Taking the curl of the Boussinesq momentum equation and using incompressibility gives the <vorticity equation>
$$
\boxed{
\partial_t\boldsymbol\omega+(\mathbf u\mathbin{\cdot}\nabla)\boldsymbol\omega
=(\boldsymbol\omega\mathbin{\cdot}\nabla)\mathbf u
+\nabla\left(\frac{\rho'}{\rho_0}\right)\times\mathbf g
+\nu\nabla^2\boldsymbol\omega}.
$$
The second term on the right is <baroclinic vorticity generation>. It is nonzero where a horizontal density gradient crosses the vertical gravitational acceleration.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 3 / a / Solution

{wiki=Baroclinity}

Vorticity is generated when density and pressure gradients are not parallel. Under the <Boussinesq approximation>, the source is $\nabla(\rho'/\rho_0)\times\mathbf g$.

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 Incoming links: Baroclinic vorticity generation