Two kinds is a plane flow's privilege

The rung two below this one proved something clean enough to be worth restating exactly. At a point where a plane incompressible flow is not moving, the local flow is its velocity gradient; that gradient is trace-free, because incompressibility is exactly the statement that the trace vanishes; its two eigenvalues therefore sum to zero; and a pair of numbers summing to zero is either real and opposite — a saddle — or purely imaginary — a centre. Incompressibility is doing the whole of that work, and it is worth remembering that it is a statement about a flow rather than about a fluid. Nodes and spirals cannot occur, and a published sketch containing one is wrong about something more basic than the physics.

That is a strong result obtained from almost nothing, and the temptation is to carry it into three dimensions unchanged. It does not go. The same sentence — the eigenvalues sum to zero — is a constraint on three numbers rather than two, and one equation on three unknowns leaves a surface of possibilities where one equation on two unknowns left a line.

The cubic, and the two numbers that decide it #

A 3×3 trace-free tensor has a characteristic polynomial with the quadratic term missing:

$$\lambda^3 + Q\lambda + R = 0, \qquad Q = -\tfrac{1}{2}\operatorname{tr}(A^2), \qquad R = -\det A.$$

Two coefficients rather than three, because the first invariant is the trace and the trace is gone. So a critical point in three dimensions is described, up to a change of basis, by exactly two numbers — and every question about what kind it is has to be answerable from those two.

It is, and the answer is the discriminant:

$$D = \frac{27}{4}R^2 + Q^3.$$

Where $D > 0$ the cubic has one real root and a complex conjugate pair, which is a rotation in one plane with a stretch or a squeeze along the axis perpendicular to it. Where $D < 0$ all three roots are real, and the point is a node in one direction with two saddle directions crossing it. The sign of the remaining real quantity then splits each case in two, by whether the odd direction is being pulled out or pressed in.

Four generic kinds, and the curve $D = 0$ between them.

Where the plane result went #

The plane result did not become false. It became a slice.

Take a plane flow and embed it: nothing moves in the third direction and nothing depends on it, so the gradient has a row and a column of zeros and its determinant vanishes identically. $R = 0$, always, exactly, with no rounding in it. That is a line through the diagram, and the whole of the plane classification is what happens on it.

On that line $D = Q^3$, so the sign of $D$ is the sign of $Q$ — and $Q$, for an embedded plane flow, is the plane gradient’s determinant. Negative is a saddle and positive is a centre, which is precisely the rung below’s rule, recovered as a special case rather than contradicted.

What is new is everything off the line, and it is most of the diagram. A plane flow has one number to be classified by; a three-dimensional one has two, and the second is the determinant a plane flow is forbidden to have.

The kind a plane cannot have, and what it is #

The interesting half of the diagram is the upper lobes, and it is worth being concrete about what a point there is doing.

The complex pair is a rotation — half the vorticity, which is not the same as the fluid going round. The real eigenvalue is a stretch along the axis of that rotation. So a critical point with $D > 0$ is a vortex being pulled out along its own axis, or squeezed along it — and that object cannot exist in a plane flow, because a plane flow has no third direction to pull along.

That last observation is worth pausing on, because it is not obvious and the arithmetic is short. Pulling hard on a vortex does not eventually overwhelm its rotation and turn it into something else. The rotation lives in the antisymmetric part of the gradient and the stretch in the symmetric part; they are different pieces of the same tensor and the stretch does not reach into the other one. So however violently the axis is drawn out, the local pattern is a spiral, and the discriminant grows rather than approaching zero.

This is the structural reason vortex stretching has no local limit. Circulation round a material tube cannot change, so a tube whose area falls must spin faster, and nothing in that argument sets a ceiling. Here is the same absence from the other side: no amount of stretch changes what kind of critical point a stretched vortex is.

What decides which kind a flow has #

If four kinds are available, something has to allocate them, and the knob is not the Reynolds number.

A gradient with no antisymmetric part is a pure strain — the object whose principal axes a parcel of dye finds in its first instant — its tensor is symmetric, and a real symmetric matrix has real eigenvalues. So a strain-only flow lands below the curve every time, with three real roots and no rotation anywhere in it. Turn the rotation up and the population moves into the lobes.

Which is a statement about a sample of a model and about nothing that has been measured, and the figure says so in its own note. What it demonstrates is the mechanism — that the allocation between the two halves of the diagram is the balance between rotation and strain, and that a flow’s Reynolds number does not enter the classification at any point. What it does not demonstrate is what a real flow’s distribution looks like.

That distribution is, as it happens, one of the more striking measurements in the subject. Every experiment and every simulation of turbulence that has plotted the joint statistics of $Q$ and $R$ finds them concentrated along the lower-right branch of exactly this curve, in a shape the literature calls the teardrop, at every Reynolds number in the range a real one does not have. The site does not compute it — that needs a resolved turbulent field, which nothing here has — and quoting the shape without producing it is as far as this essay is prepared to go. The curve is derived here; the population on it is borrowed.

The one place on the curve that is not a boundary #

There is a point of the diagram worth naming because it is where a great deal of real flow sits and because it is a case the classification declines to classify.

Axisymmetric strain — a rate $e$ along one axis and $-e/2$ across both of the others, which is what a round jet’s core does and what a stagnation flow onto a plate does — has a repeated eigenvalue. Two of the three roots coincide, and a cubic with a repeated root has discriminant exactly zero. So axisymmetric strain does not sit inside either lobe or below the curve: it sits on the curve, and the assertion behind these figures checks that it does rather than assuming it.

That is not a defect of the scheme. A point on the boundary is a genuinely marginal case, the kind that a small perturbation can push either way, and a classification that silently rounded it into one lobe would be hiding the one thing about it worth knowing. It is also the reason the teardrop’s two branches are where measurements pile up: the branches are the axisymmetric states, and a flow spends much of its time near one.

How much room the same sentence leaves #

The two cases are worth counting, because “one equation on three unknowns instead of two” understates how much was given away.

A general 2×2 matrix has four entries. Setting the trace to zero leaves three, and of those three only one combination decides the type — the determinant — so the classification is a question with a single number in it and two possible answers. A general 3×3 matrix has nine entries. Setting the trace to zero leaves eight, and the classification takes two numbers.

That is the whole difference, and it is not a difference of degree. In the plane the space of critical points has a single boundary in it: the determinant passes through zero and a saddle becomes a centre. In space the space of critical points is a plane with a curve drawn on it, and a curve in a plane can be approached from either side, run along, touched tangentially and crossed twice by a path that started and finished in the same lobe.

The strongest version of the plane theorem is therefore that it is a coincidence of low dimension. It says something true and useful about plane flows, and reading it as a fact about fluids is reading a property of the number two.

Q, and the slider in every post-processor #

There is a connection here that is not usually pointed out and that makes the whole diagram feel less abstract, because one of its two axes is a quantity a great many people have adjusted with a mouse.

Split the velocity gradient into its symmetric and antisymmetric parts — the strain rate and the spin, which is the decomposition the first rung of the deformation ladder is about. Then the second invariant works out to

$$Q = \tfrac{1}{2}\left(\lVert\Omega\rVert^2 - \lVert S\rVert^2\right),$$

the difference between the squared magnitudes of the rotation and the strain, halved. Positive where the local flow spins more than it shears, negative where it shears more than it spins.

That is the Q-criterion, and it is the default way a vortex is identified in a computed flow field: draw the surface where $Q$ takes some positive value and what appears is a set of tubes that look like vortices. Every visualisation package has it, most people who use it treat it as a heuristic with a threshold that has to be tuned, and it is in fact the second coefficient of the characteristic polynomial of the velocity gradient — the same $Q$ as the vertical axis of the hero figure, arrived at from the other end.

Which explains both its usefulness and its arbitrariness in one stroke. It is a genuine invariant, so it does not depend on how the axes were drawn; and it is one of two numbers rather than the whole classification, so a surface of constant $Q$ cuts across all four kinds of point and the threshold has to be tuned because the criterion is a projection. A reader who has spent an afternoon sliding that threshold and wondering what principle sets it now has the answer: no principle sets it, because the quantity it thresholds is one coordinate of a two-dimensional classification.

One point, worked through #

The arithmetic is short enough to do once in full, and doing it makes the abstraction concrete.

Take the gradient of a vortex spinning at unit rate about the $z$ axis while being stretched along it at rate one: the diagonal is $(-\tfrac{1}{2}, -\tfrac{1}{2}, 1)$, which sums to zero, and the off-diagonal pair $\pm1$ in the $x$–$y$ block supplies the rotation.

Then $\operatorname{tr}(A^2) = -\tfrac{1}{2}$, so $Q = \tfrac{1}{4}$ — positive, which says the rotation is winning against the strain. The determinant is $+\tfrac{5}{4}$, so $R = -\tfrac{5}{4}$. The discriminant is

$$D = \tfrac{27}{4}\left(-\tfrac{5}{4}\right)^2 + \left(\tfrac{1}{4}\right)^3 = \tfrac{169}{16} = 10.5625,$$

comfortably positive, so there is one real root and a complex pair — and solving the cubic returns exactly $1$ and $-\tfrac{1}{2}\pm i$, which is what the construction put in.

The signs are worth care and are the commonest slip in this arithmetic. $R$ is minus the determinant, so a vortex being stretched has a positive determinant and a negative $R$, and lands in the left-hand lobe of the hero figure rather than the right-hand one. Reversing the stretch reverses $R$ and moves the point across to the other lobe without changing $Q$ at all, which is the horizontal reflection the diagram is symmetric about.

That last agreement is the check, not the result. The eigenvalues were known before the invariants were computed, and the point of running it forwards is that the classification arrived at them from two scalars without ever seeing the tensor. On a real flow field the scalars are what is available: a simulation stores $Q$ and $R$ per cell far more cheaply than it stores three complex eigenvalues, and every large study of this classification has been done that way.

What a plane flow is missing, and what it buys instead #

It would be easy to read all of this as a plane flow being an impoverished version of a real one. It is worth putting the trade the other way round, because the absence buys something specific and famous.

The kind a plane flow cannot have is a vortex being stretched, and vortex stretching is the term $\boldsymbol{\omega}\cdot\nabla\mathbf{u}$ in the vorticity equation. In two dimensions the vorticity is perpendicular to every velocity gradient there is, so that term is identically zero — not small, not usually negligible, but absent. And its absence is what gives a plane flow a second inviscid invariant: enstrophy is conserved as well as energy, which forces energy the other way up the scales and produces an inverse cascade instead of a forward one.

So the missing kind of critical point and the missing term in the vorticity equation and the extra conserved quantity are three descriptions of the same absence. A plane flow is not a simplification of a three-dimensional one; it is a different subject, and the classification above is the shortest way to see why.

There is a consequence for computation that follows immediately and is not always noticed. A two-dimensional simulation cannot produce a stretched vortex however fine its grid, because the object is not in the space of fields it can represent. A three-dimensional simulation on a grid too coarse to resolve the stretching produces the term at the wrong magnitude rather than at zero, which is a different kind of error and a worse one — the first is honest about what it cannot do and the second is not.

What the picture cannot show #

No flow was solved. Every point in every figure here is a tensor rather than a place, and the whole essay is a classification of what a velocity gradient can be rather than a claim about which gradients occur. A reader wanting to know how often a real flow is in each region will not find it here.

A critical point is local and the picture is not. Everything drawn is the linearisation, in the same sense that a strain rate is the first instant of a deformation and nothing after it. The classification is exact within one gradient length of the point and says nothing whatever at a greater distance, so none of these panels is a picture of a flow. Two neighbouring critical points of different kinds are perfectly ordinary and the region between them is not described by either.

The cloud is a sample of a model. The gradients scattered across the hero figure are drawn from a seeded generator with no physics behind it beyond the trace being zero. Their density carries no information about a fluid, and the only quantity read off them is which side of the curve they fall.

And the count from the two rungs below does not extend. In three dimensions the index of an isolated zero of a vector field is still an integer and the sum is still the Euler characteristic — of the enclosing surface, which for a solid region is a sphere — but the four types above do not carry the indices $\pm1$ in the same tidy way, and a three-dimensional saddle has an index that depends on how many of its directions are outward. Nothing here computes that, and the arithmetic the previous rung ran on a body’s surface is the two-dimensional theorem applied to a two-dimensional field that happens to live on a curved surface.

Who found it, and when #

The classification of critical points by the invariants of the velocity gradient was set out by Chong, Perry and Cantwell in 1990, and it is one of the few pieces of modern fluid mechanics that is entirely linear algebra. The framework had been available since the nineteenth century; what was new was the observation that a fluid dynamicist wants it in the trace-free case and that the trace-free case is much tidier than the general one.

The measurement that made it famous came shortly after, when direct numerical simulations became large enough to accumulate statistics of $Q$ and $R$ and the teardrop appeared — in isotropic turbulence, in channel flow, in mixing layers, and at every Reynolds number anybody has looked at. That a distribution this robust exists is itself a result, and it has resisted a satisfying explanation for thirty years.

And the plane result is much older than either. That an incompressible plane flow has only saddles and centres is a remark Poincaré’s classification makes immediately once the trace is set to zero, and it can be read out of nineteenth-century work on the qualitative theory of differential equations without anybody having thought about a fluid.

Where the ladder goes next #

This anchor now has the count on a plane, the count computed on a body, and the local classification in space. Two directions are open above it and they are not the same size.

The rung directly above is the index in three dimensions, which is the one thing this essay explicitly declined. A three-dimensional saddle’s index depends on the dimension of its unstable manifold, so the tidy $\pm1$ of the plane becomes a sign that has to be worked out per point, and the resulting sum over a closed surface is a genuinely different bookkeeping from the surface one. It is the arithmetic behind counting the critical points of a three-dimensional separated flow rather than of the streaks it leaves on the wall.

The one beside it is the objectivity question, and it is the more uncomfortable one. A critical point is a place where the velocity vanishes, and which places those are depends on the observer — a pattern belongs to whoever is watching. $Q$ and $R$ are frame-indifferent under a change of constant velocity and are not under a rotating one, so the whole of this classification is objective in one sense and not in another, and the criteria built on it that are used to identify vortices in a computation inherit exactly that weakness. Stating which of them survives which change of frame, and computing an example where two observers disagree about how many vortices there are, is a rung with a real result in it.