A vortex is a waveguide

The swirl that holds a wave still finds that a swirling flow down a pipe carries waves, and that above a critical swirl one of them cannot make headway upstream. A wake ends by bending needs the rate at which a bent vortex core turns, and takes it from a cut-off chosen to reproduce “Kelvin’s long bending wave”. Both are drawing on the same object without computing it: the family of waves a vortex core carries along its length, which Kelvin worked out in 1880.

This essay computes that family for the simplest vortex with a core — a Rankine vortex, uniform vorticity inside a radius $a$ and irrotational flow outside — and asks three things of it: what the waves look like, how good the long-wave formula every vortex-filament model borrows really is, and how fast a disturbance can travel along a vortex.

Why a spinning core is stiff #

A column of fluid rotating as a solid body resists being displaced sideways in a way still fluid does not. Push a ring of fluid outward and it keeps its angular momentum; at a larger radius, where the surrounding fluid has more angular momentum, it is spinning too slowly to stay, and the pressure gradient that holds the rotating fluid in pushes it back in. Push it inward and it is spinning too fast, and is flung back out. Rotation gives the core a restoring force, and a restoring force plus inertia makes waves.

The waves inside a uniformly rotating core are inertial waves, and they can only exist at frequencies below twice the rotation rate, as seen by the rotating fluid. Their pressure varies across the core as a Bessel function $J_m(\beta r)$, where $m$ counts the pattern’s lobes round the axis and $\beta$ is set by the frequency and the wavelength along the axis. Outside the core the flow is irrotational, and a disturbance there is a potential decaying away from the core as $K_m(kr)$. Matching the two at the core’s edge — the same radial velocity and the same pressure on both sides — gives Kelvin’s dispersion relation,

$$\frac{J_m'(\beta a)}{\beta a\,J_m(\beta a)} + \frac{K_m'(ka)}{ka\,K_m(ka)} = \frac{m}{(\beta a)^2\,\nu}, \qquad \beta^2 = k^2\,\frac{1 - \nu^2}{\nu^2},$$

with $\nu$ the wave’s frequency, as the rotating core sees it, over twice the rotation rate.

The relation is solved here by scanning for sign changes and bisecting, with the Bessel functions of the first kind computed from their integral representation and those of the second kind from theirs, and multiplied through by $J_m$ so that it has no poles to confuse the scan.

A family without end #

For the axisymmetric pattern, $m = 0$ — the core swelling and thinning along its length like a sausage — the roots come in an endless sequence. The first has no node inside the core, the next one, and so on, and each has a lower frequency than the last at the same wavelength. At a wavelength of $2\pi$ core radii the first four have frequencies of 0.71, 0.35, 0.23 and 0.17 times the rotation rate; scanning further finds more, crowding towards zero frequency, and doubling the range of the scan finds twice as many. A vortex core is not a single oscillator but a waveguide with infinitely many modes, like an optical fibre.

These are the waves the swirl that holds a wave still is about, there confined by a pipe wall. Here nothing confines them but the core’s own edge, and the long ones travel without dispersion at a speed that has a closed form: $2\Omega a/j_{0,1}$, where $j_{0,1} = 2.4048$ is the first zero of $J_0$. The roots give 0.83159 times the swirl speed at the core’s edge, against 0.83166 from the formula — a part in ten thousand at a wavelength of 628 core radii.

Carried on a uniform axial stream $W$, a long axisymmetric wave stands still when $W = 0.832\,\Omega a$, a swirl ratio $\Omega a/W$ of 1.20. That is the unbounded vortex’s version of the pipe’s critical swirl, which the earlier essay put at 1.92 for a core filling the pipe. The same idea, a wave the flow can hold still, gives a different number because the pipe wall and the free potential flow outside a core confine the wave differently.

The slow bend #

The pattern with one lobe, $m = 1$, is the one that matters most. A disturbance with one lobe displaces the whole core sideways, and a helical displacement is a vortex bent into a corkscrew. Among its infinitely many branches one is different in kind: at long wavelengths its frequency goes to zero, it involves the core moving almost rigidly, and it rotates against the flow. That is the slow bending wave, the helical wobble of a vortex filament.

For long waves Kelvin found its rotation rate in closed form,

$$\omega = -\frac{\Gamma k^2}{4\pi}\left(\ln\frac{2}{ka} - \gamma + \frac14\right), \qquad \Gamma = 2\pi\Omega a^2,$$

with $\gamma$ Euler’s constant. This is what the self-induction of a bent line vortex gives when the logarithm’s divergence is cut off at the core, and it is the formula the vortex-pair instability borrows. The roots of the exact relation approach it as the wavelength grows: at $ka = 0.3$ the exact wave turns 3.3 per cent faster than the formula, at 0.1 0.3 per cent, at 0.03 three parts in ten thousand, and at 0.01 three parts in a hundred thousand.

For short waves the two part company in kind. The long-wave formula contains $\ln(2/ka)$, which falls through zero when $ka$ passes $2e^{\gamma - 1/4} = 1.44$, so the formula predicts that a short enough bend stops turning and then turns with the flow. The exact wave does nothing of the sort: at $ka = 1.44$ it turns against the flow at 0.45 times the rotation rate, and at $ka = 4$ at 0.81, approaching the core’s own rotation. Past a wavelength of about four core radii the long-wave formula describes nothing real.

That settles a question the vortex-pair essay left hanging. Its growth-rate formula, built on the cut-off, contains a narrow band of instability for bends about a third of the spacing long — about four core radii for a wake’s cores — where the cut-off self-rotation passes through zero and the partner’s strain meets no opposition. The exact wave shows the self-rotation never passing through zero there. The band is an artefact of extending the long-wave formula past $ka = 1.44$, and discarding it, as that essay did, was right for a reason that can now be computed.

Why the bend turns backwards #

That the slow bend turns against the flow is the least intuitive thing about it and the most important. A core displaced sideways as a whole might be expected to be carried round by the swirl, as a leaf on the edge of a whirlpool is. It is not, because the displaced core is not a passive object in someone else’s flow: it is the vortex, and a bent vortex line moves under its own induction. Each bent element sees the rest of the line curving away from it, and the velocity the rest induces at the element is perpendicular to the plane of the bend, along what geometry calls the binormal. For a helix, the binormal velocity carries every element round the axis in the direction opposite to the vortex’s own rotation.

So the helical wobble turns backwards, at a rate set by the vortex’s circulation, the square of the bend’s wavenumber and the logarithm of the ratio of wavelength to core — slowly for long bends and faster for short ones. At a wavelength of $2\pi$ core radii the exact wave turns at a third of the core’s rotation rate, against it. The same self-induction is what makes a ring move: a ring is a bend closed on itself, and its binormal is along its axis.

Waves that turn with the flow #

The slow bend is one branch of the one-lobed family among infinitely many. At a wavelength of $2\pi$ core radii the others turn at 0.51, 0.72 and 0.81 of the rotation rate in the flow’s direction, and a further set at 1.49, 1.28 and 1.19; each has more radial nodes than the last, and as the nodes multiply the frequencies close in on the rotation rate itself. Those are waves the core’s fluid largely carries round with it, wrinkles in the core’s interior riding on its rotation, and they are the ones a smooth core damps first, because each finds somewhere outside the core where its speed matches the local swirl and gives its energy to the fluid there.

The distinction matters for what can be observed. The slow bend displaces the whole core and is visible wherever the core is marked — by condensation in a wingtip vortex, by vapour in a cavitating one, by dye in a laboratory tank. The fast branches distort only the core’s interior, and are seen only with instruments inside it.

A vortex that sings #

The clearest place Kelvin waves show themselves outside a laboratory is under water. The low pressure in the core of a propeller’s tip vortex can fall below the vapour pressure, and the core fills with a thin tube of vapour — a cavitating vortex, visible as a silver thread trailing from each blade. Such a thread can oscillate in the bending and axisymmetric modes computed here, and when an oscillation’s frequency locks onto the rate at which the blades pass, the thread “sings”: it radiates a strong tone that ships and submarines are designed to avoid. The frequencies involved are those of this essay’s branches for a core whose radius is the vapour tube’s, which is why predicting the tone means solving Kelvin’s relation for a hollow core rather than a Rankine one — the same problem with a different boundary condition at the core’s edge.

How long a bend must be for the cut-off to be right #

The practical question is how long a bend has to be before the formula, and every vortex-filament calculation built on it, can be trusted. The ratio of exact to formula falls below 1.01 at a wavenumber of about 0.17 over the core radius, a wavelength of about thirty-seven core radii. The fastest-growing bend of a trailing vortex pair, with cores a tenth of the spacing across, has $ka$ of about 0.07, where the formula is good to a fifth of a per cent — so the vortex-pair result rests on the formula where the formula is sound.

The same comparison says what a filament model cannot do. Bends a few core radii long are neither rare nor unimportant: the elliptic instability of strained vortices, the Kelvin waves excited when a vortex is struck, and the short waves on the cores of rings and tip vortices all live there. For them the core’s own structure is the physics, and a line with a cut-off is the wrong model.

How fast news travels along a vortex #

A disturbance at one point of a vortex — a boat crossing a waterspout, a wing slicing through another aircraft’s wake, an obstacle in a swirling pipe — is carried along the core by these waves, and the fastest a disturbance can spread is the largest group speed any of them has. For the first axisymmetric branch that is at the long-wave end, 0.832 of the swirl speed at the core’s edge; shorter waves carry energy slower, at 0.71 of it for a wavelength of $4\pi$ core radii and 0.33 at $\pi$.

The distinction between the two speeds is the usual one for a dispersive medium and it matters here. A wave’s crests travel at the phase speed, but a disturbance of finite length is a packet of many wavelengths, and the packet — and the energy it carries — moves at the group speed, which for these waves is always the smaller. A sharp jolt to a vortex therefore spreads along it as a front moving at the long-wave speed, trailed by shorter ripples that arrive later and later.

For a trailing vortex behind an airliner, with a circulation of 508 m²/s and a core radius of about 4.6 metres, the swirl speed at the core’s edge is 17.6 metres a second and a disturbance runs along the core at up to 14.6. A dust devil a metre across swirling at five metres a second carries its disturbances along at four. The speeds are set by the swirl, not by any property of the fluid, which is the sense in which the core is the waveguide and the fluid only its material.

Where the waves meet the rest of the subject #

The waves appear elsewhere in the subject under other names. A ring moves because it is bent, and the ring’s speed has the same logarithm of its radius over its core as the slow bend, because a ring is a bend closed on itself. The standing axisymmetric wave is what a vortex breaks down through when its swirl passes the critical value, which is the whole of the pipe’s criticality. And the slow bend, in a pair of vortices, is what each vortex’s partner strains into the long-wave instability that ends a wake.

What unites them is that a vortex core has an inside. A point vortex has none, and so has no waves and no way of bending without an infinite self-induced velocity; every one of these effects is the core’s structure reaching out into a question the point model cannot ask.

What the picture cannot show #

A Rankine vortex. The vorticity jumps from uniform to zero at the core’s edge. A smooth vortex, such as the Lamb–Oseen vortex a real core relaxes to, has a critical layer wherever a wave’s speed matches the local swirl, and many of its Kelvin waves are damped there rather than propagating freely. The long bending wave survives nearly unchanged; many of the short ones do not.

No axial flow in the core. Real trailing vortices and tornadoes carry a jet or a wake along their axis, which Doppler shifts the waves differently at different radii and can make some of them unstable.

Linear waves. Every amplitude is infinitesimal. Large bends interact, steepen and break, and the vortex-breakdown bubble of the earlier essay is what a finite axisymmetric wave becomes.

The convention the numbers depend on #

The core radius is $a$ and the core rotates at $\Omega$, so its vorticity is $2\Omega$, its circulation $\Gamma = 2\pi\Omega a^2$ and the swirl speed at its edge $\Omega a$. Frequencies are in the laboratory frame in units of $\Omega$, negative when a pattern turns against the flow; the wavenumber is along the axis, in units of $1/a$. The long-wave formula is the one quoted above with the minus sign that makes the bending wave retrograde.

Who found it, and when #

Lord Kelvin published the waves on a columnar vortex in 1880, including the long bending wave’s rotation. The use of a cut-off to reproduce his long-wave result in a filament model was made systematic by Moore and Saffman in 1972, whose treatment is why the cut-off radius for a uniform core is $0.642a$. The damping of Kelvin waves by critical layers in smooth vortices was mapped comprehensively only in the 2000s, notably by Fabre, Sipp and Jacquin in 2006.

Still open: the waves a smooth core damps #

The Rankine vortex’s waves last for ever; a real core’s do not all. The calculation that follows solves the same eigenvalue problem for the Lamb–Oseen vortex, whose vorticity falls smoothly as a Gaussian, where the discrete waves above become a mix of genuine waves and critically damped ones, and asks which of the branches drawn here survive: whether the slow bend a wake relies on is among them, as it is thought to be, and at what wavelengths the axisymmetric waves that carry disturbances along a vortex are absorbed rather than transmitted. It would say how far along a real vortex a disturbance can travel before the core itself has soaked it up.