From rapid distortion theory to statistical closure theories of anisotropic turbulence

An overview of non-local theories and models is given, ranging from linear to nonlinear. The background principles are presented and illustrated mainly for incompressible, homogeneous, anisotropic turbulence. In that case, which includes effects of mean gradients and body forces and related structuring effects, the complete rapid distortion theory (RDT) solution is shown to be a building block for constructing a full nonlinear closure theory. Firstly, a general overview of the closure problem is presented, which accounts for both the nonlinear problem and the non-local problem. A classical spectral description is introduced for the fluctuating flow and its multi-point correlations. Applications to stably-stratified and rotating turbulence are discussed. In this particular context, homogeneous turbulence is revisited in the presence of dispersive waves, taking advantage of the close relationship between recent theories of weakly nonlinear interactions, or 'wave-turbulence', and classical two-point closure theories. Applications to weak turbulence in compressible flows are touched upon. Extensions of the frontiers of rapid distortion theory (RDT) and multi-point closures are discussed, especially developments leading towards inhomogeneous turbulence. Recent works related to zonal RDT and stability analyses for wavepacket disturbances to non-parallel rotational base flows are presented.