On the Types of Instability of a Geostrophic Current with a Vertical Parabolic Profile of Velocity

An analysis is made of unstable perturbations of a geostrophic current of finite transverse scale with a parabolic vertical velocity profile of a general form (with linear and constant velocity shear) in vertically limited layer. The model is based on the potential vortex equation in the quasi-geostrophic approximation, taking into account the vertical diffusion of momentum and mass. The equation and boundary conditions were reduced to a spectral eigenvalue problem of the Orr-Sommerfeld type. To search for eigenfunctions and eigenvalues, a high-precision analytic-numerical method was used. Particular attention was paid to the study of unstable perturbations with a phase velocity exceeding the maximum flow velocity. Such instability should be distinguished from baroclinic instability and critical layer instability. It is found that the indicated instability can develop in ocean currents when the problem parameters vary in a wide range of values. It is obtained also that with an increase in the Prandtl number, the phase velocity of such disturbances increases and can significantly exceed the maximum flow velocity. However, the occurrence of such unstable perturbations is possible only in the cases when the maximum flow velocity is located in the inner region of the layer (but not necessarily in its center). It has also been found that narrow currents (the transverse scale is equal to or smaller than the Rossby radius) with a parabolic vertical profile can be unstable. The most unstable perturbations have approximately equal scales along and across the flow, that is, they are circular perturbations. A discussion of various types of geostrophic current instability with a parabolic vertical velocity profile as applied to the ocean is presented.