A flapping follower can drift fore and aft, but not sideways

The follower beats in time with the wake, not the bird found the timing rule a flapping flock follows. A leader’s wake is a wave left in the air, its tip vortex laid along the path its wingtip traced, and a follower in the upwash beside it keeps the whole of the gliding saving — 84 per cent of its own induced drag — if it lags the leader by the time the wake took to reach it, so that its wingtip retraces the leader’s path. Half a beat off that phase it keeps only a fifth. It is the rule a flock of northern bald ibises was measured following in 2014.

The rule asks for two things at once: the right place and the right phase for that place. That essay held the follower’s place fixed, and no bird holds its place. Tracked birds in formation drift fore and aft and side to side by fractions of a span over a few wingbeats, and a bird that drifts back by a tenth of a wake wavelength has moved its right phase by 36 degrees. The question that essay left was what a bird should do about it: hold the phase that suits its average place and accept the loss, or hold its place more tightly, and how the answer depends on the swing of its wings and the core of the leader’s vortex.

Two kinds of drift, two kinds of cost #

The model is the previous essay’s, unchanged. Each bird is a horseshoe wing whose two tip vortices are a quarter of π spans apart, the spacing of an elliptically loaded wing, with cores a twentieth of a span across. The leader’s wake is frozen along the path its tip traced; its circulation swells by half on the downstroke. The follower’s saving is computed in the Trefftz plane and averaged over a beat, and — because where a wing sits along the stream does not change the pair’s total — the only thing its fore-and-aft place decides is its phase.

That is the first simplification the problem allows. A follower that drifts back by ξ with its beat unchanged meets the wake at a phase error of 2πξ/λ, where λ is the wake’s wavelength: the distance flown in one beat, two or three spans for birds of the size that fly in V’s, taken here as two and a half. Fore-and-aft drift is a phase error and nothing else.

Sideways drift is different in kind. It moves the follower’s inboard wingtip off the leader’s outboard tip vortex, which is where the upwash is, and no choice of phase undoes that. It is a loss of place, measured against the vortex’s core rather than the wake’s wavelength.

Real drift is irregular, so both are treated as Gaussian, with standard deviations σ in spans. For the phase this gives a closed form: the saving is a periodic function of the phase error with a Fourier series, the average of cos nδ over a Gaussian of spread s is $e^{-n^2s^2/2}$, and the mean saving is the series with each harmonic damped by that factor. For the sideways drift the average is a quadrature against the Gaussian.

Fore and aft is cheaper #

The first figure is the comparison. In place, the follower saves 0.837 of its own induced drag. Sideways, the saving has halved by the time the wander reaches 0.185 spans — a fifth of a span’s drift either way costs half the formation’s benefit. Fore and aft, with the tip swinging a fifth of a span, it halves only at 0.80 spans, a third of the wake’s wavelength and four times as far. With a swing of a tenth of a span it never halves at all: a bird that flaps shallowly keeps most of its saving whatever its phase, because a shallow wake is nearly the gliding one.

The reason fore-and-aft drift is cheap is that the saving’s dependence on phase has a floor. A follower half a beat off the wake’s phase still crosses the leader’s vortex twice a beat and still flies in its upwash in between, only further from it; the previous essay found it keeping a fifth of the saving there, with the tip swinging a fifth of a span. A bird whose phase is spread uniformly over the whole beat keeps the average, 0.385 of its induced drag — nearly half. Sideways there is no such floor. Moved inward by a fifth of a span, the follower’s wing overlaps the leader’s and meets its downwash, and the saving falls to 0.08.

The sideways ridge #

The second figure shows the sideways ridge itself. It peaks with the follower’s inboard tip on the leader’s outboard vortex, and it is lopsided: a tenth of a span outward the follower keeps 0.58 of its induced drag, a tenth inward only 0.43. Outward, the follower moves into weaker upwash; inward, part of its wing moves into the leader’s downwash, which is a cost. That lopsidedness is the bird at the point pays’s asymmetry seen from the other side — the region just inboard of a tip is where a wing pays for its neighbour.

A fatter core flattens the ridge and lowers it. With a core a tenth of a span across the saving in place is only 0.56, and the sideways wander that halves it is 0.21 spans rather than 0.185: a vortex that has spread tolerates more drift because it has less to lose. The previous essay made the same point about height, and the two are one fact. The upwash beside a trailing vortex is concentrated within about a core’s distance of the vortex, and a wing has to be that close to collect it.

A bigger swing makes the timing matter more #

How cheap the fore-and-aft drift is depends on how deeply the birds flap, and the third figure shows it. The wake’s wave is as tall as the tip’s swing, so a deeper beat makes a follower half a beat off further from the vortex, and the floor lower. With the tip swinging a fifth of a span, fore-and-aft wander of 0.80 spans halves the saving; with a swing of three-tenths, 0.50; with four-tenths, 0.37. Below a swing of about 0.17 span the saving never halves however the phase wanders. The sideways wander that halves it does not depend on the swing at all, because a bird drifting sideways in phase still retraces the leader’s height exactly.

The previous essay argued which end of this figure real birds sit at. Birds cruise with a Strouhal number between about 0.2 and 0.4, and since that number is the tip’s peak-to-peak swing over the wake’s wavelength, a bird with a wake two or three spans long swings its tip 0.3 to 0.45 of a span each way. At a swing of four-tenths the fore-and-aft allowance is twice the sideways one, not four times. The conclusion survives, narrowed: a cruising bird can drift fore and aft two to three times as far as sideways before it has lost half of what the formation gives it.

Re-timing the beat #

The trade the previous essay posed has a third option it did not name. A bird that drifts need not hold its average phase; it can re-time its beat towards the phase that suits where it is now, as the ibises were seen to. It cannot do so instantly. It must sense the change — through the upwash on its wings, which is the only signal it has — and adjust its beat over a few wingbeats. The model makes the adjustment a first-order lag: the bird’s phase moves towards the right one with a time constant $T_\varphi$, measured in beats.

What that lag costs depends on how fast the drift itself changes. Measured birds’ positions wander over a few beats, so the drift is treated as a Gaussian process with a correlation time $T_c$ of four beats: it has the spread σ, and it forgets where it was over about four beats. For that process the error a lagging bird leaves has a closed form. It is Gaussian with a variance that is a share $T_\varphi/(T_\varphi + T_c)$ of the drift’s own phase variance: a bird that re-times four times faster than its place changes leaves a fifth of the variance, and one that re-times as fast as its place changes leaves half.

The fourth figure shows one history. The follower’s fore-and-aft place wanders with a standard deviation of half a span. Holding its average phase, its saving swings between the full 0.84 and 0.19 as it drifts, spends about half its time below half its best, and averages 0.47 over these forty beats. Re-timing with a lag of one beat, it spends an eighth of its time there and averages 0.61. The bird does not have to be quick. It has only to be quicker than its own drift.

How fast is fast enough #

The fifth figure generalises the history. For a half-span wander, re-timing within a beat keeps 0.666 of the follower’s induced drag against 0.506 for holding the phase; within half a beat, 0.712; within four beats — as slowly as the drift itself — 0.578. For a quarter-span wander re-timing within a beat keeps 0.76 against 0.65 held, most of the way back to the 0.84 in place. For a whole-span wander even quick re-timing cannot recover it all, because the drift is then comparable with the wake’s wavelength and the error left is a sizeable fraction of a beat.

This is the answer to the trade. Holding place tighter fore and aft buys what re-timing buys, at the cost of flying harder to stay put; re-timing costs nothing aerodynamic, only attention. Sideways there is no such substitute: no timing makes up for being off the vortex, and the only remedy is to be there. A follower should spend its precision on its sideways place and its attention on its timing.

What the drift is worth in power #

The saving is a fraction of induced drag, and induced drag is only part of what a flapping bird pays. A migrating bird flies between the speed for least power, where a parabolic drag polar makes induced drag three-quarters of the aerodynamic power, and the speed for greatest range, where it is half. That converts every number here. A follower in place, saving 0.84 of its induced drag, saves two-fifths to three-fifths of its aerodynamic power. A half-span fore-and-aft wander held at the average phase gives up a third of its induced drag, 0.33, which is a sixth to a quarter of its power; re-timing within a beat takes that loss down to 0.17, a twelfth to an eighth. Sideways, a fifth of a span of wander costs half the saving, 0.42 of its induced drag and a fifth to three-tenths of its power, with no timing to take it back.

These are differences a bird can feel. Weimerskirch and colleagues found great white pelicans’ heart rates about an eighth lower flying in formation than alone, which is the whole of the formation’s benefit as the bird’s body registers it; a drift that costs a fifth of it is a few per cent of the bird’s effort, over a migration of thousands of kilometres.

The model’s timing assumption deserves its own word here. Each instant of a beat is treated as steady, which asks that a beat be long compared with the time the air takes to cross the wing — a reduced frequency well below one, the condition slow enough to be steady puts a number on. A bird’s re-timing lag of a beat is longer still, so the drift and its correction are slow on the scale the aerodynamics needs, and the quasi-steady saving is the right thing to average.

A rule the birds can follow #

The division is also a division of the information a bird has. Its sideways place relative to the vortex is sensed directly: the upwash on its inboard wing rises and falls as it drifts across the ridge, and the gradient says which way to move — the same local signal a gliding bird uses to hold a fair V’s curve. Its phase error is sensed through the same upwash, as the moment in the beat when it is strongest; a bird that times its beat to that moment is re-timing without knowing where the leader is. Neither needs the bird to see anything but the air on its own wings, and neither needs it to know the wavelength.

That is why the ibises’ record is plausible. A bird that keeps close watch on one local signal — the upwash on its inboard wing — and responds to it on the timescale of a beat would hold its sideways place tightly and let its fore-and-aft place drift, re-timing as it went. It would be doing exactly what this calculation says pays. The measurements reported the phase following the position; they did not report which of the two directions the birds held more tightly, and the calculation says which it should be.

What was checked #

The wander rests on three computations, and each has a check that does not share its method. The Fourier series’ Gaussian mean agrees with direct quadrature of the beat-averaged saving against a Gaussian to two parts in a billion, at three spreads. With no wander, both the fore-and-aft and the sideways averages give the flapping pair’s in-phase saving, 0.837, to rounding. The re-timing error’s variance, measured over forty thousand beats of seeded wander, agrees with its closed form to 1.4 per cent at three pairs of lags — against the discrete-time form, which the sampled history obeys exactly; the discrete form agrees with the continuous one used for the figures to two parts in a hundred thousand at a hundred thousand steps a beat. The checks refuse a negative spread, a correlation time of zero and a direction that is neither fore and aft nor sideways.

What the calculation leaves out #

A wake that moves. The leader’s wake is frozen where it was drawn. In the air it sinks under its own downwash and rolls up, so a follower further back meets it lower and more concentrated; the phase rule is unchanged but the sideways ridge narrows with distance.

A drift that is not Gaussian. Birds may drift in sudden steps rather than smoothly, and a bird that jumps half a span in one beat leaves a large error for the lag to work off. The Gaussian process is the simplest wander with a timescale, and it is not a measurement of how birds drift.

Two wanders at once. The fore-and-aft and sideways drifts are treated separately. A bird drifting in both loses both, and near the ridge’s edge the phase matters more, since a bird off the vortex has less upwash for its timing to collect.

The cost of holding still. Holding a place tightly costs power the calculation does not have — a bird must speed up and slow down, and climb and sink, to correct its drift. What the calculation says is how much each kind of drift costs in the formation’s saving, which is what the cost of correcting it has to be weighed against.

Who found it, and when #

The saving of flight in formation was worked out for gliding wings by Lissaman and Shollenberger in 1970 and by Hummel in the 1980s, and the flapping case by Willis, Peraire and Breuer and others in simulation. The measurement is Portugal and colleagues’ of 2014, on northern bald ibises trained to follow a microlight, which found the timing rule the frozen wake derives; the Strouhal band that fixes the swing is Taylor, Nudds and Thomas’s of 2003. What a bird’s drift costs is the calculation here, and the lift beside a wing, where the argument began with two wings tip to tip, is where the halving that everything here is measured against comes from.

Still open: a flock that wanders together #

Every number here is for one follower behind one leader who holds still. In a flock the leader wanders too, and a follower’s right phase moves with its leader’s drift as well as its own; further down the V each bird’s phase is the sum of the drifts of every bird ahead of it, and the errors add. The next calculation lets every bird in a V wander, with each re-timing to the bird directly ahead, and asks how the saving falls along the arm — whether the birds at the ends of a long V, which in a gliding flock save the most, lose the most once the whole flock is drifting, and whether there is a length of V beyond which the accumulated drift makes an extra bird cost the flock more than it saves.