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Landscape Evolution

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Mathematical Geoscience

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 36))

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Abstract

Chapter 6 describes a model for the evolution of hillslope topography by means of fluvial erosion. Coupled equations for the hillslope elevation and depth of water flow represent conservation of sediment and water mass, respectively. A uniform downhill slope is unstable to the Smith–Bretherton rilling instability, and this is regularised at large wave number by a singular term involving the ratio of water depth to hillslope elevation. The same theory is then used to describe the form and evolution of finite depth channels. These can be described by a nonlinear diffusion equation coupled with an integral constraint, the combination of which appears to give a well-posed problem for the channel depth; in addition, the channel profile selects its own width automatically. The issue of combining channel dynamics with longer term hillslope erosion is then discussed.

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Notes

  1. 1.

    The formation of the Himalayan mountains is actually more complicated than that which would arise from a simple buckling mechanism. Geological investigations indicate that the Himalayas are formed by a backflow of partially molten rock driven by its own buoyancy; the partial melting occurs in the subducting Indian crust due to extremely high levels of uranium concentration.

  2. 2.

    A fractal set is one whose dimension is non-integral. Curves have dimension one, areas have dimension two. A way of characterising the dimension of a set is to count the number of boxes N of size ε required to cover the set. If N∼ε −D as ε→0, then the set has fractal dimension D. This is consistent with our intuitive sense of dimension, but also allows the calculation of non-integer dimension for such exotica as the Koch snowflake, the Sierpinski gasket, and so on. Fractal sets typically exhibit power-law relationships in their description.

  3. 3.

    More precisely, \(\frac{d}{l}\sim(\frac{\Delta\rho_{sw}(1-\phi)}{Kf^{1/2}\rho_{w}})\frac{U_{D}}{r_{D}}\sim \frac{U_{D}}{r_{D}}\), so high mountains are (in this theory) a consequence of high uplift rate and low rainfall, which makes intuitive sense.

  4. 4.

    It is an unfortunate feature of applied mathematics that there are not enough letters in the Roman and Greek alphabets, even allowing for capitals, overhats, tildes, asterisks, subscripts and superscripts. Duplication is inevitable, and here, apologetically, we use f for the dimensionless sediment transport function, having just made use of it as the friction factor.

  5. 5.

    Note that in this case we avoid all the complications of degeneracy at x=0.

  6. 6.

    Meaning that they are non-zero only on finite interval(s).

  7. 7.

    For constant uplift U and rainfall r and the Meyer-Peter/Müller transport law (6.139), this is not true for the steady state slope, since then \(S=\frac{U}{r}+O(\delta^{1/3})\). Nevertheless it is reasonable to suppose S′=O(1), either because the base state is not in equilibrium, or because uplift and/or rainfall are not uniform.

  8. 8.

    This is easy to show, by consideration of the time derivative of \(\int_{-\infty}^{\infty}H^{2}\,dY\).

  9. 9.

    E here is unrelated to the dimensionless erosion rate used a long time ago in (6.16).

  10. 10.

    Probably only one, but if there were more, then h 0 is taken as the largest zero.

  11. 11.

    This choice is distinct from our earlier assumption in (6.20), which was that U∼A. By selecting the scale for a to be the presumed value where abrasion ceases, we cannot guarantee A∼U, and therefore A is not necessarily O(1).

  12. 12.

    Such coarsening is familiar in the equally dendritic environment of a solidifying alloy, see for example Marsh and Glicksman (1996).

  13. 13.

    See (1.163).

  14. 14.

    Using the Cauchy–Schwarz inequality, the boundedness of E implies the boundedness of the L 2(Ω) norm. This is not the same as proving the boundedness of u (or equivalently, that of the L ∞(Ω) norm).

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  1. MACSI, Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland

    Andrew Fowler

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Fowler, A. (2011). Landscape Evolution. In: Mathematical Geoscience. Interdisciplinary Applied Mathematics, vol 36. Springer, London. https://doi.org/10.1007/978-0-85729-721-1_6

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