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

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

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

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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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Venus Super-Rotation

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BECondensation to super-rotation

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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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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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Restframe energy

• For nondivergent barotropic fluid, the energy in the restframe

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Energy II

• Dropping the last – constant term – we get

Second term is proportional to angular momentum

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Energy III

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Spin Lattice coupled BV Models

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Coupled BV - constraints

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Coupled BV – partition function

= 0

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Coupled BV - BECondensation

Super-Rotation

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Coupled BV Monte-Carlo simulation results

Sub-Rotation

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Coupled BV – disordered phase

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coupledBV – MC phase transitions

Mean Nearest Neighbor Parity

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Coupled BV – transitions II

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Phase transitions III

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Coupled BV – MC entropy

Based on X. Ding’s algorithm for calculating degeneracy

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Coupled BV – MC free energy

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Coupled SW – rotating solid sphere

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Coupled SW spin lattice models

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Coupled SW – physical quantities

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Coupled SW - constraints

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Coupled SW – partition function

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Jupiter

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Transition to subrotating solid-bodymoderate spin

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Transition pic2 moderate spin

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Anticyclonic asymmetry I

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Anticyclonic asymmetry II

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Anticyclonic dominance III

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Signs of bands - large spins

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Red Spot like - very small spin; in southern hemisphere

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

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

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Red spot like IV - negative T; south

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III Exact solutions – spherical models

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Spherical model - BEC continued

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Exact spherical model soln

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Exact soln continued

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Exact soln

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Exact soln to Physical content

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Theorems with Physical content II

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Spherical models for coupled SW flows

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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.