Sunlight in, infrared out: find the temperature where they balance.
Ts – Imbalance N – W m⁻² Year 0

1. Radiation in, radiation out

The Sun delivers the solar constant $S_0\approx1361$ W m⁻² to a surface facing it at Earth's distance. Earth intercepts sunlight on a disc of area $\pi R^2$ but radiates from a sphere of area $4\pi R^2$, so the average incoming flux is $S_0/4\approx340$ W m⁻². A fraction $\alpha\approx0.30$ (the planetary albedo: clouds, ice, deserts) is reflected straight back.

Every body emits thermal radiation. A perfect emitter (black body) obeys the Stefan–Boltzmann law, and real surfaces emit a fraction $\varepsilon$ (the emissivity):

$$F=\varepsilon\sigma T^4,\qquad \sigma=5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$$

2. The zero-dimensional energy balance model

Treat the whole planet as one point with a single temperature $T_s$ and heat capacity $C$ (mostly the upper ocean). The rate of warming equals the energy imbalance $N$:

$$C\frac{dT_s}{dt}=N=\underbrace{\frac{S_0}{4}(1-\alpha)}_{\text{absorbed sunlight}}-\underbrace{\varepsilon\sigma T_s^4}_{\text{outgoing longwave}}+\Delta F$$

Without an atmosphere ($\varepsilon=1$) the equilibrium is the effective radiating temperature $T_e=\left[\frac{S_0(1-\alpha)}{4\sigma}\right]^{1/4}\approx255$ K $=-18$ °C. The observed mean surface temperature is about 288 K (15 °C). The 33 K difference is the natural greenhouse effect: water vapour, CO₂, methane and clouds absorb infrared radiation from the surface and re-emit part of it back down. In the zero-D model this is captured by an effective emissivity $\varepsilon\approx0.61$. The greenhouse effect can be measured as a flux:

$$G=\sigma T_s^4-\text{OLR}\approx390-239\approx150\ \text{W m}^{-2}$$

3. Layer models of the atmosphere

A more physical picture places one or more atmospheric layers above the surface. Each layer is transparent to sunlight but absorbs a fraction $\varepsilon$ of the infrared passing through it (Kirchhoff's law: it emits with the same $\varepsilon$), radiating both up and down. With a single fully absorbing layer, $T_s=2^{1/4}T_e\approx303$ K; with two, $T_s=3^{1/4}T_e\approx335$ K. Partly absorbing layers ($\varepsilon\approx0.49$ for two layers) reproduce today's 288 K. Each layer is colder than the one below, so the planet radiates to space from cold upper levels: this is the essence of the greenhouse effect. Adding CO₂ raises the effective emission height to colder air, reducing outgoing radiation until the surface and troposphere warm.

4. Radiative forcing from CO₂

CO₂'s strongest absorption bands are nearly saturated, so its forcing grows with the logarithm of concentration (Myhre et al., 1998):

$$\Delta F_{\mathrm{CO_2}}=5.35\,\ln\frac{C}{C_0}\ \text{W m}^{-2},\qquad \Delta F_{2\times}=5.35\ln 2\approx3.7\ \text{W m}^{-2}$$

Every doubling adds the same forcing: going from 280 to 560 ppm matters as much as going from 560 to 1120 ppm. This simplified expression gives the instantaneous-plus-stratospheric forcing; IPCC AR6 assesses the effective radiative forcing of doubled CO₂ (which includes fast atmospheric adjustments) at $3.93\pm0.47$ W m⁻². The lab uses the classic 5.35 ln(C/C₀) for simplicity, so its ECS values run about 6% lower than with the AR6 forcing for the same feedbacks.

5. Feedbacks and climate sensitivity

For small changes, linearise the outgoing radiation. The climate feedback parameter $\lambda$ (W m⁻² K⁻¹) says how much extra energy escapes per kelvin of warming:

$$C\frac{d\,\Delta T}{dt}=\Delta F-\lambda\,\Delta T,\qquad \lambda=\lambda_{\text{Planck}}-\sum_i b_i,\qquad \lambda_{\text{Planck}}=4\varepsilon\sigma T^3\approx3.3$$

(Here $\lambda$ is defined as positive for a stable climate. AR6 uses the opposite sign convention and quotes a Planck response of about −3.2 and a net feedback of about −1.2 W m⁻² K⁻¹.)

The Planck response alone would give $\Delta T_{2\times}\approx3.7/3.3\approx1.1$ K. Amplifying feedbacks $b_i$ reduce $\lambda$: water vapour (a warmer atmosphere holds about 7% more vapour per kelvin), surface albedo (melting snow and ice), and clouds, partly offset by the lapse-rate feedback. The equilibrium climate sensitivity

$$\text{ECS}=\frac{\Delta F_{2\times}}{\lambda}$$

is assessed by IPCC AR6 at a best estimate of 3 °C, likely (66%) range 2.5–4 °C and very likely (90%) range 2–5 °C. Most of the remaining uncertainty comes from cloud feedbacks. The sensitivity parameter is sometimes quoted instead as $1/\lambda$ in K per W m⁻² (about 0.8 for ECS ≈ 3 °C).

6. Ocean heat capacity and response time

The ocean's mixed layer (tens of metres deep) has heat capacity $C=\rho c_p h f_{\text{ocean}}\approx 1025\times3990\times h\times0.7$ J m⁻² K⁻¹. After a step in forcing, the linear model relaxes exponentially:

$$\Delta T(t)=\frac{\Delta F}{\lambda}\left(1-e^{-t/\tau}\right),\qquad \tau=\frac{C}{\lambda}$$

For $h=70$ m and $\lambda=1.2$, $\tau\approx5$ years. The deep ocean adds a much slower response (centuries), so today's warming is only part of the eventual equilibrium warming. The emissions tab uses a two-box model (mixed layer plus deep ocean) for this reason.

7. Multiple equilibria: the snowball

If albedo rises as the planet cools (more ice), the absorbed-sunlight curve becomes S-shaped and can cross the outgoing curve three times. The middle crossing is unstable; the outer two are stable, so the same sunlight can support a warm planet or a frozen one. Geological evidence suggests Earth went through "snowball" episodes in the Neoproterozoic, roughly 700 and 640 million years ago.

8. The carbon cycle

Carbon is exchanged between the atmosphere (about 900 GtC today, i.e. ~425 ppm, with 1 ppm ≈ 2.12 GtC ≈ 7.8 GtCO₂), land vegetation and soils (a few thousand GtC) and the ocean (roughly 38 000 GtC, mostly in the deep ocean). Of the CO₂ we emit, roughly 45% stays in the atmosphere (the airborne fraction); land and ocean sinks take the rest. A pulse of CO₂ decays on several timescales: in words, part of each tonne emitted is absorbed within decades, more over centuries, and about a fifth stays for millennia. An impulse-response function captures this:

$$C(t)=C_0+\frac{1}{2.124}\int_{-\infty}^{t}E(t')\,\Big[a_0+\sum_{i=1}^{3}a_ie^{-(t-t')/\tau_i}\Big]dt'$$

The $a_0$ term (about 20%) represents CO₂ that stays in the atmosphere for many millennia. That is why warming depends almost linearly on cumulative emissions: the transient climate response to cumulative emissions (TCRE) is about 0.45 °C per 1000 GtCO₂ (likely 0.27–0.63). A temperature limit therefore implies a finite carbon budget, and warming stops rising only once CO₂ emissions reach net zero.

9. Methane and other forcings

Methane is a much stronger absorber per molecule but has a perturbation lifetime of only about 12 years, so cutting methane slows warming quickly. Aerosols from burning coal and oil reflect sunlight and brighten clouds, masking part of the greenhouse warming (AR6: about −1.3 W m⁻² for 1750–2019, with a wide range of −2.0 to −0.6 W m⁻²; aerosols are the largest single source of uncertainty in total human forcing, assessed at 2.7 [2.0 to 3.5] W m⁻² for 2019). Cleaning up air pollution therefore removes some cooling, which the pathways model includes.

About the models. These are teaching models. The pathways tool uses round-number sector shares for 2024 (fossil CO₂ ≈ 37 GtCO₂, land-use ≈ 4 GtCO₂), a pulse-response carbon cycle based on Joos et al. (2013) with sink timescales tuned to reproduce the observed rise to about 423 ppm (2024 global mean), a one-box methane model, simple aerosol and other-gas forcings, and a two-box ocean energy balance. It reproduces the observed rise in CO₂ (the Today button's 425 ppm is the 2025 level) and the assessed human-induced warming (about 1.35 °C in 2024 relative to 1850–1900), but it is not an Earth system model. Single years scatter around that trend: 2024, boosted by El Niño, was the first calendar year about 1.5 °C above pre-industrial, which is not the same as passing the Paris 1.5 °C level (defined on 20-year averages).

Example 1: Earth's effective temperature

  1. Absorbed sunlight: $\frac{S_0}{4}(1-\alpha)=\frac{1361}{4}\times0.70=238.2$ W m⁻².
  2. Set equal to $\sigma T_e^4$: $T_e=\left(\frac{238.2}{5.67\times10^{-8}}\right)^{1/4}=(4.20\times10^{9})^{1/4}=254.6$ K.
  3. That is −18.6 °C, about 33 K colder than the observed 288 K.

Example 2: What effective emissivity gives 288 K?

  1. $\sigma T_s^4=5.67\times10^{-8}\times288^4=390.1$ W m⁻².
  2. $\varepsilon=238.2/390.1=0.611$. The lab's default 0.612 gives 287.8 K (14.7 °C).
  3. The greenhouse effect is $G=390.1-238.2\approx152$ W m⁻².

Example 3: Forcing and equilibrium warming for today's CO₂

  1. CO₂ alone, at about 425 ppm (2025): $\Delta F=5.35\ln(425/280)=5.35\times0.417=2.23$ W m⁻².
  2. Planck feedback at 288 K: $\lambda_P=4\varepsilon\sigma T^3=4\times0.612\times5.67\times10^{-8}\times288^3=3.31$ W m⁻² K⁻¹.
  3. With fast feedbacks $b=2.1$: $\lambda=3.31-2.1=1.21$ W m⁻² K⁻¹.
  4. Equilibrium warming: $\Delta T=2.23/1.21\approx1.8$ K; for doubled CO₂, $\text{ECS}=3.71/1.21=3.1$ K. Press Today in the lab and compare the equilibrium readout: about 16.5 °C, i.e. +1.8 K above the 14.7 °C starting state. (After you press Today the lab's ECS readout drops from 3.07 K to about 2.9 K. It is slightly lower than this linear estimate because the Planck response grows as $T^3$ as the planet warms.)
  5. The observed warming so far (about 1.3 °C) is smaller for two reasons: the deep ocean has not caught up (the climate is not in equilibrium), and aerosols offset part of the greenhouse forcing.

Example 4: How fast does the ocean respond?

  1. Mixed layer $h=70$ m covering 70% of the globe: $C=1025\times3990\times70\times0.7=2.0\times10^8$ J m⁻² K⁻¹.
  2. $\tau=C/\lambda=2.0\times10^8/1.21=1.66\times10^8$ s $=5.3$ years.
  3. After one $\tau$ the planet has reached $1-e^{-1}=63\%$ of its equilibrium warming; after $3\tau\approx16$ years, 95%. A deeper mixed layer (try 200 m) slows the response proportionally.

Example 5: A carbon budget in years

  1. AR6 1.5 °C budget (50%) from 2020: 500 GtCO₂. Emissions ≈ 40 GtCO₂ per year.
  2. At constant emissions: $500/40=12.5$ years, i.e. exhausted around 2032.
  3. With a straight-line decline to zero, the budget is the area of a triangle: $\tfrac12\times40\times T=500\Rightarrow T=25$ years, i.e. net zero by about 2045.
  4. Using TCRE: 500 GtCO₂ × 0.45 °C/1000 GtCO₂ ≈ 0.23 °C of extra CO₂-driven warming.
  1. No greenhouse. In the zero-D model set ε = 1 and the fast-feedback slider to 0 (CO₂ at 280 ppm, ice–albedo off). Confirm the equilibrium is about 255 K. (auto-checked)
  2. Measure ECS. Starting from pre-industrial, press Double CO₂ and run until the imbalance is below 0.05 W m⁻². Your warming should match the ECS readout. (auto-checked)
  3. Snowball and hysteresis. Turn on the ice–albedo feedback and press Snowball start. Then raise S₀ and CO₂ until the planet thaws, and bring them back to today's values. Does it freeze again? How far above 1361 W m⁻² did you need to go? (auto-checked on thaw)
  4. Hold 2 °C. In the pathways model keep peak warming below 2.0 °C with an ECS of 3 °C, using no more than 5 GtCO₂ per year of carbon capture (the Rapid net zero preset uses 10, so you will need other levers). (auto-checked)
  5. Net zero by 2050. Reach net-zero CO₂ by 2050 using no more than 5 GtCO₂ per year of carbon capture. (auto-checked)
  6. Aerosol unmasking. Choose Rapid net zero and look closely at warming in the 2030s. Why does it keep rising for a while even as emissions fall steeply? (Hint: what else does burning coal emit?)
  7. The value of methane. From Current policies, compare the peak warming change from a 60% methane cut with that from a 60% deforestation cut. Which acts faster, and why?
  8. Two-layer model. Choose Two-layer atmosphere, set the layer ε to 1 and the fast-feedback slider to 0. Check that $T_s\approx3^{1/4}\times255\approx335$ K.

Two rainy seasons, two oceans

Much of Kenya, Uganda and Tanzania has a bimodal rainfall regime: the "long rains" around March–May and the "short rains" around October–December, as the tropical rain belt passes overhead twice a year. Large parts of northern and eastern Kenya are arid or semi-arid, so a single failed season can be devastating for pastoralists and rain-fed farmers.

The short rains are strongly linked to sea-surface temperatures. A positive Indian Ocean Dipole (warmer western Indian Ocean) and El Niño tend to bring wetter short rains, as in 1997, 2019 and 2023, when flooding was widespread. La Niña and a negative IOD tend to bring drier short rains; the 2020–2022 sequence of failed seasons during a long La Niña contributed to the worst drought in the Horn of Africa in decades.

Heat

East Africa is warming at roughly the global rate or a little faster over land. Higher temperatures raise evaporative demand, stress crops during flowering and livestock, and push some highland malaria transmission upslope.

Heavier downpours

A warmer atmosphere holds about 7% more water vapour per °C, so the heaviest rainfall events tend to intensify even where total rainfall changes little, increasing flood and landslide risk.

The East African paradox

Many climate models project wetter conditions, especially in the short rains, yet the long rains declined from the 1980s to the 2010s. Resolving this mismatch is an active research question, so projections should be read with care.

Food and water

Most farming in the region is rain-fed. Variability matters as much as averages: a late onset or a dry spell at flowering can ruin a season. See the Crop & Water Balance lab for adaptation options such as water harvesting and drought-tolerant varieties.