  A number of approximations have been proposed to estimate basic hydrodynamic
quantities, in particular the frequency of a limit cycle. One of these, RZIF
(for Real Zero Imaginary Frequency), calls for linearizing the governing
equations about the mean flow and estimating the frequency as the imaginary
part of the leading eigenvalue. A further reduction, the SCM (for
Self-Consistent Model), approximates the mean flow as well, as resulting only
from the nonlinear interaction of the leading eigenmode with itself. Both RZIF
and SCM have proven dramatically successful for the archetypal case of the wake
of a circular cylinder.
  Here, the SCM is applied to thermosolutal convection, for which a
supercritical Hopf bifurcation gives rise to branches of standing waves and
traveling waves. The SCM is solved by means of a full Newton method coupling
the approximate mean flow and leading eigenmode. Although the RZIF property is
verified for the traveling waves, the SCM reproduces the nonlinear frequency
only very near the onset of the bifurcation and for another isolated parameter
value. Thus, the nonlinear interaction arising from the leading mode is
insufficient to reproduce the nonlinear mean field and frequency.
