Reciprocal-temperature ice-cap fold ($(\Gamma -\beta s{)}^2=4\alpha \beta \Gamma$)

In the idealized normalized model $\dot{M}=1-\beta z-\alpha /(\Gamma z-s)$ on the cold branch $\Gamma z>s$, snow gain and melt loss balance at two roots when $\Gamma -\beta s>2\sqrt{\alpha \beta \Gamma }$. Assume ice height increases with mass. Differentiation gives $d\dot{M}/dz=-\beta +\alpha \Gamma /(\Gamma z-s{)}^2$, so the higher root is stable and the lower root unstable. The roots merge at the displayed saddle-node bifurcation. Here $s$ is a sea-level Celsius temperature, so absolute kelvin values cannot be substituted into the reciprocal law. The singularity at the melting threshold limits this model's physical scope.