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In this paper, we consider the Boussinesq equations with magnetohydrodynamicsconvection in the domain $\mathbb{T} \times \mathbb{R}$ and establishes thenonlinear stability of the Couette flow $(\mathbf{u}_{sh} = (y,0),\mathbf{b}_{sh} = (1,0), p_{sh} = 0, \theta_{sh} = 0$). The novelty in thispaper is that we design a new Fourier multiplier operator by using theproperties of the enhanced dissipation to overcome the difficult term$\partial_{xy}(-\Delta)^{-1}j$ in the linearized and nonlinear system. Then, weprove the asymptotic stability for the linearized system. Finally, we establishthe nonlinear stability for the full system by bootstrap principle.

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