What survives being wound up

Circulation is the integral of the velocity round a closed curve, and there are two entirely different questions to ask about it. One is what happens to the circulation round a curve fixed in space as the flow changes through it. The other is what happens to the circulation round a curve made of fluid particles, carried along and deformed by the flow itself.

Only the second has a theorem, and it is the most useful one in the subject.

The statement, and what it costs #

Kelvin’s theorem: in a fluid that is inviscid, barotropic, and driven only by conservative body forces, the circulation round a material loop does not change.

The proof is three lines and each line spends one hypothesis. Differentiating $\Gamma = \oint \mathbf{u}\cdot\mathrm{d}\boldsymbol{\ell}$ following the fluid gives two terms: the acceleration of the fluid, and the rate at which the line element is stretched. The second integrates to zero round any closed curve, because it is the integral of $\mathrm{d}(\tfrac{1}{2}|\mathbf{u}|^2)$. The first is Euler’s equation, and it leaves

$$\frac{\mathrm{D}\Gamma}{\mathrm{D}t} = -\oint\frac{\mathrm{d}p}{\rho} + \oint\mathbf{F}\cdot\mathrm{d}\boldsymbol{\ell}.$$

Now: viscosity would have put a term here and did not, because the fluid is ideal. The first integral vanishes if $\rho$ is a function of $p$ alone — that is what barotropic means, and it makes the integrand an exact differential. The second vanishes if the body force is the gradient of a potential, which is what conservative means and which gravity is.

Everything interesting in this essay is what happens when one of those three sentences is false.

Testing it on something that solves the equations #

A conservation law is worth exactly as much as the test that could have caught it failing, so the flow used here is chosen carefully. It is $\psi = \sin x\sin y$, whose vorticity is $2\psi$ — a function of the stream function alone, which makes it an exact steady solution of Euler’s equations rather than a plausible velocity field. Testing Kelvin’s theorem on a field that solves nothing would prove nothing.

The loop is 240 marked particles advected by fourth-order Runge–Kutta, remeshed as it stretches, and its circulation is the line integral round the polygon they make at each instant. Nothing is compared against a formula.

The drift, and why it is quoted #

A conserved quantity computed to four figures with no error estimate beside it is a claim about arithmetic wearing the clothes of a claim about physics. The refinement test is what separates them, and it is cheap: run the same thing twice at different resolutions and see whether the discrepancy cares.

The area check is the other half. A material area in an incompressible flow is exactly conserved, and the markers hold it to 1.5 × 10⁻⁴ — so the polygon is tracking the fluid rather than drifting, and the circulation result is not being carried by an integrator that has quietly stopped following anything.

Breaking it, first: viscosity #

This is the cleanest demonstration available, because the loop does nothing at all. It is not stretched, not folded, not moved off station: it is the same circle throughout. And its circulation collapses.

What changed is that vorticity crossed it. In an ideal fluid vorticity is stuck to the fluid — that is Helmholtz’s theorem, and it is Kelvin’s theorem in another dress — so a material loop keeps whatever vorticity it enclosed at the start. Viscosity is the one mechanism that can move vorticity relative to the fluid, and the amount that leaks across is what the figure measures.

The two routes agree to 5 × 10⁻¹⁴: the line integral round the material circle, and the closed form $\Gamma(1 - e^{-r^2/4\nu t})$.

Breaking it, second: a fluid that is not barotropic #

The condition Kelvin needs is that the pressure and density surfaces coincide, so that $\oint \mathrm{d}p/\rho$ is the integral of an exact differential round a closed curve and therefore zero. Let them cross and the integral is the volume of the “solenoid” between them — which is why this term has been called the solenoidal term since Bjerknes named it in 1898.

Three things in the world are made by it and they are not small:

A sea breeze. Land warms faster than water, the air over it is lighter, and the pressure surfaces tilt against the density surfaces exactly as in the figure. The result is a circulation that starts from a still atmosphere, and the rate computed here — 2.6 m²/s per second for a thirty per cent density contrast over a metre — scales to something that spins up a coastal breeze in an hour.

Vorticity behind a curved shock. A shock of varying strength leaves entropy varying across the flow behind it, so the density and pressure gradients are no longer parallel and vorticity is generated in a flow that had none. This is why a blunt-body shock leaves a rotational flow behind it and why the potential-flow methods this site relies on stop applying there.

Every buoyancy-driven flow there is. Convection, plumes, thermals and the whole of atmospheric circulation begin as baroclinic torque, and calling it “hot air rises” hides the mechanism: what rises is a circulation generated because two families of surfaces are not parallel.

Breaking it, third: a force that is not a gradient #

Gravity drops out of Kelvin’s proof because it is the gradient of a potential. The Coriolis force is not. In a rotating frame the conserved quantity is $\Gamma + fA$, and any ring that changes the area it encloses changes its own circulation to compensate.

A kilometre-wide ring at 45° latitude, contracted to a quarter of its radius, acquires 294 m²/s of circulation and 0.19 m/s of swirl. That is the mechanism behind a tornado and a hurricane, and it is emphatically not the mechanism behind a bath vortex, where the same arithmetic gives a planetary contribution some four hundred times smaller than the residual stirring.

Two different questions about the same integral #

It is worth being explicit about the distinction the first paragraph made, because the two questions have opposite answers in the same flow.

Take the diffusing vortex again and ask about a circle of radius 20 mm fixed in the laboratory. Its circulation falls, because vorticity is leaving the region it encloses. Now ask about the material circle of the same radius — and in this particular flow they are the same curve, since nothing moves radially, so the answers agree. That coincidence is what makes the vortex such a clean demonstration and it is not general.

In the cellular flow the two questions come apart completely. A fixed circle has a circulation that never changes either, because the flow is steady and nothing about it depends on time. The material loop’s constancy is a much stronger statement: the curve is a different curve at every instant, sampling different fluid, wound into a shape with no resemblance to what it started as, and the integral over it is the same number. The Eulerian answer is a triviality about a steady field; the Lagrangian one is a theorem.

Why a material loop is hard to draw #

The numerical side of this is worth recording, because it decided which flow the figures use.

A material loop in an unbounded flow past a body is destroyed within a few chords. The markers that pass close to the body race away from those out in the free stream, and the loop becomes a filament thousands of times longer than it started — measured here, before this essay’s configuration was chosen, at a stretch factor of 9,500 by the time a loop released ahead of a cylinder had passed it. At that point the polygon is not resolving a curve at all, its enclosed area is meaningless, and its circulation is arithmetic rather than physics.

Remeshing helps and does not solve it: markers are inserted wherever an edge has grown past 1.5 times the average, which keeps the loop resolved for a while at a cost in points that grows with the stretching. It also introduces a check worth having, because inserting a marker changes the polygon and must not change the circulation — a remesh that moved the answer would be a bug that looked exactly like a physical effect, appearing suddenly and only in the runs that stretched furthest. The figures above are drawn from runs that inserted several hundred markers, and the drift they show is the same drift the un-remeshed short runs show. The general shape of the problem is that Lagrangian methods lose resolution at exactly the rate the flow stretches material, which in a turbulent flow is exponential — the reason particle-tracking approaches to turbulence are so much harder than they look.

What the theorem is for #

It is easy to read Kelvin’s theorem as a statement that nothing interesting can happen, which is the opposite of how it is used. Its real work is done by contradiction.

The starting vortex. A wing at rest has no circulation round any loop. Take a huge material loop enclosing the wing and start the wing moving: the circulation round that loop must still be zero. The wing now has circulation, because that is what lift is; therefore something with equal and opposite circulation must have been left behind, and it has — the starting vortex is a deduction from this theorem rather than an observation.

Why an aerofoil needs a sharp trailing edge. Circulation cannot be generated inside an ideal fluid, so the only way a wing gets any is through the viscous layer at the trailing edge. The Kutta condition is a statement about where Kelvin’s theorem is allowed to fail, and this is the sharpest way to say why an ideal-flow theory of lift needs viscosity to be present and does not need it to be computed.

Why vortex lines behave like elastic threads. Helmholtz’s results — that vortex lines move with the fluid, and that a vortex tube’s strength is constant along it and constant in time — follow from Kelvin’s theorem applied to loops drawn round the tube. Everything on this site about vortex rings and point vortices sits on that.

The theorem as a way of asking where the vorticity came from #

There is a habit of thought that Kelvin’s theorem supports and that is worth naming, because it is how the theorem is actually used in practice.

Given a flow with vorticity in it, the theorem says the vorticity did not appear from nowhere: either it was there at the start, or one of the three hypotheses failed somewhere along the path the fluid took. So the question “where did this vorticity come from” always has an answer, and there are only three kinds of answer.

A boundary layer, which is viscosity — the overwhelmingly common case, and the reason every vortex in an aeronautical flow can be traced back to a surface.

A density front, which is baroclinicity — the case for most of what happens in the atmosphere and the ocean, and for the vorticity behind a curved shock.

A rotating frame, which is the non-conservative body force — the case for everything at planetary scale.

Asking which of the three produced a given vortex is a sharper question than asking how the vortex formed, and it usually has a definite answer. That is what a theorem with three hypotheses is for.

The local version, which is the one a forecaster uses #

Kelvin’s theorem is about a loop, and a loop is an awkward object to carry around: it has to be followed, remeshed and integrated over, as the section above found. There is a pointwise descendant, it is stronger where it applies, and it is the central quantity of geophysical fluid mechanics.

The construction is short. Take a scalar the fluid carries unchanged — potential temperature will do, since a parcel moving without heating keeps it — and draw the Kelvin loop inside a surface of constant value of that scalar. Then the barotropic objection disappears: on such a surface the density is a function of the pressure, whatever it does elsewhere. Shrink the loop to a point and what is left is a quantity attached to each parcel,

$$q = \frac{1}{\rho}\left(\boldsymbol\omega + 2\boldsymbol\Omega\right)\cdot\nabla\theta,$$

which is materially conserved: $\mathrm Dq/\mathrm Dt = 0$. That is Ertel’s potential vorticity, and it needs only that the flow be frictionless and the scalar be carried unchanged — the two surviving hypotheses of the three above, with the rotation folded in rather than broken.

Its shallow-water form is the one to hold in mind: $(\zeta + f)/h$, the absolute vorticity divided by the layer’s depth, conserved as the column moves. Everything follows from reading that as a fraction.

Stretch a column and it must spin faster. A layer of air crossing a mountain range is squashed on the way up and stretched on the way down, so its vorticity falls and then rises, and it acquires a spin it did not have — which is the lee trough downstream of every major range, and the train of waves that follows it across the continent.

And it is invertible, which is why it is not merely a conserved label. Given the distribution of potential vorticity, a balance assumption and the conditions on the boundaries, the entire wind and temperature field can be recovered from it. One scalar carries the dynamics.

So the honest closing statement about Kelvin’s theorem is that its integral form is what this essay tested and its local form is what is used. A quantity that has to be followed round a loop became a number printed on a chart, and the step between them was choosing which surface to draw the loop in.

What the model does not contain #

The test flow is two-dimensional and steady. Kelvin’s theorem holds in three dimensions and in unsteady flow, and nothing here checks it there. The reason for a steady, bounded, cellular flow is practical: a material loop in an unbounded flow past a body is torn to shreds within a few chords — the markers nearest the body race away from those in the free stream — and the polygon stops resolving anything long before the physics does. That failure was measured before this configuration was chosen, at a stretch factor of 9,500 by the time the loop had passed a cylinder.

The three failures are computed separately and never together. A real flow can be viscous, baroclinic and rotating at once, and their effects are not additive in general.

The baroclinic calculation is a rate, not a flow. It says how fast circulation is generated at the first instant, from a fluid at rest. What happens next — the front slumping, the circulation organising into a current — is a nonlinear problem nothing here solves.

Nothing is said about turbulence. The theorem holds instantaneously in a turbulent flow too, but material loops in turbulence stretch exponentially rather than linearly, and the numerical test above would fail on resolution within a few eddy turnovers. That failure would be the arithmetic’s, and distinguishing it from a physical one is exactly the problem the refinement test exists for.

Who found it, and when #

William Thomson, later Lord Kelvin, published the theorem in 1869, in the middle of the decade he and Helmholtz spent on vortex motion — the same decade in which he proposed that atoms were knotted vortex rings in the ether, an idea that was wrong and that generated the mathematical field of knot theory as a side effect.

Helmholtz’s vortex theorems came first, in 1858, and are equivalent to Kelvin’s in an ideal fluid; Kelvin’s formulation is the one that generalises, because it isolates the three hypotheses instead of assuming them. Bjerknes added the baroclinic term in 1898 while trying to explain atmospheric circulation, which is why the meteorological literature calls the whole result the Bjerknes circulation theorem and the fluid-dynamical literature calls it Kelvin’s.

Where the ladder goes next #

This essay’s loop was carried by a steady flow and stretched linearly. Make the flow depend on time — in the simplest way possible, by switching between two steady flows — and the stretching becomes exponential, neighbouring particles separate at a rate that can be measured, and the trajectories become unpredictable in a flow field with no randomness in it anywhere.