Vortices move each other

Take a single point vortex, alone in an infinite fluid, and ask where it goes.

It goes nowhere. Not slowly, and not approximately: its own velocity field is $u_\theta = \Gamma/2\pi r$ about its own centre, which is antisymmetric about that centre — the flow on one side is the exact reverse of the flow on the other — so there is no velocity at the vortex to carry it anywhere. A vortex is the one object in this subject that is completely inert about itself.

Everything interesting therefore comes from having more than one, and two is already enough to exhaust what can be written in closed form.

The two cases that have answers #

Two vortices of strengths $\Gamma_1$ and $\Gamma_2$ a distance $d$ apart move only by carrying each other, and the arithmetic is a line each.

Opposite and equal. Each vortex sits in a field of magnitude $\Gamma/2\pi d$ pointing the same way, so the pair translates, perpendicular to the line joining them, at

$$U_{\text{pair}} = \frac{\Gamma}{2\pi d}$$

with the separation fixed. It travels forever, and it does not slow down, because nothing is dissipating anything.

Equal and the same sign. Now the two induced velocities point in opposite senses, so the pair rotates about the point between them, with period

$$T = \frac{4\pi^2 d^2}{\Gamma}$$

and again the separation is fixed. Every co-rotating pair anybody has ever seen — two smoke rings side by side, two bath vortices, the two vortices behind a delta wing — is a version of this.

Both results are computed here rather than quoted. The stepper is a fourth-order Runge–Kutta on the positions with the self-term omitted by construction, and assertPairMotion compares the distance travelled against $\Gamma t/2\pi d$: at $\Gamma = 2$, $d = 1$ and half a time unit, the closed form gives $0.15915494309190$ and the integration gives $0.15915494309191$.

Why adding the fields is allowed #

Both of those results were obtained by evaluating one vortex’s field at the other’s position and adding. That step is the whole method, and it is legal for a reason worth naming.

The velocity field of an incompressible, irrotational flow comes from a potential satisfying Laplace’s equation, and Laplace’s equation is linear — so solutions add. A vortex plus a vortex is a solution; a vortex plus a uniform stream is a solution; a vortex plus its own image in a wall is a solution. Nothing here requires the disturbances to be small, which is the usual price of a superposition argument, because nothing here is being linearised: the equation was linear to begin with.

What is not linear is the motion. Each vortex is carried by the sum of the others’ fields, and the positions then feed back into the fields, so the trajectories obey a nonlinear system even though the flow at every instant is a sum of exact solutions. That separation — a linear field equation with a nonlinear advection of the singularities — is the whole character of the subject, and it is why four vortices can be chaotic in a fluid with no viscosity, no forcing and no small parameter.

What the solver computed, and how it was checked #

The useful property of a point-vortex system is that it has four conserved quantities and they are computable from the positions alone:

$$H = -\frac{1}{4\pi}\sum_{i<j}\Gamma_i\Gamma_j \ln r_{ij}, \qquad P_x = \sum_i \Gamma_i x_i, \qquad P_y = \sum_i \Gamma_i y_i, \qquad L = \sum_i \Gamma_i |\mathbf{x}_i|^2$$

$H$ is the interaction energy, $P_x$ and $P_y$ are the two components of linear impulse, and $L$ is the angular impulse. None of them appears anywhere in the integrator, which only ever adds up the velocity each vortex induces at the others. They are computed afterwards, from the positions the stepper produced, so agreeing is a test of the integration rather than a restatement of it.

There is one subtlety in that check worth recording, because it looks like a bug and is not. A counter-rotating pair exactly one unit apart has $H = -\Gamma_1\Gamma_2\ln(1)/4\pi = 0$ exactly, and dividing a drift by zero reports a perfectly conserved run as infinitely bad. So the drift is scaled by the size of the terms that went into the sum rather than by the size of what came out of it — a distinction that costs four lines and turns a false failure into a real check.

What those four numbers actually mean #

The invariants were introduced as things to check the integrator against, which undersells them. Each one is a statement about where the vortices are allowed to go, and together they explain why the two cases at the top of this essay behave so differently.

Divide the linear impulse by the total strength and the result is a position:

$$\mathbf{x}_c = \frac{\sum_i \Gamma_i \mathbf{x}_i}{\sum_i \Gamma_i}$$

the centre of vorticity — the vortex system’s analogue of a centre of mass, with circulation in place of mass. Since the numerator is conserved and the denominator is a constant by Kelvin’s theorem, that point never moves. Not approximately, and not on average: an arbitrary number of vortices, milling about chaotically for as long as one likes, has a point in the plane they collectively cannot leave.

The exception is the case where the denominator vanishes, and that is the whole difference between the essay’s two pairs. A counter-rotating pair has $\sum\Gamma_i = 0$, so there is no centre of vorticity to be pinned to — the conserved vector $\sum\Gamma_i\mathbf{x}_i$ is then independent of where the origin is put, and what it measures is not a position at all but a genuine linear impulse, the momentum the pair carries. A system with zero net circulation may therefore translate forever, and a system with any net circulation at all may not. The translating pair and the orbiting pair are not two unrelated results; they are the two sides of whether one denominator is zero.

The angular impulse says something similar about spreading. Referred to the centre of vorticity it becomes

$$\sum_i \Gamma_i |\mathbf{x}_i - \mathbf{x}_c|^2 = \text{constant}$$

which for a cluster of vortices all of the same sign is a sum of positive terms with a fixed total. So such a cluster cannot disperse. It can rearrange itself as violently as the chaos allows, and the weighted spread of the whole set stays exactly what it started as — one vortex wandering far out is paid for by the others closing in.

Mixed signs remove that guarantee, because the sum is then no longer of one sign and a positive and a negative term can both grow without the total moving. Which is exactly what a counter-rotating pair does when it leaves: two contributions to the second moment growing without bound, cancelling perfectly, and the invariant undisturbed while the vortices disappear over the horizon.

None of this needs the equations to be solved. These are constraints on every trajectory the system has, integrable or chaotic, two vortices or two hundred — which is why they are worth more than the closed forms they are being used to check.

Positions as conjugate variables #

The reason these systems have exactly this many invariants is worth a paragraph, because it is one of the strangest facts in classical mechanics.

Kirchhoff showed in 1876 that the point-vortex equations are Hamilton’s equations, with $H$ above as the Hamiltonian — and with $\Gamma_i x_i$ and $y_i$ as the conjugate pair. The two coordinates of a vortex are canonically conjugate to each other. In ordinary mechanics position is conjugate to momentum and the phase space of $N$ particles in a plane has $4N$ dimensions; here the phase space is the plane the vortices live in, and $N$ vortices give $2N$ dimensions.

That halving is why the counting works out as it does. A system with $2N$ phase-space dimensions needs $N$ independent invariants in involution to be integrable, and the point-vortex system has three ($H$, $P_x^2 + P_y^2$, $L$) whatever $N$ is. So one vortex is trivial, two are integrable, three are integrable — and four are not.

Three, and then no further #

Gröbli worked out the three-vortex problem in 1877 in a doctoral thesis that was then forgotten for a century. It is integrable, and its solutions include collapses in finite time when the strengths satisfy a particular relation.

Four vortices are chaotic, and the demonstration is Aref’s, in 1979. That is a genuinely surprising place for chaos to appear: the system is inviscid, energy-conserving, and has no forcing, no dissipation and no small parameter anywhere in it. The equations are four coupled first-order ordinary differential equations with logarithmic interactions, and they are as unpredictable as anything in the truncation that made the same point about convection.

The field the pair carries with it #

A translating pair does something a single vortex cannot: it carries a region of fluid along with it.

That blob is the whole reason the pair is a useful object. It is what a smoke ring looks like in cross-section, it is why a pair of trailing vortices behind an aircraft descends steadily through the air, and it is the mechanism by which a vortex pair carries momentum a long way from wherever it was made.

The descent rate of an aircraft’s wake follows directly. The trailing vortices behind a wing are a counter-rotating pair of separation about $\pi b/4$ for elliptic loading and strength $\Gamma_0 = 2W/\rho U \pi b$, and the pair formula gives their sinking speed at once. For a large airliner it comes out at a metre or two per second — which is why wake turbulence sinks below the flight path, and why the separation rules that govern it are written in terms of time rather than distance.

Walls are vortices in disguise #

The one other thing that moves a vortex is a boundary, and it moves it by being an image.

A wall is made by reflection: the flow with a wall in it is the flow with an image vortex of opposite sign at the mirror position, and the image satisfies the no-through-flow condition exactly. So a vortex near a wall is half of a counter-rotating pair, and it travels parallel to the wall at $\Gamma/4\pi h$, where $h$ is its height — the pair separation being $2h$.

This is not an analogy. The image is a genuine solution and the motion is the two-vortex result applied to it, which is why ground effect has the same algebra in it as this essay.

The strengths never change, and that is a theorem #

One thing has been assumed throughout without comment: each vortex’s $\Gamma$ stayed the same number all the way through every integration above. That is not a modelling convenience.

Kelvin’s theorem says the circulation round a material circuit is constant in an inviscid fluid under conservative body forces. Draw a circuit round one vortex and follow it: the circuit is carried about by the flow, deformed, stretched — and the circulation inside it cannot change. So the strengths are constants of the motion, and the only degrees of freedom the system has are the positions.

That is why a point-vortex calculation is a dynamical system rather than a fluid simulation. Its state is $2N$ numbers, its evolution is exact given those numbers, and everything the fluid does is determined by where the singularities are. It is also why the model can be trusted for as long as the cores stay small and viscosity stays out of the argument: the moment two cores overlap, the circuit that was supposed to enclose one vortex encloses part of another, and the theorem stops separating them.

The same theorem is what makes the exact theory able to say anything at all about lift. A wing’s bound circulation is a constant of the motion in exactly this sense, which is why it had to be paid for by shedding an equal and opposite vortex on the runway.

What the picture cannot show #

A point vortex has infinite kinetic energy. The integral of $u_\theta^2$ over the plane diverges logarithmically at the centre, which is why the Hamiltonian above is an interaction energy with the self-terms dropped. Nothing here can say anything about the energy of a single vortex, and a real vortex’s energy depends on its core.

Real vortices merge. Two like-signed vortices closer than about three core radii do not orbit forever: they wrap around each other and combine into one, which the point model cannot represent because it has no core to wrap. That is a viscous and finite-amplitude process, and it is what happens to the two vortices behind a delta wing within a few chords.

Nothing here is three-dimensional. These are line vortices seen end-on, straight and infinite, and a real vortex line is curved. Curvature makes a filament move under its own influence — the one thing a straight vortex cannot do — and that is the next essay.

Where the model stops #

The point-vortex model is exact for the flow outside the cores and says nothing about what happens inside them. It is at its best when the cores are small compared with the distances between vortices, which is why it describes a pair of trailing vortices well and a pair of merging ones badly.

It also has no viscosity, so nothing in it decays. Real trailing vortices persist for minutes and then break up through an instability of the pair itself — a slow sinusoidal one, discovered by Crow in 1970, which grows because each vortex is perturbed by the other’s field. The point model can be extended to predict that instability and cannot predict the breakup that follows it.

And the whole apparatus assumes the vorticity is concentrated. A distributed vorticity field is not a sum of point vortices, and treating it as one is an approximation whose error nobody has bounded in general.

Who found it, and when #

Helmholtz laid down the vortex laws in 1858 and drew the two-vortex results immediately: he has the pair translating and the like-signed pair rotating, in the same paper that introduced the idea of a vortex line as a material object. Kirchhoff found the Hamiltonian structure in 1876. Gröbli solved the three-vortex problem in 1877; his thesis was rediscovered in the 1980s, by which time several people had solved it again.

Aref showed in 1979 that four vortices are chaotic, and the point-vortex system became a standard example of Hamiltonian chaos in a system with no obvious complexity in it. Crow’s instability of the trailing pair, from 1970, is the reason an aircraft’s wake breaks up at all — and it is the point at which this beautifully simple model starts describing something an air traffic controller cares about.

What it is good for, which is more than it looks #

A model with no viscosity, no cores and no third dimension sounds like a toy, and it is used in earnest for three things.

Wake prediction. The descent and decay of an aircraft’s trailing pair is computed this way in every airport separation study, with the cores and the atmosphere added as corrections to a two-vortex skeleton.

Vortex methods. A whole family of flow solvers represents vorticity as a large collection of these singularities, moves each by the sum of the others’ fields, and never needs a grid at all. The attraction is that empty fluid costs nothing to compute, since a region with no vorticity in it has no elements in it — the opposite bargain from the grid solvers whose cost is set by the volume.

Understanding what a wall does. The image construction turns every boundary into more vortices, and a surprising number of results — ground effect, the motion of a ring towards a surface, the spreading of a vortex blown at a plate — are the two-vortex answer applied to a reflection.

The common thread is that the model is exact where the vorticity is not, which is most of the domain, and that its errors are confined to regions whose size is known.

Where the ladder goes next #

A straight vortex cannot move itself and a pair of them can move each other. What is left is the case where a single vortex moves itself, which requires it to be curved — and which turns out to involve a logarithm nobody can get rid of.