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If plants and the ocean absorb CO₂, why does it build up in the air? Before the Industrial Revolution, what nature took from the air was given back in equal measure. Human emissions are an extra flux, with no counterpart. Plants and the ocean respond to the excess and start absorbing more, but only part of it. In the simulator, of the roughly 750 GtC emitted from 1850 to 2025, about 42% stayed in the air, 26% went into the ocean and the rest, about a third, went into plants and soils (almost as much as deforestation took from them). See the Fate graph.
If the ocean is so large, why doesn’t it absorb all the extra CO₂? For three reasons. Only the top layer (about 100 m) exchanges CO₂ with the air, and the water takes decades to reach the intermediate layer and about a thousand years to reach the deep ocean. Next, the CO₂ that dissolves reacts with the water and with the carbonate dissolved in it; the more carbon the ocean has already absorbed, the less carbonate is left for this reaction, and each extra GtC becomes harder to absorb (Revelle factor, from 9.5 to about 11 today; it is the same chemistry as ocean acidification). Finally, warmer water dissolves less gas. In the Constant emissions scenario, the ocean absorbs more carbon per year in 2100 than today (4.2 versus 3.5 GtC/yr), but, for each extra ppm of CO₂ in the air, it absorbs only half of what it absorbs today.
What does the ΔT × GtC graph show? Warming grows almost in direct proportion to the total carbon already emitted, and not to the rate of emissions. In the simulator, it is about 1.6 °C for every 1000 GtC (the IPCC estimates 1.65 °C, with a likely range of 1.0 to 2.3 °C). This is where the idea of a carbon budget comes from: to stay below a given level of warming, only a limited total amount can be emitted, and warming only stops rising when emissions get close to zero.
What happens when emissions reach zero? In the Zero in 25 years scenario, warming reaches about 1.4 °C around 2050 and then falls slowly (1.3 °C in 2100). Two effects almost cancel out: the CO₂ in the air decreases, because plants and the ocean keep absorbing it, but the deep ocean, still colder than the surface, takes up less and less heat. CO₂ does not return to 280 ppm: in the Pulse scenario, about 43% of the carbon released is still in the air after 100 years, and 21% after 1000 years.
Why does the simulator consider only CO₂? CO₂ is the gas that contributes most to warming and the one that stays longest in the air, and it is the only one that circulates between the reservoirs shown here. Other gases, such as methane and nitrous oxide, also warm the planet, and aerosols from pollution cool it. Therefore, the ΔT of the simulator is the warming caused by CO₂, and it should not be compared directly with the total observed warming. The 1.5 °C and 2 °C lines in the Temperature and ΔT × GtC graphs mark the targets of the Paris Agreement, which apply to total warming (with all these effects): they serve as a reference, not as an exact comparison.
Box model with five carbon reservoirs (stocks in GtC; 1 ppm of CO₂ ≈ 2.124 GtC) and two temperature layers, integrated with 4th-order Runge-Kutta (time step of 0.25 year).
Carbon. Atmosphere $C_A$, vegetation $C_V$, soils $C_S$, surface ocean $C_M$ (down to 100 m) and deep ocean, divided into the intermediate layer $C_I$ (100 to 1000 m) and the bottom $C_P$ (the simulator shows the sum $C_I + C_P$):
$$\begin{aligned} \dot C_A &= R_V + R_S + F_{\text{out}} - P\\ &\quad - F_{\text{in}} + E_{\text{fos}} + E_{\text{def}}\\ \dot C_V &= P - R_V - L - E_{\text{def}}\\ \dot C_S &= L - R_S\\ \dot C_M &= F_{\text{in}} - F_{\text{out}} + F_{\text{up}} - F_{\text{down}}\\ \dot C_I &= F_{\text{down}} - F_{\text{up}} + G_{\text{up}} - G_{\text{down}}\\ \dot C_P &= G_{\text{down}} - G_{\text{up}} \end{aligned}$$Adding up the six equations, only $E_{\text{fos}}$ remains: total carbon only increases through the burning of fossil fuels; deforestation ($E_{\text{def}}$) transfers carbon from vegetation to the atmosphere.
Land fluxes. Photosynthesis $P$, plant respiration $R_V$, fall of leaves and branches $L$ and soil respiration $R_S$:
$$\begin{aligned} P &= P_0\,[1 + \beta \ln(C_A/C_{A0})]\\ R_V &= P/2\\ L &= \tfrac{P_0}{2}\,C_V/C_{V0}\\ R_S &= \tfrac{P_0}{2}\,(C_S/C_{S0})\,Q_{10}^{\,T/10} \end{aligned}$$with $P_0 = 120$ GtC/yr; the factor $Q_{10}^{\,T/10}$ only applies with the soil feedback on.
Ocean fluxes. Exchange with the air ($F_{\text{in}}$, $F_{\text{out}}$), mixing of the surface with the intermediate layer ($F_{\text{down}}$, $F_{\text{up}}$) and of the intermediate layer with the bottom ($G_{\text{down}}$, $G_{\text{up}}$):
$$\begin{aligned} F_{\text{in}} &= k\,p_{\text{atm}}\\ F_{\text{out}} &= k\,p_{\text{oc}}\\ p_{\text{oc}} &= p_q(D)\,e^{0{,}0423\,T}\\ F_{\text{down}} &= \frac{C_{I0}}{\tau_{\text{mix}}}\,\frac{C_M}{C_{M0}}\\ F_{\text{up}} &= \frac{C_I}{\tau_{\text{mix}}}\\ G_{\text{down}} &= \frac{C_{P0}}{\tau_{\text{deep}}}\,\frac{C_I}{C_{I0}}\\ G_{\text{up}} &= \frac{C_P}{\tau_{\text{deep}}} \end{aligned}$$with $k = 80/280$ GtC/yr per ppm, $p_{\text{atm}} = C_A/2.124$ and $p_{\text{oc}}$ in ppm, and $\tau_{\text{deep}} = 1000$ years (age of deep water from carbon-14). The factor $e^{0.0423\,T}$ (4.2% per °C) only applies with the ocean feedback on.
Carbonate chemistry. Dissolved CO₂ reacts with water ($\text{CO}_2 + \text{H}_2\text{O} \rightleftharpoons \text{H}^+ + \text{HCO}_3^- \rightleftharpoons 2\text{H}^+ + \text{CO}_3^{2-}$), and only the small fraction that remains as CO₂ returns to the air. The dissolved inorganic carbon of the surface is $D = D_0\,C_M/C_{M0}$ ($D_0 \approx 1986$ µmol/kg). With fixed alkalinity ($A_T = 2300$ µmol/kg), $[\text{H}^+]$ is obtained from
$$\begin{aligned} A_T &= D\,\frac{K_1[\text{H}^+] + 2K_1K_2}{[\text{H}^+]^2 + K_1[\text{H}^+] + K_1K_2}\\ &\quad + \frac{B_T K_B}{K_B + [\text{H}^+]} + \frac{K_w}{[\text{H}^+]} - [\text{H}^+] \end{aligned}$$and then $p_q = D\,[\text{H}^+]^2 / \{K_0\,([\text{H}^+]^2 + K_1[\text{H}^+] + K_1K_2)\}$, with the equilibrium constants of seawater at 18 °C and salinity 35. The Revelle factor, $\xi = (\Delta p_q/p_q)/(\Delta D/D)$, comes out of the calculation: ≈ 9.5 in the pre-industrial era, and it grows as the ocean absorbs carbon (≈ 11 today); each extra GtC becomes harder to absorb. With the Revelle factor off, $p_q = 280\,C_M/C_{M0}$ (CO₂ would dissolve like any other gas, $\xi = 1$).
Emissions. $E_{\text{fos}}$ and $E_{\text{def}}$ follow the historical data up to the last year with data; after that, the values of the “Human emissions” panel, reduced in the cut scenarios. In the “Pulse” scenario, all the carbon enters in the first year.
Climate. Two-layer energy balance: surface (atmosphere + ocean mixed layer, anomaly $T$) and deep ocean ($T_p$):
$$\begin{aligned} C_s\,\dot T &= \Delta F - \lambda T - \gamma\,(T - T_p)\\ C_p\,\dot T_p &= \gamma\,(T - T_p) \end{aligned}$$with forcing $\Delta F = 5.35\,\ln(C_A/C_{A0})$ W/m², $\lambda = F_{2\times}/S$ ($F_{2\times} = 5.35 \ln 2 \approx 3.7$ W/m² and $S$ the climate sensitivity), $C_s = 0.094\,h$ ($h$ in meters), $C_p = 106$ W·yr/(m²·°C) and $\gamma = 0.73$ W/(m²·°C).
Pre-industrial equilibrium. $C_{A0} = 594.7$ (280 ppm), $C_{V0} = 450$, $C_{S0} = 1700$, $C_{M0} = 900$, $C_{I0} = 9000$ and $C_{P0} = 28\,000$ GtC, with $T = T_p = 0$: all derivatives vanish (for example, the atmosphere receives 60 + 60 + 80 and loses 120 + 80 GtC/yr).
Fossil-origin carbon is tracked as a tracer in all reservoirs (each flux carries the fossil fraction of the reservoir it leaves). Fossil reserves start at 5000 GtC, the order of magnitude of estimated resources. The model is a teaching model: it considers only CO₂ (no other greenhouse gases and no aerosols). The annual historical emissions (fossil fuels and cement, minus carbonation; land-use change) are those of the Global Carbon Budget 2025, from 1850 to 2024, plus the projection for 2025.
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