On the Influence of Boundary Conditions on the Instability of Geostrophic Currents

An analysis of the influence of boundary conditions on the instability of a geostrophic zonal current of finite transverse scale with a vertical parabolic velocity profile of a general form in a vertically limited layer has been carried out. The model is based on the potential vortex equation in the quasi-geostrophic approximation, taking into account the vertical diffusion of mass and momentum. The equation and boundary conditions are reduced to a spectral eigenvalue problem of the Orr–Sommerfeld type. A high-precision analytical–numerical method is used to calculate eigenfunctions and eigenvalues. Two types of conditions at the horizontal boundaries of the layer are considered: the equality of vertical velocity disturbances and buoyancy fluxes to zero (Problem I) and the equality of vertical velocity disturbances and horizontal velocity disturbances to zero (Problem II). It is found that the boundary conditions of Problem II, which include no-slip conditions, contribute to the stabilization of longwave unstable disturbances and narrow the range of unstable shortwave disturbances. It is noted, however, that all types of current instability obtained by solving Problem I, such as baroclinic instability, instability of the critical layer, and new instability (characterized by a phase velocity exceeding the maximum current velocity), also arise when using no-slip boundary conditions, but in a narrower range of changes in the physical parameters of the original equation.