Theory of Two-Dimensional Anisotropic Turbulence

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Abstract
Utilizing an abridged version of the test field model, we examine the relaxation of two-dimensional homogeneous turbulence back to its isotropic state. Our procedure is to represent the departure from isotropy in terms of an angular Fourier series and to derive equations governing the temporal relaxation of higher angular harmonics from the test field model. The resulting equations for the anisotropic part of the Reynolds stress tensor are linearized, and examined in some detail both analytically, and for a simple atmospheric spectrum with an enstrophy inertial-range, numerically. It is found that the relaxation back to isotropy is very non-local in wavenumber space, a result seemingly in counter-distinction to three-dimensional turbulence for which the relaxation is supposedly local. The difference is explained by the importance in two-dimensional flows of direct straining of small scales by large scales. Some preliminary direct spectral numerical simulation data in support of these ideas are also presented. Utilizing the linearized version of the theory, we give an estimate of the relaxation rate of the anisotropic part of the total Reynolds stress, similar to that originally given by Rotta for three-dimensional turbulence If the anisotropy is centered in the energy-containing range, we obtain a value for the rate coefficient of ∼0.25(E½/L), where E is the total kinetic energy, and L the turbulence integral scale. The implications of these findings for subgrid-scale parameterization are discussed, and a formalism for describing the evolution of the large scales with parameterized treatment of the small scale is sketched. Two new effects beyond those customarily represented in three-dimensional turbulence theory appear to require attention: a production of subgrid-scale turbulence energy which depends on a certain measure of the excess of (large scale) strain rate over the (large scale) vorticity, and a production of subgrid-scale anisotropy by means of the direct straining by the large scales. Formulas estimating these effects are presented.

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