# Contents

## Idea

Enstrophy is the integral of the vorticity $\vec{\omega}$, it is an important concept in fluid dynamics, e.g. for the Navier-Stokes equations, and especially for turbulence. Under certain conditions, the rate of decrease of the energy of a fluid flow is proportional to the enstrophy.

## Details

### Definition

The general definition of the enstrophy $E(\vec{u})$ of a fluid flow in a region $\Omega \subseteq \mathbb{R}^d$, $d = 2$ or $d = 3$, is

$E(\vec{u}) = \sum_{i, j = 1}^d \int_{\Omega} \| \frac{\partial u_i}{\partial x_j} \|^2 d x = \int_{\Omega} \| \nabla \vec{u} \|^2 d x$

### Enstrophy and Energy Decay

We recall the Navier-Stokes equations for a viscous, incompressible homogeneous fluid flow, the momentum balance equation is:

$\frac{\partial \vec{u}}{\partial t} - \nu \Delta \vec{u} + (\vec{u} \cdot \nabla) \vec{u} + \nabla p = \vec{f}$

The mass balance equation is:

$\nabla \cdot \vec{u} = 0$

With the density normalized to 1, the energy $e(\vec{u})$ is given by

$e(\vec{u}) = \frac{1}{2} \int_{\Omega} \| \vec{u} \|^2 d x$

That is, there is only kinetic energy in the fluid flow, we don’t describe e.g. potential or thermal energy.

If the boundary values are zero, and all integrals converge, we can use the divergence condition to perform an integration by parts in the definition of the enstrophy and get:

$E(\vec{u}) = \int_{\Omega} \| \nabla \times \vec{u} \|^2 d x$

When we assume $\Omega = \mathbb{R}^3$ and a sufficiently fast decay of pressure and velocity at infinity, we can use the Navier-Stokes equations to prove the relation

$\frac{d}{d t} e(\vec{u}) + \nu E(\vec{u}) = \int \vec{f} \cdot \vec{u} d x$

(To this end, take the momentum equation, take the scalar product with $\vec{u}$ and integrate over $\Omega$.)

In the absence of external forces $\vec{f}$ we get that the energy decreases proportionally to the enstrophy as was alluded to in the introduction:

$\frac{d}{d t} e(\vec{u}) = - \nu E(\vec{u})$

### Enstrophy and Function Spaces

From the definition we see that $L^2(\Omega)$ is the space of fluid flows that have finite energy, while the Sobolev space $H^1(\Omega)$ (square integrable functions with square integrable gradient) is the space of fluid flows with finite energy and finite enstrophy.