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.