Vorticity equation


The vorticity equation of fluid dynamics describes evolution of the vorticity of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid.
The equation is:
where is the material derivative operator, is the flow velocity, is the local fluid density, is the local pressure, is the viscous stress tensor and represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.
The equation is valid in the absence of any concentrated torques and line forces, for a compressible Newtonian fluid.
In the case of incompressible and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation
where is the kinematic viscosity and is the Laplace operator.

Physical interpretation

Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to
Alternately, in case of incompressible, inviscid fluid with conservative body forces,
For a brief review of additional cases and simplifications, see also. For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.

Derivation

The vorticity equation can be derived from the Navier–Stokes equation for the conservation of angular momentum. In the absence of any concentrated torques and line forces, one obtains
Now, vorticity is defined as the curl of the flow velocity vector. Taking the curl of momentum equation yields the desired equation.
The following identities are useful in derivation of the equation:
where is any scalar field.

Tensor notation

The vorticity equation can be expressed in tensor notation using Einstein's summation convention and the Levi-Civita symbol :

In specific sciences

Atmospheric sciences

In the atmospheric sciences, the vorticity equation can be stated in terms of the absolute vorticity of air with respect to an inertial frame, or of the vorticity with respect to the rotation of the Earth. The absolute version is
Here, is the polar component of the vorticity, is the atmospheric density,,, and w are the components of wind velocity, and is the 2-dimensional del.