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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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April 22, 2023 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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April 22, 2023 3
Venus Super-Rotation
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April 22, 2023 4
BECondensation to super-rotation
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April 22, 2023 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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April 22, 2023 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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April 22, 2023 7
Restframe energy
• For nondivergent barotropic fluid, the energy in the restframe
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April 22, 2023 8
Energy II
• Dropping the last – constant term – we get
Second term is proportional to angular momentum
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April 22, 2023 9
Energy III
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April 22, 2023 10
Spin Lattice coupled BV Models
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April 22, 2023 11
Coupled BV - constraints
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April 22, 2023 12
Coupled BV – partition function
= 0
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April 22, 2023 13
Coupled BV - BECondensation
Super-Rotation
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April 22, 2023 14
Coupled BV Monte-Carlo simulation results
Sub-Rotation
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April 22, 2023 15
Coupled BV – disordered phase
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April 22, 2023 16
coupledBV – MC phase transitions
Mean Nearest Neighbor Parity
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April 22, 2023 17
Coupled BV – transitions II
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April 22, 2023 18
Phase transitions III
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April 22, 2023 19
Coupled BV – MC entropy
Based on X. Ding’s algorithm for calculating degeneracy
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April 22, 2023 20
Coupled BV – MC free energy
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April 22, 2023 21
Coupled SW – rotating solid sphere
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April 22, 2023 22
Coupled SW spin lattice models
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April 22, 2023 23
Coupled SW – physical quantities
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April 22, 2023 24
Coupled SW - constraints
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April 22, 2023 25
Coupled SW – partition function
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April 22, 2023 26
Jupiter
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April 22, 2023 27
Transition to subrotating solid-bodymoderate spin
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April 22, 2023 28
Transition pic2 moderate spin
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April 22, 2023 29
Anticyclonic asymmetry I
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April 22, 2023 30
Anticyclonic asymmetry II
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April 22, 2023 31
Anticyclonic dominance III
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April 22, 2023 32
Signs of bands - large spins
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April 22, 2023 33
Red Spot like - very small spin; in southern hemisphere
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April 22, 2023 34
More on Red Spot like II with slightly different constraints; south
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April 22, 2023 35
Red spot like III - small spin large potential enstrophy; south
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April 22, 2023 36
Red spot like IV - negative T; south
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April 22, 2023 37
III Exact solutions – spherical models
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Spherical model - BEC continued
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April 22, 2023 39
Exact spherical model soln
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April 22, 2023 40
Exact soln continued
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April 22, 2023 41
Exact soln
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April 22, 2023 42
Exact soln to Physical content
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April 22, 2023 43
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.