An Anelastic, Scale-Separated Model for Mixing, with Application to Atmospheric Transport Phenomena

We present and analyze a simpliied, scale-separated, anelastic uid model which is designed to assess the innuence of weak compressibility in the diiusive transport of a passive scalar. Our model incorporates a slowly varying, density induced, anisotropy into a xed, rapidly varying (small scale) uid ow. 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 ows are non-solenoidal over large scales and approximately incompressible on small scales. We apply homoge-nization methods to calculate the large scale, eeective mixing experienced by a passive scalar diiusing in the presence of this small scale ow. Over large scales, the evolution of the scalar eld is governed by an eeective, variable coeecient, anelastic mixing equation. The variable coeecients entering this equation are shown to depend non-trivially upon both the large scale anisotropy as well as the structure of the small scale uid ow. We establish that anelastic eeects produce anisotropic mixing properties not shared by the analogous incompressible closure. Speciically, the mixing equation possesses exact non-trivial 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 (Figure 3) and focused (Figures 1 6,7) contaminants, behavior not possible in the analogous incompressible model. These results imply that anelastic eeects, which occur naturally in the atmosphere, provide mechanisms which locally reduce mixing and generate inhomogeneities in large scale concentration elds.