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Let $\ell =(H_m-H_0)/\sin \alpha$ be the along-slope distance to the snowline, so $H(x)=H_m-x\sin \alpha$ and $a(x)=A(1-x/\ell )$. Define $D=g\sin \alpha /(3\nu )$.

Treat ice as an incompressible Newtonian fluid, neglect inertia, and use lubrication theory with thickness measured normal to the slope. The bed has a no-slip boundary condition, the free surface has zero tangential stress, and normal pressure is hydrostatic. The small thickness slope allows its pressure-gradient contribution to be neglected against gravity along the mountain. With normal coordinate $y$, the tangential equation and boundary conditions areIntegration givesThe horizontal radius of a ring is $x\cos \alpha$, so mass conservation gives the gravity-driven ice flow on a conical slope equationNegative accumulation is ice ablation and applies only where ice exists; the ice-free region has $h=0$.

In a steady state, regularity and zero total flux at the apex require $xD{h}^3\to 0$ as $x\to 0$. Integrating givesRequiring a continuous zero-thickness steady terminus yields the steady conical ice cap with a linear accumulation gradient:The terminus lies below the snowline, allowing the ablation region to balance snowfall. The maximum thickness occurs at $x=3\ell /4$.

The volume follows from integrating the ring areas. With $s=x/\ell$,Substituting $D$ and $\ell =\Delta H/\sin \alpha$ givesThe ideal outer profile has steep slopes very close to the apex and terminus. Those small regions require local corrections to the assumed slope balance, while the bulk profile and leading volume follow from the stated approximation.

Initially, away from the apex and snowline, transport is small and local ice accumulation dominates: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 scalesThere is also an earlier local adjustment near the apex: the factor $1/x$ makes the same flux significant at distance $x_a\sim D{A}^2{t}^3$. The apex filling layer of a conical ice cap is described byAt 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 $[x/(D{A}^2){]}^{1/3}$, and its region grows out to the snowline when $t\sim t_{\ast }$.

At times comparable to $t_{\ast }$, 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 $x_N\to 3\ell /2$. For a moving ice-covered disk, mass conservation and its front condition giveThe 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.