Vorticity equation in two dimensions

In two dimensions, the incompressible velocity field ${\mbox{\boldmath$u$}}$ can be expressed in terms of the stream-function $\psi$ as:

$${\mbox{\boldmath$u$}} = (\partial_y \psi, -\partial_x \psi)$$ (1.58)

The vorticity field, defined as the curl of velocity, ${\mbox{\boldmath$\omega$}} = \nabla \times {\mbox{\boldmath$u$}}$, in two dimensions has only one non-zero component which is orthogonal to the plane of velocity and is related to the stream-function by

$$\omega= - \Delta \psi$$ (1.59)

Thus instead of giving a description of the flow in term of the two components of velocity, which are not independent because of the incompressibility condition, it is convenient to rewrite the two-dimensional Navier-Stokes equations in terms of the vorticity scalar field:

$${\partial \omega \over \partial t} + {\mbox{\boldmath$u$}} \... ...nabla^2 \omega - \alpha {\mbox{\boldmath$u$}} + f_{\omega} \;,$$ (1.60)

The linear dissipative term accounts for friction between the thin layer of fluid which is considered, and the rest of the three dimensional environment. Its effects will be discussed in the following chapter. The term $f_{\omega}$ represents the external source of energy acting on the largest scales - e.g. stirring. This term counteracts the dissipation by viscosity $\nu$ and friction $\alpha$ and allows to obtain a statistically steady state.

To solve Eq. (1.60) it is necessary to specify a set of boundary conditions which are required to solve the Poisson equation (1.59) for the stream function. In most studies on 2D turbulence, periodic boundary conditions are assumed in both the two directions. The presence of realistic no-slip boundaries gives origin to a source of vorticity fluctuations.