Bifurcation analysis of a two-dimensional magnetic Rayleigh-B\'enard problem

TL;DR

This study uses deflated continuation to analyze how increasing magnetic field strength affects bifurcation structures in a 2D magnetic Rayleigh-Bénard problem, revealing complex dynamics and potential for solution stabilization.

Contribution

It introduces a numerical approach combining deflation and continuation to explore magnetic effects on bifurcations in Rayleigh-Bénard convection at high Chandrasekhar numbers.

Findings

Bifurcation onset delays with increasing magnetic field.

Rich dynamics with multiple bifurcations observed.

Magnetic field can stabilize certain flow states.

Abstract

We perform a bifurcation analysis of a two-dimensional magnetic Rayleigh--B\'enard 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--B\'enard problem. We compute steady states at a high Chandrasekhar number of $Q=10^3$ over a range of the Rayleigh number $0\leq \Ra\leq 10^5$. 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 $Q=10^3$. 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.