Stability conditions and a p r i o r i estimates for equivalent-barotropic modons

Abstract

Linear Liapunov stability conditions are obtained for all classes of equivalent‐barotropic modons. The stability conditions are stated in terms of a parameter η called the ‘‘generalized disturbance wavenumber’’ related to the ratio of the initial values of disturbance enstrophy to energy. It is shown that c>0 modons (c is the drift speed, nondimensionalized by the long wave speed) are stable when η≤κ (κ is the modon wavenumber), and that c<−1 modons are stable when η≥κ. This dependency of the stability on the initial spectral structure of the disturbance is observed in numerical calculations. A priori L²‐type estimates bounding the growth of perturbations are derived. The instability mechanism is interpreted in terms of Fj≂rtoft’s energy cascade theorem [Tellus 5, 225 (1953)].