Maximum entropy theory and the rapid relaxation of three-dimensional quasi-geostrophic turbulence.

Turbulent flow in a rapidly rotating stably stratified fluid (quasi-geostrophic turbulence) commonly decays toward a stable pattern of large-scale jets or vortices. A formula for the most probable three-dimensional end state, the maximum entropy state (MES), is derived using a form of Lynden-Bell statistical mechanics. The MES is determined by a set of integral invariants, including energy, as opposed to a complete description of the initial condition. A computed MES qualitatively resembles the quasistationary end state of a numerical simulation that is initialized with red noise, and relaxes for a time on the order of 100 (initial) eddy turnovers. However, the potential enstrophy of the end state, obtained from a coarsened potential vorticity distribution, exceeds that of the MES by nearly a factor of 2. The estimated errors for both theory and simulation do not account for the discrepancy. This suggests that the MES, if ever realized, requires a much longer time scale to fully develop.