Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere

Superfluid high REynolds von Kármán experiment

The Superfluid High REynolds von Kármán experiment facility exploits the capacities of a high cooling power refrigerator (400 W at 1.8 K) for a large dimension von Kármán flow (inner diameter 0.78 m), which can work with gaseous or subcooled liquid (He-I or He-II) from room temperature down to 1.6 K. The flow is produced between two counter-rotating or co-rotating disks. The large size of the experiment allows exploration of ultra high Reynolds numbers based on Taylor microscale and rms velocity [S. B. Pope, Turbulent Flows (Cambridge University Press, 2000)] (Rλ > 10000) or resolution of the dissipative scale for lower Re. This article presents the design and first performance of this apparatus. Measurements carried out in the first runs of the facility address the global flow behavior: calorimetric measurement of the dissipation, torque and velocity measurements on the two turbines. Moreover first local measurements (micro-Pitot, hot wire,…) have been installed and are presented.

Nonlinear energy transfers and phase diagrams for geostrophically balanced rotating-stratified flows

Equilibrium statistical mechanics tools have been developed to obtain indications about the natural tendencies of nonlinear energy transfers in two-dimensional and quasi-two-dimensional flows like rotating and stratified flows in geostrophic balance. In this article we consider a simple model of such flows with a nontrivial vertical structure, namely, two-layer quasigeostrophic flows, which remain amenable to analytical study. We obtain the statistical equilibria of the system in the case of a linear vorticity-stream function relation, build the corresponding phase diagram, and discuss the most probable outcome of nonlinear energy transfers, both on the horizontal and on the vertical, in the presence of stratification and rotation.

Phase transition to super-rotating atmospheres in a simple planetary model for a nonrotating massive planet: exact solution

An energy-enstrophy model for the equilibrium statistical mechanics of barotropic flow on a massive nonrotating sphere is introduced and solved exactly for phase transitions to rotating solid-body atmospheres when the kinetic energy level is high. Unlike the Kraichnan theory which is a Gaussian model, we substitute a microcanonical enstrophy constraint for the usual canonical one, a step which is based on sound physical principles. This yields a spherical model with zero total circulation, microcanonical enstrophy constraint, and canonical constraint on energy, leaving angular momentum free as is required for any model whose objective is to predict super-rotation in planetary atmospheres. A closed-form solution of this spherical model, obtained by the Kac-Berlin method of steepest descent, provides critical temperatures and amplitudes of the symmetry-breaking rotating solid-body flows. The critical values depend linearly on the relative enstrophy, with proportionality constant derived from the spectrum of the Laplace-Beltrami operator on the sphere, as expected within an energy-enstrophy theory for macroscopic turbulent flows. This model and its results differ from previous solvable models for related phenomena in the sense that the model is not based on a mean-field assumption.