Leapfrogging rings end by merging, not by parting

Two rings leapfrog only if they start alike settles the ideal-fluid question completely. Two coaxial vortex rings set off in the same plane either pass through each other in turn for ever or part for ever, and which one happens is decided before they move. The pair is a Hamiltonian system with two conserved quantities, its energy and its impulse, and if no split of the impulse between two free rings could hold as much energy as the pair started with, the rings can never separate: the energy of their interaction binds them. The sign of that binding margin changes at a radius ratio of 0.340; above it, leapfrogging is permanent.

Real smoke rings leapfrog two or three times and stop. The argument above needed the energy to be conserved, and in a real fluid it is not: each core diffuses outward, the ring slows, and the kinetic energy drains into heat. The obvious guess is that the drain lowers the pair’s energy until the binding margin reaches zero and the rings drift apart. This essay computes what viscosity does to the margin and to the rings, and finds that the guess is wrong: the rings do not part. They merge.

Rings with growing cores #

The rings are the same thin filaments as before, each carried by the other’s exact field — computed from elliptic integrals — and by its own self-induced speed. What changes is the core. A thin ring whose vorticity is spread in a Gaussian of radius $\sigma$ moves, by Saffman’s formula of 1970, at

$$U = \frac{\Gamma}{4\pi R}\left[\ln\frac{8R}{\sigma} - 0.558\right],$$

and its energy is the quantity whose derivative with respect to the ring’s impulse $\pi\Gamma R^2$, at fixed core volume, is that speed: $E = \tfrac12\Gamma^2 R\,[\ln(8R/\sigma) - 2.058]$. The relation is checked numerically at three radii to six parts in $10^{11}$. The core radius changes for two reasons. When a ring is stretched to a larger radius its core thins, keeping its volume; and viscosity spreads it, exactly as it spreads a straight vortex whose circulation it cannot take away, so that

$$\sigma^2 = \frac{\sigma_0^2}{R} + 4\nu t,$$

with $\sigma_0$ the core at unit radius. With the circulation and the larger ring’s starting radius both one, the only new number is the vortex Reynolds number $\mathrm{Re} = \Gamma/\nu$, and time is in units of $R^2/\Gamma$.

The inviscid case is recovered by setting $\nu = 0$, and it must reproduce the earlier essay. It does: a pair of radius ratio 0.6 leapfrogs indefinitely — 394 changes of order in 800 time units — with its energy conserved to two parts in $10^{11}$.

The rings do not drift apart #

The first figure draws the pair at a vortex Reynolds number of 10,000, in the meridian plane: axial position against radius for each ring’s core. Each arch is a pass: the smaller ring, faster, runs forward through the larger one and is pulled wider as it goes; the larger, now behind, is drawn narrower and faster and runs through in its turn. The viscous pair follows the inviscid one closely, falling slightly behind as its thicker cores slow both rings. The orbit is not what viscosity changes.

The second figure follows the binding margin — the pair’s energy less the most that two free rings with the same impulse could hold, both reckoned with the cores as they are at that moment. With no viscosity it is constant. At a Reynolds number of 3,000 its lowest value over two hundred time units is 0.968 of where it started; at 1,000, over a hundred, 0.90. It falls, and it stays positive.

The reason is that diffusion lowers both sides of the comparison. The pair’s energy falls because its cores thicken, and each ring’s self-energy depends on its core only through $\ln(8R/\sigma)$. But the free rings the pair is compared with have cores that would have thickened in exactly the same way, so their energy falls by almost the same amount. The difference, which is nearly all interaction energy, is barely touched. Viscosity drains the energy that measures whether the rings are bound, and the energy that measures whether they could be free, at the same rate. The margin the earlier essay found is robust against it, and whatever ends a real pair’s leapfrogging, it is not escape.

The cores come to fill the gap #

What diffusion does change is the size of the cores against the distance between them. Each time one ring threads the other, the two cores pass close — for rings of radius ratio 0.6, within 0.40 of a radius of each other in the meridian plane — and the cores are growing.

The third figure follows the ratio of the larger core radius to the distance between the two cores. It oscillates with every pass, peaking as the rings thread each other, and its peaks climb steadily as the cores diffuse: from 0.19 at the start, to 0.24 after a few passes at a Reynolds number of 3,000, after a few dozen time units at 10,000, after sixty at 30,000.

The number 0.24 is not arbitrary. In a meridian plane the two cores are, locally, two parallel vortices of the same sign, and two same-signed vortices with Gaussian cores are known to merge — to wrap round each other and become one — once the core radius exceeds about 0.24 of the distance between them. Meunier, Le Dizès and Leweke measured and computed that threshold in 2002 for vortex pairs in the plane. Applied to the passing rings, it says when the thin-ring model has reached the end of its validity in a more interesting way than by losing accuracy: when the cores at their closest approach are fat enough that the two rings stop being two.

The count, in proportion to Reynolds number #

The fourth figure counts the passes each pair makes before its cores reach the threshold. With the threshold at 0.24: one pass at a Reynolds number of 3,000, three at 10,000, eleven at 30,000 and 39 at 100,000. With a looser threshold of 0.30: three, ten, 31 and 104. Either way the count is in proportion to the Reynolds number, and the proportionality has a closed form that the march reproduces. The cores reach the threshold when $\sigma^2$ reaches $(\theta d_{\min})^2$, with $\theta$ the threshold and $d_{\min}$ the closest approach of the cores; diffusion gets them there in a time $[(\theta d_{\min})^2 - \sigma_0^2/\alpha]\,\mathrm{Re}/4$; and the rings pass once every $T_{\text{pass}}$. So

$$N \approx \frac{\left[(\theta d_{\min})^2 - \sigma_0^2/\alpha\right]\mathrm{Re}}{4\,T_{\text{pass}}},$$

with $d_{\min} = 0.400$ and $T_{\text{pass}} = 2.03$ read from the inviscid pair. It predicts 11.9 passes at 30,000 and 39.6 at 100,000, against 11 and 39 marched.

Laboratory smoke rings are made at vortex Reynolds numbers of a few thousand, and the calculation gives them one to three passes before their cores merge — which is what is seen. Photographs of leapfrogging pairs at those Reynolds numbers show a pass or two and then the two rings coalescing into one larger ring; the stronger, faster rings of a water tank manage more.

Why the margin hardly moves #

The numbers show how little the drain touches what binds the rings. At the start, for the pair of radius ratio 0.6, the two rings’ own energies add up to 2.04 and the pair’s total is 2.70; the difference, 0.67, is the energy of their interaction, a quarter of the whole, and the binding margin is 0.60 of it. By the moment the cores reach the merging threshold at a Reynolds number of 10,000, the self-energies have fallen to 1.81 and the total to 2.48, because the cores have thickened — and the margin is 0.61. Diffusion has taken a tenth of the pair’s energy and none of the margin.

That is not a coincidence of these numbers. A ring’s self-energy depends on its core only through $\ln(8R/\sigma)$, the same logarithm for every ring whose core diffuses at the same rate, and the free rings the pair is compared with are two such rings. The core’s growth lowers the pair’s energy and the free rings’ ceiling by amounts that differ only through the rings’ radii, and at these radii the two amounts are nearly the same. The interaction energy, which depends on where the cores are and not on how fat they are, is untouched. A drain that acts on the logarithm cannot remove a binding that lives in the interaction — which is the same distinction that lets the energy of a line vortex fall for ever while its circulation does not.

What the logarithm can do is run out. As the cores grow towards the gap between them, the thin-core description stops being a description of two rings, and the cores, not the energy, end the dance — the same way a vortex held in a strain is ended by the loss of its shape rather than by any slow leak.

What the merged ring must be #

If the leapfrogging ends in a merger, the pair’s energy and impulse have to go somewhere, and they constrain what comes out. The impulse is conserved through any merger, since nothing outside pushes on the fluid: two rings of radii 0.824 and 0.826 — which is where the pair of radius ratio 0.6 is, at the moment its cores reach the threshold at a Reynolds number of 10,000 — become one ring of twice the circulation and the same total impulse, of radius 0.825. The energy cannot increase through the merger; it can only be dissipated.

That is enough to say something definite about the merged ring’s core. If the two cores simply combined their areas, the merged ring’s core would be 0.127, and a ring of circulation two, radius 0.825 and that core would carry an energy of 3.12 — more than the 2.48 the pair has at that moment. The merger cannot do that. Setting the merged ring’s energy no higher than the pair’s gives a least core of 0.188: the merged core must be at least 1.48 times the radius of the combined cores, with more than twice their area. Energy alone forces the merger to mix the two cores with fluid that carried no vorticity, spreading them further than simple addition would, and the minimum spreading is computed rather than guessed.

The same bound says the merged ring is fast. With the least core it is allowed, it travels at 0.58 in these units, against the 0.38 that the two separate rings averaged just before merging: one ring of doubled circulation outruns the pair it came from. A merged smoke ring leaves the site of the merger visibly faster than the pair approached it, which is the one piece of the whole story an observer with a stopwatch can check.

Which pairs merge soonest #

The closed form separates the count into two parts that depend on the radius ratio — the closest approach $d_{\min}$ and the time between passes $T_{\text{pass}}$ — and a factor that depends only on the Reynolds number. For rings of radius ratio 0.45, 0.5, 0.6 and 0.7 the closest approach is 0.55, 0.50, 0.40 and 0.30 of the larger radius, and the time between passes 6.1, 4.0, 2.0 and 1.0. The two shrink together as the rings become more nearly equal, and in the count they nearly cancel: at a Reynolds number of 10,000 the prediction is 3.9, 4.4 and 4.0 passes for the first three ratios, and 3, 4 and 3 marched. Then at 0.7 the count collapses to nothing, because the starting cores are already at the threshold when the rings first pass.

So there is a broad band of radius ratios over which the number of passes is set by the Reynolds number alone, and an edge, close to equal rings, beyond which there is no leapfrogging at all — the rings merge on their first meeting. The inviscid band runs from 0.34, where the binding margin changes sign, to one; the viscous band is narrower at both ends, bounded below by the margin and above by the cores.

What merges, and why the count depends on so little #

The count depends on only four things: the threshold, the closest approach, the starting core, and the Reynolds number. The closest approach and the time between passes are properties of the inviscid orbit, which the earlier essay’s energy argument fixes from the radius ratio alone. The starting core enters only by subtraction: a pair started with fatter cores has less distance to diffuse. So two pairs with the same radius ratio and the same Reynolds number merge after the same number of passes whatever their size.

The famous demonstration uses nearly equal rings, fired one behind the other, and the calculation says why it both works and stops. The inviscid theory says nearly equal rings are the most firmly bound; the viscous calculation says they also pass closest, and their cores touch soonest. Merger in the meridian plane is the axisymmetric cousin of the merger of two plane vortices of the same sign, the process by which co-rotating vortices move each other into one and by which two-dimensional turbulence builds its large eddies.

What was checked #

The fifth figure lists the checks. Saffman’s speed is the impulse derivative of the stated energy. The inviscid march conserves energy, which is the check that the mutual field and the time-stepping are right. The margin stays positive at a Reynolds number of 3,000 over two hundred time units. And the marched count of passes agrees with the closed form at two Reynolds numbers, which is the check that the count is what diffusion predicts rather than an artefact of when the march happened to sample.

What the picture cannot show #

The merger itself. The calculation stops when the cores reach the merging threshold; it does not follow two thin rings becoming one, which is a problem in the full Navier–Stokes equations; the cores wind round each other in the meridian plane as they merge, into a spiral of the kind that a rolled-up sheet leaves as a record. The threshold is borrowed from plane vortex pairs, and a passing pair of rings is only locally a plane pair for the short time of each pass, so the count could shift by a factor of order one — which is why two thresholds are drawn.

Thin rings. Saffman’s formula assumes the core is small against the ring’s radius. At the merging threshold the cores are about a tenth of the smaller ring’s radius, still thin but not negligibly so.

Axisymmetry. Real rings are unstable to waves round their circumference, and a ring several diameters old in a laboratory has usually begun to wobble; the calculation keeps the rings perfectly circular. A ring moves because it is bent, and a ring bent further moves unevenly.

One radius ratio in detail. The figures follow 0.6; the count’s dependence on the ratio enters through the closest approach and the time between passes.

The convention the numbers depend on #

The vortex Reynolds number is the circulation over the kinematic viscosity, and it is the natural one for rings: a smoke ring of a few centimetres in air has one of a few thousand. Lengths are in units of the larger ring’s starting radius and times in $R^2/\Gamma$. The core radius $\sigma$ is the Gaussian’s, the radius at which the vorticity has fallen by a factor of $e$. A pass is one change of order along the axis.

Who found it, and when #

Helmholtz described leapfrogging in 1858, in the paper that founded vortex dynamics. The thin-ring speed with a diffusing core is Saffman’s, from 1970. The first careful experiments on leapfrogging pairs were Yamada and Matsui’s, in 1978, and the energy argument for when a pair is bound is the earlier essay’s. The merging threshold of Gaussian vortex pairs in the plane is due to Meunier, Le Dizès and Leweke, from 2002.

Still open: the merged ring #

The calculation ends where the merger begins. What comes out is one ring of twice the circulation, and its radius and core — and so its speed, and how long it lasts — depend on how much of the pair’s energy the merger dissipates and how the two cores’ vorticity is rearranged. The next calculation follows the merger itself with the axisymmetric Navier–Stokes equations, from two Gaussian rings at the threshold, and asks what fraction of the pair’s energy survives into the single ring, whether its core is simply the two cores combined, and whether a merged ring from a pair that leapfrogged once travels further or less far than the pair would have done had it never met.