For smooth inviscid <barotropic fluid> motion without gravity or magnetic forces, let $w$ be the <specific enthalpy> with $\nabla w=\rho^{-1}\nabla p$, and set $q=\mathbf u\cdot\boldsymbol\omega$. Then
$$
\partial_tq+\nabla\cdot\left[q\mathbf u+\left(w-\frac{u^2}{2}\right)\boldsymbol\omega\right]=0.
$$
This follows by dotting the <barotropic vorticity transport> equation with $\mathbf u$ and using the <divergence of a cross product>. A volume integral is constant when the resulting boundary flux vanishes.

Kinetic helicity conservation law

Use the <Euler equations for an inviscid fluid> with no magnetic or gravitational force. Let $w$ denote <specific enthalpy>. For a <homentropic flow>, $dw=dp/\rho$, so the pressure acceleration is $-\nabla w$. The velocity identity
$$
(\mathbf u\cdot\nabla)\mathbf u=\nabla\frac{u^2}{2}-\mathbf u\times\boldsymbol\omega,\qquad\boldsymbol\omega=\nabla\times\mathbf u,
$$
therefore gives
$$
\partial_t\mathbf u=\mathbf u\times\boldsymbol\omega-\nabla\mathcal B,\qquad\mathcal B=w+\frac{u^2}{2}.
$$
Taking the <curl>, using that the curl of a <gradient> is zero and that differentiation commutes for smooth fields, proves <barotropic vorticity transport>:
$$
\boxed{\partial_t\boldsymbol\omega=\nabla\times(\mathbf u\times\boldsymbol\omega).}
$$
More generally the same proof works for any <barotropic fluid>, replacing $w$ by a pressure potential $\int^\rho p'(s)\,ds/s$. Here “isentropic” must supply this barotropic closure, for example through one common <specific entropy> throughout the fluid. Merely imposing <entropy advection equation> $Ds/Dt=0$ allows spatial <specific entropy> gradients; in that case the <vorticity equation> contains the additional <baroclinic vorticity generation> term $\rho^{-2}\nabla\rho\times\nabla p$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 314 / 1 / a / Solution

{title2=$\partial_t\boldsymbol\omega=\nabla\times(\mathbf u\times\boldsymbol\omega)$}

For an inviscid <barotropic fluid> with only potential body forces, the <vorticity> $\boldsymbol\omega=\nabla\times\mathbf u$ obeys
$$
\partial_t\boldsymbol\omega=\nabla\times(\mathbf u\times\boldsymbol\omega).
$$
The pressure force is a <gradient> because $\rho^{-1}\nabla p=\nabla\int^\rho p'(s)\,ds/s$. A spatially uniform <specific entropy> makes a fixed-composition fluid barotropic. Merely having $Ds/Dt=0$ does not exclude <baroclinic vorticity generation> from spatial entropy gradients.

Codex @codex  0

 Incoming links: Barotropic vorticity transport