Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2022 / 5 / a / Solution

The constants $R_a,R_b,R_d$ are effective reservoir capacities: stored carbon dioxide mass anomaly per unit effective concentration anomaly, in GtCO${}_2$ per ppmv. Only $C_a$ is a literal atmospheric mole fraction; $C_b,C_d$ are scaled reservoir variables chosen to share its equilibrium value. The exchange coefficients $r,s$ are conductances, in GtCO${}_2$ per ppmv per year: $r(C_a-C_b)$ is transfer from the atmosphere to the fast reservoir and $s(C_b-C_d)$ transfer from that reservoir to the deep ocean. They are not by themselves inverse times; division by a reservoir capacity gives an inverse-time rate.

Hydrostatic balance estimates total atmospheric mass as $M_{\mathrm{air}}=4\pi R_E^2\bar{p}/g$. One ppmv of a dilute ideal gas is a mole fraction $10^{-6}$, so, with dry-air molar mass $M_{\mathrm{mol},\mathrm{air}}\simeq 28.97$ g/mol and carbon dioxide molar mass $44.01$ g/mol,The final factor converts kilograms to gigatonnes. This is carbon dioxide mass, not the smaller carbon mass.