Weakly Inhomogeneous Turbulence Theory with Applications to Geophysical Flows

In this study, we investigate the problems of inhomogeneous turbulence. The main goal has been to find the connections between analytic turbulence theories, notably the formalism of the direct interaction approximation (DIA), and turbulence phenomenology. To simplify the results from an inhomogeneous DIA model, we have employed a diagonalization procedure and an assumption of weak inhomogeneity. The diagonalization procedure expands the off-diagonal correlations in terms of the diagonal correlations and the mean shear, and the weakly inhomogeneous assumption assumes a scale separation between the eddies and mean field. As a consequence, the DIA diagonalization process becomes an asymptotic expansion. It also leads to the recovery of some inhomogeneous quantities in the configuration space. Therefore useful information about spatial transfer is extracted from what is basically a spectral transfer model. We consider a simple two-dimensional shear flow and a passive tracer flow as practical examples of applications of the theory. In both cases, we find expressions almost identical to those of classical eddy diffusion equation. The eddy diffusivities in our results are positive definite and can be obtained without recourse to empirical constants. We use an abridged form of the DIA to calculate eddy diffusivities for large-scale atmospheric motions in which the energy spectrum is prescribed. Under the assumption that the inhomogenous eddy diffusivities represent the total eddy diffusivities, these calculated coefficients agree well with observed values. A directly simulated two-dimensional, barotropic turbulence model is designed to verify various assumptions made in weakly inhomogeneous theory. The experiment is especially concerns in the calculation of the averaged Green's function defined in the DIA. From this experiment the theoretical prediction of the fluctuation-dissipation theorem is tested against simulated results. The agreement is excellent. The 3 simulated energy spectrum and decorrelation rates are also compared with the theoretical results. The analytic turbulence theories hold convincingly for all our experiments.