= Apex filling layer of a conical ice cap
For an initially bare cone with accumulation $a\simeq A$ near its apex, the early outer thickness is $At$. The radial gravity flux becomes comparable to accumulation over distance $x_a\sim DA^2t^3$. Setting $h=AtF(\eta)$ and $\eta=x/(DA^2t^3)$ gives
$$
F-3\eta F'=1-\frac1\eta(\eta F^3)',\qquad F(\infty)=1.
$$
Regular apex flux requires $F\sim(\eta/2)^{1/3}$ near zero. The <boundary layer> therefore approaches the local steady thickness $(Ax/(2D))^{1/3}$ while farther regions are still filling.
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