The double cascade as a necessary machanism for the instability of steady equivalent-barotropic flows

A large class of steady solutions of the two-dimensional Euler equation and of the equation of conservation of potential vorticity for equivalent-barotropic flow, is presented; instabilities can grow only if their energy concentrates at length scales larger than of the steady state. This possible growth of energy at large scales is accompanied by a corresponding growth of enstrophy at small scales. Such a distribution of energy and enstrophy, which is well known in the different context of two-dimensional turbulence, is therefore found to constitute a necessary mechanism for instability