A streak along the jet moves the floor while the crossing is flown

A jet moves the floor that a uniform wind cannot established what reaches the diversion floor and what does not. The floor is the least allowance at which New York is joined to London by a chain of sixty-minute sets, and to first order in the wind it answers to one quantity: the difference between the integral of the along-track wind over the far half of a gap and over the near half. A uniform wind puts the same integral on both halves and cancels exactly. A jet lying across a gap, with its core over one half, does not.

Every case in that measurement produced the difference the same way — by moving a jet sideways. And the essay ended by naming the case it could not make: a jet lying along a gap, with one speed down its whole length, puts the same wind on both halves and moves the floor by ten metres.

Real jets are not one speed along their length. They carry streaks: stretches of core a thousand kilometres or so long and markedly faster than the air either side, which is an ordinary feature of a winter North Atlantic jet rather than an exotic one.

The floor moves forty kilometres #

Sliding the streak along the axis from well before the gap to well past it:

• with the fast air over the Goose Bay half the floor rises, by up to 40.6 km of still-air radius;
• with it over the Narsarsuaq half the floor falls, by up to 43.5 km;
• at either end of the sweep, with the streak far from the gap, the floor returns to the plain jet’s value, as it must.

Forty kilometres against the ten metres a uniform-speed jet along the same gap produced — and against the 745 km allowance the rule itself asks for, which a crossing is a chain of decisions prices in operational terms. The wind field is the same jet in cross-section at every point of that sweep; what changed is where along it the fast air sits.

The dashed curve is the first-order prediction, and it tracks the computed floor over the whole sweep — 40.6 against a predicted 35.1 at the positive peak, −43.5 against −45.4 at the negative one. So the integral’s reach is wider than the case that produced it. It was derived by splitting a great circle at its midpoint and integrating the along-track wind over each half, and nothing in that derivation cares whether the wind varies across the track or along it. The measurement confirms what the algebra should have made obvious and nobody had checked.

That is the cleanest form of the result. Two quite different wind fields — one varying across the flight, one along it — are not distinguishable to the floor. Only the half-integral difference reaches it, and either geometry can produce any value of that. A forecaster who knows the difference knows everything about the allowance; one who knows the jet’s position and not its along-axis structure knows a fraction of it.

What decides how much #

The movement is linear in the streak’s excess speed — 12.4 km for a 50 km/h streak, 54.9 km for a 250 km/h one — because the half-integral difference is linear in the wind and the floor is linear in that. There is no threshold below which a streak can be ignored, only a size below which the number stops mattering.

The length is more interesting, because it has an optimum and the optimum explains itself.

A streak much shorter than the gap has too little of the flight inside it to shift either half-integral much. A streak much longer covers both halves equally — which is the definition of a wind that cancels. So a long enough streak is a uniform wind as far as the floor is concerned, however fast it is, and the movement falls from 48 km at six-tenths of the gap’s half-length to 11.7 km at two and a half times it.

The case that matters is the one whose length is comparable with the gap it lies along, and the North Atlantic gaps are six hundred to a thousand kilometres. That is the same scale as a real jet streak, which is why this is a case about the weather rather than a case about a model. The rule that keeps a route near land is pinned at both ends is where the gap lengths themselves are established.

What the half-integral is, and why it has no opinion about geometry #

The result that two unlike wind fields land on one line deserves its mechanism spelled out, because the mechanism is short and it decides what else follows.

The sixty-minute set about an airfield is a circle in still air. In a wind it is that circle moved, to first order, by the wind’s own displacement over the hour — which is what a diversion allowance in a wind is the same circle, moved upwind establishes and measures. Two airfields either side of a gap therefore have their sets moved by the winds they meet, and the gap closes when the two moved circles touch.

So the quantity that decides the gap is the difference of two displacements, and a displacement is an integral of the wind along the track. Split the great circle at its midpoint, integrate the along-track component over each half, take the difference: that is the whole of the first-order formula. It contains no information about the shape of the wind field, only two numbers extracted from it by integration — and an integral is exactly the operation that discards shape.

Two consequences follow, and the second is the useful one.

A wind field can be arbitrarily complicated and still move the floor by nothing, provided its two half-integrals happen to match. A field with a strong jet over each half of the gap, blowing the same way at the same strength, is as inert as calm air. The line drawn straight on the page is a route is the reminder that the track this integral runs along is itself a choice; change the track and both halves change.

And a summary of the weather is enough, if it is the right summary. Two numbers per gap — the two half-integrals — carry everything the floor responds to at first order. A dispatcher does not need a wind field; they need those two numbers, and nothing in the standard presentation of a forecast supplies them.

Why a crossing is the natural unit #

The time-varying half of the result changes what the quantity is, not merely how large it is, and the distinction is worth keeping separate from the numbers.

A diversion floor is defined on a chain: New York to London is joined when every consecutive pair of airfields along the route is within reach, and a crossing is a chain of decisions sets out what that chain actually is. The binding gap is the weakest link, and which link binds depends on the wind — so a floor is already a minimax over gaps rather than a property of one.

Adding time makes it a minimax over gaps and over the hours the aeroplane is exposed. That is a strictly larger quantity, and the ordering matters: the worst gap at the worst moment is not in general the worst gap at departure, because the streak that binds one gap at hour one may have drifted past it by hour three while another gap comes under a different one. Nothing here measures that second effect, because the Goose Bay–Narsarsuaq gap pins this chain throughout the window swept. It is the obvious next thing, and it would make the floor a function that cannot be evaluated gap by gap at all.

What can be said from this measurement alone is the lower bound: at least thirty kilometres, from one gap under one streak, with the pin never changing hands.

The integral is exactly wrong where it is exactly confident #

The residual is a few kilometres over most of the sweep and is largest where the prediction is exactly zero.

With the streak centred on the gap’s own midpoint, the excess covers both halves symmetrically and the integral difference vanishes by construction — first order predicts no movement at all. The computed floor moves −10.1 km.

That is a second-order term surviving where the first-order term is identically zero, and it is the one place on the whole sweep where the formula’s relative error is unbounded. A term that vanishes by symmetry does not vanish; it only stops being visible to an argument that works to first order. Where the shortest route stops being the only one is the same shape of finding elsewhere in this collection: a symmetry that makes a first-order answer degenerate is exactly where the second order has to be looked at. The practical shape of that is familiar from a diversion allowance in a wind is the same circle, moved upwind, where the moved-circle approximation is excellent except where it predicts a cancellation.

Ten kilometres is small beside forty. It is not small beside zero, which is what a dispatcher applying the rule at a symmetric streak would have been told to expect.

The floor is a function of time #

This is the half of the question a sideways offset cannot reach at all, and it is the operationally important half.

A jet’s position across the ocean is a fact about the weather at a moment, and it changes slowly. A streak is carried along the jet at something like the speed of the flow, so over the four hours a crossing takes it travels a thousand kilometres — which the sweep above has just shown is the whole distance between no effect and the maximum effect.

At departure the fast air is 1,400 km short of the gap and the floor is 635.5 km. Three and a half hours in, the streak lies over the Goose Bay half and the floor reaches 665.8 km — 30.3 km higher than the value a dispatcher reading the departure analysis would have used.

An allowance is not a quantity to be evaluated once. It is the worst of what it will be while the aeroplane is over water, and the difference between that and its value at departure is a quantity no static wind field can price.

The summary is the whole sequence in one place. A uniform wind cancels exactly, whatever its speed. A jet with one speed along its length, lying along the gap, moves the floor by ten metres. The same jet moved sideways so its core covers one half moves it 12.8 km. A streak moved along the same gap, with nothing changed across the jet, moves it 44.9 km — three and a half times the sideways case, at the same core speed.

What an operator would do differently #

The measurement is about a rule that is applied every day, so it is worth stating what follows for somebody applying it — separately from the geometry, and without overstating what one gap under one stated streak can support.

The allowance is already a margin, and this is a second one. The rule asks for a stated still-air radius; the floor measured here is the least radius at which the chain closes, and the difference between the two is the operator’s own room. A streak worth forty kilometres eats a part of that room which nobody has counted, because the computation that produced the floor used a wind field valid at one time.

The cost of ignoring it is not symmetric. A streak over the far half of the gap lowers the floor, and nothing bad happens when an operator has more room than they thought. A streak over the near half raises it, and the same operator has less. So the quantity that matters is not the movement but its upper tail, and a rule of thumb built on the average of the sweep would be exactly the wrong summary — the same asymmetry a diversion allowance in a wind is the same circle, moved upwind records for the circles themselves.

And the honest form of the answer is a window rather than a number. A floor computed from the departure analysis is the value at one moment of a quantity that moves by tens of kilometres over the crossing. Reporting it as a single number is a claim the computation does not support; reporting the worst over the window is a claim it does, at the cost of one extra evaluation per forecast hour.

What each number was checked against #

A streak six thousand kilometres from the gap must leave the floor where the plain jet puts it, to within half a kilometre. It does. That is the control an along-axis field can fail in a way a sideways offset cannot: an along-axis coordinate taken with the wrong sign, or from the wrong pole, would put the streak somewhere else entirely and a sweep would still look like a sweep.

A streak centred on the gap’s own midpoint must put the same excess on both halves, so the integral cancels. It is required below six per cent of the difference the same streak makes 500 km off centre, and gives less than that. This is the stronger control, because it is the only place where a field that is over the gap has to leave the prediction alone — and it is the case whose residual the essay then reports.

A streak over one half must move the floor by something worth acting on, at least two kilometres. It moves it forty.

And the plain jet the streak sits in is the same object as the previous measurement’s, so the floor it produces is the number that measurement published. Everything here is a change from it.

Where the model stops #

The streak is a Gaussian bump on the core speed and nothing else. A real streak has structure across the jet too — it is usually a little wider as well as faster — and it is not symmetric along its length. Both would change the numbers and neither changes that an along-axis variation reaches the floor, which is what the half-integral argument already guarantees.

The drift is stated, not derived. A streak is carried at something near the flow speed, and 250 km/h is chosen as a round figure inside the range a 100-to-300 km/h jet offers. A slower drift moves less of the sweep into the crossing window and a faster one moves more; the 30 km penalty is a function of that choice and the existence of a penalty is not.

Quickest flights on a graph, as before: fifteen-kilometre cells, an eight-neighbour reach, the wind evaluated at the midpoint of each edge. The quickest route is not the shortest is where that solver is built and its discretisation priced.

One gap, and it is the one that binds. The Goose Bay–Narsarsuaq gap is what pins this chain, so a streak elsewhere on the route would move the floor by nothing until it moved enough to make another gap the binding one. That is a real limitation: the response of the floor to a wind field is piecewise, with a change of pin at each corner, and nothing here sweeps a streak along a different gap to find where the pin changes hands.

And the floor is still a still-air radius, which is the rule’s own currency and not a statement about any aeroplane. A crossing bends by a law only a conformal chart can show is where the neighbouring question — what a sharp change in wind does to the track itself rather than to the allowance — is measured.

Still open: whether a forecast places a streak’s end well enough to use this #

The measurement makes the case for looking, and it also states the precision the looking would need, which is the part worth carrying.

The floor moves from nothing to its maximum over about five hundred kilometres of streak position — the distance from the gap’s midpoint to its end. So an error of a hundred kilometres in where a forecast puts a streak’s fast part is worth roughly a fifth of forty kilometres, or eight; an error of three hundred is worth most of it. Numerical forecasts place a jet’s core to considerably better than a hundred kilometres at short range. Whether they place the along-axis structure — where a streak begins and ends — to anything like that is a different question, because an along-axis feature is advected rather than steered, and its position at four hours depends on a speed rather than on a geometry.

If they do, the gain is straightforward: an operator could plan to the worst of the crossing window rather than to the departure value, and the thirty kilometres measured here is what that is worth. If they do not, the honest consequence is the reverse one — that a floor computed from any single analysis carries an uncertainty of tens of kilometres that nothing in the computation reports, and the right response is a margin rather than a better number.