What a point vortex is not

Vortices move each other establishes the property the whole point-vortex model is built on: a straight filament induces no velocity on itself, so everything a vortex does is done by another one, and the dynamics of $N$ vortices is $N(N-1)$ mutual inductions and nothing else.

The shape a vortex keeps establishes the other half: a circular patch of uniform vorticity has exactly the exterior field of a point vortex of the same circulation, so the two are indistinguishable to anything outside.

Put the two together and the model looks not merely good but exact. This essay is about the sense in which it is exact, which is stronger than expected, and the sense in which it fails, which is more abrupt.

The exactness is a theorem, not a small parameter #

Start with the good news, because it is better than the usual statement.

A circular patch sits in the velocity field of its partner. Its centroid moves at the area average of that field over its own disc — that is what a centroid does. And each Cartesian component of the velocity field of a point vortex is a harmonic function away from the vortex.

A harmonic function’s average over a disc is its value at the centre. That is the mean value property, and it is exact.

So the centroid of a circular patch moves at exactly the velocity the point model gives, at every separation, with no correction of any order. Computed by quadrature over a 600 by 720 polar grid at separations from 2.2 radii to 12, the departure is $1.3\times10^{-13}$.

This is worth sitting with. The usual account of point-vortex dynamics is that it is a small-core approximation. It is not: while the patches are round it is not an approximation at all.

Which is why the failure has to come from the shape #

If the model is exact for round patches and fails for real ones, the failure is entirely the shape’s.

And the shape does not stay round. Each patch sits in the other’s field, which is not uniform — a point vortex at distance $D$ produces a strain of rate $\Gamma/2\pi D^2$ at the other’s position — so each patch is stretched along one axis and squeezed along the other.

Measured against its own vorticity $\omega = \Gamma/\pi a^2$, that strain is

$$\frac{e}{\omega} = \frac{a^2}{2D^2},$$

which is a number the point model does not have, because it has no $a$.

The computation, and the defect it found #

Everything below is contour dynamics: two closed polygons of sixty-four nodes each, advected by the velocity their own boundaries induce, with the singular panels integrated in closed form. There is no grid, no viscosity and no model.

The first run lost twenty-four per cent of the patch area per orbit, in a scheme whose entire claim is that the area is exactly conserved. Refining the time step did not help, which said the error was in space; redistributing the nodes along the contour did not help either, which said it was not node bunching.

It was the singular panel. When the velocity is evaluated at a panel’s midpoint the integral of $\ln|s|$ over that panel is $h(\ln(h/2) - 1)$; when it is evaluated at a panel’s end — which is what advecting the nodes does, since two panels meet at each node — it is $h(\ln h - 1)$. Using the first formula where the second belongs leaves a bias of a fixed sign that no refinement removes.

With both cases integrated correctly the area drifts by 4.5 parts in a hundred thousand over a full orbit, at the same resolution and the same step.

That is the kind of defect the site’s whole method exists to catch. It produced smooth, plausible, completely wrong pictures at every resolution, and nothing but a conservation law would have found it.

What the orbit does at five radii #

With the machinery working, the measurement.

At a separation of five patch radii the two centroids follow the exact point-vortex circle to 0.91 per cent of a separation over one complete co-rotation, and the orbit’s radius wobbles by 0.07 per cent. The point model is doing its job, and the small residual is the shape starting to matter.

The shapes tell the other half. Each patch is elongated by the other, reaching an aspect ratio of 1.21 at the widest, and it does not settle: it nutates, running from a circle out to an elongated ellipse and back, over and over.

That is not a numerical artefact and it is not damping that is missing. An inviscid flow is reversible and has nothing to relax with. A patch released as a circle in a strain does not approach the equilibrium ellipse; it oscillates about it for ever, in the way a pendulum released off-vertical oscillates rather than settling.

What is conserved while the shapes are changing #

A computation that is being asked to detect a fourteen per cent error needs to say what it is holding exactly, and contour dynamics holds three things.

The area of each patch, because the flow is incompressible and the boundary is material. That is the check the panel defect was found with, and it now holds to four parts in a hundred thousand.

The circulation of each patch, which follows from the area and the vorticity being constant. It is conserved by construction here rather than measured, so it is not evidence.

And the energy and the impulse of the whole system, which are conserved by the Euler equations and which the discretisation approximates. Neither is measured in this essay, and both would be a stronger test than the area: the area is conserved by any scheme that advects a closed curve by a divergence-free field, whereas the energy is conserved only if the induced velocity is right. Recording that as a gap is more useful than pretending the area check is decisive.

What the area check is decisive about is the defect it caught, which was a bias in the induced velocity itself.

The equilibrium, and where the ellipse model is wrong #

The mean aspect ratio over the orbit is 1.130, against the pure-strain equilibrium of 1.084 — an elongation about 1.55 times the model’s.

That discrepancy is real and it has a cause. Moore and Saffman’s relation is for a patch in a steady external strain with no rotation of the frame. A co-rotating pair supplies a strain and a frame rotation, and the second adds to the elongation. The pure-strain model is the right first statement and the wrong second one, and the factor is measured rather than smoothed over.

And then the model stops working, quite suddenly #

Close the pair from five radii to three and a half — a thirty per cent move — and the centroid error over one orbit goes from 0.91 per cent to 14 per cent: fifteen times worse.

Nothing discontinuous has happened to the circulation, the separation law or the induced velocity. What has happened is that the strain has driven the aspect ratio up, the patches are no longer round, and the mean value property — which was the entire licence for the point model — no longer applies.

The threshold the point model does not have #

Push further and there is a genuine threshold, and it is the sharpest statement in this essay.

The strain limit of the previous essay is $e/\omega = 0.150142$; above it no steady elliptical patch exists at any aspect ratio. With the strain supplied by a partner, $e/\omega = a^2/2D^2$, so the threshold is at

$$\frac{D}{a} = \frac{1}{\sqrt{2 \times 0.150142}} = 1.825.$$

Below that separation there is no steady shape at all: the patches are drawn out into filaments and merge. Two point vortices co-rotate at any separation whatever, down to zero.

The pure-strain number is not the right one — the full co-rotating calculation of Saffman and Szeto puts merger at about $D/a = 3.16$, for exactly the reason the aspect ratio was underpredicted above — but the existence of a threshold is what matters, and the point model has none.

The two failures, and why only one of them is gradual #

It is worth separating the two things that go wrong, because they are usually run together and they behave differently.

The first is gradual. As the patches deform, the mean value property stops applying and the centroid velocity acquires an error. That error grows smoothly with the aspect ratio, which grows smoothly with $a^2/2D^2$, and a calculation can be made as accurate as anybody likes by separating the vortices further. It is an approximation with a small parameter, and it behaves like one.

The second is not gradual at all. Below a separation there is no steady shape, the patches are torn into filaments and wound around each other, and two vortices become one. There is no small parameter in that: on one side of the threshold the configuration exists and on the other it does not, and the point model’s prediction — a perfectly good co-rotation, for ever, at any separation — is not wrong by a percentage. It is describing an object that is no longer there.

Distinguishing the two matters because the first is what everybody has in mind when they say “point-vortex dynamics is an approximation”, and the second is the one that ruins a calculation.

What the same argument says about a vortex near a wall #

The mean value property has one more consequence and it is worth extracting, because it explains something about image systems.

A patch near a plane wall moves in the field of its image. That field is again a point vortex’s — the image of a circular patch is a circular patch, and its exterior field is a point vortex’s — so the centroid again moves at exactly the image speed while the patch stays round.

But the strain from an image at distance $2h$ is $\Gamma/2\pi(2h)^2$, so the patch is deformed by its own reflection, and it is deformed towards the wall in the sense that the elongation is parallel to it. A patch approaching a wall therefore flattens, and a flattened patch’s centroid does not travel at the image speed any more.

That is the two-dimensional shadow of what a wall made by reflection does not have to consider, since the image method there is applied to a potential flow with no vorticity in it at all. Where the vorticity is a finite patch, the image is a real strain and the reflection has a physical consequence.

The pattern this makes with its neighbours #

Three neighbouring essays take the same shape and it is worth naming the shape once.

An idealisation removes an internal structure — a thickness, a size, a shape. The idealised object reproduces the field exactly or nearly exactly, which is what makes the idealisation attractive. And what it loses is a response: how the object reacts to being strained, sheared or accelerated.

For a sheet that cannot stay a sheet the lost response is the finite thickness that would have cut off the short waves, and the consequence is a singularity in finite time. For the part of the flow inside the body it is the distance to an interior singularity, and the consequence is an expansion that diverges on the body’s own surface. Here it is a shape’s response to strain, and the consequence is a threshold.

In all three the field is right and the response is missing, and the field is the only thing a picture shows.

What this does to the collection’s own point-vortex work #

Three of this site’s essays use point vortices for real results, and it is worth saying which of them this affects.

Three is the most that can be predicted is about the integrability of three vortices and the chaos of four. That is a statement about the point-vortex system as a Hamiltonian dynamical system, and it stands on its own terms: it is not claiming to describe patches.

The street this site cannot draw computes the Kármán spacing ratio, 0.28055, from an infinite row of point vortices. That is an equilibrium condition rather than an evolution, and the vortices in a real street are far apart compared with their cores, so the number is safe.

A ring moves because it is bent is the case where the core already had to be put back: a filament’s self-induced speed diverges logarithmically as the core is thinned, and that essay computes the coefficient of the logarithm. The three-dimensional version of this essay’s problem is worse rather than better.

The rule that comes out of it #

The condition for the point model is not “small cores”. It is round cores, and the two are not the same condition.

A pair of small patches in a strong strain is elongated and the model fails. A pair of large patches in a weak strain is round and the model is exact. What decides is $a^2/2D^2$ — the ratio of the strain they impose on each other to their own vorticity — and it is a property of the configuration rather than of any single vortex.

That is a usable rule and it is easy to evaluate. In a trailing pair behind a wing, $a/D$ is about a tenth, $e/\omega$ is about $5\times10^{-3}$, and the aspect ratio the strain forces is 1.02: the point model is excellent. In a merging pair in a mixing layer it is not.

Why the pictures do not warn anybody #

The failure has the property this collection keeps meeting: it is invisible.

At $D/a = 3.5$ the two patches look like two vortices. Their boundaries are smooth, their centroids are where a reader would put them, the flow field between them is unremarkable, and the trajectory is a circle to the eye. The fourteen per cent is a fourteen per cent in phase — the pair is going round at the wrong rate — and a drawing does not carry a rate.

That is the same reason a sheet that cannot stay a sheet needed a refinement study and a Fourier spectrum, and the same reason ideal against real exists at all. A wrong flow field is beautiful.

The number to carry #

One ratio decides everything above and it is worth stating on its own.

$$\frac{e}{\omega} = \frac{a^2}{2D^2}$$

is the strain a partner imposes, measured against a patch’s own vorticity. Below about $0.02$ — which is $D/a$ above five — the point model is good to a per cent over an orbit. Above $0.15$ there is no steady shape at all. Between them the model degrades, non-linearly, and the degradation is in the rate rather than in anything a drawing shows.

Evaluating it takes one division and it answers the question a reader actually has, which is whether the calculation in front of them is allowed.

Limits recorded rather than smoothed over #

The runs are one orbit. Nothing here says what happens over ten, and the errors quoted are the worst over a single rotation. A long integration accumulates, and the accumulation is not linear because the deformation feeds back on the motion.

The strain limit used is the pure-strain one. It is the wrong threshold by a factor of about 1.7, for a reason this essay identifies and does not fix. Computing the co-rotating equilibria properly means finding the V-states, which is a nonlinear contour problem and is not attempted.

The patches are equal and of the same sign. A counter-rotating pair translates rather than rotating and is deformed differently; an unequal pair does something else again. None of that is here.

And there is no merger. The computation is stopped well before the patches touch, because a contour-dynamics run through a merger needs surgery on the contours — filaments have to be cut off — and a scheme that cuts things off is a scheme with a parameter in it.

The residue #

The limit is the one that shrinks a patch to a point, and the residue is not what a reader would guess.

It is not an error term in the velocity: that is exactly zero while the patch is round. It is the patch’s response to strain — an aspect ratio, a nutation, and a separation below which the object does not exist. All three are properties the point vortex does not have, none of them is visible in the field a few radii away, and the last of them is the difference between two vortices and one.