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 ) and magnetic field (characterised by the Hartmann number Hai=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 Rac for the onset of convection. We outline phase diagrams showing the dominating factors determining Rac . When 0.83Haz-0.5 for vertical and 0.66Haₓ-1.01 for horizontal magnetic field, Rac is mainly determined by the geometrical confinement with Rac=502 -4.0 . When 21/6π 1/3Haz-1/3 for vertical and 5 for the horizontal magnetic field, Rac is mainly determined by the magnetic field with Rac=π ²Ha² . In the intermediate regime, both the magnetic field and spatial confinement determine Rac , 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 Rac = 9.9\, -2.0 Haₓ² , which is explained by a theoretical model. The magnetic field also modifies the length scale . For a vertical magnetic field, decreases with increasing Haz , following $