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 were reduced to a spectral eigenvalue problem of the Orr–Sommerfeld type. A high-precision analytical-numerical method was used to calculate eigenfunctions and eigenvalues. Two types of conditions at the horizontal boundaries of the layer were considered: the equality of vertical velocity disturbances and buoyancy fluxes to zero (problem I); 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 long-wave unstable disturbances and narrow the range of unstable short-wave 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, as well as 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.