Single-timescale atmospheric carbon reservoir ($\dot{C}=b_{1}E-b_0(C-C_0)$)

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.