= Ertel's theorem
{c}
{title2=$D(\rho^{-1}\boldsymbol\omega_a\cdot\nabla\alpha)/Dt=0$}
{wiki}
= Ertel theorem
{c}
{synonym}
An inviscid fluid with conservative body force, a materially conserved stratifying scalar $\alpha$, and an <equation of state> making $\nabla\rho,\nabla p,\nabla\alpha$ coplanar conserves its <Ertel potential vorticity>. The stretching of <absolute vorticity> per unit <density> cancels the deformation of the scalar <gradient>; the remaining baroclinic triple product vanishes. In an incompressible density-stratified ideal fluid, choosing the advected <density> as $\alpha$ gives that cancellation directly.
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