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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 $B_0\hat{\mathbf{y}}$ givesHere $w=u_z$ and $p_{\ast }=p^{\prime }+B_0{b}_y/{\mu }_0$ is the perturbation of magnetohydrodynamic total pressure. The gradient part of the Lorentz force has been absorbed into $p_{\ast }$; the remaining term is magnetic tension. The sign in the temperature equation follows from $d{T}_b/dz=-\Delta T/d$.

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 giveswithThese 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 ${\partial }_z{b}_x={\partial }_z{b}_y=0$.

Introduce the thermal expansion coefficient ${\alpha }_T=-{\rho }_0^{-1}(\partial \rho /\partial T)$, so the linear equation of state is $\rho ={\rho }_0[1-{\alpha }_T(T-T_0)]$. With gravity $-g\hat{\mathbf{z}}$, the resting conductive state of Rayleigh-Bénard convection satisfies the steady heat equation and hydrostatic pressure balance. Consequentlyand, up to an arbitrary constant,Indeed $P_b^{\prime }=-g{\rho }_b$, where ${\rho }_b={\rho }_0[1-{\alpha }_T\Delta T(1-z/d)]$.

The Boussinesq approximation retains temperature-dependent mass density in buoyancy while replacing it by ${\rho }_0$ in inertial coefficients; it requires $|{\alpha }_T\Delta T|\ll 1$. Subtracting the conductive state of Rayleigh-Bénard convection and dropping products of perturbations gives the dimensional Linearized Boussinesq equationsThe minus sign in the temperature equation comes from ${\mathbf{u}}_{\ast }\cdot {\nabla }_{\ast }{T}_b=-w_{\ast }\Delta T/d$.

Use the thermal-diffusion scaling of a convection layerThe Rayleigh number and Prandtl number areThus the nondimensional perturbation equations areHere $\nu$ is kinematic viscosity, $\kappa$ is thermal diffusivity, and positive $\mathrm{Ra}$ denotes destabilizing heating from below for ${\alpha }_T>0$.