The equations we use are based on the *Swift-Hohenberg* equation
__[1]__
that was introduced in 1974, and has been much studied since then. It is an
equation for a single real field that is a function of two space variables
(representing the horizontal coordinates in a convection system) and time.
The field can be thought of as the temperature field at the
midplane.

The **Swift-Hohenberg** eqaution is:

A

For domain coarsening we are particularly interested in the differences between systems far from equilibrium, and sytems approaching a thermodynamic equilibrium.

The __Swift-Hohenberg__ fails to
model the non-equilibrium nature of the system, because it is "potential". (This means
that the dynamics always decreases a particular functional integral of the field - called
the potential. From
this it can be argued that the dynamics will eventually cease.) A slight modification
yields an equation that is non-potential, so that the dynamics need not be purely
relaxational. The modification changes only the nonlinear piece, to give:

To model the

The extra terms include the effect of a slowly varying, horizontal velocity field U that advects the convection rolls (first equation). The velocity U is in turn driven by distortions in the rolls, as given through the vertical vorticity by the second and third equations:

More details can be found in our

For

The first term changes sign if the rotation sense is reversed. The second term allows us to tune the angle at which the patches of stripes grow in.

More details can be found in our
__paper [2]__.

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Last modified Thursday, September 26, 1996

Michael Cross