Motivated in part by the problem of large-scale lateral turbulent heat transport in the
Earth's atmosphere and oceans, and in part by the problem of turbulent transport
itself, we seek to better understand the transport of a passive tracer advected by
various types of fully developed two-dimensional turbulence. The types of turbulence
considered correspond to various relationships between the streamfunction and the
advected field. Each type of turbulence considered possesses two quadratic invariants
and each can develop an inverse cascade. These cascades can be modified or halted, for
example, by friction, a background vorticity gradient or a mean temperature gradient.
We focus on three physically realizable cases: classical two-dimensional turbulence,
surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence
at scales large compared to the radius of deformation. In each model we assume that
tracer variance is maintained by a large-scale mean tracer gradient while turbulent
energy is produced at small scales via random forcing, and dissipated by linear
drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting
from the inverse cascades, and use the expected velocity and length scales to predict
integrated tracer fluxes.
When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When β is
significant, we find a clear distinction between the tracer mixing scale, which depends
on β but is nearly independent of drag, and the energy-containing (or jet) scale, set
by a combination of the drag coefficient and β. Our predictions are tested via high-
resolution spectral simulations. We find in all cases that the passive scalar is diffused
down-gradient with a diffusion coefficient that is well-predicted from estimates of
mixing length and velocity scale obtained from turbulence phenomenology.