Three is the most that can be predicted

A point vortex has no velocity of its own: its field is antisymmetric about it and there is nothing at its centre to be carried by. So a system of them is a set of points, each moving in the field of the others, with no free parameters — no core size, no viscosity, no discretisation, nothing to converge.

It is the simplest system fluid mechanics has, and it is a Hamiltonian one. Writing $H = -\sum_{i<j}\Gamma_i\Gamma_j \ln r_{ij}/4\pi$, the equations of motion are

$$\Gamma_i\dot x_i = \frac{\partial H}{\partial y_i},\qquad \Gamma_i\dot y_i = -\frac{\partial H}{\partial x_i},$$

so $x$ and $y$ are canonically conjugate to each other, which is unusual and is why $N$ vortices have $N$ degrees of freedom rather than $2N$.

The invariants, and the counting #

Four quantities are conserved exactly, for any number of vortices in an unbounded plane:

$$H,\qquad P_x = \sum\Gamma_i x_i,\qquad P_y = \sum\Gamma_i y_i,\qquad L = \sum\Gamma_i(x_i^2+y_i^2).$$

The Hamiltonian is the interaction energy; the two impulses are the linear impulse; $L$ is the angular impulse. None of them involves the time step, the ordering of the sum or anything the integrator touches, which is what makes them a test of it.

Liouville’s theorem says a Hamiltonian system with $n$ degrees of freedom is integrable if it has $n$ independent invariants in involution — mutually commuting under the Poisson bracket. $P_x$ and $P_y$ do not commute with each other, but $H$, $L$ and $P_x^2 + P_y^2$ do.

So there are three invariants in involution, always. Three vortices have three degrees of freedom and are integrable. Four vortices have four, and three is not enough.

What integrable buys #

For three vortices the consequence is concrete: the motion is confined to a two-dimensional surface in the six-dimensional phase space, which is a torus, and motion on a torus is quasi-periodic. The separations $r_{12}, r_{13}, r_{23}$ evolve in a closed system of three equations that can be reduced to quadratures, and Gröbli did exactly that in 1877 — a thesis that was then lost for a century and rediscovered in the 1970s, by which time several people had done it again.

A three-vortex system can therefore do complicated things. It can have periodic orbits, near-collapses, and self-similar collapse in finite time when the strengths satisfy $\sum 1/\Gamma_i = 0$. What it cannot do is separate exponentially from a neighbour, because trajectories on a torus diverge at worst linearly.

And the fourth #

Add one more vortex — a weak one, near the middle of the other three, arranged with no symmetry — and the same integrator on the same equations produces this.

The invariant drift is the load-bearing part of that caption. A chaotic-looking trajectory from a sloppy integrator is a picture of accumulated rounding, and the way to tell the two apart is to check the quantities the integrator is not enforcing. Here they hold to $10^{-14}$ over a hundred thousand steps.

Why the fourth vortex is a small change #

It is worth dwelling on how little was added, because the size of the change is the striking part.

The fourth vortex has strength 0.8 against the others’ 1, 1 and −0.6. It is placed near the centroid of the three. Its presence changes each of the others’ velocities by a modest fraction, and the resulting picture over one or two circulation times is not obviously different from the three-vortex one.

What changed is the number of degrees of freedom against the number of invariants. Three and three is integrable; four and three is not. There is no intermediate case — no configuration of four vortices that is “slightly chaotic” in the sense of having three and a half degrees of freedom — and the transition happens at a whole number.

Special cases can restore the symmetry and with it the integrability: four vortices at the corners of a square with equal strengths, or two identical counter-rotating pairs, have extra symmetries that supply an extra invariant and stay regular. That is why the configuration used here is deliberately asymmetric — a symmetric four-vortex system is a poor demonstration of a result about the generic case, and it is exactly the arrangement anybody would draw first.

Measuring how fast it becomes unknowable #

The measurement is straightforward: run the system twice from initial conditions a billionth apart, and watch the separation.

For four vortices it grows exponentially, at a rate of about 0.033 per time unit — a factor of 140,000 over two hundred time units. For three, over exactly the same interval with the same integrator, the growth is a factor of 180, and it is a power of the time rather than an exponential of it.

The ratio between them is 780, and it is the ratio rather than either number that carries the argument, because the exponent depends on the configuration and the comparison does not.

What a Lyapunov exponent costs #

The practical content of an exponent $\lambda$ is a formula for how long a prediction lasts. An initial uncertainty $\varepsilon$ grows to an unacceptable $\delta$ after

$$t = \frac{1}{\lambda}\ln\frac{\delta}{\varepsilon},$$

so improving the initial measurement by a factor of a thousand buys $\ln(1000)/\lambda$ more time — about 208 time units here — and buys the same amount again for the next factor of a thousand, and no more.

That is the whole content of the word chaotic, and it is worth having as a table rather than as an adjective, because the shape of the trade is unintuitive. Prediction does not become expensive; it becomes logarithmically cheap and therefore useless. There is no amount of measurement that buys a long prediction, and the reason is not that measurement is hard.

The information the equations do not contain #

This rung sits among others about what the equations and the boundary conditions leave undetermined, and this is the sharpest form of it because nothing here is missing at all.

The equations are exact. The invariants are exact. The integrator conserves them to fourteen digits. Every quantity in the problem is known — and the future is not, because knowing the initial condition to any finite precision is not the same as knowing it.

The missing information is the tail of the decimal expansion of the initial position, and it is missing in the strong sense: it is not a quantity anybody has ever failed to measure carefully enough, it is a quantity that does not have a value in any physical sense. A vortex’s position is a fiction to begin with; a real one is a distributed patch, and asking where its centre is to twenty decimal places is asking a question about a model rather than about a fluid.

Where it matters in a fluid #

Point vortices are a model, and the question is which real situations inherit the result.

Vortex methods. A whole family of numerical schemes represents a flow by thousands of vortex elements moved by each other’s fields. Every such calculation is an $N$-vortex system with $N$ in the thousands, so it is chaotic, and two runs of the same simulation differing in the last bit of a coordinate will diverge. The individual trajectories in a vortex-method calculation are not predictions; the statistics are.

Wake vortices. Two trailing vortices behind an aeroplane are a two-vortex system and are entirely predictable — they descend at a rate this collection computes elsewhere. Four, from an aeroplane whose flaps are down, are not, and the Crow instability that eventually destroys the pair is a growth of exactly the kind of perturbation this rung is measuring.

And two-dimensional turbulence, where the vortices are real, long-lived, and numerous. The inverse cascade — energy moving to larger scales rather than smaller — is in part a story about vortex mergers, which is an $N$-vortex system reducing its own $N$.

Why the invariants are allowed to be evidence #

The argument above turns on a single methodological claim: the tangle is a property of the system rather than of the arithmetic, and the proof offered is that the energy and the impulses hold to fourteen digits over a hundred thousand steps. That claim deserves examining, because for most integrators it would not be available.

A general-purpose scheme conserves nothing. Its energy error accumulates step by step in the same direction, so it drifts linearly with time, and a long run of a Hamiltonian system with an ordinary Runge–Kutta method ends up on a completely different energy surface from the one it started on. A chaotic-looking trajectory from such a run is not evidence of chaos; it is a picture of a system slowly sliding through phase space under the influence of its own truncation error.

What makes the check above meaningful is that the integrator is symplectic: it preserves the geometric structure the Hamiltonian formulation has, exactly, at every step and for any step size. The consequence is not that it conserves the energy — it does not — but something better for this purpose. Backward error analysis shows that a symplectic method solves a nearby Hamiltonian system exactly, one whose Hamiltonian differs from the true one by a term of the order of the truncation error. So the computed trajectory conserves that modified energy exactly, and the true energy is therefore trapped within a fixed distance of its initial value for ever, oscillating rather than drifting.

That is what fourteen digits over a hundred thousand steps is reporting, and it is why it can be offered as evidence. A bounded error is a different kind of statement from a small one: it says the run has not left the surface it was released on, however long it is continued.

There is an honest limitation on the other side of it, and this essay’s own subject supplies it. A symplectic integrator’s trajectory still diverges from the true trajectory exponentially, at exactly the rate measured above — because the modified system is a perturbation of the true one, and a perturbation of a chaotic system separates like any other. So the picture of four vortices after two hundred time units is emphatically not where those four vortices would be. It is where four vortices very like them would be, on the same energy surface, obeying a Hamiltonian a rounding error away from the intended one.

Which is the strongest available form of what a long chaotic computation can claim, and it has a name — shadowing. The computed orbit is not the orbit that was asked for, and it is a true orbit of a nearby system, so every statistical property it exhibits is a property something real would exhibit. What cannot be claimed is any statement about a particular trajectory at a particular time.

The invariants are the part of the answer that survives, in the arithmetic exactly as in the physics — which is the same conclusion this essay reaches about the fluid, arriving from the integrator instead.

What this is not #

Three misreadings worth heading off, because this subject collects them.

It is not turbulence. Chaos in a four-vortex system is a property of a four-dimensional dynamical system; turbulence is a property of a field with a continuum of degrees of freedom and an energy cascade between them. The randomness in a turbulent flow is not in the equations either, and that is the only thing the two have in common.

It is not a failure of determinism. The system is deterministic, in the strict sense that the state now fixes the state later. What fails is the inference from approximately now to approximately later, which is a different statement and is the one every practical use of a physical law depends on.

And it is not caused by the vortices getting close. Close approaches make the exponent larger and are not necessary to it. The system is chaotic over regions of phase space where nothing dramatic happens, and the exponent above was measured on a run with no near-collisions in it.

What the picture cannot show #

The phase space. A four-vortex system lives in eight dimensions with three constraints, and every figure here is a projection onto the physical plane. Two trajectories that appear to cross do not; two that appear close may be far apart in the variables that matter.

The exponent is an average. A Lyapunov exponent is a long-time average of a local stretching rate, and the local rate varies enormously along a trajectory. A prediction horizon computed from the average is a typical horizon, not a guaranteed one.

And the vortices have no cores. Everything here is a model in which the vorticity is concentrated at points. A real vortex patch of finite size deforms, and two patches that pass close enough merge — which removes a degree of freedom, and can turn a chaotic system into a regular one by reducing four vortices to three.

The distinction in that last caption is not pedantry. Streamlines are not the paths particles take unless the flow is steady, and a four-vortex flow is as unsteady as a flow gets. A figure of vortex trajectories and a figure of streamlines are pictures of different objects, and only the first of them has anything to do with predictability.

It is worth naming the one quantity that stays predictable when nothing else does. The invariants do: the energy, the two impulses and the angular impulse are the same at the end of a chaotic run as at the beginning, to fourteen digits, and they are the same for both members of a diverging pair. So the long-run statement that can be made about a four-vortex system is not where the vortices are but which surface in phase space they are confined to — and that surface is fixed by three numbers known at the start.

Chaos removes the trajectory and leaves the constraints, which is why statistical descriptions of chaotic systems are made out of conserved quantities and why the invariants above are the first thing any such description computes.

Who found it, and when #

Helmholtz gave the point-vortex equations in 1858 and Kirchhoff put them in Hamiltonian form in 1876. Gröbli integrated the three-vortex problem in 1877, in a Göttingen dissertation that vanished so completely that Novikov re-derived it in 1975 and Aref found the original afterwards. Aref established the chaos of the four-vortex problem in the late 1970s, at the same moment the rest of physics was discovering that low-dimensional Hamiltonian systems were generically chaotic.

The surprising connection is with the age of the question. The three-body problem of celestial mechanics and the three-vortex problem of fluid mechanics are both integrable at three and chaotic at four, and for the same structural reason — a Hamiltonian system whose invariants come from the symmetries of the plane, which supply exactly three in involution however many bodies there are. Poincaré’s discovery that the gravitational three-body problem is not integrable is not a counterexample: gravitational bodies have twice as many degrees of freedom each, because position and momentum are independent, while a vortex’s two coordinates are conjugate to one another. A vortex is half a particle, and that is why three of them are soluble and three planets are not.

One last observation about what kind of result this is, because it is easy to file as a curiosity about a toy model. The point-vortex system is the only place in this collection where every approximation has been removed: there is no discretisation, no closure, no truncated series, no coarse grid, no neglected term. Whatever unpredictability appears in it is not an artefact of a method.

That is exactly what makes it worth having. Everywhere else in fluid mechanics, a computation that loses predictability can be suspected of losing it to its own numerics, and the suspicion is usually right. Here it cannot be, and the invariant drift of $10^{-14}$ is the proof. The unpredictability is in the physics, and the physics is four points moving in each other’s fields.

Where the ladder goes next #

Below this rung is the two-vortex problem that has closed forms, and what circulation is, which is the quantity the whole Hamiltonian is built out of.

Beside it are the randomness that is not in the equations, which is the same distinction made about a continuum, and a vortex near a wall, which is a two-vortex problem in disguise and is therefore predictable forever.