The Second Derivative, Part II: Reading the Curves

'Climate Change: A Timeline' — cartoon by @semi_rad



The Second Derivative, Part II: Reading the Curves
Climate · Earth System · Acceleration · Part II

Reading the Curves

Part I argued that the climate is not just changing but speeding up. Part II shows the working — ten charts, the measurements behind them, and the strongest reasons to doubt the story.

This is Part II. Part I — The Second Derivative: Why Climate Change Is Speeding Up made the argument in prose: several independent parts of the Earth system are changing at increasing rates, and the useful question has shifted from how fast to how fast is the speed changing. Several readers asked, reasonably, to see the actual numbers. This piece is that: the data room. It repeats almost none of Part I’s argument and instead does four things — explains how acceleration is measured, walks through the evidence one indicator at a time, takes the counter-arguments seriously, and states plainly what observations would prove the whole thing wrong. Every figure here is drawn from the values in the cited sources, updated through the 2025 data year.

There is a particular kind of dishonesty available to anyone who draws climate graphs. Pick your start year carefully, choose a y-axis that flatters the story, fit a curve that bends where you want it to, and the same dataset will support a shrug or a scream. The defence against this is not to avoid graphs. It is to say out loud what each one is doing: which numbers are measured, which are fitted, where the error bars live, and what the picture would look like if the claim were false.

So here is the promise for what follows. Every chart states its source and its baseline. Where I have drawn a fitted line rather than raw observations, the caption says so. Where a number is a back-of-envelope calculation of mine rather than something a scientist published, it is labelled as such — and there are exactly two of those, both in the doubling-time table, both flagged. And there is a whole section near the end devoted to the best available arguments against the acceleration reading, including one that a fair-minded person should find genuinely uncomfortable.

I. What “acceleration” actually means

Start with the vocabulary, because almost every argument about this topic is really an argument about which derivative someone is talking about.

Take any climate quantity — global sea level, say. The value is how high the sea is. The first derivative is the rate: how many millimetres it rises per year. The second derivative is how that rate is itself changing: whether this decade’s millimetres-per-year is larger than last decade’s. All three can be measured. They tell different stories, and the third one is the one that has changed most in the last twenty years of climate science.

Figure 1 · The three questions you can ask of any climate curve Schematic, not data. The same underlying system read three ways: how much, how fast, and whether the “how fast” is growing. A straight line in the middle panel is exactly what a constant, non-zero value in the right panel looks like.
Three panels showing value, rate and acceleration for an accelerating quantity Left panel: an upward-curving line labelled value. Middle panel: a straight rising line labelled rate. Right panel: a flat horizontal line labelled acceleration, constant and positive. VALUE — “how high is the sea?” bending upward time → first derivative of this ↓ RATE — “mm per year?” rising steadily time → first derivative of this ↓ ACCELERATION — “is it speeding up?” positive and constant zero line this is the number in dispute
How to read it: if the right-hand panel sat on the zero line, the sea would still be rising — steadily, forever. Everything in this essay is about whether that panel sits above zero, and by how much.

How you actually measure a second derivative

In practice nobody eyeballs the bend. The standard method is to fit a quadratic — h(t) = a + bt + ½ct² — to the observed series and report c, the acceleration term, together with its uncertainty. If the confidence interval on c excludes zero, the acceleration is statistically detectable; if it straddles zero, you have a rate that may be constant and a record too short or too noisy to tell.

For satellite-era sea level, that coefficient is 0.084 ± 0.025 mm/yr².[1] The interval is comfortably clear of zero, which is why sea-level acceleration is the least controversial claim in this entire essay. Keep that number in mind; it does a lot of work later.

The noise problem, stated up front

Second derivatives are brutally sensitive to noise. Differentiating a wobbly series once amplifies the wobble; twice, and short records can produce almost any answer you like. This is why single years are nearly worthless as evidence of acceleration — in either direction — and why the honest analyses here use records of 25 years or more.

2025 makes the point better than any argument could. Global sea level rose just 0.08 cm in 2025, against 0.59 cm in 2024 — a factor of seven smaller, because La Niña shifted water onto land as Amazon rainfall.[2] Atmospheric CO&sub2; grew 2.23 ppm in 2025, well below 2024’s record 3.33 ppm.[3] Read either as a trend and you would conclude the crisis is over. Both are weather on top of climate. The discipline this essay tries to keep is simple: trends from decades, never from years — especially when the year is convenient.

II. The dashboard

Six numbers, all from the 2025 data year or the most recent published assessment. They are the summary of everything that follows.

1.8Earth’s energy imbalance in 2023, W/m² — roughly twice the model expectationMAURITSEN 2025
23 ZJHeat added to the upper 2000 m of ocean in 2025 alone — about 37 years of world energy useCHENG 2026
0.084Sea-level acceleration, mm/yr² (±0.025) — the fitted second derivativeNASA / NEREM
408 GtGlacier mass lost in 2025 — second-worst year in the 50-year recordWGMS 2026
0.27 °CHuman-induced warming per decade over 2016–2025, up from ~0.2 °CIGCC 2025
130 GtCO&sub2; left in the 1.5 °C budget from January 2026 — about three years of emissionsIGCC 2025

III. The master dial: Earth’s energy imbalance

Everything else in this essay is downstream of one quantity. Earth’s energy imbalance (EEI) is the difference between the sunlight the planet absorbs and the infrared it radiates back to space, measured in watts per square metre of surface. If it is zero, the climate is in equilibrium at whatever temperature it currently holds. If it is positive, energy is piling up somewhere.

It is a small number in absolute terms — of order one watt per square metre against roughly 340 coming in — which is precisely why it is hard to measure and why the measurement is a genuine achievement. Two entirely independent methods agree on it: satellites (NASA’s CERES instruments) watching radiation leave the top of the atmosphere, and the Argo float array measuring heat accumulating in the sea. That the two agree is the reason to take the number seriously.[4]

Figure 2 · Earth’s energy imbalance is growing Anchor values from the cited literature, with the published CERES trend line drawn through them. This is not a continuous reconstruction — annual EEI is noisy, and only three published anchor points are plotted.
Earth's energy imbalance from 2001 to 2024 A rising trend line from about 0.4 watts per square metre in 2001 to about 1.4 in 2024, with anchor points at 0.5 in 2005, 1.0 in 2019 and an outlying 1.8 in 2023. 0 0.5 1.0 1.5 2.0 W/m² 2001 2005 2010 2015 2020 2024 CERES trend: +0.45 W/m² per decade (2001–2024) 0.5 2005 1.0 2019 1.8 (2023) NASA/NOAA: doubled in 14 years →
Sources: the 2005 and 2019 anchors are the endpoints of the NASA/NOAA study that found the imbalance roughly doubled over that window.[4] The 2023 value and the +0.45 W/m²/decade trend are from Mauritsen et al. (2025), using CERES-EBAF v4.2.1.[5] Caveat: 2023 was an exceptional year — a strong El Niño plus record-low planetary albedo — and sits well above the trend line. It is plotted here because it is the published figure, not because one year proves anything.

Two features of this chart deserve emphasis. The first is the slope: an imbalance growing at roughly 0.45 W/m² per decade means the rate of heat accumulation is itself increasing. That is the definition of acceleration, measured directly at the source rather than inferred from a downstream symptom. The second is the gap between observation and expectation. Mauritsen and colleagues note that the observed imbalance is running at roughly twice what the climate models projected.[5] This cuts in an awkward direction for everyone: it means the models are wrong, and it means they were wrong in the reassuring direction.

The models underestimated it. That is not a comfort — it is the finding.

IV. Where the heat goes

An energy surplus has to end up somewhere, and the destinations are wildly unequal. The atmosphere — the part of the system we actually live in and talk about — takes about one per cent of it.

Figure 3 · Where the surplus energy is stored Share of accumulated excess heat in the Earth system, per IPCC AR6. The ocean is not one reservoir among several; it is essentially the whole account.
Distribution of excess heat in the Earth system A horizontal bar: ocean about 91 percent, land about 5 percent, ice about 3 percent, atmosphere about 1 percent. OCEAN — about 91% land ~5% ice ~3% atmosphere ~1% Every argument about “the pause” in surface warming was an argument about that last one per cent.
Source: IPCC AR6 WGI, Chapter 7 energy budget assessment.[6] Values rounded; the ocean share is variously given as 89–91% depending on period and depth range.

This is why ocean heat content is the best single thermometer for the planet. It integrates the imbalance over time, it is far less noisy than surface temperature, and it cannot be gamed by choosing a start year in an El Niño. And it is where the acceleration signal is cleanest.

Figure 4 · The ocean is taking up heat about three times faster than it was Trend in upper-2000 m ocean heat content, zettajoules per year, over three published periods. A zettajoule is 10²¹ joules.
Ocean heat uptake rate by period Bar chart: 3.1 zettajoules per year for 1958 to 1985, 9.2 for 1986 onward, 10.8 for 2007 to 2023. 0 3 6 9 12 ZJ / year 3.1 1958–1985 9.2 1986 onward 10.8 2007–2023 ×3 Each bar is a published linear trend over its period, not an annual value.
Source: Cheng et al., IAP/CAS analyses in Advances in Atmospheric Sciences.[7][8] Note the overlap: the 1986-onward and 2007–2023 periods are not independent — the third bar is a subset of the second. It is shown because the further increase within the recent window is itself the acceleration signal, but it should not be read as a fourth independent data point.

The 2025 assessment, published in January 2026 by 55 scientists across ten research teams, found the upper 2000 m gained roughly 23 ± 8 ZJ over 2024 — another record, in a year that was globally cooler at the surface than 2024 and trending into La Niña.[8] That divergence is instructive. Surface temperature fell; ocean heat content did not. The atmosphere is where the weather is. The ocean is where the accounting is.

What 23 zettajoules means, if you want a physical picture

23 × 10²¹ joules is roughly 37 years of total global primary energy consumption at 2023 rates, absorbed by the sea in twelve months.[8] Comparisons of this kind are frequently abused — “X Hiroshima bombs per second” and the like — because they make an arithmetically true statement feel like a physical threat. The honest use is narrower: it tells you that the energy involved is enormous relative to anything human-scaled, and therefore that the momentum in the system is not something a change of policy stops quickly. It says nothing about how bad any particular consequence will be.

V. Sea level: the cleanest second derivative we have

Sea level is the flagship case, for three reasons. The satellite altimetry record is continuous since 1993 and calibrated across five successive missions; the acceleration term is statistically significant rather than marginal; and the physical causes are separately measurable, so the acceleration can be attributed rather than merely observed.

The headline: global mean sea level has risen about 10 cm since 1993, and the rate has gone from roughly 2.1 mm/yr in the early record to something in the range 3.9–4.5 mm/yr over the past decade.[9][2] But the more revealing way to see it is to ask what would have happened if the 1993 rate had simply continued.

Figure 5 · Four centimetres that a straight ruler would have missed Global mean sea level since 1993. The upper curve is the quadratic fit implied by the published acceleration coefficient; the lower dashed line is what a constant 2.1 mm/yr would have produced. The shaded wedge is the contribution of acceleration alone.
Global mean sea level rise 1993 to 2025, accelerating curve versus constant rate Two lines from zero in 1993. The constant-rate line reaches about 67 millimetres by 2025; the accelerating curve reaches about 110 millimetres. The gap between them widens over time. 0 25 50 75 100 mm above 1993 1993 2000 2010 2020 2025 110 mm observed 67 mm if the rate had held +43 mm from acceleration Acceleration coefficient: 0.084 ± 0.025 mm/yr². The interval excludes zero — the bend is real, not a fitting artefact.
What is drawn: both curves are analytic, not raw altimetry. The solid line is h(t) = 2.1t + 0.042t², using the published initial rate and acceleration coefficient;[1] it lands at 110 mm in 2025, against NASA’s observed figure of about 100 mm.[2] The 10% overshoot is the honest size of the error in treating a noisy 32-year record as a perfect parabola — and it is why the wedge should be read as “roughly four centimetres,” not as a precise accounting.

What makes the sea-level case strong is not the curve; it is that the curve can be decomposed. Roughly a third of the rise is thermal expansion — water swelling as it warms, which is a direct mechanical consequence of the ocean heat content in Figure 4. The rest is ice: mountain glaciers, Greenland, Antarctica, plus a smaller term from changes in land water storage. Each of those contributions is measured independently, by satellite gravimetry and by glaciological survey, and the sum reconciles with the altimetry to within the uncertainties. When four separate measurement systems close a budget, the result is not an artefact of any one of them.

VI. The forcing: what we are actually doing

Upstream of the imbalance is the carbon. And here the acceleration is not subtle: it is visible in a chart a schoolchild could read, because we have been measuring it continuously at Mauna Loa since 1958.

The relevant quantity is not the CO&sub2; concentration — that has been rising all along — but its annual growth rate. The concentration is the value. The growth rate is the first derivative. Whether that growth rate is itself climbing is the second.

Figure 6 · The CO&sub2; growth rate has roughly tripled since the 1960s Mean annual increase in atmospheric CO&sub2; at Mauna Loa, ppm per year, averaged by decade. The final bar covers 2020–2025 only.
Decadal mean CO2 growth rate at Mauna Loa Bars rising from 0.86 ppm per year in the 1960s to 2.56 in the 2020s, with a small dip in the 1990s. 0 1 2 3 ppm / year 0.861960s 1.281970s 1.611980s 1.511990s 1.972000s 2.422010s 2.562020–25 the 1990s dip is real Individual years within these decades range from 0.31 to 3.33 ppm. The decade is the honest unit here.
Source: computed from NOAA GML’s published annual growth rates, 1959–2025.[3] The dip matters. The 1990s came in below the 1980s — the collapse of Soviet-bloc industry, the Mount Pinatubo eruption, and a stronger land carbon sink. It is the clearest evidence in this essay that these curves are not destiny: when emissions fell, the growth rate fell. That is the whole argument for policy, sitting in plain view inside a chart about acceleration.
Two recent years, pulling opposite ways

2024: 3.33 ppm — among the largest one-year increases in the entire record, driven by El Niño, extensive wildfire, and a weakened land sink.[3] In May 2025 the seasonal peak passed 430 ppm at Mauna Loa for the first time.[10]

2025: 2.23 ppm — back below the 2010s average as La Niña restored the land sink.[3]

If you wanted to argue that CO&sub2; growth is accelerating out of control, you would quote 2024. If you wanted to argue it has peaked, you would quote 2025. Both would be the same error. The decadal bars in Figure 6 are the signal; those two years are the variance around it.

VII. Ice, which does not average nicely

Ice is where acceleration is easiest to feel and hardest to summarise, because the two great reservoirs behave differently and the sea ice at each pole behaves differently again.

Figure 7 · Land ice loss, by period Left: mountain glaciers outside the ice sheets, mean annual mass loss (WGMS). Right: the Greenland and Antarctic ice sheets combined (IMBIE). Both in gigatonnes per year; one gigatonne is a cubic kilometre of water.
Glacier and ice-sheet mass loss rates by period Glaciers: under 100 gigatonnes per year in 1976 to 1995, 230 in 1996 to 2015, 390 in 2016 to 2025. Ice sheets: 105 gigatonnes per year in 1992 to 1996, 372 in 2016 to 2020. 0 100 200 300 400 GLACIERS — Gt/yr lost <1001976–95 2301996–2015 3902016–25 quadrupled in five decades ICE SHEETS — Gt/yr lost 1051992–96 3722016–20 3.5× in under 25 years Glacier bars are running means over the stated windows; ice-sheet bars are IMBIE five-year means.
Sources: WGMS / Nature Reviews Earth & Environment global glacier mass change assessment, 2025 hydrological year;[11] IMBIE mass balance of the Greenland and Antarctic ice sheets, 1992–2020.[12] The two panels use different periods because the two measurement programmes do — they are not directly comparable, and the totals should not be added across panels without care.

The glacier record is the more striking of the two, partly because it is longer and partly because the pattern is so relentless: six of the ten worst years on record have occurred in the past seven years, and cumulative loss since 1975 stands at about 9,583 ± 1,211 Gt — some 26 mm of sea-level rise from mountain ice alone.[11] The 2025 loss of 408 ± 132 Gt was the second-highest ever measured.

And now the honest complication: sea ice

Sea ice is the indicator that does not fit the clean acceleration story, and leaving it out would be exactly the kind of selection this essay is supposed to avoid.

Arctic sea ice is in unambiguous long-term decline, and March 2025 set the lowest winter maximum in the 47-year satellite record.[13] But the September 2025 minimum came in tenth-lowest, tied with 2008 and 2010 — not a record, and roughly where it has hovered for two decades.[14] Summer Arctic ice loss looks more like a steep decline that has flattened into a noisy plateau than like something accelerating. Antarctic sea ice, meanwhile, was stable or slightly increasing until about 2016 and then fell off a cliff, with the 2025 minimum effectively tied for second-lowest and the winter maximum third-lowest on record.[15] That looks less like acceleration than like a regime shift — a system that sat in one state for decades and abruptly moved to another.

Why the awkward indicator is the interesting one

A framework that explained everything equally well would be suspicious. Sea ice shows that “the second derivative is positive” is not a universal law of the climate system — it is a finding that holds strongly for the energy-integrating quantities (imbalance, ocean heat, sea level, land ice) and holds poorly or not at all for quantities governed by threshold physics and circulation. Antarctic sea ice may be the better warning: some parts of this system do not accelerate smoothly. They jump.

VIII. Surface temperature, and the shrinking budget

Surface temperature is the indicator everyone quotes and the worst one for detecting acceleration, precisely because it lives in that one per cent of the system from Figure 3. El Niño and La Niña slosh heat between ocean and air on a two-to-seven-year cycle, and that sloshing is large compared with the underlying trend. It is the reason the “pause” of the early 2010s was arguable at all, and the reason 2025 came in cooler than 2024 while the ocean set another record.

The way around this is to separate the human-induced component from the natural variability, which is what the annual Indicators of Global Climate Change assessment does. Its 2025 edition puts human-induced warming at 1.37 °C above 1850–1900, rising at 0.27 °C per decade over 2016–2025 — against roughly 0.2 °C per decade in the preceding decades.[16]

Figure 8 · The pace of human-caused warming, with its honest uncertainty Rate of human-induced warming, °C per decade. The bracket on the right-hand bar is the published likely range — and it is wide.
Rate of human-induced warming per decade Two bars: about 0.18 degrees per decade for the late twentieth century, and 0.27 degrees per decade for 2016 to 2025, with an uncertainty range from 0.2 to 0.4. 0 0.1 0.2 0.3 0.4 °C / decade ~0.18 late 20th century (IPCC AR6 assessment) 0.27 2016–2025 (IGCC 2025) likely range 0.2 – 0.4 °C/dec The lower end of that bracket overlaps the earlier rate. Surface temperature alone would not settle this question.
Read the bracket, not just the bar. A central estimate of 0.27 with a range of 0.2–0.4 means the data are consistent with a modest speed-up and with a dramatic one. This is the weakest of the acceleration lines of evidence taken alone — which is why the argument rests on Figures 2, 4 and 5, where the ranges are tighter. Sources: IGCC 2025[16] and IPCC AR6.[6]

Two further findings from the 2025 data year are worth recording plainly. Copernicus ranked 2025 the third-warmest year, 0.13 °C cooler than 2024 — and the past eleven years are the eleven warmest on record.[18] More significantly, the WMO reported that 2023–2025 is the first three-year period whose average exceeds 1.5 °C above pre-industrial.[17] As Part I noted, a single year above 1.5 is not a breach of the Paris threshold, which is defined on multi-decadal averages. Three years is still not a breach. It is, however, the shape a breach makes on the way in.

The budget, which is the number that should actually alarm you

Figure 9 · The remaining 1.5 °C carbon budget, in years 130 GtCO&sub2; remaining from January 2026 for a 50% chance of holding below 1.5 °C, against current emissions of about 42 GtCO&sub2; per year.
Remaining carbon budget expressed as years of current emissions A bar of 130 gigatonnes divided into three blocks of 42 gigatonnes representing one year each, plus a small remainder. 2026 2027 2028 0 65 GtCO&sub2; 130 GtCO&sub2; Each block = one year at 42 GtCO&sub2;/yr On current trends, human-induced warming reaches 1.5 °C around 2030.
Source: IGCC 2025.[16] A carbon budget is a probabilistic construct with wide uncertainty, and “three years” does not mean something happens in 2029 — it means that from 2029 onward, every further tonne is spent against a lower probability of staying under the line. Exceeding the budget does not end the project; it changes what the project is.

What the surplus does when it reaches the surface

Energy accounting is abstract until it lands somewhere. The clearest recent case is marine heatwaves. In 2023 they covered roughly 96% of the ocean surface at some point in the year, with some events persisting over 500 days — four times the historical average duration.[21] Across the 2023–24 summers the world saw about 3.5 times as many marine heatwave days as any previous year on record, and nearly a tenth of the global ocean hit an all-time high temperature. The consequences were not abstract either: the fourth global coral bleaching event, fishery closures, and mass strandings.

Note the structure of that number. It is not a trend line bending gently. It is a multiple — 3.5× the previous record — and it arrived in two years. This is what accelerating energy input looks like once it passes through a system with thresholds in it: not a smooth ramp, but long quiet stretches punctuated by events that break the previous scale.

IX. Doubling times — and why I distrust my own table

Part I flagged, as rhetorical overreach, the popular claim that climate doubling times are heading “toward zero.” That flag stands. But the underlying device — asking how long an accelerating quantity takes to double — is genuinely useful, so here it is, computed honestly and with its weaknesses on display.

IndicatorEarlier rateRecent rateFactorImplied doubling
Earth’s energy imbalance
2005 → 2019
0.5 W/m²1.0 W/m²×2.0~14 yr
Ice-sheet mass loss
1992–96 → 2016–20
105 Gt/yr372 Gt/yr×3.5~13 yr
Glacier mass loss
1976–95 → 2016–25
<100 Gt/yr390 Gt/yr×3.9~18 yr
Sea-level rate
1990s → past decade
2.1 mm/yr4.5 mm/yr×2.1~20 yr
Ocean heat uptake
1958–85 → 1986–2023
3.1 ZJ/yr9.2 ZJ/yr×3.0~21 yr
CO&sub2; growth rate
1960s → 2020s
0.86 ppm/yr2.56 ppm/yr×3.0~37 yr
Warming rate
late 20th c. → 2016–25
0.18 °C/dec0.27 °C/dec×1.5~51 yr
Four reasons to hold this table loosely
  • The doubling times are mine, not the literature’s. Each is computed as T · ln2 / ln(r) from the two cited endpoint rates and the gap between their midpoints. No source publishes these figures; they are arithmetic laid over other people’s measurements.
  • Endpoint choice drives the answer. Shift a window by five years and several of these numbers move by a third. That sensitivity is the whole reason the fitted-coefficient method in Section I is preferred by actual analysts.
  • Exponential framing is an assumption, not a finding. A doubling time only means something if the process is exponential. Most of these are better described as roughly linear acceleration, for which a doubling time is a convenient summary rather than a physical constant.
  • Nothing here extrapolates. An 18-year doubling of glacier loss cannot continue indefinitely for the trivial reason that glaciers run out. Several of these curves must eventually bend the other way — not because the warming stops, but because the thing being lost is finite.

Used carefully, the table earns its place. Every indicator that integrates the energy surplus over time doubles on a timescale of roughly one to two decades — a human career, not a geological epoch. The indicators further from the energy budget, surface temperature and CO&sub2; growth, double more slowly. That ordering is not arbitrary: it is exactly what you would expect if a single growing imbalance were driving the whole system, felt most directly by the reservoirs that store it.

X. The best arguments against all of this

If you only ever encounter the case for a proposition, you have not evaluated it. So here are the four strongest objections to the acceleration reading, in ascending order of how much they worry me.

1. The records are short, and second derivatives are noisy

The satellite altimetry record is 32 years old. Reliable global ocean heat content really begins with Argo in the mid-2000s. CERES covers 2001 onward. Detecting a curvature in a record that short, in a system with decadal internal variability, is genuinely hard — and it is a standard result that plausible-looking accelerations can emerge from purely random walks over short windows.

How much weight it deserves: some, but less than it first appears. The defence is not any single record but their agreement. Altimetry, gravimetry, Argo floats, CERES radiometers, glaciological surveys and the Mauna Loa flask record are independent instruments with independent failure modes, and they bend the same way. Noise does not conspire.

2. Overlapping windows inflate the picture

Look again at Figure 4. The “1986 onward” and “2007–2023” bars share most of their data; presenting them side by side invites you to read three independent accelerating steps where there are really one and a half. Similar objections apply to any chart built from nested periods, and this essay contains several.

How much weight it deserves: a fair hit on the presentation, not on the underlying claim. I have flagged it in the caption rather than dropping the bar, because the alternative — showing only non-overlapping windows — would discard the most recent data. But a reader who mentally deletes the third bar loses very little of the argument.

3. Several 2025 indicators went the other way

Sea level rose barely at all. CO&sub2; growth fell by a third. Global surface temperature dropped 0.13 °C from 2024. The Arctic summer minimum was unremarkable. A sceptic could assemble a coherent-sounding “the acceleration has stopped” post entirely from 2025 data, and it would be no more dishonest than the alarmist posts built entirely from 2023 data.

How much weight it deserves: very little as evidence, a great deal as discipline. La Niña explains the sea-level and CO&sub2; figures directly and was expected. But this objection is the reason for the rule stated in Section I, and anyone making the acceleration argument is obliged to apply that rule symmetrically — which means not quoting 2023 either.

4. The aerosol problem — the objection that should worry you

This is the serious one, and it is not much discussed outside the literature.

Sulphate aerosols from burning coal and heavy fuel oil reflect sunlight and seed brighter clouds. They have been masking a portion of greenhouse warming for a century. Over the past fifteen years, that mask has been coming off fast: China’s air-quality programme cut East Asian sulphur emissions dramatically, and in January 2020 the International Maritime Organization’s fuel rules cut shipping sulphur worldwide, reducing observed ship tracks by about a quarter almost immediately.[20]

Recent modelling finds that East Asian aerosol cleanup has likely contributed to the observed acceleration of global warming since 2010.[19] Estimates of the IMO 2020 forcing range from about +0.07 to +0.14 W/m², and one strand of the literature argues the resulting temperature response is not yet distinguishable from internal variability.[20] The disagreement is live.

Why this cuts deep

If a meaningful share of the recent speed-up is unmasking rather than compounding, then part of what Figures 2 and 8 show is a step change dressed as acceleration — a one-off removal of a shade, not an engine revving. Steps and accelerations look identical over a short window and imply completely different futures. A step finishes; an acceleration does not.

Three things keep this from dissolving the argument. The cleanup is finite — there is only so much sulphate left to remove, so it cannot explain a continuing rise beyond the next decade. It does not touch the pre-2010 portion of the sea-level or ocean-heat acceleration at all. And it is not, in any case, good news: it means we had been receiving an unearned discount on our emissions, paid for with the air quality that was killing millions of people, and the bill for ending that public-health disaster is arriving as heat.

XI. What would change my mind

A claim you cannot imagine being wrong is not a scientific claim. So, concretely — the following observations, sustained over the intervals given, would falsify or substantially weaken the acceleration reading. I would want to see each of them, and I am recording them now so that a future reader can check.

  • Sea level. A refitted quadratic over the full altimetry record whose acceleration coefficient falls to within one standard error of zero, holding for five years of updates. Currently 0.084 ± 0.025 mm/yr².
  • Energy imbalance. A decade of CERES observations with no significant positive trend — roughly, a 2026–2035 mean statistically indistinguishable from the 2011–2020 mean.
  • Ocean heat. Ten consecutive years in which the 0–2000 m uptake rate sits at or below the 1986–2023 trend of 9.2 ZJ/yr. Two or three such years would prove nothing; ten would.
  • Land ice. A decade in which glacier loss averages below the 230 Gt/yr of the 1996–2015 period, absent a major volcanic eruption to explain it.
  • Attribution. A convergence of the aerosol literature on the conclusion that most of the post-2010 speed-up is unmasking rather than accumulated forcing — which would not make the warming less real, but would make “acceleration” the wrong word for it.

Conversely, the observations that would strengthen it: continued divergence between observed and modelled EEI; ocean heat records in La Niña years, as happened in 2025; and the sea-level acceleration coefficient rising rather than holding.

Figure 10 · Scorecard: how strong is the acceleration evidence, indicator by indicator My own assessment of confidence that the second derivative is meaningfully positive, based on record length, signal-to-noise, and independence of the measurement. Not a formal IPCC-style calibration.
Confidence scorecard by climate indicator Energy imbalance, ocean heat and sea level rated very high; land ice and CO2 growth high; warming rate and marine heatwaves medium; Arctic summer sea ice low; Antarctic sea ice flagged as a different phenomenon. weak very strong Earth’s energy imbalance Ocean heat content Global sea level Glacier mass loss Ice-sheet mass loss CO&sub2; growth rate Surface warming rate Marine heatwaves Arctic summer sea ice Antarctic sea ice regime shift, not acceleration declining, but not obviously accelerating
The pattern is the point. Confidence tracks how directly an indicator integrates the energy surplus. The three strongest are the three closest to the energy budget itself; the weakest are the ones mediated by circulation and threshold physics. A framework that scored everything “very strong” would be a framework that had stopped looking.

XII. The bottom line

Part I ended on a sentence I still stand behind: it is speeding up, we are the foot on the pedal, and we can ease off. Having spent this essay in the actual numbers, I would add three qualifications and one amplification.

The qualifications. The acceleration is unambiguous where the system integrates energy — the imbalance, the ocean, sea level, land ice — and much weaker or absent where circulation and thresholds dominate. Some of the post-2010 speed-up at the surface is probably aerosol unmasking rather than compounding, and that distinction genuinely matters for what the next twenty years look like. And every doubling time in Section IX is a summary statistic, not a law; none of them should be extrapolated past the decade they were fitted in.

The amplification. The single most robust finding in this entire body of evidence is also the least discussed in public: Earth’s energy imbalance is running at roughly twice what the models projected, and it is measured two independent ways that agree. Everything downstream — the sea level, the ice, the heatwaves — is that surplus finding somewhere to go. If you remember one number from these two essays, make it that one, because it is the only one that sits upstream of all the others.

The 1990s dip in Figure 6 is the most hopeful thing in this essay. It happened because emissions fell.

And that is where I want to leave it, because a data room full of bending curves can read as fatalism, and the data do not actually support fatalism. Look once more at Figure 6. The 1990s bar is lower than the 1980s bar. It is lower because industrial emissions fell across a large part of the world, for reasons that had nothing to do with climate policy and everything to do with economic collapse and a volcano — but it fell, and the atmosphere noticed within a decade. The curves in this essay are not a prophecy. They are a record of what a particular quantity of emitted carbon has done so far. The next set of bars has not been drawn.

References

  1. Nerem, R. S. et al. (2018). “Climate-change–driven accelerated sea-level rise detected in the altimeter era.” PNAS 115(9). Source of the 0.084 ± 0.025 mm/yr² acceleration coefficient. pnas.org
  2. NASA JPL (2026). “NASA Analysis Shows La Niña Limited Sea Level Rise in 2025.” jpl.nasa.gov
  3. NOAA Global Monitoring Laboratory. “Trends in Atmospheric Carbon Dioxide — annual mean growth rate, Mauna Loa, 1959–2025.” Decadal means in Figure 6 computed from this series. gml.noaa.gov
  4. NASA / NOAA (2021). “Joint NASA, NOAA Study Finds Earth’s Energy Imbalance Has Doubled.” Based on Loeb et al., Geophysical Research Letters. nasa.gov
  5. Mauritsen, T. et al. (2025). “Earth’s Energy Imbalance More Than Doubled in Recent Decades.” AGU Advances. CERES-EBAF v4.2.1 trend of 0.45 W/m² per decade, 2001–2024. agupubs.onlinelibrary.wiley.com
  6. IPCC (2021). Sixth Assessment Report, Working Group I: The Physical Science Basis, Ch. 7 (Earth’s energy budget) and Ch. 2 (observed warming rates). ipcc.ch
  7. Cheng, L. et al. (2025). “Record High Temperatures in the Ocean in 2024.” Advances in Atmospheric Sciences. link.springer.com
  8. Cheng, L. et al. (2026). “Ocean Heat Content Sets Another Record in 2025.” Advances in Atmospheric Sciences, doi:10.1007/s00376-026-5876-0. 55 scientists, 10 institutions; 23 ± 8 ZJ increase in 2025. link.springer.com
  9. NASA Earthdata / PO.DAAC. “The Rate of Global Sea Level Rise Doubled During the Past Three Decades.” earthdata.nasa.gov
  10. Scripps Institution of Oceanography (2025). “Annual Carbon Dioxide Peak Passes Another Milestone” — May 2025 monthly mean of 430.2 ppm (Scripps) / 430.5 ppm (NOAA). scripps.ucsd.edu
  11. WGMS / Zemp, M. et al. (2026). “Global glacier mass change in 2025.” Nature Reviews Earth & Environment. 408 ± 132 Gt lost in 2025; 9,583 ± 1,211 Gt since 1975. nature.com
  12. IMBIE Team (2023). “Mass balance of the Greenland and Antarctic ice sheets from 1992 to 2020.” Earth System Science Data 15, 1597. essd.copernicus.org
  13. NSIDC (2025). “Arctic sea ice hits record low maximum extent for the year” — 14.33 million km², 22 March 2025. nsidc.org
  14. NSIDC (2025). “2025 Arctic sea ice minimum squeezes into the ten lowest minimums” — 4.60 million km², 10 September 2025. nsidc.org
  15. Carbon Brief / NSIDC (2025). “Antarctic sea ice winter peak in 2025 is third smallest on record.” carbonbrief.org
  16. Forster, P. M. et al. (2026). “Indicators of Global Climate Change 2025.” Earth System Science Data 18, 3889. Human-induced warming 1.37 °C; 0.27 [0.2–0.4] °C/decade over 2016–2025; remaining 1.5 °C budget 130 GtCO&sub2; from January 2026. essd.copernicus.org
  17. World Meteorological Organization (2026). “WMO confirms 2025 was one of warmest years on record” — and the first three-year period (2023–2025) averaging above 1.5 °C. wmo.int
  18. Copernicus Climate Change Service (2026). “2025 was the third hottest year on record.” climate.copernicus.eu
  19. Wang, B. et al. (2025). “East Asian aerosol cleanup has likely contributed to the recent acceleration in global warming.” Communications Earth & Environment. nature.com
  20. On the 2020 shipping fuel regulation: Yuan, T. et al. (2024), “Abrupt reduction in shipping emission as an inadvertent geoengineering termination shock,” Communications Earth & Environment; and Jordan, G. & Henry, M. (2025), “Surface temperature effects of recent reductions in shipping SO&sub2; emissions are within internal variability,” Atmospheric Chemistry and Physics 25, 4443 — cited together because they disagree. acp.copernicus.org
  21. Tan, H. et al. (2025). “Record-breaking 2023 marine heatwaves.” Science. See also the 2025 Nature Climate Change analysis of the 2023–24 marine heatwave summers. science.org
On method and tools

This article was researched and written collaboratively with Claude (Anthropic): human specification and critical review, machine research synthesis and drafting, iterative refinement through structured dialogue. The research phase involved live searches of NASA, NOAA GML, NSIDC, the WGMS, IMBIE, Copernicus, the WMO and the peer-reviewed record, and every quantitative claim links to a primary source in the references.

All ten figures are hand-built SVG, drawn from the values in the cited sources rather than from redrawn published graphics. Figure 1 is schematic and labelled as such. Figure 5 plots analytic curves from the published coefficients, not raw altimetry, and the caption gives the size of the resulting discrepancy against the observed total. The decadal CO&sub2; means in Figure 6 were computed directly from NOAA’s published 1959–2025 annual series. The doubling times in Section IX and the confidence ratings in Figure 10 are the author’s own arithmetic and judgement respectively, flagged as such in place.

Where Part I separated the science from the rhetoric that prompted it, Part II tries to do something harder: state the case, then argue against it in good faith, and write down in advance what would prove it wrong.
Authored by: Luis Matos Ferreira
Physicist & Developer

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