A jet moves the floor that a uniform wind cannot

A diversion allowance in a wind is the same circle, moved upwind put a wind of one speed from one bearing over the whole North Atlantic and found the regulation’s geometry nearly untouched by it. Every airfield’s sixty-minute set became the still-air circle moved upwind by the wind’s run, and because every circle moved by the same run, no gap between two of them changed. The least radius at which New York can be joined to London by a chain of overlapping circles — 624.66 kilometres, set by the gap between Goose Bay and Narsarsuaq — hardly moved.

That essay ended on the assumption the whole result rests on. Winds over the North Atlantic are not uniform. They gather into jets a few hundred kilometres wide, fast in the core and slack outside it, and a jet lying across a gap does not blow the same way on both sides of it.

So the question is whether the floor is still a property of the airfields once the wind has structure, or whether a jet can move it — and if it can, by how much, which way, and whether the size of the move can be known without computing a single quickest flight.

A set in a jet is a circle that has been bent #

The first thing a jet takes away is the closed form. In a uniform wind, the places from which an airfield can be reached in sixty minutes are a disc for a reason that fits in three sentences. In a wind whose speed changes from place to place, the quickest flight from a point to an airfield is not a straight line held against the wind, and the time it takes has no formula. It is the problem the quickest route is not the shortest describes for a current: in a moving medium, time and length are different questions.

It is computed here as a shortest path through a fine network of places fifteen kilometres apart. Every step between two of them is timed on the sphere by solving the wind triangle along it — crabbing out the cross-wind and adding the along-track wind to what is left of the airspeed — and the least total time from each place to each airfield is found by the same kind of search the reach set takes the shape of the roads runs on a road grid. A network can only step in the directions it has, and this one has 176, so a flight wanting to run between two of them is lengthened by at most 0.19 per cent. In still air it reproduces every great-circle time to within 0.17 per cent, and the floor to within twenty metres: 624.68 kilometres against the closed form’s 624.66.

Measured against the circle that fits each of them best, the sets have kept most of their old shape. Narsarsuaq, where the jet blows at 176 km/h, has a set whose best circle has moved 143 kilometres, and whose edge departs from that circle by up to 51 kilometres outward and 50 inward. Nuuk, in 138 km/h, has moved 123 and departs by up to 57. Those departures are the bending, and they are a fifteenth of the radius.

Two things in those numbers are worth separating from the rest.

The first is that neither set has moved as far as the wind at its airfield would move it. A uniform wind of 176 km/h moves a set 176 kilometres; Narsarsuaq’s has moved 143. A sixty-minute diversion to Narsarsuaq is mostly flown somewhere other than Narsarsuaq, and the wind there is weaker, so the set moves by something closer to the wind a diversion actually meets than to the wind on the airfield.

The second is Goose Bay. The jet at Goose Bay itself is almost calm, and its set still departs from a circle by 27 kilometres, because the far edge of its set lies 745 kilometres to the east and reaches into the jet’s western flank. A set is shaped by the wind over the whole of it. An airfield’s own weather report says very little about where its sixty-minute set lies, because the set is the places from which the airfield can be reached — the second of the two sets the set that can be reached is not the set that can reach separates, which in a moving medium is not the set of places the airfield could reach — and that set is decided by the wind along every route into the airfield, not by the wind on it.

What does not change is the size. Every one of the four sets has the area of the still-air circle to within 0.21 per cent. That is an observation about this jet, whose wind runs everywhere parallel to its core and so neither gathers air into any place nor spreads it out of one, and the measurement agrees with the obvious account: a wind that does not converge anywhere moves and bends a set without growing it. The account is plausible and it is not proved here. And every one of the sets is still star-shaped about its airfield, so the outlines drawn by walking out along 180 bearings, as every reach set ever drawn is too small describes, are the whole of each set.

The floor moves, and a uniform wind’s does not #

The quantity the rule turns on is not the shape of any one set. It is the floor: the smallest allowance at which the airfields’ sets form a chain from New York to London. The rule that keeps a route near land is pinned at both ends found it pinned in still air by one gap, and in minutes it is 50.31; written as the still-air radius it corresponds to, it is the 624.66 kilometres that every figure here is measured against.

In the southerly jet the floor falls by 9.9 kilometres at a core speed of 50 km/h, 19.3 at 100, 28.4 at 150, 36.9 at 200 and 45.2 at 250. That is nearly a straight line through the origin, at 18.4 kilometres for every 100 km/h.

A uniform southerly of the same speeds raises the floor by 1.6, 2.7, 2.7, 3.0 and 3.6 kilometres. So a jet of 50 km/h moves the floor further than a uniform wind of 250 does, and in the opposite direction. The two are not different amounts of the same effect. One grows with the wind and the other barely registers it, and the reason is short enough to write down in full.

Why a uniform wind cancels and a jet does not #

Take one gap, between two airfields $A$ and $B$ a distance $D$ apart, and ask how long a straight diversion takes from a point on the great circle between them.

To first order in the wind, the time along a straight track of length $d$ is the still-air time less the tail-wind collected along the way:

$$t \approx \frac{d}{V} - \frac{1}{V^2} \int w_\parallel \, \mathrm{d}s ,$$

where $V$ is the airspeed and $w_\parallel$ is the wind along the direction of flight. The cross-wind does not appear. Crabbing into a cross-wind $c$ reduces the speed made good along the track from $V$ to $\sqrt{V^2 - c^2}$, which differs from $V$ by $c^2 / 2V$ — a second-order amount. Bending the track does not appear either, because a straight track is where the time is stationary: a small bend of size $\epsilon$ changes the time by something of size $\epsilon^2$, which is Fermat’s principle, and the section on bending below measures it.

The two sets first touch where the two flying times are equal. Moving along the gap from its midpoint shortens one diversion and lengthens the other, and the balance falls where the difference in tail-wind has been paid for. At that point the time is $D/2V$, less the sum of the tail-wind integrals of the two half-flights from the midpoint, $a_A$ and $a_B$, over $2V^2$. Written as the still-air radius it corresponds to, the floor through that gap moves by

$$\Delta r \approx -\frac{a_A + a_B}{2V} ,$$

where $a_A$ is the tail-wind collected flying from the midpoint to $A$ and $a_B$ flying from the midpoint to $B$, each measured along its own direction of flight.

A wind blowing the same way along the whole gap gives one half-flight a tail-wind and the other the same head-wind. The two integrals are equal and opposite, their sum is zero, and the floor does not move. That is the moved-circle result, rederived without any circles, and it says why: a uniform wind is invisible to a gap because each half of the gap takes back what the other half gives.

A jet breaks the symmetry. In the southerly jet the core lies over the half of the Goose Bay–Narsarsuaq gap nearer Narsarsuaq, and the gap runs north-east, so the wind has a large component along it there and a small one on the other half. A flight from the midpoint to Narsarsuaq collects an average tail-wind of 116 km/h over its 625 kilometres. A flight from the midpoint to Goose Bay pays an average head-wind of only 19.5. The sum is 60,470 kilometres times km/h, and divided by twice 745 it predicts the floor through that gap falling 40.6 kilometres. Computed as quickest flights, it falls 36.9.

That also explains why the uniform wind moves the floor at all. On a flat ocean it would not. On the sphere, a wind that keeps one compass bearing does not meet a long gap at one angle, because the gap’s own bearing turns along it as the meridians converge. The two halves then collect slightly different amounts, and for a northerly of 150 km/h the integral predicts a fall of 4.95 kilometres, against 4.91 computed.

The integral, against every wind tried #

One example agreeing proves little, and the integral is cheap enough to test widely: it needs the wind along a great circle and nothing else, while each floor computed from quickest flights needs sixteen searches over the whole ocean.

Thirty-six jets of 200 km/h, from twelve directions and at three positions across the gaps, move the floor by anything from 35.9 kilometres down to 23.3 up. The integral puts every one of them within 3.5 kilometres of its computed value, and 2.1 kilometres root-mean-square. Twelve uniform winds of 150 km/h move it by between 6.6 down and 2.7 up, and the integral puts every one of those within 0.6.

The integral’s errors have a consistent direction. It is exactly linear in the wind, and the computed floor is slightly less than linear: the integral overstates the move by 3 per cent at 50 km/h and by 10 per cent at 200. What it leaves out — the cost of the cross-wind, and the small freedom to bend — is second order and grows with the square of the speed. In the southerly jet it works against the first-order move at every speed; across the other jets it falls on either side.

What the moved circles got wrong #

The crosses in that figure are a correction, and it is owed to the essay this one continues.

That essay computed the floor in a uniform wind by moving each still-air circle upwind and finding the least radius at which the moved circles join. For a wind of 150 km/h it reported the floor moving between 16.7 kilometres down and 6.2 up, depending on direction. Computed as quickest flights through the same winds, the floor moves between 6.8 down and 3.1 up.

The moved circle is the right shape for a uniform wind’s set, and the earlier essay checked how right it is: from every point of a moved circle the flying time is sixty minutes to within one and a half per cent. One and a half per cent of 745 kilometres is eleven kilometres, which is exactly the size of the quantity it was then used to measure. The floor in a uniform wind moves by a few kilometres, and the approximation used to find it was good to about ten.

The correction strengthens the earlier conclusion rather than overturning it. The floor under a uniform wind of 150 km/h moves by less than a twentieth of the wind’s run, where that essay said a tenth, and it changes which gap pins it in exactly the directions that essay found: computed both ways, the pin is Goose Bay to Narsarsuaq and Goose Bay to Iqaluit at the same twenty-four directions. Its figure now carries both curves.

Which side of the gap the core lies on #

The integral says what matters is which half of a gap the fast wind is on. Moving the jet sideways tests that directly, because it moves the core from one half to the other without changing anything else.

With the core 400 kilometres east of the Goose Bay–Narsarsuaq midpoint, the floor falls 34.1 kilometres. With it directly over the midpoint it falls only 9.5. With the core 200 kilometres west it rises 18.6, and 400 kilometres west it rises 23.3.

The sign follows from the half-flights. A southerly on the Narsarsuaq half helps the flight towards Narsarsuaq, which is heading north-east. The same southerly on the Goose Bay half hinders the flight towards Goose Bay, which is heading south-west. So the same jet lowers the floor from one side of the gap and raises it from the other. Where it straddles the midpoint the two halves nearly cancel, as a uniform wind’s do.

Further west still, the curve turns down again: 800 kilometres west of the midpoint the floor falls 24.8. The jet has not changed. The gap has. That far west the core lies across the other candidate crossing, the gap between Goose Bay and Iqaluit, which in still air is 0.72 kilometres longer than the Goose Bay–Narsarsuaq gap. On that gap the core is on the side that helps, and the floor is pinned there instead.

That near-tie is where a jet does something no uniform wind can. The uniform-wind essay found the floor’s pin passing between those two gaps as the wind swung round the compass, a feature a forecast could move. A jet adds a third. With its core over the middle of the Goose Bay–Narsarsuaq gap, that gap’s floor falls by only 5.8 kilometres and the Goose Bay–Iqaluit gap’s rises by 6.2, and the easiest crossing becomes a third, through Nuuk, whose gap from Goose Bay falls further than either. No uniform wind of 150 or 200 km/h, from any of the directions tried, ever made that gap the binding one.

The practical reading is about what an approval computed in still air is worth. On this ocean it is worth a floor that a day’s jet can move by some 35 kilometres either way, according to which side of a gap its core happens to cross. The rule’s own sixty-minute radius sits 120 kilometres above the floor, so no jet measured here closes the ocean. What it does is change how much of that margin is left, and which gap the margin is being measured across.

The quickest diversion bends, and it hardly matters #

The integral uses straight tracks, and so did the uniform-wind essay’s diversion times: every flight there was held on the great circle, crabbing into the wind, which is the quickest track in a uniform wind on a plane. In a jet the quickest track is not straight, and the lead left by the uniform-wind essay asked how it should be computed.

The diversion drawn is the one where holding the great circle loses most to the quickest track, among places within sixty minutes of Narsarsuaq in this jet tried every ten degrees of bearing and fifty kilometres of distance. It starts 600 kilometres north-north-east of the airfield, on the jet’s eastern flank, and flies south into the southerly. The quickest track does what the wind invites: it holds the weaker wind on the flank, keeping 260 kilometres east of the core for its first 170 kilometres, and crosses into the core’s head-wind only in the last part of the way, where it can no longer be avoided. It strays at most 42 kilometres from the great circle and saves 39 seconds on a flight of an hour.

Nowhere else tried within that airfield’s allowance saves more. Along the still-air route across the whole ocean, the largest saving in this jet from any point to its quickest airfield is 8 seconds.

The saving is second order, as Fermat’s principle says it must be. At a core speed of 100 km/h it is 0.10 minutes; at 200 it is 0.64, more than six times as much for twice the speed; at 250 it is 1.26. A saving with a part in proportion to the speed would roughly double between 100 and 200. This one grows faster than the square, because at higher speeds the cross-wind cost the bend avoids is itself growing.

So the two halves of this essay are one fact seen twice. At first order in the wind, the straight track is the quickest, which is why a single integral along each gap predicts the floor, and why a diversion’s time can be computed by holding the great circle. What bending buys is a second-order refinement, worth seconds on a diversion and a few kilometres on a floor. It is a different situation from a crossing bends by a law only a conformal chart can show, where the speed jumps across a line and the bend at the line is set by a law of angles. A jet changes speed over hundreds of kilometres, and the bend it asks for is correspondingly gentle.

What each number was checked against #

The quickest times are the output of a search, and a search can be wrong in ways that look like results, so every part of it was checked against something that shares none of its arithmetic.

In calm air the network is the closed form. Its floor over the northern airfields is 624.68 kilometres against the 624.66 the still-air bottleneck gives with no routing at all, and dropping Bermuda, the Azores and Lisbon — whose links are all far longer than the northern ones — leaves it exactly unchanged. Every calm flying time beyond twenty minutes is the great-circle time to within 0.17 per cent, inside the bound the step directions allow.

In a uniform wind it is the integral, to 0.6 kilometres in every direction, and it is not the moved circles, which differ from it by up to 9.6 — the check that the correction above is a property of the approximation, not of the search.

A jet that should do nothing does nothing. A jet of the same 200 km/h, centred on the midpoint of the Goose Bay–Narsarsuaq gap and blowing straight along it, puts the same wind on both halves. The integral predicts no move. The computed floor moves by 0.01 kilometres. A jet at full strength over the widest gap that left the floor where it was is what the account requires; a large move there would have meant the floor was answering to something other than the difference between the halves.

And bending must not save at first order. The saving at twice the speed must be more than three times the saving at the lower speed, which a term in proportion to the speed would forbid. It is 6.2 times.

What the stated jet leaves out #

The jet is stated, not observed. One core along a great circle, its speed falling away as a bell curve with a half-width of 250 kilometres, the same along its whole length and steady in time. Real jets curve, meander, and carry streaks of faster wind a thousand kilometres long; and a forecast one is uncertain in position by more than the offsets that change the floor’s sign here. A crossing is a chain of decisions is the essay about planning against a forecast that turns out wrong.

The aircraft’s speed and height are fixed. The allowance is flown at 745 km/h at one level. An aircraft that loses an engine descends, and jets are weaker lower down, so 200 km/h is a strong wind for a one-engine diversion rather than a typical one.

A diversion starts on its track. A real one starts with a turn, which the shortest route a vehicle can fly prices at a fixed length whatever the leg, and a fraction of a minute on a diversion of this size.

And the set of airfields is fixed. The floor is decided by the gaps between these twenty, and a jet’s effect is decided by which of those gaps it crosses. A different set of alternates puts the gaps elsewhere, and the same integral would say which of them a jet can move.

Still open: a jet that is not the same along its length #

The integral says a gap answers only to the difference between the wind on its two halves. Everything here produced that difference by moving a jet sideways, so that its core lay over one half and not the other.

A jet can produce the same difference another way. Real jets carry streaks — stretches of core a thousand kilometres or so long, much faster than the core on either side — and a streak lying along a gap, with its fast part over one half, would put different winds on the two halves without any change across the jet at all. By the integral, a streak’s entrance or exit over the middle of a gap should move the floor as far as a sideways offset does, while the aligned jet measured above, the same speed along its whole length, moved it by ten metres.

Whether the computed floor agrees with the integral when the wind varies along a gap rather than across it, how far a streak moving along the gap during a crossing would move the floor while the crossing is being flown, and whether any forecast places a streak’s end well enough to use the answer, are questions a jet with one speed along its whole length cannot ask.