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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> parameter representing the small aspect ratio of the physical domain. In our previous paper [<xref ref-type="bibr" rid="b14">14</xref>] 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 downwind-matching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.</p>

Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier–Stokes equations*

“…Later and taking a different approach by using maximal L q t -L p x -regularity methods and quadratic estimates, we proved in [19] norm convergence of the same order for a large set of p, q ∈ (1, ∞), see also [20] for the more difficult case of Dirichlet boundary conditions. Non-periodic domains are included also in [17]. The cases of the scaled Boussinesq equations is considered in [47,48], the case of the compressible primitive equations is discussed in [21] and the inviscid case in [52].…”

Energy equality for the compressible Primitive Equations with vacuum

“…Further, Li and Titi [40] proved convergence of the weak solutions of incompressible Navier-Stokes equations to the strong solutions of PE. Based on [2,40], Donatelli and Juhasz [20] give a justification that PE model with the pollution effect is the hydrostatic limit of the Navier-Stokes equations with an advection-diffusion equation. Grenier [32] used the energy estimates and Brenier [7] used the relative entropy inequality to prove that the smooth solutions of incompressible Euler system converge to smooth solutions of inviscid PE.…”