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Jökulhlaups

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Mathematical Geoscience

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

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Abstract

Chapter 11 studies the Nye model of the subglacial floods known as jökulhlaups. First it is shown that the model can explain the observations at Grímsvötn on the ice cap Vatnajökull in Iceland. Then floods below ice sheets are discussed; in particular the mechanics of a sagging ice cauldron are analysed. The possibility of paleo-floods underneath Antarctica and the Laurentide Ice Sheet is discussed.

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Notes

  1. 1.

    There are problems with this, however: see further discussion in the notes.

  2. 2.

    At Grímsvötn, this is normally the case. Apparently the high geothermal heat levels melt ice at the caldera walls, so that water is usually present at the walls all the way to the surface. See Björnsson (1988, pp. 70–73).

  3. 3.

    There are other considerations which suggest that a useful value of γ, at least during a violent flood, may be smaller than the value in (11.21); in this case the assumption of equilibrium of (11.33) may indeed be realistic. See also the discussion in the notes.

  4. 4.

    One might wonder how the assumption that Q and N in (11.34) are functions only of t can be squared with a hydraulic potential gradient Φ(x) which depends on distance downstream. In fact, because the lake refilling condition is applied only at the channel inlet, the value of Φ in (11.34) is actually that at the inlet, i.e., Φ(0). The case where this is negative is considered below.

  5. 5.

    Reasonable estimates of Φ for Grímsvötn yield a=3.33, b=4.316, given our choice of Φ 0 and N 0. For these values, the seal is actually strong, but would be weak according to (11.58) if a<3.1: hence our modified choice of a so that the floods are weak. A likely modification to the predicted type of seal in (11.58) arises from the effects of water temperature near the lake; this is discussed further in the notes.

  6. 6.

    The hypsometry of Mars is odd: the southern hemisphere is elevated, and the northern hemisphere is relatively flat, and much lower.

  7. 7.

    The study of Martian landforms is called areomorphology.

  8. 8.

    At least during floods. In normal circumstances, wide channels may be preferred (Hooke et al. 1990). See also the discussion on hybrid channels in the notes for Chap. 10.

  9. 9.

    Our previous choice of A L =30 km2 (after (11.30)) corresponded to the maximum area before 1940, since when the lake area has declined; 10 km2 is approximately the minimum lake area in 1972. A L is an approximately linearly increasing function of lake level, and hence a decreasing function (see (11.25)) of N 0.

  10. 10.

    Note: in Fig. 1 of this paper, the two marks of 130° E should both read 163° E (David Sugden, private communication).

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

    Andrew Fowler

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Fowler, A. (2011). Jökulhlaups. In: Mathematical Geoscience. Interdisciplinary Applied Mathematics, vol 36. Springer, London. https://doi.org/10.1007/978-0-85729-721-1_11

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