= Solution
The quadratic coefficient must have units $\mathrm{W\,m^{-2}\,K^{-2}}$; the printed $\mathrm{K^{-1}}$ makes its term dimensionally incompatible with a radiative flux. Interpret its numerical value as $\lambda_2=0.05\,\mathrm{W\,m^{-2}\,K^{-2}}$, with $\lambda_1=1.5\,\mathrm{W\,m^{-2}\,K^{-1}}$. The total imbalance is
$$
N=F_{\rm ext}-\lambda_1\Delta T+\lambda_2\Delta T^2.
$$
A quadrupling gives $F_{\rm ext}=2(3.75)=7.5\,\mathrm{W\,m^{-2}}$. The two equilibrium roots are
$$
\Delta T_\pm=\frac{\lambda_1\pm\sqrt{\lambda_1^2-4\lambda_2F_{\rm ext}}}{2\lambda_2},
\qquad \boxed{\Delta T_-=6.34\,\mathrm K,\quad\Delta T_+=23.66\,\mathrm K.}
$$
For positive heat capacity, stability requires $\partial N/\partial T=-\lambda_1+2\lambda_2\Delta T<0$. Thus only the lower root is stable. A <saddle-node bifurcation> occurs at
$$
\boxed{F_{\rm crit}=\frac{\lambda_1^2}{4\lambda_2}=11.25\,\mathrm{W\,m^{-2}},\quad
\Delta T_{\rm crit}=\frac{\lambda_1}{2\lambda_2}=15\,\mathrm K.}
$$
The logarithmic forcing relation gives $C_{\rm crit}=280\,2^{11.25/3.75}=2240\,\mathrm{ppm}$. Strict linear stability holds below this concentration on the lower branch; the equality is marginal and is the supremum, not another strictly stable equilibrium. Sustained forcing above it causes warming without a balancing equilibrium in this quadratic model. At such large changes its local feedback expansion can fail, and other physics can intervene; it does not prove unlimited real-planet warming or locate a measured Earth runaway threshold.
Back to article page
