commit f956269266b08d86a1c8061f7c5717cecd257da8
parent af47cfa8b0d1094514e5e81af7daeb048862f947
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 3 Dec 2019 14:26:19 +0100
Improve results
Diffstat:
2 files changed, 40 insertions(+), 38 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -19,7 +19,7 @@
% complete documentation is here: http://trackchanges.sourceforge.net/
%%%%%%%
-%\usepackage{amsmath}
+\usepackage{amsmath}
\draftfalse
@@ -139,10 +139,10 @@ The local contribution to fluidity is defined as:
\begin{linenomath*}
\begin{equation}
g_\mathrm{local}(\mu, \sigma_\mathrm{n}') =
- %\begin{cases}
- %\sqrt{d^2 \sigma_\mathrm{n}' / \rho_\mathrm{s}} ((\mu - C/\sigma_\mathrm{n}') - \mu_\mathrm{s})/(b\mu) &\mathrm{if } \mu - C/\sigma_\mathrm{n}' > \mu_\mathrm{s} \mathrm{, and}\\
- %0 & \mathrm{if } \mu - C/\sigma_\mathrm{n}' \leq \mu_\mathrm{s}.
- %\end{cases}
+ \begin{cases}
+ \sqrt{d^2 \sigma_\mathrm{n}' / \rho_\mathrm{s}} ((\mu - C/\sigma_\mathrm{n}') - \mu_\mathrm{s})/(b\mu) &\mathrm{if}\quad \mu - C/\sigma_\mathrm{n}' > \mu_\mathrm{s} \mathrm{,}\quad\mathrm{and}\\
+ 0 & \mathrm{if}\quad \mu - C/\sigma_\mathrm{n}' \leq \mu_\mathrm{s}.
+ \end{cases}
\label{eq:g_local}
\end{equation}
\end{linenomath*}
@@ -182,6 +182,7 @@ Specifically designed laboratory experiments with various tills should inform th
\subsection{Simulation setup}
We apply the model in a one-dimensional setup with simple shear (Fig.~\ref{fig:validation}a).
+Parameter values and their references are listed in Table~S1.
The lower boundary condition for the granular phase is no slip ($v_x(L=0) = 0$).
The upper boundary condition for the granular phase is fixed shear friction $\mu(z=L_z)$ under stress controlled settings, or fixed shear velocity $v_x(z=L_z)$ for rate-controlled experiments.
The upper normal stress ($\sigma_\mathrm{n}$) is constant, but effective normal stress ($\sigma_\mathrm{n}' = \sigma_\mathrm{n} - p_\mathrm{f}$) varies if water pressure $p_\mathrm{f}$ changes.
@@ -193,7 +194,6 @@ Real glacier settings fall somewhere in between, depending on how important basa
Stress-controlled conditions approximate a setting where ice flow directly responds to changes in subglacial strain rates.
Whillans Ice Plain, West Antarctica is an example of this setting, where a low surface slope and low driving stress results in stick-slip movement \cite<e.g.,>[] {Bindschadler2003}.
A rate-controlled setup is the opposite end member, where changes in bed friction do not influence ice flow velocity.
-Parameter values and their references are listed in Table~S1.
\begin{figure*}[htbp]
\begin{center}
@@ -205,7 +205,7 @@ Parameter values and their references are listed in Table~S1.
Comparison between shear experiments on subglacial till and the non-local granular fluidity (NGF) model applied in this study.
a) Experimental setup for the NGF model in one-dimensional shear.
The upper boundary is constant friction $\mu$ (stress controlled), or constant velocity $v_x$ (velocity controlled).
- b) Rate dependence of critical-state friction in laboratory experiments on till (after \citeA{Iverson2010}), and the NGF model with material friction $\mu_\mathrm{s} = 0.5$ and effective normal stress $\sigma_\mathrm{n}' = 100$ kPa.
+ b) Rate dependence of critical-state friction in laboratory experiments on till (after \citeA{Iverson2010}), and the NGF model with material friction $\mu_\mathrm{s}$ = 0.5 and effective normal stress $\sigma_\mathrm{n}'$ = 100 kPa.
The dimensionles parameter $b$ controls frictional rate dependence (Eq.~\ref{eq:g_local}).
c) Mohr-Coulomb analysis of till samples and NGF model.
d) Modeled strain distribution under varying effective normal stress ($\sigma_\mathrm{n}'$) with the discrete-element method (DEM, \citeA{Damsgaard2013}) and the NGF model.
@@ -220,28 +220,17 @@ Parameter values and their references are listed in Table~S1.
We first compare the modeled mechanical behavior to various tills tested in laboratory settings.
Over five orders of strain-rate magnitude, some tills show slight rate weakening and others are slightly rate strengthening (Fig.~\ref{fig:validation}b).
Shear-strain rates up to $\sim5 \times 10^3$ a$^{-1}$ are realistic for natural glacier systems \cite{Cuffey2010}.
-Our model is effectively rate-independent over most of the range, but higher $b$ values provide larger frictional resistance at extreme shear-strain rates (Fig.~\ref{fig:validation}b), making the model under these conditions rate strengthening.
-The modeled friction value can be linearly scaled by adjusting $\mu_\mathrm{s}$ in Eqs.~\ref{eq:g_local} and~\ref{eq:cooperativity}.
-Our model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:mohr_coulomb}), which is useful as these parameters are often constrained from Mohr-Coulomb tests on till samples.
+The model is effectively rate-independent over most of the range, but higher $b$ values provide larger frictional resistance at extreme shear-strain rates (Fig.~\ref{fig:validation}b), making the model under these conditions rate strengthening.
+The modeled friction value can be linearly scaled by adjusting $\mu_\mathrm{s}$ in Eqs.~\ref{eq:cooperativity} and~\ref{eq:g_local}.
+The model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:validation}c), which is useful as these parameters are often constrained from Mohr-Coulomb tests on till samples.
The NGF model contains parameter $A$ for adjusting the degree of material non-locality (Eq.~\ref{eq:cooperativity}).
-Unfortunately, no laboratory experiment exists in the literature where the effects of normal stress are analysed for changes in strain distribution in the till.
-Instead, we compare the modeled strain distribution with discrete-element results from \citeA{Damsgaard2013} which allow us to calibrate $A$.
-By inserting relevant material parameters for grain size, friction, stress, and shear velocity (DEM, Table~S1), the NGF model model approximates the strain distribution well (Fig.~\ref{fig:strain_distribution}).
+However, at present no laboratory experiment exists in the literature where the effects of normal stress are analysed for changes in strain distribution in the till.
+Instead, we compare the modeled strain distribution with discrete-element results from \citeA{Damsgaard2013}.
+By inserting relevant material parameters for grain size, friction, stress, and shear velocity (DEM, Table~S1), the NGF model model approximates the strain distribution well (Fig.~\ref{fig:validation}d).
Both models show that sediment advection is pressure dependent, with low effective normal stresses producing shallow deformation, and high effective normal stresses deepening the material mobilization.
The DEM results took more than two months of computational time, whereas the continuum model is completed in a fraction of a second, albeit without detail of individual particle kinematics and adjustment towards the critical state.
-Figure~\ref{fig:parameter_test} is a systematic analysis of parameter influence under a constant shear rate.
-All experiments are at a shear rate of 300 m a$^{-1}$ and a normal stress of $\sigma_\mathrm{n}'$ = 100 kPa.
-Several observations emerge from this parameter sensitivity analysis.
-The representative grain size $d$ has a major influence on the strain distribution, where finer materials show deeper deformation.
-The material is slightly weaker with larger grain sizes.
-The shear zone is more narrow with higher material static friction coefficients ($\mu_\mathrm{s}$), as the material is less willing to fail.
-Our implementation of cohesion does not influence strain after yield.
-Static friction and cohesion both linearly scale the bulk friction, as expected with Mohr-Coulomb materials (see also Fig.~\ref{fig:mohr_coulomb}).
-The non-local amplitude $A$ slightly changes the curvature of the shear strain profile, but does not affect the overall friction.
-There is a significant strengthening when the bed thickness $L_z$ begins to constrict the shear zone thickness.
-
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.49\textwidth]{experiments/fig-stick_slip_stress.pdf}
@@ -249,23 +238,22 @@ There is a significant strengthening when the bed thickness $L_z$ begins to cons
\caption{\label{fig:stick_slip}%
Stick-slip dynamics during sinusoidal water-pressure forcing from the top.
Stress and shear velocity are measured at the top of the sediment bed.
- a) Stress-controlled setup with $\mu = 0.4$.
- b) Rate-controlled setup with $v_x = 1$ km/a.
+ a) Stress-controlled setup with applied friction $\mu$ = 0.4.
+ b) Rate-controlled setup with applied shear velocity $v_x$ = 1 km/a.
}
\end{center}
\end{figure}
-Next we vary the top water pressure and observe the shear dynamics over a simulation time of seven days Figure~\ref{fig:stick_slip}.
-We perform tests under both stress and rate-controlled configurations.
-The response during the first cycle ($t<1$ d) is slightly different from later cycles ($t>1$ d) since the model is initialized with a hydrostatic water-pressure distribution.
+Next we vary the top water pressure and observe the shear dynamics over a simulation time of seven days (Fig.~\ref{fig:stick_slip}).
+The experiments are performed under both stress and rate-controlled configurations.
+The response during the first cycle ($t<1$ d) is slightly different from later cycles ($t>1$ d) as the model is initialized with a hydrostatic water-pressure distribution.
Under stress-controlled conditions (Fig.~\ref{fig:stick_slip}a), the system shows stick-slip behavior where velocities range from 0 to $\sim$9 km/d.
The depth of maximum deformation moves into the bed as effective normal stress at the top boundary increases.
-However, under stress-controlled conditions the till flux is only significant when the bed is rapidly slipping.
-The sediment advective flux is not significant during deep deformation events, as the overall shear velocity is low.
+Under the stress-controlled conditions the till flux peaks during rapid slip as water pressure at the ice-bed interface is at its maximum.
In the rate-controlled configuration (Fig.~\ref{fig:stick_slip}b), the shear stress varies as effective normal stress oscillates, as expected from a Mohr-Coulomb material.
As in the stress-controlled configuration, deformation propagates into the bed as effective normal stress increases at the top.
-Contrary to the stress-controlled setup, the till flux is under rate-controlled shear largest during deep deformation events.
+Contrary to the stress-controlled setup, the till flux is under rate-controlled shear largest during deep deformation events, which occur when the ice-bed water pressure is at its minimum value.
\begin{figure}[htbp]
\begin{center}
@@ -279,15 +267,11 @@ Contrary to the stress-controlled setup, the till flux is under rate-controlled
\end{center}
\end{figure}
-Figure~\ref{fig:hysteresis}) demonstrates that the grain/fluid system displays strong hysteresis in shear velocity, strain distribution, and till flux under both stress and rate-controlled conditions.
-The sediment flux has opposite trends, according to the driving mode.
+Figure~\ref{fig:hysteresis} demonstrates that the grain/fluid system displays strong hysteresis in shear velocity, strain distribution, and till flux under both stress and rate-controlled conditions.
+As noted from the previous figure, the sediment flux has different maxima between the driving modes.
Stress-controlled shear produces a large sediment flux in a thin deforming layer close to the boundary during slip events when effective pressure is at its lowest value (Fig.~\ref{fig:hysteresis}a).
On the other hand, under rate-controlled conditions the majority of sediment flux occurs as a plug-like flow where deformation occurs deep in the bed during maxima in effective normal stress at the ice-bed interface (Fig.~\ref{fig:hysteresis}b).
-When water pressure drops at the ice-bed interface, there is a remnant of high fluid-pressure diffusing downwards (Fig.~\ref{fig:stick_slip_depth}).
-The granular deformation primarily occurs where the effective normal stress is at its lowest value.
-The depth of maximum shear-strain rate corresponds to the depth of minimum in effective normal stress, as long as the shear zone width can be accommodated.
-
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=15.0cm]{experiments/fig-stick_slip_rate_depth.pdf}
@@ -299,6 +283,8 @@ The depth of maximum shear-strain rate corresponds to the depth of minimum in ef
\end{center}
\end{figure*}
+Strong water pressure decrease at the ice-bed interface can cause reversal of the effective normal stress at some depth beneath the ice-bed interface (Fig.~\ref{fig:stick_slip_depth}).
+The depth of maximum shear-strain rate corresponds to the depth of minimum in effective normal stress.
Figure~\ref{fig:stick_slip_depth} shows a time-stacked series of simulation state with depth.
The experimental setup is rate-controlled and identical to Fig.~\ref{fig:stick_slip}b and~\ref{fig:hysteresis}b.
The water pressure perturbations decay exponentially with depth with a phase shift.
@@ -307,9 +293,24 @@ Deep deformation occurs when the effective normal stress is smaller at depth tha
We next perturb the top water pressure with pulses of triangular and square shape (Fig.~\ref{fig:pulse}).
Regardless of perturbation shape, the maximum deformation depth increases with increasing perturbation amplitude.
+Figure~S1 contains a systematic analysis of parameter influence in the model equations.
+All experiments are at a shear rate of 300 m a$^{-1}$ and a normal stress of $\sigma_\mathrm{n}'$ = 100 kPa.
+Several observations emerge from this parameter sensitivity analysis.
+The representative grain size $d$ has a major influence on the strain distribution, where finer materials show deeper deformation.
+The material is slightly weaker with larger grain sizes.
+The shear zone is more narrow with higher material static friction coefficients ($\mu_\mathrm{s}$), as the material is less willing to fail.
+Our implementation of cohesion does not influence strain after yield.
+Static friction and cohesion both linearly scale the bulk friction, as expected with Mohr-Coulomb materials (see also Fig.~\ref{fig:mohr_coulomb}).
+The non-local amplitude $A$ slightly changes the curvature of the shear strain profile, but does not affect the overall friction.
+There is a significant strengthening when the bed thickness $L_z$ begins to constrict the shear zone thickness.
\section{Discussion}%
\label{sec:discussion}
+
+Granular deformation occurs where the effective normal stress is at its lowest value.
+Due to sediment non-locality, the stress minima needs to be of a sufficient thickness in order for a shear zone establishment.
+
+
In this study it is assumed that there is a strong coupling between ice and bed.
However, overpressurization and slip at the ice-bed interface may cause episodic decoupling at the interface and reduce bed deformation, as observed under Whillans Ice Stream, West Antarctica \cite<e.g.,>[] {Engelhardt1998}, and in deposits from Pleistocene glaciations \cite<e.g.,>[] {Piotrowski2001}.
We see the presented framework as a significant improvement of treating sediment advection in ice-flow models, but acknowledge that a complete understanding of the sediment mass budget requires improved models of ice-bed interface physics.
diff --git a/si.tex b/si.tex
@@ -431,6 +431,7 @@ In rate-\emph{limited} experiments, the iterative procedure is only performed fo
\includegraphics[width=15cm]{experiments/fig-parameter_test.pdf}
\caption{\label{fig:parameter_test}%
Analysis of parameter influence on steady-state strain distribution and bulk friction during shear.
+ All experiments are performed under constant shear rate of 300 m a$^{-1}$ and a normal stress of $\sigma_\mathrm{n}'$ = 100 kPa.
Parameter values marked with an asterisk (*) are used outside of the individual parameter sensitivity tests.
}
\end{center}
