The term Dω/Dt on the left-hand side is the material derivative of the vorticity vector ω. It describes the rate of change of vorticity of the fluid particle. This change can be attributed to unsteadiness in the flow (∂ω/∂t, the unsteady term) or due to the motion of the fluid particle as it moves from one point to another (u ∙ (∇ω), the convection term).
The term (ω ∙ ∇) u on the right-hand side describes the stretching or tilting of vorticity due to the flow velocity gradients. Note that ∇u is an order-2 tensor with nine components.
where v = 1/ρ is the specific volume of the fluid element. One can think of ∇ ∙ u as a measure of flow compressibility. Sometimes the negative sign is included in the term.
The term 1/ρ2∇ρ × ∇p is the baroclinic term. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces.
The term ∇ × (∇ ∙ τ/ρ), accounts for the diffusion of vorticity due to the viscous effects.
The term ∇ × B provides for changes due to external body forces. These are forces that are spread over a three-dimensional region of the fluid, such as gravity or electromagnetic forces. (As opposed to forces that act only over a surface (like drag on a wall) or a line (like surface tension around a meniscus).
For a barotropic fluid, ∇ρ × ∇p = 0. This is also true for a constant density fluid (including incompressible fluid) where ∇ρ = 0. Note that this is not the same as an incompressible flow, for which the barotropic term cannot be neglected.
For inviscid fluids, the viscosity tensor τ is zero.
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.
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 ω are the components of wind velocity, and ∇h is the 2-dimensional (i.e. horizontal-component-only) del.