= Volume-conserving conical ice-current similarity solution
After <ice accumulation> and <ice ablation> cease, the <similarity solution> conserving $M=\int xh\,dx$ is
$$
h(x,\tau)=\sqrt{\frac{x}{5D\tau}},\qquad 0<x<x_N(\tau),\qquad x_N=\left(\frac{125}{4}DM^2\tau\right)^{1/5}.
$$
It is dry outside this interval. At its terminus the thickness $h_N$ is nonzero: the <Rankine-Hugoniot condition> gives $\dot x_N=q_N/h_N=Dh_N^2=x_N/(5\tau)$. This is an <entropy solution> of the gravity-only <kinematic wave> model and a long-time approximation for general finite-volume initial profiles. Local pressure-gradient effects smooth the idealized discontinuous front.
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