Why the Planet Has Not Finished Warming Yet

Nobody Disputes the Exponentials

The physics in this article is about as settled as physics gets. Two coupled linear differential equations, constant coefficients, a closed-form solution that any second-year undergraduate can derive in an afternoon. It has been in the literature since the 1970s and in its modern form since Held and colleagues wrote it down cleanly in 2010 [1]. There is nothing contested about it.

And yet the consequence is misunderstood more or less constantly. Stop emitting today, the reasoning goes, and the warming stops. Or its mirror image: stop emitting today and the warming continues for centuries no matter what. Both of those are claims about a quantity called committed warming, and both of them are stated with more confidence than the quantity deserves, usually by people who have not written down what they mean by it.

So we wrote the model. It has four parameters and it runs in eight seconds. The question we put to it is the one in the title: how much of the warming that past forcing has already bought is still on its way, and what sets how long it takes to arrive. Hansen and colleagues called this warming "in the pipeline" twenty years ago [7], and Wigley [8] and Meehl and colleagues [9] both published estimates of it in the same week of 2005. Our job here is not to improve on them. It is to derive the thing from first principles, check the arithmetic six ways, and be clear about which parts of the answer are physics and which are a modelling choice.

Two Boxes and a Pipe Between Them

The model is two well-mixed reservoirs. The upper one is the atmosphere, the land surface and the ocean mixed layer, all lumped together, because on the timescales that matter they warm together. Call its temperature anomaly \(T\) and its heat capacity \(C\). The lower one is everything below the mixed layer, temperature \(T_0\), heat capacity \(C_0\). Radiative forcing \(F(t)\) lands on the upper box. The climate feedback \(\lambda\) removes heat from it in proportion to how warm it has become. And a pipe of conductance \(\gamma\) carries heat down to the lower box, at a rate proportional to the temperature difference between them.

That is all of it. \(\gamma\) is usually called the ocean heat uptake efficiency, and it plays a role very like a feedback while the deep box is still cold: heat leaving through the pipe cools the surface exactly as heat radiating to space does. The difference is that the pipe fills up. Gregory [4] worked out what vertical transports do to time-dependent warming; Raper, Gregory and Stouffer [5] showed that the spread in model warming is set as much by \(\gamma\) as by \(\lambda\). The two-layer form we use is the one Geoffroy and colleagues calibrated against 16 CMIP5 models [2].

What the model leaves out is a longer list than what it keeps. There is no spatial structure at all, which removes the Southern Ocean, polar amplification and any change in ocean circulation in one stroke. There is no carbon cycle, so forcing is something we prescribe rather than something the system produces. Feedbacks are linear and constant, which is the assumption that fails hardest and which section 10 is about. Heat uptake efficacy is fixed at one, where Winton, Takahashi and Held [6] and the second Geoffroy paper [3] both find it nearer 1.3. And two boxes are a caricature of an ocean that actually ventilates on a continuum of timescales. We are fitting a spectrum with two exponentials and calling it a model.

It works anyway, over a century or two, which is the interesting thing about it.

The Algebra Before the Arithmetic

Write the system as \(\dot{\mathbf{x}} = A\mathbf{x} + \mathbf{f}\) with \(\mathbf{x} = (T, T_0)\). Then

and the eigenvalues of \(A\) are \(-1/\tau_f\) and \(-1/\tau_s\). Solving the characteristic polynomial, with \(b = (\lambda+\gamma)/C + \gamma/C_0\) and \(\delta = b^2 - 4\lambda\gamma/(C C_0)\):

For a step forcing \(F\) applied at \(t = 0\) to a system at rest, the surface temperature is a sum of exactly two exponentials,

with \(a_f + a_s = 1\). That identity is not decoration. It says the two modes between them account for the whole journey from zero to equilibrium, and it is the first thing we check.

Put in the CMIP5 multimodel means from Geoffroy and colleagues [2]: \(C = 7.3\) and \(C_0 = 106\) W yr m⁻² K⁻¹, \(\lambda = 1.13\) and \(\gamma = 0.70\) W m⁻² K⁻¹. Then \(\tau_f = 3.9486\) yr, \(\tau_s = 247.75\) yr, \(a_f = 0.60493\), \(a_s = 0.39507\). The paper itself says the fast constant is of order 4 years and the slow one of order 250 years, so we are within 1.3% and 0.9% of the published statement, which is as close as an order-of-magnitude claim allows anyone to get.

The shape in Figure 1 is the whole argument. A steep rise, a knee at about a decade, and then two and a half centuries of slow creep. Anyone who has watched a thermostat fight a cold house has seen it. The room warms quickly; the walls take all night.

Working Notes From the Club Table

Six Ways to Catch Ourselves Being Wrong

A model with a closed-form solution is a gift, because it can be made to grade its own homework. We integrated the same system three independent ways and compared them at 61 time points spanning 0.05 to 3000 years.

None of that makes the model true. It makes the model consistent, which is a much smaller claim and the only one this section is entitled to.

Reading the Timescales Off the Decay

Everything so far solved algebra. Nothing measured anything. So we did the thing you would do if you did not already know the answer: hit the model with a pulse of heat and watch it relax.

One watt-year per square metre delivered over the first 0.05 years, then the forcing switched off, then 4000 years of integration. Peak surface response 0.1344 K, reached at year 0.1. Then the decay, which on a logarithmic vertical axis is two straight lines with a bend between them. Fit the far end, where the fast mode has been dead for four hundred e-foldings, and the slope gives \(\tau_s\). Subtract that mode. Fit what is left near the start and the slope gives \(\tau_f\).

The fits return \(\tau_s = 247.745427\) yr and \(\tau_f = 3.948631\) yr. The eigenvalues say \(\tau_s = 247.745427\) yr and \(\tau_f = 3.948631\) yr. The differences are 1.4×10⁻¹³ and 7.7×10⁻¹⁴ years, which is to say the measurement and the theory are the same number and the only reason they differ at all is that a double-precision float has 52 bits of mantissa.

The Arithmetic

Equilibrium climate sensitivity is the easy one. Forcing divided by feedback:

Transient climate response is the temperature at year 70 of a 1%-per-year CO₂ ramp. Because forcing goes as the logarithm of concentration and concentration goes up exponentially, the forcing rises linearly, reaching \(F_{2\times}\) at year 70. Our model gives 1.8982 K. The ratio is 0.6217, so at the moment of doubling the planet has delivered 62% of the warming that doubling will eventually cause and owes the other 1.1549 K.

For comparison, the IPCC AR6 assessment gives 3.0 K for ECS and 1.8 K for TCR as best estimates [14], a synthesis that rests in turn on the multiple-lines-of-evidence assessment of Sherwood and colleagues [15]. We are 0.05 K and 0.10 K above those. We are also not entitled to call that agreement a confirmation of anything, because our parameters came from calibrations against the same generation of models that fed the assessment. It is a consistency check on the two-box reduction and nothing more.

Now the commitment. After an abrupt doubling, the fraction of equilibrium realized at time \(t\) is \(1 - a_f e^{-t/\tau_f} - a_s e^{-t/\tau_s}\). At \(t = 100\) years:

So 73.61% realized, and \(3.0531 \times 0.263861 = 0.8056\) K still owed. Look at the two subtracted terms. The fast one is six parts in a trillion. After a century the pipeline is the deep ocean and nothing else, and every decimal place of the answer comes from one exponential with a 248-year time constant.

The last column is the honest bookkeeping. Heat stored rises to 345.91 W yr m⁻² and stops there, and \(345.91 = 7.3 \times 3.0531 + 106 \times 3.0531\), which is what \(CT + C_0T_0\) has to be when both boxes have finished. Solving the same expression for the times when the realized fraction crosses round numbers gives 6.55 years to reach half of equilibrium, 113 years to reach three quarters, 340 years to reach nine tenths and 911 years to reach 99%. The distance between those last two is the point of the article.

Turning the Two Knobs

The model has two parameters that matter here, and they do different jobs. \(\lambda\) sets where the system ends up. \(\gamma\) sets how long it takes to get there. Hold \(\lambda\) fixed and sweep \(\gamma\) and the equilibrium never moves by a thousandth of a kelvin; only the route changes, and it changes a lot.

The right panel carries a result that surprises people. A less sensitive climate is also a faster one. Raising \(\lambda\) from 0.60 to 2.00 drops ECS by a factor of 3.33 and drops \(\tau_s\) from 335 years to 205, because the same feedback that limits the destination also shortens the journey. Transient response varies by only a factor of 2.07 over that range, which is why the transient number is so much better constrained by observation than the equilibrium one, and why an energy-budget estimate of ECS is such a hard thing to pin down.

Across a 5×5 grid of the two parameters spanning their published ranges, the realized fraction at 100 years runs from 60.8% to 87.4%, and TCR/ECS from 0.454 to 0.809. Neither knob dominates. They contribute roughly comparable amounts of the spread, which is the result Raper, Gregory and Stouffer got from full general circulation models a quarter of a century ago [5], and it is mildly reassuring that four equations reproduce it.

Four Hundred Thousand Climates

One set of parameters gives one answer, and the parameters are not known. So we sampled them. \(\lambda\) from a normal distribution with the published intermodel mean and standard deviation, 1.13 ± 0.31. \(\gamma\) uniform over the published range 0.5 to 1.2, uniform because we have the range and not the shape, and we would rather say that than invent a shape. \(C_0\) normal at 91 ± 27, the published figures with the outlying INM-CM4 model removed. \(C\) held fixed, because we could not find a published intermodel spread for it and because section 7 showed that \(C\) governs only the fast mode, whose contribution to the century-scale pipeline is six parts in a trillion.

Four hundred thousand draws, master seed 20260315, streams spawned through numpy's SeedSequence. Truncation to physically admissible ranges rejected 6,242 of them, 1.56%, leaving 393,758.

Every output here is right-skewed, because ECS goes as \(1/\lambda\) while \(\lambda\) is sampled symmetrically, and a reciprocal has a long right tail. The committed warming has a mean of 0.957 K and a median of 0.772 K, and the gap between those two numbers is the skew doing its work. We quote the median. The standard error on the mean is 0.0011 K, which is 0.11% of it, so the sampling noise is not what limits this result. The width of the 5th-to-95th interval, 0.35 to 2.17 K, is.

These are not a posterior distribution and we are not going to call them one. They are the spread of a set of calibrated models, sampled independently, which throws away the real correlations between the parameters. A model with a strong feedback tends to have a particular heat uptake efficiency for reasons of shared physics, and our sampler does not know that. Cummins, Stephenson and Stott [17] do this properly, with a Kalman filter fitted to observed series rather than a scatter of independent draws.

The Strongest Objection We Can Make

Here is the part where the model fails, and the failure is more interesting than any of the successes above.

Everything so far is a thought experiment about step functions. Nothing in it touches the actual Earth. So we built the closest thing we could to a historical run. Forcing grows exponentially from 1750, reaching 2.72 W m⁻² in 2019, which is the total anthropogenic effective radiative forcing assessed in AR6 chapter 7 [14]. The exponential shape is our choice. Its one free parameter, the growth time, we set by requiring that the model's top-of-atmosphere imbalance in 2019 come out at 0.87 W m⁻², the value reported by von Schuckmann and colleagues for 2010 to 2018 [11]. The fit gives a growth time of 87.35 years and hits the target to six decimal places.

One published number in, one published number matched, and then the model makes a prediction it was not told about: how much the surface should have warmed. It says 1.41 K between 1850 and 2019. The observational assessment says 1.09 K [14]. We are 29% high, and the gap is 0.32 K.

Our numerical error is of order 10⁻¹³ K, so none of that gap is arithmetic. There is no standard error to quote either, because with the parameters fixed the model is deterministic and gives 1.4077 K every time. The nearest thing to a sigma available is the intermodel spread, and against that the discrepancy is 1.83 standard deviations, with 3.34% of our Monte Carlo trials drawing a \(\lambda\) large enough to reproduce the observed record. So the observation is not impossible under our model. It is just a long way from where the multimodel mean sits.

Take the same three published numbers and rearrange them. Effective feedback is forcing minus imbalance, over warming:

The historical record, read through this model, implies a climate sensitivity of 2.03 K. The CMIP5 calibrations imply 3.05 K. That gap has a name and a large literature: it is the pattern effect. Feedbacks depend on where the warming is, and the spatial pattern of warming over the twentieth century, with the eastern Pacific and Southern Ocean lagging, produced a more strongly stabilising cloud response than the equilibrium pattern will. Armour [12] showed that an effective feedback diagnosed from the historical period is biased toward low sensitivity for exactly this reason. Zhou and colleagues [13] estimate that accounting for it raises committed warming substantially. Mauritsen and Pincus [10] came at the same question from the observations and got a commitment of about 1.3 K under constant composition.

Rerun our historical forcing with \(\lambda = 1.697\) instead of 1.13 and the model gives 1.0482 K by 2019 against an observed 1.09 K, so the forcing history and the lag are fine. The feedback is what is wrong. And notice what that substitution does to the answer this article is about: it raises the realized fraction at 2019 from 68.0% to 76.2% and cuts \(\tau_s\) from 248 years to 215. A model tuned to match the historical record says there is less in the pipeline, which is precisely the inference Armour and Zhou argue is an artefact.

We could have fitted \(\lambda\) to the record and published an article whose historical run matched observations to 4%. We decided not to. Fitting the one parameter that the whole result depends on, to the one observation that could have tested it, would have produced a better-looking article and a worse one.

What Would Have Changed the Answer, and What It Still Cannot Say

Several choices in this study were not forced, and it is worth saying where a different one would have moved the numbers.

Efficacy. We set the ocean heat uptake efficacy to 1. Winton, Takahashi and Held [6] find values near 1.3, meaning a watt leaving through the pipe suppresses surface warming about 30% more than a watt radiating to space. Geoffroy and colleagues build this into the second paper of the pair [3] and get a longer slow timescale, around 250 years rather than our 248, and a lower TCR. Adopting efficacy would have widened the TCR/ECS gap, made the committed warming larger, and strengthened the article's headline. We left it out because it adds a fifth parameter that we would have had to take on trust.

The forcing history. Our exponential is an idealisation with one adjustable number. The real AR6 forcing series is not exponential; it has a volcanic spike every few decades, an aerosol plateau from the 1950s to the 1980s, and a methane wiggle. Using it would have changed the 2019 realized fraction, though not by much, because the surface integrates forcing over a two-hundred-year memory and short excursions average out. What it would have changed is the imbalance, which responds to the last decade or two and would then no longer match von Schuckmann and colleagues [11] on our single fitted parameter.

Two boxes rather than three. Adding a third box with a millennial timescale would leave everything before year 200 nearly untouched and would add a long tail beyond it. Our model says 99% of equilibrium is reached at year 911. A three-box model would push that out, possibly by centuries. Anyone quoting our 911 should treat it as a lower bound set by the model's structure.

The definition of commitment. We hold composition constant. Hold emissions at zero instead and carbon dioxide starts leaving the atmosphere, which cuts the forcing at roughly the rate the ocean delivers the pipeline. MacDougall and colleagues [18], running eighteen models, find a zero-emissions commitment centred near zero with a spread of about ±0.3 K. The two processes very nearly cancel, and the cancellation is a coincidence of magnitudes rather than a law. Our 0.81 K at 100 years is not that number and should never be quoted as though it were.

Half of \(F_{4\times}\). We took \(F_{2\times} = 3.45\) W m⁻² as half the published quadrupling forcing of 6.9 W m⁻² [2], which assumes the logarithmic relation between concentration and forcing holds exactly across a factor of four. It holds well. It does not hold perfectly. An error here scales ECS, TCR and the committed warming all by the same factor and leaves every ratio and every timescale in this article untouched, which is why we have not worried about it.

What this does and does not tell you

It tells you that the delay is real, that it has a measurable size, and that the size is set by the deep ocean's heat capacity divided by the rate at which heat reaches it. Roughly a quarter of the response to a step change is still absent a century later. The first half arrives in under seven years; the last tenth takes six hundred more. Those two sentences are the entire physical content and they follow from four numbers and two equations.

It does not tell you what will happen. Having no carbon cycle, it cannot say what the forcing does next, and having no geography it cannot say where the warming lands. Worse than either, its feedbacks are constant, which section 10 showed is wrong in a way that costs us 29% on the one comparison we can make against the real record. Rugenstein and colleagues [16], by running climate models out to genuine equilibrium rather than extrapolating, found that sensitivity itself drifts upward over millennia. A model with constant coefficients cannot represent that at all, and ours does not try.

What it does do is make one thing hard to misunderstand. The temperature you see is not the temperature the forcing has bought. It is the temperature the forcing has bought so far, and the difference between those is a quantity you can write down, integrate, and check against conservation of energy at every step. We got a residual of 2.5×10⁻¹¹ W yr m⁻² on that check. The physics is not the uncertain part.

Reproducing this

Everything in this article comes from one file and one command. You need Python 3.12 and numpy; no other package is used, and there is no data file to download because there is no data.

python ocean-heat-lag.py > ocean-heat-lag-output.txt

Expect about eight seconds. Ours took 7.7 s on Python 3.12.3 and numpy 2.4.2, and repeat runs on the same laptop land between 7 and 9 seconds. The master seed is 20260315, hard-coded at the top of the file, and the three Monte Carlo streams are spawned from it through numpy's SeedSequence.spawn, so the output is deterministic to the last printed digit. The longest single computation is the 600,000-step Runge-Kutta run behind validations V1 and V3, which takes about two seconds; the four hundred thousand Monte Carlo trials are evaluated through the closed form in vectorised numpy and cost a small fraction of that. Setting the environment variable SJC_FIGDIR to a directory makes the script also write the SVG fragments used for the five figures above, which is how they were made; it changes nothing that is printed. Numbers differing from ours by more than the printed standard errors mean something is wrong, and we would like to hear about it. The interactive model runs the same two equations in your browser and reproduces the headline numbers on its default settings.

References