Enstrophy

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In fluid dynamics, the enstrophy $\mathcal{E}$ can be described as the integral of the square of the vorticity $η$ given a velocity field $\mathbf{u}$ as,

$\mathcal{E}(\mathbf{u}) =\frac{1}{2} \int_{S} \eta^{2}dS.$

Here, since the curl gives a vector field in 2-dimensions (vortex) corresponding to the vector valued velocity component in the Navier-Stokes equations, we can integrate its square over a surface S to retrieve a continuous linear operator, known as a current. This equation is however somewhat misleading. Here we have chosen a simplified version of the enstrophy derived from the incompressibility condition, which reduces to vanishing divergence,

$\nabla \cdot \mathbf{v} = 0.$

More generally, when not restricted to the incompressible condition, the enstrophy may be computed by:

$\mathcal{E}(\mathbf{u}) = \int_{S} |\nabla (u)|^{2}dS.$

The enstrophy can be interpreted as another type of potential density (ie. see probability density); or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particularly useful in the study of turbulent flows, and is often identified in the study of thrusters as well as the field of flame theory.