= Solution
Initially, away from the apex and <snowline>, transport is small and local <ice accumulation> dominates:
$$
\boxed{h(x,t)\simeq At\left(1-\frac x\ell\right)\quad(0<x<\ell),\qquad h=0\quad(x>\ell).}
$$
The early ice volume is $V(t)\simeq\pi A\ell^2\cos\alpha\,t/3$. Balancing radial transport $D(At)^3/\ell$ with $A$ identifies the global transition scales
$$
\boxed{t_*\sim\left(\frac{\ell}{DA^2}\right)^{1/3},\qquad H_*\sim\left(\frac{A\ell}{D}\right)^{1/3}.}
$$
There is also an earlier local adjustment near the apex: the factor $1/x$ makes the same flux significant at distance $x_a\sim DA^2t^3$. The <apex filling layer of a conical ice cap> is described by
$$
h=AtF(\eta),\qquad \eta=\frac{x}{DA^2t^3},\qquad F-3\eta F'=1-\frac1\eta(\eta F^3)',\qquad F(\infty)=1.
$$
At the apex, $F\sim(\eta/2)^{1/3}$ gives the local steady profile $h\sim(Ax/(2D))^{1/3}$. Thus gravity-driven redistribution reaches a fixed position $x$ on a time of order $[x/(DA^2)]^{1/3}$, and its region grows out to the <snowline> when $t\sim t_*$.
At times comparable to $t_*$, ice transport becomes important across the mountain's accumulation region, carries ice below the <snowline>, and establishes an ablation zone. The cap then approaches the <steady conical ice cap with a linear accumulation gradient>, with $x_N\to3\ell/2$. For a moving ice-covered disk, <mass conservation> and its front condition give
$$
\frac{dV}{dt}=2\pi A\cos\alpha\left(\frac{x_N^2}{2}-\frac{x_N^3}{3\ell}\right).
$$
The total input vanishes at $x_N=3\ell/2$, explaining the final extent. The narrow apex and front regions are governed by local adjustments to the bulk approximation; they do not create a further distinct large-scale time regime in this scaling description.
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