The exponents that stop being thirds

Kolmogorov’s 1941 argument is a statement about every moment at once. In the inertial range the statistics of the velocity difference across a separation $r$ depend only on $\varepsilon$ and $r$, and dimensions then fix

$$\langle |\delta u(r)|^p\rangle \propto (\varepsilon r)^{p/3}, \qquad \zeta_p = \frac{p}{3},$$

one exponent per moment, all of them on one straight line.

Read as a statement about the distribution rather than about the moments, it is even stronger: the probability distribution of $\delta u/r^{1/3}$ does not depend on $r$ at all. Rescale, and the histogram is the same histogram.

That is what is false, and this essay is about what the falsehood measures.

The one exact anchor, which every model must hit #

Before comparing models it is worth being clear about what is actually known, because it is one number.

The one exact result is Kolmogorov’s four-fifths law: the third moment of the longitudinal velocity difference is exactly $-\tfrac45\varepsilon r$, with no adjustable constant anywhere, derived from the Navier–Stokes equations under homogeneity, isotropy and high Reynolds number — the one place in the subject where where the energy goes’s dimensional argument is replaced by a consequence of the equations.

So $\zeta_3 = 1$, exactly, and it is the only exponent that is known rather than modelled. Every model below is checked against it here and every one satisfies it to $10^{-12}$, which is not a coincidence: a model that missed it would be wrong about the one thing that is not in dispute.

The second constraint, which is where a famous model fails #

There is a second thing that is known, and it is a constraint rather than a value.

Novikov’s argument: the velocity difference across any separation is bounded by the large-scale velocity, so a higher moment cannot fall off faster with $r$ than a lower one. Therefore $\zeta_p$ must be non-decreasing in $p$, and convexity of the moments makes it concave.

K41’s straight line satisfies it. The $\beta$-model’s straight line satisfies it. She–Lévêque satisfies it. The log-normal model does not:

$$\zeta_p = \frac{p}{3} - \frac{\mu p(p-3)}{18}$$

has $d\zeta/dp = \tfrac13 - \mu(2p-3)/18$, which is zero at $p = 3/2 + 3/\mu$ — 13.5 for $\mu = 0.25$, found here by bisection on the computed derivative — and negative beyond.

Thirteen and a half is not an academic moment. Structure functions to $p = 12$ and beyond are routinely measured, and the model says something impossible there. That is the refutation this essay carries.

What the departures actually are #

Comparing the models at the moments people quote:

At the second moment everything agrees to three per cent, which is why the spectrum — a second-order object — cannot distinguish between them. At the twelfth moment they span a factor of 1.6, and the log-normal model has already turned round.

Why the low moments are useless for choosing #

The table repays a second look, because it explains why this is settled at high $p$ and not at low.

At $p = 2$ the four values are 0.667, 0.733, 0.694 and 0.696 — a spread of ten per cent, and three of the four within half a per cent of each other. At $p = 6$ the spread is twelve per cent. Only past $p = 10$ do they separate decisively, and only past 13 does one of them become impossible.

That is the ordinary situation when models are being distinguished by a curve rather than by a mechanism: they were all built to agree where the data are good, so the discrimination has to happen where the data are bad. The two constraints — $\zeta_3 = 1$ and monotonicity — are more useful than any amount of curve fitting, because they are exact and they are checkable without data.

What the departure means: a dimension #

The interesting content is not that the exponents fall below $p/3$; it is what the shape of the fall says.

Every one of these sets becomes a straight line at large $p$, and the intercept of that line is a codimension — the codimension of the set of points carrying the most intense events.

$$\zeta_p \to p\,s + C.$$

Computed here as $\zeta_p - p\,d\zeta/dp$ at $p = 400$:

• K41 gives $C = 0$, which is a space-filling set. The intense events are everywhere.
• The $\beta$-model with $D = 2.8$ gives $C = 0.2$, which is $3 - D$ by construction.
• She–Lévêque gives $C = 2.000000$, which is a set of dimension one — filaments.

She–Lévêque’s asymptotic slope is $1/9$ to $10^{-6}$, and the codimension of two is the whole content of the model: the most dissipative structures in a turbulent flow are one-dimensional, and everything about the high moments follows from that.

That is a geometric fact about a flow, extracted from a sequence of statistical moments. It is the most satisfying thing in this essay.

Where the flow produces this, and what it looks like #

It is worth attaching the arithmetic to a picture, because the codimension of two has a physical referent.

High-resolution simulations of homogeneous turbulence show intense vorticity organised into long thin tubes — filaments a few Kolmogorov lengths across and an integral scale long, with vorticity magnitudes many times the mean, occupying a tiny fraction of the volume. Those are the structures She–Lévêque’s codimension is describing: dimension one, embedded in three, so codimension two.

Where a vortex stops is the site’s essay on the difficulty of saying where such an object begins and ends — four criteria, three of which coincide in two dimensions, one of which is a knob, and the one that is not a knob failing to be objective. The exponents here are a way of measuring the dimension of a set without having to define its boundary, which is exactly the difficulty that essay records.

That is a real advantage of the statistical route: it extracts a geometric number from moments, and moments do not require anybody to decide where a structure ends.

The flatness, which is the same statement in one number #

There is a way to see intermittency without any of the machinery, and it is worth having.

The flatness of the velocity difference — its fourth moment over the square of its second — scales as $r^{\zeta_4 - 2\zeta_2}$. K41 makes that exponent exactly zero, so the distribution’s shape is scale-independent and its flatness is a constant.

Every intermittency model makes the exponent negative: $-0.200$ for the $\beta$-model, $-0.111$ for the log-normal, $-0.112$ for She–Lévêque. So the flatness grows as the separation shrinks, by a factor of 2.8 to 6.3 over four decades.

A distribution whose flatness grows is one whose tails are getting heavier. Small separations have occasional enormous velocity differences that large separations do not, and that is intermittency stated without a single exponent.

Turbulent some of the time is the same phenomenon at the largest scale rather than the smallest: at the edge of a jet a probe is inside turbulent fluid part of the time and in smooth flow for the rest, and most of the fluctuation it records is the switching rather than the turbulence. The intermittency here is the internal version — the switching is between quiescent regions and thin intense ones, and it happens at every scale.

Why the spectrum cannot see any of this #

It is worth being explicit about which measurements are blind to intermittency, because it explains why the argument lasted so long.

The spectrum is a second-order object. It contains $\zeta_2$ and nothing else, and every model here agrees about $\zeta_2$ to a few per cent — a difference of 0.03 in an exponent, over the decade of inertial range the range a real Reynolds number does not have says is available, is a difference of seven per cent in amplitude at the far end. That is inside the scatter of any measurement.

So intermittency is invisible to the spectrum, invisible to the correlation function, and invisible to the second-order structure function, which are the three most commonly measured objects in the subject. It shows only in the higher moments, which are dominated by rare events and therefore need enormous sample sizes, or in the flatness, which needs the same.

The moment a spectrum cannot hold makes the general version of the point: a synthesised field with the right amplitudes and random phases reproduces every second-order measurement and contains no cascade at all.

Measuring high moments, which is harder than it sounds #

The practical difficulty deserves a paragraph, because it is the reason the exponents above $p = 10$ are argued about rather than settled.

$\langle|\delta u|^{12}\rangle$ is dominated by the largest velocity differences in the sample. If the distribution has a tail, the moment is set by the few most extreme events, and the number of samples needed for a stable estimate grows very fast with $p$. A record with $10^7$ points may give a clean $\zeta_4$ and a meaningless $\zeta_{12}$, and the meaninglessness is not visible in the fit’s residual — it shows as a value that changes when the record is extended.

The standard defence is extended self-similarity: plot $\langle|\delta u|^p\rangle$ against $\langle|\delta u|^3\rangle$ rather than against $r$, which uses the exactly-known third moment as the abscissa and gives much straighter lines over a much wider range. It works, it is used everywhere — and it is the same trick as measuring a quantity against another measured quantity rather than against a coordinate, which the frequency a wake chooses uses to collapse three bodies onto one curve. It measures $\zeta_p/\zeta_3$ rather than $\zeta_p$ — which is the same thing only if $\zeta_3 = 1$, so the exact result is doing work there too.

The anomaly is stronger for something merely carried #

The exponents discussed here are the velocity’s, and there is a case that is more anomalous still and that changes what the anomaly can be about.

Take a passive scalar — a dye, or a temperature small enough not to affect the flow — and measure its structure functions. Its exponents fall further below the straight line than the velocity’s, and its increments have heavier tails: a scalar field carried by turbulence develops long smooth ramps ending in abrupt cliffs, and those fronts are sharper than anything in the velocity field advecting them.

That is a strong constraint on any explanation. A passive scalar obeys a linear advection–diffusion equation — there is no nonlinearity in it at all — and it is more intermittent than the nonlinear field carrying it. So the anomaly cannot be a property of the momentum equation’s own nonlinearity. It is a property of the advection: of stretching and folding acting on a field, whichever field that is.

Which is why the passive-scalar problem is where exponents were first derived from a model rather than fitted to data, and why an argument about intermittency that appeals only to the vortex stretching term is arguing about the harder half of a phenomenon that shows up in the easier one.

Where the models come from, in one paragraph each #

They are worth distinguishing because they are three different kinds of argument.

The $\beta$-model is the crudest and the most transparent. Suppose the cascade is not space-filling: at each step the active region occupies a fixed fraction $\beta$ of the volume it came from. Then the active set is a fractal of dimension $D$, the energy is concentrated in it, and the exponents follow immediately. Its prediction is a straight line, which is its weakness — real exponents are curved.

The log-normal model is Kolmogorov’s own 1962 refinement. Suppose the dissipation averaged over a ball of radius $r$ is log-normally distributed with a variance growing as $\ln(L/r)$. That gives the parabola, one parameter $\mu$, and a curve that fits the low moments well. Its failure at high $p$ is the failure of the log-normal assumption’s tail, and it was pointed out almost immediately.

She–Lévêque is a log-Poisson model: the dissipation is built from a hierarchy with a most-intense limiting structure, and the codimension of that structure is the only parameter. Setting it to two — filaments — gives $\zeta_p = p/9 + 2(1 - (2/3)^{p/3})$ with no free parameter at all, and it fits measured exponents to within their scatter out to $p = 16$.

That last point is worth the emphasis. A model with no fitted constant that reproduces a curve is evidence about a mechanism; a model with one fitted constant that reproduces the same curve is evidence about the constant.

What the limit is here #

The limit in this essay is $p \to \infty$, and it is a limit over moments rather than over a physical parameter, which makes it unusual among its neighbours and instructive for the same reason.

K41’s answer is that the exponents grow at a rate of one third for ever. She–Lévêque’s is that the rate falls and saturates at one ninth. The residue of the limit is the difference between those two slopes, and it is not a small correction to a value — it is a geometric statement about the flow.

Saturation of the slope means that beyond some order, higher moments carry no further information: they are all dominated by the same structures, the same filaments, and asking for a higher moment tells nothing new. That is a strong statement and it is testable in principle, and it is one of the things the highest-order measurements are trying to see.

What the shell model produces, without being told #

There is one piece of evidence in this collection that is not a fitted model, and it belongs here.

A flux that runs both ways integrates a shell model — a geometric ladder of wavenumbers with one complex amplitude each and a quadratic nonlinearity that conserves energy — and measures the scaling of its own moments. Its second-order exponent comes out at 0.745 and its sixth at 1.849, against K41’s 0.667 and 2.

The sixth being below two is anomalous scaling produced by a dynamical system rather than assumed by a statistical hypothesis. The system has no space in it, no geometry, no vortex tubes and no dissipation field to be log-normal — and it is intermittent anyway.

That is worth more than it might seem. It says the departure from self-similarity does not require the geometrical picture the models are built on; a nonlinear cascade with the right conservation laws produces it on its own. What the geometrical picture supplies is an interpretation of the number, not the number.

What would settle it #

It is worth naming what a decisive measurement would look like, since eighty years of them have not been.

The models differ decisively only above $p \approx 10$, and a moment of order ten is dominated by events several standard deviations out. The number of independent samples needed for a stable estimate grows roughly exponentially with the order, and the number of independent samples in a record is the record length divided by the integral scale — not by the sampling interval.

So the requirement is a very long record at a very high Reynolds number, and the two pull against each other: a high Reynolds number needs a large apparatus, and a large apparatus has a long integral scale and therefore fewer independent samples per unit time.

The atmospheric surface layer is where the compromise is best, and it brings its own difficulty, which is that it is neither homogeneous nor stationary over the hours a long record takes. Five numbers one name is the collection’s account of how much stratification complicates any atmospheric measurement, and it applies here too.

Limits recorded rather than smoothed over #

Every model here is a model. None of the four is derived from the Navier–Stokes equations; each is a hypothesis about the statistics of the dissipation field, with the exponents following. The only thing derived is $\zeta_3 = 1$.

No data is used. This essay compares models with each other and with the two constraints, and it does not compare any of them with a measurement. Published exponents are quoted nowhere here, because the point being made is about the internal consistency of the models rather than about which fits best.

The codimension is read off an asymptote. Evaluating at $p = 400$ is a device for extracting an intercept; nobody measures a four-hundredth moment, and the geometric interpretation is a property of the model’s functional form rather than of anything observed.

And the constraint used is the weak one. Novikov’s inequality is the requirement that $\zeta_p$ be non-decreasing; there are stronger constraints — from the existence of moments, and from the transversal exponents — and none of them is applied here.

The residue #

The limit is the one K41 takes: a range of scales in which nothing survives but $\varepsilon$, so that the distribution of velocity differences keeps its shape and one exponent describes every moment.

What survives is a whole function. Not a corrected exponent, not a term in an expansion, but a sequence $\zeta_p$ with a shape of its own — falling below $p/3$, curving, and approaching a straight line whose intercept is a dimension.

The limit throws away the shape of a distribution, and the residue is the shape of a distribution.