Chapter 3 · The hierarchy of models

Part I — What predictability means Knob: which rung, initial-error amplitude live notebook

Overview

This book spends most of its length on systems with three variables, or forty, or one. An operational forecast model has of order $10^9$ . So the question the whole book depends on: what exactly is a small model telling you about a large one?

“It builds intuition” is not a claim that can be checked. The claim this book actually makes is narrower and testable:

Dimensionless relationships transfer between rungs. Dimensional constants do not. And some properties exist at no rung below a particular structure, so they cannot be learned by going smaller at all.

Each is measured here. The first two hold cleanly. The third is the one that matters, because it says where the method fails — and the property it fails on is the most famous claim in the subject.

The rungs

rung	variables	time	what it is for
logistic map	1	discrete	the route to chaos, and universality (ch. 5)
Lorenz 63	3	continuous	sensitive dependence, the strange attractor (ch. 6)
Lorenz 96 (12)	12	continuous	spatiotemporal chaos, small enough to see (ch. 11)
Lorenz 96 (40)	40	continuous	the standard assimilation testbed (ch. 19)

They have almost nothing in common — a map on the unit interval, a flow on a fractal attractor of dimension 2.06, a flow on a 27-dimensional one — and their leading exponents are not in the same units. What they all have is an error that grows and then saturates, so there is one quantity to compare. The machinery is checked against an exact identity at every rung: the exponents sum to the divergence of the flow, $\sum_i\lambda_i = \mathrm{tr}\,\mathbf{J}$ , reproduced to parts in $10^5$ .

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 transfers: one law, no fitted constant

If error grows as $\delta_0 e^{\lambda_1 t}$ until it saturates at $\delta_\infty$ , the time to reach a fraction $f$ of saturation satisfies

\lambda_1 T = \ln \frac{f\,\delta_\infty}{\delta_0} .

That form is dimensionless on both sides, has no fitted constant, and each rung supplies its own $\lambda_1$ and $\delta_\infty$ . Measured at $f = 0.5$ over four rungs and four initial-error amplitudes spanning six decades:

$\lambda_1 T$	$d=8$	$d=6$	$d=4$	$d=2$
logistic map	17.07	12.55	8.03	4.02
Lorenz 63	16.86	12.69	7.86	3.63
Lorenz 96 (12)	17.26	12.59	8.04	3.55
Lorenz 96 (40)	17.22	12.96	8.74	4.07
the law predicts	17.73	13.12	8.52	3.91

Worst departure anywhere: 9.2 %. A one-variable discrete map and a forty-variable continuous flow, with leading exponents differing by a factor of three and not even in comparable units, agree with a constant-free law to within a tenth.

That licenses using Lorenz 63 to reason about how forecast error responds to better initial conditions — the shape of that dependence is the same at every rung, and each decade of accuracy buys $\ln 10/\lambda_1$ of lead time, which is what chapter 20 measures on a cycling assimilation system. It licenses transferring no number at all: $\lambda_1$ is 0.50 per iteration for the map against 1.63 per time unit for Lorenz 96.

The departures are not zero for two identifiable reasons, both pushing the same way: a random perturbation is not aligned with the leading Lyapunov vector and spends a while becoming one, and the local growth rate varies around the attractor by a factor of several (chapter 7), so averaging over 48 base states does not average over everything.

What does not transfer: how many directions the error lives in

Section 2 could be read too generously, as though getting $\lambda_1$ right were enough. Lorenz 96 at $F=8$ is the same system at every $N$ — same equations, same forcing, a longer chain — so climb it slowly.

Across a factor of five in $N$ , $\lambda_1$ varies by 13 %: it is a property of the local dynamics and does not care how long the chain is. Meanwhile the number of unstable directions goes 2 → 13, growing at 0.34 per variable, and the attractor dimension at 0.68 per variable. Both are extensive.

So a small model can have exactly the right growth rate and be wrong about the size of the problem by orders of magnitude — and the extensive quantity is the one that sets operational cost. An ensemble must span the unstable subspace, so its size is set by the unstable dimension and not by $\lambda_1$ ; that is why chapter 19 finds localisation compulsory in a large system and why its rank problem does not appear in Lorenz 63 at all. The observation count scales the same way, and so does the number of singular vectors that matter (chapter 16).

Lorenz 63, with one unstable direction, cannot pose those questions. It does not answer them wrongly — it does not contain them.

Where the ladder breaks

The third category is worse than the second: a property present at no rung below a particular structure, so going smaller cannot teach it.

Lorenz (1969) argued the atmosphere has a finite predictability limit — a horizon better observations cannot push back. The argument needs a spectrum of scales. But the law above says the horizon grows without bound as $\delta_0 \to 0$ , every decade buying another $\ln 10/\lambda_1$ for ever. Both cannot be right about one system.

Measured on 24 octave bands with band growth rates scaling as $\ell^{-\alpha}$ , error seeded in the smallest band, over ten decades of initial accuracy:

$\alpha = 0$ (every single-scale system in this book): the horizon gains 7.29 time units and is still climbing;
$\alpha = 2/3$ (Kolmogorov, the value Lorenz’s argument needs): it gains 0.001, against a limit of 1.45 that the first decade already reached.

So the finite predictability limit is a property no system in this book has, except the cascade built to have it. The logistic map does not. Lorenz 63 does not. Lorenz 96 does not. Two-scale Lorenz 96, which looks as though it should, does not either: chapter 12 measures its horizon gain at 0.145 time units per decade with error seeded in the fast variables against 0.148 in the slow ones — a 2 % difference, so it makes essentially no difference where the error is put, which a real upscale cascade would never permit. Two rungs of scale separation are not a spectrum.

This is the sharpest limitation on everything else here, and it cuts in a specific direction. Chapter 20’s logarithmic return and chapter 22’s day-per-decade skill record are both correct about their systems, and neither can settle whether the real atmosphere’s horizon is bounded — because that depends on structure the low-order models lack.

Exercises

At which rung and which initial-error amplitude is the departure from the law largest, and does its sign tell you which of the two spoiling effects dominates?
The unstable dimension grows at 0.34 per variable. Estimate it for a model with $10^7$ variables, and say what that implies for an ensemble anyone can afford.
The law and the cascade disagree about the limit as $\delta_0 \to 0$ , and both are measured here. Explain precisely which assumption of the law fails in the cascade.
Sort a result from a later chapter into the three categories — chapter 9’s saturation, chapter 11’s $m^* = 8$ , chapter 27’s Airy constant. Which transfer?