The noise that warns a tube is about to sing

A heated Rijke tube, a gas turbine’s combustor and a rocket’s thrust chamber share a habit. Under some conditions they run quietly, apart from the broadband roar of their turbulent flow, and under others they hum — one acoustic mode of the chamber grows from nothing into a large, steady oscillation fed by the heat it modulates. The tube that sings when heated low down followed the mechanism in Rijke’s tube: heat released while the pressure is high adds energy to the sound, and when that driving outweighs the mode’s acoustic losses, the quiet state is no longer stable and the sound takes off.

That account describes the boundary. A machine operator needs something else: some way to tell, while the machine is still quiet, how close it is to the boundary. Changing the fuel mix, the load or the inlet temperature moves the boundary around, and nobody can afford to find it by crossing it. The question is whether the quiet state carries any sign of the instability it is approaching. It does, and the sign is in the noise.

A margin that shrinks to zero #

Below onset, the resonant mode behaves like any damped oscillator. Disturb it and it rings down; the rate at which its amplitude decays is the margin of stability. In a resonator fed by heat, that rate is the difference of two things: the acoustic losses, which take energy away in proportion to the mode’s energy, and the heat driving of Rayleigh’s criterion, which puts energy back in the same proportion. Write the net rate as $\sigma$, negative while the losses win. As the operating point drifts towards onset the driving grows and $\sigma$ climbs towards zero; at zero the losses and the driving balance exactly, and beyond it the quiet state is unstable.

A turbulent flame does not leave the mode alone to ring down. Every eddy that burns a little early or a little late releases a small, random pulse of heat, and each pulse kicks the mode. The mode’s amplitude, followed in a frame that turns with its own frequency so that the fast oscillation is removed, then obeys the simplest equation a noisy resonance can obey: it decays at rate $|\sigma|$ and is kicked randomly at a strength $D$. Close to onset the same equation gains a cubic term, the saturation that will eventually cap the oscillation’s amplitude — the normal form of what dynamicists call a supercritical Hopf bifurcation, the generic way a steady state turns into an oscillation.

The three traces use identical kicks. Only the margin differs, and the difference is plain. At a decay rate of 20 per second each burst of sound dies in a few hundredths of a second, before the next set of kicks can add much to it, and the trace is a thin, ragged band. At 6 per second the bursts are larger and survive long enough to pile up. At 2 per second the mode is ringing for half a second after every kick, the kicks add in long runs before the damping catches up, and the result sounds like a weak version of the oscillation that onset will bring. An ear placed at the chamber would hear the hiss acquire a pitch.

The kicks add, the damping subtracts #

The variance of the mode — the mean square of its amplitude — comes from a balance that can be written in one line. Each kick adds a random amount to the amplitude, and random additions add their squares: over a time $t$ the kicks raise the mean square by $4Dt$ in the convention used here, two components each kicked with strength $2D$. The damping removes the mean square at a rate $2|\sigma|$ times itself, because energy decays at twice the rate amplitude does. The two settle where they balance:

$$\langle |A|^2 \rangle = \frac{2D}{|\sigma|}.$$

The kicks did not change; the variance grows because the damping that drains them weakens, and it grows without limit in the formula as $\sigma$ goes to zero.

The simulated points follow the linear law across most of the range, and that is the useful part: a factor of four in the margin is a factor of four in the variance, whatever the mode’s frequency or the source of its kicks. Close to onset the points bend away below the curve. They bend because the fluctuations have become large enough to feel the cubic saturation, which pulls large amplitudes back harder than the linear damping would. The exact stationary distribution includes that term, and it carries the variance smoothly through onset at a finite value and on along the growing amplitude of the self-sustained tone. Where the bend begins is set by the noise: the linear law holds while $2D/|\sigma|$ is small compared with $|\sigma|$, which is to say while the margin is larger than about $\sqrt{2D}$. For weak noise the linear law holds almost to onset; for strong noise the bend begins early, and the sharpest feature of the warning is smoothed away.

Why a thermometer does not do this #

The variance law looks as though it should apply to every resonator in the world, and it does not. The noise a high Q moves out of the way followed an oscillator in thermal equilibrium — a mirror on a spring, a quartz tuning fork — and found its mean square displacement fixed at $kT$ over the stiffness however lightly it was damped. Reducing the damping there narrows the noise peak but leaves its area alone.

The difference is where the kicks come from. In thermal equilibrium the damping and the kicks are one mechanism: the same molecular collisions that drag on the oscillator also jostle it, and the fluctuation–dissipation theorem ties their strengths together, so that halving the damping halves $D$ as well and the ratio $2D/|\sigma|$ does not move. Half a kT in a piece of wire is the same theorem applied to a resistor, where the noise voltage is set by the resistance that dissipates it, and the jiggle that proved atoms is the same balance between drag and kicks in a pollen grain.

A combustor is not in equilibrium, and its margin is not eroded by reducing its friction. The acoustic losses stay what they were; what changes is the flame’s coherent feedback, which pushes energy back into the mode in step with it and cancels part of the losses. The turbulent kicks are a separate process, set by the flow and the burning, and they carry on at the same strength while the margin closes. That separation is what lets the variance grow. Critical slowing down needs a system driven away from equilibrium by something that can cancel its damping without silencing its noise, and that is precisely the description of a machine approaching a self-excited oscillation.

A memory that lengthens #

The variance is one measurement. A second, often more robust, is how long the mode remembers a kick. A disturbance decays as $e^{-|\sigma| t}$, so the correlation between the mode’s amplitude now and its amplitude a time $\tau$ later falls as $e^{-|\sigma|\tau}$, and the time at which it drops to $1/e$ is simply $1/|\sigma|$. This is the property that gives the phenomenon its name: near onset the system takes longer and longer to recover from small disturbances, and the recovery time measures the margin directly, in seconds, without needing to know $D$.

The memory carries the same physics as the variance but needs one less number. The variance depends on both the margin and the strength of the kicks, so a rising variance could mean a closing margin or a noisier flame; the memory depends on the margin alone. A combustor whose turbulence grows louder at higher load would show a rising variance with no change in its margin, and its correlation time would correctly say so. That is the reason practical schemes lean on the memory, or on its equivalent in the spectrum.

The memory also near onset falls below its linear value. A fluctuation large enough to feel the saturation is pulled back faster than the linear rate, and at onset itself the memory, like the variance, is finite — set by the noise and the nonlinearity rather than by a margin that has reached zero.

The hiss narrows towards a tone #

A decay in time is a width in frequency, a point the width that is a lifetime made for a bell: ring it and time the decay, or drive it and measure its response, and the two numbers are one number in two coordinates. The noisy mode is that bell, struck continually and at random. Its spectrum is a Lorentzian line centred on the mode’s frequency, whose full width at half height is $|\sigma|/\pi$ hertz and whose total area is the variance.

On an engine’s pressure sensor this is what the approach to onset looks like: a broad bump at the frequency of a chamber mode, standing above the turbulent background, that narrows and rises as the operating point moves. The height grows faster than the area because both the narrowing and the growing variance push the peak up, so the peak is the most visible of the three signs. Its width is the most useful, because it is the decay rate in another unit: fit a Lorentzian to the bump and the margin of stability can be read off in reciprocal seconds.

The width also explains why the hiss in the hero acquired a pitch. A peak 6 Hz wide on a 200 Hz carrier is a sound whose frequency wanders by a few per cent within each tenth of a second: a hiss with a colour. At 0.6 Hz the wander is a third of a per cent, and the ear hears a tone, swelling and fading. Nothing about the mode’s frequency has changed; only its coherence has.

Measuring a margin without crossing it #

The engineering use is old, and the reason it is attractive is that it is passive. The conventional way to measure a mode’s damping is to excite it — a loudspeaker, a pulse, a bump — and time the decay. In a running gas turbine that is impractical, and close to onset it is dangerous, since the test is a push towards the very state it is trying to avoid. The turbulence is already pushing, all the time, at every frequency. Reading the decay rate from the mode’s response to its own background noise needs only a pressure sensor and patience.

In the early 2000s Tim Lieuwen and colleagues at Georgia Tech showed that the damping of a combustor’s mode could be estimated in exactly this way, from the autocorrelation of the dynamic pressure recorded during normal running, and that the estimate fell steadily towards zero as the combustor was brought towards instability. The same idea, under the name output-only modal identification, is used for bridges and aircraft wings, whose flutter is another self-excited oscillation whose onset is approached by a damping that falls to zero as speed rises: flight-flutter testing tracks the damping of a wing’s modes from their response to turbulence and stops before it reaches zero.

The approach is not limited to engineered machines. Since around 2009 ecologists and climate scientists have looked for the same statistics — rising variance, lengthening autocorrelation — before a lake turns turbid, before a fishery collapses, in the records of past abrupt climate changes. The mathematics is the same wherever a quiet state loses stability slowly under a constant background of random disturbance, and it is that generality that makes the signals both attractive and, as the last figure shows, treacherous.

When the noise starts the song #

The tube that sings when heated low down found that Rijke’s tube does not start singing gently. Raise its heating and it stays silent until the linear threshold is passed, then jumps straight to a loud note; lower the heating and the note persists well below the threshold. Between the two, the quiet state and the loud state are both stable, separated by a barrier: an amplitude below which disturbances decay and above which they grow to the loud state. This is a subcritical onset, and most combustors that have been studied carefully show one.

In the bistable range the noise does not just reveal the margin; it can cross the barrier. A run of kicks that happens to push the amplitude past the unstable threshold carries the mode into the loud state, and the quiet state is abandoned before it has lost stability at all.

The jump is a barrier-crossing, and its timing is governed by the same arithmetic as a molecule escaping over an activation energy: the mean waiting time grows roughly as the exponential of the barrier height divided by the noise strength. The noise that helps a weak signal through used exactly that arithmetic for a particle hopping between two wells. With the stronger noise, the waiting time falls below a few hundred natural times as soon as the loud state exists; with the weaker, the quiet state holds until the barrier has shrunk considerably.

The awkward point is what the warning signs of the earlier figures say at that moment. At a growth rate of −0.24 the quiet state’s decay time is still about four natural times, and with the stronger noise its variance is about 0.042 in the same units. Neither is anywhere near the dramatic growth that comes as $\sigma$ approaches zero. The quiet state has only begun to slow down, and the jump is already likely. Critical slowing down measures the distance to the point where the quiet state loses stability; a subcritical onset with noise does not wait for that point. Peter Ditlevsen and Sigfús Johnsen made the same objection about climate records in 2010, in a paper whose title called early-warning signals in noise-driven transitions wishful thinking, and the objection applies with full force to a combustor in its bistable range.

What the measurement can still say #

The case is not hopeless. A subcritical onset does leave marks in the noise, of a different kind. As the barrier shrinks, the quiet state’s fluctuations become increasingly non-Gaussian: occasional large excursions, which climb part of the way up the barrier and fall back, become more frequent than the linear picture predicts. In turbulent combustors this shows as intermittency — bursts of nearly periodic oscillation, lasting a few cycles, scattered through otherwise aperiodic noise — and R. I. Sujith’s group at IIT Madras reported in 2014 that the fraction of time spent in such bursts rises steadily as a combustor approaches instability, giving an earlier warning than the variance does. Measures that respond to the shape of the noise’s distribution, or to the loss of its irregularity, can see a barrier the second moments miss.

The other remedy is to measure the margin in a way that does not depend on the noise reaching it. A small deliberate perturbation — a pulse from a fuel valve already being modulated for control — gives a decay rate directly; it is the active test the passive method was meant to avoid, but at a size far below the barrier it is safe. The passive and active estimates of $|\sigma|$ should agree; when the passive variance rises faster than the active decay rate falls, the excess is a sign that the noise is reaching into the nonlinear region, which in a subcritical system is the region nearest the barrier.

What the figures leave out #

The model is one acoustic mode with constant-strength white noise, written in its normal form with a cubic or quintic nonlinearity, integrated with a fixed time step from a seeded random sequence. A real combustor has several modes, close enough in frequency that their noise peaks can overlap, and the turbulent forcing is coloured rather than white, stronger at some frequencies than others; the mode’s frequency drifts with temperature, so a fixed carrier is an idealisation. Turbulence can also multiply rather than add — modulating the mode’s own damping, not just kicking it — which changes the stationary distribution and can shift the onset itself. The amplitudes and noise strengths are in the model’s units, the decay rates of the first four figures are chosen to be plausible for a chamber mode, and the times of the last are in units of the mode’s natural time; the barrier figure’s onset is drawn at zero and its loud state from −0.25 by the choice of normal form. The correlation times are the 1/e lags of long simulated records, interpolated between samples, and carry sampling error of a few per cent.

Still open: how much warning a real engine gives #

Whether the statistics of a combustor’s noise can be trusted to warn of an approaching instability, in a machine whose fuel, load and ambient conditions all vary, is an open engineering question, and so is the deeper one of which statistic to trust when they disagree. The second moments — variance and correlation time — are well founded where the onset is supercritical and the noise weak; measures of intermittency and of the noise’s shape look further ahead in subcritical systems but are harder to calibrate. Real engines also move: the margin may close faster than any statistic computed over a window of seconds can follow. And on the same timescale the noise itself may change, as the turbulence intensity rises with load. The calm that is the ghost of a cycle and the map a dripping tap turns out to be are reminders that the neighbourhood of a bifurcation holds other behaviours besides slowing down, each with its own statistics, and a combustor can show more than one. The chatter a stiffer holder removes and the frequency that gets an answer belong to the same family of oscillators that are pushed by what they do; in each, how close the quiet state is to giving way is a number that the noise is measuring all the time.

Below the onset of a self-excited oscillation, a resonant mode kicked by noise of strength $D$ and decaying at rate $|\sigma|$ carries a variance $2D/|\sigma|$, a correlation time $1/|\sigma|$ and a spectral peak $|\sigma|/\pi$ wide — 0.051, 25 ms and 12.7 Hz at a decay rate of 40 per second, 0.72, 374 ms and 0.48 Hz near 1.5 — so the margin can be read from the quiet running, as long as the noise is independent of what erodes the margin; where the onset is subcritical, noise can throw the mode into the loud state while those signs are still modest. A resonator announces its instability in its hiss, unless the hiss starts it first.