Two-reservoir carbon adjustment ($t_p\ll t_C$)

A coupled atmosphere–ocean mass conservation model can have a fast atmospheric exponential relaxation time and a much slower ocean-carbon time. For atmospheric inventory coefficient $I$, exchange coefficient $h$, ocean water mass $M_o$, removal coefficient $b$ and linear surface-pressure relation $p_s=C/K$, atmospheric balance gives $p\simeq C/K+(v-A_{L}W)/h$. Substitution into the ocean equation cancels the weathering and exchange terms, giving $(M_o/b)\dot{C}=v/(M_{{\mathrm{CO}}_2}b)-C$. Sustained extra emissions therefore first raise pressure at nearly fixed ocean concentration and then increase both reservoirs over the ocean removal time.