Introduction

When the velocities of particles change slowly with time the geostrophic approximation to the horizontal wind becomes more accurate. While at first sight this seems a good thing, because it makes the equations simpler, but it is also worrying, for the momentum equations lose some of their ability to be predictive. Thus the term in Dυ/Dt is a small residual between two large terms. If horizontal gradients of pressure are largely balanced by the Coriolis force, how are we to find what is left over to make the momentum evolve? This dilemma can be tackled by eliminating the pressure term from the momentum equations. This gives, of course, the vertical component of the vorticity equation.

Scale analysis

We use the equations of motion to establish a set of order-of-magnitude relations between variables. This process is analogous to solving the equations, except that it has the lesser aim of seeing that the numbers involved could possibly represent a solution. While we aim to be deductive, we find that it is easier to justify some approximations only after they have been made. Thus our end point is really one of plausible consistency.

Many weather systems have a much longer transverse than longitudinal scale. This is consistent with the notion that they are there in order to transfer properties like heat in the transverse direction.