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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?

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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.

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