= Solution
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 $-g\widehat{\boldsymbol z}$, the resting <conductive state of Rayleigh-Bénard convection> satisfies the steady <heat equation> and <hydrostatic pressure> balance. Consequently
$$
\boxed{\boldsymbol u_b=0,\qquad T_b(z)=T_0+\Delta T(1-z/d),}
$$
and, up to an arbitrary constant,
$$
\boxed{P_b(z)=P_b(0)-\rho_0g\left[(1-\alpha_T\Delta T)z+\frac{\alpha_T\Delta T}{2d}z^2\right].}
$$
Indeed $P_b'=-g\rho_b$, 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 $|\alpha_T\Delta T|\ll1$. Subtracting the <conductive state of Rayleigh-Bénard convection> and dropping products of perturbations gives the dimensional <Linearized Boussinesq equations>
$$
\partial_{t_*}\boldsymbol u_*=-\rho_0^{-1}\nabla_*p_*+g\alpha_T\theta_*\widehat{\boldsymbol z}+\nu\nabla_*^2\boldsymbol u_*,\qquad
\partial_{t_*}\theta_*-\frac{\Delta T}{d}w_*=\kappa\nabla_*^2\theta_*,\qquad \nabla_*\cdot\boldsymbol u_*=0.
$$
The minus sign in the temperature equation comes from $\boldsymbol u_*\cdot\nabla_*T_b=-w_*\Delta T/d$.
Use the <thermal-diffusion scaling of a convection layer>
$$
\boldsymbol x_*=d\boldsymbol x,\qquad t_*=\frac{d^2}{\kappa}t,\qquad
\boldsymbol u_*=\frac\kappa d\boldsymbol u,\qquad
\theta_*=\Delta T\theta,\qquad p_*=\frac{\rho_0\kappa^2}{d^2}p.
$$
The <Rayleigh number> and <Prandtl number> are
$$
\boxed{\mathrm{Ra}=\frac{g\alpha_T\Delta T d^3}{\nu\kappa},\qquad \mathrm{Pr}=\frac\nu\kappa.}
$$
Thus \b[the nondimensional perturbation equations are]
$$
\boxed{\partial_t\boldsymbol u=-\nabla p+\mathrm{Ra}\,\mathrm{Pr}\,\theta\widehat{\boldsymbol z}+\mathrm{Pr}\nabla^2\boldsymbol u,\qquad
\partial_t\theta-w=\nabla^2\theta,\qquad \nabla\cdot\boldsymbol u=0.}
$$
Here $\nu$ is <kinematic viscosity>, $\kappa$ is <thermal diffusivity>, and positive $\mathrm{Ra}$ denotes destabilizing heating from below for $\alpha_T>0$.
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