2 - Scale analysis of the governing equations

Summary

Equations 1.9a, 1.9b and 1.9c (more commonly presented without rotational effects) are known as the Navier–Stokes equations, and are a notoriously difficult mathematical problem. Demonstrating that they have a smooth, physically reasonable solution is one of the Clay Mathematics Institute's Millennium Problems. There are two major difficulties with these equations. In a strange sense, the equations are too complete! They contain as solutions, and cannot discriminate between, the multiplicity of diverse processes that make up the wide spectrum of atmospheric phenomena. Furthermore, there are strong non-linearities embodied in the advective parts (terms of the form u∂u/∂x) of the material derivative. This non-linearity results in broad-spectrum or multi-scale solutions, and underlies the existence of chaotic behaviour. Our task is to find a way of simplifying the equations through identification of approximations that may render the equations at least partially tractable. As will be seen, the approximate forms will be applicable only to a limited range of scales, which conveniently helps limit the set of phenomena that must be dealt with. In effect the approximations act as a band-pass filter, thereby narrowing our field of view. As will be demonstrated, some of the approximations can be identified with the three scales illustrated in Figure 1.1. While there exist no usefully applicable solutions to the full equations, it is nonetheless possible to make enormous strides in understanding atmospheric phenomena by a careful consideration of scales of phenomena involved, and an analysis of approximate forms of the full equations. In a very real sense, this is an admission that while we do have a complete, unified theory of atmospheric motion, it is not a very helpful theory because of the intractability of the full equations. Fortunately, the equations are enormously helpful, if we consider phenomena according to their scale.

Order of magnitude analysis

The first approach is to investigate the possibility that some of the terms (hopefully the troublesome ones) in Equations 1.9a, 1.9b and 1.9c can be neglected because of their size relative to the remaining terms. This procedure is conducted by an order of magnitude analysis.