  The original Oberbeck-Boussinesq (OB) equations which are the coupled two
dimensional Navier-Stokes(NS) and heat conduction equations have been
investigated by E.N. Lorenz half a century ago with Fourier series and opened
the way to the paradigm of chaos. In our former study-Chaos, Solitons and
Fractals78, 249 (2015)-we presented fully analytic solutions for the velocity,
pressure and temperature fields with the aim of the self-similar Ansatz and
gave a possible explanation of the Rayleigh--B\`enard convection cells. Now we
generalize the Oberbeck-Boussinesq hydrodynamical system, going beyond the
first order Boussinesq approximation and consider a non-linear temperature
coupling. We investigate more general, power law dependent fluid viscosity or
heat conduction material equations as well. Our analytic results obtained via
the self-similar Ansatz may attract the interest of various fields like
meteorology, oceanography or climate studies.
