A diversion allowance in a wind is the same circle, moved upwind

The rule that keeps a route near land is pinned at both ends drew a circle of 745 kilometres round each of twenty North Atlantic airfields and found the shortest route from New York to London that never leaves them: 5,698 kilometres, 158 longer than the great circle. It found that the crossing becomes possible at all only once the circles reach 624.66 kilometres, where Goose Bay’s and Narsarsuaq’s first touch, and that the price falls to nothing at 1,250 kilometres, where the great circle is covered. It ended on the assumption every one of those numbers rests on: the air is still.

That assumption is not the essay’s. It is the rule’s. The regulation that for decades kept twin-engined airliners within sixty minutes of an adequate airport measures those minutes at the one-engine cruise speed in still air, and the approvals for extended operations that relaxed it kept the same convention with more minutes. A diversion allowance is a time; the rule converts it into a distance by setting the wind to zero; and an aircraft that has just lost an engine over the ocean does not get to set the wind to anything.

The set a wind describes is a circle, moved #

The first question is what shape replaces the circle. The expectation written into the last diversion essay was a lopsided region stretched downwind of its airfield. It is not, for a wind that blows at one speed from one direction, and the reason takes three sentences.

An aircraft flying at airspeed V on a heading u, in a wind W, moves over the ground at V u + W, so after a time t it has moved V t u + W t. The places from which it can arrive at an airfield after exactly t are therefore the points t times W upwind of the airfield and then V t away from that point in any direction — a circle of radius V t, centred a distance W t upwind. And every such circle for a shorter time lies inside the one for the whole allowance T, because its centre is less far upwind by exactly as much as its radius is smaller, and the wind is slower than the aircraft.

So the set of places within sixty minutes of an airfield in a uniform wind is one disc of the still-air radius, V T, moved upwind by the wind’s run W T. It is not stretched and not lopsided. It has moved, and it has moved without changing size.

That is special to a uniform wind, and it is worth seeing why by contrast. The reach set takes the shape of the roads found a reach set with corners, because on a road grid the speed depends on the direction of travel in a way no shift can remove. A uniform wind also makes the ground speed depend on direction, but in the one special way that is a translation: every direction gains the same vector. The circle survives because the dependence on direction is exactly a shift and nothing else.

That is the same geometry the set that can be reached is not the set that can reach found for a current: what a vehicle can reach from a point drifts downstream, and what can reach the point sits upstream. A diversion rule is entirely about the second set, because the aircraft is going to the airfield, and the second set is the one the circles in a diversion figure are supposed to be.

The derivation is for a flat plane. On the Earth, a wind that blows from one compass bearing everywhere is not the same vector everywhere, and a diversion follows a great circle whose bearing changes along the way. So the moved circle is checked rather than trusted: from every point on it, the flying time to the airfield is computed by solving the wind triangle every ten kilometres along the great circle — crabbing out the cross-wind, adding the along-track wind to what is left of the airspeed — and summing.

From the still-air circle round Keflavík, a 150 km/h westerly makes the flying time anything from fifty minutes to seventy-five, depending on which side of the circle the aircraft starts from. From the moved circle, every point is between 59.1 and 60.8 minutes. The flat-plane result holds on the sphere to one and a half per cent, and what is left over is the Earth’s curvature turning a wind of constant bearing into a wind that is not quite uniform.

From the same route, the quickest diversion in three kinds of air #

The still-air route is the one the rule accepts. The useful measurement is what the wind does to the diversions from that route, because that is the route an operator would plan to the rule.

In still air the profile sits at or below sixty minutes everywhere, by construction, touching the allowance along the stretches where the route runs on the rim of a circle — that is what the shortest route inside a union of circles does. Put a wind on it and every stretch that ran along a rim is either inside the allowance or outside it, depending on which side of its airfield the wind has moved the circle to.

A westerly of 150 kilometres an hour, the direction the North Atlantic’s jet streams usually blow from, does very little. The worst diversion from the still-air route becomes 61.7 minutes. A northerly of the same speed is a different matter: the worst point, in the gap between Greenland and Iceland, needs 68.7 minutes to reach Keflavík, nearly nine minutes over the allowance. That is a route the rule accepted, flown in a wind a forecast would not remark on, with an engine-out diversion fifteen per cent longer than the rule says it can be.

The quickest airfield is also not always the nearest one. Which airfield a point should turn to is decided by time, and time in a wind is a cost that depends on direction: a partition under a directed cost has two versions is the essay about what that does to the question “which one is closest”. Here it means the boundaries between the regions served by each airfield move with the wind too. In the 150 km/h northerly, 39 of the 116 points along the still-air route, taken every fifty kilometres, are quicker to a different airfield than the nearest one: twenty of them turn to Gander rather than Goose Bay, seven to Shannon rather than Prestwick, and three to Shannon rather than Keflavík — in each case to the airfield further south, downwind.

The error has a direction #

The difference between the westerly and the northerly suggests that direction, rather than speed, decides how wrong the still-air rule is. It does.

Round the compass the worst diversion swings from 68.7 minutes in a northerly to 54.0 in a southerly. The rule is too generous in winds from the north half of the compass and too strict in winds from the south, and between the two it crosses the allowance twice, with winds from a little south of east and from a little south of west.

The reason is where the airfields are. The shortest crossing is pinned against the edges of the Greenland and Iceland circles, which lie north of it; the route runs along their southern rims, so the diversions that matter from the pinned stretches head north. A wind from the north blows straight into those diversions and a wind from the south blows them along. The westerly jet, the wind the North Atlantic actually has most, happens to blow across them, which is why the rule survives it so well. That is a fact about this route and these airfields, not about the rule: a crossing pinned against airfields to its south would reverse the pattern.

A dead headwind is the bound #

There is a limit to how wrong the still-air rule can be, and it has a closed form. The worst a wind can do to a diversion that the rule allows is to blow straight down its whole length from the airfield: a diversion of the full 745 kilometres at a ground speed of V − W takes 60 V / (V − W) minutes.

At 150 km/h the bound is 75.1 minutes, and the northerly’s 68.7 is more than half of the way to it. At 250 km/h, a strong jet, the bound is 90.3 and a northerly reaches 77.8. A westerly stays far below either, at 62.8 minutes even at 250.

A rule that allowed for any wind would forbid the crossing #

The bound suggests an obvious repair: write the rule so that it holds in the worst wind, whatever its direction. A still-air circle drawn with the radius a dead headwind leaves, (V − W) T, would guarantee sixty minutes from every point inside it in any wind up to W. At 150 km/h that radius is 595 kilometres.

At 595 kilometres the North Atlantic cannot be crossed at all. The least radius at which a chain of circles joins New York to London is 624.66 kilometres, so a direction-proof rule allows the crossing only if it is written for winds below 745 − 624.66 = 120.3 km/h, and every stronger design wind closes the ocean. That is a sharp threshold rather than a trade-off: the price of the route does not rise smoothly as the design wind grows, it rises to its floor and then the route is gone.

So the still-air convention is not merely convenient. A rule honest about every wind a crossing might meet is, on this ocean, a rule against crossing it, and the still-air rule is what makes an allowance of sixty minutes compatible with the airfields the North Atlantic has. What it gives up is exactly the time error measured above.

So a route planned to the still-air rule carries an error that is bounded by the wind speed alone, and realised by the geography: how close a route comes to the bound depends on whether its binding diversions point into the wind. An operator who knew only the wind speed could bound the error; knowing the route’s pinned airfields and the wind’s direction tells how much of the bound is spent.

The honest route, and the floor that hardly moves #

A rule honest about time would draw the moved circles instead of the still-air ones, and the shortest route inside them is found exactly as before — through the corners where two moved circles cross.

The honest route changes by tens of kilometres: 68 longer in a northerly, which pushes the circles north away from the great circle and forces a new bend at the corner of Gander’s and Narsarsuaq’s, and 63 shorter in a southerly, which pushes them towards it. In the northerly the honest route’s own worst diversion is 60.75 minutes rather than sixty, which is the sphere’s one and a half per cent showing again.

The floor behaves quite differently, and the moved-circle result says why before anything is computed. On a flat ocean a uniform wind moves every circle by the same vector, and moving every circle by the same vector changes no distance between any two of them. Two circles overlap when their centres are within twice the radius, so if no distance between centres changes, the least radius at which a chain of overlapping circles joins New York to London cannot change either. The floor of the rule is invariant under a uniform wind.

On the sphere it is not quite. A wind from one bearing moves Goose Bay, at 53 degrees north, and Narsarsuaq, at 61, by the same distance but not by the same vector, because the meridians converge between them. Measured, the floor moves between 16.7 kilometres down and 6.2 up for a wind that moves every circle 150 kilometres — a tenth of the run at most.

The floor’s pin is a near-tie, and the wind breaks it #

One thing about the floor does change, and it is worth noticing because the still-air essay called both walls of the price “pinned to single features”. The floor is set by the longest gap in the best chain of airfields across the ocean, and in still air that gap is Goose Bay to Narsarsuaq, half of which is 624.66 kilometres. The next candidate is Goose Bay to Iqaluit, half of which is 625.38. The two alternative chains across the Labrador Sea are 0.72 kilometres apart in their longest links.

A wind breaks the tie. In winds from anywhere between the south-south-west and just east of north, the floor stays pinned to Goose Bay and Narsarsuaq. In winds from anywhere between the north-east and just west of south, the sphere’s small distortion of the gaps favours the other chain, and the floor is pinned to Goose Bay and Iqaluit instead. The floor’s value hardly moves, and the feature it is pinned to changes with the weather.

That is the degenerate case the still-air figures could not show: a floor pinned to a single gap in still air is pinned to whichever of two nearly equal gaps the wind happens to favour, and a gap that is the binding one by 1.4 kilometres in 1,250 is a pin a forecast can move.

What the still-air convention buys, and what it costs #

A rule in still air has a virtue that a rule in forecast winds does not: it can be checked once, for a route, without knowing the weather. The circles are fixed, the corners are fixed, and an operator’s approval does not have to be re-derived every morning. A crossing is a chain of decisions measured what planning a crossing on a wind forecast and re-planning it as the forecast is corrected is worth; a diversion rule that depended on the forecast would carry all of that into the question of whether a flight may depart at all.

The price of that convention can now be stated precisely. On this crossing the still-air rule’s diversions are wrong by at most the head-wind bound, 60 V / (V − W) minus sixty, and in the winds this ocean actually has — westerlies, often strong — they are wrong by a minute or two. In a northerly of 150 km/h they are wrong by nearly nine minutes, and in a southerly they are conservative by six. The error is not a percentage of the allowance applied everywhere. It lives on the few stretches where the route is pinned against a rim, and its sign is set by whether the wind blows towards the airfields that pin it.

Two things the measurement leaves out both run the same way and both are small. Every flying time here starts on track towards the airfield, and a real diversion starts with a turn; the shortest route a vehicle can fly found that bounding a vehicle’s curvature adds a fixed length to a route whatever the leg, which on a diversion of 745 kilometres is a fraction of a minute. And the one-engine speed is held constant, where a real aircraft descends to an altitude it can hold on one engine and meets a different wind there. Neither changes the shape of the result: the circles move, and the error follows the wind’s direction.

The quickest route is not the shortest made the point that once the medium moves, distance and time part company. A diversion rule is a place where the regulation chose distance, knowingly, as a stand-in for time, and the moved circle is the exact measure of what that stand-in costs: nothing about the rule’s geometry, a floor that holds within a tenth of the wind’s run, and a time error whose size is bounded by the wind’s speed and whose sign is set by its direction.

Still open: a wind that is not the same everywhere #

Everything here assumed one wind speed from one bearing across the whole ocean, and that assumption is what made every set a moved circle. Real winds over the North Atlantic are concentrated in jets a few hundred kilometres wide, strong in the core and weak outside it, with the strongest winds often lying across exactly the stretch of ocean where the airfields are furthest apart.

In a wind that varies across a diversion, the places within sixty minutes of an airfield are no longer a moved circle. The quickest diversion across a jet is not a great circle either: where the wind changes sharply across the edge of the core, a track bends there by a law of angles, which is what a crossing bends by a law only a conformal chart can show measured for a speed that jumps across a line. A point on the far side of a jet can reach an airfield only by crossing the core, and the set bulges where the wind helps and pinches where it does not, which is the lopsided region the still-air essay expected and a uniform wind does not produce. Whether the floor is still nearly invariant when the circles stop being circles, whether a jet lying across the Labrador Sea moves the pin between Narsarsuaq and Iqaluit by more than a uniform wind can, and how the quickest diversion should be computed when its best track is not a great circle, are questions a uniform wind cannot ask.