Chjan Lim RPI, Troy, NY, US http://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 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

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

Venus Super-Rotation November 20, 2018

BECondensation to super-rotation November 20, 2018

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

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

Restframe energy For nondivergent barotropic fluid, the energy in the restframe November 20, 2018

Energy II Dropping the last – constant term – we get Second term is proportional to angular momentum November 20, 2018

Energy III November 20, 2018

Spin Lattice coupled BV Models November 20, 2018

Coupled BV - constraints November 20, 2018

Coupled BV – partition function = 0 November 20, 2018

Coupled BV - BECondensation Super-Rotation November 20, 2018

Coupled BV Monte-Carlo simulation results Sub-Rotation November 20, 2018

Coupled BV – disordered phase November 20, 2018

coupledBV – MC phase transitions Mean Nearest Neighbor Parity November 20, 2018

Coupled BV – transitions II November 20, 2018

Phase transitions III November 20, 2018

Coupled BV – MC entropy Based on X. Ding’s algorithm for calculating degeneracy November 20, 2018

Coupled BV – MC free energy November 20, 2018

Coupled SW – rotating solid sphere November 20, 2018

Coupled SW spin lattice models November 20, 2018

Coupled SW – physical quantities November 20, 2018

Coupled SW - constraints November 20, 2018

Coupled SW – partition function November 20, 2018

Jupiter November 20, 2018

Transition to subrotating solid-body moderate spin November 20, 2018

Transition pic2 moderate spin November 20, 2018

Anticyclonic asymmetry I November 20, 2018

Anticyclonic asymmetry II November 20, 2018

Anticyclonic dominance III November 20, 2018

Signs of bands - large spins November 20, 2018

Red Spot like - very small spin; in southern hemisphere November 20, 2018

More on Red Spot like II with slightly different constraints; south November 20, 2018

Red spot like III - small spin large potential enstrophy; south November 20, 2018

Red spot like IV - negative T; south November 20, 2018

III Exact solutions – spherical models November 20, 2018

Spherical model - BEC continued November 20, 2018

Exact spherical model soln November 20, 2018

Exact soln continued November 20, 2018

Exact soln November 20, 2018

Exact soln to Physical content November 20, 2018

Theorems with Physical content II November 20, 2018

Spherical models for coupled SW flows November 20, 2018

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. November 20, 2018