A wandering flock passes no error down the V

A flapping follower can drift fore and aft, but not sideways let one bird wander behind a leader that held perfectly still. A follower flying in a flapping leader’s upwash keeps most 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, and every span it drifts backwards moves that right phase by $2\pi/\lambda$, a wake wavelength of 2.5 spans being one beat. A bird that holds a fixed phase pays for its drift; a bird that re-times its beat, following its own right phase through a lag of about one beat, pays much less. With half a span of wander correlated over four beats, re-timing kept 0.666 of the bird’s induced drag as a saving where holding the phase kept 0.506.

That essay ended on the worry that makes a flock different from a pair. In a V every bird wanders, the leader included, and each follower re-times to the wake in front of it — the wake of a bird that is itself only approximately on time. Bird two follows bird one’s imperfect phase, bird three follows bird two’s, and further down the arm each bird’s beat carries the drift of every bird ahead of it. The errors ought to add. If they do, the long V that a gliding flock favours would be the wrong shape for a flapping one, and there would be a length beyond which another bird costs more than it saves.

They do not add, and the reason is one line of algebra about the way the birds re-time.

The chain #

Each bird $j$ wanders fore and aft by $\xi_j$, an Ornstein–Uhlenbeck process with a standard deviation of half a span and a correlation time of four beats, independent of every other bird’s. The leader, bird zero, wanders too and keeps its own beat. Bird $j$ flies in the wake of bird $j-1$, and the right phase for it is the phase that wake actually carries — bird $j-1$’s beat $\psi_{j-1}$ — shifted by the fore-and-aft gap between them:

$$\theta_j = \psi_{j-1} + \frac{2\pi}{\lambda}(\xi_j - \xi_{j-1}).$$

The bird re-times through a filter, $\psi_j = L\,\theta_j$, and its error is $\delta_j = (1 - L)\,\theta_j$. Everything is linear, so it can be followed in frequency. The previous essay’s re-timing was a first-order lag with a time constant of one beat, $L = 1/(1 + i\omega T_\varphi)$: the bird’s beat relaxes towards the right one at a rate set by how quickly it notices and responds.

Unrolling the chain gives every bird’s right phase in terms of every bird’s wander, and the error’s spectrum is the wander’s spectrum times $|1-L|^2$ times a bracket,

$$B_j = 1 + |1-L|^2\sum_{p=0}^{j-2}|L|^{2p} + |L|^{2(j-1)},$$

which counts how much of each bird ahead reaches bird $j$. If $B_j$ grew with $j$, the errors would accumulate.

Every bird is off by the same amount #

For the first-order lag the bracket is exactly two for every bird. The first follower’s error is 0.632 squared radians, and so is the second’s, the tenth’s and the thirtieth’s, to the last digit the calculation carries. A bird holding a fixed phase has an error set by its offset from the bird ahead, 3.16 squared radians at every place, which is flat for the obvious reason. The re-timed V is flat for a reason that is not obvious at all.

The reason is that the lag and its complement are power-complementary. For $L = 1/(1+ix)$,

$$|L|^2 + |1 - L|^2 = \frac{1}{1+x^2} + \frac{x^2}{1+x^2} = 1$$

at every frequency. Then $B_{j+1} = 1 + |1-L|^2 + |L|^2(B_j - 1)$, and if $B_j = 2$ so is $B_{j+1}$; the first follower’s bracket is two because it sees two wanders, its own and the leader’s. The part of the leader’s wander that the first follower fails to follow lands in its error; the part it does follow passes into its beat and on to the second follower, where it is again split, and the shares always add to the whole. What the arm transmits is exactly what each bird already had to cope with, and no more.

So every follower in the V has twice the error of the previous essay’s lone follower, whose leader held still and contributed nothing, and none has more than that.

What the flock saves #

Translated into savings, through the previous essay’s Fourier series of the saving against phase error — each saving computed in the Trefftz plane, where induced drag is the price of having ends and a neighbour’s upwash pays part of it — a follower with no wander saves 0.837 of its induced drag, a lone follower re-timing behind a still leader 0.666, and every follower in a wandering V 0.601 — the first and the thirtieth alike. A V that holds its phase saves 0.438 a bird. The wandering leader costs every bird in the arm about six and a half per cent of its induced drag, once, and nothing compounds.

That answers the previous essay’s last question in the simplest way. There is no length of V beyond which an extra bird costs the flock more than it saves, at least not on account of timing. Each added bird at the end of an arm finds the same expected error as every bird before it, and saves the same expected share.

Why the leader’s wander enters once #

It is worth seeing the cancellation bird by bird rather than in a bracket. Suppose the leader lurches back by a tenth of a span and stays there. The first follower’s right phase jumps by a quarter of a radian; through its one-beat lag it closes the gap over the next beat or two, so for a while it is off phase, and then it is on. The second follower sees the first follower’s beat move — smoothly, not as a jump, because the first follower’s lag smoothed it — and closes on that more gently, with a smaller error. Each bird down the arm sees a smoother version of the same step and makes a smaller error following it. The leader’s lurch is felt along the whole arm and its cost shrinks at every bird.

Meanwhile each bird’s own lurches start a new disturbance of the same kind at its own place. The sum of a shrinking tail from every bird ahead, and a fresh disturbance from the bird itself and from the one directly ahead, is the bracket — and for a single lag the shrinkage and the replenishment balance exactly. What the first follower suffers from the leader, the tenth suffers spread over nine predecessors, each contributing a little, and the total is the same.

The physics underneath is the one the follower beats in time with the wake established: a follower times itself to the wake, not to the bird, and the wake carries the bird’s actual beat. Because where a wing sits along the stream does not change the pair’s total, the fore-and-aft wander matters only through that timing, and the timing is a linear filter of the positions — which is what makes the chain solvable at all.

A thirty-bird skein #

The numbers carry over to a real flock with rough figures for a goose. At its cruising speed a large bird’s induced drag is a substantial part of its total — of order half, for a goose near its minimum-power speed. A follower in a perfectly held formation would then save about four-tenths of its power; a lone re-timing follower behind a steady leader about a third; every follower in a wandering V about three-tenths. The leader’s wander costs each of twenty-nine followers about three per cent of its total power, and costs the bird at the end of the arm no more than the bird behind the leader.

A smoother re-timing would cost the end bird about one and a half per cent more than the first follower. That is small, and it is bounded, so even then the argument against long V’s from timing alone is weak. What limits a flapping V’s length, if anything does, is somewhere else: the leader’s own unshared effort, the price the bird at the point pays, and the sideways wander that pairwise offsets do not average.

The identity, drawn #

The result is not a property of birds but of the filter, and it fails for the next simplest one. Re-time through two lags of half a beat in series — the same mean delay of one beat, but a smoother response, the shape a bird might produce if its correction itself takes a moment to build up — and the passed and missed powers no longer add to one. In the middle of the range, at about twice the reciprocal of the re-timing time, they add to one and a half. What the bird misses is no longer exactly what it fails to pass on; part of the wander is passed on and missed, and that part is new error that the next bird inherits.

So the same flock, with the same wander and the same mean re-timing delay, accumulates. The first follower’s error rises to 0.819 squared radians — the smoother filter is worse even for one bird — the tenth’s to 1.05, and the thirtieth’s to 1.08. The accumulation is not unbounded: the bracket’s sum is a geometric series in $|L|^2 < 1$ and converges, so a long arm settles at about a third more error than its first follower. Its savings fall from 0.575 to 0.545 along the arm.

A simulation of the arm #

The spectral calculation hides nothing, but it is worth seeing the chain run. A ten-bird arm is simulated for twenty thousand beats, sixty-four steps a beat: each bird’s position an exact Ornstein–Uhlenbeck step, each bird’s right phase taken from the bird ahead’s current beat, each beat advanced through its filter exactly for an input held through the step. The simulated one-lag variances stay flat down the arm; the two-lag variances climb as predicted. The largest disagreement with the spectrum is 2.2 per cent, and it is the right kind of error: a run of finite length has a sampling error of about that size in a variance, and a time step of a sixty-fourth of a beat leaves a first-order bias of about one and a half per cent, which an earlier run at an eighth of a beat showed at twelve.

That earlier run is worth recording as a warning. Stepped at eight samples a beat, the simulated one-lag arm did accumulate — from twelve per cent above the spectrum at the first bird to twenty-two at the eighth — because a lag discretised coarsely is no longer exactly power-complementary. An analysis of a flock that stepped its birds’ responses at the wingbeat would have found an accumulation that the birds do not have.

How fast the birds re-time #

The re-timing time decides how large every error is and, for the single lag, nothing about how it is shared along the arm. A bird re-timing in a tenth of a beat has a tiny error and a bird taking ten beats an error approaching the fixed phase’s, and at every one of those the first and thirtieth birds with a single lag are identical. With two half lags the thirtieth bird’s error exceeds the first’s by about a third over most of the range — 1.31 times at a tenth of a beat, 1.32 at one — and by less, 1.21, at ten beats, where the birds have nearly stopped re-timing and both errors are approaching the fixed-phase value, which does not accumulate. Accumulation needs a filter that is neither instantaneous nor absent.

Where the error lives #

The error’s spectrum says where in time it comes from. It sits between the wander’s own frequency, a quarter of a radian a beat, and the re-timing’s, one radian a beat. Slower wander than that is followed; faster wander is too small to matter because the Ornstein–Uhlenbeck spectrum has little there. In that middle band the two-lag filter’s excess compounds bird by bird, and it is there that the thirtieth bird’s spectrum stands above the first’s. A bird watching its neighbour’s wingtip over two or three beats is measuring exactly that band.

Five checks on the chain #

The identity is checked at five frequencies to rounding, and the bracket is checked to be two for birds one to twenty; the two-lag filter’s excess is checked to be real, up to 0.43 at the frequencies sampled. The first follower’s variance computed from the spectrum is exactly twice the previous essay’s closed form, $s^2T_\varphi/(T_\varphi + T_c)$ with $s = 2\pi\sigma_x/\lambda$, to rounding — the leader’s wander adds as much as the follower’s own. The simulated arm agrees with the spectrum within the run’s sampling error for both filters. The tests also refuse a bird at place zero, a third-order filter the calculation does not define, and a re-timing time of zero.

What a pairwise flock leaves out #

Every bird but the one ahead. Each follower’s saving comes from the bird directly in front, through the lift beside a wing that its neighbour’s tip vortex makes. In a real V a bird also sits in the outer edges of the upwash of birds two and three places ahead, which is what makes a gliding V’s end birds save more than its first followers; a fair V is a curved V and the bird at the point pays are that calculation for gliding wings. Those second-neighbour wakes carry the phases of birds further ahead, whose errors are correlated with the nearer bird’s, and the identity here does not cover them.

Sideways wander. It is a loss of place rather than of phase and is pairwise by construction: each bird’s sideways error is its offset from the bird ahead, of the same size at every place, as for the fixed phase.

A bird’s choice of what to follow. The model’s birds re-time to the wake they are in. A bird that watched the leader instead, several places ahead, would not inherit the intermediate birds’ errors but would mis-time against the wake it actually flies in.

Gaussian, stationary wander. The wander is a statistical stand-in for flight that turns, climbs and gusts. A flock manoeuvring together has correlated wander, which a pairwise offset partly cancels.

The convention: phase error in radians, savings in induced drag #

Phase errors are in radians of the wake’s own cycle, so an error of one radian is a sixth of a beat. Wander is in wingspans with a standard deviation of half a span and a correlation time of four beats, the previous essay’s case; the wake wavelength is 2.5 spans. Savings are fractions of the follower’s own induced drag in the frozen-wake Trefftz-plane model the previous essays used, with a tip swing of a fifth of a span and a lift swing of a half. “One lag” means the bird’s beat relaxes towards the right phase at a rate of one per beat; “two half lags” means that relaxation is applied twice in series at twice the rate, the same mean delay.

Who timed the ibises #

The phasing of flapping birds in formation was measured on a flock of northern bald ibises fitted with data loggers by Portugal and colleagues in 2014, and the flapping pair’s aerodynamics was computed by Willis, Peraire and Breuer and by others since. Power complementarity of a first-order section is a textbook property of filters, used in audio crossover networks to make a signal split two ways add back to itself, and the telescoping of a chain of identical complementary stages is the same property applied repeatedly. That the timing errors of a flapping V do not accumulate along its arm for exactly this reason, and do for a smoother response, is what the calculation adds.

Still open: the second neighbour’s phase #

A bird in a V does not sit only in the upwash of the bird ahead; the outer part of its wing is in the weaker upwash of the bird two places ahead, whose beat is timed for somewhere else. The next calculation gives each bird the induced field of its two nearest predecessors, each carrying its own actual phase, and asks whether a bird can still find one beat that suits both — whether the second neighbour’s contribution, arriving at a phase error equal to the first neighbour’s re-timing error plus its own, reintroduces the accumulation the identity removed, and so whether the end birds of a long flapping V keep the advantage they have in a gliding one.