General stability conditions for zonal flows in a one-layer model on the β-plane or the sphere

Abstract

Sufficient stability conditions are derived for a zonal flow on the β-plane or the sphere. Two conditions guarantee both shear stability (to perturbations with vanishing zonal average) and inertial stability (to longitude-independent perturbations). These conditions are not restricted to normal-mode disturbances, and are derived without making use of the quasi-geostrophic approximation. The main limitation of the model is to have only one layer.On the β-plane, the conditions are: (i) that the product of the meridional gradient of potential vorticity and the difference between an arbitrary constant and the zonal velocity be everywhere non-negative; and (ii) that the absolute value of this difference be nowhere larger than the local phase speed of long gravity waves. Inertial stability is independently assured if the Cariolis parameter and the potential vorticity are everywhere of the same sign (this well-known condition can be easily violated near the equator, but the flow may nonetheless be stable).If the meridional gradient of potential vorticity has everywhere the same sign, then conditions (i) and (ii) can be shown to be consequences of the conservation of a total pseudo-energy E0 and pseudomomentum P0, defined so that their lowest-order contribution is quadratic in the deviation from the fundamental state (even in the case that the perturbation is longitude-independent). Thus, if there exists a value of α such that the integral of E0 − αP0 is positive-definite, then the flow is stable. In this case, the stability conditions are valid for small, rather than infinitesimal, perturbations.The parameters of stable flows, as guaranteed by these conditions, are investigated for the family of Gaussian jets centred at the equator; both the cases of an unbounded ocean and a semi-infinite ocean, poleward from a zonal wall, are considered. Easterlies with the width of a Kelvin wave and westerlies with that width or wider may be unstable, even though the gradient of potential vorticity is positive for any strength of the jet.