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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
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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 November 20, 2018
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Venus Super-Rotation November 20, 2018
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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 November 20, 2018
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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 November 20, 2018
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Restframe energy For nondivergent barotropic fluid, the energy in the restframe November 20, 2018
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Energy II Dropping the last – constant term – we get
Second term is proportional to angular momentum November 20, 2018
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Energy III November 20, 2018
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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 November 20, 2018
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Coupled BV - BECondensation
Super-Rotation November 20, 2018
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Coupled BV Monte-Carlo simulation results
Sub-Rotation November 20, 2018
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Coupled BV – disordered phase
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coupledBV – MC phase transitions
Mean Nearest Neighbor Parity November 20, 2018
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Coupled BV – transitions II
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Phase transitions III November 20, 2018
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Coupled BV – MC entropy Based on X. Ding’s algorithm for calculating degeneracy November 20, 2018
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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 November 20, 2018
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Transition to subrotating solid-body moderate 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 November 20, 2018
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Exact soln November 20, 2018
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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, 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
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