commit 479cc97e83f83f54ab751cb3cf4b88d84fcd71ef
parent 976b9545d218a275fbe2d38a9c749f1adbf24acc
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Thu, 10 Oct 2019 16:01:16 +0200
Write conclusion
Diffstat:
2 files changed, 25 insertions(+), 16 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -147,6 +147,7 @@ However, the NGF model is dry, and in the context of subglacial mechanics, dry m
In this paper we expand the steady-state NGF continuum model for granular flow by \citet{Henann2013} with cohesion and a coupling to pore-pressure diffusion, and analyze how fluid-pressure perturbations affect strain distribution and material stability.
+
\section{Methods}%
\label{sec:methods}
@@ -323,8 +324,6 @@ In rate-\emph{limited} experiments, the iterative procedure is only performed fo
Importantly, the resultant shear velocities are in this setup not limited by anything but sediment kinematics.
The simulated velocities are for the most part far greater than any glacial setting, where horizontal stresses keep ice masses in place over weak beds.
-
-
\subsection{Simulation setup}
Parameter values and their references are listed in Table~\ref{tab:params}.
For the first experiment with variable water pressure, we apply a water-pressure forcing amplitude of 50 kPa that modulates effective stress at the top around 100 kPa (Fig.~\ref{fig:stick_slip}).
@@ -378,6 +377,7 @@ For the first experiment with variable water pressure, we apply a water-pressure
}
\end{table*}
+
\section{Results}%
\label{sec:results}
@@ -505,24 +505,22 @@ Deep deformation occurs when top water pressure is at a minimum, and the effecti
We next perturb the top water pressure with pulses of triangular and square shape (Fig.~\ref{fig:pulse}).
\textbf{ANALYSIS OF AMPLITUDE EFFECT}
+
\section{Discussion}%
\label{sec:discussion}
-The stress-dependt sediment advection observed here is relevant for instability theories of subglacial landform development \citep{Hindmarsh1999, Fowler2000, Schoof2007, Fowler2018}.
+The stress-dependt sediment advection observed in Fig.~\ref{fig:strain_distribution} is relevant for instability theories of subglacial landform development \citep{Hindmarsh1999, Fowler2000, Schoof2007, Fowler2018}.
From geometrical considerations, it is likely that bed-normal stresses on the stoss side of subglacial topography are higher than on the lee side.
-With all other physical conditions being equal, Figure~\ref{fig:strain_distribution} indicates that sediment advection through shear is stress dependent.
-Topography of non-planar ice-bed interfaces (proto-drumlins, ribbed moraines, etc.) may be modulated or amplified through the variable transport capacity, unless normal stress variations are overprinted by spatial variations in water pressure \citep[e.g.][]{McCracken2016, Iverson2017b, Hermanowski2019}.
-
+With all other physical conditions being equal, our results indicate that shear-driven sediment advection would be larger on the stoss side of bed perturbations than behind them.
+Topography of non-planar ice-bed interfaces (proto-drumlins, ribbed moraines, etc.) may be transported and modulated through the variable transport capacity, unless stress differences are overprinted by spatial variations in water pressure \citep[e.g.][]{McCracken2016, Iverson2017b, Hermanowski2019}.
Previously, \citet{Iverson2001} modeled the subglacial bed as a series of parallel Coulomb-frictional slabs.
They demonstrated that random perturbations in effective stress at depth can distribute deformation away from the ice-bed interface.
\citet{Tulaczyk1999} and \citet{Tulaczyk2000} demonstrated that Darcian diffusion of pore-pressure variations into the bed can distribute strain away form the ice-bed interface, without a lengthscale controlling deformation.
-We couple the water-pressure diffusion with a more complex sediment rheology.
+We couple the water-pressure diffusion with a more complex sediment rheology than the above studies.
The slight rate dependence (Fig.~\ref{fig:rate_dependence}) makes it relatively trivial to couple to ice-flow models, while retaining realistic sediment physics.
-
-
-The water pressure variations vary with the same periodocity as the forcing, but with exponential decay in amplitude and increasing lag at depth.
-The skin depth is defined as the distance where the fluctuation amplitude decreases to $1/e$ of its surface value \citep[e.g.][]{Cuffey2010}.
+At depth, the water pressure variations display exponential decay in amplitude and increasing lag.
+The skin depth is defined as the distance where the fluctuation amplitude decreases to $1/e \approx 37\%$ of its surface value \citep[e.g.][]{Cuffey2010}.
As long as fluid and diffusion properties are constant, an analytical solution to skin depth $d_\text{s}$ [m] in our system follows the form \citep[after Eq.~4.90 in][]{Turcotte2002},
\begin{linenomath*}
\begin{equation}
@@ -539,7 +537,6 @@ Figure~\ref{fig:skin_depth} shows the skin depth for water under a range of perm
The stick-slip experiments (Fig.~\ref{fig:stick_slip}) correspond to a skin depth of 2.2 meter.
Practically all of the shear strain through a perturbation cycle occurs above the skin depth (magenta line in Fig.~\ref{fig:stick_slip_depth}).
-
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.5cm]{experiments/fig8.pdf}
@@ -550,18 +547,30 @@ Practically all of the shear strain through a perturbation cycle occurs above th
\end{center}
\end{figure}
-
-
-
+We find that skin depth calculations can be a useful starting point for determining scenarios where deep deformation is possible.
+It is worth noting that the water pressure deviations need to exceed the lithostatic and hydrostatic gradients with depth.
+This means that minima in effective normal stress are increasingly difficult to create at larger depths through pure diffusion from the ice-bed interface.
+Deep deformation is observed in glacier settings with coarse subglacial tills \citep[e.g.][]{Truffer2000, Kjaer2006}.
+Due to higher hydraulic permeability, coarse tills are more susceptible to deep deformation, but require longer-lasting perturbations in water pressure (Fig.~\ref{fig:skin_depth}).
+Contrarily, fine-grained tills are unlikely to cause deep deformation.
+Instead, lateral water input at depth is a viable mechanism for creating occasional episodes of deep slip, in particular when horizontal bedding decreases vertical permeability \citep[e.g.][]{Kjaer2006}.
\section{Conclusion}%
\label{sec:conclusion}
+We present a new model for coupled computation of subglacial till and water.
+The model is based on the concept of non-local granular fluidity \citep{Henann2013}, but is extended with cohesion and pore-pressure diffusion.
+The mechanics adhere to Mohr-Coulomb plasticity, with a weak and highly non-linear rate dependence governed by stress and sediment properties.
+In agreement with laboratory results, the material is effectively rate-independent at glacial shear velocities.
+The rate dependence is only significant as kinematics approach a landslike-like state.
+A simple shear experimental setup is adapted for analyzing the mechanical response under different stresses and water-pressure variations.
+With cyclical water-pressure variations at the ice-bed interface, deep deformation occurs when remnant high water pressures at depth overcome the lithostatic gradient.
+Deep deformation may be common in coarse-grained subglacial tills with strong annual water-pressure differences.
\section*{Appendix}%
\label{sec:appendix}
-The source code for the grain-water model is available at \url{https://src.adamsgaard.dk/1d_fd_simple_shear}.
+The grain-water model is written in C and is available under free-software licensing at \url{https://src.adamsgaard.dk/1d_fd_simple_shear}.
All results and figures can be reproduced by following the instructions in the experiment repository for this publication, available at \url{https://src.adamsgaard.dk/.manus_continuum_granular1_exp}.
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diff --git a/experiments/fig8.pdf b/experiments/fig8.pdf
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