Two rings leapfrog only if they start alike

A ring moves because it is bent ended its account of the vortex ring with the leapfrog: two coaxial rings, one behind the other, the rear one drawn narrower and faster by the front one’s field, passing through it, and the roles exchanging, over and over. It drew two equal rings doing it and checked that the total impulse did not move. That is the picture in every textbook and every smoke-ring video, and it invites a reading that is false: that leapfrogging is simply what two rings travelling the same way do.

It is what two similar rings do. Start a pair in one plane with the smaller ring well under the larger in radius and something else happens. The smaller ring, being faster, draws ahead. The larger ring’s field widens it and slows it, and the smaller ring’s field narrows the larger and speeds it up — exactly the mechanism of the leapfrog. But the adjustment may not be enough. The front ring can end up still faster than the one behind, and then the two simply part company.

This essay finds the ratio at which that changes, and finds it twice: once by marching the two rings, and once without marching them at all.

Two starts a few per cent apart #

The model is the one the earlier essay used, with one change. Each ring is a filament of circulation $\Gamma$ carried at Kelvin’s self-induced speed,

$$U = \frac{\Gamma}{4\pi R}\left(\ln\frac{8R}{a} - \frac14\right),$$

and by the other ring’s exact field, written in elliptic integrals. The change is in the core. The earlier essay held the core radius $a$ fixed as a ring stretched; here the core keeps its volume, $2\pi^2 a^2 R$, as an inviscid core must, so a ring that widens also thins. The rings have equal circulation, and every length is in units of the larger ring’s starting radius. The core starts at six hundredths of that.

On the left of the figure the smaller ring starts at 0.3 of the larger’s radius and at twice its speed — 0.75 against 0.37 in units of $\Gamma$ over the larger radius. As it draws ahead it passes into the part of the larger ring’s field where the flow spreads outwards, and it widens, from 0.30 to 0.52. The larger ring, now behind a ring, sits where that ring’s field converges, and it narrows from 1.00 to 0.90. Both changes are in the direction of a leapfrog. Neither is enough. When the rings are far enough apart to stop feeling each other, the front ring is still the smaller and the faster, and nothing is left to change that.

On the right the smaller ring starts at 0.5. The same widening and narrowing happen, but now they carry on until the rings’ radii cross, the rear ring becomes the faster, and it comes through. A full exchange takes 8.3 time units, and the pair repeats it for as long as the run lasts.

Somewhere between 0.3 and 0.5 the outcome changes. Bisecting on the march finds it, and that is one answer. The more interesting answer comes from asking what the march is conserving.

Two conserved quantities, one degree of freedom #

A coaxial ring is described by two numbers, its radius and its position along the axis, and those two are a conjugate pair in the sense of classical mechanics. The same structure is what makes a pair of point vortices so easy to predict: each plane vortex’s two coordinates are conjugate to each other, and the invariants follow from symmetries. The momentum conjugate to position is the ring’s impulse, $\pi\Gamma R^2$; the Hamiltonian is the kinetic energy of the flow. That the ring moves at $U = \partial H/\partial P$ is the statement that a ring’s speed is the rate at which its energy grows with its impulse.

Two consequences follow. The total impulse is conserved, because nothing in the problem depends on where along the axis the pair is — this is the invariant the earlier essay checked, and it is the momentum the pair has given the fluid, the quantity a control volume round the rings would measure the way the momentum theorem measures a wing’s lift. The energy is conserved, because nothing depends on time. And with two rings, those two conservation laws leave the pair with one degree of freedom: fix the impulse and the larger ring’s radius follows from the smaller’s, and fix the energy and the separation follows from both. So the orbit of a pair is not something to integrate. It is a curve of constant energy, and the march only says how fast the pair travels round it.

The energy has two parts. Each ring has its own, which for a volume-conserving core is

$$E(R) = \tfrac12\,\Gamma^2 R\left(\ln\frac{8R}{a} - \frac74\right),$$

in units with the density taken out. Its derivative with respect to impulse is Kelvin’s speed with its quarter exactly, which is the check that the two formulas belong together; a centred difference of the energy reproduces the speed to a part in ten billion. For a fixed core the constant is $5/4$ instead. And the pair has an interaction energy, $2\pi\Gamma$ times the Stokes stream function of one ring at the other, which is positive for two rings turning the same way and falls to zero as they separate.

The march conserves both to about a part in a hundred billion over thirty changes of order, for either core. That confirms the model is exactly Hamiltonian, and it makes the next step available.

What escape would need #

Suppose the pair does separate for ever. Then at the end the interaction energy is gone, and what remains is two free rings. They must still share the pair’s impulse and still hold the pair’s energy, since neither has gone anywhere. So escape needs a way of dividing the impulse between two free rings that holds exactly the starting energy.

The figure draws, for three starting ratios, the energy two free rings would have for every such division. Each curve has a maximum, and the maximum is always at the middle, where the rings are equal. That is not a coincidence of the numbers. A ring’s speed is the slope of its energy against its impulse, and a larger ring is slower, so the slope falls as impulse is added: the energy is concave in the impulse. The sum of two concave functions with a fixed total is largest when the total is split evenly. So the most energy two free rings can carry, for a total impulse $I$, is $2E\big(\sqrt{I/2}\big)$ in closed form, and the scan lands on it to rounding.

Now compare that with the pair’s starting energy. If the pair starts with more energy than two equal free rings of the same impulse could hold, no separated state is available and the rings cannot escape. They are bound, the way a planet on an ellipse is bound: not by any force holding them in, but by having too much of a conserved quantity to be anywhere else. The circulation of each ring, the other invariant the problem holds fixed, is vorticity added up over its core, and it is what sets the scale of every energy here. Here the conserved quantity that binds is an excess of energy rather than a deficit, because the interaction energy of two same-signed rings is positive and parting with it has to be paid for by the rings’ own energies, which have a ceiling.

The difference, the pair’s energy less that ceiling, is a binding margin, and it is a function of the starting state alone.

The sign of one number #

The margin rises steadily with the starting ratio, because a smaller ring closer in size to its partner shares more interaction energy with it, and it changes sign at 0.3398. Bisecting on the marched outcome — a pair counted as escaped when its gap passes eight radii within four hundred time units — puts the change at 0.3402, with the search’s bracket 0.0008 wide. The two numbers are the same to the width of the search.

Strictly, the argument only proves one direction. A positive margin forbids escape, full stop. A negative margin only permits it; nothing in the energy alone says a permitted escape must happen. That it does happen, at every ratio marched, is what the orbit picture explains, because with one degree of freedom there is nowhere else for the orbit to go.

Orbits are contours #

Fix the impulse, and draw every pair as a point: its separation along one axis, the radius of one ring along the other. The energy is a height over that plane, and every orbit is a contour line. Near the middle of the picture the contours close — a pair that starts there goes round a loop, the separation swinging positive and negative as the rings take turns in front and the radius rising and falling with it. That loop is the leapfrog. Far out the contours are open, running off to infinite separation with the radius levelling out, and a pair on one of those escapes.

Between them lies one contour, the separatrix, at the energy of two equal free rings. Its arms reach out to infinite separation at a radius of $\sqrt{I/2}$ — two equal rings, at equal speeds, drifting apart infinitely slowly. A pair starting inside it is enclosed. A pair starting outside it has a road to infinity and, having one degree of freedom, has nothing to do but follow it. That is why the energy threshold is not merely a bound but the threshold.

The system being integrable is not a detail. Three point vortices are integrable too, and four are not: one more degree of freedom and the level sets stop being curves, and nothing like this picture can be drawn. Two coaxial rings sit on the right side of that line. Two rings that are not coaxial, or a third ring, would not.

Slower and slower at the edge #

The separatrix leaves a mark on the pairs just inside it. An orbit close to the separatrix follows it a long way out before it turns back, and out there the rings are nearly equal, nearly free and nearly the same speed, so the separation changes very slowly. The time for a full exchange grows accordingly: 0.20 time units at a ratio of 0.9, where the rings barely separate at all, 8.3 at 0.5, 24 at 0.4, 64 at 0.36, 116 at 0.35 and 206 at 0.345. Halving the distance from the threshold nearly doubles the exchange time.

That has a practical reading. A pair of real smoke rings just inside the threshold would exchange once, then travel a very long way apart, and whatever the ideal-fluid picture says about a second exchange, viscosity and the rings’ own instabilities would have long since decided it. The leapfrogs that are actually filmed are the brisk ones, far inside the separatrix, with rings made alike.

Thinner cores need closer rings #

The threshold is not a universal number, and the way it depends on the core is the last thing the energy argument explains. A core a fifth of the ring’s radius binds a pair down to a ratio of 0.26. At six hundredths the threshold is 0.34; at two hundredths, 0.38; at a thousandth, 0.45; at a hundred-thousandth, 0.51; at a hundred-millionth, 0.56, and still climbing.

The reason is in the two parts of the energy. A ring’s own energy carries the logarithm of its thinness, $\ln(8R/a)$, and grows without bound as the core is thinned. The interaction energy between two rings is a property of their radii and separation and has no core in it at all. So as the cores thin, the ceiling on what two free rings can hold rises with the logarithm while the interaction that must exceed it does not, and the rings have to start closer in size to make up the difference. In the limit of an ideal filament, with no core at all, only equal rings would be bound — the same logarithm that made the ring’s own speed diverge in the essay before, reaching into the pair.

The core model moves the threshold by a few points. Holding the core radius fixed, as the earlier essay did, binds slightly less easily — 0.37 instead of 0.34 at six hundredths — because a ring that widens without thinning gains less energy. The two curves converge as the core thins, since both are dominated by the same logarithm. The three marched thresholds sit on the volume-conserving curve.

The checks #

Four checks hold the argument up, and each could have failed. The derivative of the self energy with respect to impulse matches Kelvin’s speed to a part in ten billion, for both core models, which is what makes the march Hamiltonian at all. The march conserves the energy built from those pieces to a part in a hundred billion, which would not happen if the interaction energy had the wrong normalisation or sign. The scanned maximum of the free rings’ energy lands on the closed form of two equal rings to rounding. And the energy threshold and the marched threshold agree at three core sizes, to within the width of the marched search every time.

What the model leaves out #

Viscosity. A real core diffuses, its radius growing like the square root of time, and viscosity dissipates the energy the whole argument rests on. It also breaks the conservation of circulation that Kelvin’s theorem guarantees only in an ideal fluid, so even the equal circulations of the two rings are not safe. The energy is no longer conserved, so a pair’s margin changes as it travels. Thickening the cores lowers the threshold, which binds pairs; losing energy lowers the margin, which frees them. Which wins is not something this calculation can say.

The core’s shape. The thin-core formula treats a core as circular and small, and in two dimensions that is exactly right only for a patch far from anything else. When two rings pass close to each other the cores are strained into ellipses, and at close approach the thin-core picture is no longer accurate, for the reason a strained patch keeps an elliptical shape — which matters most for the brisk leapfrogs of nearly equal rings, not for the threshold, where the rings are far apart for most of the orbit.

Stability. Coaxial is a special arrangement. A pair slightly off axis, or rings whose cores carry the waves every real ring carries, may not stay coaxial through an exchange, and the one-degree-of-freedom picture is exactly what would be lost. A ring that is stretched as it widens also has its vorticity amplified, the mechanism that feeds turbulence, and a volume-conserving core is the simplest statement of that; a wavy core stretches unevenly and the picture here has nothing to say about it.

Starting in one plane. Every pair here starts level. A pair started with the smaller ring behind the larger has a different starting energy and a different threshold; the margin is just as easy to compute for it.

Who worked it out #

Helmholtz described the leapfrog in 1858 as a thing coaxial rings do. Augustus Love showed in 1894 that for the two-dimensional analogue — two pairs of point vortices travelling together — the exchange happens only above a ratio of the pairs’ widths, and gave that ratio as $3 - 2\sqrt2$, about 0.17. Hicks and Dyson in the 1880s and 1890s worked out the energy and interaction of coaxial rings in elliptic integrals. The modern interest came from experiments and from integrable-systems theory: Borisov, Kilin and Mamaev in 2013 treated the thin-cored coaxial problem as the Hamiltonian system it is and mapped where leapfrogging happens. What is added here is the margin as a number computed from the start, checked against the march at three core sizes, and the reason the threshold climbs as the core thins.

Still open: a core that grows #

The argument above depends entirely on the energy being conserved, and in a real fluid it is not. The next calculation lets each core grow as $a^2 = a_0^2 + 4\nu t$ and lets viscosity drain energy at the rate a diffusing core dissipates it, then follows the binding margin along a march. A pair started just inside the threshold would then either drift out through the separatrix and escape after a few exchanges, or be bound more tightly as its cores thicken, and the answer would say how many leapfrogs a real pair of smoke rings of given Reynolds number should manage before it parts. The earlier essay put that number at two or three from experiment. Whether it follows from the margin is a matter of computing it.