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Stability of advective flow in a rotating layer of a conducting fluid placed in a permanent uniform magnetic field

An exact solution of the Navier-Stokes equations in the Boussinesq approximation is presented. It describes advective flow in a flat rotating layer of an incompressible conducting fluid, on the horizontal boundaries of which a linear temperature distribution is set. The rotation axis is perpendicular to the fluid layer. The layer is placed in a constant homogeneous magnetic field oriented opposite to the force of gravity. The behavior of the flow velocity and temperature depending on the Taylor and Hartmann numbers is investigated. The stability of advective flow under normal perturbations is investigated within the framework of linear theory using a new numerical method. The problem is reduced to a one-dimensional system of partial differential equations with unknown functions depending on time and the vertical coordinate, which is solved using the grid method. Calculations have shown that a weak magnetic field increases the stability of the flow without changing the oscillatory character of instability. For the nonlinear formulation of the problem, the finite-amplitude perturbations in the supercritical region near the minima of neutral curves are studied numerically. Calculations have shown that thermal spots moving in opposite directions appear near the upper and lower boundaries of the layer. Near the horizontal boundaries of the layer forms a system of running helical vortices, arising in the region with unstable temperature stratification. The magnetic field strength perturbation has a spiral shape.