Chapter 12 · Scale-dependent error growth and the intrinsic limit

Part IV — Many scales, many degrees of freedom Knob: $\alpha$, the number of resolved octaves, where the error is seeded live notebook

Overview

Every predictability estimate so far in this book has had the same shape. An error of size $\delta_0$ grows at rate $\lambda_1$ , so the useful forecast lasts $\lambda_1^{-1}\ln(\delta_{\rm sat}/\delta_0)$ — and as $\delta_0 \to 0$ that grows without bound. Chapter 8 put a number on the exchange rate: $\ln 10/\lambda_1 = 3.8$ days per decimal digit of initial precision, and the next digit buys 3.8 days again. Slow, but unlimited.

Lorenz (1969) argued that a system with a spectrum of scales behaves differently, and the conclusion is far more troubling than the butterfly effect. If small scales grow faster than large ones, then error introduced at the smallest scale reaches the largest in a finite time, no matter how small the scale you start from. Improving the resolution of an observing system then stops buying lead time — not because the improvement is slow, but because the limit is bounded.

This chapter separates the two cases, and the whole thing turns on one exponent.

The model

Discretise into octave bands: band $n$ has scale $L\,2^{-n}$ and growth rate

\lambda_n = \lambda_0\,2^{\alpha n},

so $\alpha$ measures how much faster small scales grow. Kolmogorov scaling gives the eddy turnover time $\tau \sim \varepsilon^{-1/3}\ell^{2/3}$ , hence $\alpha = 2/3$ . Lorenz 63 and Lorenz 96 have $\alpha = 0$ : one growth rate, scale-independent.

Each band’s error is measured against its own saturation level and grows logistically, forced by the band one octave smaller:

\frac{de_n}{dt} = \lambda_n\bigl(e_n + \kappa e_{n+1}\bigr)\bigl(1 - e_n\bigr).

With a single band the second term vanishes and this is exactly the logistic model of chapter 9 — a reduction the tests check to $10^{-8}$ .

The critical point is what “improving the observations” means here. An observing system has a resolution, and about scales finer than it we know nothing at all, so the error there starts at saturation. Adding a band therefore represents seeing one octave further down, not reducing an amplitude — and those two kinds of improvement turn out to behave completely differently.

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.

Four results

The limit is finite if and only if $\alpha > 0$ . Seeding the finest resolved band at saturation and adding octaves, the time for the largest scale to be contaminated converges to 2.3035 at $\alpha = 1/3$ , 1.4466 at $\alpha = 2/3$ and 1.1220 at $\alpha = 1$ — while at $\alpha = 0$ it grows without bound, at a settled 0.281 per octave out to 128 bands. So a finite predictability horizon follows from small scales growing faster than large ones, and from nothing else.

The increments die as $2^{-2\alpha}$ , not $2^{-\alpha}$ . Measured per-octave increment ratios are 0.630, 0.397 and 0.250 against $2^{-2/3}, 2^{-4/3}, 2^{-2}$ = 0.630, 0.397, 0.250 — exact to three decimals. That is the square of what the naive “sum the band timescales” argument predicts, which is also why that argument gets the convergence right and the constant wrong: it gives 2.70 at $\alpha = 2/3$ against the measured 1.4466, because the bands overlap in time rather than waiting for one another.

Where you improve matters far more than by how much. At $\alpha = 2/3$ , reducing the initial error at the finest band by sixteen orders of magnitude moves the horizon from 1.4453 to 1.4795 — 2%. Reducing it at the coarsest band instead obeys the familiar law, $\ln 10/\lambda_0$ per decade, and keeps paying indefinitely. Small-scale accuracy is nearly worthless because small-scale error saturates almost immediately; large-scale accuracy is what buys time.

Two scales are not a cascade, and the two-scale Lorenz 96 shows how completely they are not. Coupling 8 slow variables to 256 fast ones ten times quicker gives a genuine dynamical upscale cascade: perturb only the fast variables and the slow error appears shortly after, growing at 2.96–2.99 per time unit — a rate set by the slow dynamics and independent of how small the fast perturbation was, across eight decades of it. That is the mechanism, cleanly.

But the finite limit is absent. Averaged over 32 base states, the return on initial accuracy is 0.145 time units per decade when the error is seeded in the fast variables against 0.148 when seeded in the slow ones — a 2% difference, so it makes essentially no difference where the error is put. The limit needs a spectrum, not two rungs of one, and this is worth stating plainly because two-scale Lorenz 96 is widely used and easily over-read.

The model does deliver one sharp warning of its own. The leading Lyapunov exponent of the coupled system is set by the fast subsystem: $\lambda_1 = 24.7$ per time unit, a doubling time of 0.028 time units, or 0.14 days at the conventional five days per unit. Read as “the” error-doubling time that would put weather predictability at a few hours. The slow variables — the ones a forecast is about — actually double every 0.232 time units, 1.16 days. So $\lambda_1$ overstates large-scale error growth by a factor of 8.3, and in a multiscale system it and the predictability of the large scales are different questions. That is the correction chapter 7’s single number needs.

The honest caveat

Whether the atmosphere actually has $\alpha > 0$ across the relevant range of scales is an empirical question, not a mathematical one, and it is the question the whole “two-week limit” rests on. The cascade model shows that if the spectrum of growth rates behaves as Kolmogorov scaling suggests, the limit is finite; it cannot show that it does. Measurements of atmospheric error growth as a function of scale, and the resulting estimates of the intrinsic limit, remain actively argued over [citation needed: on observational estimates of scale-dependent error growth]. Chapter 13 takes up what operational forecast errors actually do, and chapter 22 the difference between the horizon we have and the horizon we could have.

Exercises

Analytic. Show that $\sum_n \lambda_n^{-1}$ converges for $\alpha > 0$ and diverges for $\alpha = 0$ , and evaluate the sum. Then explain why the measured contamination time at $\alpha = 2/3$ is 1.4466 rather than the 2.70 that sum predicts.

Computational. Reproduce the $2^{-2\alpha}$ scaling of the per-octave increment for three values of $\alpha$ , and state what it implies about the number of octaves worth resolving.

Exploratory. Set $\alpha = 0$ and add octaves until the contamination time doubles. Now do the same at $\alpha = 2/3$ . Explain what an observing-system designer should conclude from the difference.