The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity).
where D/Dt is the material derivative operator, u is the flow velocity, ρ is the local fluid density, p is the local pressure, τ is the viscous stress tensor and B 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 flow (i.e., low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation: where ν is the kinematic viscosity and
Under the further assumption of two-dimensional flow, the equation simplifies to: 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.
[2] For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.
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 vorticity equation can be expressed in tensor notation using Einstein's summation convention and the Levi-Civita symbol eijk: 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, η is the polar (z) component of the vorticity, ρ is the atmospheric density, u, v, and w are the components of wind velocity, and ∇h is the 2-dimensional (i.e. horizontal-component-only) del.