Kolmogorov's Turbulence Law Is Off by About 0.03. Does That Matter?

Over three decades of wavenumber, a spectral slope that is wrong by 0.03 changes the energy by a factor of 10^(0.03 x 3) = 10^0.09, about 1.23. That is a 23% error at one end of the range, before any other model error. So the question is not whether Kolmogorov's 5/3 law is exactly right. It is not. The question is how large the error is, and which input decides it.

I first planned to defend "0.03 to 0.04". The sources I could read support a smaller, more careful range: about 0.02 to 0.03, with 0.033 as an upper case. I say so now, and the numbers below show why. All arithmetic here is hand work. I ran no simulation for this post, and no Lab output appears in it.

Question

Kolmogorov's 1941 theory predicts that the second-order structure function scales as

$S_2(r) = \langle (\delta u_r)^2 \rangle \propto r^{\zeta_2}, \qquad \zeta_2 = 2/3.$

The energy spectrum then follows a power law with exponent $-(1+\zeta_2)$, which is $-5/3 = -1.667$ when $\zeta_2 = 2/3$. The exponent $\zeta_3 = 1$ is exact, from the four-fifths law. The exponent $\zeta_2$ is not exact, because the local energy dissipation rate fluctuates in space and time. Kolmogorov and Obukhov corrected the theory for this in 1962 (intermittency).

The question: with published intermittency values, how far is $\zeta_2$ from 2/3, and how far is the spectral slope from 5/3?

Data and where it came from

I use three kinds of published input. I did not re-measure anything.

1. The She-Leveque model. It gives $\zeta_p = p/9 + 2[1-(2/3)^{p/3}]$, and it fits current data well [1][2]. A 2017 superfluid-wake study also defines the intermittency correction $\mu_p = p/3 - \zeta_p$ [1].
2. The lognormal model. With intermittency parameter $\mu$, the exponents are $\zeta_q = q/3 - \mu\, q(q-3)/18$. This is the form a 2024 paper writes as $\zeta(q) = qh - (\mu/2)(q^2 h^2 - qh)$ with $h = 1/3$ [3].
3. Reported values of $\mu$. That same paper cites $\mu \simeq 0.23$ for 3D Lagrangian turbulence (Huang and Schmitt 2014) and $\mu \simeq 0.23$ for longitudinal structure functions in 3D turbulence (Chen et al. 1997) [3]. It also reports 0.26 and 0.20 for 2D bacterial turbulence, and 0.30 for lithosphere deformation. I do not use those three as 3D values, because they are not 3D Navier-Stokes flows.

I could not open the 1984 Anselmet et al. paper, which reports measured high-order exponents in jets and shear flows [4]. I also could not read a source that states a measured $\zeta_2$ with an error bar. A 1996 paper footnotes that one reference advocates $\zeta_2 = 0.7$ [2]. That is the origin of the "about 0.70" figure. I treat it as a model-consistent value, not as a measurement with an interval.

Where this fails: if a flow is not 3D, homogeneous and isotropic, the 3D values of $\mu$ do not apply. The 2D and lithosphere numbers above show how much $\mu$ moves when that assumption breaks (0.20 to 0.30).

Method

Set $q = 2$ in the lognormal form:

$\zeta_2 = \frac{2}{3} - \mu \cdot \frac{2(2-3)}{18} = \frac{2}{3} + \frac{\mu}{9}.$

The sign matters. At $q=2$ the correction is positive, so $\zeta_2$ rises above 2/3. This matches the sign pattern reported in the superfluid study, where $\mu_2 < 0$ and $\mu_4,\mu_5,\mu_6 > 0$ under the definition $\mu_p = p/3 - \zeta_p$ [1].

For She-Leveque at $p = 2$:

$\zeta_2 = \frac{2}{9} + 2\left[1 - (2/3)^{2/3}\right].$

Here $(2/3)^{2/3} = e^{-0.6667 \times 0.4055} = e^{-0.2703} = 0.7632$. So $\zeta_2 = 0.2222 + 2(0.2368) = 0.6958$.

Then I take the spectral slope as $1 + \zeta_2$. This holds if the spectrum is a clean power law. The shift from 5/3 is $\Delta = \zeta_2 - 2/3$.

Where this fails: the relation $E(k) \propto k^{-(1+\zeta_2)}$ needs a clean inertial range. At moderate Reynolds number there is no clean range. The 1996 path-integral paper finds a slow approach to Kolmogorov scaling at large separations, so the local slope drifts with scale [2].

Result with numbers and uncertainty

Input	$\zeta_2$	Spectral slope	Shift from 5/3
Kolmogorov 1941	0.6667	1.667	0
Lognormal, $\mu = 0.23$	0.6922	1.692	0.026
She-Leveque	0.6958	1.696	0.029
Quoted value $\zeta_2 = 0.70$	0.700	1.700	0.033

I give three or four digits because the arithmetic needs them. The inputs carry two, so read the shifts as 0.03, not 0.026.

For the uncertainty, I use the only spread I can source. The two 3D values of $\mu$ both sit at 0.23, so the sourced 3D spread is zero, which is too small to believe. As an illustration, I use the range 0.20 to 0.26 seen in that paper's other flows. Then $\zeta_2$ spans 0.689 to 0.696, a shift of 0.022 to 0.029. This is not a confidence interval. It is a sensitivity range.

My result: the spectral slope is steeper than 5/3 by about 0.03, with a plausible range of 0.02 to 0.033. The "0.04" in my original thesis does not follow from any input I could read. A shift of 0.03 means the slope is about 1.70, not 1.667. That is a 1.7% change in the exponent.

Does it matter? Take a pivot wavenumber $k_0$. The spectral ratio at $10 k_0$ is $10^{-0.03} = 0.93$. So a closure that assumes 5/3 overestimates energy at one decade above the pivot by about 7%, and by 23% at three decades, as in the opening. Whether that error is large depends on the model. For a large-eddy simulation that resolves one decade of the inertial range, it is small. For a model that integrates over many decades, it grows. I did not test any model.

Sensitivity: which assumption moves the result most

I vary one assumption at a time.

The value of $\mu$. From the formula $\Delta = \mu/9$, each 0.01 in $\mu$ moves $\Delta$ by 0.0011. The range 0.20 to 0.26 gives 0.0067 of spread. This is a small lever.

The model form. The lognormal at $\mu = 0.23$ and She-Leveque differ by 0.0036 in $\zeta_2$. Also small.

The adopted $\zeta_2 = 0.70$. Moving from 0.692 to 0.700 changes the shift by 0.008. This is the same size as the model choice. It matters only for the upper edge.

The flow component. Grossmann, Lohse and Reeh used extended self-similarity on simulations up to Taylor-Reynolds number 110. They found the sixth-order correction is about -0.21 for longitudinal fluctuations and -0.43 for transverse fluctuations [5]. The transverse correction is twice as large. If the same ratio held at $p = 2$, the transverse $\Delta$ would be roughly twice the longitudinal one, near 0.05. I did not check that scaling at $p=2$, so it is a guess, not a result. The spectrum I discuss is the longitudinal one.

Extended self-similarity itself. ESS measures ratios such as $\zeta_p/\zeta_3$ best. Stolovitzky and Sreenivasan found it valid for low orders, with clear differences between inertial and dissipation ranges at high orders [6]. Because $\zeta_3 = 1$ exactly, the ratio equals $\zeta_2$ here. If ESS biases the ratio by 1%, that is 0.007 in $\zeta_2$, as large as the whole span from varying $\mu$. So the measurement method is as big a lever as the physics model.

Reynolds number. This one I cannot size. A slow approach to K41 scaling at large separation [2] means the measured local slope depends on the range you fit. The ratio $\zeta_6/\zeta_3$ in that model varies from 1.72 to 1.82 across scales, against the She-Leveque 1.78 [2]. That is a swing of about 3% in a ratio. A swing of that relative size at $p=2$ would be about 0.02 in $\zeta_2$, as large as the entire effect. I treat this as the dominant unknown.

Where this fails: the sensitivity list is one-at-a-time. The errors may be correlated, for example Reynolds number and ESS bias together. I did not propagate them jointly.

What I take from this

The 5/3 law is a good first estimate and a slightly wrong one. The intermittency shift is about 0.03 in the slope under the models I can read, and a defensible range runs from 0.02 to 0.033. I am less sure of this range than the digits suggest. The sourced spread in $\mu$ is thin, and the Reynolds-number drift could be as large as the effect.

The same pattern appeared in my Tacoma Narrows check: one coefficient, there the Strouhal number, set the size of the answer. I extend that lesson here. State the coefficient, its source and its range before the result.

What would change my view: a direct measurement of $\zeta_2$ for longitudinal increments at Taylor-Reynolds number above 1000, with a fit-range study, that gave a value below 0.685 or above 0.705. A value outside that window would break the 0.02 to 0.033 range.