Forced, dissipative generalizations of finite-amplitude wave- activity conservation relations for zonal and nonzonal basic flows

Abstract

The method used is an extension of the momentum-Casimir and energy-Casimir methods that have been applied elsewhere to prove nonlinear stability theorems such as that of Arnol'd, and to generate finite amplitude wave-activity conservation relations for nondissipative flows. The wave activity density and flux, and the source or sink term associated with forcing and dissipation, are all second-order disturbance quantities which, for a large class of flows, may be evaluated in terms of Eulerian quantities. Explicit forms of the wave-activity relation are given for disturbances to zonally uniform and zonally varying basic states, for two-dimensional flow on a β-plane and for three-dimensional flow on a sphere described by the primitive equations in isentropic coordinates. -from Author