Chapter 26 · Earth system prediction

Part VI — Predictability of the second kind Knob: reservoir memory, feedback strength live notebook

Overview

Chapter 24 initialised one slow variable and asked what that bought. Chapter 25 ramped a forcing and asked what survived. Both took the forcing as given — prescribed from outside, known exactly, identical in every member. An Earth system model does not get that. Emissions are prescribed; concentrations are not, because the sinks that remove carbon are themselves part of the climate. So the forcing is a state variable with a feedback loop through it, and predicting the forcing is part of the forecast.

Three consequences, each measured here: there is not one slow component but a hierarchy, and which one you ask about decides what “predictable” means; the response to given emissions is uncertain in a way the response to given forcing is not; and the parameters governing that uncertainty are not observable — which is what emergent constraints exist to work around.

The model

Chapter 24’s coupling, extended: Lorenz 63 as the weather, four reservoirs spanning a factor of 64 in memory, and a carbon reservoir that emissions fill and a climate-dependent sink drains.

\dot S_k = \frac{-S_k + \lambda(z - z_{\rm ref})}{T_k}, \qquad \dot C = E(t) - \frac{C}{\tau_0\,(1 + \alpha S_1)}, \qquad \rho_{\rm eff} = \rho_{\rm ref} + \kappa_C\,C .

The reservoirs are driven but passive, deliberately. Chapter 24 measured a wall in this model family — loop gain and feedback amplitude both scale as $\lambda\kappa$ , so any reservoir coupling strong enough to move the atmosphere has already destroyed the reservoir’s memory. Routing the only feedback through carbon steps around it, because the carbon loop’s gain is set by $\alpha$ and its amplitude by $\kappa_C$ , two independent parameters.

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.

Memory is a parameter; amplitude is a consequence

Undriven, each reservoir decays as $e^{-t/T_k}$ exactly — a test asserts it to $10^{-10}$ . So the memory is not a discovery. What Hasselmann’s argument actually predicts is the amplitude: a reservoir integrating white weather has $\sigma_S = \sigma_z\sqrt{\tau_{\rm int}/T}$ , so $\sigma_S\sqrt{T}$ should be constant.

The spread falls from 0.954 to 0.0588 across the hierarchy, very nearly as $1/\sqrt T$ — the trade-off at the heart of the subject, since a long-memory component has a lot of predictability and very little amplitude. But the law is only approached: measured amplitudes exceed it by factors of 2.23, 1.55, 1.19, 1.10 at $T = 2, 8, 32, 128$ , converging towards 1 as the reservoir slows.

The reason is the approximation, not the model. Hasselmann’s result needs the driving to look white on the reservoir’s own timescale, and Lorenz 63’s $z$ decorrelates in a couple of tenths of a time unit — negligible against $T = 128$ , not against $T = 2$ . That matters practically: a lag-1 autocorrelation fitted to the fastest reservoir returns a memory three times too short, because it measures the direct un-integrated response rather than the Hasselmann part. Which is why the memory below is measured from forecasts instead.

The useful lead is $1.141\,T$ , and essentially nothing else

Two hundred forecast cases, each run twice from the same weather analysis — once with the reservoirs initialised (analysis error 30 % of climatological spread), once with them drawn from climatology.

The algebra is exact. For an Ornstein–Uhlenbeck reservoir, forecast and truth share their predictable part up to $\epsilon$ , and their unpredictable parts are independent:

\operatorname{var}\varepsilon_{\rm init}(\ell) = \epsilon^2 e^{-2\ell/T} + 2\sigma^2\!\left(1 - e^{-2\ell/T}\right), \qquad \operatorname{var}\varepsilon_{\rm uninit} = 2\sigma^2 ,

so the advantage is a pure exponential $(2\sigma^2 - \epsilon^2)e^{-2\ell/T}$ and the lead at which initialised RMSE reaches 95 % of uninitialised is $\ell^\ast = -\tfrac{T}{2}\ln\!\big[2\sigma^2(1-f^2)/(2\sigma^2-\epsilon^2)\big]$ — strictly proportional to the memory. (The saturation is $2\sigma^2$ , not $\sigma^2$ ; getting that wrong shifts $\ell^\ast$ by a fifth of a memory time, which is how it was caught.)

Measured: the fitted decay rate over $2/T$ comes out 1.24, 0.87, 1.01, 0.89, and the useful lead over its prediction 1.32, 1.05, 1.04, 1.16. For the two middle reservoirs that is 4–5 % on a law with no fitted constant. The two departures are the two ends and both are identifiable: at $T=2$ the reservoir is not an Ornstein–Uhlenbeck process at all, and at $T=128$ the control run is only 150 memory times long, so the climatological spread it is scored against is itself uncertain.

The observing system barely matters. The coefficient depends on the analysis error only logarithmically — improving the analysis a hundredfold, from 30 % of the climatological spread to 0.3 %, moves it from 1.141 to 1.164. A 2 % gain. You cannot observe your way to decadal skill in a seasonal component; you can only find a slower one, and by the section above the slower ones have the smallest signals.

When the forcing is a state variable

More carbon raises the forcing, which warms the fast system, which warms $S_1$ , which weakens the sink, which leaves more carbon. A loop with gain $g = E\tau_0\,\alpha\,\kappa_C\,\lambda\,\mathrm{d}\langle z\rangle/\mathrm{d}\rho$ and an exactly solvable equilibrium, $C/C_0 = (1+\alpha S_1^{\rm base})/(1-g)$ .

For $g \le 0.5$ the measured amplification matches $1/(1-g)$ to 0.67 %. Beyond that it degrades to 13.5 % at $g = 0.80$ — and for a reason worth naming, because it is not the obvious one. The derivation linearises $\langle z\rangle$ in $\rho_{\rm eff}$ using $\mathrm{d}\langle z\rangle/\mathrm{d}\rho = 1.0007$ , measured over $\rho \in [26,40]$ ; at $g = 0.80$ the equilibrium has carried $\langle z\rangle$ to 57. The feedback formula fails first not because the feedback is strong but because the climate sensitivity it was built on is no longer the right number.

And past $g = 1$ the word runaway is wrong. The residence time grows linearly in $C$ , so the sink term approaches the constant $1/(\tau_0\alpha\kappa_C)$ and carbon accumulates at the fixed rate $E - 1/(\tau_0\alpha\kappa_C)$ for ever — measured at 0.96 of that asymptote at $t = 12{,}000$ and still approaching. It is a saturation, the sink ceasing to work, not a divergence. The two have different signatures and different remedies.

Three sources of spread, and their ranking depends on the question

With the forcing prescribed, a projection divides into forced response and internal variability, and that is the whole story. With emissions prescribed a third term appears, because two models given the same emissions realise different forcings. Decomposing a 216-member ensemble — 3 feedback strengths × 3 emissions scenarios × 24 start states:

reservoir	internal share at lead 600	falls below half at
$T = 2$	10.85 %	100 TU
$T = 8$	1.35 %	30 TU
$T = 32$	0.20 %	26 TU
$T = 128$	0.05 %	33 TU

For the fastest reservoir internal variability never stops mattering; for the slowest it is gone. Same ensemble, same forcing, same everything — only the question changed. A projection of a fast component is permanently a statement about a distribution, exactly as chapter 25 found, and no narrowing of parameters helps because most of the spread is not parametric. A projection of a slow component is a statement about a number whose uncertainty is almost entirely the scenario chosen and the feedback assumed. Those need different research programmes, and conflating them is how a genuine disagreement about parameters gets mistaken for irreducible noise, or the reverse.

An emergent constraint, measured exactly, that tells you nothing

If the feedback strength contributes a quarter of the long-lead spread and is not observable, can it be inferred from something that is? That is the logic of an emergent constraint, and the natural observable here is the real one — the sensitivity of the carbon growth rate to climate variability [citation needed: Cox et al. (2013)] — whose expectation follows in closed form:

\frac{\partial \dot C}{\partial S_1} = \frac{C\,\alpha}{\tau_0\,(1 + \alpha S_1)^2}.

The measurement is excellent. Across the ensemble the observable matches that expectation to 5.7 % at worst, with a within-record correlation between climate anomaly and carbon growth rate of 0.96–0.996.

It constrains nothing. Across a one-parameter ensemble the correlation between observable and response is only +0.37, and the standard constraint calculation — regression, prediction interval, observational error — reports a narrowing of −1 %. The arithmetic is not lying; there is no information in the predictor. The cause is in the formula: the observable goes as $C\alpha/(1+\alpha S_1)^2$ and $S_1$ grows faster than linearly in $\alpha$ , so the denominator eventually wins and the observable turns over at $\alpha \approx 0.070$ while the response keeps accelerating. A non-monotone predictor cannot be inverted, however precisely it is measured.

With a second parameter it is worse than uninformative. Letting the sink time vary as well — every member still with a genuine equilibrium — the correlation with the response becomes −0.44, the wrong sign, and the correlation with the amplification factor $1/(1-g)$ , the quantity the constraint nominally targets, is −0.005. Zero.

Concretely: two members 1.6 % apart in the observable differ by a factor of 6.3 in response (5.31 against 33.68). No observational precision separates them, because the observable depends on $C$ , $\tau_0$ and $\alpha$ jointly while the response depends on a different combination of the same three.

And a measurement trap sits on top of the physics one. All of the above uses detrended anomalies. Measured on a transient window instead — a growing carbon reservoir, which is the situation of any real record — the undetrended observable has the wrong sign for every one of the 17 members, because rising $C$ means falling $\dot C$ while $S_1$ rises, so the regression measures the trend rather than the sensitivity. Removing a straight line is not enough either: the transient is curved, so even detrended the estimate recovers at most 0.70 of the exact value and still has the wrong sign for 3 of 17.

What this model cannot do

Worth stating, because a low-order model invites over-reading. There is no permanent airborne fraction — cut emissions and the sink removes all the carbon, the forcing returns to its unforced value, and every reservoir relaxes back; a test asserts exactly that. So nothing here speaks to zero-emissions commitment, which needs an irreversible pathway this carbon cycle does not have. The reservoirs are passive, for the reason chapter 24 measured. There is no cryosphere, no vegetation, no ocean circulation, and $\rho$ is not a greenhouse gas. What transfers is the structure: a hierarchy of memories, a forcing that is part of the state, and a three-way uncertainty budget.

Exercises

The useful lead is $1.141\,T$ . Work out what analysis error would be needed to double it, and say whether that number is physically meaningful.
The amplification formula fails at $g = 0.8$ because $\langle z\rangle$ has left the range where $\mathrm{d}\langle z\rangle/\mathrm{d}\rho$ was fitted. Design a measurement that would fix it, and say what it costs.
The ranking of uncertainty sources inverts between the fastest and slowest reservoirs. At what memory do the two terms cross, and what sets that timescale?
The emergent constraint fails because the observable turns over. Propose a second observable that would break the degeneracy, and state what it must be sensitive to.