Enstrophy

In fluid dynamics, the enstrophy 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 the field of combustion theory.
Given a domain formula_1 and a once-weakly differentiable vector field formula_2 which represents a fluid flow, such as a solution to the Navier-Stokes equations, its enstrophy is given by:
formula_3
Where formula_4. This is quantity is the same as the squared seminorm formula_5of the solution in the Sobolev space formula_6. 
In the case that the flow is incompressible, or equivalently that formula_7, the enstrophy can be described as the integral of the square of the vorticity formula_8,
or, in terms of the flow velocity,
formula_10
In the context of the incompressible Navier-Stokes equations, enstrophy appears in the following useful result
formula_11
The quantity in parentheses on the left is the energy in the flow, so the result says that energy declines proportional to the kinematic viscosity formula_12 times the enstrophy.