Carbon cycle

Carbon is exchanged among atmospheric, oceanic, biological and geological reservoirs. A model that only redistributes carbon among closed reservoirs cannot permanently remove its total excess burden.

A conservative two-reservoir model separates fast atmospheric exchange from slow ocean adjustment. Let atmospheric carbon-dioxide inventory be $Ip$, where $p$ is atmospheric partial pressure and $I$ is mass per pressure; let ocean dissolved carbon inventory be $M_{o}C$ in moles, with ocean water mass $M_o$ and dissolved inorganic carbon concentration $C$ in moles per mass. Write $p_s=C/K$ for a linear air-sea equilibrium approximation, and take exchange $h(p-C/K)$ positive into the ocean.

Let emissions $v$ and weathering transfer $W$ be carbon-dioxide-equivalent mass rates, $M_{\mathrm{C}{\mathrm{O}}_2}$ the corresponding molar mass, and $bC$ an ocean carbon-removal rate in moles per time. The balances areMultiplying the second by $M_{\mathrm{C}{\mathrm{O}}_2}$ and adding the first gives total-inventory change $v-M_{\mathrm{C}{\mathrm{O}}_2}bC$: exchange and weathering transfer cancel, as required by mass conservation in this chosen model.

If atmospheric relaxation $t_p=I/h$ is short compared with the ocean response, set the first derivative approximately to zero. ThenThe reduced ocean timescale is $t_C=M_o/b$. A sustained emissions increase first raises atmospheric pressure at nearly fixed $C$, then both adjust as the ocean inventory changes. This separation requires $t_p\ll t_C$ and slow forcing relative to $t_p$; the full coupled equations must be retained when those conditions fail. Weathering is represented here as an atmosphere-to-ocean transfer, not automatically as permanent burial.

Dissolved inorganic carbon is the total dissolved carbon in inorganic species, principally aqueous carbon dioxide, bicarbonate and carbonate. A reservoir model may express its concentration in moles of carbon per kilogram of water. Air–sea exchange, river inputs, carbonate chemistry and burial affect its inventory; a single linear concentration-pressure relation is a model simplification.

Transfers of carbon among chemical forms and reservoirs.

Carbon held in organic matter, distinguished from carbonate and dissolved inorganic reservoirs.

A linear climate-carbon model with emissions $E$, concentration $C$, effective injection coefficient $b_1$, baseline $C_0$ and relaxation rate $b_0$. Constant emissions have equilibrium $C_0+b_{1}E/b_0$, and all transients about that equilibrium decay on time $b_0^{-1}$. Stopping emissions returns concentration to the baseline in this model. Real multiple reservoirs and long-lived carbon uptake tails prevent treating that one-time-constant assumption as universal.

Equal-capacity reservoirs exchanging carbon symmetrically preserve their total burden. Their difference decays at rate $2k$, so a pulse into one reservoir leaves one half there permanently unless an external sink is added. Coupling this response to a faster first-order thermal model produces an initial warming followed by relaxation to a nonzero asymptote.