Equilibration of a simple baroclinic flow in a beta channel and on the sphere

The statistical equilibration of baroclinic waves in a two-level quasigeostrophic model subject to forcing and dissipation is studied. The model employed may be formulated in either spherical or Cartesian geometry and is restricted to a midlatitude channel. Parameters are chosen so that only up to three waves can become supercritical (one planetary- and two synoptic-scare waves). It is found that both geometries exhibit essentially two equilibration regimes as the forcing temperature gradient varies. At low forcing, the planetary-scale wave is not excited while the two synoptic-scale waves equilibrate with steady finite amplitude. In this regime, the equilibrated temperature gradient is sensitive to forcing; the authors argue that this is due to the barotropic governor effect, At higher forcing, the planetary wave becomes active and the solution is aperiodic. In this regime, the planetary wave acts to reduce the barotropic shear spun up by the synaptic waves, thereby limiting the role of the barotropic governor; the equilibrated temperature gradient then becomes much less sensitive to forcing. The Cartesian and spherical cases differ both in the structure of the equilibrated state and in the strength of the barotropic governor (which is greater on the sphere). These differences are related to the geometric curvature terms and not to the meridional variation of beta.