1. MODELLING AND COMPUTATIONAL SIMULATION OF EULERIAN FLOW Wayne Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 874-2749

2. RIGID BODIES Euler’s equation inertia operator (from mass distribution) angular velocity in the body for their inertial motion Theoria et ad motus corporum solidorum seu rigodorum ex primiis nostrae cognitionis principiis stbilita onmes motus qui inhuiusmodi corpora cadere possunt accommodata, Memoirs de l'Acad'emie des Sciences Berlin, 1765.

3. IDEAL FLUIDS Euler’s equation pressurevelocity in space for their inertial motion Commentationes mechanicae ad theoriam corporum fluidorum pertinentes, M'emoirs de l'Acad'emie des Sciences Berlin, 1765. outward normalof domain

4. GEODESICS Moreau observed that these classical equations describe geodesics, on the Lie groups that parameterize their configurations, with respect to the left, right invariant Riemannian metric determined by the inertia operator (determined from kinetic energy) on the associated Lie algebra Une method de cinematique fonctionnelle en hydrodynamique, C. R. Acad. Sci. Paris 249(1959), 2156-2158

5. LIE GROUPS, LIE ALGEBRAS, AND THEIR REPRESENTATIONS Lie groupLie algebra define the adjoint linear dual For and coadjoint representations

6. CARTAN-KILLING OPERATOR Theorem 1.The Cartan-Killing operator is self-adjoint and satisfies B is nonsingular iff G is semisimple (Cartan), B is positive semidefinite iff G is compact (Weyl) defined by

7. TRAJECTORIES AND VELOCITIES These trajectories are called the (angular) velocities in the body and in space. and Definition Ifis a smooth trajectory are trajectories in its Lie algebra in a Lie group G then In the sequel we will let and define

8. INERTIA OPERATORS AND KINETIC ENERGY and a left, right invariant Riemannian metric on G is an inertial operatorDefinition if it is self-adjoint and positive definite. Then A defines an inner product Kinetic energy of rigid bodies and fluids defines an inertial operator

9. EULER’S EQUATION ON LIE GROUPS Theorem 2. The motion of a physical system whose configuration space is a Lie group G is a trajectory in G. If the kinetic energy E is left, right invariant then the trajectory is a geodesic (shortest path between any two points) with respect to the left, right invariant Riemannian metric induced by E Arnold, V. I., Mathematical Methods of ClassicalMechanics, Springer, New York, 1978 Theorem 3. A trajectory is a geodesic with respect to the leftrightinvariant Riemannian metric on G iff u satisfies Euler’s eqn.

10. GLOBAL ANALYSIS based on this geometric formulation provides a powerful tool for studying fluid dynamics Arnold used it to explain sensitivity to initial conditions in terms of curvature Ebin, Marsden, and Shkoller used it to derive existence, uniqueness and regularity results for both Euler’s and Navier-Stokes equations These ideas are fundamental for the study of a large class of nonlinear partial differential equations and have developed into the extensive field of topological hydrodynamics

11. VORTICITY FORMULATION If Theorem 4. Let be the Cartan-Killing operator, and then be an inertial operator, with satisfies the equation satisfies the vorticity equation Ifis nonsingular then the converse holds

12. VORTICITY FORMULATION is a geodesic for the left Corollary. Let defined on a semisimple Lie group be an inertial operator Then invariant Riemannian metric defined right byiffwhere for and is the vorticity Under these conditions the enstrophy is constant

13. LAGRANGIAN FORMULATIONS andTheorem 5. Let andfor thensatisfies iff Corollary A trajectoryis a geodesic invariant Riemannian metric iff with respect to the leftright

14. LAGRANGIAN FORMULATIONS andTheorem 6. Let and for thensatisfies iff Corollary If G is semisimple then is a geodesic with respect to the left invariant Riemannian metric right induced by an inertial operator A iff the vorticity satisfies

15. EULER SYMPLECTIC INTEGRATOR Preserves Error

16. IMPROVED SYMPLECTIC INTEGRATOR Preserves Error

17. SPECTRAL BASIS forThere exist a basis such that and If and then

18. SPECTRAL REPRESENTATION The vorticity equation has the spectral representation where the structure constants are defined by

19. STREAM FUNCTION (orthogonal coordinates Then The operator with the Green’s function is constant along particles in the flow, therefore the momentsare invariant is convolution

20. EULER FLOW ON Identified with ideal flows onthat are periodic with respect to the subgroup with average value zero, for the spectral basis of the complexified Poisson Bracket Lie algebra

21. FAIRLIER, FLETCHER AND ZACHOS for odd defined the map

22. ZEITLIN used the approximation to approximate flow onby flows on

23. NUMERICAL EXPERIMENTS Simplest model is the 8 dim Lie group Vorticity is represented as a real valued function on the group Rotations, translations, and reflections (later must be combined with multiplying the vorticity by –1) are Lie group automorphisms that also commute with the inertial operator A and therefore with Euler flow Vorticities that are invariant under these operations are either fixed points or periodic points of Euler flow

24. WAVELET BASES Neither the canonical Fourier basis nor the canonical sparse matrix basis provides a sparse representation of Euler’s equation on SU(n) Wavelet vorticity bases provide nearly sparse representations for Euler’s equations because (i)Green’s operator is Calderon-Zygmund (ii) Poisson bracket is exponentially localized Wavelet bases provide simple approximations for invariant moments and energy We are using wavelet bases to study Okubo-Weiss criteria for two-dimensional turbulence

25. FUTURE STUDIES Determine if Euler flow on SU(3) is integrable Explore gauge theoretic issues including instantons Analyze the effects of curvature Investigate vorticity cascade starting with SU(5) Study the role of symmetry