commit 4b45b2c20ee90c421e248df474d70892c3490039
parent a05860a75cc642137e6ff9727498f1a35bbe42f8
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 22 Jul 2019 12:18:14 +0200
Begin adding limitations
Diffstat:
2 files changed, 146 insertions(+), 14 deletions(-)
diff --git a/BIBnew.bib b/BIBnew.bib
@@ -3932,7 +3932,6 @@
@article{Iverson1995,
doi = {10.1126/science.267.5194.80},
- url = {https://doi.org/10.1126%2Fscience.267.5194.80},
year = 1995,
month = {jan},
publisher = {American Association for the Advancement of Science ({AAAS})},
@@ -6267,7 +6266,6 @@
number = {8},
publisher = {John Wiley & Sons, Ltd},
issn = {1096-9837},
- url = {http://dx.doi.org/10.1002/esp.2138},
doi = {10.1002/esp.2138},
pages = {1105--1112},
keywords = {ribbed moraine, instability model, numerical computation},
@@ -8735,7 +8733,6 @@ Winton and A. T. Wittenberg and F. Zeng and R. Zhang and J. P. Dunne},
@article{Ugelvig2018,
doi = {10.1016/j.epsl.2018.03.022},
- url = {https://doi.org/10.1016%2Fj.epsl.2018.03.022},
year = 2018,
month = {may},
publisher = {Elsevier {BV}},
@@ -8747,7 +8744,6 @@ Winton and A. T. Wittenberg and F. Zeng and R. Zhang and J. P. Dunne},
}
@article{Hermanowski2019,
doi = {10.1002/esp.4630},
- url = {https://doi.org/10.1002%2Fesp.4630},
year = 2019,
month = {apr},
publisher = {Wiley},
@@ -8757,7 +8753,6 @@ Winton and A. T. Wittenberg and F. Zeng and R. Zhang and J. P. Dunne},
}
@article{Iverson2017,
doi = {10.1016/j.geomorph.2017.10.005},
- url = {https://doi.org/10.1016%2Fj.geomorph.2017.10.005},
year = 2017,
month = {dec},
publisher = {Elsevier {BV}},
@@ -8781,7 +8776,6 @@ Winton and A. T. Wittenberg and F. Zeng and R. Zhang and J. P. Dunne},
@incollection{Roux1998,
doi = {10.1007/978-94-017-2653-5_13},
- url = {https://doi.org/10.1007%2F978-94-017-2653-5_13},
year = 1998,
publisher = {Springer Netherlands},
pages = {229--236},
@@ -8792,7 +8786,6 @@ Winton and A. T. Wittenberg and F. Zeng and R. Zhang and J. P. Dunne},
@article{Pouliquen2006,
doi = {10.1088/1742-5468/2006/07/p07020},
- url = {https://doi.org/10.1088%2F1742-5468%2F2006%2F07%2Fp07020},
year = 2006,
publisher = {{IOP} Publishing},
volume = {2006},
@@ -8805,7 +8798,6 @@ Winton and A. T. Wittenberg and F. Zeng and R. Zhang and J. P. Dunne},
@article{Iverson2012,
doi = {10.1130/g33079.1},
- url = {https://doi.org/10.1130%2Fg33079.1},
year = 2012,
month = {aug},
publisher = {Geological Society of America},
@@ -8838,3 +8830,129 @@ Winton and A. T. Wittenberg and F. Zeng and R. Zhang and J. P. Dunne},
title = {A finite element implementation of the nonlocal granular rheology},
journal = {Int. J. Num. Meth. Eng.}
}
+
+@article{Evans2010,
+ doi = {10.1002/nag.877},
+ year = 2010,
+ publisher = {Wiley},
+ pages = {n/a--n/a},
+ author = {T. M. Evans and J. D. Frost},
+ title = {Multiscale investigation of shear bands in sand: Physical and numerical experiments},
+ journal = {Int. J. Numer. Anal. Meth. Geomech.}
+}
+@article{Pellitero2016,
+ doi = {10.1016/j.cageo.2016.06.008},
+ year = 2016,
+ month = {sep},
+ publisher = {Elsevier {BV}},
+ volume = {94},
+ pages = {77--85},
+ author = {R. Pellitero and B. R. Rea and M. Spagnolo and J. Bakke and S. Ivy-Ochs and C. R. Frew and P. Hughes and A. Ribolini and S. Lukas and H. Renssen},
+ title = {{GlaRe}, a {GIS} tool to reconstruct the 3D surface of palaeoglaciers},
+ journal = {Comput. {\&} Geosci.}
+}
+
+@article{Kirchner2016,
+ doi = {10.1016/j.quascirev.2016.01.013},
+ year = 2016,
+ month = {mar},
+ publisher = {Elsevier {BV}},
+ volume = {135},
+ pages = {103--114},
+ author = {N. Kirchner and J. Ahlkrona and E.J. Gowan and P. Lötstedt and J.M. Lea and R. Noormets and L. von Sydow and J.A. Dowdeswell and T. Benham},
+ title = {Shallow ice approximation, second order shallow ice approximation, and full Stokes models: A discussion of their roles in palaeo-ice sheet modelling and development},
+ journal = {Quat. Sci. Rev.}
+}
+@article{Locke1995,
+ doi = {10.1016/0169-555x(95)00053-5},
+ year = 1995,
+ month = {nov},
+ publisher = {Elsevier {BV}},
+ volume = {14},
+ number = {2},
+ pages = {123--130},
+ author = {W. W. Locke},
+ title = {{Modelling of icecap glaciation of the northern Rocky Mountains of Montana}},
+ journal = {Geomorphology}
+}
+
+@article{Pelletier2010,
+ doi = {10.1016/j.geomorph.2009.10.018},
+ year = 2010,
+ month = {mar},
+ publisher = {Elsevier {BV}},
+ volume = {116},
+ number = {1-2},
+ pages = {189--201},
+ author = {J. D. Pelletier and D. Comeau and J. Kargel},
+ title = {Controls of glacial valley spacing on earth and mars},
+ journal = {Geomorphology}
+}
+@article{Egholm2011,
+ doi = {10.1029/2010jf001900},
+ year = 2011,
+ month = {may},
+ publisher = {American Geophysical Union ({AGU})},
+ volume = {116},
+ number = {F2},
+ author = {D. L. Egholm and M. F. Knudsen and C. D. Clark and J. E. Lesemann},
+ title = {Modeling the flow of glaciers in steep terrains: The integrated second-order shallow ice approximation ({iSOSIA})},
+ journal = {J. Geophys. Res.: Earth Surf.}
+}
+@article{Hindmarsh2004,
+ doi = {10.1029/2003jf000065},
+ year = 2004,
+ month = {mar},
+ publisher = {American Geophysical Union ({AGU})},
+ volume = {109},
+ number = {F1},
+ author = {R. C. A. Hindmarsh},
+ title = {A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modeling},
+ journal = {J. Geophys. Res.: Earth Surf.}
+}
+@article{Cohen2000,
+ doi = {10.3189/172756500781832747},
+ year = 2000,
+ publisher = {Cambridge University Press ({CUP})},
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+ year = 2010,
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+ title = {An {ExcelTM} spreadsheet program for reconstructing the surface profile of former mountain glaciers and ice caps},
+ journal = {Comput. {\&} Geosci.s}
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+ doi = {10.5194/esurf-4-159-2016},
+ year = 2016,
+ month = {feb},
+ publisher = {Copernicus {GmbH}},
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+ number = {1},
+ pages = {159--174},
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+ title = {Basal shear stress under alpine glaciers: insights from experiments using the {iSOSIA} and Elmer/Ice models},
+ journal = {Earth Surf. Dynamics}
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+
+@article{Iverson2000,
+ doi={10.1126/science.290.5491.513},
+ year={2000},
+ author={Iverson, R. M. and Reid, M. E. and Iverson, N. R. and LaHusen, R. G. and Lo- gan, M. and Mann, J. E. and Brien, D. L.}
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diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -77,7 +77,9 @@ The local fluidity is defined as:
\label{eq:g_local}
\end{equation}
where $d$ [m] is the representative grain diameter, $\mu_\text{s}$ [-] is the static Coulomb yield coefficient, $C$ [Pa] is the material cohesion, and $b$ [-] is the non-linear rate dependence beyond yield.
-For steady flow the non-locality is determined by a Poisson-type equation where strain is spread in space, as scaled by the cooperativity length $\xi$:
+The failure point is determined by the Mohr-Coulomb constituent relation.
+Beyond failure, the flow is governed by a Poisson-type equation that distributes strain in space according to material properties and stress state.
+The flow non-locality is summarized by the cooperativity length $\xi$:
\begin{equation}
\nabla^2 g = \frac{1}{\xi^2(\mu)} (g - g_\text{local}),
\label{eq:g}
@@ -88,7 +90,7 @@ where
\label{eq:cooperativity}
\end{equation}
The non-locality scales with nonlocal amplitude $A$ [-].
-It is worth noting that the above formulation distributes strain in space based on material properties and stress, as observed in simple granular materials \citep[e.g.][]{Damsgaard2013}.
+In the above mathematical framework, the material slightly strengthens when the shear zone size is restricted by thickness of the granular bed.
\subsection{Fluid-pressure evolution}%
\label{sub:fluid_pressure_evolution}
@@ -101,13 +103,24 @@ where $\mu_\text{f}$ denotes dynamic fluid viscosity [Pa s], $\beta_\text{f}$ is
The sediment is assumed to be in the critical state throughout the domain, as in the original formulation by \citet{Henann2013}.
The fluid pressure is used to determine the effective normal stress used in the granular flow calculations (Eq.~\ref{eq:shear_strain_rate} and~\ref{eq:g_local}).
+\subsection{Limitations of the continuum model}%
+\label{sub:limitations_of_the_continuum_model}
+The presented model considers the material to be in the critical (steady) state throughout the domain.
+Consequently, porosity is prescribed as a constant and material-specific parameter.
+For that reason the model is not able to simulate uniaxial compaction or shear-induced dilation \citep[e.g.][]{Iverson2000, Iverson2010-2, Damsgaard2015} or compaction \citep[e.g.][]{Dewhurst1996}.
+A transient granular model with state-dependent porosity is currently under development.
+
+The strain distribution is in the presented model dependent on the representative grain size $d$.
+However, it is uncomfortable to describe the grain size distribution with a single value for diamictons such as most subglacial tills \citep[e.g.][]{Hooke1995}.
+Future research will investigate how wide grain-size distributions affect strain distribution.
+
+
\subsection{Numerical solution procedure}%
\label{sub:numerical_solution_procedure}
-These parameters do not change over the course of a simulation, and are kept constant everywhere in the domain where the material is of identical origin.
-
-The above formulation is applicable to any spatial dimensionality, for the purposes of this study we apply it in a 1D spatial reference system.
-Shear deformation is restricted to occur in horizontal (x) shear zones.
+The presented formulation is applicable to any spatial dimensionality.
+For the purposes of this study we apply it in a 1D spatial reference system.
The axis $z$ is pointed upwards with a domain length of $L_z$.
+Shear deformation is restricted to occur in horizontal (x) shear zones.
We assign depth coordinates $z_i$ and fluidity $g_i$ to a regular grid with ghost nodes and cell spacing $\Delta z$.
The normal stress is assumed to increase with depth due to lithostatic pressure from the overburden ($\sigma_\text{n}(z) = \int^{z'=L_z}_{z'=z} \rho_\text{s} \phi G dz' + \sigma_\text{n,t}$), where G is the magnitude of gravitational acceleration and $\sigma_\text{n,t}$ is the normal stress applied on the top of the domain.
@@ -310,6 +323,7 @@ The stick-slip experiments (Fig.~\ref{fig:stick_slip} to~\ref{fig:stick_slip_dep
Practically all of the shear strain through a perturbation cycle occurs above the skin depth (magenta line in Fig.~\ref{fig:stick_slip_depth}).
+
\section{Conclusion}%
\label{sec:conclusion}
