### Conservation of Vorticity in 2D

Starting from the rotating momentum equations in 2D,

$$\frac{D\vec{u}}{Dt} + f \hat{k}\times \vec{u} = -\frac{1}{\rho}\nabla p$$

prove that the relative vorticity is materially conserved in barotropic, incompressible flow. That is,

$$\frac{D\zeta}{Dt} = 0$$

where $$\zeta = \frac{\partial v}{\partial x} – \frac{\partial u}{\partial y}$$.

From Wikipedia.