The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

Abstract

A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain -and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an ǫ parameter representing the small aspect ratio of the physical domain. In our previous paper [15] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwindmatching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.

On the hydrostatic approximation of compressible anisotropic Navier-Stokes equations -- rigorous justification

“…Then Giga, Hieber and Kashiwabara et al [26,27] extended the results into maximal regularity spaces. Recently, Donatella and Nora [17] proved the convergence in downwind-matching coordinates. For the stationary case, readers can refer to [3,8].…”