Chapter 14 · From chaos to turbulence

Part IV — Many scales, many degrees of freedom Knob: resolution, error-seeding scale live notebook

Overview

Chapter 12 established that whether the predictability horizon is bounded depends on one exponent, $\alpha$ , describing how much faster small scales grow than large ones — finite for $\alpha > 0$ , unbounded for $\alpha = 0$ . But it postulated its octave bands. A real fluid comes with an energy spectrum instead. So what sets $\alpha$ in an actual flow?

If $E(k) \sim k^{-p}$ , an eddy of size $1/k$ has $u_k^2 \sim kE(k)$ and turnover time $\tau \sim 1/(k u_k)$ , so

\tau(k) \sim \bigl[k^3 E(k)\bigr]^{-1/2} \sim k^{(p-3)/2} \qquad\Longrightarrow\qquad \alpha = \frac{3-p}{2}

and the two turbulent cases fall on opposite sides of chapter 12’s boundary:

	$p$	$\alpha$	chapter 12’s verdict
three-dimensional (Kolmogorov)	$5/3$	$2/3$	finite: horizon 1.4466
two-dimensional (enstrophy range)	$3$	$0$	unbounded: 0.281 per octave

The model

chaoslib.turbulence is a two-dimensional pseudospectral solver for Navier–Stokes in vorticity form, dealiased by the two-thirds rule. Its warrant is that two-dimensional Euler conserves two quantities exactly: after 800 inviscid steps energy has drifted $1.9\times10^{-7}$ and enstrophy $1.8\times10^{-6}$ .

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

What the chapter measures

A fluid differs from a lattice by range, not by having no characteristic scale. Both spectra have a peak — a tempting claim to the contrary is simply not what the figure shows. The fluid’s falls 3.55 decades over 3.4 octaves above its peak; Lorenz 96’s falls 0.71 over 1.2. And the fluid’s peak moves: from $k=9$ to $k=4$ over 15 time units, energy travelling upscale as like-signed vortices merge, with enstrophy falling from 1.000 to 0.315 while energy falls only 33%. That selective decay — dissipation acts as $k^2$ , energy weights modes by $1/k^2$ — is what permits the upscale transfer. Lorenz 96’s peak cannot move: chapter 11 derived it from a linear instability with $m^*=8$ fixed by the dispersion relation, and a system whose scale is pinned by its own linear physics cannot have a gradient of growth rates across scales.

There is no inertial range at any affordable resolution, and the chapter says so. The widest stretch of spectrum whose local slope stays within 0.4 of $-3$ is 0.00, 0.04 and 0.05 octaves at $64^2$ , $128^2$ and $256^2$ . Quadrupling the grid does not help. A fit window chosen after looking at the plot gives $p = 3.011$ over $k\in[6,30]$ — and $1.53$ over $[5,25]$ , or $5.52$ over $[10,40]$ , which is what makes the 3.011 worthless. This is the discipline of chapter 8 applied to a spectrum: the local slope shows a plateau if there is a power law, and here there is not. A decade of inertial range needs $N \gtrsim 1024$ with sustained forcing, which is outside what a browser runs.

The cascade, though, is real. Error placed in a single wavenumber shell moves upscale monotonically: the band $k\in[8,18)$ goes from 0.000 to 0.536 of the total error while the seeded band falls from 1.000 to 0.394, and the error-weighted mean wavenumber falls throughout. The total amplitude barely changes, because shell 25 at $96^2$ sits in the dissipation range — the redistribution is what chapter 12’s argument needs, and it is what is shown.

One input borrowed, and said so

The relation $\alpha = (3-p)/2$ is algebra, not a measurement: given this estimate of $\tau(k)$ , its slope against $k$ is fixed by the slope of the spectrum it was computed from, so agreement between them is a tautology. What this chapter measures is the cascade; what it takes from theory and observation is the exponent $p$ .

Which is the atmosphere?

Both, at different scales. At synoptic scales the atmosphere is strongly stratified and rotating, quasi-two-dimensional, with an observed spectrum near $k^{-3}$ — giving $\alpha \approx 0$ and no intrinsic limit. Below a few hundred kilometres the spectrum shallows toward $k^{-5/3}$ , giving $\alpha \approx 2/3$ and a finite one [citation needed: on the observed atmospheric spectral transition].

So chapter 12’s question has no single answer for the atmosphere: one answer for the scales carrying most of the energy, another for those carrying most of the enstrophy, and the intrinsic limit depends on how strongly the second contaminates the first. That is why the two-week figure has been argued over for fifty years, and why chapter 12’s caveat is the honest position rather than a hedge.

Exercises

Analytic. Derive $\tau(k) \sim [k^3E(k)]^{-1/2}$ from $u_k^2 \sim kE(k)$ , obtain $\alpha = (3-p)/2$ , and evaluate it for $p = 1, 5/3, 3, 5$ .

Computational. Step through the three resolutions watching the local slope. At which wavenumbers is it near $-3$ ? Then satisfy yourself that a fitted line through a chosen window would have looked convincing.

Exploratory. Section 4’s error barely amplifies because shell 25 is dissipative. Predict what changes at shell 8 and what does not.