= Solution
Write the background <mass density> as $\bar\rho(z)=\rho_0+\rho'_0z$ and its <hydrostatic pressure> as $\bar p(z)$, where
$$
\bar p'(z)=-g\bar\rho(z),\qquad
\bar p(z)=p_* -g\rho_0z-\frac12g\rho'_0z^2.
$$
Apply an <Eulerian perturbation of a fluid variable> to the background fields:
$$
\rho_{\rm tot}=\bar\rho+\varrho,\qquad p_{\rm tot}=\bar p+\pi.
$$
Thus $\pi$ is the Eulerian <pressure> perturbation, in pascals; it is not the background <hydrostatic pressure>.
For <incompressible flow>, <mass conservation> becomes $D\rho_{\rm tot}/Dt=0$. Linearizing about rest gives
$$
\partial_t\varrho=-u_z\rho'_0.
$$
The <Euler equations> are $\rho_{\rm tot}D\mathbf u/Dt=-\nabla p_{\rm tot}-\rho_{\rm tot}g\widehat{\mathbf z}$. The background <pressure gradient> and weight cancel. Dropping products of perturbations gives
$$
\bar\rho\,\partial_t\mathbf u=-\nabla\pi-\varrho g\widehat{\mathbf z}.
$$
Under $|\rho'_0z|\ll\rho_0$, replace $\bar\rho$ by $\rho_0$ in inertia but retain the background <mass density> gradient in the displacement-induced <buoyancy>. This is the <Boussinesq approximation>. Its linear <Boussinesq equations> here are
$$
\boxed{\begin{aligned}
\partial_xu_x+\partial_zu_z&=0,\\
\rho_0\partial_tu_x&=-\partial_x\pi,\\
\rho_0\partial_tu_z&=-\partial_z\pi-g\varrho,\\
\partial_t\varrho&=-\rho'_0u_z.
\end{aligned}}
$$
The absolute value makes explicit that the small-density-variation condition concerns magnitude, including the usual negative $\rho'_0$ of stable <density stratification>.
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