Long-wave instabilities in the SQG model with two boundaries

Surface quasi-geostrophic (SQG) flows with a much larger horizontal scale than the Rossby radius of deformation are considered. A new version of the SQG model with two boundaries, which is reduced to a nonlinear system of partial differential equations, is proposed to describe the dynamics of such flows. This system describes the interaction between the barotropic and baroclinic components of the stream function and generalises the two-dimensional Euler equations for flows with a vertical velocity shear. The laws of conservation of both total and surface potential energies, which follow from this system, have been formulated. The solutions of a number of problems in the theory of baroclinic instability, which are in agreement with already known solutions, have been obtained within the framework of this system. It is shown that vertical shear flows are absolutely unstable, i.e. their instability is independent of the horizontal velocity profile structure. A generalised system of the SQG model equations, which additionally takes into account the beta-effect and the Ekman bottom friction, has also been proposed. The transformation of jet flows due to the bottom friction and the influence of the beta-effect on the stability of shear flows have been studied based on this system.