A theory for the forcing and dissipation in stochastic turbulence models

Recent studies reveal that randomly forced linear models can produce realis tic statistics for inhomogeneous turbulence. The random forcing and linear dissipation in these models parameterize the effect of nonlinear interactio ns. Due to lack of a reasonable theory to do otherwise, many studies assume that the random forcing is homogeneous. In this paper, the homogeneous ass umption is shown to fail in systems with sufficiently localized jets. An al ternative theory is proposed whereby the rate of variance production by the random forcing and dissipation are assumed to be proportional to the varia nce of the response at every point in space. In this way, the stochastic fo rcing produces a response that drives itself. Different theories can be for mulated according to different metrics for measuring "variance.'' This pape r gives a methodology for obtaining the solution to such theories and the c onditions that guarantee that the solution is unique. An explicit hypothesi s for large-scale, rotating flows is put forward based on local potential e nstrophy as a measure of eddy variance. This theory, together with conserva tion of energy, determines all the parameters of the stochastic model, exce pt one, namely, the multiplicative constant specifying the overall magnitud e of the eddies. Comparison of this and more general theories to both nonli near simulations and to assimilated datasets are found to be encouraging.