Chapter 23 · Boundary-forced predictability and the S2S window

Part VI — Predictability of the second kind: from S2S to climate Knob: forcing period and amplitude live notebook

Overview

Weather forecasts are useful for about a week. Seasonal forecasts of the coming winter are useful too — issued months ahead, when no trace of today’s weather can possibly survive. Both cannot be initial-value problems. What is the seasonal forecast actually using?

Chapter 1 measured the answer’s shape: information carried by today’s state decays to nothing while information carried by the forcing does not decay at all, and the two cross. That chapter held the forcing fixed. This one lets it move.

The model

Lorenz 63 with a slowly oscillating Rayleigh number, $\rho(t) = 28 + 6\sin(2\pi t/40)$ , with a period an order of magnitude longer than the trajectory’s own predictability time. That separation is what makes $\rho$ a boundary condition rather than part of the fast dynamics — the cheapest caricature of ENSO, of the seasonal cycle, or of any other slow driver.

The whole argument is one decomposition: measure each forecast distribution against two climatologies — the one pooled over all forcing phases, and the one conditioned on the phase at verification time.

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

Forecast information decays to a floor, not to zero. Against the pooled climatology it falls from 2.93 nats and settles at 0.146; the forcing phase alone is worth 0.120. At long lead the forecast has stopped being a forecast in the ordinary sense — it has become a statement about the forcing.

And the two contributions add up. The floor plus what remains of the initial condition, $0.120 + 0.027 = 0.147$ , against a measured plateau of 0.146 — agreement to 1 %. Relative entropy is not additive in general, so this is a check rather than an identity; that it holds this closely says the two sources are, here, close to independent. The forcing overtakes the initial state at lead 17.

Windows of opportunity are real. Resolved by the phase a forecast was launched at, the lead at which the initial-state signal is spent runs from 9 to 17 TU — a factor of 1.9. With fifteen launches per bin that could be sampling noise, so it is checked: splitting the launches into two independent halves and asking whether they agree about which phases are the predictable ones gives a correlation of +0.96.

Amplitude is what the forcing is worth; period mostly is not. Phase information rises monotonically with amplitude (0.0015 → 0.252 nats). Across the slow range, an eightfold change in period moves it by about 10 %. Once the forcing is slow enough for the system to equilibrate to whatever $\rho$ currently is, making it slower adds essentially nothing.

The predictability desert, and why it is not empty

Between the two regimes lies the well-known awkward gap: too far ahead for the initial state to help much, not far enough for the boundary signal to be the whole story. Here it runs from about lead 10 to lead 17.

The useful point is that the desert has a floor under it. Skill there is small but not zero, and what is left is boundary-forced — which is exactly why subseasonal forecasting is worth attempting rather than a category error. The gap is thin, not empty.

A result that came out backwards

At a forcing period of 2.5 TU the phase information rises to 0.167 nats — above every slow value. That is not the forcing becoming more informative. It is the decomposition breaking down.

With a period shorter than the trajectory’s own predictability time, “the phase” is no longer a slowly-varying boundary condition the state has forgotten about; it is strongly correlated with where the state is right now. Conditioning on it is then partly conditioning on the initial condition, and the split into first-kind and second-kind information stops meaning what it says.

The clean separation this chapter relies on is a property of the timescale gap, not of the mathematics, and it is worth knowing that it fails rather than assuming it holds. Real slow drivers — ENSO at three to seven years against synoptic weather at days — sit very far from that failure, which is precisely why the framing is useful for them.

Care taken

The zero-amplitude control matters: with no forcing the phase label is meaningless, so the phase-conditioned climatology must equal the pooled one. Measured, boundary information collapses to 0.0015 nats — the estimator’s floor — against 0.120 at the working amplitude, a factor of 80. Without that control the whole decomposition could have been measuring an artefact of the binning.

The windows-of-opportunity claim is not simply a restatement of which phases are most informative: across the eight bins the two correlate at +0.66, which with eight points is suggestive rather than established. They are different questions — one asks what the boundary condition is worth, the other how long the initial condition survives.

Exercises

The two curves in section 2 start at nearly the same value and separate. At what lead do they first differ by more than 10 %, and what does that lead correspond to?
The floor-plus-residual check agrees to about one per cent. Construct a case where it would fail badly — what would have to be true of the two information sources?
The phases with the most informative forcing are not exactly the phases with the longest-lasting forecasts. Which of the two would you rather know about, as a user deciding whether to act on a forecast?
Section 4 shows the decomposition failing at short forcing period. Estimate, from chapter 1’s numbers, the shortest period for which you would trust it.