Closure theory for the effective diffusivity tensor in forced beta-plane turbulence

Theoretical and numerical results for the stirring of a tracer both along and across turbulent jets is presented. Such a system can be taken as a model for the stirring of true tracers along jets in either the atmosphere and ocean, or, less obviously, for the stirring of baroclinic potential vorticity by non-zonal flow in the ocean. The model flow considered is two-dimensional turbulence with a mean vorticity gradient and forced randomly and isotropically at small-scales. The tracer transported by the flow is forced by a mean tracer gradient that is arbitrarily oriented with respect to the mean vorticity gradient. Such a tracer can be decomposed into two independent tracers, one forced by a gradient that is parallel to the vorticity gradient (and so is stirred across jets), and another that is forced by a mean gradient that is perpendicular to the mean vorticity gradient (and so is stirred along jets). The effective diffusivity tensor for the full tracer can be computed from the eddy correlations of the two deomposed tracers. The across-jet diffusion is well-described by mixing length theory, while the diffusivity of the tracer stirred along the jets is the result of shear dispersion. At only moderate levels of anisotropy in the flow, the along-jet diffusivity is two orders of magnitude larger than the across-jet diffusivity. The skew flux of the tracer is of the same magnitude as, but opposed to, the mean flow.