Two-dimensional shear flow turbulence: A statistical theory of vortex filaments

A theory of two-dimensional free shear flow turbulence is proposed. In particular, the role of interactions of fluctuations with the mean flow is distinguished from that of interactions among fluctuations. The former are responsible for the extraction of free energy necessary for maintaining the turbulence, while the latter lead to the enstrophy cascade which generates small-scale eddies. On the length scales of greatest interest, turbulence is anisotropic because of the influence of mean flow shear. At smaller scales, self-similar enstrophy transfer occurs and yields the familiar power law spectrum. Free energy extraction is accompanied by mean flow relaxation. Two distinct mechanisms are involved. In addition to the usual quasilinear diffusion, there is a nonlinear drift which acts to enhance localized fluctuations and flatten the mean flow profile. The saturated state is characterized by collective resonances which must be nonlinearly damped in order to balance their generation by seed-eddy induction. This balance gives rise to a condition for the determination of the nonlinear damping rate, and thus the turbulence level. The spatial structure of the saturated fluctuations is also predicted