The Inviscid 3D Quasi-Geostrophic System on Bounded Domains

TL;DR

This paper derives a new inviscid 3D quasi-geostrophic model on bounded domains, introducing novel boundary conditions and proving the existence of global weak solutions through variational methods.

Contribution

It provides a formal derivation of the QG system with new boundary conditions and establishes the existence of solutions using elliptic regularity theory.

Findings

New boundary conditions for the inviscid 3D QG system.

Existence of global weak solutions proved.

Elliptic regularity results for the variational problem.

Abstract

We present a formal derivation of the inviscid 3D quasi-geostrophic system (QG) from primitive equations on a bounded, cylindrical domain. A key point in the derivation is the treatment of the lateral boundary and the resulting boundary conditions it imposes on solutions. To our knowledge, these boundary conditions are new and differentiate our model from closely related models which have been the object of recent study. These boundary conditions are natural for a variational problem in a particular Hilbert space. We construct solutions and prove an elliptic regularity theorem corresponding to the variational problem, allowing us to show the existence of global weak solutions to (QG).