Initially, this paper reviews the mathematical issues surrounding hydrostatic primitive equations (HPEs) and non-hydrostatic primitive equations (NPEs) that have been used extensively in numerical weather prediction and climate modelling. A new impetus has been provided by a recent proof of the existence and uniqueness of solutions of viscous HPEs on a cylinder with Neumann-like boundary conditions on the top and bottom. In contrast, the regularity of solutions of NPEs 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 HPEs, the creation of small-scale structures is allowed locally in space and time at sizes that scale inversely with the Reynolds number.