A transition that needs a second number

Two concentric cylinders, the inner one turning, the fluid between them. It is the simplest shear flow with curvature in it, it has an exact steady solution, and above a threshold that solution gives way to a stack of toroidal vortices that has been photographed more often than almost anything else in this subject.

The number attached to the threshold is a Taylor number,

$$\mathrm{Ta} = \frac{4\Omega_1^2 d^4}{\nu^2}\cdot\frac{1-\mu}{1-\eta^2}, \qquad \eta = \frac{r_1}{r_2},\quad \mu = \frac{\Omega_2}{\Omega_1},$$

and the value quoted for it is 1707.762. Which is, to every digit, the critical Rayleigh number of a layer heated between two rigid walls.

That is not a coincidence and it is not an analogy. In the narrow-gap limit the two linearised problems are the same sixth-order eigenvalue problem, with centrifugal force where buoyancy was, so this essay imports the answer from that one rather than recomputing it. Writing the operator twice would be writing the same operator twice; the point is that there is only one.

The base flow, which is exact #

Before any stability question there is a steady solution, and it is one of the few exact ones this collection has.

For a steady, axisymmetric, purely azimuthal flow the momentum equation collapses to $\frac{\mathrm d}{\mathrm dr}\left[\frac1r\frac{\mathrm d(rv)}{\mathrm dr}\right] = 0$, whose general solution is $v = Ar + B/r$, with $A$ and $B$ fixed by no slip at each cylinder:

$$A = \Omega_1\frac{\mu - \eta^2}{1 - \eta^2},\qquad B = \Omega_1 r_1^2\frac{1-\mu}{1-\eta^2}.$$

Substituting that back into the equation by central differences leaves a residual of order 10⁻⁵ on a step of 10⁻⁴, which is the differencing rather than the solution. The check is worth running because $A$ vanishes at $\mu = \eta^2$ and a solution checked with the algebra that produced it is not checked.

Rayleigh’s criterion, which has no Taylor number in it #

The first thing the collapse discards is not small.

Rayleigh’s inviscid criterion is that a rotating flow is stable when the angular momentum increases outwards. For circular Couette flow it comes out exactly: with $r^2\Omega = Ar^2 + B$,

$$\Phi(r) = \frac{1}{r^3}\frac{\mathrm d}{\mathrm dr}\left(r^2\Omega\right)^2 = \frac{4A(Ar^2 + B)}{r^2},$$

and since $Ar^2 + B = r^2\Omega > 0$ for co-rotation, $\Phi \ge 0$ everywhere if and only if $A \ge 0$, which is $\mu \ge \eta^2$.

No viscosity. No gap width. No Taylor number. A machine with $\mu > \eta^2$ cannot be made unstable to an axisymmetric disturbance by spinning it faster, because spinning it faster raises the Taylor number and does not change the sign of $\Phi$.

Bisection on the discriminant at seven radius ratios returns $\mu$ within 10⁻¹⁶ of $\eta^2$ every time. That is a boundary in a plane the Taylor number does not have an axis for, and it decides whether there is a transition at all.

The eigenvalue, borrowed rather than rewritten #

Below Rayleigh’s line the viscous problem decides where, and that is where the imported operator comes in.

Its minimum is at $\mathrm{Ta} = 1707.757$ and a wavenumber of $a = 3.1158$, against the published 1707.762 and 3.117. Those are the convection essay’s own numbers, produced by its own discretised operator with two rigid walls, called here with no modification whatever.

The correspondence is worth stating precisely because it is easy to hear as a family resemblance. In the narrow-gap limit the linearised Taylor problem reduces to

$$(D^2 - a^2)^2 W = \mathrm{Ta}\,a^2\,\Theta,\qquad (D^2 - a^2)\Theta = -W,$$

with $\Theta$ the azimuthal velocity perturbation instead of the temperature. The buoyancy that drives the convection problem and the centrifugal imbalance that drives this one enter the same slot. Two different physical mechanisms, one operator, one number.

The vortex wavelength is the same too. $2\pi/a = 2.017$ gap widths, so a Taylor vortex is square in cross-section to within two per cent, exactly as a Bénard cell is — and for the same reason, which is that the operator does not know which problem it is solving.

What a real machine is allowed to choose #

The 1707.762 is a minimum over a continuum of axial wavenumbers. A finite column does not have one.

A Taylor vortex occupies half an axial wavelength, so $N$ vortices in a column of height $\Gamma d$ need $a = N\pi/\Gamma$, and the machine’s threshold is the smallest marginal value over the integers.

The curve has to be computed against a floor measured on the same grid, and that is not fastidiousness. The Richardson-extrapolated critical value is about a thousandth below what the N = 64 operator returns, which is enough to make a constrained minimum appear to beat an unconstrained one — a machine going unstable below its own threshold, which is arithmetic and reads as physics.

At an aspect ratio of 2.5 the column pays 5.66 per cent for having ends. At an integer aspect ratio it pays a hundredth of that, because $a = \pi$ is within one per cent of the critical wavenumber. And above Γ = 10 the worst case has fallen to 0.27 per cent, because the allowed wavenumbers have got closer together.

The count is decided by the height #

The other half of the same fact is that the number of vortices is an integer chosen by the geometry. Near each riser of the staircase two states have nearly equal thresholds — the sweep records gaps below one per cent for Γ above about 8 — and that near-degeneracy is where the hysteresis everybody who has run this experiment has seen comes from.

Coles’ 1965 experiments found dozens of distinct stable states at one Taylor number, differing in the vortex count and in the azimuthal wavenumber, reachable by different histories of acceleration. A number that names a threshold cannot name a state, and this machine has many at once.

Counter-rotation, where the criterion changes character #

The rotation-ratio axis has a left-hand end as well, and it behaves differently enough to be worth separating.

With the outer cylinder turning backwards, μ is negative, and Rayleigh’s discriminant is negative over part of the gap and positive over the rest: there is a radius at which $r^2\Omega$ has a maximum, and outside it the angular momentum increases outwards. So the flow is inviscidly unstable in an inner annulus and inviscidly stable in an outer one, and the disturbance that grows is confined to the first.

That has two consequences the collapse cannot express. The critical Taylor number rises with counter-rotation, because the unstable region is a fraction of the gap and the viscous damping is set by the whole of it. And the vortices that appear do not fill the gap: they sit against the inner cylinder, with a nearly quiescent layer outside them, which is visible in every photograph of a counter-rotating rig and is invisible in any number formed from Ω₁ alone.

Taylor measured that in 1923 and it is one of the more striking parts of the agreement he obtained: the threshold curve bends upward as μ goes negative, by a factor of several by μ = −1, and his hand computation follows it.

The general moral is the one the Richardson number’s local form also makes: a criterion evaluated pointwise across a domain can be satisfied in one part of it and not another, and a single number formed for the whole domain has to choose which part to be about.

The criterion that stopped a star from forming #

The discriminant computed above is a laboratory statement about two brass cylinders, and it has been the central obstacle in a completely different subject for thirty years. It is worth following, because it is the strongest demonstration that Rayleigh’s line is a real boundary rather than a technicality.

Matter orbiting a star follows Kepler’s law: $\Omega \propto r^{-3/2}$. So the specific angular momentum, $r^2\Omega$, goes as $r^{1/2}$ — it increases outwards, everywhere, at every radius. By the exact criterion of this essay the discriminant is positive throughout and an accretion disc is Rayleigh-stable, with no viscosity and no Taylor number able to change it.

And an accretion disc has to be turbulent, or nothing works. For matter to spiral inwards onto the star it must shed angular momentum outwards, and molecular viscosity in a disc of gas is so feeble that the inward drift it produces would take vastly longer than the age of the universe. Discs manifestly do accrete — young stars are visibly assembling — so something is transporting angular momentum, and the flow’s own hydrodynamic instability is ruled out by the sign of one derivative.

The way out was to change what the criterion is about. Thread the disc with a weak magnetic field and the stability condition is no longer about the angular momentum but about the angular velocity: instability now requires only that $\Omega$ fall outwards, which every Keplerian disc does. The mechanism is a spring. A field line joining two fluid elements at slightly different radii is stretched by the differential rotation and pulls back — slowing the inner element, which therefore loses angular momentum and falls further in, where it orbits faster still and stretches the line further. It runs away.

That is the magnetorotational instability, and the field it needs is arbitrarily weak: its role is to couple neighbouring radii, not to supply energy.

And the laboratory has been used to check the negative half of the argument. Taylor–Couette rigs have been built with rotation ratios chosen to be quasi-Keplerian — Rayleigh-stable by the exact line above — and run to Reynolds numbers in the millions to see whether the flow becomes turbulent anyway through some finite-amplitude route. It does not. Two brass cylinders, in a basement, testing whether stars need magnetism to form.

Three numbers, and the Taylor number is one #

Putting it together, the honest specification of the problem needs at least four quantities and the Taylor number is one of them.

The radius ratio η, which decides where Rayleigh’s line is and, at finite gap, changes the critical Taylor number itself — the narrow-gap value is a limit, and for a gap of a tenth of the inner radius the true threshold is a per cent or two higher. The rotation ratio μ, which decides whether there is a transition. The aspect ratio Γ, which decides the threshold’s scallop and the state. And the Taylor number, which decides how far above the threshold the machine is.

That is a familiar shape by now. The Keulegan–Carpenter number needs β beside it; the capillary number needs the viscosity ratio and the flow type; and here the residual is two ratios of lengths and one of speeds. What is unusual is that one of the residuals — Rayleigh’s line — is not a correction at all. It is a region of the parameter space in which the collapse has no content.

What happens above the threshold #

The essay so far is about one bifurcation, and the machine has a great many, which is worth laying out because the Taylor number does order them.

Just above 1,708 the state is steady axisymmetric Taylor vortices — the stack of tori, with no azimuthal structure at all. Somewhere above that, at a Taylor number a few times critical and depending on every one of the three ratios, the vortices develop a travelling azimuthal wave and become wavy vortices, which is a second bifurcation with its own eigenvalue and its own azimuthal wavenumber. Then modulated waves, with a second incommensurate frequency. Then chaos.

That sequence — steady, periodic, quasi-periodic with two frequencies, chaotic — is the Ruelle–Takens route, and Taylor–Couette is where it was first observed cleanly in a fluid. Gollub and Swinney’s 1975 measurements are the ones usually cited, and they are a laser-Doppler spectrum with a discrete set of peaks appearing and then broadening, which is as direct a picture of a route to chaos as this subject has.

Far above, the vortices survive as turbulent Taylor vortices: the flow inside them is turbulent and the large-scale toroidal pattern is still there and still has a wavelength near two gap widths. That is worth noticing beside the Lorenz truncation, which is a different fluid problem’s route to the same place and which stops describing its own fluid long before it gets there.

Why the ends matter more than they should #

It is worth asking why a per cent or two of threshold is worth a section, when most of this collection’s model errors are larger.

The answer is that the threshold is an eigenvalue, and eigenvalues are what a transition is. Below it nothing happens; above it a pattern grows. So a five per cent error in the threshold is a five per cent error in the speed at which a machine changes state, and it is measurable to far better than that — a Taylor–Couette rig is one of the most accurately reproducible experiments in fluid mechanics, with onsets repeatable to a fraction of a per cent.

It is also the reason the end conditions are a research subject rather than a nuisance. Real end plates drive Ekman circulations that are present at every Taylor number, so a real machine has a weak vortex pair at the ends before the bulk goes unstable at all, and the bifurcation is imperfect: what should be a sharp onset is a smooth growth. The scalloped curve above is the idealised statement of a real effect, computed for stress-free ends that no apparatus has.

Where this leaves the number #

None of it makes the Taylor number useless. It orders the problem correctly, it collapses the fluid and the speed and the gap into one quantity, and above the threshold the sequence of transitions — wavy vortices, modulated waves, turbulent Taylor vortices, featureless turbulence — is laid out along it in the same way the Keulegan–Carpenter regimes are laid out along theirs.

What it does not do is decide. It is a number about the strength of the driving, in a problem whose answer is set by three ratios of geometry and one of the two rotation rates.

The narrow gap, and what widening it costs #

Every number above is a narrow-gap number, and the limit is worth pricing.

The narrow-gap reduction throws away the curvature terms in the linearised equations, keeping only their leading effect through the centrifugal driving. For $\eta = 0.95$ that is excellent. For $\eta = 0.5$ — an inner cylinder half the outer’s radius, which is an ordinary laboratory geometry — the critical Taylor number computed with the full equations is several per cent from the narrow-gap value and the critical wavenumber has moved as well, because the vortex is no longer square when the two walls have different circumferences.

More importantly the shape of the neutral surface changes: at wide gaps the unstable region does not extend all the way to Rayleigh’s line, because the viscous problem is decided by a region near the inner cylinder rather than by the gap as a whole. So the exact inviscid boundary and the viscous threshold stop being two independent statements about the same plane.

This file does not compute any of that. It computes the exact criterion, the narrow-gap eigenvalue and the quantisation, and says which of the three is a limit.

Where the work came from #

G. I. Taylor’s 1923 paper is one of the founding documents of the subject and is remarkable for a reason that has nothing to do with the number: it is the first place where a linear stability calculation and an experiment agreed. Rayleigh’s criterion was 1917 and is inviscid; Taylor added the viscosity, solved the eigenvalue problem by hand, built the apparatus, and measured thresholds within one per cent of his own predictions across a range of rotation ratios including counter-rotation.

Chandrasekhar’s 1961 book is where the narrow-gap correspondence with Bénard convection is set out at length, and it is also where the 1707.762 comes from. Coles’ 1965 paper is the one that found the multiplicity of states.

The order is worth noticing. The exact criterion is oldest, the eigenvalue is next, and the discovery that the machine has many states at one Taylor number is the most recent by forty years — which is the usual order in this subject, where the sharpest results arrive first and the complications that limit them arrive later.

What this leaves #

Two residuals: a line in the rotation-ratio plane above which no Taylor number is enough, and an integer chosen by the height of the apparatus.

The next essay is about a number whose velocity nobody supplied — a speed nobody imposed — where the flow makes its own scale and the Reynolds number of the result is a consequence rather than a setting.