What's the corresponding symmetry of enstrophy conservation? In fluid mechanics, especially 2D turbulence study, people talk about conservation of enstrophy. But I can't really understand enstrophy very well, and what's the corresponding symmetry of enstrophy conservation?
 A: The enstrophy of a flow is defined as follows, where $\boldsymbol\omega$ is the vorticity field
$$\mathcal{E} = \iint \frac{\boldsymbol \omega^2}{2} \, d\mathbf{S}$$
Enstrophy is conversed only in 2-dimensions. This is because the vortrex stretching term $\boldsymbol \omega \cdot \nabla \mathbf{u} = 0$ only in 2d. In 3d it can be used to blow up or diminish the enstrophy of the flow.
We read conservation of enstrophy as follows 
 $$\frac{\partial}{\partial t}\iint \frac{\boldsymbol \omega^2}{2} \, d\mathbf{S} = 0$$ 
The conservation of enstrophy is a result of skew-symmetry in the convective operator $c$ in its trilinear form as follows 
$$c((u, v), w) = - c((u, w), v)$$ 
This is because in the full equation for the evolution of enstrophy, the trilinear term disappears in 2d.
$$\frac{\partial}{\partial t}\iint \frac{\boldsymbol \omega^2}{2}d\mathbf{S} = -\frac{\nabla \boldsymbol\omega^2}{\mathrm{Re}} + c((\boldsymbol\omega, \boldsymbol u), \boldsymbol\omega)$$
Which ensures that in 2d the enstropy of a flow is conserved.
