Let $u=psi_z$ and $w=-psi_x$, so <mass conservation> for the two-dimensional <incompressible flow> is automatic. Write $b$ for buoyancy and introduce the diffusive operators
$$
D_\nu=\partial_t-\nu\nabla^2,
\qquad
D_\kappa=\partial_t-\kappa\nabla^2.
$$
The <Linearized Boussinesq equations> are
$$
D_\nu u=-\frac1{\rho_0}p_x,
\qquad
D_\nu w=-\frac1{\rho_0}p_z+b,
\qquad
D_\kappa b=-N^2w.
$$
Taking the curl of the <momentum conservation> equations eliminates the <pressure> and gives
$$
-D_\nu\nabla^2\psi=b_x.
$$
Apply $D_\kappa$ and use $D_\kappa b=N^2\psi_x$. Since the constant-coefficient <linear partial differential operators> commute,
$$
\boxed{\left[D_\nu D_\kappa\nabla^2+N^2\partial_x^2\right]\psi=0.}
$$
This is the viscous-diffusive <internal gravity wave> equation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 345 / 1 / a / Solution

{c}

The linearized Boussinesq equations describe small velocity, pressure, and buoyancy perturbations about a motionless, stably stratified reference state. With <kinematic viscosity> $\nu$, <mass diffusivity> $\kappa$, and <buoyancy frequency> $N$, they are
$$
(\partial_t-\nu\nabla^2)\mathbf u=-\frac1{\rho_0}\nabla p+b\mathbf e_z,
\qquad
(\partial_t-\kappa\nabla^2)b=-N^2w,
\qquad
\nabla\mathbin\cdot\mathbf u=0.
$$

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 Incoming links: Linearized Boussinesq equations