves

403

Similarly, exp(im'41) = exp(im'4,)

hm'p

+ 7exp(im&).

(3.41)

We now substitute (3.35) and (3.38) into (3.36), use (3.40) and (3.41), and equate the coefficients of the two basic components exp(im4,) and exp(im'4,). The results are, after a good deal of straightforward calculations, a1 = -az sinh m + hBB2rn' sinh m'

fil

=

).

(3.43)

sinh m ) exp(im'42). + hPaz m2 m'(a - m)

(3.44)

-b2 sinh m' - h/3a2m sinh m

Using (3.40) and 3.41), we have, for

=

-

(

ct2

sinh m

- (pz sinh m'

4; at the interface,

" sinh m' + hPp2 mm(a - m')

This will be used in the first term of (3.34). The next step is to calculate the second term in (3.34). Since it contains the factor q1 - q 2 , which is of O(D), we need only to use the terms of zeroth order in /?in (3.44) for 4; at the interface. A straightforward calculation then gives

[

rL2 (ql - q2)q2

,#I'~= - hpe-iar[p2m'(m- uo)sinh m' exp(im4,) a42

+ a,m(m'

- g o ) sinh m exp(im'4,)],

with all components but the basic ones neglected. The third term in (3.34) is e-'"'FL2

(I

a2m sinh m - B -/lzm'sinh m') exp(im&) 4

P2m' sinh m' - f a 2 m sinh m 4

(3.45)

Chia-Shun Yih

404

Putting (3.44)-(3.46) into (3.34) and equating terms of the same wave number, we have [see Eqs. (2.27), (3.29), and (2.30) for comparison] a2(1

+ r tanh m)(a - m)2 cosh m + $B28(1+ r tanh m’)m’(2a - m - m’) cosh m’ + B2rhbm‘ sinh m’ m’(oo - m) a2m sinh rn - !/3,mr sinh in‘ 4

(3.47)

and

f12( 1 + r tanh m’)(a - m’)’ cosh m‘

+ 4a2/l(1 + I tanh m)m(2a - m - m‘) cosh m + a2rhflm sinh m = F;’

(B2m’sinh m’ - -B4a 2 m sinh m).

(3.48)

Let (cr - m)’ = (ao - m)2

+ Alp + O(B2),

Then cr=ao+

2(00 - m)

+ O(P2),

( 3.49)

and, upon neglect of terms of O(B2),we can write (3.47) as ct,il cosh m

S2 [2(1 + I tanh m’) + 4( 1 +mI ttanh m)

x (2ao - m

- m’) cosh m’ + F ;

+ H1P2 cosh m‘ = 0, where

Hi =

sinh m’] (3.50)

405

Instability of Surface and Internal Waves

Similarly, (3.48) can be written as no- m‘ a. - m

&Al cosh m‘-

ma2 [2( 1 + r tanh m)(200 - m - m’) cosh m + 4( 1 + r tanh m‘)

+ F;’

sinh m] + H 2 a z cosh m = 0,

(3.51)

where

The sign of H 1is determined by the sign of m‘(oo - m ) + 1, m(ao - m’) which is equal to

- .m’J(m) __ + 1 mJ(m‘) by virtue of (3.21) and (3.22). It is a simple matter to show that

“W) 1, since m’ > m. Thus the sign of HI is negative. Similarly, the sign of H, is also negative . We now evaluate 2a0 - m - m‘ by the method described in Section I1 [Eqs. (2.35)-(2.39)]. Equation (3.21) can now be written as (o0- m)’

+ F;’(o0

- m)K(m)- F;’a,K(m)

= 0.

(3.52)

where tanh m 1 + r tanh m

(3.53)

ao-rn=i[-F;’K(m)+f(m)],

(3.54)

K(m)= Solving (3.52), we have

406

Chia-Shun Yih

in which

+

f ( m ) = [ F ; 4 K Z ( m ) 4F;2aoK(m)]”z.

(3.55)

The positive sign is taken in (3.54) because a. - m is positive. Similarly, a. - m’ satisfies (3.52), with m replaced by m‘. Hence a.

- m‘ = t[- F;’K(m‘) - f ( m ’ ) ] ,

(3.56)

the negative sign before f ( m ’ ) being taken because oo - m’ is negative. The sum of (3.54) and (3.56) is, after multiplication by 2,

2(2a0 - m - m’) = -F;’[K(m)

+ K(m’)] - M ,

(3.57)

where

M = f ( m ’ ) - f ( m ) > 0.

(3.58)

Returning to (3.50),we see that the term in brackets becomes -D cosh m’, with

D

=~

;

1 + r tanh m‘ 2 [ K ( m ) Mlm’. 1 + r tanh m

+

(3.59)

Similarly, the term in brackets in (3.51) becomes - E cosh m, with E =~

;

1 + r tanh m 2 [K(m’)+ M ] m . 1 + r tanh m’

Therefore we can write (3.50) and (3.51)as a221

=

;(

-

cosh M’

aO-m E a z n 1 = a ( % - H 2 )

7

a2 cosh m

coshm!

9

in which D and E are positive and H I and H, negative. From these two equations we obtain

(3.61)

-=

Since g o - m is positive and a. - m’ is negative, At 0, and the flow is unstable for the disturbances under consideration. Letting

1 = 14 1/2(ao - m),

Instability of Surface and Internal Waves

407

k

FIG.13. Variation of

-A:

with k for r = 1 and various values of FZ.

the growth rate is then AS. In Figs. 13-15, -A: is plotted against k for r = 1, 0.5, and 0.25, and for various values of Ff. We note that both E and H 2contain the factor rn. Therefore so long as rn is not zero the primary flow is unstable. As mentioned before, if F f is not less than unity rn is always positive. If F f is less than unity rn is positive only if k k,, k, being the root of (3.25).

=-

k

FIG.14. Variation of -A: with k for r = 0.5 and various values of F f .

Chia-Shun Yih

408

k

FIG.15. Variation of -1; with k for r = 0.25 and various values of Ff.

In conclusion,stationary internal waves in the fluid system under consideration are unstable for all nonzero wave numbers if F; 2 1, and for all wave numbers greater than k, if F; < 1. We note also that (3.21) and (3.22) show that the m-waves travel with the stream and the m’-waves travel against it.

D.

STATIONARY WAVES IN A CONTINUOUSLY STRATIFIED

FLUID

We shall consider the flow of an incompressible stratified fluid with a general velocity U oover a wavy bottom. If U ois independent of y and p ( y ) denotes the mean density in the absence of wave motion, the stream function is uoy

+ Af(y) cos kx,

where, for the time being, x and y are dimensional, k is the dimensional wave number, andfsatisfies the equation [see Yih, 1965, p. 20, Eq. (31), with U o replacing the c there, or p. 175, Eq. (104), with U oreplacing U and c equal to zero for steady flow]

(pf’y -

(

k2p

+$)f = 0,

(3.62)

with the accent indicating differentiation with respect to y . If the flow is supposed to be confined between two rigid boundaries at distance d apart, the upper one being flat and horizontal and the lower one wavy, we can use d

Instability of Surface and Internal Waves

409

as the length scale. The density on the lower boundary, p o , will be used as the density scale. If we write

then dropping the circumflexes, we have

(pf’)’ - ( k 2 p + F - ’ P ’ ) f = 0,

(3.63)

where the primes indicate differentiationwith respect to the dimensionless y. For simplicity we shall use the Boussinesq approximation, which we expect to yield sufficiently accurate results if the maximum density variation is small in comparison with the mean density. Thus p will be assumed constant and equal to 1 unless it is associated with gravity. If we adopt Boussinesq’s approximation, then whether we assume p to be linear in y or to be an exponential hnction of y the results are the same. We shall then assume p to be a linear function of y, and [note that the present /?, chosen to conform to established usage, differs from that in (3.19)]

p’

-/?.

=

(3.64)

Then (3.63) becomes

f”- (kZ- F r 2 ) f = 0,

(3.65)

where (3.66)

Ff = F2/?.

The condition for f at the upper boundary, where y = 1, is f(1) = 0. The solution of (3.65) satisfying this condition is f(y) = sin ~ (-ly),

I C = ~ FT

- k2.

(3.67)

If the lower boundary is given in dimensionless terms by y = y cos kx,

then since the dimensionless stream function is

h = W ( y ) cos kx, (3.68) $ = Y + h(x, Y ) , and $ can be taken to be zero at the lower boundary, A is related to y by - A sin

Thus, for the primary flow to exist, integer.7

K

K

= y.

is not equal to nn, where n is any

?This condition can be removed by considering nonlinearity.

Chia-Shun Yih

410

We have given the solution for the stationary waves only by a linear theory. As we have stated before, only the first harmonic is of importance, and a nonlinear theory would give substantially the same results. The amp& tude A, being arbitrary (so long as the primary flow exists), can be regarded as the A for the nonlinear theory. The only difference that a nonlinear theory will bring about is the function f ( y ) . But this difference will be of the order O(A),and a glance at (3.68) convinces us that if only terms of the orders O(1) and O ( A )are retained, it is sufficient to use (3.67) and (3.68).

E. INSTABILITY OF STATIONARY WAVESIN A CONTINUOUSLY STRATIFIED FLUID Since the flow is rotational because the fluid is stratified, we must use the Euler equations for the determination of the perturbation flow. If we denote the velocity components of the primary flow by u and u and those of the perturbation by u’ and u’, and write

v=u+ul,

U=u+uI,

the Euler equations are, under the Boussinesq approximation and in dimensionless terms,

au av -at+ u - +axv - = ay Px av av = - p y - F - y p + p’), - + u- + vat ax ay aul

(3.69)

3

a U I

(3.70)

where p is the (total) pressure and p’ the perturbation in density. The equation of incompressibility is

aP’

-

at

a ( p + p’) + v-a / p + p’) = 0, + uax aY

(3.71)

and the equation of continuity for the perturbation flow is aul au’ -+-=o, ax

ay

which permits us to use a stream function $’, in terms of which u’ = 9’Y ’

uI=

-9’

(3.72)

X )

with the subscripts indicating partial differentiation. The velocity components u and u of the primary flow are given by u = $, =

1 + A f ’ ( y ) cos kx,

u=

-i,bx

= A k f ( y ) sin

kx.

(3.73)

Instability of Surface and Internal Waves

411

Elimination of p between (3.69) and (3.70) gives (3.74)

where 1 is the vorticity of the primary flow and flow, so that

-

5 that of the perturbation

5 = -vzI+v.

= -v2*,

The vorticity equation for the primary flow is (3.75)

Taking the difference between (3.74) and (3.75) and neglecting quadratic terms in perturbation quantities, we have L V ~ I ++VE = F - ~ ~ ; ,

(3.76)

where

a

a

at

ax

L=-+--,

E = hy V2$!,- h, V 2 P y+ rl/l V2hx- PxV Z h y.

(3.77)

We now evaluate the right-hand side of (3.76). Using (3.71), we obtain

Lp’ = H,

(3.78)

where

H

= -dpX - v’PY - hy P ’x

+ h p’ x

(3.79)

y ’

In (3.79), the p differs from the p in (3.63) by a term of order O(A),for after the wave motion represented by (3.63), p for the primary flow is a function of both x and y . Indeed,

+ VPy =0

or P J P Y = *XI$, > so that p is a function of I) only. If terms of orders higher than A are neglected, we must have, as the linear theory demands, UP,

P

= 1 - B$,

for otherwise, in the absence of wave motion (i.e., A = 0 and $ = y ) , we should not have (3.64). Thus,

p,

=

-p@x,

p Y = -D$

y

=

- P(1 + hY).

(3.80)

+ Ap; +

(3.8 1)

Let $’ = I+vo + Alpl

+

..a,

p’ = p b

**..

412

Chia-Shun Yih

Then from (3.72), (3.76), and (3.78), we have

L2, v2qo

+ Ff(qo),, = 0,

(3.82)

where L,,, is L when only terms of order O(1) are retained. Let us try the following form for q0:

9o(Y) e x p m - 0 0 t)l. Then

To the order O(l), the boundary conditions are

go(0) = 0 = go(1). The solution of the eigenvalue problem for go is then (a,

- a)'

+

= FT2a2/(a2 n2x2),

(3.83)

where n indicates the mode. Given k, we choose a oosuch that the two roots of a in (3.83), m and m', satisfy

m' - m = k.

(3.84)

As in Section III,C, m is less than ooand m' greater than a, . This can be seen by plotting the two sides of (3.82) against a for any a. and Fi . Thus a.

- m = F;'rn(m2

D 0 - m' =

+ n2x2)-'/',

-F;1m'(m'2

+n n ) 2

2 -112.

(3.85) (3.86)

The difference of the left-hand sides of (3.85) and (3.86) is k. From the right-hand sides we see then that it is impossible to satisfy (3.84) unless

k < 2F;'.

(3.87)

If (3.87) is satisfied, however, it is possible to find m and m' satisfying (3.83) and (3.84), and the quantities m and a. are plotted against k in Figs. 16 and 17 for n = 1, Figs. 18 and 19 for n = 2, and Figs. 20 and 21 for n = 3. These curves are constructed by first assuming m, finding a, by (3.85), and solving (3.83) for m' with the 0, just found. Then k is known, and we have for this k the value for m and the value for ao. All the curves in Figs. 16-21 have vertical asymptotes at k = 2F; We shall then take

$6

= (ael

+ be2) sin nxy,

(3.88)

with el = exp[i(mx - at)],

e, = exp[i(m'x - at)].

(3.89)

Instability of Surface and Internal Waves

413

k

FIG.16. Variation of rn with k for n = 1 and various values of F t .

Note that we use CJ in (3.89) rather than o0, for we anticipate that the terms of order O ( A )in (3.76) and (3.78) will change goby an amount of order O ( A ) . The use of an exponential time factor is certainly justified by the forms of (3.76) and (3.78). Let pb = (a, el

+ b , e 2 )sin nny.

k

FIG. 17. Variation of uo with ,k for n = 1 and various values of F f .

(3.90)

Chiu-Shun Yih

414 E

4

m

?

2

I

0 k

FIG.18. Variation of m with k for n = 2 and various values of Ff.

Then the terms of order 0(1)in (3.78) give a, = pm/(oo- m),

bl

= pm’b/(a0

- m’).

(3.91)

Now, from (3.76) and (3.78), we obtain

L? V2* = -LE

+ F-’H,.

k

FIG. 19. Variation of oo with k for n = 2 and various values of FZ,

(3.92)

415

Instability of Surface and Internal Waves

5

4

m 3

2

I

c

0.5

1.5

I

2

k

FIG.20. Variation of m with k for n = 3 and various values of F:.

Let (0

- m)’ = (oo- m)’ +

A

+ o(A’).

(3.93)

When terms of the zeroth order in A are collected in (3.92),we have (3.82), which is satisfied by (3.88). When collecting terms oforder O(A)in (3.92),we ,I and pb for p’ in E and H except in the term - v’pyin H , because use I& for P

k

FIG.21. Variation of uo with k for n = 3 and various values of F’

Chiu-Shun Yih

416

py contains the term -/?,which is of the zeroth power in A. Furthermore, we use Lo E for L E because E = O(A). With all this in mind, (3.92) becomes

A V2$/'o + L; Vz$/'l

+ Ff(li/;),, = -Lo E + F; ' K ,

(3.94)

where K = /?-'(H - flu'), = [$/'oyhx - $/'oxhy - B-'(pb,h, - &h,)],

.

Straightforward calculation gives

+ Eze,,

LoE = Elel

where E i = E L I+ E i z

Ez = Ezi

9

+ E22

9

with E l l = bA(2m')-'(mfZ + n2n2)oo(oo - m)(oo- 2m')f' sin nny, El, = bAkn7r(2m'2)-1(m'2 + n2a2)oo(oo - m)(oo - 2m')fcos nay, E,, = aA(2m)-'(m2 + ~ ~ a ~ ) -o m')(oo ~ ( o-~2m)f' sin nay,

+

E,, = -aAknn(2m2)-'(mZ n2a2)oo(oo - m')(oo- 2m)fcos nny. As for K, we have

FL2K = K l e l

+ K,e,,

where Kl = K l l

+ K12

9

K, = K , ,

+ K,,

7

with

K l l = bA(2m')-'m(mf2 + n21r2)oo(oo - m')f' sin nay, K , , = bA(2m'2)-1knnm(m'2 + n2d)oo(oo- m')fcos nny,

+ n2a2)oo(oo- m ) f ' sin nay, K,, = -aA(2m2)-'knxm'(m2+ n27r2)oo(oo - m)fcos K Z t = aA(2m)-'m'(m2

nay.

We now multiply (3.94) by 2 sin nlry and integrate from y = 0 to y = 1. [Integrating from the lower boundary? would introduce terms of O(AZ).] The terms involving JI; vanish upon integration by parts. The rest of the t Note that A@\ is of order O ( A )everywhere.It is zero on the lower boundary,at which y is also of order O(A).Hence ,4q1is of order O ( A 2 )and A$',? = O ( A )between y = 0 and the lower boundary. Integration in this interval gives terms of O(A2).

Instability of Surface and lnternai Waves

417

terms, upon separating terms containing el from those containing e2, integrating by parts, and neglecting terms of O(A2),gives -aA(m2

+ n2n2)1,= bA(Zm')-'(m'' + n2n2)aoM1 (3.95)

-bA-

oo - m' (m" oo - m

+ n2n2)1,= aA(2m)-'(m2 + n2n2)ooM2 (3.96)

where 1

I =

jof' sin2 nny dy, + m(oo- m'), - m')(oo- 2m) + m'(a0 - m).

M , = -(oo - m)(oo - 2m') M , = -(go

Now since K is not equal to any integral multiple of n,1 is not zero. Furthermore, M1 and M 2 are equal, as we can see by a glance at their definitions. Hence

- ___ oo oo - m'm 0;(4mrn')-~M;(l -&)(l

1 21 -

+&)12.

(3.97)

Since oo is greater than m but less than m',and since k is less than m', we see that 1;