Enstrophy Explained

In fluid dynamics, the enstrophy

l{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:^[1] where |\nabla u | ^{2} = \sum ^{n} _{i,j =1} \left| \partial _{i} u ^{j} \right| ^{2} . This quantity is the same as the squared seminorm | u | ^{2} _{H ^{1(\Omega)} ^{n}} of the solution in the Sobolev space H ^{1(\Omega)} ^{n} .

Incompressible flow

In the case that the flow is incompressible, or equivalently that

\nabla ⋅ u = 0 , the enstrophy can be described as the integral of the square of the vorticity \omega :^[2] l{E}(\boldsymbol \omega) \equiv \int _{\Omega} |\boldsymbol \omega| ^{2} dx

or, in terms of the flow velocity:

l{E}( u ) \equiv \int _{\Omega} |\nabla x u| ^{2} dx

In the context of the incompressible Navier-Stokes equations, enstrophy appears in the following useful result:

(d)/(dt) \left( (1)/(2) \int _{\Omega} | u | ^{2} \right) = - \nu l{E}( u ) \nu times the enstrophy.

See also

• Atmospheric circulation
• Turbulence

Further reading

• Arakawa . A. . Lamb . V.R. . A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations . Monthly Weather Review . January 1981 . 109 . 1 . 18–36 . 10.1175/1520-0493(1981)109<0018:APEAEC>2.0.CO;2 . 1520-0493 . free .
• Umurhan . O. M. . Regev . O. . Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results . Astronomy and Astrophysics . December 2004 . 427 . 3 . 855–872 . 10.1051/0004-6361:20040573 . astro-ph/0404020 . 2004A&A...427..855U . 15418079.
• Weiss. John. The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D: Nonlinear Phenomena. March 1991. 48. 2–3. 273–294. 10.1016/0167-2789(91)90088-Q. 1991PhyD...48..273W .

Notes and References

1. Book: Navier-Stokes equations and turbulence. 2001. Cambridge University Press. Ciprian Foiaş. 0-511-03936-0. Cambridge. 28–29. 56416088.
2. Doering, C. R. and Gibbon, J. D. (1995). Applied Analysis of the Navier-Stokes Equations, p. 11, Cambridge University Press, Cambridge. .