Bursts and memory pull a cloud both ways

A pair’s memory shapes the cloud it spreads into took Richardson’s diffusion of pair separations, which has no memory, and gave each pair a relative velocity that it keeps for β turnover times of eddies its own size. The cube law of separation survived, because it is dimensional. Almost nothing else did. The constant in front of the cube fell to about a third of Richardson’s by β = 0.1 and to a twenty-fourth by β = 1, and the cloud’s shape — its kurtosis, the ratio $\langle r^4\rangle/\langle r^2\rangle^2$ — fell from Richardson’s 3.76 towards a Gaussian’s, to about 1.9 at the memory of 0.7 turnover times that Kolmogorov’s constants suggest real pairs have.

That essay drew a practical conclusion. Once a cloud has forgotten where it was released, its kurtosis depends on the memory alone, not on the dissipation rate or the time, so the shape of a measured cloud is a reading of β that needs none of the hard-to-measure quantities the constant depends on. It ended by naming the weak point of its own model: the relative velocity it gave the pairs was Gaussian, and measured relative velocities are not.

What a measured velocity difference looks like #

The difference in velocity across a separation r in the inertial range has a distribution with heavy tails. Its flatness — $\langle \delta v^4\rangle/\langle \delta v^2\rangle^2$, which is 3 for a Gaussian — is about 4 across most of the inertial range and climbs higher towards the smallest eddies. The cause is the one Kolmogorov added to his theory in 1962. The rate of dissipation that sets the size of velocity differences is not uniform: it is concentrated in sheets and filaments and varies by orders of magnitude from place to place, and averaged over a region of size r it is a random quantity whose logarithm is roughly Gaussian. The velocity difference across r is a Gaussian-ish fluctuation multiplied by the local dissipation’s cube root, and a Gaussian with a random amplitude is heavier-tailed than a Gaussian. The exponents that stop being thirds are the same fact seen in the moments, and a dissipation correlated across every scale is where the local rate’s lognormal structure comes from.

That structure says how to add intermittency to the pair model without changing anything else. Multiply the variance of the relative velocity by a positive amplitude Y of mean one, lognormal with a spread s chosen so that the velocity’s flatness $3e^{s^2}$ is whatever is wanted, and let that amplitude wander with a correlation time of α turnover times of the pair’s current separation. The direction of the velocity is remembered for β turnover times as before; its size is now remembered for α. Because Y has mean one, the mean diffusivity is still $r^{4/3}$. Nothing has been added to the budget. Only the way it is spent has changed.

Two limits the answer must reach #

Two limits bracket the answer, and both can be stated exactly.

If the amplitude is re-drawn quickly, much faster than the direction, each pair sees a velocity whose size flickers around its average many times before it has moved appreciably, and by the central limit theorem it spreads as if the velocity had been Gaussian with the mean variance. The flatness of the velocity disappears from the cloud entirely. The model reproduces that: with the amplitude re-drawn every fiftieth of a turnover time, its cloud matches the Gaussian pairs’ constant and kurtosis to within three per cent.

If instead the amplitude is frozen for each pair, the cloud is a mixture of Richardson clouds, each with its own diffusivity Y K. Richardson’s mean-square separation goes as the cube of the diffusivity, so the mixture’s constant is Richardson’s times $\langle Y^3\rangle$, and its kurtosis is Richardson’s times $\langle Y^6\rangle/\langle Y^3\rangle^2$. For a lognormal those are exponentials of the spread, and at a velocity flatness of 4 the frozen, memoryless cloud has a kurtosis of 50 — thirteen times Richardson’s. A velocity flatness of 4 becomes a cloud flatness of 50 because the cube law raises the amplitude to the third power and the fourth moment of the cloud raises it again.

Real pairs sit between these limits in both memories at once, and the question is where.

Bursts put the tails back #

The figure repeats the earlier calculation’s main result — the cloud’s kurtosis against the direction memory β — for a velocity of flatness 4, with the amplitude remembered for 0.3, 1 and 3 turnover times, and frozen. The lowest curve is the Gaussian velocity of the earlier essay. Every intermittent curve lies above it, and the longer the amplitude is remembered the higher it lies.

At the direction memory real pairs are estimated to have, 0.7 turnover times, the Gaussian cloud’s kurtosis is 1.89. With bursts remembered for a turnover time it is 3.13; for three turnover times, 3.41; with the amplitude frozen, about four. Richardson’s memoryless value, 3.76, sits inside that range. The shape the earlier essay read as a clean measurement of memory has been pushed most of the way back to where memory had moved it from, by a property of the velocity that the memory has nothing to do with.

The two effects work in opposite directions for a reason that is simple to state. Memory narrows the cloud because a pair that keeps its velocity cannot string lucky pushes together into a run: it gets one push, for longer. The fastest pairs in a memoryless cloud are the ones that happened to be pushed outward many times in a row; memory takes away the independence that made such runs possible. Bursts widen the cloud because a pair that happens to sit in a region of intense dissipation for a while gets a larger velocity for that whole while, and at larger separations it spreads faster still. Memory removes one source of lucky runs and bursts supply another.

The cube law amplifies the flatness #

How strongly bursts act depends on the velocity’s flatness, and the dependence is steep. At a direction memory of 0.7, with the amplitude remembered for one or three turnover times, a flatness of 3.5 leaves the cloud at about 2.5; a flatness of 4 brings it to Richardson’s neighbourhood; a flatness of 5 takes it past 5, and a flatness of 6 to between eight and ten, more than twice Richardson’s.

This is the frozen limit’s arithmetic showing through the memory. Separation grows as the cube of diffusivity, so the cloud’s second moment weights a pair’s amplitude as its cube and the fourth moment as its sixth. A modest flatness in the velocity, which is a statement about the amplitude’s fourth moment, becomes a large flatness in the separation. And since the velocity’s flatness itself varies across the inertial range — close to 3 at the largest scales, 4 in the middle, higher towards the dissipation range — pairs released at different separations, or followed over different ranges, will see very different clouds even with the same memory.

A line of accidental Richardsons #

For each amplitude memory there is one direction memory at which the two effects cancel and the cloud has Richardson’s kurtosis exactly. The figure plots that line for a flatness of 4. Amplitudes remembered for three turnover times cancel a direction memory of 0.59; amplitudes frozen for good cancel a direction memory of about one. The 0.7 estimated for real pairs lies between them.

That is an uncomfortable result, because it offers an explanation for something the earlier calculations did not. The earlier essay noted that measured clouds have usually been narrower than Richardson’s, and attributed some of the shortfall to memory. On this model, memory with a Gaussian velocity makes clouds much narrower than any measurement reports, while memory with realistic bursts makes clouds that look like Richardson’s. A tracking experiment that measures a kurtosis near 3.7 has not found memoryless pairs. It may have found pairs with the memory Kolmogorov’s constants predict and the intermittency the velocity statistics measure, in a cancellation that no one designed.

What still reads the memory #

The shape alone cannot separate the two effects. The constant can help.

The figure places each cloud by both numbers: its $t^3$ constant as a fraction of Richardson’s 1144/81, and its kurtosis. With a Gaussian velocity the two fall together as the memory grows, and either reads the memory. With bursts the curves lift away from the Gaussian one. Bursts restore the kurtosis a great deal and the constant only a little — at a direction memory of 0.7 the constant rises from 0.89 to about 1.1 with bursts, still under a tenth of Richardson’s 14.1 — so an intermittent cloud with Richardson’s shape sits far to the left of Richardson’s own point.

So the reading the earlier essay proposed survives in a corrected form. A cloud whose kurtosis is near 3.76 and whose constant is near 1144/81 is memoryless pairs. A cloud with the same kurtosis and a constant a tenth as large is memory and bursts together. The constant is the hard number to measure, as that essay said, because it multiplies a dissipation rate known only to tens of per cent and is cubed in time; but an error of tens of per cent does not turn a tenth into one. The value of Richardson’s constant taken from measurement and simulation is about half a unit, as the further apart, the faster they part recorded, far below 1144/81 — which, on this picture, was the memory speaking all along while the shape was hidden by the bursts.

How many pairs a shape needs #

Bursts also make the shape harder to measure. A cloud’s fourth moment is carried by its outermost pairs, and in a bursty cloud those are the few pairs that caught a long, intense burst. The figure draws twelve clouds of 1,500 pairs each at a direction memory of 0.7, seed by seed: with a Gaussian velocity the sample kurtosis scatters by 0.06 about its mean, and with bursts of flatness 4 remembered for three turnover times by 0.4. Reading the bursty cloud’s shape to the same precision takes about forty times as many pairs. That is the same lesson as a record being as long as its integral scales: the statistic that converges slowly is the one weighted towards rare events, and in turbulence the rare events are where the statistic lives.

For a particle-tracking experiment, which follows thousands of pairs rather than tens of thousands, this matters directly. A kurtosis reported to two figures from a few thousand pairs in a flow with realistic intermittency has an uncertainty that is itself a substantial fraction of the gap between the memoryless and remembering predictions. The constant, which depends on the second moment, is steadier by the same argument.

The single parcel, again #

How far a parcel gets followed one parcel with Taylor’s theory, where a parcel’s spreading depends on its velocity’s correlation function and nothing else. Intermittency barely enters there. A parcel’s mean-square displacement depends only on its velocity’s correlation function, which a mean-one amplitude leaves nearly unchanged, and a frozen amplitude leaves exactly unchanged; its displacement’s flatness gains the velocity’s own excess and no more, because the parcel’s diffusivity does not depend on where it has got to. Pairs are different, because the rate at which a pair separates depends on its separation, which turns a temporary burst into a permanent head start. The more strongly a process feeds back on its own state, the more of the velocity’s distribution it sees, and Richardson’s is about as strong a feedback as turbulent transport offers.

What it does to a spreading plume #

The cloud of separations is not an abstraction. It is the shape of a puff of smoke or pollutant released at a point and followed as it grows: the mean-square separation of pairs is the puff’s size, and the tails of the separation distribution are the fraction of the material that has got far from the centre sooner than the average. For anyone asking how soon a concentration above some threshold reaches a distance, the tails are the whole question.

On the Gaussian-memory picture those tails are thin, and a model calibrated on the puff’s mean size would predict the far edge conservatively only by luck. With bursts the far edge is carried by the pairs that sat in intense dissipation, and a model that matches the mean size and uses a Gaussian velocity underestimates how much material arrives early and far. The error goes the dangerous way. It is the same structural lesson as a closure with no memory at all: a single number fitted to the average of a process with memory and rare events describes the average and nothing else, and the rest of the distribution is where the consequences are.

The scalar the puff carries has its own cascade, and its intermittency is stronger than the velocity’s. A calculation that follows concentration rather than separation would meet both at once.

What was checked #

The amplitude-mixed velocity was sampled 400,000 times and has the flatness asked of it to a quarter of a per cent. With the flatness set to 3, the model must be the Gaussian pairs of the earlier essay, and it is to within the two ensembles’ sampling error, 2.3 per cent; with the amplitude re-drawn every fiftieth of a turnover time it must forget the flatness, and it does to 3.2 per cent. The frozen, memoryless limit is a closed form and is quoted rather than simulated, for the reason the scatter figure gives: its kurtosis of 50 is carried by pairs so rare that no ensemble of tens of thousands samples them fairly. The tests also refuse a flatness below 3, which no mixture of Gaussians can produce, an amplitude memory of zero, and a negative tolerance.

What the model leaves out #

A lognormal amplitude with one correlation time. Kolmogorov’s refined similarity makes the amplitude at scale r depend on the dissipation averaged over r, which is correlated with the amplitudes at every larger scale; a pair growing through the inertial range sees an amplitude with memory at every scale it passes. One correlation time in units of the current turnover time is the simplest stand-in, and the figures show that the answer is sensitive to it.

A flatness that does not depend on separation. The measured flatness rises as the separation falls. Pairs released small start in the burstiest part of the range and grow out of it, which would make early times more tailed than late ones.

Independence of amplitude and direction. A burst in real turbulence is a structure — a vortex sheet or tube — whose geometry favours some directions of relative velocity. Correlating the two would change the separation’s growth as well as its shape.

Who worked it out #

Richardson’s law is from 1926 and Kolmogorov’s refined similarity from 1962, with Obukhov’s lognormal model of the same year. Stochastic models of pair separation with a relative velocity rather than a diffusivity are Thomson’s, from 1990, and Borgas and Sawford’s; Sawford’s review of 2001 discusses the non-Gaussian relative velocity and the difficulty of choosing its statistics consistently. Superstatistics — a Gaussian whose variance is itself random — is Beck and Cohen’s name, from 2003, for the device used here. The measurement of pair-separation statistics from particle tracking, and its sensitivity to the rare fast pairs, is the subject of Bourgoin and colleagues’ work from 2006 on.

Still open: a flatness that grows as the pair shrinks #

The calculation uses one flatness for every separation, and measured flatness is not constant: it is near 3 at the integral scale and rises steadily towards the Kolmogorov scale, roughly as a small power of the inverse separation. A pair released close together starts where the bursts are fiercest and separates into calmer statistics, so its cloud should be most tailed early and relax towards the Gaussian-memory shape later — a kurtosis that falls in time even after the release separation is forgotten. The next calculation makes the lognormal spread a function of r, with the scaling Kolmogorov’s 1962 theory gives, and asks whether the cloud’s kurtosis then drifts with time, how fast, and whether that drift is what makes measured shapes depend on where in the inertial range the pairs were released.