A cloud grows out of its bursts and keeps their shape

Bursts and memory pull a cloud both ways gave a pair of fluid particles a relative velocity whose size comes in bursts, set by a local rate of dissipation that varies, and found that bursts push a cloud’s tails out as hard as memory pulls them in. At the memory real pairs are estimated to have — a direction remembered for 0.7 of a turnover time of eddies the pair’s own size — a velocity with the flatness of 4 measured in the inertial range put the cloud’s kurtosis back near Richardson’s 3.76, from the 1.89 that a pair’s memory had given it with a Gaussian velocity.

That calculation used one flatness for every separation, and said so as its weakest assumption. Measured flatness is not one number. It is close to 3, the Gaussian value, across the largest eddies of a flow, and it rises steadily as the separation shrinks, reaching 6, 8 or more near the smallest eddies. A pair released a few Kolmogorov lengths apart begins its life where the bursts are fiercest and spends it growing into calmer statistics. This essay follows the cloud through that change.

The law the flatness follows #

The scale dependence has a standard form, and it comes from the same place the bursts do. In Kolmogorov’s and Obukhov’s 1962 refinement, the rate of dissipation averaged over a region of size r is a random quantity whose logarithm is Gaussian, with a variance that grows as the region shrinks:

$$\sigma^2(r) = \mu \ln\frac{L}{r},$$

where L is the integral scale — the size of the largest eddies — and μ is the intermittency exponent, measured between about 0.2 and 0.3. The velocity difference across r goes as the cube root of that dissipation times r, and for a lognormal the moments are exponentials of the log-variance, so the flatness of the velocity difference is

$$F(r) = 3\,\frac{\langle \varepsilon_r^{4/3}\rangle}{\langle \varepsilon_r^{2/3}\rangle^2} = 3\,e^{4\sigma^2/9} = 3\left(\frac{L}{r}\right)^{4\mu/9}.$$

With μ = 0.25 that is a power of about a ninth. The figure draws it for three exponents. It is 3 at the integral scale by construction, reaches the earlier calculation’s 4 at about a thirteenth of L, and is 6.46 at a thousandth of L and 13.9 at a millionth. A small power still covers a great deal of ground across the six decades a large flow’s inertial range spans; the range a laboratory flow actually has is narrower, and the range a real Reynolds number does not have is about how much narrower.

The earlier model already had what this needs. Its velocity is a Gaussian whose variance is multiplied by a lognormal amplitude of mean one and spread s, so that its flatness is $3e^{s^2}$; the spread is the only thing to change. Here it follows the pair’s current separation, $s^2(r) = (4\mu/9)\ln(L/r)$ below the integral scale and zero above it. The amplitude’s underlying random process keeps its own memory of α = 3 turnover times, and the direction its memory of β = 0.7, as before. Because the amplitude always has mean one, the mean diffusivity is still Richardson’s $r^{4/3}$ at every separation: the budget is unchanged and only its spending varies with scale.

A shape that never settles #

The figure plots each cloud’s kurtosis against its own rms size, in units of the integral scale, which is the natural clock for a cloud growing through a range of statistics. The three fixed-flatness clouds behave as the earlier essays found. Released with every pair at the same separation, each starts with a kurtosis near one, forgets its release within a few turnover times, and then keeps one shape: about 1.9 for a Gaussian velocity, about 3.5 for a flatness of 4, about 5.3 for a flatness of 5. That settled value is what the earlier calculations reported, and it is what made a cloud’s shape look like a measurement of anything at all.

The cloud whose flatness follows the law does not settle. Released at a ten-thousandth of the integral scale, where the law’s flatness is 8, it rises to a kurtosis of 6.85 by the time its rms size is two thousandths of L, and then falls, steadily and without a plateau, to 5.7 at 0.016 L, 3.95 at an eighth and 2.9 by 0.57 L. On the way it crosses all three fixed-flatness clouds. There is no size at which it could be said to have its shape.

The mechanism is the one the earlier essay found, running in reverse. A pair’s separation grows as the cube of its diffusivity, so a burst that lasts gives a pair a head start it never loses, and the cloud’s tails are made of pairs that caught such bursts. At small separations, where bursts are fierce, those heads start are large and many; as the cloud grows, the pairs it is made of see calmer and calmer velocities, and new head starts are smaller. The tails stop being replenished at the rate they were built.

The shape lags the statistics #

The obvious guess is that a cloud simply has the shape of its current statistics — that a cloud whose rms size is r looks like a settled cloud of flatness F®. That guess can be computed: run fixed-flatness clouds at 3, 3.5, 4, 5, 6 and 7, record the kurtosis each settles to, and read off the value at the flatness the law gives at the growing cloud’s size. The dashed curve is that instantaneous shape.

From about a hundredth of the integral scale on — once the cloud has forgotten its release and turned over — it sits above that curve. At an eighth of the integral scale, where the law gives a flatness of 3.77, a settled cloud would have a kurtosis of 3.23; the growing cloud has 3.94. The difference is 0.6 at a twentieth of L and 0.8 at three-tenths: it does not shrink as the cloud grows. The cloud carries the shape of the burstier scales it grew through, because its outermost pairs are outermost by virtue of bursts they caught when they were smaller, and nothing in the calmer statistics they now see takes those head starts away. Richardson’s own diffusion has no memory; a cloud of pairs has the memory its own history puts into its tails, whether or not any single pair remembers anything.

This matters for how a measurement is read. A particle-tracking experiment that measures a cloud’s kurtosis at some size and compares it with a model evaluated at that size is comparing a quantity with a history to one without. The right comparison is with a model that has passed through the same scales, which means the release separation, the integral scale and the intermittency exponent all have to be stated alongside the number.

Where the pairs started, and where they are #

The earlier essay’s still-open question asked whether the drift in shape is what makes measured shapes depend on where in the inertial range the pairs were released. The figure releases four clouds, spaced a decade apart from a hundred-thousandth of L to a hundredth, and draws each against its own size.

Each begins the same way: all pairs alike, a kurtosis near one, a rise over a few turnover times of the release scale as the release is forgotten. Then each joins a single falling curve. By three-tenths of the integral scale the four read 3.39, 3.21, 3.31 and 3.05 — the first three within their sampling scatter of one another, the last still finishing its rise. The cloud released at a hundredth of L never reaches the high kurtosis the others pass through, because it was never small enough to catch the fierce bursts that build it.

So the answer is yes, in a specific form. A release separation matters twice: briefly, while the cloud forgets it, and permanently, through which part of the falling curve the cloud can reach. Two experiments measuring the same flow at the same cloud size will agree once both clouds have forgotten their releases; two experiments measuring at the same time after release will not, because at a fixed time a cloud released smaller is smaller, and sits further up the curve. A shape reported against time since release carries the release separation inside it, and a shape reported against size does not.

The cube law, approached #

The falling shape has a companion that bears on the most quoted result in the subject. Richardson’s $\langle r^2\rangle = g\,\varepsilon t^3$ is dimensional: once the release is forgotten, the only combination of separation, time and dissipation rate available makes the mean-square separation grow as the cube of time. In a model with a fixed flatness that is exactly what happens: the local exponent $d\ln\langle r^2\rangle/d\ln t$ climbs from near one — the ballistic start — and settles at three, to within a per cent, from a hundredth of the integral scale on.

With the flatness following the law the dimensional argument still holds at each separation, and the exponent still does not reach three. It peaks at 2.94 near a thirteenth of L and falls back. The reason is the constant. The $t^3$ law’s constant g is larger for a burstier velocity, because the head starts that bursts give add up in the mean square as well as in the tails, and as the cloud grows into calmer statistics g falls: from 1.75 when the cloud is two thousandths of L across to 1.45 at a hundredth and 1.16 at 0.57 L. A mean square that grows as $g(t)\,t^3$ with g falling grows more slowly than $t^3$, by exactly the rate at which g falls.

This is a different correction from the one the exponents that stop being thirds describe. Those are genuine anomalous exponents of the structure functions, fixed numbers at every scale in a flow with an infinite inertial range. The pair model here has no anomalous exponent at all: its mean diffusivity is exactly $r^{4/3}$, and in a range that went on for ever its cloud would approach a steady fall of g rather than a new power law. What it shows is that a local exponent can sit visibly below three for the whole of a realistic range without anything being wrong with the dimensional argument, and that is what particle-tracking data have generally shown: local slopes that climb towards three and rarely hold a plateau there. A plateau needs the statistics to stop changing with scale, which is precisely what the 1962 refinement says they never do.

How much rests on the exponent #

Every number so far used μ = 0.25, the middle of the measured range. The exponent is not well known — values from 0.15 to 0.35 have been reported, depending on the flow, the Reynolds number and whether it is read from the dissipation’s correlation or its moments — and the figure shows how much the cloud cares. With μ = 0.15 the cloud peaks at 4.4 and has fallen to 2.5 by half the integral scale; with 0.35 it peaks at 12.7 and is still at 3.2 there. At a twentieth of the integral scale the three clouds read 3.5, 4.66 and 6.3.

The spread between them, 2.8, is larger than the whole of memory’s effect in the Gaussian calculation, which moved the kurtosis from 3.76 to 1.89. The earlier essay found that a cloud’s shape reads a combination of two memories and a flatness. This one adds that the flatness is itself a function of scale set by an exponent known to about forty per cent. Reading a pair memory off a measured shape would need that exponent measured in the same flow to a precision measurements of it have not usually claimed.

What was checked #

The model changes one line of the earlier one, and two checks make sure that one line does only what it says. With the exponent set to zero the spread is zero everywhere and the model must be the Gaussian pairs; with the spread held at the constant that gives a flatness of 4 it must be the earlier essay’s fixed-flatness run. Both share the random sequence of the runs they are compared with, so the comparison is exact, and both agree to the last bit. A third check samples 400,000 velocities at a separation of 5·10⁻⁴ L and compares their flatness, 7.16, with the law’s 6.98: agreement to 2.6 per cent, which is the sampling error of a fourth moment at that flatness. The fourth releases pairs at the integral scale itself. There the law’s spread is zero and stays zero as the cloud grows, so the cloud must be the Gaussian pairs’ cloud, and at late times it matches that cloud’s kurtosis to 1.1 per cent. The tests also refuse a negative exponent, which would make the small scales calmer than the large ones, a release outside the integral scale, and a tolerance of zero.

Every cloud is averaged over three seeds of 8,000 pairs. The scatter between seeds reaches 0.8 in the kurtosis where the cloud is burstiest and is about 0.2 further out, for the reason a record is as long as its integral scales gives for any statistic weighted towards rare events.

What the picture cannot show #

The curves stop at a little over half the integral scale, and that is not a choice of window. The model keeps Richardson’s diffusivity $r^{4/3}$ at every separation, including separations larger than the largest eddies, where a real pair’s separation stops accelerating and grows diffusively instead, as a single parcel does in how far a parcel gets. By half the integral scale a substantial fraction of the cloud’s outer pairs are beyond L, and the cloud’s shape there is made partly of pairs the model is describing wrongly. The fall is real before that point; its end is not computed.

The lognormal law is itself a model. It gets the measured flatness roughly right across the inertial range, but the dissipation’s statistics are not exactly lognormal — the log-Poisson and multifractal descriptions that followed fit the high moments better — and the details of the tail are what a cloud’s kurtosis is most sensitive to. The earlier essay’s other simplifications remain: one correlation time for the amplitude, and an amplitude independent of the velocity’s direction, where a real burst is a structure with a geometry.

And no figure here shows a single pair. The shape is a property of thousands of pairs together, which is why it can carry a history that no individual pair’s velocity remembers.

Where the pieces came from #

Richardson’s law is from 1926. The lognormal refinement is Kolmogorov’s and Obukhov’s, both published in 1962, and its intermittency exponent has been measured ever since, with Sreenivasan and Kailasnath’s 1993 survey giving the 0.25 used here. Stochastic models of pair separation with a velocity rather than a diffusivity are Thomson’s, from 1990, and Borgas and Sawford’s; the dependence of a cloud’s statistics on the release separation was brought into focus by Bourgoin, Ouellette, Xu, Berg and Bodenschatz’s 2006 measurements, and Salazar and Collins’s 2009 review discusses why a clean $t^3$ range has been so hard to see. The construction used here — a Gaussian whose variance is a lognormal with a scale-dependent spread — joins Beck and Cohen’s superstatistics to the 1962 law, and a dissipation correlated across every scale is where the lognormal’s scale dependence comes from.

Why one law is protected and the other is not #

The model keeps Richardson’s diffusivity exactly, and the cube law’s exponent still drifts, which invites the question of why intermittency can move one dimensional result and not another. The one exact result of turbulence, the four-fifths law, is protected because the third moment of a velocity difference is linear in the dissipation rate: averaged over a flow, a dissipation that comes in bursts gives the same third moment as one that does not, and the law holds with its constant exact at every scale. Richardson’s constant has no such protection. A pair’s mean-square separation goes as the cube of its diffusivity, so the constant weights a burst by its amplitude cubed, and the frozen-amplitude limit of the earlier essay put that in closed form: g grows as $\langle Y^3\rangle$, an exponential of the lognormal’s spread. A spread that depends on scale makes g depend on scale, and a quantity that is dimensionally $g\,\varepsilon t^3$ with g drifting is fitted by a power law of exponent a little under three. The same argument is why the structure functions of order other than three acquire anomalous exponents while the third does not. Linearity in the dissipation is what an exact result in this subject needs, and the pair law does not have it.

Still open: whether memory can be read against a drifting shape #

The earlier essay proposed that a cloud’s kurtosis and its constant, read together, separate memory from bursts. With the flatness depending on scale both numbers drift, and they drift together, so the reading becomes a pair of curves against size rather than a point. The next calculation fits the direction memory β at each size, with the intermittency exponent fixed at a measured value, to the kurtosis and constant a tracking experiment reports — Bourgoin’s data give both against size — and asks whether one β fits the whole curve, as it should if the memory is a property of the turbulence, or whether the fitted memory itself drifts with scale, which would say the model has put into the memory something that belongs to the bursts. The scalar a puff carries has its own cascade, with an intermittency stronger than the velocity’s, and it would be the next place to look for the same drift.