A theory for stochastic models of large-scale turbulence

The idea of parameterizing nonlinear interactions in a turbulent flow by a random forcing and dissipation, while retaining linear dynamics of eddies evolving on a shear-flow, has now been explored extensively. These stochastic models will be reviewed and the basis of their success, due primarily to the non-normal nature of linear dynamical operators, will be explained. Nevertheless, due to lack of a reasonable theory to do otherwise, many studies assume that the forcing is statistically isotropic. In this talk, an alternative theory is proposed whereby the variance of the forcing and dissipation, introduced to parameterize nonlinear interactions, are assumed to be proportional to the eddy variance at every point in space. In this way the stochastic forcing produces a response that drives itself. Remarkably, this theory, together with conservation of energy, constrains all of the parameters of the stochastic model except one, namely, the multiplicative factor specifying the overall magnitude of the eddies. The predictions of this theory are compared with the results of numerical simulations and climate analyses.