A strained vortex loses its ring at the limit of what it holds

A strained vortex holds until it has no shape to hold put a uniform patch of vorticity into a pure strain and settled what happens. Up to a strain of 0.150 times the patch’s vorticity it takes a steady elliptical shape, one of two Moore and Saffman found, and the rounder of the two is stable all the way to the limit. Beyond the limit there is no steady shape, and the patch is stretched out into a filament. So for a uniform patch the existence limit is the whole answer.

A real vortex is not uniform. Its vorticity is greatest at its centre and falls off outward, smoothly in a vortex shed from a wing or spun up in the atmosphere. In a strain such a vortex is not torn apart all at once. Its outer layers, where the vorticity is weak, are drawn off into filaments, and its core, where the vorticity is strong, survives — the vortex shrinks to the size the strain allows. That essay ended on the natural way to predict that size: apply the uniform limit layer by layer. Each layer would go when the strain reached 0.150 times its own vorticity, and the core that survives would be everything whose vorticity is more than the strain over 0.150.

That reading is wrong, and by a large factor. The way to see it is to build the simplest vortex with more than one layer — a core inside a ring — and watch which strain takes the ring.

A core inside a ring #

The vortex here has two uniform layers: a ring of vorticity one, out to a radius of one, round a core of stronger vorticity out to a smaller radius. Each layer is uniform, so the shape a vortex keeps’s method of contour dynamics still applies: the whole flow is set by the two edges, the outer one carrying a jump in vorticity from the ring’s value to nothing and the inner one a jump from the core’s value to the ring’s, and each edge moves with the velocity both edges induce plus the strain. Seven vortices were built this way, with cores from 0.4 to 0.8 of the radius and core vorticities from 1.5 to 4 times the ring’s.

The strain is not switched on at once. It is raised steadily from nothing over forty times the ring’s rotation time and then held, so that the vortex is led along its family of steady strained shapes rather than shaken into one. An edge counts as stripped when some part of it reaches more than three times its starting radius from the vortex’s centre: a circle is at one, the steady ellipse at the uniform patch’s limit at 1.7, and an edge past three has been pulled into an arm. For each vortex the strain that strips the ring is found by bisection, to four thousandths.

The method is calibrated on the uniform patch, run the same way. It strips at 0.146, three per cent below Moore and Saffman’s exact 0.150: a ramp of finite length and a finite run let the patch go slightly early, at the strain where the stretching becomes irreversible within the run. Every stepped result is read against that 0.146 rather than against 0.150, so that the method’s own bias cancels.

The first figure shows what stripping looks like for the vortex with a core of twice the ring’s vorticity out to six-tenths of the radius, in a strain of 0.245. At first both edges lean into ellipses at forty-five degrees to the stretching, as a strained vortex does. Then the ring’s edge is drawn out at its two tips into arms along the stretching direction, and the arms lengthen while the core inside barely changes: it stays close to an ellipse of modest aspect. The ring is being stripped and the core is not.

The second figure follows the two edges’ reach through time either side of the threshold. In a strain of 0.215 both edges nod as the strain rises and settle, neither reaching past one and a half times its starting radius: the vortex has found a steady shape. In a strain of 0.24 the ring’s edge runs: it passes three radii at forty-nine time units and keeps going, until its filaments outgrow the calculation’s budget of points eight time units later. The core’s reach over the same time never passes 1.55. The threshold between the two behaviours for this vortex is a strain of 0.227.

The ring holds far past its own limit #

That 0.227 is the number that settles the question. The ring’s own vorticity is one, so the layer-by-layer reading says it should go at 0.146, the calibrated uniform limit. It holds to more than one and a half times that.

The third figure shows the same comparison for all seven vortices. It plots the ring’s stripping strain as a multiple of the uniform limit of its own vorticity, against the mean vorticity enclosed by the ring’s edge — the vortex’s whole circulation divided by its area — in units of the ring’s vorticity. On the layer-by-layer reading every point would lie at one. None does. They run from 1.3 for the weakest core to 3.35 for the strongest and largest, and they rise in step with the mean enclosed vorticity, lying close to the diagonal rather than the horizontal.

The reason is that a layer of a vortex does not rotate at its own vorticity. A fluid particle on the ring’s outer edge goes round the vortex at a speed set by all the circulation inside its orbit, not by the vorticity of the fluid immediately around it. The strain tries to pull the edge out along one axis; the vortex’s rotation carries the edge round before the strain can finish the job; and how fast it is carried round is the enclosed circulation over the radius. A uniform patch’s limit is a statement about that competition between strain and rotation, and for a uniform patch the rotation is set by its vorticity only because its own vorticity and its mean enclosed vorticity are the same number. For a stepped vortex they are not, and the rotation follows the mean.

The limit of the vorticity enclosed #

The fourth figure plots the ring’s stripping strain directly against the uniform limit applied to that mean enclosed vorticity. The seven points lie on a line, and just above the diagonal: from 1.11 to 1.16 times the prediction. The uniform patch, the calibration, sits on the diagonal by construction.

So the uniform patch’s limit does apply layer by layer, if the layer’s “vorticity” is read as the mean vorticity it encloses. Read that way it predicts each ring’s stripping strain to within sixteen per cent across a range in which the own-vorticity reading is off by a factor of up to three and a third. The remaining error has a consistent sign. A stepped vortex holds its ring a little longer than a uniform patch of the same mean vorticity would hold its edge, and the likely reason is the core’s stiffness: a strong core deforms less in the strain than the uniform patch’s interior would, and a ring wrapped round a rounder core is held to a rounder shape.

What survives in the middle #

The core is the other half of the question, and it is harder to follow, because by the time the strain takes the core the ring’s arms are long filaments that the calculation must carry with ever more points.

The fifth figure shows the strains that took the core, for the five vortices where the calculation could follow it. For a core, its own vorticity and its mean enclosed vorticity are the same number, so the two readings agree. In four of the five the core goes within a tenth of its own limit, from 0.91 to 1.03 times it — the core, left behind by its ring, behaves very nearly as a uniform patch of its own vorticity. The fifth, a large core of four times the ring’s vorticity out to eight-tenths of the radius, held to 1.3 times its limit; its thin ring was stripped early and part of the ring’s fluid stayed wrapped round the core, adding to what it encloses. The two vortices with strong small cores are missing: their rings’ filaments exhausted the calculation’s budget of points before the core went. That is an honest limit of contour dynamics without surgery, which would cut the filaments off and carry on.

What stripping leaves behind #

The stepped vortex also shows something about the shape of what survives. Before the strain, the vortex had two edges and two vorticities. After the ring is stripped, what remains is the core with, at most, a thin skin of ring fluid wrapped round it, and the jump in vorticity at its edge is the full jump from the core’s value to the near-nothing of the stripped filaments. Stripping does not smooth a vortex. It sharpens it: whatever the profile began as, the surviving vortex ends with a steep edge, at the radius where the enclosed rotation just beats the strain.

That is a qualitative difference from what viscosity does to a vortex, which is to spread it, and it is why strained vortices in two-dimensional turbulence are seen with sharp edges and smooth cores rather than as the Gaussians a diffusing vortex becomes. The filaments themselves are the other half of the picture. Each arm of stripped ring is a thin strip of vorticity being stretched along the strain, a shear layer in all but name, and a sheet that cannot stay a sheet found what a thin layer of vorticity does when it is left alone: it rolls up. In a strong enough strain it is not left alone, and the stretching thins it faster than it can roll — which is why the filaments in the first figure grow long and straight.

The threshold itself has the character the drop falls at a fold and a lost steady state still holds the film described for other flows. Below the stripping strain the vortex has a steady shape to settle into; above it, that shape has met its twin and vanished, and the ring lingers near where the shape used to be before it runs. The second figure’s run at 0.24 shows the lingering: the ring’s reach creeps from 1.3 to 1.5 radii over the last ten time units of the ramp and the first few after it, and only then accelerates, passing three radii at forty-nine — the ghost of the lost steady state slowing the escape, as it did for the uniform patch.

What the reading predicts for a smooth vortex #

The sixth figure carries the two readings over to a smooth vortex, to show how much the difference matters. A Lamb–Oseen vortex has vorticity falling from $\omega_0$ at its centre as a Gaussian of unit width. Read layer by layer, a strain of $0.05\,\omega_0$ strips every radius whose own vorticity is below a third of the peak, which is everything beyond 1.05 widths. Read by the mean vorticity enclosed, with the factor of 1.15 the stepped vortices showed, the same strain strips only beyond 1.82 widths: the surviving core is nearly twice as wide, and holds three times the area. At the strongest strains the two readings agree, since at the centre own and mean are the same; at weak strains the gap grows without limit, because the mean enclosed vorticity of a Gaussian falls off as the inverse square of the radius while its own vorticity falls off as a Gaussian.

The figure is an extrapolation and is labelled as one. Two steps are not a continuum, and the factor of 1.15 is measured on two-step profiles. What the stepped vortices establish firmly is which quantity the limit reads: the vorticity enclosed, not the vorticity present. That much does not depend on the number of steps, because it is a statement about how fast an edge is carried round.

What the calculation was checked against #

With no strain the stepped vortex is a nested Rankine vortex, which is steady: run for thirty time units, both edges stay circles to rounding, and the enclosed areas, which contour dynamics should conserve exactly, drift by four parts in a million. With the core’s vorticity equal to the ring’s, the inner edge carries no jump and the calculation is the uniform patch’s; it strips at 0.146, the calibration above. The seven stripping strains were each found by bisection to four thousandths and are stored in a table, because each costs a minute or two of contour dynamics. One of them is re-bracketed whenever the calculation is re-run: the vortex of the first figures holds its ring at 0.006 below its stored threshold and loses it at 0.006 above. The checks refuse a core larger than the vortex, a core weaker than its ring and a negative strain.

What the calculation leaves out #

Surgery. Contour dynamics carries filaments for ever, and they take ever more points. Dritschel’s contour surgery cuts filaments finer than a set scale, which is what would let the cores of the two missing vortices be followed.

Smooth profiles. The rule is measured on two-step vortices. A three-step vortex, or a smooth one approximated by many steps, would test whether the factor of 1.15 is a constant or creeps with the profile, which is the next thing to establish.

How the strain arrives. The strain here rises over forty rotation times. A strain switched on suddenly overshoots and strips more, and a strain that rotates — as a neighbouring vortex’s does — is a different problem, which two strainings and the order they came in began on.

Three dimensions. A real vortex in a strain can also be stretched along its own axis, which strengthens it. The two-dimensional calculation has no axial stretching.

The rotation decides #

Past three, an ellipse is a shear layer found that a free elliptical vortex is stable up to an aspect of three and that its instability is a shear layer’s. Vortices move each other showed how one vortex’s field is another’s strain. What this calculation adds is the reading that connects a vortex’s profile to its survival: an edge survives a strain if the vortex inside it rotates fast enough to carry the edge round before the strain can pull it out, and how fast it rotates is the circulation it encloses, not the vorticity it touches.

That is why a vortex’s core is so much tougher than its weakest layers suggest. Every snapshot says vortex found how ambiguous the word is in a turbulent field, where strained vortices are everywhere; the useful size of such a vortex is the radius within which its enclosed rotation beats the strain it sits in. A layer-by-layer estimate shrinks every vortex by too much, because it lets each layer fight the strain alone, when every layer has the whole of the vortex inside it on its side.

Who worked it out #

Moore and Saffman found the strained ellipse’s limit in 1971 and Kida its evolution in 1981. Zabusky, Hughes and Roberts introduced contour dynamics in 1979, and Dritschel made it, with surgery, the standard tool for vortex stripping. Legras and Dritschel’s work of the early 1990s on strained and stripped vortices with smooth profiles, and Mariotti, Legras and Dritschel’s on vortex stripping and the erosion of coherent structures, established that a strained vortex is eroded to a sharp-edged core whose edge vorticity is set by the strain, and that the erosion depends on the profile through more than its local vorticity.

Still open: a vortex of three steps #

Two steps establish which quantity the limit reads, but not whether the factor of 1.15 above it is universal. The next calculation builds vortices of three nested layers with the same total circulation and different profiles — a steep one, a gentle one, a Gaussian sampled at three radii — strips them in the same ramped strain, and asks whether each layer still goes at a fixed multiple of the uniform limit of the vorticity it encloses, whether that multiple depends on how much stiffer the vorticity inside it is, and so whether a single rule, fitted on steps, predicts the core a smooth vortex keeps.