Last week the line going around was that the deep ocean's heat uptake has "nearly quadrupled" in three decades. It comes from a real paper, open access, by a careful oceanographer, and the arithmetic behind the sentence is correct.
That is what makes it interesting. Nothing goes wrong in the calculation. Something goes wrong in the grammar.
What the paper says
Gregory C. Johnson, Observed Multi-Decadal Acceleration of Globally Averaged Abyssal Ocean Warming, Geophysical Research Letters 53, e2026GL124104 (2026), doi 10.1029/2026GL124104. Open access, CC BY-NC. I read the ten pages off the NOAA repository copy, because the publisher's PDF sits behind a bot wall.
From the abstract, verbatim:
We find a statistically significant increase of the heating rate of the abyssal (4,000–6,000 dbar) ocean, from 5.4 (±4.9) TW in 1988 to 20.2 (±3.9) TW in 2018.
The plain language summary then says the abyssal heat uptake "has approximately quadrupled from 1988 to 2018", and the results section says it "nearly quadrupled in three decades ... with the increase being statistically significant".
Divide one by the other and you get 3.74. Correct.
The division
Two estimates, each with an uncertainty, one divided by the other. So the thing you get is not a quantity with a range unless you carry both uncertainties through. Let me do it, in a script, on the numbers as printed:
point ratio 3.74 baseline 1988 in half-widths of its own stated uncertainty: 1.10 R2 (+/- is the plotted 5-95% band, the reading most generous to the authors) baseline 90%: [0.5, 10.3] ratio lower edge 20.2/10.3 = 1.96 ; upper edge unbounded (the interval contains zero) R1 (+/- is 1 sigma) baseline 90%: [-2.7, 13.5] ratio lower edge 1.50 ; upper edge unbounded The paper prints ± on those TW values without saying whether it is one sigma or the plotted 5–95% band. The Figure 4 caption does say the shading is the 5%–95% range, so I computed the ratio under both conventions. The reading most favourable to the "quadrupled" claim is the one where ± is the 90% band, because it makes the denominator as narrow as possible. Under that most generous reading, the ratio's lower edge is 1.96 and its upper edge does not exist.
That is the whole finding. "Nearly quadrupled" is compatible with the numbers. So is "it tripled". So is "it rose by 40 times". The point value 3.74 is the only thing the sentence can be about, and the sentence is doing the work of a fact.
Why the upper bound vanishes: the 1988 estimate is 5.4 ± 4.9. It sits 1.1 half-widths away from zero. Its plausible range includes zero, and a denominator that reaches zero cannot produce a finite ratio.
For scale, here is what the baseline would have had to be for "nearly four times" to have a lower bound of 3 at 90% confidence: 11.6 TW instead of 5.4. More than twice the reported value. The headline is not a small overreach on the edge of the interval. It is a quantity that needs a denominator this data does not have.
What the paper actually established
This is where I want to be careful, because the paper is more honest than its summary.
The statistically significant object is not the ratio. It is the change, and the acceleration term behind it. Put the change in absolute units instead of multiplicative ones:
absolute change: +14.8 TW +/- 6.3 (independent combination), 2.36 half-widths from zero That is a legitimate result. The heating rate of the abyssal ocean went up by roughly fourteen terawatts over thirty years, and the increase is about 2.4 half-widths from zero. If you want to say something strong about this paper, that is the sentence. It is less quotable than "quadrupled" precisely because it carries its own precision instead of hiding it inside a multiplication.
The paper is also candid about its own limits, in a way I wish were more common. It samples only about 18% of the ocean volume well enough to fit an acceleration at all (29% for trends). Deep Argo is at about 20% of its design strength. And it says, in the discussion, that the small sampled fractions "may reduce confidence in the reported statistical significance of their global integrals". An author writing that sentence knows what his data can carry.
Which raises the question of where the multiplier came from. My guess, and it is a guess: a plain-language summary is written to be understood, and "5.4 to 20.2 terawatts" is not understandable, so the author reached for the one transformation that makes it a story. He reached for a fold change. It is the correct arithmetic, and that is exactly why nobody checked it.
The other half of the story
The same figure contains a second thing worth looking at, and the paper reports it plainly. The layer above (2,000–4,000 dbar) went from 29.4 (±22.1) TW to 25.0 (±17.5) TW. A slight decrease. The uncertainty on that change, combined, is about 28 TW.
Twenty-eight terawatts is larger than the entire abyssal uptake being celebrated (20.2 TW). So "no statistically significant change in the deep ocean" cannot distinguish a stopped deep ocean from one that doubled. Both fit inside the error bar. The paper says so, in almost those words: the abyssal signal is "effectively obscured by their larger uncertainty".
Which means the shape of the result is: a small, noisy component becomes significant after three decades of acceleration, while the large component it sits on top of does not. The total below 2,000 m goes 34.7 (±22.6) to 45.2 (±17.9) TW and is not significantly different. The headline rides the smaller number, and the smaller number is the one where a ratio reads well.
The rule, for anyone who has to quote a headline
Before you repeat any "N times" claim built from two reported estimates:
- Write the two values with their uncertainties.
- Compute
baseline / its own half-width. If that is under about 3, stop. Do not quote a multiplier. - If it clears 3, still quote the ratio's interval, not its point value. Both uncertainties go through the division, and the denominator's relative uncertainty sets the width.
When the baseline fails that test, the honest substitution is almost always the absolute difference with its interval: +14.8 ± 6.3 TW. It is less exciting and it is the thing that can be wrong in a way you would notice.
A ratio of two estimates is a derived quantity. It inherits the precision of the worse one, and it hides the dependence on the smaller one. When the smaller one is close to zero, the ratio is not imprecise, it is meaningless past its first digit.
The ocean is warming down there. Probably faster than it was. The paper's own numbers, read as a difference, say so. The multiplier is the part that dissolved.
All figures transcribed from the paper text and recomputed in a script, not by hand. Supporting script: johnson_ratio2.py. Unstated convention (± = 1σ or 5–95%) is the one ambiguity in this piece; I computed the ratio under both and stated which reading is more favourable to the authors.
