The count computed on a body

The rung below this one ended with an admission. It had proved the index theorem in the plane, checked it four ways on solved fields, and then reached the version an experimentalist actually uses — the one about the pattern of streaks a film of oil leaves on a body — and stopped. Its own words: nothing here solves for one on a three-dimensional body.

So the surface rule arrived as a citation. Nodes minus saddles equals two on anything shaped like a ball, zero on anything with a hole through it, and a reader was asked to take it.

This essay computes it. What the computation turned out to be worth is not the confirmation, which was never in doubt — Poincaré–Hopf is a theorem and this site is not going to refute one. It is that putting a real field on a real surface and turning a parameter makes the invariance visible as an event: the number of stagnation points on the body falls from six to four and then to two, at two stream speeds that have closed forms, and the quantity the theorem is about does not move through either.

What had to be built, and what it deliberately is not #

The obstacle in the rung below was real and it is worth naming precisely, because it is the reason the surface version is rarer in a textbook than the plane one.

A velocity field in a plane is a map from two numbers to two numbers, and every tool this site owns works on one. A skin-friction field is a map from a point of a curved surface to a vector in that surface’s tangent plane — the limiting direction of the velocity as the wall is approached, which is what the layer’s own profile leaves behind at its foot, and neither end of that is a pair of numbers until somebody chooses coordinates. Choosing them is where the trouble starts.

The field used here is the surface gradient of a quadratic form. Take $\tfrac{1}{2}\,\mathbf{p}\cdot M\mathbf{p}$ on the unit sphere with $M$ symmetric and traceless, and let the pattern be its gradient along the surface. That is not decoration: near a wall the wall shear of a slow flow past a smooth body is the tangential gradient of a scalar to leading order, and for an ellipsoid it is exactly of this form. The critical points are then the eigenvectors of $M$ — six of them, at plus and minus each axis — and in each sign they are a maximum, a saddle and a minimum. Four nodes, two saddles, difference two.

What this is not is a solve. No Navier–Stokes solution produced this field and none was asked for. The essay is about a constraint that holds whatever the flow is, so using a prescribed field is not a weakness of the demonstration but the point of it — a constraint proved on one solution would be a much smaller claim. The same discipline governs the separation pattern in the rung below, which is two vortices and their images rather than a computed wake, and says so.

The coordinates are where the critical points hide #

The obvious way to find the zeros of a field on a sphere is to write it in the $(\theta, \varphi)$ rectangle and scan. It fails, and it fails at exactly the places that matter most.

The nose and the tail of the body are one point each. In the chart they are lines: the entire top edge of that rectangle is the single point $\theta = 0$, drawn six hundred pixels wide. A scan that sweeps the interior of the rectangle and stops at its boundary has excluded the two critical points a body is most reliably going to have, and a scan that includes the boundary finds each of them several hundred times.

So the search here is done in the tangent plane. At each starting point the code builds an orthonormal pair spanning the tangent plane, takes a Newton step in it, renormalises back onto the surface, and repeats. The poles of the chart are ordinary points of that search because the search never uses the chart. The index of each zero is then measured by walking a small circle round it in that same tangent basis and counting how many times the field’s direction turns — which is the definition of an index rather than a consequence of one, and it works on a degenerate zero where the eigenvalue story does not.

That distinction is load-bearing later, so it is worth stating now: the classification into node and saddle comes from eigenvalues and can fail; the index comes from a winding number and cannot.

The scan that found one point six times #

The first version of this reported the count on a sphere as six, with every one of the six a node and none of them a saddle. Six is not two, and a theorem had not been broken.

What had happened is worth carrying, because it is a general hazard of any search of this kind. With a strong stream added, the field near the rear of the body becomes very flat — a lot of the surface sits at a very small velocity — and forty Newton starts converged to forty points about a thousandth of a radian apart, every one of them satisfying $|\boldsymbol{\tau}| < 10^{-9}$ honestly. The deduplication threshold was tighter than that spread, so one critical point was recorded five times, each contributing $+1$.

The repair is not a tighter tolerance, because the numbers were not wrong. Two zeros closer together than the circle the index is measured on cannot be told apart by this method, and the honest answer is to merge them and report one point carrying the sum of their indices. That is also the right answer physically: a merged pair is what a bifurcation looks like from a small distance away, and reporting it as one point of index zero is a statement rather than a rounding.

This is the class of defect this site’s ledger records three times under a different heading — a distance chosen by a rule of thumb rather than measured against the thing it has to separate. Here the distance that matters is the index loop’s own radius, and once the two are tied together the count is right at every stream speed tested.

Six, then four, then two #

With the search trustworthy, a stream can be added to the pattern and turned up. This is the part that was not visible from the theorem.

Adding a uniform stream along the long axis is adding $Ux$ to the potential, so the whole field remains a surface gradient and the critical points remain those of

$$\Phi = \tfrac{1}{2}\!\left(m_1x^2 + m_2y^2 + m_3z^2\right) + Ux$$

restricted to the sphere. Those are where $\nabla\Phi$ is normal to the surface, which is $\nabla\Phi = \lambda\mathbf{p}$, and that is three equations with an unknown multiplier rather than anything requiring a solver. Working through the cases: the pair displaced in $y$ exists only while $U < m_1 - m_2$, the pair displaced in $z$ only while $U < m_1 - m_3$, and the two on the axis exist always.

With the coefficients used here, $(1, 0, -1)$, those thresholds are exactly 1 and exactly 2. Below one there are six critical points; between one and two there are four; above two there are two. The numerical scan agrees at $U = 0.999$ and $U = 1.001$, and again either side of two, which is what turns the scan from something self-consistent into something checked.

A pair cannot leave alone, and that is the whole content of the invariance in its most usable form. What annihilates is always a node and a saddle together, because their indices are $+1$ and $-1$ and the sum is fixed by the surface. If a picture appears to have lost a node with no saddle going with it, either a saddle was there and was missed or a node has appeared elsewhere — and the arithmetic says which of those to go looking for.

The change that alters every streak and no number #

There is a second parameter worth turning, and it separates the count from everything about the picture that a reader would naturally think it was measuring.

A focus and a node have the same index. The eigenvalues have gone from real to complex and the streaks now spiral, which changes the appearance of the pattern completely and changes its interpretation substantially — and a spiral in the streaks is not the spin of the fluid, which a shear and a rotation can share while looking nothing alike — a focus in an oil-flow picture is the footprint of fluid leaving the surface, a vortex lifting off, and finding one is how three-dimensional separation is identified in the first place — the same object a delta wing keeps deliberately over its upper surface.

None of that is in the index. The winding number round a focus is one turn and the winding number round a node is one turn, and a continuous deformation from one to the other passes through no integer in between because there is no integer in between to pass through.

So the count knows about existence and nothing about kind, and reading it as though it graded the picture is the misuse it invites. It cannot tell a separation from an attachment. What it can do is insist that the ledger balance, which is a weaker statement and one that no amount of looking harder at a photograph will supply.

The body that need have no stagnation point at all #

The negative case is where the theorem stops being bookkeeping and says something about the world.

The sum of the indices must be the Euler characteristic. For a sphere that is 2, and 2 cannot be reached by a sum with no terms in it, so a ball in a stream must carry at least one stagnation point on its surface. That is true at the Reynolds number where the wake is two steady eddies and at the one where it is shedding, and it does not care which. For a torus it is 0, and 0 is exactly what a sum with no terms comes to — so nothing forbids a field with no zeros, and the one drawn here is one.

The minimum of its magnitude over a quarter of a million sample points is 1 to fifteen decimal places, which is the constant component and is what a nowhere-vanishing field looks like when it is checked rather than asserted.

This has an engineering reading that is not a curiosity. A nacelle, an annular cowl, a ducted fan shroud, a wing with a through-slot — the last of which this collection has already argued about from the lift side, where the gap does not act as a nozzle: each of these is a torus, its surface pattern is required to sum to nothing rather than to two, and an experimentalist checking an oil-flow photograph against the arithmetic is checking it against a different number than for a fuselage. The rule that a technician learned on a fuselage gives the wrong answer on a cowl, and it gives it silently.

The arithmetic in front of a photograph #

The rung below described how the count is used in practice and this rung can put numbers on it, which turns a piece of advice into an instruction.

Suppose an oil-flow photograph of a fuselage shows, plainly, three places where the streaks radiate outward or spiral inward, and no place where two streaks cross. The reader’s tally is three nodes and no saddles. The surface requires the difference to be two.

Three is not two, so the photograph is incomplete, and the arithmetic says by exactly how much: one saddle is missing. Not “look more carefully” — one, and it is a saddle rather than anything else, because a node would make the discrepancy worse. That is a genuinely different instruction, and it is worth knowing that a saddle is the hardest of the three to see. A node is a starburst and a focus is a spiral, and both catch the eye; a saddle looks like nothing at all, two families of lines passing each other with a gap between them in a region where nothing appears to happen.

The same arithmetic run the other way is a warning. If a reader counts five nodes and one saddle, the difference is four rather than two, and the reading is that two of the five are not nodes — most likely one of them is two points too close together to separate, which is exactly the failure this essay’s own scan made and had to be repaired for.

It is also a different instruction from the one a photograph usually invites, where the eye is asked to trust a smooth picture and generally does.

And it is a check on an argument as well as on a picture. A proposed separation pattern, sketched on a napkin to explain a measurement, can be refused by counting it before anybody builds a model. That is not an appeal to intuition, and it is the reason the technique survives in a subject where almost every other constraint has a Reynolds number in it.

Where else the same sum forbids something #

The result is not about aerodynamics and it shows in the range of places it turns up, which is worth a paragraph because it is the surest way to remember what the hypothesis actually is.

A planet’s wind field. Whether the planet’s rotation matters to a given flow at all is a ratio rather than an opinion, and it is a separate question from this one: The surface winds of a rotating planet are a tangent field on a sphere, so they must have critical points and the indices must sum to two. There is no arrangement of global circulation, at any rotation rate, with wind everywhere — which is a statement about the atmosphere that contains no atmosphere in its derivation.

A cyclone’s eye. The same requirement acting locally: a closed circulation on a curved surface has to carry its centre with it.

And the case that is not a fluid at all. Comb a hairy ball and a parting or a crown appears somewhere; the theorem is indifferent to whether the field is hair, wind or wall shear, because its only hypotheses are that the field is continuous and that the surface is closed. Nothing in it knows what a fluid is, and that is precisely why it cannot be improved by knowing more about one.

What the hypothesis does require, and what is easy to lose, is that the field be tangent and continuous everywhere. A field with a genuine discontinuity — a sharp edge, where the wall shear direction jumps — is outside the theorem, and a body with sharp edges is most bodies. The usual repair is to round the edge and take a limit, which is legitimate and is where the half-integer weights in the experimentalists’ version come from.

What the picture cannot show, and one thing it should not be asked #

Three limits, and the third is the one most likely to be mistaken for a result.

The field is prescribed and no fluid was moved. Nothing here is the kind of solved field the viscous machinery produces, and no Reynolds number appears anywhere in the argument. Every figure here is a pattern rather than a solution, and the argument is about what patterns are permitted rather than about which one occurs. Any claim about where the separation on a real ellipsoid sits is outside this essay entirely and would need a solver this site does not have.

The surface is a sphere and a real body is not. Poincaré–Hopf depends on the surface only through its Euler characteristic, so an ellipsoid, a fuselage and a rugby ball share the sphere’s answer exactly; but the positions of the critical points depend on the shape in every detail, and nothing here computes them for anything but the quadratic used.

The count is not a measure of complexity. A pattern with sixteen nodes and fourteen saddles and a pattern with two nodes and no saddles satisfy the same equation. The theorem places a floor under how few points there may be and no ceiling at all on how many, so a photograph is never wrong for having too much in it — only for having an unbalanced amount.

And the assertion that has to be able to fail: the search is fed a quadratic form with a repeated coefficient, whose critical points are not isolated and cannot be counted, and it must refuse rather than return a number. It is also fed a form with a trace, which is not the field of a divergence-free flow, and must refuse that too. A count that came back from either would be a count that means nothing.

Who found it, and when #

Poincaré established the index of a plane vector field in the 1880s in his work on the qualitative theory of differential equations, and Hopf extended it to closed surfaces of any dimension in 1926 — which is where the theorem’s second name comes from and where the Euler characteristic enters. The result about combing a hairy ball is the sphere case stated for a schoolroom.

Its arrival in fluid mechanics is much later and came from experiment rather than from theory. Legendre in 1956 and Lighthill in 1963 set out the classification of surface-flow patterns; Hunt and co-workers in the 1970s produced the form with half-nodes and half-saddles that is used when the pattern is on a section through a body rather than on a closed surface. That the technique spread through experimental aerodynamics rather than through analysis is not an accident: it answers a question an experimentalist has, in front of a photograph, and it answers no question a theoretician had.

What was not available to any of them is the third figure in this essay. Watching the count hold through a merge requires turning a parameter and recomputing, and recomputing a surface pattern forty times is a thing a person with a wind tunnel and a can of oil cannot do.

Where the ladder goes next #

The rung below proved that a plane incompressible flow has only two kinds of critical point, from one line of arithmetic about a trace. This rung has left that result completely alone, because everything in it happens on a surface where the field has no incompressibility constraint of its own.

The rung above is where the two-kinds result stops being true, and it is the interesting direction. In three dimensions the velocity gradient is still trace-free and the eigenvalues still sum to zero, and that is a far weaker constraint on three numbers than on two: it permits one real eigenvalue and a complex pair, which is a spiral being stretched along its own axis and is exactly the structure that makes three-dimensional turbulence different from two-dimensional. Four generic kinds rather than two, separated by a curve with a closed form, and a plane flow turns out to be one line of the resulting diagram.

The one beside it is what happens when the parameter turning is a physical one rather than an invented stream — a Reynolds number, say — and the pattern’s change is a bifurcation of the solution rather than of a prescribed field. That is the same arithmetic with a solver behind it, and the solver is what this site would need to build.