Onset and length scales of wall modes in confined magnetoconvection with a vertical magnetic field

Using linear stability analysis, we study the onset and formation mechanism of wall modes in confined magnetoconvection cells with the degree of confinement characterised by the cell aspect ratio $\varGamma$ . We first outline the phase diagram of the dominating factors that determine the critical Rayleigh number $Ra_c$ for the onset of convection in the $\varGamma -Ha$ phase space, with $\textit{Ha}$ being the Hartmann number. Our study shows that $Ra_c$ is primarily determined by geometrical confinement, and bulk convection onset occurs with $Ra_c = 1090 \varGamma ^{-4.0}$ for $\varGamma \lt \varGamma _{c_1} = 1.21 \textit{Ha}^{-0.48}$ . No wall modes form and $Ra_c$ depends on the strength of both the confinement and magnetic field for $\varGamma _{c_1} \leqslant \varGamma \lt \varGamma _{c_2} = 4.07 \textit{Ha}^{-0.53}$ . For $\varGamma _{c_2}\leqslant \varGamma \lt \varGamma _{c_3}=0.99 \textit{Ha}^{-0.10}$ , wall modes emerge and $Ra_c$ drops below the bulk onset Rayleigh number for magnetoconvection. When $\varGamma \geqslant \varGamma _{c_3}$ , wall modes become fully developed with an onset Rayleigh number for wall modes $Ra_{c,w} \approx 65 \textit{Ha}^{1.5}$ . In this fully developed regime, the radial velocity profile and $Ra_{c,w}$ become independent of $\varGamma$ . Through analysing the length scales of wall modes and their interaction with spatial confinement, we show dynamically how wall modes emerge in confined cells: while the first layer with a characteristic length scale $\ell _1 = 1.04 \textit{Ha}^{-0.56}$ forms when $\varGamma \geqslant 5.39 \textit{Ha}^{-0.58}$ , the second layer with a characteristic length scale $\ell _2 = 4.94 \textit{Ha}^{-0.56}$ emerges when $\varGamma \geqslant 9.07 \textit{Ha}^{-0.53}$ . These scaling relations provide practical guidelines for experimental and numerical studies of the wall-mode dynamics.