Nearly symmetric and nearly baroclinic instabilities in the presence of diffusivity. Part 1. Growth rate patterns

Abstract

Computations are performed to examine the instabilities of baroclinically sheared Eady basic flows with respect to banded normal-mode perturbations in three-dimensional space in the presence of eddy diffusivity with two (free-slip and non-slip) types of boundary conditions. The non-dimensional model system contains four external parameters: the Richardson number, the Ekman number, the Prandtl number and the ratio between inertial and buoyancy frequencies. The solutions are controlled mainly by the first three parameters. Growth rate patterns are computed for unstable modes as functions of the horizontal wavelength, $l$, and tilt angle $\alpha $ of the band orientation with respect to the basic shear (measured negative clockwise from the basic-shear direction). It is found that the main growth rate pattern (for non-propagating modes with respect to the middle-level basic flow) has only one maximum unless the Ekman number is sufficiently small. The growth rate pattern obtained with the free-slip boundary conditions has a slightly larger global maximum and is more symmetric with respect to the symmetric axis in the ($l$, $\alpha$) space than that obtained with the non-slip boundary conditions. When the Richardson number is increased from 0.25 to 1.0, the maximum growth rate decreases and the associated instability changes gradually from a nearly symmetric type to a nearly baroclinic type as manifested by the continuous increase of $l$ (from mesoscale to synoptic scale) and continuous change of $\alpha $ (from nearly zero to nearly ${-}90^{\circ})$. When the Ekman number is sufficiently small, the main growth rate pattern can have two local maxima if the Richardson number is within a subrange $0.8 < {\hbox{\it Ri}} < 1.0$. One of the local maxima is near the symmetric axis and the other is near the baroclinic axis in the wavenumber space. When the Richardson number increases through a transitional value in the subrange, the global maximum growth rate decreases continuously but the maximum point jumps from one local maximum to the other and the associated instability switches from a nearly symmetric type to a nearly Eady baroclinic type. The subrange depends on the smallness of the Ekman number and it diminishes as the Ekman number increases to 0.0025 (for the non-slip case). The computed growth rates and ($l$, $\alpha $) are compared with the nearly inviscid results of Miller & Antar and the inviscid results of Stone.