The theory of energy and enstrophy conserving stochastic models for shear flow turbulence is discussed. A basic point is that discretized models should be consistent with the idea of an upscale energy cascade and downscale enstrophy cascade. This idea implies that the terms in the discretized model associated with the nonlinear interactions should conserve (or possibly amplify) energy but dissipation enstrophy. Thus, the basic problem is to design a random forcing and systematic dissipation that also conserve energy but dissipation enstrophy. This problem is solved in a comprehensive manner by introducing the concept of "optimal forcing functions"-- structures that produce the optimal extraction of mean enstrophy (minus the dissipation by large-scale eddies) for fixed enstrophy injection by the random excitation. The resulting constrained model exhibits more realistic behavior than unconstrained stochastic models. For instance, in accordance with observations, the constrained stochastic model cannot sustain turbulence in sufficiently weak shears or in the presence of sufficiently strong dissipation. Other special properties of the optimal forcing functions suggest that they can be used to reduce the dimensionality of the turbulence parameterization problem.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics