  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.
