01 · The idea
A box is just bookkeeping
A box model is the least clever idea in science, and that is exactly its strength. You draw a box around some part of the world, count what goes in, count what comes out, and whatever is left over is what is inside.
A box has no interior geography. Everything in it is treated as perfectly mixed, one number for the whole thing, so the model can tell you how much carbon is in the ocean but never whereabouts. As a result of this simplicity, we can build a model that runs a million years of planetary chemistry in under a minute and whose every term you can point at and defend.
- dN/dt · how fast the amount in the box is changing
- Fin · everything arriving, per year
- Fout · everything leaving, per year
That is the whole equation. Every box in every box model ever built is this one sentence. The rest of this page is about what you put into Fin and Fout, and how you know.
02 · One box
The bathtub
Give the outflow one property, that it grows in proportion to how full the box is, and the box acquires a personality. A fuller bathtub pushes harder on the drain. Written down, the outflow becomes N/τ, where τ ("tau") is the residence time.
- N/τ · the fuller the box, the faster it drains
- τ · residence time, in years
Two things follow, and both matter later. The box settles at N = Fin × τ, inflow times residence time, wherever it started from. And if you dump in a slug of extra carbon, the excess decays away as a clean exponential, e−t/τ.
One box, live
Drag either control. The dashed line is where the box will settle.
Hold on to that exponential. The whole argument of this page is that the atmosphere behaves nothing like this box, and the way you find out is that a single exponential cannot be made to fit what really happens.
03 · Two boxes
Air and ocean
One box cannot represent a planet, because a planet has places for carbon to go. Add a second box for the ocean and connect the two. The connecting flux is driven by a difference in partial pressure: gas moves from wherever its pressure is higher toward wherever it is lower, at a rate set by how quickly the sea surface can exchange.
- Fas · net carbon crossing the sea surface
- pCO₂ · the pushing pressure of CO₂ on each side
- kas · how fast the surface can exchange, giving a timescale of about a year
Everything now hangs on one question. When you push carbon into seawater, how hard does its pCO₂ push back? Answer that and you know how much the ocean will take. Get it wrong and you will be off by a factor of four, in a direction that has fooled a lot of people.
04 · Provenance
Where the numbers come from
The fair objection to any model is that its numbers were chosen to make the answer come out right. The defence is to make the choosing visible, so here it is in full.
ANEMONE prescribes only quantities that have been measured, then derives everything else from a single requirement: that the preindustrial world sat still. If carbon was neither accumulating nor draining away in 1750, then the interior of the ocean has no freedom left. Its concentrations follow by arithmetic.
You do not have to take my word for any of this. The model is about a thousand lines of Python and the calibration routine that produces the right-hand column below is one function you can read in a few minutes: github.com/nems808/ANEMONE.
Prescribed, because measured
Derived, as outputs
Two of those derived numbers are the tell. Nobody asked the model to produce an export production of 10 GtC/yr or a rain ratio of 0.12. They fall out of the steady-state requirement, and they happen to match what the ocean is observed to do.
05 · The certain part
Chemistry is not up for debate
Models get argued about because they contain judgement. This part contains none of it. When CO₂ dissolves in seawater it reacts, and the proportions it settles into are fixed by thermodynamics, the same law that decides which way heat flows. Those proportions have been measured in laboratories to four decimal places, published with their uncertainties, and re-measured by people who would have been delighted to prove the first group wrong.
Everything in the panel below is computed live from those published constants,56789 running the same equations the model runs,10 including the boron that quietly does part of the buffering. Nothing here has been tuned. There is nothing here that could be tuned.
Seawater carbonate system, live
Dissolved carbon and alkalinity set everything else. Move them.
The number to watch is the Revelle factor. It answers the question from the last section. Raise dissolved carbon by 1% and pCO₂ rises by about 10%, so the ocean pushes back roughly ten times harder than you pushed. That is the buffer, and it is arguably the most important single number in the carbon cycle.
Now drag DIC to the right and watch the Revelle factor climb. The buffer weakens as you use it. Hold that thought until section 14.
06 · Stage one
The ocean as soda water
Start with the simplest defensible ocean. CO₂ dissolves in it the way it dissolves in a fizzy drink, following Henry's law, and nothing further happens. Instant mixing, no chemistry, no biology.
Two intuitions compete here, and it is worth committing to one before you read on. The reservoir argument says the ocean holds about sixty times the carbon the atmosphere does, so it ought to swallow around 98% of whatever we emit. The soda-water argument says run Henry's law and see.
76% still in the air, forever
Both intuitions miss, and they miss from opposite sides. As plain soda water the ocean holds only about 200 GtC of dissolved CO₂, which is less than the atmosphere contains. The sixtyfold reservoir is real enough, but hardly any of it is dissolved gas. It is something else, and that something else is the next section.
07 · Stage two
The reaction that does the work
Dissolved CO₂ does not stay as CO₂. It reacts with water and with the carbonate ion already present in seawater, and almost all of it ends up as bicarbonate. That is where the ocean's enormous carbon inventory actually lives, as ions rather than as gas.
- Incoming CO₂ gets consumed, so it stops pushing back
- It eats a carbonate ion in the process
- Fewer carbonate ions left means the next molecule is harder to absorb
Adding this one reaction moves the answer by a factor of four, from 76% airborne down to 18%. It is the largest single correction anywhere in the model, and it comes straight out of thermodynamics with nothing fitted.
08 · Stage three
The ocean is not one box
Stage 2 assumed the whole ocean mixes instantly. It does not. Surface water exchanges with the deep over centuries, and until it does, the deep ocean's enormous capacity is simply unavailable.
So the ocean gets split into three: surface, then the thermocline, then the deep, with carbon moved between them at prescribed ventilation rates. Why three rather than two? Because the real decay has both a decades-long mode and a centuries-long one, while a two-box ocean has exactly one mixing timescale. Tune it to match the century and it overshoots the millennium. That was tried first, and it does not work.
Notice what this does to the hundred-year answer. It goes up, from 18% to 44%. Slowing the ocean down leaves more carbon in the air on human timescales, even though the thousand-year answer barely shifts.
09 · Stage four
The seafloor answers
Acidify the deep ocean and it starts dissolving its own floor. The seabed holds a reactive layer of calcium carbonate, mostly the shells of things that died and sank, and when the water above turns corrosive enough that carbonate goes back into solution, handing back the alkalinity the incoming CO₂ consumed.1
This is the ocean partially repairing its own buffer. It takes thousands of years and it pulls the airborne fraction down to roughly 8% at a hundred thousand years, but it can never finish the job, because all it does is redistribute carbonate that was already in the ocean system.
- Ω · the saturation state of the deep water; below 1 it dissolves carbonate
- Msed · reactive carbonate available on the seafloor
- The further below saturation, the faster it dissolves
10 · Stage five
The thermostat
The last process is the slowest and the strangest of them: rock. CO₂ dissolved in rainwater attacks silicate minerals on land, and the products wash down to the sea where they are eventually buried as carbonate. Carbon leaves the ocean and atmosphere altogether and becomes rock again.
What makes this a thermostat is that the reaction runs faster when the planet is warmer and when there is more CO₂ around to dissolve. Push CO₂ up and weathering speeds up, drawing it back down. Let CO₂ fall and weathering slows, letting volcanoes refill the air.
That second half matters as much as the first, and it is easy to skip past. Weathering is only one arm of the loop. The other is volcanic outgassing, which returns carbon from the crust to the atmosphere at a rate that does not care what the climate is doing. A one-way sink would strip the air bare; a one-way source would cook the planet. It is the pairing of a constant source with a temperature-sensitive sink that makes a thermostat, in exactly the way a house needs both a furnace running flat out and a valve that responds to the room.
- More CO₂ in the air gives faster weathering
- Warmer gives faster weathering, roughly an e-folding per 12 K
- Both terms push the planet back toward the point where volcanoes and weathering balance
Left alone, the model settles at 279.99 ppm. Nobody set it there. That is simply the CO₂ level at which silicate weathering consumes exactly what volcanoes emit, and the thermostat finds its own setpoint.
11 · The slow budget
What the thermostat actually costs
A feedback is not interesting until you know how strong it is. The previous section described a loop; this one puts the numbers on it, and the numbers are the reason the loop matters over millions of years and not over centuries.
Here is the entire geological budget of the model, in and out of the surface system, meaning the atmosphere and ocean taken together. Every figure is the model's own calibrated steady state.
| Flux | GtC/yr |
|---|---|
| Volcanic outgassing Carbon out of the crust. Does not respond to climate. | +· |
| Carbonate weathering Carbon dissolved out of limestone and carried to the sea. | +· |
| Carbonate burial Carbon locked back into rock on the seafloor. | −· |
| Net | 0.000 |
Something is missing from that table, and its absence is the whole point: silicate weathering does not appear in it. Silicate rock contains no carbon, so weathering it adds none to the sea. What it delivers instead is alkalinity, and alkalinity is what forces carbon to precipitate. The reason burial (·) exceeds what carbonate weathering supplied (·) is precisely the · GtC/yr of burial that silicate weathering paid for out of the atmosphere.
That is the difference between the two weatherings, and it is worth being slow about, because it is the thing most often gotten backwards. Follow one atom of calcium through each path:
The carbonate path
- Takes 1 CO₂ from the air, and one carbon out of the limestone
- Buries 1 carbon, and hands the CO₂ straight back
- Net effect on the air over a full circuit · zero. A round trip, not a sink.
The silicate path
- Takes 2 CO₂ from the air, and nothing from the rock — silicate has no carbon in it
- Buries 1 carbon, and hands 1 CO₂ back
- Net effect on the air · one CO₂ removed, permanently
Notice that the second line is identical in both. The burial step does not know or care which rock the calcium came from; the entire difference between a sink and a round trip is made on land, in how much CO₂ the first line had to spend to free that calcium.
So the only true geological sink in the model is silicate weathering, and in steady state it must equal the volcanic source exactly. The model does not discover this; it imposes it, in one line, when it calibrates: Fsil,0 = Fvolc. That single identity is what gives the thermostat a setpoint to find.
Divide the surface system's entire carbon inventory, · GtC of atmosphere plus ocean, by that · GtC/yr, and you get a turnover time of about · years. That is how long it takes to cycle every carbon atom in the air and the sea through rock, and it is why the charts in the next sections run to a million years rather than stopping at a thousand.
The restoring force, live
Weathering against outgassing. Where the curve crosses the flat line, the planet is in balance — drag away from it and read what pulls it back.
Two things are worth doing with that panel. The first is to drag CO₂ far to the right and watch the imbalance grow: the thermostat pushes harder the further you shove it, which is what makes the setpoint stable rather than merely a place the system happens to sit. The second is to drag CO₂ below 280 and watch the imbalance reverse. Weathering falls behind the volcanoes and the air refills. The thermostat has two directions, and the recovery from a Snowball Earth runs on the second one.
The exponent slider is there because that number is not well known. Move it from 0.1 to 0.5 and watch what does and does not change. The setpoint does not move at all, not even slightly: at 280 ppm the ratio pCO₂/pCO₂0 is exactly 1, and 1 raised to any power is still 1, so the crossing sits where the steady-state identity puts it no matter what the exponent is. What the exponent changes is the slope through that crossing, and with it the restoring time, which more than doubles across the range.
That split is the honest state of the field, and it is why the equation is worth trusting further than its own coefficients. The existence of the feedback follows from bookkeeping that has to balance. Its strength is an empirical question that is still open, which is what section 17 returns to.
Now the number that should stop you. Silicate weathering removes · GtC/yr. Fossil fuel and cement emissions in · were · GtC/yr — about · times faster. The thermostat is real, it works, and it is running roughly two orders of magnitude too slowly to be of any use to anyone now alive.
Which is also why stage five is the only stage that ever finishes. Stage four hands back alkalinity from the seafloor and stalls with · of the pulse still airborne, forever, because it is only rearranging carbonate the ocean already had. Stage five is the one process that takes carbon out of the system altogether, and it drives the airborne fraction to · — but it needs 400,000 years to do it.
12 · The licence
Why we are allowed to assume that
A negative feedback mechanism for the long-term stabilization of Earth's surface temperature
Journal of Geophysical Research 86, 9776–9782 · 1981 4
Every term in the weathering law of section 10 is an assumption, and assumptions in a model need a source. This is the source, known universally as WHAK after its three authors.
The problem it solved was the faint young Sun. Early in Earth's history the Sun was about 30% dimmer, which should have left the planet frozen solid. It was not frozen: there is liquid water in the rock record throughout. Something held the temperature up while the Sun brightened, and kept holding it as conditions changed underneath.
WHAK's answer was that the silicate weathering feedback does this automatically, with no biology required and no coincidence required. The way weathering responds to CO₂ and to temperature, which is to say the exponents in the law two sections back, is what makes Earth's climate self-correcting over millions of years.
This is what standing on the shoulders of others looks like in practice. Stage 5 is four lines of code. Those four lines are licensed by forty years of work testing whether the feedback is real, and the honest caveat is that its exponents remain poorly constrained, which is why section 17 asks for a band rather than a line.
13 · The result
Five stages, five answers
Here is every stage on one plot: the airborne fraction of a 1,000 GtC pulse against time, from one year out to a million. Time runs on a logarithmic axis, which is the only way to fit five distinct timescales onto one page.
Toggle the stages on and off. Each one adds a single process, and each one changes the answer somewhere different along the axis.
There is no single number here. The decay is a sum of exponentials spanning five orders of magnitude in time, so "the lifetime of CO₂" describes a choice about which timescale you care about rather than a fact about the world. The impulse response function used by the IPCC contains a constant term with no time constant attached to it at all, 21.7% of a pulse that simply never decays.3 This model is an attempt to say what that 21.7% really is.
14 · The twist
The ocean gets worse at its job
One last thing, and it is the part people find hardest to believe. Everything so far has been described in fractions, as though the ocean took a fixed share. It does not work that way. Emit twice as much and the ocean takes a smaller share of a bigger number.
You already know why. The Revelle factor climbs as carbon goes in. Every molecule absorbed eats a carbonate ion and makes the next one harder to absorb. Drag the slider.
For scale, cumulative human emissions so far come to roughly 700 GtC. Every curve here is a separate run of the full model rather than an interpolation of one.
15 · The confrontation
Does any of this match the real world?
Everything so far has been an idealised pulse dropped into a preindustrial ocean. Reality ran a different experiment, and we have the records. So take the actual fossil-fuel and cement emissions from 1750 onward,11 feed them into the model, and see where the atmosphere ends up.
Nothing in the model was fitted to the curve below. The parameters were all set from section 4 and never touched again.
The observations come from two sources spliced together: Law Dome ice cores up to 1958,12 then direct measurement at Mauna Loa.13 The temperature record is HadCRUT5.14 The ocean side of the model has its own check against the Hawaii Ocean Time-series at Station ALOHA:15 over 1988 to 2024 the model acidifies at 2.0 × 10⁻³ pH units per year against 1.9 × 10⁻³ observed.
The model lands a few ppm low, and there is a reason for it that was decided in advance. ANEMONE has no land biosphere, so emissions from deforestation and land-use change are left out of the forcing. Those come to roughly 200 GtC historically, and they very nearly cancel against the carbon the land has taken back up over the same period. Comparing fossil-only emissions against an atmosphere-plus-ocean model is the like-for-like test.
The warming envelope is wider than the CO₂ envelope on purpose. It spans the IPCC AR6 likely range for climate sensitivity, 2.5 to 4.0 K per doubling,18 because how much warming you get per unit of CO₂ is much less certain than where the CO₂ goes. The model warms instantly, with no ocean thermal lag, so it should sit above the observed transient during a ramp-up. It does, and the observations track the lower edge.
16 · The independent test
The tracer nobody meant to release
Matching CO₂ is a weaker test than it looks. The ventilation rates in section 8 were tuned against the observed impulse response,3 so getting the airborne fraction right is partly circular. A model earns trust by predicting something it was never shown.
Atmospheric nuclear testing provides exactly that. Bomb tests through the 1950s and early 60s nearly doubled the amount of radiocarbon in the air,17 a global tracer-release experiment nobody would have been allowed to run on purpose. Watching where that spike went, and how fast, tests the ocean mixing directly.
Two separate things get tested here, and the model passes one of them.
The decadal air-sea exchange holds up. From the 1965 peak onward, the modelled decline tracks the observations16 closely. The peak itself overshoots, for reasons that were known before the run: a one-box atmosphere has no stratosphere to delay the tropospheric peak, and no land biosphere, which took up something like a quarter of the bomb carbon.
The deep ocean comes out too young. The preindustrial radiocarbon gradients are predictions, never inputs, so they are a clean test. Here is what the model says against what the ocean says.
The deep box is roughly 40‰ too young. Tuned to reproduce the century-scale airborne fraction, this ventilation scheme cannot also reproduce the radiocarbon age of the deep ocean. Two timescales are not enough to stand in for the real ocean's continuum of them.
This is what a genuine validation result looks like. The summary is that the model is correct for the timescale it was calibrated for, and too vigorous for the millennial deep ocean, and you should trust its century-scale answers more than its ten-thousand-year ones.
17 · Honesty
What this model gets wrong
“All models are wrong, but some are useful.” George Box 19
Box was writing about statistics, and the line has been worn smooth by repetition, but it earns its keep here. A model you cannot criticise has stopped being a model and become a sales pitch. So these are the places where ANEMONE is known to be wrong, roughly in the order they would change an answer you actually care about.
Ventilation is imposed
The rates at which water moves between ocean layers were tuned to match the observed impulse response. There is no fluid dynamics in a box model, so it cannot derive them.
Two mixing timescales, not a spectrum
The real ocean ventilates across a continuum. Three boxes give two timescales. Section 15 shows what that costs.
No land biosphere, no permafrost
Both matter most in the first century, which is exactly the period most people are asking about.
Warming is instantaneous
Equilibrium climate sensitivity is applied with no ocean thermal inertia. Adequate for the weathering feedback, wrong for anything about committed warming.
Weathering exponents are loose
Sweeping the silicate exponent across its plausible range visibly moves the 400,000-year tail. An honest presentation shows a band.
Sediment inventory is approximated
CaCO₃ compensation depends on an assumed reactive sediment inventory of about 1,560 GtC. Full sediment models do real work that this approximates away.2
CO₂ lingering for millennia does not imply that warming keeps climbing for millennia. If emissions stopped, temperature would stay roughly flat rather than continue rising, because falling CO₂ roughly cancels the lag from ocean thermal inertia. Long CO₂ lifetime and continued warming are separate claims, and running them together is the most common mistake made about this science.
18 · The answer
How long until it is gone
Which brings us back to the question this page opened with. Here is the model's answer for a 1,000 GtC pulse, about what another half-century of business as usual would add on top of what has already been emitted. It is the same stage-five curve from section 13, read a different way: not how much is left at time t, but how long until a given fraction has gone.
Read down that column. Every step takes five to ten times longer than the step before it, and the pattern holds all the way down. There is no point where the curve turns and finishes — it only gets flatter, because each process that takes over from the last is slower than it was and has less left to work on.
This is the hidden assumption in the question, and it is why methane is not a useful analogy. Methane has a lifetime because methane has one destruction process. Carbon dioxide has five, spread across five orders of magnitude in time, and any single number that summarises them has thrown away the thing you wanted to know.
What we emit decides which end of that staircase we stand on. Between the smallest and largest pulses the model was run with, a factor of ten in emitted carbon, the fraction still airborne after a century moves from · to ·. After 400,000 years it moves from · to ·, which is to say it does not move at all. The far end is fixed, because silicate weathering does not care how much carbon it has to remove, only how fast it can remove it. The near end is not fixed, and the near end is the part measured in centuries.
None of which tells anyone what to do about it, and the model has no opinion to offer. Read it alongside the caveat that closes section 17: carbon staying up for millennia is not the same claim as temperature climbing for millennia. All this model does is say how long the carbon stays, and say it in a shape that no single number can hold.
References
Where this comes from
Numbers in the text link here. Every curve on this page was generated by running the model itself, and the carbonate chemistry in section 5 runs the same published equilibrium constants live in your browser.
github.com/nems808/ANEMONE the model, the data, and the scripts that made every figure here
- Archer, D. (2005) "Fate of fossil fuel CO2 in geologic time". J. Geophys. Res. 110, C09S05.
- Archer, D. et al. (2009) "Atmospheric lifetime of fossil fuel carbon dioxide". Annu. Rev. Earth Planet. Sci. 37, 117–134.
- Joos, F. et al. (2013) "Carbon dioxide and climate impulse response functions for the computation of greenhouse gas metrics: a multi-model analysis". Atmos. Chem. Phys. 13, 2793–2825.
- Walker, J. C. G., Hays, P. B. & Kasting, J. F. (1981) "A negative feedback mechanism for the long-term stabilization of Earth's surface temperature". J. Geophys. Res. 86, 9776–9782.
- Weiss, R. F. (1974) Mar. Chem. 2, 203–215. CO₂ solubility in seawater.
- Lueker, T. J., Dickson, A. G. & Keeling, C. D. (2000) Mar. Chem. 70, 105–119. Carbonic acid dissociation constants K₁ and K₂.
- Dickson, A. G. (1990) Deep-Sea Res. 37, 755–766. Boric acid dissociation.
- Millero, F. J. (1995) Geochim. Cosmochim. Acta 59, 661–677. Water dissociation and pressure corrections.
- Mucci, A. (1983) Am. J. Sci. 283, 780–799. Calcite solubility.
- Uppström, L. R. (1974) "The boron/chlorinity ratio of deep-sea water from the Pacific Ocean". Deep-Sea Res. 21(2), 161–162. Total boron in seawater.
- Friedlingstein, P. et al. (2025) "Global Carbon Budget 2024". Earth Syst. Sci. Data 17, 965–1039. Fossil fuel and cement emissions, retrieved via Our World in Data.
- MacFarling Meure, C. et al. (2006) "Law Dome CO2, CH4 and N2O ice core records extended to 2000 years BP". Geophys. Res. Lett. 33, L14810.
- Keeling, C. D. and NOAA Global Monitoring Laboratory. Mauna Loa atmospheric CO₂ record.
- Morice, C. P. et al. (2021) "An updated assessment of near-surface temperature change from 1850: the HadCRUT5 data set". J. Geophys. Res. Atmos. 126, e2019JD032361.
- Dore, J. E. et al. (2009) "Physical and biogeochemical modulation of ocean acidification in the central North Pacific". PNAS 106, 12235–12240. Station ALOHA, Hawaii Ocean Time-series.
- Hua, Q. et al. (2021) "Atmospheric radiocarbon for the period 1950–2019". Radiocarbon 64(4), 723–745.
- Naegler, T. & Levin, I. (2006) "Closing the global radiocarbon budget 1945–2005". J. Geophys. Res. 111, D12311.
- IPCC (2021) Climate Change 2021: The Physical Science Basis. Working Group I contribution to the Sixth Assessment Report. Cambridge University Press. Equilibrium climate sensitivity likely range, 2.5 to 4.0 K.
- Box, G. E. P. (1976) J. Am. Stat. Assoc. 71, 791–799, "Science and statistics", where the argument first appears. The exact wording most people quote, "essentially, all models are wrong, but some are useful", is from Box & Draper (1987), Empirical Model-Building and Response Surfaces, p. 424.