Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field with vertical dissipation

In this paper, we establish the nonlinear stability of Couette flow in a uniform magnetic field for the Boussinesq equations with magnetohydrodynamics convection in the domain T×R with only vertical dissipation. This extends the result for full dissipation case in the reference Bian et al. (2020) to partial dissipation case and the result for Boussinesq system in the reference Deng et al. (2020) to the full system in the presence of the magnetic field.