A Theory for the Forcing and Dissipation in Stochastic Turbulence Models

Recent studies reveal that randomly forced linear models can produce realistic statistics for inhomogeneous turbulence. The random forcing and linear dissipation in these models parameterize the effect of nonlinear interactions. Due to lack of a reasonable theory to do otherwise, many studies assume that the random forcing is homogeneous. In this paper, the homogeneous assumption is shown to fail in systems with sufficiently localized jets. An alternative theory is proposed whereby the rate of variance production by the random forcing and dissipation are assumed to be proportional to the variance of the response at every point in space. In this way, the stochastic forcing produces a response that drives itself. Different theories can be formulated according to different metrics for measuring “variance.” This paper gives a methodology for obtaining the solution to such theories and the conditions that guarantee that the solution is unique. An explicit hypothesis for large-scale, rotating flows is put forward based on local potential enstrophy as a measure of eddy variance. This theory, together with conservation of energy, determines all the parameters of the stochastic model, except one, namely, the multiplicative constant specifying the overall magnitude of the eddies. Comparison of this and more general theories to both nonlinear simulations and to assimilated datasets are found to be encouraging.