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| In [[fluid dynamics]], the '''enstrophy''' <math>\mathcal{E}</math> can be described as the integral of the square of the [[vorticity]] <math>\eta</math> given a velocity field <math>\mathbf{u}</math> as,
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| :<math> \mathcal{E}(\mathbf{u}) =\frac{1}{2} \int_{S} \eta^{2}dS. </math>
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| Here, since the curl gives a [[scalar field]] in 2-dimensions ([[vortex]]) corresponding to the vector-valued [[velocity]] solving in the incompressible [[Navier–Stokes equations]], we can integrate its square over a surface S to retrieve a [[continuous linear operator]] on the space of possible velocity fields, known as a ''current''. This equation is however somewhat misleading. Here we have chosen a simplified version of the enstrophy derived from the [[Incompressible fluid|incompressibility condition]], which is equivalent to vanishing divergence of the velocity field,
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| :<math> \nabla \cdot \mathbf{u} = 0. </math>
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| More generally, when not restricted to the incompressible condition, or to two spatial dimensions, the enstrophy may be computed by:
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| :<math> \mathcal{E}(\mathbf{u}) = \int_{S} |\nabla (\mathbf{u})|^{2}dS. </math>
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| where
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| :<math> |\nabla (\mathbf{u})| </math>
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| is the [[Frobenius norm]] of the gradient of the velocity field <math>\mathbf{u}</math>. | |
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| The enstrophy can be interpreted as another type of [[potential density]] (''ie''. see [[probability density function|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 [[turbulence|turbulent flows]], and is often identified in the study of [[Electrostatic ion thruster|thruster]]s as well as the field of [[flame theory]].<ref>[http://courseware.mech.ntua.gr/ml22058/pdfs/combustion-flame.pdf Overview of Flame Theory] (class notes, National Technical University of Athens, Greece)</ref>
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| == External links ==
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| * [http://aanda.u-strasbg.fr:2002/articles/aa/full/2004/45/aa0573-04/aa0573-04.right.html Hydrodynamic stability of rotationally supported flows]
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| * [http://www2.appmath.com:8080/site/frisch/frisch.html The dynamics of enstrophy transfer in two dimensional hydrodynamics]
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| == References ==
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| {{reflist}}
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| [[Category:Fluid dynamics]]
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| [[Category:Spacecraft propulsion]]
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| {{fluiddynamics-stub}}
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