= Solution
With both <ice accumulation> and <ice ablation> removed, the equation is $h_t+x^{-1}(xDh^3)_x=0$ and the conserved volume is $V_0=2\pi\cos\alpha\,M$, where $M=\int_0^{x_N}xh\,dx$.
If the thickness and extent scales are $H(t)$ and $X(t)$, <mass conservation> gives $HX^2\sim M$, while the flow equation gives $H/t\sim DH^3/X$. Therefore $X\propto t^{1/5}$ and $H\propto t^{-2/5}$. Set $h=\tau^{-2/5}f(\xi)$, $\xi=x\tau^{-1/5}$, with a possible virtual time origin $\tau=t+t_0$. The <similarity solution> satisfies
$$
-\frac25 f-\frac15\xi f'+\frac D\xi(\xi f^3)'=0.
$$
Integration and regular zero total flux at the apex give $D\xi f^3=\xi^2f/5$, hence the positive profile is
$$
\boxed{h(x,\tau)=\sqrt{\frac{x}{5D\tau}},\qquad 0<x<x_N(\tau),}
$$
with dry bed beyond the front. Volume normalization gives
$$
M=\frac{2x_N^{5/2}}{5\sqrt{5D\tau}},
$$
so the <volume-conserving conical ice-current similarity solution> has
$$
\boxed{x_N=\left(\frac{125}{4}DM^2\tau\right)^{1/5}=\left(\frac{125g\sin\alpha\,V_0^2\tau}{48\pi^2\nu\cos^2\alpha}\right)^{1/5},\qquad h_N=\sqrt{\frac{x_N}{5D\tau}}.}
$$
The terminus has finite thickness and is a <shock wave> in the gravity-only <kinematic wave> equation. The <Rankine-Hugoniot condition> gives $\dot x_N=q_N/h_N=Dh_N^2=x_N/(5\tau)$, exactly agreeing with the similarity extent. The characteristic speed behind the front is $3Dh_N^2$, larger than its speed, while the dry-bed characteristic speed is zero, so the front is compressive and gives an <entropy solution>.
This solution describes the long-time spreading, rather than exactly matching the earlier steady profile at the instant snowfall stops. The initial transient can be described by the <characteristic transformation for conical ice drainage>: with starting point $s$, put $K(s)=s^{1/3}h_s(s)$; then
$$
h(x,t)x^{1/3}=K(s),\qquad x^{5/3}=s^{5/3}+5DK(s)^2t
$$
where characteristics remain smooth. Subsequent crossings are resolved by the same conservation and entropy conditions. Restoring the neglected local pressure gradient would smooth the idealized front.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-332-conical-ice.png]
{title=Steady accumulation profile and volume-conserving spreading on a cone}
{description=The steady ice cap ends at three halves of the snowline distance. After accumulation and ablation cease, the long-time gravity-only similarity profile spreads outward and has a finite-thickness front. Both panels use the same conserved ice volume.}
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