1. Chjan Lim
RPI, Troy, NY, US
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
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3. Venus Super-Rotation
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4. BECondensation to super-rotation
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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
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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
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7. Restframe energy
For nondivergent barotropic fluid, the energy in the restframe
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8. Energy II Dropping the last – constant term – we get
Second term is proportional to angular momentum
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9. Energy III
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10. Spin Lattice coupled BV Models
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11. Coupled BV - constraints
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12. Coupled BV – partition function
= 0
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13. Coupled BV - BECondensation
Super-Rotation
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14. Coupled BV Monte-Carlo simulation results
Sub-Rotation
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15. Coupled BV – disordered phase
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16. coupledBV – MC phase transitions
Mean Nearest Neighbor Parity
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17. Coupled BV – transitions II
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18. Phase transitions III
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19. Coupled BV – MC entropy
Based on X. Ding’s algorithm for calculating degeneracy
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20. Coupled BV – MC free energy
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21. Coupled SW – rotating solid sphere
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22. Coupled SW spin lattice models
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23. Coupled SW – physical quantities
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24. Coupled SW - constraints
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25. Coupled SW – partition function
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26. Jupiter
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27. Transition to subrotating solid-body moderate spin
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28. Transition pic2 moderate spin
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29. Anticyclonic asymmetry I
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30. Anticyclonic asymmetry II
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31. Anticyclonic dominance III
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32. Signs of bands - large spins
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33. Red Spot like - very small spin; in southern hemisphere
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34. More on Red Spot like II with slightly different constraints; south
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35. Red spot like III - small spin large potential enstrophy; south
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36. Red spot like IV - negative T; south
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37. III Exact solutions – spherical models
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38. Spherical model - BEC continued
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39. Exact spherical model soln
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40. Exact soln continued
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41. Exact soln
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42. Exact soln to Physical content
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43. Theorems with Physical content II
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44. Spherical models for coupled SW flows
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45. References:
M. Kac and I. Berlin, 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.
November 20, 2018