Weathering feedback with fast thermal adjustment ($ϵ\dot{\theta }=-\theta +\Gamma (P-1),\dot{P}=v-P^{\mu }{e}^{\theta }$)

In this idealized greenhouse-weathering model, temperature adjusts rapidly when $ϵ\ll 1$, giving $\theta \simeq \Gamma (P-1)$. The remaining carbon balance has equilibrium $\mu \log P_{\ast }+\Gamma (P_{\ast }-1)=\log v$. For positive $\mu ,\Gamma$, the left side has derivative $\mu /P+\Gamma >0$ and ranges from minus infinity to infinity, so there is exactly one positive equilibrium for every sustained positive source $v$.

Linearization of the reduced carbon equation gives dimensionless decay rate $v(\Gamma +\mu /P_{\ast })$, hence dimensional relaxation time $t_C/[v(\Gamma +\mu /P_{\ast })]$. Its negative feedback is increased weathering when either temperature or carbon pressure rises. The two-variable equilibrium is also linearly stable: its Jacobian has negative trace and determinant $v(\mu /P_{\ast }+\Gamma )/ϵ>0$. A fast radiative response therefore does not imply a fast carbon-cycle response. Linearized radiative coefficients must be calibrated at a consistent reference state before numerical use.