Thinking in Waves Episode 42 / Part XIII — The Frontier, as Waves

When the order stops being a single number

Intermittency in Turbulence —
The order was a distribution Kolmogorov's $-5/3$, in this series' language, makes turbulence a "5/6-order" wave — 5 dB/oct, phase 75°. Exactly between 1/f (order 1/2) and Brownian motion (order 1).
★★ But measurements depart from it systematically, and the departure is this — the order is not one number but a distribution of width 1.222.
★ And the mean does not move: $\zeta_3=1$ stays exactly 1 no matter how intermittent the flow.

Tools needed: Episodes 1, 2, 13, 14, 22, 30, 39Computation: kenshou/ranryu.js

Episode 14 sorted "three reasons the order leaves the integers"; Episode 39 added a fourth (the bulk mass).
★ Here comes the fifth, and the most awkward: the order is not a single number.

01Reading $-5/3$ as an order

All you have is $\varepsilon$ (dissipation rate per unit mass) and $k$ $$E(k)\ [\mathrm{m^3/s^2}]=\varepsilon^a k^b\ \Longrightarrow\ \text{length}:2a-b=3,\ \ \text{time}:-3a=-2\ \Longrightarrow\ a=\tfrac23,\ b=-\tfrac53$$

Quantity	Value	In this series' words
power slope $\beta$	1.666667	$E(k)\propto k^{-\beta}$
★ order $\alpha=\beta/2$	0.833333	★ amplitude $\propto k^{-\alpha}$
dB/oct $=6\alpha$	5.000000	Ep. 1
phase $=90\alpha$ [deg]	75.000000	Ep. 1
★ Hurst $H=\alpha-1/2$	0.333333	★ exactly $1/3$

The point of §01

Turbulence is a "5/6-order" wave.
Exactly between 1/f (order 1/2, Episode 2) and Brownian motion (order 1).

★ And $H=1/3$ is K41's $\zeta_p=p/3$ at $p=1$ — "order," "dB/oct," "Hurst exponent" and "structure-function exponent" are all the same number said four ways.
★★ But only if the order is one number.

02If the order is one number, every exponent agrees

We synthesize $f^{-\beta}$ noise and measure the exponents of the structure functions $S_p(r)=\langle|\delta u|^p\rangle\propto r^{\zeta_p}$ for each $p$. With a single order, $\zeta_p/p$ must not depend on $p$.

$\beta$	$H=(\beta-1)/2$	$\zeta_1$	$\zeta_2/2$	$\zeta_4/4$	$\zeta_6/6$
1.2	0.100000	0.15906	0.15932	0.15928	0.15833
5/3	0.333333	0.34639	0.34697	0.34732	0.34747
2.0	0.500000	0.50260	0.50269	0.50277	0.50352
2.5	0.750000	0.75138	0.74998	0.74798	0.74887

The point of §02

★ $\zeta_p/p$ is constant in $p$ — that is what "a single order" means (a monofractal).
★ Only the $\beta=1.2$ row is off (0.159 rather than 0.100) — for $H<0.2$ the finite sampling interval still bites; a limit of our synthesis method. The other three rows agree to within 0.3 %.

03But real turbulence splits the dissipation multiplicatively

Energy passes from large eddies to small ones, but never evenly. In a model that splits it into $p$ and $1-p$ at each stage (the p-model), we measure the $q$-th moment $Z(q,r)=\sum\mu_i^q\propto r^{\tau(q)}$.

$q$	$\tau$ (measured)	$\tau$ (exact, $-\log_2(p^q+(1-p)^q)$)	difference	if monofractal: $q-1$
0	−1.000000	−1.000000	0	−1.000
1	−0.000000	0.000000	1.7e−14	0.000
2	0.785875	0.785875	2.4e−14	1.000
4	2.010425	2.010425	4.4e−14	3.000
5	2.552156	2.552156	3.2e−14	4.000

The point of §03

Measurement agrees with the exact result to $10^{-14}$ ($p=0.7$, 18 levels = 262144 boxes).
★ And $\tau(q)$ is not linear in $q$ — at $q=4$ it is 2.010, against 3.000 for a monofractal. That bend is intermittency.
★★ At $p=0.5$ (an even split) it returns to the straight line $\tau(q)=q-1$ (confirmed numerically: 0, 1.000000, 3.000000). The bend comes from splitting unevenly.

04★★ Legendre transform, and the distribution appears

$q$ is the dial that selects which strength of region you look at $$\alpha(q)=\frac{d\tau}{dq}\quad(\text{the local exponent that }q\text{ picks up}),\qquad f(\alpha)=q\alpha-\tau(q)\quad(\text{dimension of the set with that exponent})$$

$q$	$\alpha$ (local exponent)	$f(\alpha)$	Meaning
−4	1.697073	0.207439	picks up the weak regions
0	1.125769	1.000000	★ $f=1$ = dimension of the support
1	0.881291	0.881291	★ the "representative" value
4	0.554466	0.207439	picks up the strong regions
8	0.515963	0.012758	the very strongest

The point of §04 — ★★★ the core of this episode

$\alpha$ runs from 0.514573 to 1.736966 (width 1.222392).
Not one number, but a set with width.

Exactly, from $-\log_2 p=0.514573$ to $-\log_2(1-p)=1.736966$ — a width of $\log_2\frac{1-p}{p}$.
★ At $p=0.5$ the width is zero (a single point, $\alpha=1$). The unevenness becomes the width of the distribution.
★★ In this series' words: a wave whose order differs from place to place. Measure it locally with Episode 22's wavelets and you get a different dB/oct at each location.

05To the velocity structure functions — bending away from $p/3$

A box's measure is $\mu=\varepsilon_r\cdot r$, so the dissipation's own exponent is $\tau_\varepsilon(q)=1-q+\tau(q)$ $$\zeta_p=\frac p3+\tau_\varepsilon\!\left(\frac p3\right)\qquad(\text{refined similarity hypothesis})$$

$p$	K41: $p/3$	★ p-model ($p=0.7$)	difference	measured (literature)
1	0.333333	0.360919	+0.027586	0.37
2	0.666667	0.693721	+0.027055	0.70
3	1.000000	1.000000	★ 0.000000	1.00
4	1.333333	1.282153	−0.051180	1.28
5	1.666667	1.543081	−0.123586	1.53
6	2.000000	1.785875	−0.214125	1.78

The point of §05

★★ One dial ($p=0.7$) reproduces the literature values of $\zeta_1$ through $\zeta_6$ to within 1–2 %.
★ The larger $p$, the further it bends down — high moments pick out the very strongest eddies, and those are rare.
★ Episode 2 made 1/f from adding many relaxation processes. Here the superposition is multiplicative — addition tends to a Gaussian by the central limit theorem; multiplication tends to a log-normal. Intermittency happens because the cascade multiplies.

06★ And yet $\zeta_3=1$ does not move

$p$	★ $\zeta_3$	$\zeta_2$	$\zeta_6$	$\beta=1+\zeta_2$	dB/oct
0.5	1.00000000	0.666667	2.000000	1.666667	5.0000
0.7	1.00000000	0.693721	1.785875	1.693721	5.0812
0.8	1.00000000	0.732441	1.556393	1.732441	5.1973
0.9	1.00000000	0.801364	1.286304	1.801364	5.4041

The point of §06

$\tau_\varepsilon(1)=1-1+\tau(1)=0$ regardless of $p$ (mean dissipation is conserved) — so $\zeta_3=1$ exactly, for every $p$. That is the 4/5 law.
★★★ And here is the crucial part: even at $p=0.7$, $\beta=1.6937$, a difference of 0.0271 from K41's 1.6667. In dB/oct that is a difference of 0.081.
★ Which is why intermittency is nearly invisible in a power spectrum. "$-5/3$ fits well" is really the second moment being insensitive.

07In this series' language — a fifth reason

Why the order leaves the integers	Episode
① superposition of many time constants	Ep. 2
② dimension ($\alpha=(d-3)/2$)	Ep. 30
③ continuous variation at a critical point	Ep. 13
④ the bulk mass ($\alpha=2\nu$)	Ep. 39
★ ⑤ the order is not a single number (it is a distribution)	★ here

How you measure	The order you get	Episode 1's quantity
slope of the power spectrum	one number, from $\zeta_2$	dB/oct
first-order structure function	one number, from $\zeta_1$	—
★ higher moments	★ a different value for each $p$	★ they disagree
★ measuring locally	★ a different value at each place	★ a distribution

The point of §07

★★★ The very act of "measuring dB/oct" depends on which moment you measure with.
★ For a monofractal every method gives the same number (confirmed in §02). Turbulence is not one.
★★ The single axis of "order," carried since Episode 1, stops being single here.

08What wave language can and cannot say

Item	Verdict	Basis
$-5/3$ in one line of dimensions	◎ analytic	order 5/6, 5 dB/oct, 75°
$H=\alpha-1/2=1/3$	◎ numerics	confirmed on synthetic noise
Monofractal → $\zeta_p/p$ constant	◎ numerics	agrees for $p=1,2,4,6$
★ $\tau(q)$ is not linear	◎ numerics	★ exact to $10^{-14}$
★ The order is a distribution	◎ analytic	★ width $=\log_2\frac{1-p}{p}$
$\zeta_p$ bends below $p/3$	◎ analytic	$p=0.7$ fits literature to 1–2 %
$\zeta_3=1$ independent of $p$	◎ analytic	the 4/5 law survives
★ The real value of $p$	× cannot say	★ a fitted parameter
★ Why the cascade multiplies	× cannot say	★ assumed here

Exercises

§06 explained why "$-5/3$ fits well." Does that mean K41 was right?
Show answer
No — it means the second moment was insensitive. The shift in $\zeta_2$ is only 0.027, or 0.081 dB/oct. It is buried in measurement error.
★ Where K41 is wrong shows up clearly in the sixth moment — predicted 2.000 against a measured 1.78, an 11 % discrepancy.
★★ The lesson: seeing one exponent fit well is not a test of a theory. This is why every episode of this series records which moment was measured. ★ The same structure as Episode 5's "anything longer than the age of the universe is indistinguishable from a constant" — through an insensitive quantity, a difference is the same as no difference.
In §04, $\alpha$ spanned 0.51 to 1.74. Why no negative $\alpha$, and none above 2?
Show answer
Because each stage offers only two splits, $p$ and $1-p$. The strongest point is the one that drew $p$ every time, with $\mu=p^n$, i.e. $\alpha=-\log_2p$; the weakest is $\alpha=-\log_2(1-p)$.
★ So the ends of the distribution are set by the most extreme path. Change the model and the ends change — a log-normal cascade formally has no ends.
★★ In real turbulence, though, the cascade stops at the viscous (Kolmogorov) length, so it cannot become arbitrarily extreme. ★ As in Episodes 5 and 22 — how finely you look decides the answer.
Episode 2's 1/f was also a superposition. How does it differ from intermittency here?
Show answer
Addition versus multiplication. Episode 2 added many relaxation processes to make $1/f$ — additive superposition tends to a Gaussian by the central limit theorem, and the order comes out as a single number.
★ Here each stage multiplies. A multiplicative superposition adds in the logarithm, so it tends to a log-normal — heavy tails, where rare large values dominate. Hence the bending of high moments.
★★ Whether the order is one number or a distribution was decided by whether the superposition adds or multiplies. ★ Episode 28 sorted "two sources of exponentials," and one of them — additive becoming multiplicative — is at work here too.

Summary — the order was a distribution

Kolmogorov's $-5/3$ comes out of one line of dimensional analysis. Translated: turbulence is a "5/6-order" wave — 5 dB/oct, phase 75°, Hurst $H=1/3$. Exactly between 1/f (order 1/2) and Brownian motion (order 1).

First we checked what a single order looks like — for synthetic $f^{-\beta}$ noise, $\zeta_p/p$ is constant in $p$ and equals $(\beta-1)/2$.

But a real cascade multiplies. In the p-model our measured $\tau(q)$ matches the exact result to $10^{-14}$, and $\tau(q)$ is not linear in $q$ (2.010 at $q=4$, against 3.000 for a monofractal). At $p=0.5$ it straightens out — the bend comes from splitting unevenly.

★★★ The core — after a Legendre transform, $\alpha$ runs
from 0.514573 to 1.736966 (width $\log_2\frac{1-p}{p}=1.222$).
The order is not one number but a set with width.

In terms of velocity structure functions, $\zeta_p$ bends below $p/3$ — and one dial ($p=0.7$) reproduces the literature values of $\zeta_1$ through $\zeta_6$ to 1–2 % (0.361/0.37, 0.694/0.70, 1.000/1.00, 1.282/1.28, 1.543/1.53, 1.786/1.78).

★ And yet $\zeta_3=1$ does not move — mean dissipation is conserved, so $\tau_\varepsilon(1)=0$ and $\zeta_3=1$ exactly, for every $p$ (the 4/5 law).
★★★ The crucial part: even at $p=0.7$, $\beta=1.6937$, differing from K41 by only 0.081 dB/oct. "$-5/3$ fits well" was really the second moment being insensitive.

★ After Episode 14's three and Episode 39's fourth — ⑤ the order is not a single number. The act of "measuring dB/oct" depends on which moment you measure with. ★ The single axis carried since Episode 1 stops being single here.

Next time For 42 episodes we have lined physics up along the single axis of order.
★ Last time a different kind of quantity appeared — integers (winding and Chern numbers) — and this time the order became a distribution.
★★ Next time we take stock — as in Episode 15, we score the 27 episodes from 16 to 42. What was right, what was wrong, and where we were only translating.

This document is Episode 42 of the "Thinking in Waves" series. Computations are in kenshou/ranryu.js. On the numbers here: the structure-function exponents of the synthetic noise, the p-model cascade and the measurement of $\tau(q)$, the Legendre transform, and the values of $\zeta_p$ and $\beta$ were all computed here. Kolmogorov's $-5/3$ law and $\zeta_p=p/3$, the 4/5 law ($\zeta_3=1$), Mandelbrot's multiplicative cascades and the p-model, $\tau(q)=-\log_2(p^q+(1-p)^q)$, the multifractal formalism (the Legendre transform), the refined similarity hypothesis, and the measured values of $\zeta_p$ (such as $\zeta_6\approx1.78$) are all well-known results and none are discoveries of this article — what this article did was place them on this series' axis of "order," add "the order becomes a distribution" as a fifth entry to Episode 14's list, and put a number on how little intermittency moves the dB/oct: 0.081. In §05, $p=0.7$ reproduces the literature well, but that is one fitted parameter and does not make the p-model the correct model. Several models of intermittency exist (log-normal, She–Lévêque and others) and this article simply took the simplest. The $\beta=1.2$ row in §02 is off by 0.06 because of a limitation of the synthesis method. Literature values of $\zeta_p$ depend on measurement conditions and Reynolds number; those quoted are representative. — To print, use your browser's Print and "Save as PDF."

01Reading \(-5/3\) as an order