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The original Oberbeck-Boussinesq (OB) equations which are the coupled twodimensional Navier-Stokes(NS) and heat conduction equations have beeninvestigated by E.N. Lorenz half a century ago with Fourier series and openedthe way to the paradigm of chaos. In our former study-Chaos, Solitons andFractals78, 249 (2015)-we presented fully analytic solutions for the velocity,pressure and temperature fields with the aim of the self-similar Ansatz andgave a possible explanation of the Rayleigh--B\`enard convection cells. Now wegeneralize the Oberbeck-Boussinesq hydrodynamical system, going beyond thefirst order Boussinesq approximation and consider a non-linear temperaturecoupling. We investigate more general, power law dependent fluid viscosity orheat conduction material equations as well. Our analytic results obtained viathe self-similar Ansatz may attract the interest of various fields likemeteorology, oceanography or climate studies.

Rayleigh-B\`enard convection in the generalized Oberbeck-Boussinesq system