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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 ${\alpha }_T>0$.