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Based on joint work with X. Ding Cf. Physica A 2006, J. Math Phys june 2007,

Chjan Lim RPI, Troy, NY, US http://www.rpi.edu/~limc. Statistical Equilibrium in large scale atmospheric and planetary flows - exact solns for phase transitions to super-rotation in barotropic and divergent flows coupled to rotating sphere. Based on joint work with X. Ding

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Based on joint work with X. Ding Cf. Physica A 2006, J. Math Phys june 2007,

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  1. Chjan Lim RPI, Troy, NY, UShttp://www.rpi.edu/~limc Statistical Equilibrium in large scale atmospheric and planetary flows - exact solns for phase transitions to super-rotation in barotropic and divergent flows coupled to rotating sphere Based on joint work with X. Ding Cf. Physica A 2006, J. Math Phys june 2007, and new book Vorticity, Stat Mech and MC Simulations, Springer Oct 2006

  2. Collaborators and acknowledgement • Xueru Ding, PhD student at RPI • Dr. Joseph Nebus, lecturer at NUS Singapore • Support provided by US ARO and DOE

  3. Venus Super-Rotation

  4. BECondensation to super-rotation

  5. Coupled fluid – solid sphere system • Thin fluid shell – nondivergent and divergent – envelopes rotating infinitely massive solid sphere • Fluid assumed inviscid but able to exchange energy and angular momentum with sphere • This complex torque mechanism is not resolved

  6. Energy of the Fluid • Due to above modeling assumptions, rest frame energy of the fluid is not a Hamiltonian • No need for that in generalized path-integral approach used here • There is no need for a local in time governing PDE in the statistical mechanics approach in use

  7. Restframe energy • For nondivergent barotropic fluid, the energy in the restframe

  8. Energy II • Dropping the last – constant term – we get Second term is proportional to angular momentum

  9. Energy III

  10. Spin Lattice coupled BV Models

  11. Coupled BV - constraints

  12. Coupled BV – partition function = 0

  13. Coupled BV - BECondensation Super-Rotation

  14. Coupled BV Monte-Carlo simulation results Sub-Rotation

  15. Coupled BV – disordered phase

  16. coupledBV – MC phase transitions Mean Nearest Neighbor Parity

  17. Coupled BV – transitions II

  18. Phase transitions III

  19. Coupled BV – MC entropy Based on X. Ding’s algorithm for calculating degeneracy

  20. Coupled BV – MC free energy

  21. Coupled SW – rotating solid sphere

  22. Coupled SW spin lattice models

  23. Coupled SW – physical quantities

  24. Coupled SW - constraints

  25. Coupled SW – partition function

  26. Jupiter

  27. Transition to subrotating solid-bodymoderate spin

  28. Transition pic2 moderate spin

  29. Anticyclonic asymmetry I

  30. Anticyclonic asymmetry II

  31. Anticyclonic dominance III

  32. Signs of bands - large spins

  33. Red Spot like - very small spin; in southern hemisphere

  34. More on Red Spot like II with slightly different constraints; south

  35. Red spot like III - small spin large potential enstrophy; south

  36. Red spot like IV - negative T; south

  37. III Exact solutions – spherical models

  38. Spherical model - BEC continued

  39. Exact spherical model soln

  40. Exact soln continued

  41. Exact soln

  42. Exact soln to Physical content

  43. Theorems with Physical content II

  44. Spherical models for coupled SW flows

  45. References: • M. Kac and I. Berlin, 1953 spherical model paper Phys Rev 1952 • Feynman, PhD thesis 1943 Princeton U. • Polyakov, Gauge fields and Strings • C. Lim and J. Nebus, Vorticity, Stat Mech and MC simulations, Springer-Verlag book Oct 2006 • Kraichnan JFM 1975 • Work of J. Sommeria 1986 • Work of Tabeling et al • Work of G. van Heijst on 2d no slip square domains • X. Ding and C. Lim, Physica A 2006 • C. Lim, J Math Phys 2007 • C. Lim, SIAM J. Applied Math 2005 • C. Lim, Exact solutions for a statistical equilibrium theory of SW flows coupled to massive rotating sphere, preprint 2006.

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