enstrophy

In fluid dynamics, the enstrophy \mathcal{E} can be interpreted as another type of potential 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 in combustion theory and meteorology.

Given a domain \Omega \subseteq \R^n and a once-weakly differentiable vector field u \in H^1(\R^n)^n which represents a fluid flow, such as a solution to the Navier-Stokes equations, its enstrophy is given by:{{Equation box 1|cellpadding|border|indent=:|equation= \mathcal{E}({\bf u}) := \int_\Omega |\nabla \mathbf{u}|^2 \, d \mathbf x |border colour=#0073CF|background colour=#F5FFFA}}where |\nabla \mathbf{u}|^2 = \sum_{i,j=1}^n \left| \partial_i u^j \right|^2 . This quantity is the same as the squared seminorm |\mathbf{u}|_{H^1(\Omega)^n}^2of the solution in the Sobolev space H^1(\Omega)^n.