Past three, an ellipse is a shear layer

The shape a vortex keeps built a patch of uniform vorticity out of nothing but its boundary and checked Kirchhoff’s ellipse against it: an elliptical patch of semi-axes $a$ and $b$ turns rigidly, at $\Omega = \omega ab/(a+b)^2$, for ever. It then named a second threshold beside Moore and Saffman’s strain limit and left it uncomputed — Love’s result that a free ellipse is stable only up to an aspect ratio of three — with the remark that existence and stability are two questions and the second usually has a lower answer.

This essay computes it. The threshold at three turns out to be easy to derive and exact, and not alone. It is the first of a sequence of thresholds, one for each way the boundary can be disturbed, and following the sequence to very long ellipses lands on a result from a different part of the subject. An elongated vortex does not come apart in some way peculiar to ellipses. It comes apart the way a shear layer does.

Disturbing a boundary #

The boundary of a patch is all there is to disturb. The vorticity inside is uniform and moves with the fluid, so any change in the flow is a change in where the edge is. Love described the edge in elliptic coordinates, in which the ellipse is a line of constant $\xi$ and the angle $\eta$ runs round it, and disturbed it as $\xi = \xi_0 + \varepsilon\cos m\eta$: a bump with $m$ crests round the boundary.

Each such bump, in the frame turning with the ellipse, grows or oscillates as $e^{st}$ with

$$\frac{s^2}{\omega^2} = \frac14\left[q^{2m} - \left(\frac{2m\lambda}{(1+\lambda)^2} - 1\right)^2\right], \qquad q = \frac{\lambda - 1}{\lambda + 1},$$

where $\lambda = a/b$ is the aspect ratio. The two terms have simple readings. The first is the disturbance’s own strain field, which falls off as a power of $q$ because a bump with more lobes has a field that dies away faster from the boundary; the second is the rate at which the rotation of the ellipse and the flow along its boundary carry the bump round. When carrying wins the bump oscillates, and when strain wins it grows.

Two cases can be read off at once. For $m = 2$ the two terms are identical at every aspect ratio, since $4\lambda/(1+\lambda)^2 - 1 = -q^2$, and $s = 0$: a two-lobed bump on an ellipse is just a slightly different ellipse, which also rotates steadily, so it neither grows nor decays. The calculation returns this to within rounding, $3\times10^{-17}$, at five aspect ratios. For $m = 3$ at $\lambda = 3$, $q = \tfrac12$ and $q^6 = \tfrac1{64}$, while $18/16 - 1 = \tfrac18$ squares to the same $\tfrac1{64}$. The threshold is three exactly, and bisection on the sign finds it to twelve figures.

A sequence of thresholds #

The three-lobed disturbance is not the only one that turns. Every mode with more lobes crosses zero too, at a larger aspect ratio: four lobes at 4.61, five at 6.20, six at 7.77, seven at 9.35. Below its threshold each oscillates, and the three-lobed mode’s frequency falls to zero as its threshold is approached — $0.105\,\omega$ at an aspect ratio of 2.5, $0.043\,\omega$ at 2.9 — which is the usual way a stable equilibrium announces that it is about to stop being one.

The thresholds lie almost on a straight line, a new one every 1.57 or so in aspect ratio, the gaps shrinking slowly with the number of lobes. So an ellipse of aspect ratio 5 is unstable in two ways, one of 8 in four, and one of 14 in seven. The more elongated a patch, the more ways it has to come apart, and — as the last figure of this essay shows — the faster the fastest of them is.

The picture this makes is worth holding against the strain limit of the essay before. That limit, at 2.890, was about existence: above it no ellipse survives a given external strain at all. This one is about stability, and it is a whole sequence of thresholds rather than one. The two are close in value, and it would be easy to read them as the same result reached twice. They are not: one says there is no steady shape to be in, and the other says the steady shape is there and cannot be kept.

Why rotation holds a short ellipse together #

The reason a nearly round ellipse is stable, and a long one is not, is visible in the two terms of Love’s formula, and it is the same competition that decides most instabilities of rotating things.

A bump on the boundary of a patch makes its own flow. Where the boundary bulges outward, vorticity has been added outside the old edge; where it dents inward, vorticity has been taken away. The induced flow from those two changes pushes the bulges further out along one diagonal and the dents further in along the other — a strain, and a strain amplifies a bump that lines up with it. That is the first term, and it is weak for a round patch, because on a circle the bumps’ own fields cancel round the boundary, and strong for a flat one, whose two long sides feel each other across a short distance.

Against that, everything on the boundary is being carried. The ellipse turns, and the fluid on its edge runs round it faster than the ellipse turns, so a bump is swept along the boundary and through the bump’s own strain field before it can be amplified much. That is the second term. On a nearly round patch it wins easily and every disturbance merely travels round, which is exactly what a point vortex’s neighbours do to one another: orbit, rather than merge or separate. On a long patch the carrying slows — the rotation rate falls as $\lambda/(1+\lambda)^2$ — while the two long sides draw closer, and past three the strain catches the three-lobed bump before the flow can carry it away.

Nothing about this needs viscosity, and nothing in it changes the circulation. The filament that forms carries vorticity away from the core without any of it being created or destroyed, which is Kelvin’s theorem holding while the shape it holds for is lost.

The second route: contour dynamics #

Love’s formula is a linear analysis done with the boundary in elliptic coordinates, and it can be checked against a calculation that knows nothing about it. The contour dynamics of the earlier essay represents the patch by a polygon of nodes, moves each node with the velocity the whole boundary induces there, and conserves the area exactly in principle and to a few parts in ten thousand in practice. Started from an ellipse with a small three-lobed bump, it is simply marched.

The bump is then measured the way Love describes it. At each moment the nodes are rotated back into the frame of the undisturbed ellipse, each is given its elliptic coordinates, and the three-lobed Fourier component of the displacement $\xi - \xi_0$ round the boundary is taken. Nothing in the march or the measurement contains the formula it is being compared with.

At an aspect ratio of 4 the bump grows, and after an initial adjustment its logarithm is a straight line. Fitted between $t = 20$ and $60$ in units of $1/\omega$, its slope is $0.1045\,\omega$, against Love’s $0.1061\,\omega$ — 1.6 per cent apart, from a 160-node polygon whose area drifts by six parts in ten thousand over the run. At an aspect ratio of 2.5 the same bump oscillates, its size swinging between about 0.7 and 1.8 times its start with no trend, as a mode below its threshold should.

The growth rate means more when set against the ellipse’s own rotation. At an aspect ratio of 4 the ellipse turns at $0.16\,\omega$, once every $39/\omega$, and in that time the three-lobed bump grows by $e^{4.2}$, a factor of 65. An ellipse just past its threshold does not survive one turn with a visible disturbance. At an aspect ratio of 8 the fastest bump grows by $e^{9.5}$ per turn.

Three lobes, then a filament #

Started with a larger bump, three thousandths, the linear growth runs into the nonlinear stage within a turn, and the shape shows what the three lobes do. A three-lobed displacement on an ellipse does not look like three bumps. Superposed on the two ends of a long shape, it fattens one end and thins the other, and the pear-shaped patch at $t = 30$ is the result. By $t = 42$ the thin end has been pulled round by the rotation into a long curved filament of vorticity, trailing from a fatter core.

The march is stopped there on purpose. A filament thinner than the spacing between nodes cannot be represented by a polygon that does not add nodes, and this one does not: at $t = 42$ the area is still conserved to two parts in a thousand, and ten time units later it is off by six per cent. What happens next — the filament wrapping round the core and thinning further, and the core relaxing to a rounder, stable shape with less than its original aspect ratio — is what contour dynamics with node insertion and contour surgery shows, and it is not claimed here. The claim is only that the three-lobed growth leads to filamentation rather than to some other steady shape, which the drawing shows directly.

Rayleigh’s strip at the end of the sequence #

Following the fastest disturbance to very long ellipses gives the last result, and it is the one that explains all the others. The fastest rate rises steeply past three, reaches $0.15\,\omega$ at an aspect ratio of 8 and $0.19\,\omega$ at 50, and then levels off: $0.2008\,\omega$ at 1000, where the fastest disturbance has 399 lobes. The level it approaches is $0.2012\,\omega$, which is the growth rate Rayleigh found in 1880 for the most unstable wave on an infinite strip of uniform vorticity between two regions of irrotational flow.

That is not a coincidence, and the formula shows why. As $\lambda$ grows with $m/\lambda$ held fixed, $q^{2m} = \big((\lambda-1)/(\lambda+1)\big)^{2m}$ tends to $e^{-4m/\lambda}$, and $2m\lambda/(1+\lambda)^2$ tends to $2m/\lambda$. With $k = m/\lambda$ standing for a wavenumber along a strip of half-width one, that is Rayleigh’s dispersion relation for the strip, $s^2/\omega^2 = \tfrac14[e^{-4k} - (1 - 2k)^2]$, term for term. Its maximum is where $e^{-2x} = 1 - x$ with $x = 2k$, at $x = 0.797$, giving the rate $\tfrac12\sqrt{x(1-x)} = 0.2012$, and the fastest mode’s $m/\lambda$ tends to 0.3985 in the calculation.

So an elongated elliptical vortex is a shear layer, seen end to end: its two long sides are the two edges of a strip of vorticity, and its instability past three is the strip’s instability, bent round into a closed shape. The strip’s fastest wave is about eight strip widths long, and an ellipse has to be very long indeed before that many lobes fit round it; short of that, what grows is the longest wave that fits, three lobes, at a rate the rotation and the curvature of the ends hold well below the strip’s.

This also connects to the vortex sheet, which is the same strip taken to zero thickness. A sheet is unstable at every wavelength, with the rate growing without bound as the wavelength shrinks, which is why its computations blow up at the grid scale. A strip of finite thickness caps the rate at $0.2012\,\omega$ and selects a wavelength, because its thickness is a length the sheet does not have. The ellipse’s thresholds are the same regularisation seen from the other end: a patch short enough has no room for any wave to grow.

What a patch of vorticity is for #

The patch is an idealisation that keeps what the point vortex throws away — a shape — and what a point vortex is not set out how little of a patch’s motion depends on its shape. This essay adds the counterpart. How long a patch keeps a shape depends on nothing else. Two patches with the same circulation and centroid move identically in the far field, and one of aspect ratio 2.5 will do so indefinitely while one of aspect ratio 4 will have thrown off a filament and changed its shape within one rotation.

That has a use in reading real flows. Vortices strained into elongated shapes by their neighbours are common — in the wake of a bluff body, in a mixing layer, in two-dimensional turbulence — and whether such a vortex relaxes back to a round core or is torn into filaments is decided by whether it has been stretched past a threshold like this one. A vortex that is merely elliptical is a stable object. A vortex that is more than three times as long as it is wide is a shear layer waiting to roll up.

The checks #

The two-lobed mode’s $s^2$ is zero to $3\times10^{-17}$, which is the check that the formula is written correctly, because that cancellation happens only if every factor is in the right place. The three-lobed threshold is 3 to twelve figures. The thresholds rise with the number of lobes. Contour dynamics grows the bump at Love’s rate to 1.6 per cent and holds the stable ellipse’s bump below twice its size, with the area conserved to six parts in ten thousand. And the long-ellipse limit reproduces Rayleigh’s strip rate, found here by solving its own condition rather than quoted.

What this leaves out #

Anything but uniform vorticity. Real vortices have smooth profiles, and a smooth profile is not a patch. Elliptical vortices with smooth edges have their own stability thresholds, generally different, and a smooth edge adds a continuous spectrum that damps some disturbances through the mechanism that also smooths a vortex sheet into a layer.

Viscosity. Every result is inviscid. A real filament is thinned by the flow until viscosity diffuses it, and the relaxation of the core after filamentation depends on how fast that happens.

The nonlinear end state. The march stops at a filament. Where the core ends up is computed elsewhere, not here.

Three dimensions. A column of vorticity in three dimensions has instabilities along its axis as well, the elliptical instability of strained vortices among them, and none of that is in a two-dimensional patch — nor is vortex stretching, which in three dimensions can strengthen the very filament that in two dimensions only thins.

Predictability. A single patch is a problem with one boundary and a well-defined answer. Several patches interacting are not: even four point vortices are chaotic, and patches that shed filaments into one another’s fields add to that rather than subtract from it.

Who worked it out #

Kirchhoff gave the rotating ellipse in 1876. Augustus Love found its normal modes and the threshold at three in 1893. Rayleigh’s analysis of a strip of vorticity dates from 1880, and it is the first calculation of the instability of a shear layer of finite thickness — the same paper whose inflection-point theorem clears flow between two plates of any inviscid instability at all. Norman Zabusky, Hughes and Roberts introduced contour dynamics in 1979, and David Dritschel’s work in the 1980s, with contour surgery, followed unstable ellipses past filamentation to their relaxed states. What is added here is the check of Love’s rate by a march that does not contain it, and the reading of the whole sequence of thresholds as a strip’s instability wrapped round a closed shape.

Still open: an ellipse in a strain #

Every ellipse here was free. The essay before put an ellipse in an external strain and found a steady shape up to a maximum strain, with two aspect ratios for each strain below it, and it argued that the elongated one is unstable without computing that either.

The next calculation takes Love’s analysis to the strained ellipse — Moore and Saffman’s family, with the strain entering the disturbance’s equation through the changed boundary velocity — and asks where along each branch the three-lobed mode turns unstable, and whether the lower branch, the one real vortices sit on, is stable all the way to the strain limit or loses stability before it. If it loses stability first, the limit a vortex in a strain can actually reach is lower than 0.150, and every estimate of when a strained vortex is torn apart that uses the existence limit is too generous. Whether it does is a matter of computing it.