Eddy transport of reacting substances

We examine an exact formulation of eddy fluxes\footnote{c.f. Kraichnan, R.H. (1987) Eddy Viscosity and Diffusivity: Exact Formulas and Approximations. {\sl Complex Systems}, {\bf 1}, 805-820.} but extended to tracers which react with each other. The resulting formula is evaluated using the lattice model approach,\footnote{Pierrehumbert, R.T. (2000) Lattice models of advection-diffusion. {\sl Chaos}, {\bf 10}, 61-74.} allowing not only control (including elimination) of sub-grid-scale diffusion and efficient enough computation to generate an adequate ensemble. The theory predicts that the flux is a non-local average of the mean gradients, even for passive scalars, and we can calculate the averaging kernel. The reaction terms alter the effective transport for a single scalar depending on decay time scale compared to that of the Lagrangian covariance. But, in addition, the eddies produce ``cross-fluxes'' whereby the transport of each tracer depends on the gradients of all of them.