Strongly nonlinear topographic instability and phase transitions

The climatic states or equilibrium solutions produced when initial solid body rotation zonal currents flow over topography are obtained for an inviscid spherical barotropic model using the methods of statistical mechanics. A wide range of magnitudes of initial zonal currents and topographic heights are considered. Initial eastward flows are shown to exhibit nonlinear topographic instability even in the limit as the topography h vanishes; relatively large amplitude transient waves are generated which break the rotational symmetry of the initial solid body rotation currents. The limit h = 0 thus produces different behaviour from h = 0 for which angular momentum conservation and the Rayleigh stability criterion both indicate that the solid body rotation flows are stable. Westward solid body rotation flows in contrast are little disturbed by the topography and, generate only relatively small amplitude transients for h ≠ 0 and none as h↓0. Thus, h = 0 is a regular limit for westward initial flows and a singular limit for eastward initial flows. The transition point between eastward and westward flows is shown to have many of the properties of a statistical mechanical critical point. This is demonstrated by analyzing the equilibrium eddy energy, the change in the solid body rotation wind component, the partition function, the “canonical free energy” function and the energy spectra near the critical point.