Two-dimensional cascades and mixing: a physical space approach

Abstract

We develop an analysis of the two-dimensional cascade of a tracer (passive or active Lagrangian-conserved scalar), locally in space and time, and establish connections with the modelling of turbulent mixing. We define a local scale-to-scale flux of tracer variance based on the dynamics of tracer increments. This flux reduces at small scales to the production or destruction rate of tracer gradients by stirring as a function of their local orientation with respect to the compressional axis of the strain-rate tensor. The local detailed budget of tracer variance on which this approach is based is compared to the global statistical budget expressed by Yaglom's equation. The spatial pattern of the local transfers produced by a numerical simulation as well as their statistical distribution are discussed.

We then address the problem of the parameterization of turbulent mixing. We consider an anisotropic tensor diffusivity proportional to the velocity gradients. In this model the tracer dissipation involves the axes of the strain-rate tensor and we shall refer to it as strain diffusivity. We show analytically that it locally matches the scale-to-scale flux through the cutoff scale. This matching is studied numerically in decaying two-dimensional turbulence. A comparison is made with eddy diffusivity and hyperdiffusivity. The presence of a numerical instability and ways to suppress it are discussed from numerical and fundamental points of view.

We consider the special case of vorticity, an active scalar in two dimensions. When applied to vorticity, models affect the energy budget. The two-dimensional inverse energy cascade requires that parameterizations conserve energy and we show that strain diffusivity conserves energy. We finally study the sensitivity of the large scales of the flow to the operator used on vorticity in forced stationary simulations. Strain diffusivity is found to produce more realistic spectral features than hyperviscosity.