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Closure Relations for the Depth-Averaged Modelling Equations

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Shallow Geophysical Mass Flows down Arbitrary Topography

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Abstract

The depth-averaged modelling equations in Sects. 4.5, 4.6 apply to flowing materials for which the constitutive properties are only required to satisfy ordering approximations on the components of the stress tensor and on the erosion/deposition rate.

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Notes

  1. 1.

    The matrix \(({\mathcal D}_{\;\,\beta }^{\alpha })\) in the representation \( {\varvec{D}}_{\mathcal S} ={\mathcal D}_{\;\,\beta }^{\alpha }{\varvec{\tau }}_{\alpha }\otimes {\varvec{\tau }}^{\beta }\) is needed to obtain the eigenvalues and eigenvectors of \( {\varvec{D}}_{\mathcal S}\), and it has been derived in Sect. 3.2, formula (3.53).

  2. 2.

    \( \displaystyle \delta _{\beta \gamma }=\varvec{f}_{\beta }\cdot \varvec{f}_{\gamma }= (C^{\alpha }_{\;\;\beta }{\varvec{\tau }}_{\alpha }) \cdot (C^{\lambda }_{\;\;\gamma }{\varvec{\tau }}_{\lambda })= C^{\alpha }_{\;\;\beta }C^{\lambda }_{\;\;\gamma } M^{-1}_{0\alpha \lambda } \displaystyle \Longleftrightarrow \; \mathbf {I}=\mathbf {C}^T\mathbf {M}_0^{-1}\mathbf {C}\)

    \(\Longleftrightarrow \; \mathbf {C}^{-T}\mathbf {C}^{-1}=\mathbf {M}_0^{-1} \;\Longleftrightarrow \;\mathbf {C}\mathbf {C}^{T}=\mathbf {M}_0{.} \)

  3. 3.

    A symmetric second order tensor \(\varvec{\sigma }\) on a three-dimensional Euclidean real vector space \(\mathcal V\) has the spectral decomposition

    $$\varvec{\sigma }=\lambda _1\varvec{f}_1\otimes \varvec{f}_1+\lambda _2\varvec{f}_2\otimes \varvec{f}_2 +\lambda _3\varvec{f}_3\otimes \varvec{f}_3\,,$$

    where \(\lambda _1,\lambda _2,\lambda _3\) are the (real) eigenvalues of \(\varvec{\sigma }\), and \(\{\varvec{f}_1,\varvec{f}_2,\varvec{f}_3\}\) is an orthonormal basis of eigenvectors for \(\mathcal V\).

  4. 4.

    To avoid this step one could have worked on the form (A.3) of the depth-averaged linear momentum balance equation, written for the case \({\mathcal H }=O(\epsilon ^{\gamma ^{\prime }})\).

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Authors and Affiliations

  1. Department of Mathematical Methods and Models, University Politehnica of Bucharest, Bucharest, Romania

    Ioana Luca

  2. Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan

    Yih-Chin Tai

  3. Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan

    Chih-Yu Kuo

Authors
  1. Ioana Luca
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  2. Yih-Chin Tai
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  3. Chih-Yu Kuo
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Correspondence to Ioana Luca .

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Luca, I., Tai, YC., Kuo, CY. (2016). Closure Relations for the Depth-Averaged Modelling Equations. 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_5

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