The Vorticity Transport Equation

Derivation

The Vorticity Transport Equation

D-321 Home PU-307 Threads fields · force · symmetry Depends on Navier-Stokes from the Newtonian Constitutive Law

Statement

For an incompressible Newtonian fluid of constant density \(\rho\) and constant kinematic viscosity \(\nu\), subject to a conservative body force, the vorticity \(\vec{\omega}=\nabla\times\vec{u}\) obeys \[ \frac{D\vec{\omega}}{Dt}=\frac{\partial\vec{\omega}}{\partial t}+(\vec{u}\cdot\nabla)\vec{\omega}=(\vec{\omega}\cdot\nabla)\vec{u}+\nu\nabla^2\vec{\omega}, \] obtained by taking the curl of the Navier-Stokes momentum equation; the term \((\vec{\omega}\cdot\nabla)\vec{u}\) encodes vortex stretching and tilting and \(\nu\nabla^2\vec{\omega}\) encodes viscous diffusion of vorticity.

Why it matters

The vorticity equation removes the pressure field entirely from the dynamics of incompressible flow: pressure enters Navier-Stokes as a gradient, and the curl of a gradient vanishes. What remains is a transport equation for a purely kinematic, locally measurable quantity — the local spin rate of fluid elements — so the equation isolates how rotation is created, amplified, reoriented, and destroyed.

Its structure explains the deepest qualitative fact about three-dimensional turbulence. The stretching term \((\vec{\omega}\cdot\nabla)\vec{u}\) has no counterpart in two dimensions; its presence in 3D is the engine that intensifies vorticity down to ever-smaller scales, driving the forward energy cascade, while its absence in 2D forces energy to flow the other way. The same equation governs tornado spin-up, the tightening of a bathtub vortex, the roll-up of shear layers, and the persistence of aircraft wake vortices.

Assumptions

Incompressible flow, \(\nabla\cdot\vec{u}=0\).If dropped, the identity \(\nabla\times(\vec{u}\times\vec{\omega})=(\vec{\omega}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{\omega}\) acquires a \(-\vec{\omega}(\nabla\cdot\vec{u})\) term, so expansion or compression of a fluid element dilutes or concentrates its vorticity.

Constant, uniform density \(\rho\).If dropped, the pressure force is no longer a pure gradient of a scalar times \(1/\rho\), and its curl leaves the baroclinic term \(\frac{1}{\rho^2}\nabla\rho\times\nabla p\), the source of vorticity in stratified and combusting flows.

Newtonian constitutive law with constant \(\nu\).If dropped, the viscous force is not \(\mu\nabla^2\vec{u}\), and its curl is not the clean Laplacian diffusion \(\nu\nabla^2\vec{\omega}\); non-Newtonian stresses produce extra vorticity source terms.

Conservative body force, \(\vec{f}=-\nabla\Phi\).If dropped, the curl of the body force \(\nabla\times\vec{f}\) survives as a vorticity source; e.g. the Lorentz force in MHD or non-conservative forcing injects vorticity directly.

Inertial (non-rotating) reference frame.If dropped, working in a frame rotating at \(\vec{\Omega}\) adds the Coriolis contribution, giving a stretching term \((2\vec{\Omega}+\vec{\omega})\cdot\nabla)\vec{u}\) — planetary vorticity that dominates geophysical flows.

Derivation

\[ \frac{\partial\vec{u}}{\partial t}+(\vec{u}\cdot\nabla)\vec{u}=-\frac{1}{\rho}\nabla p+\nu\nabla^2\vec{u}-\nabla\Phi \]

Incompressible Newtonian Navier-Stokes with a conservative body force; the assumed prior result. A

\[ (\vec{u}\cdot\nabla)\vec{u}=\nabla\!\left(\tfrac{1}{2}\,u^2\right)-\vec{u}\times\vec{\omega} \]

The Lamb form of the advective acceleration, from the identity \(\nabla(\vec{u}\cdot\vec{u})=2(\vec{u}\cdot\nabla)\vec{u}+2\vec{u}\times(\nabla\times\vec{u})\) with \(\vec{\omega}=\nabla\times\vec{u}\). This isolates a pure gradient. B

\[ \frac{\partial\vec{u}}{\partial t}-\vec{u}\times\vec{\omega}=-\nabla\!\left(\frac{p}{\rho}+\tfrac{1}{2}u^2+\Phi\right)+\nu\nabla^2\vec{u} \]

Substitute Step 2 and collect every gradient term (pressure head, kinetic head, potential) into one Bernoulli scalar \(H=p/\rho+\tfrac12 u^2+\Phi\). Constant \(\rho\) lets \(1/\rho\) move inside the gradient. B

\[ \nabla\times\!\left[\frac{\partial\vec{u}}{\partial t}-\vec{u}\times\vec{\omega}\right]=\nabla\times\!\left[-\nabla H+\nu\nabla^2\vec{u}\right] \]

Apply the curl operator to both sides. Curl and \(\partial/\partial t\) commute (independent variables); curl and \(\nabla^2\) commute (constant-coefficient linear operators). A

\[ \frac{\partial\vec{\omega}}{\partial t}-\nabla\times(\vec{u}\times\vec{\omega})=\nu\nabla^2\vec{\omega} \]

\(\nabla\times\nabla H=\vec{0}\): the curl of any gradient vanishes, eliminating pressure and all conservative forces. Also \(\nabla\times\partial_t\vec{u}=\partial_t\vec{\omega}\) and \(\nabla\times\nabla^2\vec{u}=\nabla^2\vec{\omega}\). B

\[ \nabla\times(\vec{u}\times\vec{\omega})=\vec{u}(\nabla\cdot\vec{\omega})-\vec{\omega}(\nabla\cdot\vec{u})+(\vec{\omega}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{\omega} \]

The vector identity for the curl of a cross product. Each term must be kept before invoking any constraint. C

\[ \nabla\cdot\vec{\omega}=\nabla\cdot(\nabla\times\vec{u})=0,\qquad \nabla\cdot\vec{u}=0 \]

The divergence of any curl is identically zero (solenoidal vorticity), and incompressibility kills the second term. Only \((\vec{\omega}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{\omega}\) survives. C

\[ \frac{\partial\vec{\omega}}{\partial t}-\big[(\vec{\omega}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{\omega}\big]=\nu\nabla^2\vec{\omega} \]

Insert the reduced identity from Steps 6-7 back into Step 5. B

\[ \frac{\partial\vec{\omega}}{\partial t}+(\vec{u}\cdot\nabla)\vec{\omega}=(\vec{\omega}\cdot\nabla)\vec{u}+\nu\nabla^2\vec{\omega} \]

Move the advective term \(+(\vec{u}\cdot\nabla)\vec{\omega}\) to the left; the two left-hand terms are the material derivative \(D\vec{\omega}/Dt\). A

\[ (\vec{\omega}\cdot\nabla)\vec{u}=\mathbf{S}\,\vec{\omega},\qquad S_{ij}=\tfrac{1}{2}\!\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) \]

Split \(\partial_j u_i\) into symmetric strain-rate \(S_{ij}\) and antisymmetric \(\Omega_{ij}=-\tfrac12\epsilon_{ijk}\omega_k\). The antisymmetric part gives \(\omega_j\Omega_{ij}=-\tfrac12\epsilon_{ijk}\omega_j\omega_k=0\). Only pure strain stretches and tilts vorticity. C

Result

\[ \frac{D\vec{\omega}}{Dt}=(\vec{\omega}\cdot\nabla)\vec{u}+\nu\nabla^2\vec{\omega}=\mathbf{S}\,\vec{\omega}+\nu\nabla^2\vec{\omega} \]

Reading. Following a fluid element, its vorticity changes only through two mechanisms. First, the local velocity gradient acting on the existing vorticity: the component of \(\mathbf{S}\vec{\omega}\) along \(\vec{\omega}\) is stretching (a vortex line pulled taut spins faster, conserving angular momentum), and the component perpendicular to \(\vec{\omega}\) is tilting (a vortex line reoriented by shear). Second, viscous diffusion, which smooths vorticity gradients and ultimately dissipates rotation. Pressure and conservative body forces are absent — they cannot create vorticity in a constant-density fluid.

Units check. \(\vec{\omega}\) has units \(\mathrm{s^{-1}}\), so \(D\vec{\omega}/Dt\) is \(\mathrm{s^{-2}}\). Stretching: \([\vec{\omega}][\nabla][\vec{u}]=\mathrm{s^{-1}}\cdot\mathrm{m^{-1}}\cdot\mathrm{m\,s^{-1}}=\mathrm{s^{-2}}\). Diffusion: \([\nu][\nabla^2][\vec{\omega}]=\mathrm{m^2\,s^{-1}}\cdot\mathrm{m^{-2}}\cdot\mathrm{s^{-1}}=\mathrm{s^{-2}}\). All three terms agree.

Limiting cases

Two-dimensional flow. With \(\vec{u}=(u,v,0)\) and \(\vec{\omega}=\omega\hat{z}\), the stretching term \((\vec{\omega}\cdot\nabla)\vec{u}=\omega\,\partial_z\vec{u}=\vec{0}\). Vorticity is materially conserved up to diffusion: \(D\omega/Dt=\nu\nabla^2\omega\) — the origin of the inverse energy cascade.
Inviscid limit, \(\nu\to 0\). The equation becomes \(D\vec{\omega}/Dt=(\vec{\omega}\cdot\nabla)\vec{u}\), the Helmholtz vorticity equation; vortex lines are frozen into the fluid (Kelvin's circulation theorem).
Pure diffusion, negligible advection and stretching. \(\partial\vec{\omega}/\partial t=\nu\nabla^2\vec{\omega}\): vorticity spreads like heat, e.g. the Lamb-Oseen decay of a line vortex.
Solid-body-like alignment. If \(\vec{\omega}\) is an eigenvector of \(\mathbf{S}\) with positive eigenvalue, stretching is maximal and exponential; if the eigenvalue is negative (compression), vorticity is squeezed and weakened.

Breaks when

Variable-density / stratified flow. When \(\rho\) is not uniform, the curl of the pressure term does not vanish; the baroclinic source \(\frac{1}{\rho^2}\nabla\rho\times\nabla p\) generates vorticity wherever density and pressure gradients are misaligned (sea breezes, Rayleigh-Taylor, flame fronts).
Compressible flow. With \(\nabla\cdot\vec{u}\neq0\) the retained term \(-\vec{\omega}(\nabla\cdot\vec{u})\) makes expanding elements lose vorticity and compressing elements gain it; the clean transport form is lost.
Rotating frame / geophysical scales. Coriolis effects add planetary vorticity \(2\vec{\Omega}\); the equation must carry absolute vorticity \(\vec{\omega}+2\vec{\Omega}\), and the derived form underestimates the true stretching source.
Non-conservative or electromagnetic body forces (MHD). The Lorentz force \(\vec{J}\times\vec{B}\) is not a gradient, so \(\nabla\times\vec{f}\neq0\) survives as an extra vorticity source.

Failure modes

Keeping the pressure term. Students write a \(-\frac{1}{\rho}\nabla\times\nabla p\) term; forgetting that curl of a gradient is exactly zero for constant \(\rho\) leaves a spurious source.
Sign or term errors in the curl identity. Dropping a term of \(\nabla\times(\vec{u}\times\vec{\omega})\), or getting the sign of \((\vec{\omega}\cdot\nabla)\vec{u}\) versus \((\vec{u}\cdot\nabla)\vec{\omega}\) wrong, flips stretching into advection.
Forgetting \(\nabla\cdot\vec{\omega}=0\). Carrying the \(\vec{u}(\nabla\cdot\vec{\omega})\) term because one confuses it with \(\nabla\cdot\vec{u}\); vorticity is solenoidal by construction.
Expecting stretching in 2D. Believing a planar flow can amplify its own vorticity; \((\vec{\omega}\cdot\nabla)\vec{u}\) is identically zero there.
Treating \((\vec{\omega}\cdot\nabla)\vec{u}\) as a scalar multiple of \(\vec{\omega}\). It is the full tensor \(\mathbf{S}\vec{\omega}\); its perpendicular part tilts vorticity into new directions and cannot be ignored.
Miscounting the diffusion coefficient. Writing \(\mu\) instead of \(\nu=\mu/\rho\) in the curled equation, mixing dynamic and kinematic viscosity.

Discussion

The vorticity equation is best read as a competition of timescales. Stretching amplifies vorticity on the strain timescale \(\tau_S\sim 1/|\mathbf{S}|\), while diffusion erodes it on \(\tau_\nu\sim\ell^2/\nu\) for a structure of size \(\ell\). Where strain wins, vortex tubes thin and intensify until the two balance at the Burgers scale \(\ell_B\sim\sqrt{\nu/a}\) (with \(a\) the local strain rate). This self-limiting balance is why turbulent vorticity organizes into thin, intense filaments rather than growing without bound, and it fixes the smallest dynamically active scale of the flow.

Integrating the equation over a material surface bounded by a closed loop recovers Kelvin's circulation theorem: in the inviscid, barotropic, conservative-force limit the circulation \(\Gamma=\oint\vec{u}\cdot d\vec{\ell}\) is constant following the flow, because stretching merely redistributes vorticity along frozen-in vortex lines without changing their total flux. Viscosity is the sole agent that changes circulation, which is why lift generation on an airfoil ultimately traces back to a viscous boundary layer shedding the starting vortex.

The absence of a pressure term is more than a computational convenience: it is why vorticity is the natural variable for a mathematically well-posed formulation of incompressible flow. Because \(\nabla\cdot\vec{u}=0\) and \(\vec{\omega}=\nabla\times\vec{u}\), the velocity is recovered from vorticity by the Biot-Savart law, making \(\vec{\omega}\) a complete, pressure-free description. The regularity question for the 3D Navier-Stokes equations reduces, via the Beale-Kato-Majda criterion, to whether \(\int_0^T\|\vec{\omega}\|_\infty\,dt\) stays finite — that is, whether vortex stretching can drive the vorticity to blow up in finite time. The stretching term \(\mathbf{S}\vec{\omega}\) is precisely the nonlinearity that makes this one of the central open problems in mathematics.

Common misconceptions. Vortex stretching does not violate energy conservation: a stretched tube spins faster because its cross-section shrinks, conserving angular momentum, and the kinetic energy it gains is supplied by the straining flow doing work — the total energy budget is set by Navier-Stokes, not by the vorticity equation alone. Likewise, "diffusion of vorticity" does not transport a substance; it is the viscous smoothing of the velocity field re-expressed in terms of its curl.

Worked examples

Vortex stretching in an axisymmetric strain. Impose the extensional strain field \(\vec{u}=\left(-\tfrac{1}{2}a\,x,\,-\tfrac{1}{2}a\,y,\,a\,z\right)\) with \(a>0\), which satisfies \(\nabla\cdot\vec{u}=0\), and let vorticity align with the stretching axis, \(\vec{\omega}=\omega_z(t)\,\hat{z}\).

Set-up: a stretching flow acting on an axially aligned vortex; neglect diffusion and axial advection to isolate stretching. B

\[ (\vec{\omega}\cdot\nabla)\vec{u}=\omega_z\frac{\partial\vec{u}}{\partial z}=\omega_z\,(0,0,a)=a\,\omega_z\,\hat{z} \]

Only the \(z\)-derivative survives because \(\vec{\omega}\) has only a \(z\)-component. A

\[ \frac{d\omega_z}{dt}=a\,\omega_z\quad\Longrightarrow\quad \omega_z(t)=\omega_0\,e^{a t} \]

The reduced vorticity equation integrates to exponential growth at the strain rate \(a\). A

Numbers: \(a=2\ \mathrm{s^{-1}}\), \(\omega_0=10\ \mathrm{s^{-1}}\), \(t=1\ \mathrm{s}\): \(\ \omega_z=10\,e^{2}=10\times7.389=73.9\ \mathrm{s^{-1}}\).

Insert values; \(e^2=7.389\). A

\[ \omega_z(1\,\mathrm{s})=73.9\ \mathrm{s^{-1}} \]

Reading. Stretching multiplies the spin rate by \(e^{a t}\); a tube stretched for one strain time nearly octuples its vorticity. Units: \(a\) in \(\mathrm{s^{-1}}\), so \(a t\) is dimensionless and \(\omega_z\) stays in \(\mathrm{s^{-1}}\).

Viscous decay of a line vortex (Lamb-Oseen). A concentrated line vortex of circulation \(\Gamma\) diffuses under \(\partial\vec{\omega}/\partial t=\nu\nabla^2\vec{\omega}\), giving the exact axisymmetric solution \[ \omega(r,t)=\frac{\Gamma}{4\pi\nu t}\exp\!\left(-\frac{r^2}{4\nu t}\right). \]

Set-up: pure diffusion of vorticity, no stretching (2D); this Gaussian is the fundamental solution of the heat equation in the plane. C

\[ \omega_{\max}(t)=\omega(0,t)=\frac{\Gamma}{4\pi\nu t} \]

Peak vorticity sits on the axis \(r=0\) and decays as \(1/t\). B

Core radius from \(\exp(-r_c^2/4\nu t)=e^{-1}\): \(\ r_c(t)=\sqrt{4\nu t}\), spreading diffusively.

The \(1/e\) radius grows like \(\sqrt{\nu t}\), the diffusion length. B

Numbers (water): \(\Gamma=0.10\ \mathrm{m^2\,s^{-1}}\), \(\nu=1.0\times10^{-6}\ \mathrm{m^2\,s^{-1}}\), \(t=10\ \mathrm{s}\).

Aircraft-wake-scale circulation in water; insert values. A

\[ \omega_{\max}=\frac{0.10}{4\pi(1.0\times10^{-6})(10)}=\frac{0.10}{1.257\times10^{-4}}=7.96\times10^{2}\ \mathrm{s^{-1}} \]

Denominator \(4\pi\nu t=1.257\times10^{-4}\ \mathrm{m^2}\). A

\[ r_c=\sqrt{4(1.0\times10^{-6})(10)}=\sqrt{4.0\times10^{-5}}=6.3\times10^{-3}\ \mathrm{m} \]

The core has spread to about \(6.3\ \mathrm{mm}\) after \(10\ \mathrm{s}\). A

\[ \omega_{\max}=796\ \mathrm{s^{-1}},\qquad r_c=6.3\ \mathrm{mm} \]

Reading. With no stretching to feed it, vorticity spreads and its peak falls as \(1/t\) while the core grows as \(\sqrt{\nu t}\). Units: \(\Gamma/(\nu t)=\mathrm{m^2\,s^{-1}}/(\mathrm{m^2\,s^{-1}}\cdot\mathrm{s})=\mathrm{s^{-1}}\); \(\sqrt{\nu t}=\mathrm{m}\).

Problems

Show explicitly, using index notation and the identity \(\nabla\times(\vec{a}\times\vec{b})=\vec{a}(\nabla\cdot\vec{b})-\vec{b}(\nabla\cdot\vec{a})+(\vec{b}\cdot\nabla)\vec{a}-(\vec{a}\cdot\nabla)\vec{b}\), that \(\nabla\times(\vec{u}\times\vec{\omega})=(\vec{\omega}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{\omega}\) for incompressible flow.
Solution
Set \(\vec{a}=\vec{u}\), \(\vec{b}=\vec{\omega}\): \(\nabla\times(\vec{u}\times\vec{\omega})=\vec{u}(\nabla\cdot\vec{\omega})-\vec{\omega}(\nabla\cdot\vec{u})+(\vec{\omega}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{\omega}\). Now \(\nabla\cdot\vec{\omega}=\partial_i(\epsilon_{ijk}\partial_j u_k)=\epsilon_{ijk}\partial_i\partial_j u_k=0\) since \(\epsilon_{ijk}\) is antisymmetric in \(i,j\) while \(\partial_i\partial_j\) is symmetric. Incompressibility gives \(\nabla\cdot\vec{u}=0\). Both middle terms vanish, leaving \((\vec{\omega}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{\omega}\).
A vortex tube of circulation \(\Gamma\) and cross-sectional area \(A\) is stretched so its length doubles at constant volume. Using conservation of circulation and \(\Gamma=\omega A\) for a thin tube, find the factor by which the mean vorticity \(\omega\) changes.
Solution
Constant volume with length doubling means the cross-section halves: \(A\to A/2\). Kelvin's theorem (inviscid) keeps \(\Gamma\) fixed. Since \(\Gamma=\omega A\), \(\omega=\Gamma/A\to\Gamma/(A/2)=2\Gamma/A\). The vorticity doubles. This is vortex stretching: halving the area to conserve volume, at fixed circulation, doubles the spin.
For the strain field of Worked Example 1 with \(a=3\ \mathrm{s^{-1}}\) and \(\omega_0=5\ \mathrm{s^{-1}}\), how long until the vorticity reaches \(100\ \mathrm{s^{-1}}\)? Ignore diffusion.
Solution
\(\omega_z=\omega_0 e^{a t}\Rightarrow t=\frac{1}{a}\ln(\omega_z/\omega_0)=\frac{1}{3}\ln(100/5)=\frac{1}{3}\ln 20=\frac{1}{3}(2.996)=0.999\ \mathrm{s}\approx1.0\ \mathrm{s}\). About one second of stretching amplifies the vorticity twenty-fold.
Estimate the viscous diffusion time for vorticity in a boundary layer of thickness \(\delta=2\ \mathrm{mm}\) in air, \(\nu=1.5\times10^{-5}\ \mathrm{m^2\,s^{-1}}\). Compare with the same thickness in water, \(\nu=1.0\times10^{-6}\ \mathrm{m^2\,s^{-1}}\).
Solution
Diffusion time \(\tau\sim\delta^2/\nu\). Air: \(\tau=(2\times10^{-3})^2/(1.5\times10^{-5})=4\times10^{-6}/1.5\times10^{-5}=0.27\ \mathrm{s}\). Water: \(\tau=(2\times10^{-3})^2/(1.0\times10^{-6})=4\times10^{-6}/1.0\times10^{-6}=4.0\ \mathrm{s}\). Vorticity diffuses about \(15\times\) faster in air, consistent with \(\nu_{\text{air}}/\nu_{\text{water}}\approx15\).
Explain, using the vorticity equation, why the drain vortex in a bathtub can spin up dramatically as water converges toward the plughole, and estimate the amplification if the radius of a fluid ring shrinks from \(r_1=0.20\ \mathrm{m}\) to \(r_2=0.01\ \mathrm{m}\).
Solution
Radial inflow provides a converging strain that stretches the vertical vortex lines; \((\vec{\omega}\cdot\nabla)\vec{u}\) amplifies \(\omega_z\). Quantitatively, conservation of angular momentum for a fluid ring, \(\omega r^2=\text{const}\) (equivalently constant circulation \(\Gamma=2\pi r v_\theta\) with \(v_\theta=\tfrac12\omega r\) for solid-body cores, giving \(\omega\propto r^{-2}\)), yields \(\omega_2/\omega_1=(r_1/r_2)^2=(0.20/0.01)^2=20^2=400\). The vorticity increases roughly four-hundred-fold, which is why the surface swirl becomes visible only near the drain. Viscous diffusion eventually caps this by spreading the core.