We present and analyze a simplified, scale-separated, anelastic fluid model which is designed to assess the influence of weak compressibility in the diffusive transport of a passive scalar. Our model incorporates a slowly varying, density induced, anisotropy into a fixed, rapidly varying (small scale) fluid flow. This anisotropy is physically motivated through the anelastic mass balance which retains vertical density variations occurring over the atmospheric scale height (∼8 km). Consequently, these steady flows are nonsolenoidal over large scales and approximately incompressible on small scales. We apply homogenization methods to calculate the large scale, effective mixing experienced by a passive scalar diffusing in the presence of this small scale flow. Over large scales, the evolution of the scalar field is governed by an effective, variable coefficient, anelastic mixing equation. The variable coefficients entering this equation are shown to depend nontrivially upon both the large scale anisotropy as well as the structure of the small scale fluid flow. We establish that anelastic effects produce anisotropic mixing properties not shared by the analogous incompressible closure. Specifically, the mixing equation possesses exact nontrivial bounded steady states, whereas the incompressible regime has only constant (i.e., spatially homogeneous) steady states. Furthermore, the anelastic mixing equation is shown numerically to possess local regions of trapped (Fig. 3) and focused (Figs. 6 and 7) contaminants, behavior not possible in the analogous incompressible model. These results imply that anelastic effects, which occur naturally in the atmosphere, provide mechanisms which locally reduce mixing and generate inhomogeneities in large scale concentration fields.