Exact model of a nongeostrophic baroclinic instability

A nongeostrophic version of the classical problem of zonal flow instability with constant shear (the Eady problem) is considered. The linearized set of dynamic equations for two-dimensional disturbances is reduced to a single wave-type second-order equation relative to modified pressure (a linear combination of pressure and stream function). Dynamic features of disturbances with zero potential vorticity are studied in the framework of the equations formulated. Asymptotic solutions of the spectral problem of hydrodynamic stability theory are derived. The initial-value problem at large Richardson numbers is considered using multiple-time-scale expansions. The solution to the problem is represented as the sum of fast (wave) and slow (quasi-geostrophic) components. In the unstable regime, the slow component describes baroclinic waves (cyclones and anticyclones) generated by inhomogeneous initial buoyancy (potential temperature) distributions at the boundaries.