Kelvin's Circulation Theorem

Derivation

Kelvin's Circulation Theorem

D-322 Home PU-307 Threads symmetry · fields Depends on Euler's Equations as the Inviscid Limit, The Vorticity Transport Equation, stokes-theorem

Statement

In a barotropic, inviscid fluid subject only to conservative body forces, the circulation \(\Gamma=\oint_{C(t)}\vec{u}\cdot d\vec{l}\) around any closed material loop \(C(t)\) — a loop composed of the same fluid particles at all times — is conserved: \(\dfrac{D\Gamma}{Dt}=0\).

Why it matters

Kelvin's theorem is the conservation law that underpins the whole edifice of ideal-flow theory. It tells us that vorticity cannot be created or destroyed inside a barotropic, inviscid fluid: it can only be advected, stretched, and tilted. A flow that starts irrotational stays irrotational, which is precisely the licence that justifies potential-flow theory, the Kutta condition, and the entire aerodynamics of thin airfoils.

It also sharpens our intuition about where vorticity does come from in the real world. Because the theorem fails exactly when the flow is baroclinic, or viscous, or forced non-conservatively, its breakdown clauses form a checklist of the true sources of rotation: density fronts, boundary layers, and body-force curl. It is the fluid-mechanical cousin of angular-momentum conservation.

Assumptions

Inviscid flow (Euler dynamics).With viscosity the momentum equation carries a \(\nu\nabla^2\vec{u}\) term whose loop integral does not vanish; viscous diffusion of vorticity across the material loop makes \(D\Gamma/Dt\neq0\).

Barotropic fluid, \(\rho=\rho(p)\).If \(\rho\) depends on more than pressure (e.g. also on temperature), \(\frac{1}{\rho}\nabla p\) is no longer a gradient, the pressure loop integral survives, and baroclinic torque generates circulation.

Conservative body force, \(\vec{f}=-\nabla\Phi\).A body force with non-zero curl (e.g. an unbalanced Coriolis or Lorentz force) contributes \(\oint\vec{f}\cdot d\vec{l}=\int(\nabla\times\vec{f})\cdot d\vec{A}\neq0\), directly forcing the circulation.

The loop is a material loop.If \(C\) were fixed in space rather than moving with the fluid, the second (loop-deformation) term would be absent and one would instead track \(\partial\Gamma/\partial t\), which is not conserved even in ideal flow.

Single-valued, smooth potentials on \(C\).If \(\Phi\) or the pressure function \(\mathcal{P}\) is multi-valued (branch cuts, sources threading the loop), \(\oint\nabla\Phi\cdot d\vec{l}\) need not vanish and the "perfect-differential" arguments fail.

Derivation

\[ \Gamma(t)=\oint_{C(t)}\vec{u}\cdot d\vec{l},\qquad \frac{D\Gamma}{Dt}=\frac{D}{Dt}\oint_{C(t)}\vec{u}\cdot d\vec{l}. \]

Define circulation on the material loop and take its material (Lagrangian) rate of change, following the same fluid particles. A

\[ \frac{D\Gamma}{Dt}=\oint_{C}\frac{D\vec{u}}{Dt}\cdot d\vec{l}\;+\;\oint_{C}\vec{u}\cdot\frac{D(d\vec{l})}{Dt}. \]

Product rule on the integrand; the domain \(C(t)\) is carried by the flow, so differentiation acts on both the velocity and the material line element. B

\[ \frac{D(d\vec{l})}{Dt}=(d\vec{l}\cdot\nabla)\vec{u}=d\vec{u}. \]

A material line element is stretched and rotated by the local velocity gradient; the endpoints move at their own velocities, so the element changes at the rate of the velocity difference across it. C

\[ \oint_{C}\vec{u}\cdot\frac{D(d\vec{l})}{Dt}=\oint_{C}\vec{u}\cdot d\vec{u}=\oint_{C} d\!\left(\tfrac{1}{2}\,|\vec{u}|^2\right)=0. \]

The second term is the loop integral of a perfect differential; the integral of any exact differential around a closed curve vanishes. B

\[ \frac{D\vec{u}}{Dt}=-\frac{1}{\rho}\nabla p-\nabla\Phi. \]

Substitute the Euler equation for inviscid flow with conservative body force \(\vec{f}=-\nabla\Phi\); this is the assumed prior result. A

\[ \frac{D\Gamma}{Dt}=-\oint_{C}\frac{1}{\rho}\nabla p\cdot d\vec{l}\;-\;\oint_{C}\nabla\Phi\cdot d\vec{l}. \]

Insert Step 5 into the surviving first term of Step 2 (the second term is already zero by Step 4). A

\[ \oint_{C}\nabla\Phi\cdot d\vec{l}=\oint_{C} d\Phi=0. \]

The body-force term is the loop integral of the exact differential \(d\Phi\); it vanishes for a single-valued potential around a closed loop. B

\[ \rho=\rho(p)\;\Rightarrow\;\frac{1}{\rho}\nabla p=\nabla\mathcal{P},\qquad \mathcal{P}(p)\equiv\int^{p}\frac{dp'}{\rho(p')}. \]

Barotropy makes \(1/\rho\) a function of \(p\) alone, so \(\frac{1}{\rho}\nabla p\) is the gradient of a single-valued pressure function \(\mathcal{P}\). C

\[ \oint_{C}\frac{1}{\rho}\nabla p\cdot d\vec{l}=\oint_{C} d\mathcal{P}=0. \]

The pressure term is now also a loop integral of a perfect differential, hence zero. B

\[ \boxed{\;\frac{D\Gamma}{Dt}=0\;} \]

Both surviving terms vanish; the circulation on any material loop is constant in time. A

Result

\[ \frac{D}{Dt}\oint_{C(t)}\vec{u}\cdot d\vec{l}=0 \qquad\Longleftrightarrow\qquad \frac{D}{Dt}\!\int_{S(t)}\vec{\omega}\cdot d\vec{A}=0 \]

Reading. The velocity circulation around a loop of marked fluid particles never changes, no matter how violently the loop is stretched, twisted, or advected. By Stokes' theorem the same statement is that the vorticity flux through any material surface bounded by the loop is frozen — the geometric content behind "vortex lines move with the fluid" (Helmholtz's theorem).

Units check. \(\Gamma=[\text{velocity}]\times[\text{length}]=\mathrm{m\,s^{-1}}\cdot\mathrm{m}=\mathrm{m^2\,s^{-1}}\). Then \(D\Gamma/Dt=\mathrm{m^2\,s^{-2}}\), which matches each dropped term: \(\oint\frac{1}{\rho}\nabla p\cdot d\vec{l}=\frac{1}{\mathrm{kg\,m^{-3}}}\cdot\frac{\mathrm{Pa}}{\mathrm{m}}\cdot\mathrm{m}=\frac{\mathrm{kg\,m^{-1}s^{-2}}}{\mathrm{kg\,m^{-3}}}=\mathrm{m^2\,s^{-2}}\). Consistent.

Limiting cases

Incompressible flow (\(\rho=\text{const}\)): a trivial special case of barotropy, \(\mathcal{P}=p/\rho\); the theorem holds exactly.
Initially irrotational flow (\(\vec{\omega}=0\) at \(t=0\)): every material loop has \(\Gamma=0\) forever, so \(\vec{\omega}\equiv0\) persists — the persistence of irrotationality.
Vortex-tube stretching: as a material loop encircling a tube contracts in area, conserved \(\Gamma\) forces the enclosed vorticity to rise, \(\omega_2/\omega_1=A_1/A_2\).
Steady flow: the theorem still refers to the moving loop, not a fixed one; \(\Gamma\) is conserved along particle paths even when \(\partial\vec{u}/\partial t=0\).
Weak baroclinicity: for small \(\nabla\rho\times\nabla p\) the circulation drifts slowly, \(D\Gamma/Dt=-\oint\frac{dp}{\rho}\approx\int_S\frac{\nabla\rho\times\nabla p}{\rho^2}\cdot d\vec{A}\), recovering Bjerknes' theorem.

Breaks when

Baroclinic flow. When \(\rho\) is not a function of \(p\) alone (e.g. a horizontal temperature gradient at constant pressure), \(\nabla\rho\times\nabla p\neq0\). The pressure loop integral no longer vanishes and \(D\Gamma/Dt=-\oint\frac{dp}{\rho}=\int_S\frac{1}{\rho^2}(\nabla\rho\times\nabla p)\cdot d\vec{A}\). This is the sea-breeze and thermal-wind engine.
Viscous flow. The added \(\nu\nabla^2\vec{u}\) term gives \(D\Gamma/Dt=\nu\oint\nabla^2\vec{u}\cdot d\vec{l}=\nu\oint\nabla\times\vec{\omega}\cdot(-d\vec{l})\), so vorticity diffuses across the material loop, especially in boundary layers where circulation is generated at solid walls.
Non-conservative body forces. A force with \(\nabla\times\vec{f}\neq0\) — an unbalanced Coriolis force in a rotating frame, a Lorentz force \(\vec{J}\times\vec{B}\), or surface-tension-driven Marangoni forcing — injects circulation directly: \(D\Gamma/Dt=\int_S(\nabla\times\vec{f})\cdot d\vec{A}\).
Loops crossing shocks or discontinuities. Across a curved shock, entropy jumps vary along the front, the flow becomes baroclinic (Crocco's theorem), and vorticity is produced — Kelvin's theorem does not survive the discontinuity.

Failure modes

Fixed-loop error: applying \(D\Gamma/Dt=0\) to a loop pinned in space. The theorem holds only for material loops that move with the fluid; a spatially fixed loop has flux entering and leaving it.
"Circulation is always conserved": forgetting the barotropic and inviscid hypotheses and asserting conservation in a real, stratified, viscous atmosphere or ocean — where baroclinic and viscous generation are the whole point.
Dropping the second term prematurely: assuming \(\oint\vec{u}\cdot D(d\vec{l})/Dt\) is "obviously" zero without recognising it is the integral of an exact differential \(d(\tfrac12 u^2)\); the reason matters, and it fails if \(\vec{u}\) is multi-valued.
Confusing conservation of \(\Gamma\) with conservation of \(\vec{\omega}\): vorticity is emphatically not constant — it intensifies under stretching. Only the flux \(\oint\vec{u}\cdot d\vec{l}\) is fixed.
Barotropic vs. isentropic muddle: claiming any adiabatic flow is barotropic. Isentropy gives \(\rho=\rho(p,s)\); it reduces to barotropy only when \(s\) is uniform.
Sign slip in the pressure term: writing \(+\frac{1}{\rho}\nabla p\) in Euler's equation, which flips the (already vanishing) contribution and confuses the baroclinic generation sign when the theorem is generalised.

Discussion

Kelvin's theorem is best read as the statement that vorticity is a material property of an ideal barotropic fluid: vortex lines are painted onto fluid particles and carried with them, never diffusing and never being cut. This is the symmetry-thread content of the result. Circulation is the Noether-like invariant associated with the particle-relabelling symmetry of ideal-fluid dynamics; it is the topological glue that keeps vortex tubes intact and gives knotted vortex lines their conserved helicity. When we later study Helmholtz's vortex theorems, they are simply geometric restatements of this one conservation law.

The fields thread appears through Stokes' theorem, which converts the line invariant into a surface-flux invariant. Because \(\int_S\vec{\omega}\cdot d\vec{A}\) is frozen while the surface \(S\) deforms with the fluid, a shrinking cross-section must be compensated by growing vorticity. This is exactly the mechanism by which a tornado, a bathtub vortex, or a stretched vortex filament in turbulence spins up — the fluid analogue of a skater pulling in her arms, but here it is a field flux, not a rigid-body angular momentum, that is conserved.

The most illuminating physics is often in the breakdown. Bjerknes generalised Kelvin's theorem to \(D\Gamma/Dt=-\oint dp/\rho\), and this baroclinic term is the source of nearly all large-scale rotation in the atmosphere and ocean: sea breezes, monsoon circulations, and the thermal wind all arise because isobars and isopycnals are misaligned. The theorem thus does double duty — where it holds, it forbids spontaneous rotation; where it fails, it names the culprit precisely.

At the deepest level the theorem reflects the Hamiltonian, symplectic structure of the Euler equations. Circulation is a Casimir-like invariant of the ideal-fluid Poisson bracket, conserved for any Hamiltonian of the barotropic form because it commutes with the entire particle-relabelling gauge group. This is why viscosity — a genuinely dissipative, non-Hamiltonian addition — is required to break it, and why the theorem is so robust to the details of the equation of state so long as barotropy holds. Crocco's theorem, \(\vec{u}\times\vec{\omega}=\nabla h_0-T\nabla s\), exposes the same structure from the energetic side: uniform stagnation enthalpy and uniform entropy are exactly the barotropic conditions under which no vorticity is generated.

Common misconceptions. Students often believe Kelvin's theorem says vorticity is conserved — it does not; it says circulation (the flux of vorticity through a co-moving surface) is conserved, which permits and indeed requires vorticity intensification under stretching. A second frequent error is to think the theorem applies to real air and water "well enough" everywhere; in fact its most important consequence for geophysics is the size of its failure, the baroclinic term, without which weather would have no engine.

Worked examples

\[ \text{Vortex spin-up: a fluid ring of radius }R_1\text{ swirls at }u_1\text{, then contracts to }R_2. \]

Barotropic, inviscid ring; circulation on this material loop is conserved. Set up symbolically before numbers. A

\[ \Gamma=\oint\vec{u}\cdot d\vec{l}=2\pi R\,u_\theta,\qquad \Gamma_1=\Gamma_2\;\Rightarrow\;2\pi R_1 u_1=2\pi R_2 u_2. \]

Axisymmetric loop, so \(\vec{u}\cdot d\vec{l}=u_\theta\,dl\); apply \(D\Gamma/Dt=0\). A

\[ u_2=\frac{R_1}{R_2}\,u_1. \]

Solve for the final swirl speed; rearrange symbols before substituting. A

\[ R_1=1\ \mathrm{m},\ u_1=0.5\ \mathrm{m\,s^{-1}},\ R_2=0.2\ \mathrm{m}:\quad \Gamma=2\pi(1)(0.5)=\pi\approx3.14\ \mathrm{m^2 s^{-1}},\quad u_2=\frac{1}{0.2}(0.5). \]

Insert numbers with units; \(\Gamma\) is the conserved quantity. A

\[ \Gamma\approx3.14\ \mathrm{m^2\,s^{-1}}\ \text{(conserved)},\qquad u_2=2.5\ \mathrm{m\,s^{-1}}. \]

Reading. Shrinking the ring fivefold quintuples the swirl speed while the circulation stays pinned at \(\pi\ \mathrm{m^2 s^{-1}}\) — the vortex spins up exactly as Kelvin's theorem demands.

Units check. \(u_2=\frac{\mathrm{m}}{\mathrm{m}}\cdot\mathrm{m\,s^{-1}}=\mathrm{m\,s^{-1}}\). Correct.

\[ \text{Starting vortex: an airfoil (chord }c\text{, span-wise 2-D) accelerates from rest to }U\text{ in still fluid.} \]

Take a large material loop enclosing the airfoil and the fluid it will disturb; initially at rest so \(\Gamma_{\text{tot}}=0\). B

\[ \frac{D\Gamma_{\text{tot}}}{Dt}=0\;\Rightarrow\;\Gamma_{\text{tot}}=\Gamma_{\text{bound}}+\Gamma_{\text{start}}=0\;\Rightarrow\;\Gamma_{\text{start}}=-\Gamma_{\text{bound}}. \]

Kelvin's theorem on the material loop; any bound circulation must be balanced by an equal, opposite shed vortex. B

\[ L=\rho U\,\Gamma_{\text{bound}}=\tfrac12\rho U^2 c\,C_L\;\Rightarrow\;\Gamma_{\text{bound}}=\tfrac12 U c\,C_L. \]

Kutta–Joukowski lift equated to the definition of \(C_L\); solve symbolically for bound circulation. B

\[ U=30\ \mathrm{m\,s^{-1}},\ c=1.5\ \mathrm{m},\ C_L=0.6:\quad \Gamma_{\text{bound}}=\tfrac12(30)(1.5)(0.6). \]

Insert numbers with units. B

\[ \Gamma_{\text{bound}}=13.5\ \mathrm{m^2\,s^{-1}},\qquad \Gamma_{\text{start}}=-13.5\ \mathrm{m^2\,s^{-1}}. \]

Reading. The lift-generating bound circulation of \(13.5\ \mathrm{m^2 s^{-1}}\) is paid for by an equal and opposite starting vortex shed into the wake, keeping the total circulation on the material loop exactly zero — Kelvin's theorem is the reason wings shed a starting vortex at all.

Units check. \(\tfrac12 U c C_L=\mathrm{m\,s^{-1}}\cdot\mathrm{m}\cdot(\text{dimensionless})=\mathrm{m^2\,s^{-1}}\). Correct.

Problems

(A) A fluid in rigid-body rotation has \(u_\theta=\Omega r\) with \(\Omega=2\ \mathrm{s^{-1}}\). Find the circulation around a circle of radius \(r=3\ \mathrm{m}\).
Solution
\(\Gamma=\oint\vec{u}\cdot d\vec{l}=u_\theta(2\pi r)=(\Omega r)(2\pi r)=2\pi\Omega r^2=2\pi(2)(3^2)=36\pi\approx113\ \mathrm{m^2\,s^{-1}}\). (Equivalently \(\Gamma=\omega\cdot\pi r^2\) with \(\omega=2\Omega=4\ \mathrm{s^{-1}}\): \(4\cdot9\pi=36\pi\), agreeing.)
(B) An incompressible vortex tube is stretched so its length doubles; by continuity its cross-sectional area halves from \(A_1=0.4\ \mathrm{m^2}\) to \(A_2=0.2\ \mathrm{m^2}\). If the initial axial vorticity is \(\omega_1=5\ \mathrm{s^{-1}}\), find \(\omega_2\).
Solution
Circulation on the material loop bounding the tube is conserved: \(\Gamma=\omega_1 A_1=\omega_2 A_2\). So \(\omega_2=\omega_1 A_1/A_2=5\times(0.4/0.2)=10\ \mathrm{s^{-1}}\). Halving the area doubles the vorticity while \(\Gamma=\omega_1A_1=2\ \mathrm{m^2 s^{-1}}\) stays fixed.
(B) A sea-breeze circuit rises through cold air (\(T_c=283\ \mathrm{K}\)) and sinks through warm air (\(T_w=293\ \mathrm{K}\)) between pressure levels \(p_0=1000\ \mathrm{hPa}\) (bottom) and \(p_1=900\ \mathrm{hPa}\) (top). Using \(1/\rho=RT/p\) with \(R=287\ \mathrm{J\,kg^{-1}K^{-1}}\), estimate the baroclinic circulation-generation rate \(D\Gamma/Dt=-\oint dp/\rho\).
Solution
\(-\oint\frac{dp}{\rho}=-\oint RT\,d(\ln p)=R(T_w-T_c)\ln\!\frac{p_0}{p_1}\). Numerically \(\ln(1000/900)=\ln1.111=0.1054\), so \(D\Gamma/Dt=287\times(293-283)\times0.1054=287\times10\times0.1054\approx302\ \mathrm{m^2\,s^{-2}}\). Positive, i.e. a thermally direct (rising-warm, sinking-cold in the sense of the circuit) circulation is spun up — exactly the sea breeze, and a case where Kelvin's theorem fails by baroclinicity.
(C) Prove that a barotropic, inviscid flow which is irrotational everywhere at \(t=0\) remains irrotational for all time.
Solution
At \(t=0\), Stokes' theorem gives \(\Gamma=\oint_C\vec{u}\cdot d\vec{l}=\int_S\vec{\omega}\cdot d\vec{A}=0\) for every closed loop, since \(\vec{\omega}=0\). Kelvin's theorem gives \(D\Gamma/Dt=0\), so \(\Gamma(t)=0\) on every material loop for all \(t\). If some region had \(\vec{\omega}\neq0\) at a later time, a small loop around it would enclose nonzero flux, contradicting \(\Gamma=0\). Hence \(\vec{\omega}\equiv0\) persists, which is the licence for potential-flow theory.
(C) Show explicitly that the loop-deformation term vanishes, i.e. \(\oint_C\vec{u}\cdot\frac{D(d\vec{l})}{Dt}=0\), using \(\frac{D(d\vec{l})}{Dt}=(d\vec{l}\cdot\nabla)\vec{u}\).
Solution
Write \(\vec{u}\cdot\frac{D(d\vec{l})}{Dt}=\vec{u}\cdot(d\vec{l}\cdot\nabla)\vec{u}=\sum_{i}u_i\,(d\vec{l}\cdot\nabla)u_i=d\vec{l}\cdot\nabla\!\left(\tfrac12\sum_i u_i^2\right)=d\!\left(\tfrac12|\vec{u}|^2\right)\), because \((d\vec{l}\cdot\nabla)f\) is precisely the differential \(df\) along the element \(d\vec{l}\). Then \(\oint_C d\!\left(\tfrac12|\vec{u}|^2\right)=0\), as the loop integral of an exact differential of a single-valued function. This confirms Step 4 of the derivation and isolates the circulation change entirely into the momentum (Euler) term.