Wave energy-momentum and pseudoenergy-momentum conservation for the layered quasi-geostrophic instability problem

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Abstract
Evolution equations and conservation laws are derived for a quite general layered quasi-geostrophic model: with arbitrary thickness and stratification structure and with either a free or a rigid (including the possibility of topography) boundary condition, at the top and bottom. The system is shown to be Hamiltonian, and Arnol'd stability conditions are derived, in the sense of both the first and second theorem, i.e. for pseudowestward and pseudoeastward basic flows, respectively, and for arbitrary perturbations of potential vorticity and Kelvin circulations.Two examples of parallel basic flow in a channel are analysed: the sine profile in the so-called equivalent barotropic model (one layer with a free boundary) and Phillips’ problem (uniform flow in each of two layers with rigid boundaries). Using the second theorem with the optimum combination of pseudoenergy and pseudomomentum it is shown that, in both cases, the basic state is nonlinearly stable if the channel width L is small enough, namely, ΛL < π and , respectively. (In the first problem, Λ is the wavenumber of the sine profile; in the second one, g′ is the reduced gravity, H1 and H2 are the layer thicknesses, and f0 is the Coriolis parameter). The stability condition of either problem is found to be also a necessary one: as soon as it is violated a grave mode becomes unstable. It is shown explicitly that the second variation of the pseudoenergy and pseudomomentum of a growing (decaying) normal mode is identically zero, defining the direction of the unstable (stable) manifold.