Gravity-driven ice flow on a conical slope

For a thin Newtonian fluid film on a conical slope at angle $\alpha$, neglecting the thickness-gradient contribution to the driving pressure gives volume flux per unit width $q=D{h}^3$, where $D=g\sin \alpha /(3\nu )$. Radial mass conservation isThe distance $x$ is measured along the slope; the horizontal radius is $x\cos \alpha$, so total volume is $2\pi \cos \alpha \int xhdx$.

For ice accumulation $a=A(1-x/\ell )$, regular zero flux at the apex givesThe steady zero-thickness terminus is at $3\ell /2$, below the snowline $x=\ell$. Accumulation above the snowline exactly balances ice ablation below it.

After ice accumulation and ice ablation cease, the similarity solution conserving $M=\int xhdx$ isIt is dry outside this interval. At its terminus the thickness $h_N$ is nonzero: the Rankine-Hugoniot condition gives ${\dot{x}}_N=q_N/h_N=D{h}_N^2=x_N/(5\tau )$. 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.

Putting $y=x^{5/3}$ and $w=x^{1/3}h$ transforms $h_t+x^{-1}(xD{h}^3{)}_x=0$ into the scalar conservation lawAlong smooth characteristic curves, $w$ is constant and $y$ grows at speed $5D{w}^2$. Starting from $h_0(s)$ givesAfter characteristic crossing, the Rankine-Hugoniot condition and entropy solution select the physical front.

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 $x_a\sim D{A}^2{t}^3$. Setting $h=AtF(\eta )$ and $\eta =x/(D{A}^2{t}^3)$ givesRegular 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.