A strained vortex holds until it has no shape to hold

The shape a vortex keeps puts a uniform patch of vorticity into a pure strain — the flow that a neighbouring vortex, or a larger eddy, exerts on it — and finds with contour dynamics what Moore and Saffman found in 1971: the patch can sit steady as an ellipse, tilted at 45 degrees to the stretching direction so that the strain turns it back exactly as fast as its own vorticity turns it on, but only while the strain is less than 0.150 of the vorticity. Below that there are two steady ellipses for each strain, a rounder and a longer one; above it there is none. Past three, an ellipse is a shear layer then computes Love’s normal modes of the free ellipse and finds it unstable to three lobes beyond an aspect ratio of three, and to more lobes beyond that.

That leaves a question the two essays set up between them and neither answers. The rounder strained ellipse at the limit has an aspect ratio of 2.89 — just short of Love’s three. If the strain lowered the threshold for three-lobed instability even slightly, the rounder branch would lose its stability before it reached the limit, and a real vortex in a slowly increasing strain would be torn apart at a strain below 0.150. Every estimate of when a strained vortex dies that uses the existence limit would then be too generous. This essay computes the stability of both branches.

Two routes to the answer #

Kida’s equations. In 1981 Shigeo Kida showed that an elliptical patch in a linear flow stays an ellipse forever, so its whole motion is two ordinary differential equations, for its aspect ratio $\lambda$ and its orientation $\theta$. In a pure strain $e$, with the vorticity set to one,

$$\frac{d\lambda}{dt} = 2e\lambda\cos 2\theta, \qquad \frac{d\theta}{dt} = \frac{\lambda}{(\lambda + 1)^2} - e\,\frac{\lambda^2 + 1}{\lambda^2 - 1}\sin 2\theta,$$

with $\theta$ measured so that the steady ellipses sit at $\theta = \pi/4$. Their steady states are exactly Moore and Saffman’s relation, $e = \lambda(\lambda - 1)/(\lambda + 1)(\lambda^2 + 1)$, recovered here to $10^{-16}$ at six strains. Linearised about a steady state, they give the exact behaviour of a disturbance that keeps the patch elliptical — the two-lobed disturbance, in Love’s numbering.

Contour dynamics. Anything else — three lobes, four, five — needs the boundary itself. The patch’s edge is marched as a polygon of 128 to 256 nodes, each moving with the velocity the whole patch induces, computed as a contour integral, plus the strain’s. It starts as the steady ellipse with a small disturbance of $m$ lobes in elliptic coordinates, and the disturbance’s amplitude is followed.

One correction was needed before the answers meant anything, and it is worth stating because it looked at first like a discovery. A patch sitting at the centre of a pure strain is at a hyperbolic stagnation point, and if it is displaced, the whole patch slides away along the stretching direction at the strain rate — exactly as a floating object does. A three-lobed disturbance of an ellipse contains a small displacement, so the first marches reported the three-lobed amplitude growing on the rounder branch at nearly the strain rate. That growth is not a change of shape. Since a linear strain is the same field about every point up to a uniform translation, the strain is now measured from the patch’s own centroid, which removes the drift and changes nothing about the shape’s dynamics; the growth disappeared.

The rounder branch holds #

The first figure is the answer for the rounder branch. At aspect ratios of 1.5, 2.2 and 2.7 — the last at a strain of 0.1496, within a thousandth of the limit — disturbances of two, three, four and five lobes were each followed for fifty vorticity times, several rotation periods of the patch. None grows exponentially. The three- and five-lobed disturbances stay within about twice their starting amplitude. The two-lobed one swings further as the limit approaches, to twelve times its start at 2.5 and 26 times at 2.7, and the four-lobed one, which it drives through the ellipse’s shape, follows it. That swing is not growth: Kida’s equations make the two-lobed disturbance exactly neutral along the whole rounder branch, and what the figure is catching is a slow, large nutation, whose period near the limit is about seventy vorticity times — longer than the time followed.

So the rounder branch holds all the way to the limit. The three-lobed disturbance that destroys a free ellipse beyond an aspect ratio of three does not destroy the strained one below 2.89, and the strain, far from lowering Love’s threshold, holds the patch in a shape that never reaches it. The existence limit is the stability limit, and the estimate that a vortex survives any strain below 0.150 of its vorticity is not too generous.

The elongated branch is a watershed #

The second figure is the two-lobed disturbance on both branches. On the rounder branch its growth rate is zero; past the limit, on the elongated branch, it rises steeply to a maximum of 0.19 of the vorticity near an aspect ratio of five and falls slowly beyond. The marched contour dynamics, at 256 nodes, gives rates within 3 per cent of Kida’s at aspect ratios of 3.6, 4.5 and 6.5 — the difference being the polygon’s own small departure from a steady ellipse, which shrinks as the nodes are added.

The elongated branch’s instability is not a lobe growing out of the edge; it is a saddle. Kida’s linearisation there has one positive and one negative real eigenvalue, of equal size, because the equations have no damping: whatever grows in one direction shrinks at the same rate in another.

The third figure shows what that means. The same elongated ellipse, nudged by a small two-lobed change of shape in one sense, is drawn out along the stretching direction and thinned towards a filament; nudged in the other sense, it turns and contracts towards the rounder shape that the same strain allows, about which it will then nutate for ever. The elongated steady shape is the boundary between the two fates — a watershed, not a state — and no real vortex is ever found on it, which is why the essay before could say that real vortices sit on the rounder branch without computing why.

A nod that slows to nothing #

On the rounder branch a disturbed ellipse nods about its equilibrium shape — its aspect ratio and its tilt oscillating together — and the fourth figure gives the frequency of that nod against the strain. For a nearly circular patch in a weak strain it is half the vorticity. It falls as the strain rises, to 0.41 at a strain of 0.10 and 0.25 at 0.14, and then falls steeply to zero at the limit, as the fourth root of the distance to it; at a strain of 0.1499 the period is seventy vorticity times. That is the restoring force of the shape disappearing: at the limit the rounder and elongated branches meet, the stable centre and the saddle merge, and there is nothing left to nod about — the same fold at which a pendant drop falls, met here from the conservative side.

A close call #

It is worth seeing how narrow the margin is. A free elliptical patch is stable to every disturbance below an aspect ratio of exactly three, and above it three lobes grow first. The rounder strained ellipses run from a circle at zero strain to an aspect ratio of 2.89 at the limit — four per cent short of three. If the strain had shifted Love’s three-lobed threshold down by more than that, the branch would have crossed it.

There was reason to expect some shift. In a free Kirchhoff ellipse the fluid runs round the boundary at a speed that varies smoothly with position, and a lobe riding on the boundary is carried round with it; that circulation is what holds a short ellipse together, as the essay on Love’s modes found. The strain changes the circulation — it speeds the flow along the stretched flanks and slows it at the compressed ends — and a lobe that lingers at a slow end has more time to grow. The computation says that this is not enough. Near the limit the three-lobed disturbance on the rounder branch stays within twice its starting size over fifty vorticity times, as it does at an aspect ratio of 1.5, and it is the two-lobed nutation, not any lobe, that shows the approach of the limit.

What the limit means for a pair of vortices #

The strain a vortex feels most often comes from another vortex. A vortex of circulation $\Gamma$ a distance $b$ away induces a strain of about $\Gamma/2\pi b^2$, and a patch of radius $a$ carrying the same circulation has vorticity $\Gamma/\pi a^2$. The ratio is $a^2/2b^2$, and setting it to the limit of 0.150 gives a core radius of 0.55 of the spacing: in this crude estimate a vortex is torn by an equal partner only when its core is more than half as wide as the distance between them.

Measured and computed mergers of two co-rotating patches happen much sooner, when the core radius exceeds about 0.3 of the spacing — the two patches roughly 3.3 radii apart centre to centre. The difference is the estimate’s crudeness, and it says what the uniform-strain limit leaves out. A partner close enough to matter does not exert a uniform strain: its field varies across the patch, pulling the near side harder than the far side, and the strained patch also turns as the pair orbits, so the strain rotates. Both lower the threshold. The limit computed here is the answer for a vortex in a strain that is uniform and steady; the partner’s strain is neither, and the essay on how a trailing pair ends is one place where the difference is the whole story.

For the trailing vortices of an aircraft, whose cores are a few per cent of their spacing, both estimates put the pair far from tearing each other. Their end comes from bending, not straining.

Past the limit, the vortex lingers #

Above the limit Kida’s ellipse has no steady state, and it is stretched without limit into a filament — a sheet of vorticity that cannot stay a sheet, and will roll up in its turn. But a vortex whose strain has crept up past the limit does not come apart at once, and how long it takes says something about the kind of transition this is.

The fifth figure starts the ellipse at the limiting shape, aspect ratio 2.89 at 45 degrees, and raises the strain above the limit by a small overshoot. The time for it to be stretched to twenty times as long as it is wide is 70 vorticity times for an overshoot of $10^{-4}$, 124 for $10^{-5}$ and 221 for $10^{-6}$ — growing as the overshoot to the power −0.2505. The vortex lingers near the shape that no longer exists, and the closer the strain to the limit, the longer it lingers.

A lost steady state still holds the film found the same lingering past the fold of a self-heating oil film, with a different exponent: there the time grows as the overshoot to the power −½. The difference is the damping. The film’s temperature relaxes: its equation is first order, and near a fold a first-order system crawls through the ghost of its lost equilibrium as $\dot x = \delta + x^2$, taking a time of order $\delta^{-1/2}$. Kida’s ellipse has no damping — its Jacobian has zero trace at every steady state — so its fold is a saddle–centre, the ellipse passes the ghost with the dynamics of a particle, $\ddot x \approx \delta + x^2$, and the time goes as $\delta^{-1/4}$. The fourth root is the signature of a conservative system losing an equilibrium, and a vortex in a strain lingers for less time than a heated film does for the same small overshoot.

Where this limit is used #

The strain limit is not only a curiosity of ideal flow. In two-dimensional turbulence the strong vortices that emerge from the inverse cascade sit in the strain of their neighbours, and a vortex whose own vorticity is more than about seven times the strain around it survives while a weaker one is drawn out into filaments; the rule of thumb in that literature is this limit, applied patch by patch. Two co-rotating vortices close enough together each put the other in a strain, and the pair that moves each other merges when that strain crosses a threshold of the same kind. The result here says the threshold can be taken at face value: there is no earlier, subtler instability waiting on the rounder branch.

What was checked #

The sixth figure lists the checks. Kida’s steady states are Moore and Saffman’s to rounding. The steady ellipse under contour dynamics, marched with no disturbance, moves by three parts in $10^4$ in ten vorticity times, which is the polygon’s resolution. The marched two-lobed growth on the elongated branch agrees with Kida’s to within 3 per cent. Every marched disturbance on the rounder branch stays bounded. And the exponent of the lingering time is 0.2505 against the saddle–centre’s 0.25.

What the picture cannot show #

A uniform patch. Real vortices have smooth vorticity profiles, and a smooth vortex in strain is stripped of its outer layers long before its core is threatened: the patch’s sharp edge makes it all-or-nothing, where a real vortex erodes. That stripping is a different calculation.

A steady strain. The strain a real vortex feels from its neighbours changes in time and turns as they move. A rotating strain can resonate with the nutation drawn in the fourth figure, and a strain whose axes turn changes what counts as a vortex at all.

Small disturbances. The stability here is linear, followed for fifty vorticity times. Finite disturbances on the rounder branch near the limit can carry the ellipse across the elongated branch’s watershed, which is one more reason a vortex near the limit is fragile even though it is stable.

Two dimensions. In three dimensions a strained vortex is also subject to the elliptical instability of its core along its axis, which a planar patch cannot show.

The convention the numbers depend on #

The vorticity inside the patch is one, so strains and growth rates are fractions of the vorticity and times are in units of its inverse. The strain is a pure strain of rate $e$, stretching along one axis and compressing along the other. The aspect ratio $\lambda$ is the ratio of the ellipse’s semi-axes. The disturbances are displacements of the boundary in the elliptic coordinate of the steady ellipse, $\cos m\eta$, and their amplitude is the Fourier coefficient at $m$ measured from the patch’s centroid.

Who found it, and when #

Kirchhoff found the rotating elliptical vortex in 1876 and Love its stability in 1893. Derek Moore and Philip Saffman found the steady strained ellipses and their limit in 1971, in work on aircraft trailing vortices. Shigeo Kida’s equations for an ellipse in a general linear flow date from 1981, and David Dritschel studied the stability of the strained ellipse with contour dynamics over a wide family of strains and rotations in 1990. Contour dynamics itself is Zabusky, Hughes and Roberts’s, from 1979.

Still open: the smooth vortex that is stripped #

A vortex with a smooth profile does not wait for the limit. In a strain its outer layers, where the vorticity is weak, are drawn off into filaments while its core, where the vorticity is strong enough for the local strain to fall below the local limit, survives; the vortex shrinks until its edge vorticity matches the strain. The next calculation gives the patch a stepped profile — two or three nested patches, each uniform — and follows it into a strain, asking whether each layer is stripped at the strain the uniform-patch limit predicts for its own vorticity, and so whether the limit computed here can be applied layer by layer to predict the size of the core that survives.