Thinking in Waves Episode 14 ／ Part VI · Extending the order into the complex plane
Turbulence's \(-5/3\) was not a fractional derivative
The ladder of spectral exponents ──
there are only three ways to leave the integers
Line up every slope in this series and almost all of them are integers or half-integers.
Only turbulence gives \(-5/3\) and convection \(-11/5\) ── denominators of 3 and 5.
★ Compute it and those denominators turn out to be the determinant of the dimensional matrix.
The dictionary built in Episode 2 ── an operation of order \(\alpha\) gives \(6\alpha\) dB/oct, and noise exponents obey \(\beta=-2\alpha\) ── puts every slope so far on one scale.
And on that scale, exactly three outliers appear.
01First, everything on one page
| Phenomenon | Exponent \(\beta\) | In the language of order | Where it appeared | Verdict |
|---|---|---|---|---|
| \(\omega^4\) (Rayleigh scattering) | \(+4\) | second derivative, squared | Ep. 12 | ◎ exact |
| \(+12\) dB/oct (acceleration) | \(+2\) | second derivative | Ep. 1 | ◎ exact |
| \(+6\) dB/oct (velocity) | \(+1\) | first derivative | Ep. 1 | ◎ exact |
| White noise | 0 | do nothing | Ep. 2 | ◎ by definition |
| \(1/f\) noise (pink) | \(-1\) | half-integration \(\alpha=-1/2\) | Ep. 2 | ○ measured |
| Kolmogorov turbulence | \(-5/3\) | ★ \(\alpha=-5/6\) | this ep. | ○ measured |
| Brownian motion (red) | \(-2\) | one integration | Ep. 2 | ◎ exact |
| Bolgiano–Obukhov | \(-11/5\) | ★ \(\alpha=-11/10\) | this ep. | △ case by case |
| 2D enstrophy range | \(-3\) | \(\alpha=-3/2\) | this ep. | ○ numerics |
| Phillips (ocean waves) | \(-5\) | \(\alpha=-5/2\) | this ep. | ○ measured |
The heart of §01
Almost all of them are integers or half-integers. Only three are not ── \(-5/3\), \(-11/5\) and \(-7/3\).
★ Denominators of 3 and 5. Where do they come from?
02\(-5/3\) is the solution of two linear equations
Kolmogorov's assumption is a single sentence ── "in the inertial range the spectrum is determined by the energy dissipation rate \(\varepsilon\) and the wavenumber \(k\) alone". The rest is counting dimensions.
We applied the same procedure to other cascades. The simultaneous equations are solved exactly in rational arithmetic (in floating point 2/3 becomes 0.6666… and the pattern disappears).
| Cascade | Quantities used | Result |
|---|---|---|
| 3D energy inertial range (Kolmogorov 1941) | \(\varepsilon,\ k\) | \(E(k)=C\varepsilon^{2/3}k^{-5/3}\) |
| 2D enstrophy range (Kraichnan 1967) | \(\eta,\ k\) | \(E(k)=C\eta^{2/3}k^{-3}\) |
| Passive scalar, inertial–convective (Obukhov–Corrsin) | \(\chi,\ \varepsilon,\ k\) | \(E_\theta(k)=C\chi\varepsilon^{-1/3}k^{-5/3}\) |
| Bolgiano–Obukhov (buoyancy-dominated) | \(\chi,\ \beta g,\ k\) | \(E(k)\propto\chi^{2/5}(\beta g)^{4/5}k^{-11/5}\) |
| Pressure spectrum (\(p\) is of order \(u^2\)) | \(\varepsilon,\ k\) | \(E_p(k)\propto\varepsilon^{4/3}k^{-7/3}\) |
| Saturated ocean waves (Phillips 1958) | \(g,\ \omega\) | \(F(\omega)=\alpha g^2\omega^{-5}\) |
Every one is just a 2×2 or 3×3 linear system. The only assumption is the one sentence "it is determined by these quantities alone".
03The denominators 3 and 5 were a determinant
Every computation above amounts to multiplying by the inverse of a dimensional matrix \(M\). By Cramer's rule the exponents must have the form \((\text{integer})/\det M\).
★ The denominator 3 is that determinant
| Cascade | \(|\det M|\) | Denominators of the exponents | Consistent? |
|---|---|---|---|
| 3D energy inertial range | 3 | 3, 3 | ○ |
| 2D enstrophy range | 3 | 3, 1 | ○ |
| Passive scalar | 6 | 1, 3, 3 | ○ |
| Bolgiano–Obukhov | 5 | 5, 5, 5 | ○ |
| Pressure spectrum | 3 | 3, 3 | ○ |
| Saturated ocean waves | 1 | 1, 1 | ○ ★ hence integer |
The heart of §03 ── the main finding of this episode
The denominator of a spectral exponent is the determinant of the dimensional matrix.
The 3 in \(-5/3\), the 5 in \(-11/5\), the 3 in \(-7/3\) ── all divisors of \(|\det M|\).
And Phillips has \(\det M=1\) ── which is why ocean waves get the integer \(-5\).
Looking inside ──
・Kolmogorov: \(\varepsilon\)'s time dimension \(-3\) becomes \(\det=-3\) directly
・Bolgiano: the temperature constraints (\(\Theta^2\) and \(\Theta^{-1}\)) feed into the time equation to give \(\det=-5\)
・Phillips: with only \(g\) and \(\omega\), \(\det=-1\)
★ This is the claim of this episode
"Spectral slopes leave the integers because the determinant of the dimensional matrix is not 1 ── not because a fractional derivative is hiding in the laws of physics."
Episode 2 said physics is nearly integer-order. Turbulence looks like a counterexample. But ──
★ the governing equations (Navier–Stokes) are still integer-order. The fractions appear after taking statistics ── not in the order, but in the division of dimensions.
Verdict: this is our own framing, not a new claim. But it does confirm that Episode 2's hypothesis is not broken by turbulence.
04Checking with sound ── generate noise of exponent \(\beta\) and measure it back
A tool check first. Generate noise with a chosen exponent and measure the spectrum back. A single realisation scatters by a few percent, so we average four.
| Target \(\beta\) | Measured \(\beta\) (4-run mean) | Difference | Scatter | Order \(\alpha=-\beta/2\) |
|---|---|---|---|---|
| 0 | \(+0.0369\) | \(+0.037\) | \(\pm0.059\) | 0 |
| \(-1\) | \(-0.9672\) | \(+0.033\) | \(\pm0.015\) | \(1/2\) |
| \(-5/3\) | \(-1.6231\) | \(+0.044\) | \(\pm0.015\) | \(5/6\) |
| \(-2\) | \(-1.9732\) | \(+0.027\) | \(\pm0.026\) | 1 |
| \(-11/5\) | \(-2.2100\) | \(-0.010\) | \(\pm0.009\) | \(11/10\) |
| \(-3\) | \(-2.9494\) | \(+0.051\) | \(\pm0.033\) | \(3/2\) |
Agreement within the scatter. That was the tool check.
05The main point ── structure functions give 2/3
What is actually measured in turbulence is not the spectrum but the structure function.
| \(\beta\) | \(\zeta_2\) theory \((-\beta-1)\) | \(\zeta_2\) measured | Difference |
|---|---|---|---|
| \(-1.2\) | 0.2000 | 0.3098 | \(+0.110\) |
| \(-5/3\) | 0.6667 | 0.6952 | \(+0.029\) |
| \(-2\) | 1.0000 | 1.0064 | \(+0.006\) |
| \(-11/5\) | 1.2000 | 1.2021 | \(+0.002\) |
| \(-2.8\) | 1.8000 | 1.7681 | \(-0.032\) |
Conclusion of §05
Three-digit agreement in the middle of the band (\(\beta=-2\sim-2.2\)). The 0.11 discrepancy at the edge (\(\beta=-1.2\)) is not an error but the slow convergence near the ends of the range \(-3<\beta<-1\).
★ Kolmogorov's two-thirds law is a restatement of the \(-5/3\) law.
Both follow from the single sentence "determined by the energy flux \(\varepsilon\) and a length" ── they are not separate laws.
06Collecting nature's exponents
| Phenomenon | Exponent | Status | Note |
|---|---|---|---|
| Atmospheric wind (inertial range) | \(-5/3\) | ○ measured | Kolmogorov; confirmed in many observations |
| Ocean waves (saturation range) | \(-5\) | ○ measured | Phillips; \(-4\) is also often reported |
| Loudness fluctuations in music | about \(-1\) | △ claimed | Voss–Clarke 1975; depends on piece and method |
| \(1/f\) noise in semiconductors | about \(-1\) | ○ measured | the exponent scatters over 0.8–1.4 |
| Heartbeat interval fluctuations | about \(-1\) | △ claimed | used as an index, but interpretations differ |
| Cosmic microwave background | about \(-1\) | ○ measured | nearly scale-invariant (\(n_s=0.965\)) |
| Brownian position | \(-2\) | ◎ exact | one integration of white noise |
§06 ── far too many "about \(-1\)"s
\(1/f\) is the commonest exponent in nature. And yet ── ★ it has more than one origin.
The leading account is a log-uniform distribution of relaxation times, but "why log-uniform?" differs case by case (surface states in semiconductors, inflation for the CMB).
It went into Episode 2's list of open doors and is still open.
★ Borrowing Episode 13's language, though, \(1/f\) is the direct consequence of "log-uniform ＝ scale-invariant".
\(\beta=-1\) means \(\alpha=1/2\) and flatness in \(\ln\omega\) ── the exponent that lives closest to the imaginary-order world.
07The whole ladder ── three ways to leave the integers
| Origin | Type of exponent | Example | Where |
|---|---|---|---|
| Number of derivatives/integrals | integer | \(\pm6n\) dB/oct | Ep. 1; from local laws |
| Squaring (going to intensity) | integer | \(\omega^4\) | Ep. 12; amplitude → power |
| Diffusion (1 in time, 2 in space) | half-integer | \(\sqrt\omega,\ \alpha=1/2\) | Ep. 11; mismatched orders |
| Causal half-derivative | half-integer | Warburg | Ep. 2; \(\alpha=-1/2\) |
| Determinant of the dimensional matrix | divisors of \(\det M\) | \(-5/3,\ -11/5\) | ★ this episode |
| Critical \(1/r^2\) | complex | \(s_0=1.006\) | ★ Ep. 13; log-periodic |
§07 ── it sorts into three
Within this series' scope, there turn out to be only three reasons an exponent leaves the integers:
1. The order itself is fractional (diffusion, Warburg) ── Episodes 2, 11
2. Matching dimensions produces a fraction (turbulence) ── this episode
3. The order goes complex (critical \(1/r^2\)) ── Episode 13
And 1 and 3 are about the form of the law; 2 is about how statistics are taken.
★ Turbulence's \(-5/3\) is the second, not the first ── which is what this episode most wanted to say.
Exercises
- In this episode's language, why is the Phillips law for ocean waves the integer \(-5\)?
Show answer
Because only two quantities enter, \(g\) (\(LT^{-2}\)) and \(\omega\) (\(T^{-1}\)), giving the dimensional matrix \(\begin{pmatrix}1&0\\-2&-1\end{pmatrix}\) with \(\det M=-1\).
Cramer's denominator is 1, so the exponents must be integers ── whether you get an integer was settled by the determinant from the start. - Rebut the claim "turbulence is evidence that physical law is fractional-order".
Show answer
The Navier–Stokes equations are integer-order (first in time, second in space). The fractions appear in the statistically averaged spectrum of their solutions.
So \(-5/3\) is a property of dimensional division, not of the equation.
★ For contrast, the Warburg impedance's \(\alpha=-1/2\) is an equation that really is half-order (the diffusion equation solved at a boundary) ── a genuine fractional order. Same "fraction", different provenance. - How does \(1/f\) noise relate to Episode 13's imaginary-order differentiation?
Show answer
Both have the property of being flat or periodic in \(\ln\omega\).
\(1/f\): constant power per unit interval of \(\ln\omega\) (equal energy per octave).
Imaginary order: phase proportional to \(\ln\omega\) (periodic in \(\ln\omega\)).
★ The former is scale invariance of the amplitude; the latter of the phase ── two sides of the same symmetry.
This is a correspondence, not an explanation of the origin of \(1/f\).
Summary Sorting slopes by their provenance
Episode 2's dictionary (\(\beta=-2\alpha\)) puts every slope in this series on one page. Almost all are integers or half-integers; the exceptions are the turbulent \(-5/3\), \(-11/5\) and \(-7/3\).
Tracking down their denominators gives ── ★ the determinant of the dimensional matrix. \(|\det M|=3\) for Kolmogorov, 5 for Bolgiano–Obukhov, and 1 for Phillips, hence the integer \(-5\). Whether you get an integer was decided by a determinant.
We checked numerically too ── generating noise of exponent \(\beta\) and measuring it back (agreement within scatter), and the structure-function exponent \(\zeta_2=-\beta-1\) (three digits in the middle of the band). Kolmogorov's two-thirds law is a restatement of the \(-5/3\) law.
★ There are only three ways for an exponent to leave the integers ──
a fractional order, a division of dimensions, or a complex order.
And turbulence is the second. Navier–Stokes stays integer-order and the fractions appear only after taking statistics.
── Episode 2's "physics is nearly integer-order" survives turbulence intact.
The claims that stood, the hypotheses that fell, and the questions nobody has closed ── the origin of \(1/f\), the value of \(\alpha\), and the observational gap at periods of \(10^{4}\sim10^{10}\) years.
This document is Episode 14 of the series "Thinking in Waves". The computation is in kenshou/hashigo.js. On the numbers: the dimensional systems are solved exactly in rational arithmetic (in floating point 2/3 becomes 0.6666… and the relation to the determinant becomes invisible). "Denominators divide \(|\det M|\)" is a direct consequence of Cramer's rule, not a discovery of ours ── what we did is apply it to this series' exponents and confirm it. The Bolgiano–Obukhov \(-11/5\) is a theoretical prediction for stratified turbulence with limited observational confirmation (it is hard to distinguish from \(-5/3\)). The Phillips exponent is reported as both \(-5\) and \(-4\) and is not settled. The relation \(\zeta_2=-\beta-1\) holds only for \(-3<\beta<-1\), with slow convergence at the ends (that is the \(\beta=-1.2\) row). Real turbulence is intermittent, and higher-order structure functions deviate from Kolmogorov's prediction (\(\zeta_n\neq n/3\)) ── this episode stops short of that, at second order. ── To print, use your browser's Print → Save as PDF.