A wake ends by bending, not by fading

Where the wake ends up settles where the two rolled-up cores behind a wing must sit, and a wake that says what made it reads the aircraft’s weight and span back out of them. Both essays name the thing that eventually destroys the pair, and neither computes it. The second lists that absence among its limits, because the instability that ends a wake in calm air acts on the same time scale as the decay it did model.

This essay computes it. The instability was found by Crow in 1970, and it has a feature unusual for anything in turbulence: it is linear, it has a closed form, and its prediction can be checked against photographs of condensation trails taken from the ground. The pair does not wear away. It bends, and each vortex’s own partner makes the bend grow.

The strain one vortex puts on the other #

Take two straight vortices of equal and opposite circulation, a distance $b$ apart. Each carries the other downward at $\Gamma/2\pi b$, which is why a wake sinks below the flight path. Seen from the frame that sinks with them, the partner’s velocity field near each core is more than a uniform drift. It also varies across the core, and a velocity that varies from place to place is a strain: a stretching along one diagonal and a squeezing along the other, at forty-five degrees to the line joining the pair.

Now bend one of the vortices, giving it a small sinusoidal wave along its length. The partner’s strain acts on the bend. The part of the displacement lying along the stretching diagonal is pulled further out, which is growth. Meanwhile the bend itself does something a straight vortex does not: a curved vortex line induces a velocity on itself, perpendicular to the plane of its curvature, and that velocity rotates the bend about the core’s axis. The rotation carries the displacement out of the stretching diagonal into the squeezing one.

The two effects are the whole of the calculation. The partner’s influence, linearised in the displacements, reduces to two functions of $\beta = kb$ that come out of the Biot–Savart integral in modified Bessel functions:

$$\psi(\beta) = \beta^2 K_0(\beta) + \beta K_1(\beta), \qquad \chi(\beta) = \beta K_1(\beta).$$

Both tend to one for very long bends, which says that a displacement much longer than the spacing simply moves the whole partner, and both fall away for short ones, because a bend much shorter than the spacing averages out before its influence reaches across. The self-rotation is the term the figure shows rising steeply. With time measured in units of $2\pi b^2/\Gamma$ — the time the pair takes to sink through its own spacing — the growth rate of a bend in which the two vortices are displaced as mirror images is

$$\alpha^2 = \left[1 - \psi + \beta^2\varpi\right]\left[1 + \chi - \beta^2\varpi\right],$$

where $\beta^2\varpi$ is the self-rotation. The first factor is small for long bends, because $\psi$ is close to one there; the second is large. The self-rotation makes the first grow and the second shrink, and the product is positive — the bend grows — only while the second has not yet reached zero.

A band of bends that grow #

The band is long-wave. The mirror-image bend grows at every wavelength longer than 5.73 spacings and fastest at 8.54 spacings, where it multiplies by $e^{0.827}$ — a factor of 2.29 — in each descent time. Behind a wing with elliptic loading the core spacing is $\pi/4$ of the span, so the fastest bend is about 6.7 spans long. Behind an airliner of sixty-metre span it is a little over four hundred metres, which is why the waving of a condensation trail is visible from the ground at all.

The bend in which the two vortices move in step — the antisymmetric mode — has the signs of both mutual terms reversed. Its first factor, $1 - \chi - \beta^2\varpi$, is negative across the whole band, so it does not grow anywhere the mirror-image bend does. That is the reason a wake is seen to pinch rather than to snake: a sideways sway of the whole pair would need the antisymmetric mode, and the pair’s own induction damps it.

A number that seems to need more explanation than it gets is the time unit. A pair descending one spacing is the natural clock because the descent speed and the strain are the same induction seen at two distances: the velocity $\Gamma/2\pi b$ carries the pair down, and its variation across the core, of order $\Gamma/2\pi b^2$, is the strain. A growth rate of order one in that unit is the statement that the pair destroys itself in a few of its own descents, whatever the aircraft, and the whole arithmetic of wake lifetime follows from it.

Mirror images, tilted at forty-eight degrees #

The growing bend has a shape as well as a rate, and the shape is fixed by the ratio of the two factors. Each vortex is displaced along a line tilted from the horizontal by an angle whose tangent is the square root of the second factor over the first. At the fastest wave that angle is 47.7°.

The angle is where the argument about strain and rotation lands. A bend exactly on the stretching diagonal, at forty-five degrees, would gain most from the strain if nothing rotated it. The self-rotation turns it continually towards the squeezing diagonal, and the bend that survives is one set slightly beyond the stretching direction, so that the rotation carries it back through the favourable orientation rather than out of it. The difference from forty-five degrees is the rotation’s signature.

Seen from above, the mirror-image bend makes the two vortices approach at every other crest and separate at the crests between. The approaching sections come within a core’s width of each other, where the two vortices’ opposite vorticity annihilates, and the vortex lines reconnect across the gap: the pair becomes a train of vortex rings, each about one fastest wavelength long, which is the “string of sausages” in photographs of old contrails. What happens after that — the rings bobbing, tilting and breaking up — is outside a linear calculation. What the linear calculation has fixed is the length of the rings and the time at which they form.

The core hardly enters #

The only place the vortices’ internal structure appears is the self-rotation, and there it enters through a single length, the cut-off $d$. A line vortex of zero thickness induces an infinite velocity on itself when bent, because the Biot–Savart integral diverges logarithmically near the point where it is evaluated. Crow’s device is to leave out a length $d$ either side of that point. The self-rotation function is then

$$\varpi(\delta) = \tfrac12\left[\frac{\cos\delta - 1}{\delta^2} + \frac{\sin\delta}{\delta} - \operatorname{Ci}(\delta)\right], \qquad \delta = kd,$$

which for long bends reduces to $\tfrac12[\ln(1/\delta) + \tfrac12 - \gamma]$. A logarithm is the reason the core hardly matters.

The cut-off is not a fitting parameter. For a vortex whose vorticity is uniform across a core of radius $a$, the long bending waves of the core are known exactly from Kelvin’s 1880 analysis, and the cut-off that reproduces their rotation rate is $d = a\,e^{1/4}/2 = 0.642a$. The two expressions agree to a part in a hundred thousand at $ka = 10^{-3}$; a vortex is a waveguide computes Kelvin’s waves in full and shows where this long-wave form of them stops being enough. The core of an elliptically loaded wing’s rolled-up wake has a radius of about 0.098 of the core spacing, which gives $d/b = 0.063$ and the 8.54 spacings above.

Changing the core tenfold smaller puts the fastest bend at 11.3 spacings; threefold larger, at 6.6. The growth rate moves from 0.868 to 0.791. An aircraft designer who changed the core size of a wake by an order of magnitude — which is roughly the difference between a clean wing and one with flaps and spoilers deployed — would change the wake’s lifetime by a few per cent. That is disappointing for anybody hoping to engineer a short-lived wake by fattening its cores, and it is the reason the deliberate schemes all work on the pair’s arrangement instead.

Two routes to the same kernels #

The two mutual-induction functions are the load-bearing part of the result, and their derivation uses three integrals of the form $\int e^{iks}(s^2 + b^2)^{-n/2}\,ds$ that are easy to get wrong by a sign or a factor of two. So they were computed a second way that shares none of that algebra.

The partner is built as a polyline: 256 straight segments a wavelength, over a hundred and twenty wavelengths each side, with its two far ends closed by straight semi-infinite lines. A straight segment’s induced velocity has an exact closed form with no series in it. Summing thirty thousand of them at the displaced point of the first vortex, once with the partner straight and once with it bent by a ten-thousandth of the spacing, gives the perturbation velocity directly. Divided by the amplitude it must be $\psi$ in one direction and $\chi$ in the other, and it is, to $10^{-4}$ at four wavenumbers from 0.3 to 3. The residual falls as the square of the segment length, which is the error of replacing a curve by chords. The same sum returns the terms in which the first vortex’s own displacement moves it through the partner’s unbent field, which must be exactly minus one and are.

The self-rotation function was checked the same way, by integrating its defining integral numerically over four thousand radians, and it agrees with the closed form to $2 \times 10^{-6}$ at cut-offs from 0.01 to 1.2. No quoted number above depends on a formula that was not recovered by a second route.

Two to three minutes, almost whatever the air does #

The airliner of the essay before carries a circulation of 508 m²/s on a core spacing of 47.1 metres. Its pair sinks at 1.72 metres a second and takes 27.4 seconds to sink through its own spacing, so the fastest bend’s amplitude e-folds every 33 seconds. The cores meet when the lateral part of the displacement reaches half the spacing — a total displacement of 0.742 of the spacing along the tilted line. From a starting bend of one per cent of the spacing, which is half a metre, that takes 5.2 descent times: 143 seconds, by which time the pair has sunk about 245 metres below the flight path.

The linear rate has certainly stopped being accurate before the cores touch, and the number is an estimate of when linking happens rather than a measurement of it. But the estimate has a property the details cannot change. The time is proportional to the logarithm of the ratio between the final and starting amplitudes, so a starting bend ten times smaller costs only $\ln 10/0.827 = 2.8$ descent times more. From a hundredth of a per cent the pair lasts 296 seconds; from ten per cent, 67. The quietest air measured is not quiet enough to make the answer much longer than five minutes, and ordinary atmospheric turbulence, which supplies a disturbance of a few per cent, makes it about two.

That is the calculation the phrase “a wake’s lifetime” is really a report of. It predicts a lifetime set by the aircraft’s own numbers, weakly modified by the weather. Crow and Bate’s 1976 comparison with measured wakes found that shape, with turbulence mattering through a logarithm while it is weak and through direct erosion of the cores only once it is strong.

What a contrail’s shape records #

The prediction is unusually easy to check against the sky. A condensation trail is the pair made visible by the water frozen in the cores, and in calm air it is seen to develop a slow, regular waviness a minute or two after the aircraft passes, then to pinch into a row of loops. Measuring the loop spacing in a photograph of known scale gives the wavelength directly, and the calculation says what it must be: between about six and eleven core spacings for any plausible core, and 8.5 for the core an elliptic loading rolls up into. A loop spacing far outside that band would mean the pinching had some other cause.

The pinching also explains a detail of the photographs that looks as if it needs a separate cause: the loops form nearly all at once along the trail. That is what an instability with a single fastest wavelength does when it grows from a broad spread of small disturbances — the fastest one outruns the rest by a factor that increases every descent time, and after five of them it is the only one visible. A trail that pinched at irregular intervals would be a trail whose growth was being driven from outside, by a gust or a thermal, rather than by the pair itself.

Shortening a wake on purpose #

Airport separation rules exist because wakes last, and a wake that could be made to end sooner would let aircraft land closer together. Since the growth rate is fixed by the pair, the idea that follows is to start the fastest bend at a larger amplitude rather than to make it grow faster: oscillate a control surface at the frequency with which the aircraft flies through one fastest wavelength. At an approach speed of 70 metres a second and a wavelength of 402 metres, that is 0.17 hertz — a gentle rocking of the ailerons or spoilers every six seconds.

The logarithm is what limits the idea. Raising the starting bend from one per cent to ten per cent saves 76 seconds out of 143, and raising it further runs into the fact that an aircraft cannot rock its own wake by a spacing without rocking itself. The schemes that have worked better in water tunnels use a wing loading that sheds four cores rather than two. Pairs of unequal vortices orbiting each other have instabilities whose growth rate is set by the stronger strain of the closer neighbour, and those can be several times faster than Crow’s.

Short bends, where the model has nothing to say #

The growth-rate formula also has a narrow band of instability at a wavenumber of about 16.8 — a bend 0.37 spacings long, about four core radii. It lies where the cut-off $\delta = kd$ is close to one and the self-rotation function passes through zero, and it is not physics: at wavelengths comparable to the core, the cut-off model of a core no longer represents what the core does, and the real short-wave instability of a strained vortex is a different mechanism — the core’s own internal waves resonating with the strain, found by Widnall, Bliss and Tsai and by Moore and Saffman in the 1970s. The long-wave band is safe from this objection by a factor of more than ten in wavelength. The short band is reported here because the formula contains it, and discarded because the formula has left its domain there.

What the picture cannot show #

Linear growth. Every rate is the rate of an infinitesimal bend. The time to linking extrapolates that rate to a displacement of three-quarters of the spacing, where the linear theory is certainly wrong in detail. What survives the extrapolation is the logarithmic dependence on the starting amplitude, because nonlinear growth near the end is fast and takes a comparable time whatever the start.

No reconnection. The rings form by viscous annihilation where the cores meet, and nothing here contains viscosity or the cores’ vorticity distributions. The lengths of the rings are the linear wavelength; their subsequent behaviour is not computed.

An unbounded, unstratified atmosphere. Near the ground the pair spreads apart instead of sinking and the instability changes; in a stably stratified atmosphere the pair’s descent is slowed by buoyancy and the spacing shrinks, which changes the clock itself. Axial flow in the cores, which real trailing vortices carry, shifts the self-rotation and is absent here.

A single cut-off standing for the core. It is exact for long bends of a uniform core and a good approximation for other profiles with the right effective radius. It says nothing about the core’s own short waves, as the last section explains.

The convention the numbers depend on #

The spacing $b$ is the distance between the rolled-up cores, not the wingspan; for elliptic loading it is $\pi/4$ of the span. The time unit is $2\pi b^2/\Gamma$, the time the pair takes to sink one spacing, and a growth rate of one is one e-fold in that time. The core radius is taken as 0.0982 of the spacing, Spreiter and Sacks’ estimate for an elliptically loaded wing’s rolled-up wake, and the cut-off is 0.642 of the core radius, which is Kelvin’s for a uniform core. “Symmetric” means the two vortices are displaced as mirror images about the plane midway between them. The starting disturbance is the amplitude of the fastest bend along its tilted line, as a fraction of $b$, and linking is taken as the moment the lateral part of the displacement reaches half the spacing.

Who found it, and when #

Sheldon Crow published the analysis in 1970, and set it beside a sequence of photographs of a jet’s contrail waving and pinching into loops; the cut-off method for a curved vortex’s self-induction is older, and its connection to the exact waves on a vortex core is Kelvin’s 1880 dispersion relation, made explicit for the cut-off by Moore and Saffman in 1972. Crow and Bate compared the prediction with measured wake lifetimes in 1976. Spreiter and Sacks estimated the rolled-up core size in 1951, using the same conservation of centroid and impulse that places the cores. The short-wave instability of a strained core, which the cut-off model cannot describe, is due to Widnall, Bliss and Tsai in 1974 and to Moore and Saffman in 1975.

The shape of the argument is the same as the one that makes two vortices move each other — each carried by the field of the other — carried one order further, from how a straight pair moves to how a bent one deforms. It is also the reason a point vortex is not a vortex: the moment a line vortex bends, its own thickness appears in the answer, and the only question is whether it appears through a logarithm, as here, or through something worse.

Still open: four vortices instead of two #

A wing with flaps down sheds four cores, not two, and the rolled-up wake’s centroid is indifferent to that while the instability is not. Two pairs of unequal, counter-rotating vortices on each side orbit one another, and each vortex then sits in the strain of a close neighbour rather than a partner a full spacing away. The strain is larger, so the growth rates can be several times Crow’s.

The calculation that follows extends the two-by-two problem above to four vortices — an eight-by-eight eigenvalue problem built from the same kernels — and asks which arrangements of flap and tip vortices destroy a wake fastest, and whether any can be reached by a real flap schedule without costing the lift that the pair’s circulation records. Beside it is the ground, where the image vortices below the runway add a second partner to each core, and a wall’s mirror turns the descending pair into two that part sideways.