Extreme events in solutions of hydrostatic and non-hydrostatic climate models

TL;DR

This paper reviews mathematical issues in hydrostatic and non-hydrostatic climate models, focusing on solution regularity and the potential for spontaneous extreme events like intense fronts.

Contribution

It provides new insights into the existence, uniqueness, and regularity of solutions for hydrostatic equations and explores the conditions under which extreme events can spontaneously form.

Findings

Viscous HPE solutions can develop small-scale structures inversely related to Reynolds number.

Regularity of NPE solutions remains an open question.

Extreme events like intense fronts can arise spontaneously in viscous HPE solutions.

Abstract

Initially this paper reviews the mathematical issues surrounding the hydrostatic (HPE) and non-hydrostatic (NPE) primitive equations that have been used extensively in numerical weather prediction and climate modelling. Cao and Titi (2005, 2007) have provided a new impetus to this by proving existence and uniqueness of solutions of viscous HPE on a cylinder with Neumann-like boundary conditions on the top and bottom. In contrast, the regularity of solutions of NPE remains an open question. With this HPE regularity result in mind, the second issue examined in this paper is whether extreme events are allowed to arise spontaneously in their solutions. Such events could include, for example, the sudden appearance and disappearance of locally intense fronts that do not involve deep convection. Analytical methods are used to show that for viscous HPE, the creation of small-scale structures is allowed locally in space and time at sizes that scale inversely with the Reynolds number.