## Abstract

A “most probable state” equilibrium statistical theory for random distributions of hetons in a closed basin is developed here in the context of two-layer quasigeostrophic models for the spreading phase of open-ocean convection. The theory depends only on bulk conserved quantities such as energy, circulation, and the range of values of potential vorticity in each layer. For a small Rossby deformation radius typical for open-ocean convection sites, the most probable states that arise from this theory strongly resemble the saturated baroclinic states of the spreading phase of convection, with a rim current and localized temperature anomaly. Furthermore, rigorous explicit nonlinear stability analysis guarantees the stability of these steady states for a suitable range of parameters. Both random heton distributions in a basin with quiescent flow as well as heton addition to an ambient barotropic flow in the basin are studied here. Also, systematic results are presented on the influence of the Rossby deformation radius compared to the basin scale on the structure of the predictions of the statistical theory.

*Corresponding author address:* Dr. Andrew J. Majda, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012-1185.

Email: jonjon@cims.nyu.edu