Wall mode dynamics and transition to chaos in magnetoconvection with a vertical magnetic field

Quasistatic magnetoconvection of a low Prandtl number fluid ($\textrm{Pr}=0.025)$ with a vertical magnetic field is considered in a unit aspect ratio box with no-slip boundaries. At high relative magnetic field strengths, given by the Hartmann number $\textrm{Ha}$, the onset of convection is known to result from a sidewall instability giving rise to the wall mode regime. Here, we carry out 3D direct numerical simulations of unprecedented length to map out the parameter space at $\textrm{Ha} = 200, 500, 1000$, varying the Rayleigh number ($\textrm{Ra}$) between $6\times10^5 \lesssim \textrm{Ra} \lesssim 5\times 10^8$. We track the development of stable equilibria produced by this primary instability, identify bifurcations leading to limit cycles, and eventually to chaotic dynamics. At {$\textrm{Ha}=200$}, the steady wall mode solution undergoes a symmetry-breaking bifurcation producing a state featuring a coexistence between wall modes and a large-scale roll in the centre of the domain which persists to higher $\textrm{Ra}$. However, under a stronger magnetic field at $\textrm{Ha}=1000$, the steady wall mode solution undergoes a Hopf bifurcation producing a limit cycle which further develops to solutions that shadow an orbit homoclinic to a saddle point. Upon a further increase in $\textrm{Ra}$, the system undergoes a subsequent symmetry break producing a coexistence between wall modes and a large-scale roll, although the large-scale roll exists only for a small range of $\textrm{Ra}$, and chaotic dynamics primarily arise due to a mixture of chaotic wall mode dynamics and arrays of cellular structures.