The shape a vortex keeps

A point vortex is a patch of uniform vorticity taken to zero size at fixed circulation. That sentence is a definition, and like most definitions in this subject it hides a limit.

The limit looks free, and the reason it looks free is a genuinely surprising fact: outside a circular patch the velocity field is the point vortex’s exactly. Not approximately, not to leading order — the same function. A Rankine vortex of radius $a$ and circulation $\Gamma$ has $u_\theta = \Gamma/2\pi r$ at every $r > a$, and the disagreement measured here on a discretised contour is $3.2\times10^{-6}$, which is the discretisation.

So a computation can carry a circulation about as a point and be right, and this collection does it repeatedly. What it cannot do is carry a shape about as a point, and this essay is about the three things the shape turns out to be.

The machinery, which is a boundary and nothing else #

In two dimensions a region of uniform vorticity moves with the fluid and stays uniform: the vorticity equation for an inviscid plane flow is $D\omega/Dt = 0$, so a patch’s interior cannot develop structure. That makes the edge the whole state, and the velocity a contour integral:

$$u = -\frac{\omega}{2\pi}\oint \ln|x - x'|\,dx', \qquad v = -\frac{\omega}{2\pi}\oint \ln|x - x'|\,dy'.$$

That is contour dynamics. There is no grid in it, no viscosity and no model — it is the exact consequence of the vorticity being constant inside a curve, and the curve being material.

The one delicacy is the panel containing the evaluation point, where the logarithm is singular and integrable. That panel is done in closed form, and there are two closed forms rather than one: $h(\ln(h/2) - 1)$ when the evaluation point is at a panel’s midpoint and $h(\ln h - 1)$ when it is at a panel’s end. Using the first where the second belongs leaves a bias which is invisible in any single picture and which showed up, in the neighbouring essay, as a co-rotating pair losing a quarter of its own area per orbit — in a scheme whose entire claim is that the area is exactly conserved.

Kirchhoff’s ellipse, which turns without changing #

The first thing a shape has that a point does not is a rotation rate.

An ellipse of semi-axes $a$ and $b$ filled with uniform vorticity $\omega$ is an exact steady solution in a rotating frame: it turns rigidly, without changing shape, for ever. Its interior flow is exactly linear —

$$u = -\frac{\omega a}{a+b}\,y, \qquad v = \frac{\omega b}{a+b}\,x$$

— which the contour integral reproduces to 8.2 parts in ten million, and its angular velocity is

$$\Omega = \frac{\omega ab}{(a+b)^2} = \omega\,\frac{\lambda}{(1+\lambda)^2}.$$

The test that this is really the rotation rate is the boundary’s normal velocity: a tangential difference is a relabelling of the same curve, so what has to match is the component along the outward normal. It matches to 1.1 parts in a million of $\Omega a$.

Read the curve for what it says about the limit. $\Omega/\omega$ is one quarter for a circle — where it is unobservable, because the shape is axisymmetric — and falls monotonically as the patch is elongated. A point vortex has no entry on this axis at all, because it has no $\lambda$.

Why the interior is linear, which is worth a paragraph #

The interior flow being exactly linear is not an accident of the ellipse and it is the reason the whole family exists. Inside a region of uniform vorticity the stream function satisfies $\nabla^2\psi = -\omega$ with $\omega$ constant, so $\psi$ is a particular quadratic plus a harmonic function. For an ellipse the boundary condition can be met with the quadratic alone, which makes the velocity linear in position and therefore a rigid rotation plus a pure strain — and a rigid rotation plus a pure strain acting on an ellipse produces another ellipse.

That is why the shape is preserved, and it is why no other simple closed curve is. A patch bounded by anything but an ellipse has a harmonic part in its interior stream function, the interior flow is nonlinear, and the shape changes. The Kirchhoff family is the complete set of steady uniform patches in an otherwise still fluid, and it is two-parameter: an aspect ratio and a size.

The check that this is more than algebra is the one run above. The contour integral knows nothing about ellipses — it is a sum over panels of a logarithm — and it returns the linear field.

Moore and Saffman’s limit, which the point vortex does not have #

The second thing is a threshold.

Put the patch in an external strain of rate $e$. There is a steady elliptical patch for every strain up to a maximum, and the relation between them is

$$\frac{e}{\omega} = \frac{\lambda(\lambda-1)}{(\lambda+1)(\lambda^2+1)}.$$

That curve rises, turns over and falls. Above its maximum there is no steady shape of any aspect ratio: the patch is drawn out into a filament and the model has nothing to offer.

The maximum is found here by golden section on the curve, at

$$\frac{e}{\omega} = 0.150142 \quad\text{at}\quad \lambda = 2.890,$$

which are the numbers usually quoted as 0.15 and 2.9. A point vortex has no such limit. It survives any strain whatever, because it has no shape to lose, and a calculation built on point vortices will therefore go on producing a plausible answer in exactly the circumstances where the real vortices in it have been destroyed.

That is the refutation this essay carries, and it is not an academic one: what a point vortex is not takes the same machinery to a co-rotating pair, where the strain each vortex feels is the other’s, and finds the separation below which the pair cannot exist as two vortices at all.

How far away a patch has to be to be a point #

The third thing is a rate of convergence, and it is worth measuring because the usual answer — “far enough” — is not a number.

For an elliptical patch of aspect ratio 3 and unit area, the worst departure on a circle is 27 per cent at two radii, 4.6 per cent at four, and 0.26 per cent at sixteen. The exponent measured across the sweep is $r^{-2.03}$ — the quadrupole, which is the first multipole a circle and an ellipse of the same circulation do not share.

What the far field remembers makes the general version of this argument: three numbers survive the journey to infinity and nothing else about a body does, so two shapes with nothing in common can produce the same flow a few radii away. The patch is that argument with the roles reversed. The far field is the same because the circulation is the same; everything about the shape is in a term that dies as the square.

The two roots, and which of them a real vortex sits on #

The steady-state relation has a feature worth reading carefully: for every strain below the maximum there are two aspect ratios that solve it, one on each side of 2.890.

The lower root is the one a real vortex sits on. Start a round patch in a weak strain and it settles near $\lambda$ slightly above one, which is the lower root; the upper root is a strongly elongated equilibrium that exists mathematically and is unstable, in the way the upper branch of a specific-energy curve is. The depth that costs least is the same shape of statement in a channel — two depths carry the flow at any energy above a floor, one at the floor, and none below it — and the two problems have nothing in common except the arithmetic of a turning point.

What happens at the maximum is that the two roots meet. There is one shape, it is marginally stable, and a strain a fraction of a per cent larger has no steady solution at all. That is not the patch becoming very elongated; it is the patch ceasing to be a patch.

A second threshold, at a rounder number #

The rotation-rate curve runs to any aspect ratio at all, and there is a second place the family gives out — for a different reason, and at a value close enough to the first to be worth keeping apart from it.

A freely rotating Kirchhoff ellipse, with no external strain anywhere, is linearly stable only up to an aspect ratio of exactly three. Love established it in 1893: perturb a more elongated one and a disturbance with three-fold symmetry grows, the patch pinches, and the steady rotation is over. Below three every disturbance merely oscillates.

So there are two thresholds and they are not the same statement. 2.890 is where a strained ellipse ceases to exist — the turning point of the strain relation, above which no aspect ratio solves it. 3 is where a free ellipse ceases to be stable — an equilibrium that is still perfectly well defined and no longer survives a nudge.

They sit consistently. Everything on the lower branch is below both, so a patch that settled there is safe on both counts; the upper branch is above the first and running towards the second, which is one more reason nothing is found on it. Existence and stability are two questions, and the second usually has the smaller answer.

The rotation rate that diverges in the limit #

Now take the limit properly. Shrink the patch at fixed circulation — which is the limit the point vortex is defined by — and watch what happens to the quantity the shape has.

The circulation is $\Gamma = \pi\omega ab$, so holding it fixed while $a$ and $b$ shrink means $\omega$ rises as the inverse area. Kirchhoff’s rate is then

$$\Omega = \frac{\Gamma}{\pi(a+b)^2},$$

which goes to infinity as the patch goes to a point.

Measured over a hundredfold shrink it rises by a factor of ten thousand, exactly as the inverse square. So the idealisation does not merely discard the shape’s rotation rate; it discards a quantity that was on its way to being infinite. That is the sharpest available statement of what is being thrown away, and it is the reason a point-vortex model has no internal time scale: the one it would have had is unbounded.

Two curves and one point. The rotation rate says what a shape does when nothing is straining it; the strain relation says what shape a given strain forces; and the aspect ratio 2.890 is where the second runs out. Beyond that the first curve is still perfectly well defined and describes nothing, which is the ordinary situation for a formula outside its own hypothesis.

What the patch keeps that the sheet did not #

It is worth putting this beside a sheet that cannot stay a sheet, because the two idealisations fail in opposite ways.

The sheet’s failure is dynamical: the model is well posed at $t = 0$, the evolution produces a singularity in finite time, and the computation needs a length put back before it can be integrated at all. The patch’s failure is structural: the model is perfectly integrable for ever, and what it has lost is a set of properties — a rotation rate, an aspect ratio, a strain limit — that the object it replaced actually had.

Neither failure is visible in the flow field a few radii away, which is where a reader looks.

What the limit is good for, which is a great deal #

None of this is an argument against point vortices, and the collection would be much poorer without them.

Vortices move each other establishes the property that makes the model work at all — a vortex never moves itself, so the whole dynamics of $N$ vortices is $N(N-1)$ mutual inductions — and three is the most that can be predicted takes it to the point where the system becomes chaotic. Both are statements about positions, and positions are what the point model gets right: the exterior field being exact means the induced velocity at another vortex’s centroid is exact too, as long as the patches stay round.

Circulation is vorticity added up is the other half of the licence. Circulation is an integral of the vorticity over an area, so a patch and a point with the same circulation are indistinguishable to any loop enclosing both — and every force in the exact theory is an integral round a loop.

An arithmetic that recurs #

There is a habit of thought here that the collection meets in several places and it is worth naming.

A quantity is computed in the idealised model. It is finite, and it is used. Then the idealisation is examined and the quantity turns out to be the limit of something that was diverging — so what looked like a value was a cancellation, and the model’s own answer is the residue of an infinity.

The patch’s rotation rate is one case. The momentum with no value is another and a sharper one: the momentum of an unbounded ideal flow is conditionally convergent, so it is zero over a disc, plus or minus the impulse over long boxes, and finite and different for every region — and Kelvin’s impulse is the well-defined quantity that replaces it. And the energy a vortex cannot have is the third: the kinetic energy of a point vortex diverges logarithmically, and what makes point-vortex dynamics work at all is that the differences between configurations are finite while the self-energies, which are infinite, are the same in every configuration and cancel.

A patch’s self-energy is finite. For a Rankine vortex of radius $a$ it is $\rho\Gamma^2/16\pi$ inside the core and $(\rho\Gamma^2/4\pi)\ln(R/a)$ outside, so the total carries the same logarithm the point vortex’s does plus a constant of one quarter. The point limit sends the logarithm to infinity and leaves the quarter behind, and the quarter is the whole of what the core contributes.

Why contour dynamics is exact and rare #

A note on the method, because its exactness is unusual and is worth knowing the reason for.

Almost every computational method in this collection approximates something: a grid discretises a field, a panel method approximates a boundary by straight segments, a marching scheme truncates a Taylor series. Contour dynamics approximates only the curve. The physics — that the region’s vorticity is uniform, that its edge is material, that the velocity is a contour integral — is exact, and refining the polygon converges to the exact evolution.

That is possible only because of the two-dimensional vorticity equation’s peculiar property: $D\omega/Dt = 0$ means a patch of uniform vorticity stays uniform for ever, so the interior needs no representation at all. Nothing analogous holds in three dimensions, and nothing analogous holds for a continuously varying vorticity.

The price is that the method describes exactly one thing. It is not a general solver; it is the exact solver for a very small class of flows, and the class happens to include the questions this essay asks.

Limits recorded rather than smoothed over #

The strain here is external and steady. Moore and Saffman’s relation is for a patch in a uniform straining field held fixed. A co-rotating pair supplies a strain and a frame rotation, and the second is not in this relation; the aspect ratio it predicts for a pair is measurably too small, by about half again, and the correction is the subject of the neighbouring essay.

The contour is a polygon. Everything here is computed with a few hundred straight panels, and the accuracies quoted — a part in a million on the rotation rate, eight in ten million on the interior — are the discretisation’s rather than the theory’s. The theory is exact.

Nothing here is three-dimensional. A patch of uniform vorticity stays uniform because $D\omega/Dt = 0$, and that equation is two-dimensional: in three dimensions the stretching term is there and the vorticity of a material element changes. The spin that feeds itself is the essay about that term, and it is the reason contour dynamics is a plane method and not a general one.

And the strain limit is a limit of the elliptical family. What is shown is that no steady ellipse exists above 0.150142. The full contour-dynamics answer allows shapes that are not ellipses, and its threshold is close but not identical.

The pattern, one more time #

A limit was taken to make a hard object into an easy one. The easy object reproduces the far field exactly, which is what made the limit look free, and it has lost three quantities — a rotation rate that diverges, a shape that responds to strain, and a threshold beyond which the object does not exist.

None of the three is visible in the thing that was kept. That is the rule this group of essays runs on, and the patch is the cleanest case of it: everything the reader can see is identical, and everything the model can do is different.