Subcritical transition and multistability in liquid metal magnetoconvection with sidewalls

TL;DR

The paper demonstrates that liquid-metal magnetoconvection in a bounded domain with insulating sidewalls undergoes a subcritical transition to turbulence, despite a supercritical linear onset of wall modes. By combining direct numerical simulations with symmetry-based amplitude equations, it identifies three coexisting wall-mode equilibria (LB, SB, MB) and shows that turbulent dynamics are organized by the subcritical SB, undergoing Hopf and Neimark-Sacker bifurcations to chaos. The mean flow in turbulence inherits the SB symmetry, illustrating how nonlinear equilibria govern large-scale structure even far from onset. These results imply that experimental and theoretical interpretations of magnetoconvection in bounded containers must account for subcriticality and symmetry, especially at higher Hartmann numbers $Ha$.

Abstract

The motionless conducting state of liquid metal convection with an applied vertical magnetic field confined in a vessel with insulating side walls becomes linearly unstable to wall modes through a supercritical pitchfork bifurcation. Nevertheless, we show that the transition proceeds subcritically, with stable finite-amplitude solutions with different symmetries existing at parameter values beneath this linear stability threshold. Under increased thermal driving, the branch born from the linear instability becomes unstable and solutions are attracted to the most subcritical branch, which follows a quasiperiodic route to chaos. Thus, we show that the transition to turbulence is controlled by this subcritical branch and hence, turbulent solutions have no connection to the initial linear instability. This is further quantified by observing that the subcritical equilibrium solution sets the spatial symmetry of the turbulent mean flow and thus, organises large-scale structures in the turbulent regime.

Figures (4)

• Figure 1: (a) Bifurcation diagram at $\hbox{\rm{Ha}}=500$ showing stable/unstable equilibria (filled/open markers), limit cycles (diamonds), invariant tori (asterisks) on the linear onset branch (LB) stemming from the linear instability at $\hbox{\rm{Ra}}_{c,L}$, the mixed symmetry branch (MB) and the subcritical branch (SB). The mean of time-dependent solutions is shown. The grey area represents $\hbox{\rm{Ra}}<\hbox{\rm{Ra}}_{c,L}$. (b) Upflow (pink) and downflow (blue) vertical velocity isosurfaces and streamlines (black) of solutions on the various branches shown from the top view at $\hbox{\rm{Ha}}=500$ at $\hbox{\rm{Ra}}=8\times10^5$ (top) and $\hbox{\rm{Ra}}=2\times10^6$ (bottom) with $w=\pm0.005$ and $\pm0.01$, respectively.
• Figure 2: Bifurcation diagram of the amplitude equations \ref{['eq:amplitude_sys']}, showing the norm of the amplitudes $\|\boldsymbol{A}\|_2$ as a function of the reduced Rayleigh number $R$ for the various stable/unstable equilibria denoted by solid/dotted lines. Single mode solutions corresponding to the LB, MB and SB states are shown in green, blue and orange respectively. Mixed mode solutions are shown in grey. Markers show the bifurcation points.
• Figure 3: Phase portrait at $\hbox{\rm{Ha}}=500$ of the (a) invariant 2-torus at $\hbox{\rm{Ra}}=4\times10^6$ and (c) chaotic solution at $\hbox{\rm{Ra}}=10^8$ constructed using time-delay embedding with $\tau\approx4$. The colour map corresponds to $\|w(t+3\tau)\|_2$. The unstable equilibrium point (LB) is shown in black. (b,d) Corresponding power spectral density of $\|w(t)\|_2$ respectively as a function of the frequency $f$. Fundamental frequencies are labelled $f_i$. Zoomed out spectra are shown in the insets.
• Figure 4: (a) Instantaneous Q-criterion isosurfaces (Q=3) coloured by the vertical vorticity for the flow at $\hbox{\rm{Ha}}=500$, $\hbox{\rm{Ra}}=10^9$ and (b) the corresponding mean flow ($w=\pm0.1$ isosurfaces (pink/blue)). (c) Equilibrium solution on the subcritical branch (SB) at $\hbox{\rm{Ra}}=2\times 10^6$ ($w=\pm0.01$ isosurfaces).