A computing project on climate physics

The 3-Kelvin Baseline

Strip the climate down to a single column of air. Infrared light, convection, and water vapor are enough to explain 3 degrees of warming. Here’s the whole argument — scroll to descend.

The one mechanism you need

Greenhouse gases don’t trap heat. They raise the exit.

Nothing about the greenhouse effect requires a blanket, a lid, or a trap. It only requires that heat leaves from higher, colder air.

Start here

Earth sheds heat as infrared light. Most of it leaves not from the ground but from somewhere up in the air — the effective emission level.

CO₂ multiple2.0× · 7.0 km · 246 K

Drag, or use the arrow keys, to move the emission level yourself.

The effective emission level rises and cools as CO₂ increases. Anchored to the textbook 5.5 km / 255 K pair at present-day CO₂.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

Effective emission altitude and temperature against CO2 multiple.
CO₂ multiple	Emission altitude (km)	Emission temperature (K)
1×	5.5	255
1.5×	6.35	249.5
2×	6.95	245.6
2.5×	7.42	242.5
3×	7.8	240.1
3.5×	8.12	238
4×	8.4	236.2

The instrument

Fifty layers of air, and real quantum spectroscopy.

The column runs from the ground to 0.1 hPa in 50 layers. Measured absorption line data is precomputed into a lookup table, so thousands of experiments are cheap.

HITRAN2020 line data

100–2500 cm⁻¹

σ(ν, P, T) lookup table

precomputed, so a run is cheap

moist-convective adjustment

caps the lapse rate at 6.5 K/km

The 50-layer grid, spaced logarithmically in pressure from 1000 hPa at the surface to 0.1 hPa at the model top.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

The model's vertical grid: layer index, pressure, and radiative-convective equilibrium temperature.
Layer	Pressure (hPa)	Altitude (km)	Temperature (K)
1	1000	0	281.7
6	391	7.05	235.9
11	153	14.1	197.4
16	59.6	21.15	198.6
21	23.3	28.19	199.2
26	9.1	35.24	199.6
31	3.56	42.29	199.8
36	1.39	49.34	199.9
41	0.543	56.39	199.9
46	0.212	63.44	200

Test 1 · Does it look like Earth?

The model’s infrared spectrum has Earth’s bite taken out of it.

This is the light leaving the planet, wavelength by wavelength. Every notch is a gas, and the depth of each notch tells you how cold the air is where that light escapes.

Outgoing longwave radiance and brightness temperature against wavenumber.
Wavenumber (cm⁻¹)	Radiance (mW m⁻² sr⁻¹ (cm⁻¹)⁻¹)	Brightness temperature (K)
100	13.3011	225
200.17	38.5759	231.15
300.33	63.7396	239.79
400.5	82.4172	247.25
500.67	93.2791	254.12
600.83	112.4694	272.12
701	43.0497	220.82
801.17	113.9912	288
901.34	97.6931	288
1001.5	80.8955	288
1101.67	65.1093	288
1201.84	51.1502	287.98
1302	36.0873	284.16
1402.17	14.3028	260.67
1502.34	6.176	246.03
1602.5	4.0115	245
1702.67	2.8803	246.85
1802.84	3.8698	264.68
1903.01	5.6621	285.75
2003.17	4.3142	288
2103.34	3.0284	288
2203.51	2.111	288
2303.67	0.3704	257.3
2403.84	0.5875	275.62

Outgoing longwave radiation across 100–2500 cm⁻¹. The window at 800–1200 cm⁻¹ rides the 288 K surface envelope; the CO₂ band at 667 cm⁻¹ bottoms out near 216 K.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

matches MODTRAN reference

Zoom in 2,500×

Saturation is settled in the wings of a single line.

At the band centre the atmosphere is already opaque — adding CO₂ changes nothing there. The absorption that still has room to grow lives out in the faint edges of each line.

Absorption line shape at 667 cm⁻¹ for three pressures. Lorentz half-width scales with pressure; line area is fixed.
Offset (cm⁻¹)	1000 hPa	500 hPa	10 hPa
-0.5	0.087417	0.0443515	0.000974122
-0.0131669	4.39143	7.96562	1.39617
-0.000346736	4.54662	9.09078	314.689
0	4.54673	9.09167	346.956
0.000346736	4.54662	9.09078	314.689
0.0131669	4.39143	7.96562	1.39617
0.5	0.087417	0.0443515	0.000974122

Altitude10 hPa · stratosphere1000 hPa · surface

1000 hPasurface

A single CO₂ line at 667 cm⁻¹, on a logarithmic scale. Pressure broadening widens the Lorentzian wings near the surface; aloft the line collapses to a Doppler needle. The area under the line is held fixed.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

The wings are why the greenhouse effect doesn’t saturate.

pressure broadening resolved

Test 2 · Radiative-convective equilibrium

Radiation alone gets Earth wrong.

Strip convection out of the column and the physics still runs — it just produces a planet nobody would recognise.

0.0K

surface temperature

Radiation alone

Let infrared radiation set the temperature on its own and the surface reaches 303.8 K — and the lowest 5 km becomes so steeply stratified it cannot physically stand.

matches the standard atmosphere

Replay the comparison at your own pace.

Temperature profiles: radiative equilibrium, radiative-convective equilibrium, and the US Standard Atmosphere 1976.
Altitude (km)	Radiative equilibrium (K)	Radiative-convective (K)	US Standard 1976 (K)
0	303.8	281.7	288.15
5	258.8	249.2	255.65
10	227.353	216.7	223.15
15	212.725	197.63	216.65
20	205.919	198.438	216.65
25	202.754	198.97	221.65
30	201.281	199.321	226.65
35	200.596	199.552	237.05
40	200.277	199.705	251.05
45	200.129	199.805	265.05
50	200.06	199.872	270.65
55	200.028	199.915	259.45
60	200.013	199.944	245.45
65	200.006	199.963	231.45
70	200.003	199.976	217.45

Radiative equilibrium (303.8 K surface) against radiative-convective equilibrium (281.7 K), with the US Standard Atmosphere 1976 dashed. This is the classic Manabe–Wetherald (1967) result, reproduced from first principles.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

Test 3 · The greenhouse signature

Doubling CO₂ warms below and cools aloft.

This is the detail that settles the argument. A greenhouse gas and a brighter sun both warm the surface — but only one of them cools the upper atmosphere at the same time.

Temperature change against altitude for a doubling of CO2, with fixed relative humidity.
Altitude (km)	Temperature change (K)
0	3.2
5	3.19
10	3.17
15	2.815
20	1.8
25	1.15
30	0.72
35	0.42
40	0.2
45	0.044
50	-0.18
55	-0.42
60	-0.49
65	-0.42
70	-0.38

Temperature change against altitude for a CO₂ doubling. Warming is uniform through the troposphere, reverses sign at 46.5 km, and reaches −0.49 K near 60 km.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

Two ways to warm a planet

Stronger sun → warms everywhere

70 km

63 km

56 km

49 km

42 km

35 km

28 km

21 km

14 km

7 km

0 km

Extra sunlight heats the whole column. Nothing cools.

More CO₂ → warms below, cools aloft

70 km

63 km

56 km

49 km

42 km

35 km

28 km

21 km

14 km

7 km

0 km

The stratosphere radiates more efficiently than it absorbs, and loses heat.

The brighter-sun column is illustrative, drawn to contrast the shape of the response — the model itself was not run with a solar perturbation.

The amplifier

CO₂ opens the door. Water vapor walks through it.

Warmer air holds more moisture, and moisture is itself a greenhouse gas. Let humidity respond and the same CO₂ doubling produces far more warming.

CO₂ alone

humidity frozen

CO₂ + water vapor

humidity responds

the blue cap is +1.22 K from water vapor

0.00K

fixed absolute humidity

0.00K

fixed relative humidity

Gain

×1.61

Feedback factor

0.38

warm air holds ≈7% more moisture per kelvin (Clausius–Clapeyron)

Surface warming for a CO₂ doubling with humidity held fixed (1.98 K) and with relative humidity held fixed (3.2 K). Bars are drawn on a shared 0–3.6 K axis.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

Surface warming from doubling CO2, with and without the water vapor feedback.
Humidity treatment	Surface warming (K)
Fixed absolute humidity (no feedback)	1.98
Fixed relative humidity (with feedback)	3.2
Water vapor contribution	1.22
Gain	1.61
Feedback factor	0.38

Push it further

Each doubling adds about the same — but not quite.

Warming grows roughly with the logarithm of CO₂, so every doubling costs a similar amount. Look closely and the line bends upward: a warmer planet holds more water vapor, which strengthens the feedback.

0.00K

surface warming at 2× CO₂

Surface warming for successive CO2 doublings, with fixed relative humidity.
CO₂ multiple	Doublings	Surface warming (K)
1×	0	0.00
2×	1	3.20
4×	2	7.39
8×	3	13.07

Surface warming against CO₂ on a base-2 logarithmic axis, with the least-squares fit over the three forced cases: 4.94 K per doubling, R² = 0.993.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

2× CO₂

3.20 K

4× CO₂

7.39 K

8× CO₂

13.07 K

Can you trust it?

The arithmetic is tight. The omissions are not.

Numerical error in this model is measured in hundredths of a kelvin. The real uncertainty is everything the column was never asked to represent.

Spectroscopy±0.01 K

from perturbing line parameters

Grid resolution0.2 K

50 layers against 100

Missing physicsunbounded →

clouds, ozone, circulation

Uncertainty budget for the modelled climate sensitivity.
Source	Magnitude
Spectroscopy	±0.01 K
Grid resolution	0.2 K
Missing physics (clouds, ozone, circulation)	Unbounded

Uncertainty budget. The first two bars are drawn to scale against each other; the third runs off the panel because omitted physics has no bound to draw.Curves are schematic reconstructions anchored to the model’s published values; all quoted numbers are taken directly from the paper.

Table I · Verification

Verification results reported in Table I of the paper.
Check	Result	Note
Top-of-atmosphere imbalance	< 0.01 W m⁻²	convergence threshold
Grid resolution	−0.2 K	Ts at 50 layers minus Ts at 100
Lookup table vs line-by-line	RMS < 0.5 W m⁻²	flux error from the tabulated cross-sections
Spectroscopic perturbation · σ(Ts)	0.01 K	surface temperature spread
Spectroscopic perturbation · σ(ECS)	0.01 K	climate sensitivity spread
Forcing target	3.7 W m⁻²	per CO₂ doubling

What this model deliberately leaves out

The honest part of the answer.

This is a clear-sky, one-dimensional, longwave-only column. Its value comes from what it excludes as much as what it includes.

Clouds

The largest wildcard, and the main reason big models disagree with each other.

Sunlight & ozone

No shortwave heating, so the stratosphere here is structurally incomplete.

Oceans & circulation

One column. No weather, no transport, no heat taken up by the sea.

3 K is the baseline physics hands you. Everything above it is a claim about clouds.