= Reciprocal-temperature ice-cap fold
{title2=$(\Gamma-\beta s)^2=4\alpha\beta\Gamma$}
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.
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