At higher orders the bottleneck hides intermittency

A relation with no turbulence in it took away one piece of evidence for anomalous scaling. The second-order structure function of turbulence is measured, at laboratory Reynolds numbers, to rise as $r^{0.70}$ rather than Kolmogorov’s $r^{2/3}$. The excess is usually read as the first sign of intermittency. That essay produced the same excess from a model spectrum with no intermittency in it at all. The cascade’s two ends overlap rather than meeting cleanly, the overlap puts a bottleneck bump in the spectrum, and the bump steepens the structure function over the separations an experimenter fits. At a Taylor Reynolds number of 200 the local slope at its flattest is 0.7093. At 10,000 it is still 0.6737. The same calculation found that the transverse function’s ratio to the longitudinal one, the 4/3 of the fraction that is really four thirds, is fixed by that slope at every separation, so the two numbers experiments report as separate discrepancies are one.

It ended on the obvious worry. If a finite Reynolds number steepens the second-order exponent by as much as intermittency is supposed to flatten it, it does something at every order. The higher-order exponents are where the evidence for intermittency is convincing, because that is where the departures from $n/3$ grow large. How much of that evidence is the Reynolds number? The answer turns out to run the other way from the worry, and it is not small.

A distribution, because a spectrum cannot hold one #

A spectrum fixes the second-order structure function and nothing else. The moment a spectrum cannot hold built two fields with identical spectra and showed that their higher moments need not agree at all. To ask about fourth and sixth order the calculation needs a stated distribution for the velocity difference at each separation, and the one used here is the simplest that carries both effects separately.

The velocity difference across a separation $r$ is written as $\delta u = \sigma(r)\, e^{\omega}\, \xi$. Here $\xi$ is a unit Gaussian. The factor $e^{\omega}$ is a random amplitude whose logarithm $\omega$ is Gaussian with a variance $\lambda^2(r)$ that grows as the separation shrinks, and its mean is set to $-\lambda^2$ so that it leaves the variance alone. The variance $\sigma^2(r)$ is the longitudinal structure function from the earlier essay’s fitted spectrum, bottleneck and all. This is Castaing’s form of the log-normal model. It says the velocity difference at a small separation is a Gaussian whose width is itself uncertain, and more uncertain the smaller the separation, which is what a cascade that passes energy down in uneven pieces should produce. A dissipation correlated across every scale built that cascade as a product of random factors, one per step, and the logarithm of a product is a sum — which is why its width grows with the number of steps between the integral scale and the separation, that is, with $\ln(L/r)$.

The width’s spread grows logarithmically:

$$\lambda^2(r) = \frac{\mu}{9}\ln\frac{L}{r},$$

from the integral scale $L$ down to the Kolmogorov scale $\eta$, below which it stops growing. With that choice the absolute moments have a closed form. They are the Gaussian’s moments of the variance times a factor $\exp[\lambda^2 n(n-2)/2]$, and in a pure power-law range their exponents depart from self-similarity by

$$\delta_n = \zeta_n - \tfrac{n}{2}\zeta_2 = -\frac{\mu\, n(n-2)}{18}.$$

That is Kolmogorov’s 1962 correction, with the intermittency coefficient $\mu$ that experiments put near 0.25. At sixth order the distribution carries a departure of a third. Setting $\mu$ to zero gives a Gaussian field, in which every moment follows the variance exactly and there is no intermittency at all.

The construction keeps the two effects apart on purpose. The Reynolds number lives entirely in $\sigma(r)$, through the spectrum. The intermittency lives entirely in $\lambda^2(r)$, through the shape. Any measured exponent is then a known mixture of the two, and each instrument can be asked how much of each it sees.

A field with no intermittency reads steep at every order #

The first figure is the control: a Gaussian field, no intermittency, read the way an experiment reads it. That means the local logarithmic slope of each absolute moment at the separation where the second-order ratio is flattest, the nearest thing the curve has to a plateau. Every moment of a Gaussian is a fixed multiple of a power of its variance, $\langle|\delta u|^n\rangle \propto \sigma^n$. So every order’s local slope is exactly $n/2$ times the second order’s, and the second order’s is the bottleneck-steepened 0.7093.

The apparent exponents fan out above Kolmogorov’s line. At a Taylor Reynolds number of 200 the sixth order reads 2.13 against Kolmogorov’s 2, and the eighth 2.84 against 2.67. The excess is $(n/2)(s - 2/3)$, with $s$ the second-order slope, so it grows in proportion to the order: 0.04 at second order, 0.13 at sixth, 0.17 at eighth. At 10,000 the sixth order still reads 2.02. A field with no intermittency at all, measured at laboratory Reynolds numbers, shows exponents that rise faster than $n/3$.

That is the whole mechanism, and the rest of the essay is its consequence. Intermittency pushes the higher exponents below $n/3$, by an amount that grows as the square of the order. The Reynolds-number excess pushes them above it, by an amount that grows linearly. The two act on the same measured slope in opposite directions, and the measured slope reports their difference.

The sixth-order slope has no plateau #

The second figure follows the sixth-order slope across separations at a Taylor Reynolds number of 200, for the Gaussian field and the intermittent one. Neither has a flat stretch. Both fall from 6 in the smooth range, where every difference is proportional to $r$, through the inertial range and down to zero beyond the integral scale. The intermittent curve sits below the Gaussian one by a constant $|\delta_6| = 1/3$ wherever its distribution is widening, because its widening adds a fixed amount to the slope at every separation between $\eta$ and $L$.

What an experimenter reports is the value where the curve is flattest. Read at 84 Kolmogorov scales, where the second-order ratio is flattest, the intermittent field gives 1.795. Against Kolmogorov’s 2 that is an anomaly of 0.205. The distribution carries 0.333. The other 0.128 has been cancelled by the bottleneck’s excess, $3 \times (0.7093 - 2/3)$, which is exactly what the Gaussian field read on its own.

The figure also shows why this is hard to see from data. A measured sixth-order slope at a Reynolds number of a few hundred is a curve with no plateau and a gentle shoulder, and the shoulder’s height is the only number it offers. Nothing in the curve says how much of the shoulder’s height is the bottleneck. That has to come from somewhere else — from the second-order slope at the same separation, which an experiment measures but rarely subtracts at the higher orders.

Fourth order shows almost nothing #

The third figure sets the three instruments side by side at a Taylor Reynolds number of 200, order by order, against the departure the distribution actually carries. The local slope reports the least, and at low orders the shortfall is severe. At fourth order the distribution carries an anomaly of 0.111 and the local slope shows 0.026, less than a quarter of it. At third order the absolute moment’s local slope reads 1.022, above one, where its distribution carries 0.958. At second order it reads 0.709, the earlier essay’s number, an exponent above Kolmogorov’s where intermittency would put it a little below. By sixth order the slope shows 62 per cent of the anomaly, and by eighth 74 per cent, because there the quadratic growth of intermittency is beginning to outrun the linear growth of the bias.

This changes what the commonly quoted numbers mean. Sixth-order exponents near 1.8 have been reported for decades from laboratory flows at Taylor Reynolds numbers of a few hundred, and 1.8 is what this field reads at 200. But this field’s distribution carries a sixth-order departure of a third, not the 0.2 that such a reading implies against Kolmogorov. If the reading is taken at face value, the intermittency coefficient comes out at about 0.15 rather than the 0.25 the field was built with. At higher orders the finite-Reynolds-number bias does not manufacture intermittency; it hides it. The direction is the opposite of the second-order case, where the same bias mimicked an anomaly that was not there.

The two statements are consistent, and the consistency is the point. The bias is one positive excess in the second-order slope, multiplied by $n/2$. At second order it is the whole signal, because intermittency’s quadratic term is close to zero there. At higher orders it subtracts from a signal that intermittency’s quadratic term makes large. A bias that inflates every exponent looks like intermittency only where intermittency is absent.

How slowly the local slope recovers #

The fourth figure follows the sixth-order share upwards in Reynolds number. The local slope reports 62 per cent at a Taylor Reynolds number of 200, 66 at 500, 74 at 1,000, 86 at 3,000 and 94 at 10,000. That last number belongs to the largest wind tunnels and the atmospheric surface layer. Most simulations and most laboratory flows sit in the lower half of the range, where a third of the sixth-order anomaly is hidden.

The rate is the earlier essay’s. The excess in the second-order slope falls roughly as the reciprocal square root of the Reynolds number, and here it is multiplied by three. Closing the sixth-order bias by a factor of ten costs a factor of a hundred in Reynolds number. The range a real Reynolds number does not have measured how little inertial range a laboratory flow has to fit in. This is the same shortage seen through a moment that magnifies it threefold.

There is a parallel in an earlier essay on the same exact law. The decay inside the four-fifths law found that a finite Reynolds number biases the one exact third-order result, and biases it in a direction opposite to the effect the measurement was looking for. The four-fifths law and the intermittency exponents are the two main supports of the evidence on small-scale turbulence. At the Reynolds numbers where both are measured, both carry a bias of the same origin, and in each case it leans against the effect being sought.

The two instruments that see through it #

The other two curves in the third and fourth figures are the remedy. Both are already in use, and this calculation says how well each works.

The normalised moment divides out the variance before taking the slope: $\langle|\delta u|^n\rangle / \langle\delta u^2\rangle^{n/2}$, which at fourth order is the flatness. Its local slope is $\delta_n$ exactly, at every Reynolds number and every separation in the inertial range, because the spectrum cancels out of it entirely. In the figures it lies on the truth at every order. A flatness that grows as the separation shrinks is intermittency, and the rate it grows is the coefficient, whatever the bottleneck does to the variance.

That cancellation is exact here because the model puts the Reynolds number in the variance and the intermittency in the shape, and the normalised moment is the one instrument that removes the first and keeps the second. In a real flow the bottleneck might also change the distribution’s shape a little near the dissipation range. The model cannot see that, and to that extent the normalised moment’s perfection is a property of the construction. What survives is the direction: of the three instruments, the normalised moment is the one the spectrum’s shape cannot reach.

The same reasoning gives a correction an experiment can apply to data it already has. The local slope of the $n$-th moment is $(n/2)s + \delta_n$, and $s$ is measured at the same separation from the same record. Subtracting $(n/2)(s - 2/3)$ from the measured slope returns $n/3 + \delta_n$, the exponent with the Reynolds number taken out, and that is algebraically the normalised moment’s slope again. It costs nothing beyond reporting the second-order slope at the separation where the higher one was read, which the published tables of exponents almost never do. Without it, the number in the table mixes two effects that the table’s reader has no way to separate.

Extended self-similarity, and what makes it extended #

Extended self-similarity plots one moment against another instead of against the separation, and reads the exponent ratio $\zeta_n/\zeta_3$ off the slope. It was introduced in 1993 because it showed straight lines over a much wider range of separations than the ordinary plot, reaching well down towards the dissipation range, and it has been the standard way of measuring the higher exponents since.

The fifth figure shows why it is extended. For a Gaussian field the slope of the sixth moment against the third is exactly 2 at every separation, in the smooth range, the inertial range, and everything between. Both moments are fixed powers of one variance, so whatever the variance does with separation cancels in the ratio. The famous extension into the near-dissipation range is what any field gives whose distribution keeps its shape. It says nothing about the cascade.

For the intermittent field the slope is not flat. It reads 1.91 at two Kolmogorov scales, 1.76 where the second-order ratio is flattest and 1.70 at half the integral scale. The distribution widens at a fixed rate in $\ln r$, while the variance’s own slope falls from 2 to nothing, so the intermittency’s share of the slope grows with separation. Read at the usual place, the ratio shows 94 per cent of the departure it should at a Taylor Reynolds number of 200, and 99 per cent at 10,000 — far better than the local slope’s 62 per cent. And the share is the same at every order, because the bias enters through the third moment in the denominator, once, rather than through each moment separately.

So the instrument the field adopted for its long range turns out to work for a different reason: it divides most of the Reynolds-number bias out. The extended range is a side effect. The correction is the substance.

What the calculation is checked against #

The last figure is the ledger. The Gaussian absolute moments used in every curve agree with a direct quadrature of the Gaussian density to four parts in a hundred million, which is the quadrature’s own error. The log-normal factor agrees with a quadrature over $\omega$ to a part in ten million million. Every slope in the figures was obtained by differencing the moments along the separation, not from the formulas above. Those formulas were then the test: the differenced local slope equals $(n/2)s + \delta_n$, and the normalised slope equals $\delta_n$, to rounding everywhere between twenty Kolmogorov scales and half the integral scale. The Gaussian field’s ESS slope is 2 to five parts in ten thousand million at every separation. The model refuses an order that is not a whole number, a negative variance, an intermittency coefficient of one or more, and a Reynolds number too small to have a range.

What the model leaves out #

No skewness. The distribution is symmetric, so its signed third moment is zero, and the energy transfer that the one exact result pins to $-\tfrac45\varepsilon r$ is not in it. The third moment used here is the absolute one, which is what extended self-similarity uses. Skewness would add a small odd-order part and leave the even orders, which carry the argument, untouched.

Second order without its own correction. The variance is the model spectrum’s, so the second-order exponent tends to 2/3 rather than to K62’s $2/3 + \mu/9$. The intermittency sits wholly in the higher orders’ departure from the second. That is a choice of bookkeeping rather than physics, and it is why the departures are written against $(n/2)\zeta_2$.

A log-normal shape. The log-normal model’s high-order moments eventually grow faster than any real cascade allows. The exponents that stop being thirds compared the candidate models and their limits. The log-normal and the She–Lévêque forms differ by a few hundredths at sixth order and by a tenth at eighth. The bias computed here does not depend on the choice: for any distribution whose shape is set apart from its variance, it enters through the variance alone.

Kolmogorov, Castaing and the extended range #

Kolmogorov’s refined similarity hypothesis and the log-normal correction are from 1962, with Obukhov. The anomalous exponents themselves were measured by Anselmet, Gagne, Hopfinger and Antonia in 1984. Castaing, Gagne and Hopfinger wrote the distribution in the form used here in 1990. Benzi and collaborators introduced extended self-similarity in 1993, and Falkovich explained the bottleneck in 1994. What the calculation here adds is the sign of the bias at each order and the share of the anomaly each instrument sees.

Still open: the measured flatness, read as the coefficient #

The normalised moment sees through the Reynolds number in this model, and the flatness of the longitudinal velocity difference has been measured across separations in a great many flows. The next calculation reads μ from those flatness slopes directly — the one estimate this model says the bottleneck cannot bias — and compares it with the values obtained from fitted sixth-order exponents at the same Reynolds numbers. If the two disagree by the share computed here, the scatter in the published intermittency coefficient is partly the Reynolds number of the flow it was measured in, and the coefficient itself is larger and more universal than the fitted exponents make it look.