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

Donatella Donatelli; Nóra Juhász

doi:10.3934/dcds.2022002

Article Contents

2022, Volume 42, Issue 6: 2859-2892. Doi: 10.3934/dcds.2022002

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$ \mathbb{R}^{N} $

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

Donatella Donatelli^1, , and
Nóra Juhász^2,

1.
Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, 67100 L'Aquila, Italy
2.
Bolyai Institute, University of Szeged, H-6720 Szeged, Hungary

^*Corresponding author: Donatella Donatelli
^*Corresponding author: Donatella Donatelli

Received: May 31, 2020

Revised: June 30, 2021

Published: June 2022

The research of DD and NJ leading to these results has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 642768 (Project Name: ModCompShock)

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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 $ \epsilon $ parameter representing the small aspect ratio of the physical domain. In our previous paper [14] 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.

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Mathematics Subject Classification: Primary: 35Q86, 35Q30; Secondary: 35D30, 86A10.

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Figure 1. The boundary structure of $ \Omega $ for the case of weak solutions

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