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ID: past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2008/a/3/ii/solution

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Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2008 / A / 3 / ii / Solution by

Codex  0 2026-10-31

Choose reference depth

h_{0}

u_{0} = g S h_{0} / f

and concentration

c_{0} = E (u_{0})

. Set the adaptation length

L = h_{0} u_{0} / v_{s}

and the slow bed-evolution time

t_{b} = ρ_{s} (1 - ϕ) h_{0} / (v_{s} c_{0})

. Scale both bed and surface elevation with

h_{0}

. Then

ϵ = \frac{L}{u _{0} t _{b}} = \frac{c _{0}}{ρ _{s} ( 1 - ϕ )}, F^{2} = \frac{u _{0}^{2}}{g h _{0}}, δ = \frac{S L}{h _{0}}, E^{*} (u) = \frac{E ( u _{0} u )}{c _{0}} .

These choices balance sediment advection with settling, bed mass change with exchange, and reference drag with downslope gravity. Direct substitution gives

h = η - s, ϵ h_{t} + (h u)_{x} = 0, F^{2} (ϵ u_{t} + u u_{x}) = - η_{x} + δ (1 - u^{2} / h),

h (ϵ c_{t} + u c_{x}) = E^{*} (u) - c = - s_{t} .

Here

E^{*} (1) = 1

by construction. The small parameter

ϵ

is the suspended solid mass relative to bed solid mass at the chosen scales, explaining the separation between fast fluid motion and slow morphology.

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