Vorticity

Vorticity is a vector field that measures the local rotation of a fluid. It is defined as the curl of the velocity field: ω = ∇ × v. Physically, it describes how fluid elements spin as they move. In three-dimensional flows, ω is a true vector with components along all axes; in two-dimensional flows confined to a plane, only the component perpendicular to the plane (often called the out-of-plane vorticity) is nonzero and represents the signed rotation rate in that plane.

Vorticity is closely connected to the concept of circulation. The circulation around a closed curve moving with the fluid equals the surface integral of ω over any surface bounded by the curve (Stokes’ theorem). A region with high vorticity has strong local rotation, and vortex filaments or tubes are regions where ω is concentrated.

The evolution of vorticity is described by the vorticity equation. In general form for a Newtonian fluid with variable density, ∂ω/∂t = ∇ × (v × ω) + (1/ρ²) ∇ρ × ∇p + ν ∇² ω, where ν is the kinematic viscosity. In incompressible, inviscid flow, this reduces to ∂ω/∂t = ∇ × (v × ω) = (ω · ∇) v, highlighting vorticity stretching and twisting. Vorticity can be generated by boundary layers, shear, baroclinic effects (density and pressure gradients not aligned), and viscous diffusion, and it is transported with the flow.

Applications include atmospheric and oceanic dynamics, aerodynamics, and turbulence research, where vorticity helps describe rotation, mixing, and the behavior of vortical structures such as vortex tubes and street patterns.