The state a machine was started into

A transition that needs a second number computes the Taylor-Couette instability: rotation between cylinders becomes unstable above a critical Taylor number, and the disturbance that first grows has a definite wavelength.

That is the statement at the threshold. This essay is about what happens above it, and the answer is that the flow acquires a choice — one it makes on the way up and then keeps.

Why there is a band #

The neutral curve gives, for each disturbance wavelength, the driving needed to sustain it. It has a minimum, and the minimum is the critical condition: below it nothing grows and at it exactly one wavelength is neutral.

Above it a whole band grows. At any driving above critical there is an interval of wavenumbers for which the neutral curve lies below the driving, and every member of that interval is unstable.

That is a completely general consequence of the curve having a minimum, and it applies to Rayleigh-Bénard convection, to Taylor-Couette flow and to every pattern-forming instability in the subject. A threshold that is a minimum over a continuous parameter is a threshold that admits a band.

And a narrower band that survives #

Not every wavelength in the unstable band is a stable final state. A pattern at a wavelength near the edges of the band is itself unstable to a slow adjustment of its own wavelength — the Eckhaus instability — and drifts towards the middle.

The stable band is narrower than the unstable one by exactly one over the square root of three — 1.997 against 3.459 at twice critical, a ratio of 0.5774 — which is the standard result for a supercritical bifurcation and comes out of the amplitude equation rather than out of any particular fluid.

So there are two bands, one inside the other, and the outer one is the set of patterns that can grow while the inner one is the set that can persist.

Which makes several flows available at once #

A real apparatus has a finite length, so it holds a whole number of vortex pairs. That quantises the admissible wavenumbers, and the question becomes how many of the quantised values fall inside the stable band.

The critical wavenumber is 2.2214 and the critical Taylor number 657.51. Above it:

At five per cent above critical there is one. At twice critical there are seven — seven distinct steady flows, each with a different number of vortex pairs, each stable, all at exactly the same Taylor number in exactly the same apparatus. At three times critical there are eight.

How wide the choice is, in numbers #

The two bands are worth quantifying, because the width is what turns a mathematical possibility into an experimental nuisance.

At five per cent above critical the unstable band spans wavenumbers from about 1.9 to 2.6 and the stable one from 2.0 to 2.4 — a range of about twenty per cent in wavelength. At twice critical the unstable band runs from 0.98 to 4.44, a factor of four and a half, and the stable one from 1.50 to 3.50.

Two features of that are worth noticing. The band opens as the square root of the distance above critical near the threshold, which is the standard scaling and means it opens quickly at first. And it is asymmetric: at twice critical the neutral band runs in wavenumber from 0.984 to 4.443 about a critical value of 2.221, which in wavelength is 1.414 to 6.386 about a critical 2.828. The band reaches 1.414 towards short wavelengths and 3.558 towards long — two and a half times as far.

That asymmetry has an experimental consequence. A pattern pushed out of the band is more likely to adjust by losing a vortex pair than by gaining one, which biases the drift and is visible in careful measurements as a preference for longer wavelengths at high driving.

What decides which one #

Nothing in the final conditions does, which is the point. The Taylor number, 1,315.02, the geometry, the fluid and the boundary conditions are identical between the seven, so the selection has to come from somewhere else — and it comes from the history.

Ramp the driving slowly and the flow adopts the wavelength that is critical at each moment, which is the minimum of the curve, and it stays near there as the band opens around it.

Start impulsively and the disturbance that happens to be present grows fastest, which need not be the critical one, and the resulting pattern is whatever the initial noise favoured — the same sensitivity to what happened to be there that every mode decays and it grows anyway is about in a different instability.

Ramp up and then down and the pattern does not retrace its path: a wavelength that was stable on the way up remains stable on the way down past the point at which it would not have been selected, which is hysteresis.

So the number of vortices in a Taylor-Couette apparatus is a record of its start-up, and it is one of the very few macroscopic quantities in fluid mechanics that is genuinely undetermined by the conditions.

What the solver computed, and how it was checked #

The neutral curve is the analytically tractable free-free case, whose closed form is (k² + π²)³/k² and whose minimum is at k = π/√2. Taylor-Couette’s own curve has no such closed form and the same structure, and the essay says so rather than pretending otherwise.

The stable band is the unstable one divided by root three in the distance from the critical wavenumber, which is the amplitude-equation result. The state count is the number of quantised wavenumbers inside it for an apparatus of a stated aspect ratio.

Three checks. That the two bands are in the ratio one over root three, exactly — 1.997 against 3.459, a ratio of 0.5774 — which would catch an error in the construction. That the state count rises with the driving, monotonically: 1, 2, 4, 4, 7, 8. And that there are at least three states at the top of the sweep; there are eight, so the multiplicity is real rather than marginal.

How the states differ #

It is worth saying what an experimenter would actually see, because eight states sounds more dramatic than it looks.

They differ in the number of vortex pairs — five to eleven at twice critical — so the vortices are of slightly different heights. At twice critical the seven states span a range of wavelength from 1.79 to 4.17 in units of the gap, which is a factor of 2.33, which is visible in a photograph and easy to miss in a measurement.

They also differ in the torque the inner cylinder requires, by a per cent or two — which is a measurable quantity that depends on a fact about the start-up, and is therefore the cleanest possible demonstration of the essay’s claim. That is a small difference and it is measurable, and it is the reason a torque measurement in a Taylor-Couette apparatus has to be reported with the vortex count.

And they can differ in which way the end vortices turn, which is a discrete choice on top of the wavelength one and multiplies the count. Coles’ famous experiment found twenty-five distinguishable states at one Reynolds number by counting the azimuthal wave number as well.

What kind of memory this is #

Placed beside the others in this collection, the multiplicity is an unusual species and it is worth naming the difference.

It is discrete. Most of the memories here are continuous quantities — a stress, a thickness, a circulation — that vary smoothly with the history. This one is an integer: the number of vortex pairs, which cannot change without a defect passing through the pattern.

It does not decay. A stress relaxes, a wake convects away, a layer diffuses. A vortex count sits where it is for as long as the driving is maintained, because moving it requires crossing a barrier rather than climbing down a gradient.

And it is a memory of a moment rather than of a history. What the flow retains is which state it fell into during the transient, and everything before and after that moment is irrelevant.

That combination — discrete, permanent, and set at an instant — makes it the closest thing in this collection to a memory in the everyday sense: a fact about the past that the system simply has, rather than a residue that is fading. The nearest neighbour is the linking number of the knot a flow cannot untie, which is also an integer and also cannot change continuously.

Why this is unusual #

Most flows in this collection are determined: given the geometry, the fluid and the driving, there is one answer. It is worth being clear about why this one is not.

The instability is supercritical, so the amplitude grows continuously from zero — there is no jump. But the wavelength is a second degree of freedom, it is not fixed by the amplitude, and the equations have a family of solutions parameterised by it. Selecting a member requires something outside the steady problem.

That is the same structure as nothing in the present picks the flow, where the circulation round a body is a free constant that the history supplies — with the difference that there the family is continuous and here the geometry has already quantised it.

A free parameter in a steady solution is an invitation to a memory, and this collection has now met the invitation three times: a circulation, a vorticity on a closed streamline, and a wavelength.

The other way the state gets decided is by forcing it, which is the essay after this one and the same solver.

What this does to an experiment #

Report the protocol. Two laboratories at the same Taylor number with different ramp rates are measuring different flows, and a disagreement between them is not necessarily an error in either.

Do not treat scatter between runs as noise. A quantity that takes one of eight values will look like a distribution when it is a set, and averaging over it produces a number that no run ever produced — the same complaint what a mean profile cannot tell anybody makes about a mean.

Ramp slowly if a reproducible state is wanted. Quasi-static ramping selects the critical wavelength and is the closest thing to a canonical protocol.

And check the aspect ratio. The quantisation depends on the length of the apparatus, so a container that is long enough to hold many pairs has a denser set of admissible wavenumbers and therefore more states — while a very short one may have only one, which makes it reproducible and unrepresentative.

The general warning is one this collection has repeated: a measurement in a system with multiple stable states is a measurement of the state, and the state is a fact about the history.

What a slow ramp actually selects #

The claim that a quasi-static ramp selects the critical wavelength deserves a closer look, because it is the only protocol that gives a reproducible answer and it is worth knowing how reproducible.

At the moment the driving crosses critical, exactly one wavelength is neutral and every other decays. The pattern that forms therefore has that wavelength, to whatever precision the noise allows.

As the driving continues to rise, the band opens around the pattern that is already there, and there is no mechanism forcing the pattern to move: it is inside the stable band, so it stays. The final wavelength is therefore the critical one, not the one that would be selected by a fresh start at the final driving.

A slowly ramped system is carrying its own threshold behaviour upwards, which is a memory of the crossing rather than of the ramp. That distinction matters: two slow ramps at different rates give the same answer, and a slow ramp and a sudden start do not.

The exception is if the ramp is slow enough for the pattern to drift within the band — which happens over very long times through the Eckhaus mechanism, and which is why a “steady” state in this system is only steady on the time scale it has been watched over.

Where the same multiplicity appears #

Rayleigh-Bénard convection. The same neutral curve, the same band, and rolls of a wavelength set by the start-up. It is the case the closed form above actually belongs to.

Turbulent flows with more than one attractor. A pipe above the critical Reynolds number, in the sense of a puff that does not know how old it is, has a laminar state and a turbulent one at the same conditions, and which one is realised depends on the disturbance history.

A rotor’s inflow and a wake’s roll-up. Neither is a pattern in the same sense, and both have the property that the final state is reached along a path that matters — a spiral is a legible record being the clearest case, where the end state’s structure is the path written down.

And any pattern-forming system. Stripes, hexagons, spirals and targets in convection, in reaction systems and in granular flow all coexist over ranges of parameter, and the selection is a start-up question in every case.

Where this leaves a computation #

There is a consequence for anybody simulating a flow of this kind, and it is not the obvious one.

A computation selects a state too. Its initial condition is whatever the code was started from, its domain length quantises the wavenumbers exactly as an apparatus does, and its answer is one member of the family. Two codes agreeing on the equations and differing in their start-up produce different steady answers and neither is wrong.

And a periodic domain makes it worse. A doubly periodic box with a chosen length admits only the wavenumbers that fit, so the domain length is selecting the wavelength directly — which is a boundary condition masquerading as a result, and it is why pattern-forming simulations are run in domains many wavelengths long and their sensitivity to that length is reported.

The honest procedure is to sweep. Compute in several domain lengths, start from several initial conditions, and report the set of states found rather than the one that came out. That is more work and it is the only way to distinguish a result from an artefact of the setup — the same discipline a duct that forgets everything but one number recommends about a development length and its threshold.

What the picture cannot show #

The neutral curve is drawn for the free-free problem and labelled as such, and the real Taylor-Couette curve is qualitatively identical and quantitatively different — a critical wavenumber of about 3.12 rather than 2.22, and a critical value that depends on the radius ratio.

Nothing here draws a vortex. The states differ in how many pairs fit, and a figure showing two of them side by side is the picture this essay wants; it requires solving the nonlinear problem, which the amplitude-equation argument above deliberately avoids.

Who found it, and when #

Taylor’s 1923 paper is one of the foundations of hydrodynamic stability and gets the critical condition right to within experimental error. The multiplicity was established by Coles in 1965, who found twenty-five distinct states at one condition and showed that the selection depended on the path taken.

The Eckhaus instability and the root-three band are from the amplitude-equation literature of the 1960s and 1970s, and they are what turned an experimental curiosity into a general statement about pattern-forming systems.

Limits recorded rather than smoothed over #

The free-free neutral curve. Chosen because it has a closed form. The real problem’s curve is computed numerically, its critical values differ, and every number in this essay that depends on the curve is therefore illustrative.

The root-three band is an amplitude-equation result. It holds near the threshold, where the amplitude equation is derived, and its accuracy degrades as the driving rises — which is exactly where the state count is largest.

The quantisation ignores the ends. A real apparatus has end plates, which force vortices near them and change the effective length. That shifts the admissible wavenumbers and is a large effect in a short container.

Steady states only. Above the first few multiples of critical, Taylor-Couette flow develops wavy vortices, modulated waves and eventually turbulence, and the counting argument here applies to the steady axisymmetric regime alone. The multiplicity persists into those regimes and is harder to state.

And nothing here computes a selection. The essay establishes that several states are available and says the history chooses; which one a given ramp produces is a nonlinear question this calculation does not answer.