= Solution
A typical net <ice accumulation> curve decreases downstream: snow gain at high elevation, zero at the <equilibrium> line, and negative values in the <ice ablation> zone. In a steady glacier with zero <volume flux> at its head, $Q(x)=\int_0^xa(X)\,dX$. If its terminus also has zero <volume flux>, a natural glacier length $L$ is the positive root of $\int_0^La(X)\,dX=0$, with the integral positive between the endpoints. Such a root requires a suitable given accumulation profile; it is not guaranteed by <dimensional analysis> alone.
Let $a_0$ be a representative accumulation rate and $K=A(\rho g\sin\alpha)^n$. Choose
$$
H_*=\left(\frac{a_0L}{K}\right)^{1/(n+2)},\qquad Q_*=a_0L,\qquad t_*=H_*/a_0.
$$
Using $x=L\hat x$, $H=H_*\hat H$, $Q=Q_*\hat Q$, $t=t_*\hat t$, $a=a_0\hat a$, both terms in <mass conservation> have scale $a_0$. The dimensionless <flux> is
$$
\hat Q=(1-\mu\hat H_{\hat x})^n\frac{\hat H^{n+2}}{n+2},\qquad\boxed{\mu=\frac{H_*\cot\alpha}{L}.}
$$
Dropping hats gives the required <partial differential equation>. The parameter measures the surface-thickness slope correction relative to the bed slope. For the linear example $a=1-2x$, accumulation changes sign at $x=1/2$ and its integrated balance vanishes at $x=1$.
\Image[/past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2010-glacier-characteristics.png]
{title=Accumulation, the steady ice profile, and the domain of formal glacier characteristics}
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