Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 77 / 4 / i / Solution

Let ${\mathbf{\omega }}_a=2\mathbf{\Omega }+\nabla \times \mathbf{u}$ be absolute vorticity, where $\mathbf{\Omega }=(0,0,f/2)$ is constant. Let $\rho$ be density and $\alpha$ a materially conserved stratifying scalar, so $D\alpha /Dt=0$. For the incompressible density-stratified ideal fluid one may choose the advected density itself; in a thermodynamic description one may use an entropy-like scalar with $\rho =\rho (p,\alpha )$. The Rossby–Ertel potential vorticity isErtel's theorem states that it is conserved following a parcel under inviscid, adiabatic motion with conservative body force and the stated stratifying-scalar condition.

Start from the rotating momentum equationHere $\Phi$ includes gravity and any conservative centrifugal potential. Taking its curl givesMass conservation is $D\rho /Dt=-\rho \nabla \cdot \mathbf{u}$, specializing to $D\rho /Dt=0$ here. Combining these equations yieldsDifferentiating the material conservation of $\alpha$ gives $D({\partial }_i\alpha )/Dt=-({\partial }_i{u}_j){\partial }_j\alpha$. Dotting these two equations, the velocity-gradient terms cancel after relabelling their indices. ThereforeIf $\alpha =\rho$, the triple product vanishes immediately. More generally, an equation of state $\rho (p,\alpha )$ makes $\nabla \rho$ a linear combination of $\nabla p$ and $\nabla \alpha$, so it also vanishes. HenceThis proves the conservation law and specifies why $\alpha$ must be a suitable stratifying scalar: an arbitrary unrelated passive tracer need not annihilate the baroclinic triple product. No viscosity or scalar diffusion is permitted in this theorem.