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CPY Document ; \' L /' WHOI-94-12 l.(/o 1993 .. ~::) : ~"" r;"v''-" / ().'- t:'~~ .' // ,/ cew\ ./ ~ -:,f';:;" 'L T L_/~" ", '_~f //// ,;/ f'/-,~," '" ' \?':'/. ~:. / ,/ c'" ~.::~d./ ~" ".;,./ / ~'/'./ Course Lectures Abstracts of Participants Fellows Project Reports WHOI-9412 "Geometrical Methods In Fluid Dynamics" 1993 Summer Study Program in Geophysical Fluid Dynamics ¡j',,, "...~. ! r ~~--~ i'/ l,~.'J / l t~,- j. by ",~.." ~. Ii. -'., "'::'' Rick Salmon, Director I' , and Barbara Ewing-Deremer, Staf Assistant l / " /i I '" ( / ';'-';":( ..' / .::t~:' i/ i Woods Hole Oceanographic Institution /~:~. ~... - i/ Woods Hole, Massachusetts 02543 ..:¡ ." /r (.""'~(~" /! Summer 1993 .Qt/ / .;'" / Tecca Report ~ Funding was provided by the National Science Foundation under Grant No. OCE-8901012. Reproduction in whole or in part is permitted for any purpose of the United States Government. This report should be cited as Woods Hole Oceanog. Inst. Tech. Rept., WHOI-94-12. Approved for public release; distribution unlimited. Approved for Diution: ru - I"LI :: Õ~:i_ 0I"ri ::ID- 0 ::-0 .. John W. rio Associate D' ducation -_0 Dean of Graduate Studies -~ "'.: Contents Group Photograph 6 Director's message 7 1993 GFD Participants 8 1993 Lecture Schedule 10 Photo: Principal Lecturer Phil Morrison 16 Principal Lectures by Philip J. Morrison: HAMILTONIAN DESCRIPTION OF THE IDEAL FLUID Introduction 1 7 i. Rudiments of few degree-of-freedom systems illustrated by passive advection in two-dimensional fluids A. Model for two-dimensional fluid motion 21 B. Passive advection 23 C. Integrable systems: one degree of freedom 24 D. Chaotic dynamics: two degrees of freedom 26 E. "Diffusion": three degrees of freedom 3 1 II. Functional differentiation, two action principles of mechanics and the action principle and canonical Hamiltonian description of the ideal fluid A. Functional calculus 3 4 B. Two action principles of mechanics 41 C. Action principle and canonical Hamiltonian description of the ideal fluid in Lagrangian or material variables 4 4 III. Noncanonical Hamiltonian dynamics - Examples A. Noncanonical Hamiltonian dynamics 50 B. Examples 1. Free rigid body 57 2. Korteweg-de Vries equation 58 3. I-D pressureless fluid 59 4. I-D compressible fluid 60 5. 2-D Euler scalar vortex dynamics 61 6. 3-D ideal fluid 63 7. General comments 66 iv. Tutorial on Lie groups and algebras, reduction-realization, and Clebsch variables A. Tutorial on Lie groups and algebras 67 B. Reduction-realization 1. Reduction of finite dimensional systems 75 2. Standard reduction 76 3. Reduction of the free rigid body 78 4. Reduction for the ideal fluid: Lagrangian to . Eulerian varables 80 C. Clebsch variables 1. Clebsch variables for finite systems 83 2. Clebsch variables for infinite systems 84 3. Fluid examples 86 4. Semi direct product relations 88 5. Other Clebsch reductions: the ideal fluid 88 V. Stability and Hamiltonian systems A. Stability and canonical Hamiltonian systems 9 1 B. Stability and noncanonical Hamiltonian systems 100 C. Dynamical accessibility 1 05 Photo: Principal Lecturer Ted Shepherd 112 Principal Lectures by Theodore G. Shepherd: .";¡,~ APPLICATIONS OF HAMILTONIAN THEORY TO GFD j, -r¡ 1. Generalized1.1 HamiltonianIntroduction dynamics 1 1 3 1.31.2 Steady Dynamics states and conditional extrema1 1 114 5 1.4 Example: barotropic vorticity equation 1 1 6 1.5 Symmetries and conservation laws 11 8 1.6 Steadily translating solutions 120 2. Hamiltonian2.1 The structure two-layer of quasi-geostrophic model flow 121 2.2 Continuously stratified flow over topography 1 23 2 3. Pseudoenergy and available potential energy 3.1 Disturbances to basic states 125 3.2 APE of internal gravity waves 126 3.3 Pseudoenergy 127 3.4 Example: stratified Boussinesq flow 129 3.5 Nonlinear static stability 131 3.6 Nonlinear saturation of instabilities 132 4. Pseudoenergy and Arnol'd's stability theorems 4.1 Arnol'd's stability theorems 134 4.2 Andrews' theorem 137 4.3 Available energy 138 4.4 Nonlinear saturation of baroclinic instability 139 5. Pseudomomentum 5.1 General construction 142 5.2 Example: barotropic vorticity equation 143 5.3 Wave, mean-flow interaction 145 5.4 Wave action 147 5.5 Instabilities 148 Reports of the Fellows A nonlinearOliver wave In rotating Bühlershallow water 153 Drop formation from viscous fingering 1 60 Silvana S. S. Cardoso A study of a shell model of fully developed turbulence 1 70 Diego del -Castillo- Negrete Gravity wave generation by quasi-geostrophic point vortices 185 Geoffrey T. Dairiki A periodic orbit expansion of the Lorenz system 199 Sean McNamara Elliptical vortices in shear: Hamiltonian formulation, vortex merger and chaos 2 11 Keith Ngan 3 A Hamiltonian weak-wave model for shallow water flow 224 Caroline L. Nore Dissipative quantum chaos 239 Dean Petrich Moisture and available potential energy 248 Kyle Swanson On aspects of Benney's equation 257 Rodney A. Worthing Other Lectures Coherent structures and homoc1inic orbits 266 N. J. Balmforth andIntergyre waterbuoyancy exchange of combined forcing wind 27 1 Liang Gui Chen Fast dynamos near integrability 275 Steve Childress Pressure fluctuations in swirling turbulent flows 277 Stephan Fauve Gulf Stream meandering: Some physics and some biology 282 Glenn Flierl and Cabell Davis Generation of solitary waves by external forcing 283 Roger Grimshaw asymptoticsSolitary waves with oscillatory Roger tails and Grimshaw exponential 29 2 inVertical the temperature Gulf gradient of finestructure California spectra 299 Juan A. Rodrigues-Sero and Myrl C. Hendershott On the representation of a magnetic field as a vector product of two gradients Louis N. Howard 3 00 4- Why do synoptic scale eddies like to go to the tropics? 304 R. Kimura and H. B. Cheong Short wavelength instabilities of Riemann ellipsoids 309 Norman R. Lebovitz forV orticity balanced coordinates and a canonical models form 3 1 2 Gudrun Magnusdottir Necessary conditions for statistical stability: with applications to large scale flow due to convection 3 1 7 Wilem V. R. Malkus Three-dimensional, Hamiltonian vortices 322 Steve Meacham Higher order models for water waves 327 Peter J. Olver Hamiltonian GFD and stability 332 Pedro Ripa Structure of the canonical transformation 337 Ian Roulstone and Michael J. Sewell On the Hamiltonian structure of semi-geostrophic theory 342 Ian Roulstone and John Norbury Linear and nonlinear baroclinic wave packets: atmospheric cyclogenesis and storm tracks 347 Kyle Swanson Instability in extended systems 351 L. Tao and E. A. Spiegel The fluid dynamics of crystal slurries 3 60 Andrew W. Woods and Richard A. Jarvis A laboratory model of cooling over the continental shelf 3 6 2 J. A. Whitehead s- o Front Row: Shepherd, Salmon, Morrison, Bokhove Middle Row: Spiegel, Frazell, Tao, Ewing-DeRemer, Petrich, Keller, Veronis, Cardoso, Nore, Dairiki, Buhler, Ngan, Worthing Back Row: Whitehead, McNamara, Woods, Del-Castilo, Swanson, Lebovitz, lerley, Balmforth, Meacham, Malkus, Vallis Director's message "Geometrical methods in fluid dynamics" were the subject of the 1993 GFD session, a mathematical summer in which the inhabitants of Walsh Cottage explored applications of Hamiltonian fluid mechanics and related ideas about symmetry and conservation laws to practical problems in geophysical fluid dynamics. Experts Phil Morrison and Ted Shepherd set the pace with an intensive introductory course, leaving themselves exhausted and the rest of us greatly stimulated. Subsequent lecturers "fanned out" to cover the spectrum of currently interesting topics, from slow manifolds to fast dynamos. The 1993 session attracted an extraordinary number of visitors, forcing the director to abandon his early promise of "no afternoon lectures after the first few weeks." But despite the heavy schedule, lots of work got done, especially by our fellows, several of whom apparently discovered the secret of going entirely without sleep. This year we stretched our meager resources to accept 10 fellows from a group of 45 outstanding applicants. However, I must sincerely apologize to the many excellent young scientists who had to be turned down. This year, in an effort to decrease the 1993 volume's size while increasing its archival value, I mildly discouraged participants from submitting abstracts of work that would soon be published in a regular journal. Therefore, readers should be sure to scan the "1993 Lecture Schedule" to get a complete picture of what went on at the cottage. Once again, it is a great pleasure to thank the Woods Hole Oceanographic Institution for its warm hospitality, and Jake Peirson and the staff of the WHOI Education office for their indispensable help. Very special thanks go to our administrative assistant, Barbara Ewing-DeRemer; to our crisis-management specialist, Bob Frazell; and to Steve Meacham and Glenn Flierl, who kept things running smoothly in the computer trailer. We gratefully acknowledge the continuing support of the National Science Foundation and the Office of Naval Research. Rick Salmon 1993 Director - -l - 1993 GFD Participants The Fellows Oliver Bühler Germany Diego del-Castillo-Negrete Mexico Silvana Cardoso Portugal Geoffrey Dairiki USA Sean McNamara USA Keith Ngan Canada Caroline Nore France Dean Petrich USA Kyle Swanson USA Rodney Worthing USA The Staff and Visitors J ames Anderson Stevens Institute of Technology Neil Balmforth Columbia University Janet Becker University of California, San Diego Onno Bokhove University of Toronto Sally Bower University of Cambridge Paola Cessi University of California, San Diego
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