= Solution
Let $\rho_0$ be the reference <mass density>, $\nu$ the <kinematic viscosity>, $\kappa$ the <thermal diffusivity>, $\eta$ the <magnetic diffusivity>, and $\alpha_T$ the <coefficient of thermal expansion>. The <conductive state of Rayleigh-Bénard convection> is $T_b=T_0+\Delta T(1-z/d)$. For the <Boussinesq approximation>, density variation is retained only in buoyancy. Linearizing about this state and the imposed <magnetic field> $B_0\hat{\mathbf y}$ gives
$$
\begin{aligned}
\partial_t\mathbf u&=-\frac1{\rho_0}\nabla p_*
+\frac{B_0}{\mu_0\rho_0}\partial_y\mathbf b
+g\alpha_T\theta\hat{\mathbf z}+\nu\Delta\mathbf u,\\
\partial_t\theta&=\frac{\Delta T}{d}w+\kappa\Delta\theta,\\
\partial_t\mathbf b&=B_0\partial_y\mathbf u+\eta\Delta\mathbf b,\qquad
\nabla\cdot\mathbf u=\nabla\cdot\mathbf b=0.
\end{aligned}
$$
Here $w=u_z$ and $p_*=p'+B_0b_y/\mu_0$ is the perturbation of <magnetohydrodynamic total pressure>. The gradient part of the <Lorentz force> has been absorbed into $p_*$; the remaining term is <magnetic tension>. The sign in the temperature equation follows from $dT_b/dz=-\Delta T/d$.
Use the <thermal-diffusion scaling of a convection layer>: length $d$, time $d^2/\kappa$, velocity $\kappa/d$, temperature perturbation $\Delta T$, field perturbation $B_0$, and total-pressure perturbation $\rho_0\nu\kappa/d^2$. Dropping dimensionless-variable decorations gives
$$
\begin{aligned}
\sigma^{-1}\partial_t\mathbf u&=-\nabla p+\zeta Q\,\partial_y\mathbf b
+R\theta\hat{\mathbf z}+\Delta\mathbf u,\\
\partial_t\theta&=w+\Delta\theta,\qquad
\partial_t\mathbf b=\partial_y\mathbf u+\zeta\Delta\mathbf b,\\
\nabla\cdot\mathbf u&=\nabla\cdot\mathbf b=0,
\end{aligned}
$$
with
$$
\boxed{R=\frac{g\alpha_T\Delta T\,d^3}{\nu\kappa},\quad
Q=\frac{B_0^2d^2}{\mu_0\rho_0\nu\eta},\quad
\sigma=\frac{\nu}{\kappa},\quad
\zeta=\frac{\eta}{\kappa}.}
$$
These are respectively the <Rayleigh number>, <Chandrasekhar number>, <Prandtl number>, and <magnetic-to-thermal diffusivity ratio>. In particular, the magnetic coefficient is $\zeta Q$, not $Q$ alone, in these thermal-time units. At $z=0,1$, the <boundary conditions> are $\theta=w=b_z=0$, $\partial_z u_x=\partial_z u_y=0$, and $\partial_zb_x=\partial_zb_y=0$.
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