We investigate the interaction of concentration fields of passive tracer with velocity fields characterizing geostrophic turbulence. We develop and compare results from equilibrium statistical mechanics, from turbulence-closure theory and from numerical simulation. A consistent account emerges. Among the results we show (1) that velocity fields efficiently scatter tracer variance to all scales, (2) that tracer variance evolves toward an equilibrium spectrum which is different from the equilibrium spectrum for vorticity variance, and (3) that intermittency of the tracer field is characteristic of a cascade of tracer variance across wavenumber space. The greater efficiency of the cascade of tracer variance relative to a vorticity cascade is due to wavenumber-local advective terms which affect tracer but not vorticity. We suggest that the more efficient tracer cascade results in shorter Lagrangian autocorrelation times for tracer than for vorticity.We investigate the spatial flux of tracer when a uniform gradient of average tracer concentration is imposed. We show (1) that the spatial flux has dominant contributions from fluctuations on scales slightly larger than the dominant energetic scales, (2) that an effective eddy-diffusivity formulation is valid and that the diffusivity agrees with simple mixing-length estimates, and (3) that eddy diffusivity is significantly anisotropic if Rossby-wave propagation occurs. Meridional diffusivity is suppressed relative to zonal diffusivity.We complement the study of stirring down from a uniform gradient with a numerical investigation of the stirring out of an initially concentrated spot. We see that eddy diffusivity can be a dangerous concept for such problems.
