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Based on joint work with X. Ding

Chjan Lim RPI, Troy, NY, US http://www.rpi.edu/~limc. Phase Transitions in Planetary atmospheres - a Shallow-Water Model and applications to Venusian super-rotation and giant spots on Jupiter. Based on joint work with X. Ding

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Based on joint work with X. Ding

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  1. Chjan Lim RPI, Troy, NY, UShttp://www.rpi.edu/~limc Phase Transitions in Planetary atmospheres - a Shallow-Water Model and applications to Venusian super-rotation and giant spots on Jupiter Based on joint work with X. Ding Cf. Physica A 2006, J. Math Phys june 2007, Phys Fluids 08, SIAP 06 and book Vorticity, Stat Mech and MC Simulations, Springer Oct 2006

  2. Collaborators and acknowledgement • Dr. Xueru Ding, PhD student at RPI • Support provided by US ARO and DOE • Based on paper submitted to Phys Fluids In april 2008 (with Xueru Ding) and Conf Proc of IUTAM Symp. Steklov Moscow 2006 (Plenary Talk)

  3. Outline of talk on a unified theory • Major physical result 1- factors for anticyclonic dominance in coherent spots on Gas Giants • Major physical result 2 – factors for absence of sub-rotating nearly solid-body barotropic flows in slowly-rotating terrestrial planets • Orientation asymmetry in the Lagrangians • Exact solns by M. Kacs’ spherical model method

  4. Recent debates • Marcus-Sommeria-Swinney 1980s – 1990 supports QG model which does not have anticyclonic asymmetry but provide a closer fit to rim velocities in GRS • Williams-Yamagata-Antipov 1980 – 1990 favors the IG regime in SWE due to this asymmetry.

  5. Detail of GRS, Ovals, Barge etc

  6. Prograde from QG

  7. Cyclonic vs Anticyclonic

  8. Derivation of SWE

  9. Lagrangian • Lagrangian is : L = KE + PE = KE_r + AM + IM + PE “=“ (u . u) + (u . u_p) + (u_p . u_p) + gh^2

  10. Euler-Lagrange of L = SWE del (L) = 0

  11. Helmholtz, stream fn and vel pot

  12. Shallow Water (SWE)

  13. From Invariants to Constraints– invariants of unforced inviscid SWE H = KE_r + PE conserved; not L, so not AM + IM = L – H; must not fix L

  14. Extension to freely-decaying SWE • All enstrophies (including higher vorticity moments) decay; Reduced energy / Hamiltonian H decays • For bdd (periodic) 2D domains, rigorous results for selective decay of quadratic enstrophy to minimum Dirichlet Quotient –related to dual cascades first proved by Foias and Saut 84, much later Majda-Wang 01 • Tea cup and Sommeria- van Heijst experiments suggest more than 2 separated time-scales in nearly-inviscid quasi-2D bdd flows – viscous time, Ekman pumping time and up-scale energy transfer time in inverse cascade • Rigorous results of Chemin, Grenier et al for quasi-2D rotating flows in bdd domains confirm at least 3 time-scales and that interior flow is inviscid.

  15. Formulation of constraints • After choosing canonical –in-action L microcanonical constraints on circulations and enstrophies follow: • (a) enstrophies nearly const in fast time • (b) canonical on enstrophies give Gaussian – not good for phase transitions

  16. Constraints Choose Action = L in path-integral form for Gibbs partition function; so H, AM, IM changes but not sum Fix 3 circulations – height h(x), rel vort, divergence; first by mass conservation and incompressibility; last 2 by Stokes on sphere Fix 2 quadratic enstrophies – rel vort and divergence instead of potential vorticity enstrophy

  17. Classical energy-enstrophy models • Canonical in both energy and enstrophy (Kraichnan 1975) is a Gaussian model that is not well-defined for low temperatures. • Miller-Robert-Sommeria theories (1990s) conserves infinite number of enstrophies • Majda-Turkington (2000s) uses finite inequality constraints and apriori distributions on small scales

  18. Mixed Gibbs partition fn

  19. Lattice models

  20. Lattice model constraints – circulat^n

  21. Lattice constraints - enstrophies

  22. Pictures from MC

  23. More pictures

  24. More pics

  25. Anticyclonic low energy spot

  26. Venus Super-Rotation

  27. BECondensation to super-rotation

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

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

  30. Energy III

  31. Spin Lattice coupled BV Models

  32. Coupled BV - constraints

  33. Coupled BV – partition function = 0

  34. Coupled BV - BECondensation Super-Rotation

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

  36. Coupled BV – disordered phase

  37. Phase transitions III

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

  39. Coupled BV – MC free energy

  40. III Exact solutions – spherical models

  41. Spherical model - BEC continued

  42. Exact spherical model soln

  43. Exact soln continued

  44. Exact soln

  45. Exact soln to Physical content

  46. Theorems with Physical content II

  47. Spherical models for coupled SW flows

  48. Jupiter

  49. Conclusions • Possible extensions to include time dynamics from one most probable state to another along the lines of S. Wang and T. Ma’s new work – looks promising from the GL example (JMP 08) • Many rigorous math results on existence of free energy minimizers for the Shallow Water Model – Direct methods of the C. of Variations

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