Forced, Dissipative Generalizations of Finite-Amplitude Wave-Activity Conservation Relations for Zonal and Nonzonal Basic Flows

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Abstract
The effects of forcing and dissipation are incorporated into finite amplitude, local wave-activity relations for disturbances to zonal and nonzonal flows. 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 he 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.

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