# Invariance of direction of vorticity

1. Oct 30, 2016

### spaghetti3451

1. The problem statement, all variables and given/known data

The vorticity vector $\vec{\omega} = \text{curl}\ \vec{v}$, defined as usual by $\omega^{2}=i_{{\vec{\omega}}}\text{vol}^{3}$, is $\textit{not}$ usually invariant since the flow need not conserve the volume form.

The mass form, $\rho\ \text{vol}^{3}$, however, $\textit{is}$ conserved.

From ${\bf{\omega}}^{2}=i_{\vec{\omega} / \rho}\rho \text{vol}^{3}$, it can be shown that the vector $\vec{\omega}/\rho$ should be invariant; that is, $\mathcal{L}_{\vec{v}+\partial / \partial t} (\omega^{2}/ \rho)=0$.

How can you use $\mathcal{L}_{\bf{X}}\circ i_{\bf{Y}}-i_{\bf{Y}}\circ\mathcal{L}_{\bf{X}}=i_{[{\bf{X}},{\bf{Y}}]}$ to prove that the vector $\vec{\omega}/\rho$ is invariant?

2. Relevant equations

3. The attempt at a solution

$\mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(\frac{\omega^{2}}{\rho}\right)$

$= \mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(\frac{i_{(\vec{\omega}/\rho)}\ \rho\ \text{vol}^{3}}{\rho}\right)$

$= \frac{1}{\rho}\bigg(\mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(i_{(\vec{\omega}/\rho)}\ \rho\ \text{vol}^{3}\right)\bigg)$

$= \frac{1}{\rho}\bigg(i_{(\vec{\omega}/\rho)}\left(\mathcal{L}_{{\bf{v}+\partial / \partial t}}\ \rho\ \text{vol}^{3}\right)+i_{[{\bf{v}+\partial / \partial t},(\vec{\omega} / \rho)]}\rho\ \text{vol}^{3}\bigg)$

$= \frac{1}{\rho}\bigg(0+i_{[{\bf{v}+\partial / \partial t},(\vec{\omega} / \rho)]}\rho\ \text{vol}^{3}\bigg).$

How do you proceed next?

2. Nov 5, 2016