Diagonalizing Approximation for Inhomogeneous Turbulence

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Abstract
An approximation for shear and Boussinesq turbulence is introduced which can greatly lessen the computational task associated with spatial inhomogeneity. It yields statistical equations for inhomogeneous flow which are of simplicity comparable to the direct‐interaction equations for homogeneous flow. The new approximation presupposes a representation of the turbulent fields by expansion in some appropriate set of orthonormal functions which obey the boundary conditions. The covariances which are off‐diagonal in the chosen representation, as well as the triple correlations, are approximated in terms of diagonal covariances and diagonal average response functions by a procedure resembling the direct‐interaction scheme. The final equations involve only the mean‐field amplitudes and the diagonal covariance and response functions. The approximation is illustrated by applying it to infinite‐Prandtl‐number Boussinesq convection between infinite horizontal planes. The range of applicability and the limitations of the approximation are discussed.

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