t t t t Venus as an Equilibrium System Fluid Movement We are interested in stationary states of fluid flow on the sphere. Atmospheres of planets are dynamic chaotic systems, yet there are interesting examples of time invariant behavior (e.g. Jupiter’s Red Dot and the Super Rotational Venusian atmosphere). Following recent approaches in atmospheric mechanics we consider a model with canonically constrained angular momentum and bounded enstrophy. To enact this work, it is essential that we move from the beta and f plane constructions to a spherical geometry. In addition to the more realistic geometry for large scale phenomena, it also enables the detection of global behavior. Stationary flow is described by the stream function. Equilibrium Thermodynamics tells us that the probability weight of any point in state space (any stream function) is given as Acknowledgments: This work was funded by grants from the ARO and DOE The atmosphere of Venusian atmosphere circulates in a super rotational movement, with winds at the equator traveling at a rate of 300 m/s. The convection prevents significant variation in the temperature. The immense density and angular momentum of the atmosphere creates a system which interacts significantly with the planet surface. The point wise vorticity is approximated by a family of ensembles on the sphere. Each ensemble consists of a lattice X on the sphere where each point is a positive or negative vortex. The state space of point vortices is equivalent to the ensemble of stream functions. In a one step renormalization, we recover the continuous vorticity on the sphere. Ensembles are evaluated by Gibbs free energy to find emergent order. Phase Space The description by stream function is equivalent to a description by point wise vorticity. Which allows us to exchange the ensemble of stream functions for that of approximating configurations of vortices on the sphere at N points x on a sphere. The transformation of energy for any point in state space Remarkably we now have a lattice problem with global interactions and an extra term, due to rotation of the planet. This is similar to solid state models of interacting species with a spatially inhomogeneous external field. The inhomogeneous term makes this problem difficult to approach analytically. Therefore we employ an approximation / renormalization technique developed by Bragg to examine properties of solid state materials. Rajinder Singh Mavi, Chjan C. Lim Phase Transitions of barotropic flow on the sphere by the Bragg method The preferred vorticity is the solution of the fixed point equation. Subject to restricting net vorticity to zero. Solutions in different thermodynamic regimes are pictured below. A two domain renormalization of the sphere into northern and southern hemispheres allows us to find conditions leading to phase transitions. We find solutions to the fixed point equation. Below, on the left we have the fixed point in positive temperature, and on the right fixed points for increasing negative beta.