A dissipation correlated across every scale

The exponents that stop being thirds computes what intermittency does to the structure-function exponents: the higher moments scale with exponents that depart from Kolmogorov’s q/3, and the departure is quadratic.

That essay is about the exponents. This one is about the field they come from, and about a property of it that is easy to miss because the word “dissipation” suggests the opposite: the dissipation field is correlated over enormous distances.

Where the correlation comes from #

The cascade picture is that a large eddy hands its energy to smaller ones, which hand it on again, and so on down to the dissipative scales. The multiplicative version of it says that each handover multiplies the energy flux by a random factor: some eddies pass on more than their share and some less.

That has an immediate consequence. The flux reaching a small eddy is a product of all the factors along the chain from the largest scale down to it, so the logarithm of the flux is a sum of the logarithms of the factors. A sum of many independent contributions is Gaussian, which is why the dissipation is log-normal — and its variance grows as the number of steps in the chain, which is the logarithm of the ratio of scales.

Two nearby points share most of their chain — they differ only in the last few steps — so their fluxes are correlated. Two distant points share fewer steps, but they still share the ones near the top, and the number of shared steps falls only as the logarithm of the separation.

The correlation is therefore a ratio of logarithms, and a ratio of logarithms is very nearly one over a very long way.

How far it reaches #

In a flow with four decades of inertial range, two points a thousand Kolmogorov lengths apart have a correlation of 0.25 in the logarithm of their dissipation. At a hundred it is 0.5, and at ten it is 0.75.

The halving separation is the geometric mean of the smallest and largest scales — half way between them on a logarithmic axis. With four decades of inertial range it sits at two decades, which is a separation of a hundred smallest scales, and the log-flux variance across the range is 2.303 at an intermittency exponent of 0.25. That is a striking place for it to be, because it means the answer is not a distance at all: a flow with more decades of inertial range correlates over more of them rather than fewer, so improving the Reynolds number does not localise the dissipation, it delocalises it.

What that does to a measurement #

The consequence is a warning about independence, and it is a large one.

Samples taken close together are not independent. Two points ten smallest scales apart share 75 per cent of their cascade; a probe recording dissipation at a sampling rate fine enough to resolve the smallest scales is producing a series whose successive values are strongly correlated — not because the instrument is slow, but because the field is. Treating them as independent samples and dividing an error bar by the square root of their number overstates the precision by a large factor.

A converged mean needs an enormous record. With four decades of inertial range and a correlation still at 0.25 a thousand smallest scales out, the effective sample count is the record length divided by something of order the integral scale rather than the Kolmogorov one — a factor of 10⁴ fewer samples than the sampling rate suggests. The number of effectively independent samples in a record is its length divided by the correlation length, and the correlation length here is comparable with the integral scale rather than with the Kolmogorov scale. A record that looks like ten million samples may contain a few hundred independent ones.

And a local dissipation is not a local quantity. The name suggests something happening at a point and it is a coarse-graining of a field whose structure spans the whole inertial range. That is the same warning turbulent some of the time gives about intermittency at the largest scales, arriving from the small ones.

The exponents, which are the visible consequence #

The measurable signature of all this is in the structure-function exponents, and it is worth putting beside the correlation because they are the same statement.

Kolmogorov’s 1941 argument gives the qth-order exponent as q/3. The log-normal cascade gives q/3 − (μ/18)q(q−3), which agrees at q = 3 and departs quadratically either side — by a quarter at the sixth moment.

The agreement at three is not a coincidence and it is the most important fact in the picture. The third-order law is exact: it follows from the Navier-Stokes equations with no model in it, which is the one exact result. Any cascade model has to reproduce it, and the log-normal one does — which is a genuine constraint on μ and is the reason the model survives.

What the solver computed, and how it was checked #

The log-normal cascade is a model rather than a solution, and everything here is algebra on it: the variance, the correlation, the halving separation and the exponents follow from two assumptions — independent multipliers and a log-normal limit — and one fitted number, the intermittency exponent μ = 0.25.

Three checks. That the halving separation is at exactly half the decades, which is a statement about the ratio of logarithms and would catch an algebra error. That the correlation at a thousand smallest scales exceeds two tenths — it reads 0.25 in a flow spanning four decades — a quarter of the shared cascade still there at a separation a thousand times the smallest scale, which is a tenth of the flow’s own size — which is the essay’s headline. And that the sixth-moment exponent departs from two by more than a tenth; it departs by 0.25, and the tenth-order one by 0.97.

Why this is a memory #

The word fits here in an unusual sense and it is worth being explicit about which one.

Nothing here is a memory in time. The correlation is between two places at one instant, and the cascade being described is a hierarchy of scales rather than a sequence of events.

But it is the same structure. The dissipation at a point is a product over its whole ancestry — every eddy that handed energy down to it — and that ancestry is shared with its neighbours. Two points are correlated because they have a common past in the cascade, exactly as two parcels of fluid are correlated because they have a common past in the flow.

The formal statement is the same too. The correlation falls as the number of unshared steps, which is the logarithm of the separation; in a scalar is a record of where its fluid was the correlation falls as the number of unshared time steps, which is the elapsed time. A logarithm in one case and a linear measure in the other, which is why the cascade’s correlations are so much longer-ranged than a flow’s.

Why a ratio of logarithms decays so slowly #

The functional form is the whole result and it is worth appreciating rather than merely quoting.

An ordinary correlation falls exponentially with separation: it has a length scale, and beyond a few of them there is nothing left. That is what a diffusive or a convective process produces, and it is what the word “correlation length” usually means.

A ratio of logarithms has no length scale at all. It is one at the smallest separation, zero at the largest, and in between it depends only on where the separation sits between the two ends on a logarithmic axis: at ten smallest scales it is 0.75, at a hundred 0.50, at a thousand 0.25 — a straight line in the logarithm, losing exactly a quarter of the correlation for each decade the separation grows. Doubling the separation costs the same amount of correlation whether the doubling happens at the small end or the large one.

That is a scale-free decay, and it is the signature of a scale-free process. The cascade has no preferred scale by construction — that is the whole content of an inertial range — so nothing in it can supply a correlation length, and the only decay available is the one the counting of shared steps produces.

The practical consequence is that quoting a correlation length for the dissipation is a category error. There is a correlation function, it is a ratio of logarithms, and summarising it by a single length throws away the property that makes it interesting.

What is actually being assumed #

Since the whole essay rests on a model, it is worth stating what the model claims and where it is known to be wrong.

That the multipliers are independent. Each step of the cascade is taken to be statistically independent of every other. That is the assumption which makes the logarithm a sum and everything else follows from it. It is not exactly true: measurements of the multipliers show correlations between adjacent steps.

That the limit is log-normal. With independent multipliers and enough steps the central limit theorem gives a Gaussian for the logarithm. Real inertial ranges have a few decades rather than many, so the convergence is incomplete and the tails are the part that is furthest off — which matters, because the high moments are precisely the tails.

And that the cascade is local in scale. Energy is taken to pass from an eddy to one somewhat smaller, rather than jumping decades. That is the assumption Kolmogorov’s whole framework rests on and it is the one with the most evidence behind it.

The log-normal model is known to be quantitatively wrong at high orders — the exponents in the table above turn over, reaching a maximum of 2.528 at q = 13 and falling after it, which is impossible — and the models that repair it, She-Lévêque and the log-Poisson family, change the exponents and keep the long-range correlation. The correlation is the robust part, because it follows from the cascade being multiplicative rather than from the distribution being log-normal.

The delivery those exponents are moments of is the neighbouring argument, on the same machinery.

What this does to the exact result #

There is a tension worth resolving, because two of this collection’s essays appear to disagree.

The one exact result establishes that the four-fifths law is exact: the third-order structure function is −(4/5)εr with no model in it and no adjustable constant. That looks incompatible with a dissipation field so variable that its higher moments have anomalous exponents.

It is not, and the resolution is precise. The four-fifths law involves the mean dissipation, and the mean is exactly what a multiplicative cascade preserves: the product of the multipliers has unit mean by construction, so ⟨ε⟩ is the same at every scale. Every other moment is not preserved, and every other moment is where the anomaly lives.

So the exact result and the intermittency are statements about different moments of the same field, and the third order is where they meet — which is why the log-normal exponents agree with Kolmogorov’s exactly at q = 3 and nowhere else.

That is the general shape of an exact result in a variable field, and this collection has met it before: a conservation law constrains a total and says nothing about a distribution, which is exact in the total, free in the profile. The four-fifths law is the total; the intermittency is the profile.

What a reader should do with this #

Three practical statements come out, and they are about measurement rather than about theory.

Report the correlation length of whatever is being averaged. For a velocity it is the integral scale; for a dissipation it is comparable, not the Kolmogorov scale. That number, divided into the record length, is the effective sample count.

Expect high moments to be badly converged. The qth moment is dominated by the tail, the tail is the rarest events, and the rarest events are the least independent. Structure-function exponents above about the sixth order are quoted with error bars that are usually optimistic for this reason.

Use the third moment where possible. It is the one with an exact law attached and the one whose convergence is best, which is why the collection’s own dissipation estimates are made from it rather than from a spectral fit — a preference the limit that is not the value argues for on separate grounds.

And be suspicious of a locally measured dissipation. A single-point estimate of ε — from a one-dimensional surrogate, from a spectrum, from a structure function — is an estimate of a coarse-grained quantity whose coarse-graining scale matters, and it is correlated with the estimate at the next station.

The same structure, in two other places #

A quantity built as a product over a hierarchy has these correlations wherever the hierarchy exists, and two other instances are worth naming because neither is about fluids.

Financial volatility. The variance of a price series is well described by a multiplicative cascade in time, and its logarithm has exactly this long-range correlation — which is why volatility clusters and why estimating it from a short window is unreliable in a way that is often mistaken for non-stationarity.

And rainfall. Precipitation fields are multifractal for the same reason, and the correlations in the logarithm of the rate span the range from a shower to a front.

What the three share is a multiplicative construction rather than an additive one. Sums of random contributions produce Gaussian fields with short correlations; products produce log-normal fields with long ones, and the difference is that a product’s logarithm is a sum over a hierarchy whose top is shared.

That is worth carrying out of fluid mechanics. Whenever a quantity is built by repeated multiplication down a hierarchy, expect its logarithm to be correlated across the whole hierarchy — and expect any error bar computed on the assumption of independence to be too small.

What the picture cannot show #

The correlation curve is drawn against separation on a logarithmic axis, which is the only readable form and which makes the decay look faster than it is. On a linear axis the curve would be flat across the whole plot and then fall at the very end, which is a fairer picture and an unreadable one.

Nothing here draws a dissipation field. The characteristic picture — thin filaments and sheets of intense dissipation in a background of very little — is what the log-normality describes statistically, and this collection’s solver cannot produce one.

Why the correlation is not a failure of the cascade picture #

It would be easy to read the surviving correlation as evidence that the cascade idea is wrong, and it is worth saying why it is not.

The cascade is a statement about energy flux, and about the fact that the flux at one scale does not care about the details of the scales far above it. That statement survives here intact: the mean flux is scale-independent over the range computed, which is what the cascade asserts.

The correlation is a statement about fluctuations, and the cascade never asserted anything about those. A multiplicative process built from independent factors has exactly this property — the logarithm accumulates, so two distant scales share every factor between them and the correlation falls only logarithmically.

So the two coexist, and what fails is a stronger claim that was never part of the original argument: that the small scales are statistically independent of the large ones. That claim is what local isotropy is usually taken to mean, and it is the one this measurement contradicts.

Who found it, and when #

The refined similarity hypothesis and the log-normal model are Kolmogorov’s and Obukhov’s, from 1962, and were a response to Landau’s objection that the 1941 theory took no account of the variability of the dissipation. The intermittency exponent has been measured many times since and sits between 0.2 and 0.3, with no consensus tighter than that.

The long-range correlation is implicit in the model from the start and is stated explicitly in the multifractal literature of the 1980s. It is one of the more counter-intuitive consequences of a picture that was introduced to make the theory more local rather than less.

Limits recorded rather than smoothed over #

A model, throughout. Nothing here is computed from the Navier-Stokes equations. The cascade is a statistical construction, the log-normality is an assumption, and μ is fitted.

Four decades of inertial range. That is a large laboratory flow or a modest atmospheric one. The numbers scale with the number of decades and the shape of the conclusion does not.

The exponent formula is known to fail at high order. Beyond about q = 12 the log-normal prediction turns over, which is unphysical. The correlations quoted here are second-order statements and are not affected.

The Reynolds number never appears. Everything is expressed as a ratio of the largest scale to the smallest, and the Reynolds number enters only through how many decades that is. A flow at ten times the Reynolds number has about a decade and a half more, so its correlations reach further in units of the smallest scale and no further in units of the largest — which is the same statement the range a real Reynolds number does not have makes about the inertial range itself.

And the correlation quoted is of the logarithm. The correlation of the dissipation itself is different and smaller, because a log-normal variable’s correlation is not its logarithm’s. That distinction is real and the logarithm is the natural variable for a multiplicative process, which is why it is the one reported.