Earth Cubed

Distributed Climate Science and Computing

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.

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August 15, 2009 - Posted by | Navier Stokes

2 Comments »

  1. Since this thread seems somewhat Related to coordinate systems I’m going to post the following two links here for now:

    http://en.wikipedia.org/wiki/Coriolis_effect
    http://en.wikipedia.org/wiki/Fictitious_force

    The second link contains a derivation of the Coriolis force.

    Comment by s243a | August 24, 2009 | Reply

  2. […] All the new coordinates are dimensionless and the scaling quantities above transform the vector operators in to the associated operators for the new dimensionless coordinate system (see nondimentionalization of Navier Stokes). […]

    Pingback by Vector Operations in Hoskins and Simmons Coordinates « Earth Cubed | August 29, 2009 | Reply


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