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

## TL;DR

This paper generalizes the classical Oberbeck-Boussinesq system for Rayleigh-Bénard convection by incorporating nonlinear temperature coupling and variable material properties, providing new analytic solutions relevant to geophysical and climate sciences.

## Contribution

It extends the OB system beyond first-order approximation, introducing nonlinear temperature effects and variable material properties, with analytic solutions derived via self-similar Ansatz.

## Key findings

- Analytic solutions for generalized convection equations.
- Inclusion of nonlinear temperature coupling.
- Potential applications in meteorology and climate modeling.

## Abstract

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.