Abstract
This chapter is devoted to the presentation of the curvilinear coordinates that we use for the description of a shallow mass flow down arbitrary topography.
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Notes
- 1.
The pair \(({\mathcal {S}},\varvec{n})\) is called an oriented surface. Of course, \(({\mathcal {S}},-\varvec{n})\) is another oriented surface.
- 2.
\(\displaystyle \varvec{n}\cdot \varvec{n}=1\;\Longrightarrow \; \frac{\partial }{\partial \varDelta ^\alpha }(\varvec{n}\cdot \varvec{n})=0 \;\Longrightarrow \; \frac{\partial \varvec{n}}{\partial \varDelta ^\alpha } \cdot \varvec{n}=0\,. \)
- 3.
For slow motion of a large (ice) mass on a deforming lithosphere, this case is important.
- 4.
Bouchut and Westdickenberg [3] noticed this useful decomposition.
- 5.
- 6.
For instance, \(A= {\partial x_1}/{\partial \xi ^1},\; U = {\partial x_1}/{\partial \lambda }\,, \) so that, \( {\partial A}/{\partial \lambda }= {\partial U}/{\partial x_1}, \) and once U is known, this represents an evolution equation for A.
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Luca, I., Tai, YC., Kuo, CY. (2016). A Topography-Fitted Coordinate System. In: Shallow Geophysical Mass Flows down Arbitrary Topography. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-02627-5_2
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DOI: https://doi.org/10.1007/978-3-319-02627-5_2
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