Chapter 25 · Climate prediction and projection

Overview

Nobody claims to know the weather on a particular day in 2080. Everybody is willing to say the day will be warmer than one in 1980. Both statements are made about the same system, by the same models, and they are not in tension — but saying exactly why they are not requires being precise about which question is being answered. When the trajectory is unpredictable, what is left that can be predicted?

The model

Lorenz 63 with a Rayleigh number that ramps: $\rho(t) = \rho_0 + \gamma t$. Everything else is untouched, so the chaos is the same chaos and only the forcing is new. The zero-rate limit is asserted as a bitwise identity with the unforced system — the ramped right-hand side is grouped term-for-term with lorenz63 so that $\gamma = 0$ reproduces it exactly, not to tolerance.

Two questions of one forecast

One ensemble of 200 members from a tight blob, measured two ways.

Asked the weather question — how far is an individual member from the truth? — it saturates at 1.48 climatological spreads, the $\sqrt2$ of two independent draws from the same distribution, and every member is thereafter worthless.

Asked the climate question — how well does the ensemble’s windowed mean match the truth’s? — it tracks to 2.8 % of a climatological spread, and keeps doing so at leads where the individual members have long since decorrelated.

Both curves come from the same integration. Predictability did not partly survive; a different functional of the same forecast was never unpredictable in the first place.

(The individual-error curve starts near 1 rather than 0 because everything here is averaged over a 20-time-unit window — climate is a time average — so the window centred on lead zero already spans ten time units of error growth.)

The variability does not shrink

The standard decomposition: two ensembles from the same start states, one ramped and one not. The difference of the means is the forced response; the control’s spread is internal variability.

Over the run the response grows from 0.48 to 15.4. Internal variability stays between 8.574 and 8.639 — a spread of 0.8 %, flat to the eye.

So the signal-to-noise ratio rises because the signal outgrows a noise that stays put, not because the system becomes quieter under forcing. The shaded band in the figure never narrows, and that is the whole reason a projection can be confident about a mean and permanently silent about any particular year.

Two times of emergence, differing by $\sqrt K$

“Time of emergence” is when the signal becomes detectable. Against whose noise?

Both are legitimate measurements. The ensemble number is the smaller and more flattering one, and it is the wrong answer to the question most people are asking, which is about the one realisation that actually happened. A statement that does not say which it means is not a statement.

The two scalings are laws, not fits. Signal grows as $\gamma t$ and ensemble noise falls as $\sigma/\sqrt K$, so emergence should go as $1/\gamma$ and as $1/\sqrt K$:

• $\mathrm{ToE} \times \gamma$ = 1.20, 1.22, 1.21, 1.24 — constant to 3 % across a factor of eight in rate;
• $\mathrm{ToE} \times \sqrt K$ = 325, 327, 344, 342, 341, 355 from $K = 10$ upward.

The dashed lines in the figure are those laws with one constant, not regressions through the points. The $\sqrt K$ law breaks at $K = 2$ and $5$, and visibly: the measured emergence there is later than the law and not even monotone in $K$. That is not a failure of the law but of the measurement — with a handful of members the noise estimate is itself made from a handful of members, and the crossing time of a ragged S/N curve is one noisy sample.

Initialisation buys a projection nothing

Chapter 24 found that initialising a slow component was worth about eight time units. Here: two ensembles from entirely different sets of climatological start states, sharing nothing, each with its own control.

Their forced responses differ by at most 0.093 — 1 % of one internal-variability standard deviation — over the whole run. Plotted on an axis spanning a full $\pm 1$ sd, the difference is a flat line at zero.

That is not a defect; it is the definition:

It also explains a practical asymmetry. A decadal system needs an observing system, an assimilation system and a re-forecast archive, and chapter 24 showed how each can go wrong. A projection needs none of them, and pays for that by being unable to tell you about any particular decade.

Exercises

1. Section 1’s two panels use the same forecast. Which would change if the ensemble had 20 members instead of 200, and which would not?
2. Internal variability stays flat here. Name a physical mechanism that would make it grow under forcing, and say what that would do to the emergence time.
3. Single-realisation emergence at S/N > 2 never arrives within the run. Using the $1/\gamma$ law, estimate the rate at which it would arrive by $t = 200$.
4. Initialisation is worthless for the forced response. At what lead does it stop being worth anything — and how would you measure that crossover with chapter 24’s machinery?

Further reading

• Hawkins & Sutton (2009), on the sources of uncertainty in climate projections [citation needed]
• Lorenz (1975), on predictability of the second kind [citation needed]
• Deser et al., on internal variability and large ensembles [citation needed]
• Palmer & Hagedorn (2006), Predictability of Weather and Climate [citation needed: chapter]