Nondimensionalization of the Navier-Stokes Equation

Most GCMs (global Circulation models) use, Nondimensionalization of the Navier-Stokes Equation. The most basic nondimenionalized Navier Stockes Eqauations can be expressed where the constants are in terms of dimensionless numbers such as:

Strouhal Number, Euler Number, Froude Number, Reynolds Number

http://www.mne.psu.edu/cimbala/me33web_Fall_2005/Lectures/Nondimensionalization_of_NS_equation.pdf

In a simplified climate model someone introduced to me the following dimensionless coordinates are used:

mu=sin(theta)
sigma=pressure/P*
lambda=longitude
theta=lattitude

http://www.mi.uni-hamburg.de/216.0.html?&L=1

This coordinate system is also used in:
A multi-layer spectral model and the semi-implicit method, By B. J. HOSKINS and A. J. SIMMONS

The prognostic equations, defined by these coordinates can be derived by doing a coordinate transform on the vortex equation

http://en.wikipedia.org/wiki/Vorticity_equation

http://en.wikipedia.org/wiki/Barotropic_vorticity_equation

I’ll try to post a derivation later.