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

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.