Linear stability analysis of magnetoconvection in finite-sized systems: competition between spatial confinement and magnetic field

We present a linear stability analysis of two-dimensional magnetoconvection considering the effects of spatial confinement (characterised by the aspect ratio $\varGamma$ ) and magnetic field (characterised by the Hartmann number $\textit{Ha}_{i=x,y,z}$ with subscript representing its direction). It is found that when the magnetic field is perpendicular to the convection domain ( $y$ -direction), it does not affect the onset of convection due to zero Lorentz force. With a magnetic field in the $z$ (vertical) or $x$ (horizontal) directions, the onset of convection is delayed, resulting in a larger critical Rayleigh number $Ra_c$ for the onset of convection. We outline phase diagrams showing the dominating factors determining $Ra_c$ . When $\varGamma \leqslant 0.83\textit{Ha}_z^{-0.5}$ for vertical and $\varGamma \leqslant 0.66\textit{Ha}_x^{-1.01}$ for horizontal magnetic field, $Ra_c$ is mainly determined by the geometrical confinement with $Ra_c=502\varGamma ^{-4.0}$ . When $\varGamma \geqslant 2^{1/6}\pi ^{1/3}\textit{Ha}_z^{-1/3}$ for vertical and $\varGamma \geqslant 5$ for the horizontal magnetic field, $Ra_c$ is mainly determined by the magnetic field with $Ra_c=\pi ^2\textit{Ha}^2$ . In the intermediate regime, both the magnetic field and spatial confinement determine $Ra_c$ , and a horizontal magnetic field is found to suppress convection more than a vertical magnetic field. In addition, under a horizontal magnetic field, there exists a subregime characterised by $Ra_c = 9.9\,\varGamma ^{-2.0} \textit{Ha}_x^2$ , which is explained by a theoretical model. The magnetic field also modifies the length scale $\ell$ . For a vertical magnetic field, $\ell$ decreases with increasing $\textit{Ha}_z$ , following $\ell