Bifurcation analysis of a two-dimensional magnetic Rayleigh–Bénard problem

We perform a bifurcation analysis of a two-dimensional magnetic Rayleigh–Bénard problem using a numerical technique called deflated continuation. Our aim is to study the influence of the magnetic field on the bifurcation diagram as the Chandrasekhar number Q increases and compare it to the standard (non-magnetic) Rayleigh–Bénard problem. We compute steady states at a high Chandrasekhar number of Q=103 over a range of the Rayleigh number 0≤Ra≤105. These solutions are obtained by combining deflation with a continuation of steady states at low Chandrasekhar number, which allows us to explore the influence of the strength of the magnetic field as Q increases from low coupling, where the magnetic effect is almost negligible, to strong coupling at Q=103. We discover a large profusion of states with rich dynamics and observe a complex bifurcation structure with several pitchfork, Hopf, and saddle–node bifurcations. Our numerical simulations show that the onset of bifurcations in the problem is delayed when Q increases, while solutions with fluid velocity patterns aligning with the background vertical magnetic field are privileged. Additionally, we report a branch of states that stabilizes at high magnetic coupling, suggesting that one may take advantage of the magnetic field to discriminate solutions.