__Mean
flow corrections of secondary instabilities of Faraday waves__

** S. Ruediger**, J. M. Vega, J. Vinals,

Florida State University

Parametrically
driven surface waves (also known as
Faraday waves) can be excited on the free surface of a fluid layer that is
periodically vibrated in the direction normal to the surface. We built on
earlier research to study
Faraday waves well above onset. Specifically, we studied traveling wave equations including mean flow
effects, and used this augmented set of equations to calculate secondary
instabilities of basic periodic
patterns, and the transition to spatio-temporal chaos. We focus on a completely
phenomenological order parameter model of Faraday waves. The advantage of the model, as compared
to other approaches based on amplitude or phase equation, is
that it retains the original rotational invariance of the fluid layer
while remaining simple enough to
allow an analytic
determination of the stability of standing waves
to various perturbations. To
improve the previous treatment
we explicitly allow for mean flows, long wavelength modes that
can be excited non-linearly by surface vibration. We have determined the
secondary instabilities and identified the usual Eckhaus and TAM lines. We have
found that both lines are shifted and reduce the region of stable ideal
solutions with increasing strength of the mean flow. We also address
the transition to spatio-temporal chaos, and its relation to the
secondary instabilities will be
described.