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Vorticity Equation

\[\frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega}\cdot\nabla)\mathbf{v} + \nu\nabla^2\boldsymbol{\omega}\]

Variables

Symbol Meaning Units
\(\boldsymbol{\omega} = \nabla\times\mathbf{v}\) vorticity s⁻¹
\(\mathbf{v}\) velocity m/s
\(\nu\) kinematic viscosity m²/s
\(D/Dt\) material derivative —

Assumptions

  • Incompressible, constant density (otherwise a baroclinic term \(\nabla\rho\times\nabla p/\rho^2\) appears)
  • Newtonian viscous stress
  • Conservative body forces (gravity drops out as the curl of a gradient)

Derivation sketch

Take the curl of the Navier–Stokes equation. The pressure gradient vanishes identically because \(\nabla\times\nabla p = 0\) — an exact elimination, not an approximation. Gravity likewise. The advective term \((\mathbf{v}\cdot\nabla)\mathbf{v}\) splits into transport of \(\boldsymbol{\omega}\) plus the stretching/tilting term.

Notes

  • \((\boldsymbol{\omega}\cdot\nabla)\mathbf{v}\) vanishes identically in two dimensions, which is why 2-D turbulence cascades energy to large scales rather than small.
  • The component along \(\boldsymbol{\omega}\) is stretching (amplifies \(|\boldsymbol{\omega}|\), conserving \(\Gamma = \omega A\)); the perpendicular component is tilting (rotates vortex lines).
  • With \(\nu = 0\) this is exactly the statement that vortex lines are frozen into the fluid — the Helmholtz theorems, and the direct analogue of the induction equation in ideal MHD.