  We study the dynamics of a system defined by the Navier-Stokes equations for
a non-compressible fluid with Marangoni boundary conditions in the two
dimensional case. We show that more complicated bifurcations can appear in this
system for a certain nonlinear temperature profile as compared to bifurcations
in the classical Rayleigh-B\'enard and B\'enard-Marangoni systems with simple
linear vertical temperature profiles. In terms of the B\'enard-Marangoni
convection, the obtained mathematical results lead to our understanding of
complex spatial patterns at a free liquid surface, which can be induced by a
complicated profile of temperature or a chemical concentration at that surface.
In addition, we discuss some possible applications of the results to turbulence
theory and climate science.
