Symmetric and asymmetric inertial instability of zonal jets on the -plane with complete Coriolis force

Abstract

Symmetric and asymmetric inertial instability of the westerly mid-latitude barotropic Bickley jet is analysed without the traditional approximation which neglects the vertical component of the Coriolis force, as well as the contribution of the vertical velocity to the latter. A detailed linear stability analysis of the jet at large Rossby numbers on the non-traditional $f$-plane is performed for long waves in both the two-layer rotating shallow-water and continuously stratified Boussinesq models. The dependence of the instability on both the Rossby and Burger numbers of the jet is investigated and compared to the traditional case. It is shown that non-traditional effects significantly increase the growth rate of the instability at small enough Burger numbers (weak stratifications) for realistic aspect ratios of the jet. The main results are as follows. (i) Two-layer shallow-water model. In the parameter regimes where the jet is inertially stable on the traditional $f$-plane, the symmetric inertial instability with respect to perturbations with zero along-jet wavenumber arises on the non-traditional $f$-plane. Both non-traditional symmetric and asymmetric (small but non-zero wavenumbers) inertial instabilities have higher growth rates than their traditional counterparts. (ii) Continuously stratified model. It is shown that by a proper change of variables the linear stability problem for the barotropic jet, on the non-traditional $f$-plane, can be rendered separable and analysed along the same lines as in the traditional approximation. Neutral, weak and strong background stratifications are considered. For the neutral stratification the jet is inertially unstable if the traditional approximation is relaxed, while its traditional counterpart is not. For a sufficiently weak stratification, both symmetric and asymmetric inertial instabilities have substantially higher growth rates than in the traditional approximation. The across-jet structure of non-traditional unstable modes is strikingly different, as compared to those under the traditional approximation. No discernible differences between the two approximations are observed for strong enough stratifications. The influence of dissipation and non-hydrostatic effects upon the instability is quantified.