Bifurcation analysis of two-dimensional Rayleigh--Bénard convection using deflation

We perform a bifurcation analysis of the steady state solutions of Rayleigh--B\'enard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialisation strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this non-linear problem, including disconnected branches of the bifurcation diagram, without the need of any prior knowledge of the dynamics. One of the disconnected branches we find contains a s-shape bifurcation with hysteresis, which is the origin of the flow pattern that may be related to the dynamics of flow reversals in the turbulent regime. Linear stability analysis is also performed to analyse the steady and unsteady regimes of the solutions in the parameter space and to characterise the type of instabilities.