commit 09b71369ea5df67fdf7b767018ea2636727864e0
parent 29380028bf88e3afe1d8a6935a9a3006bc3281e1
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 12 Nov 2019 20:28:12 +0100
Incorporate more of Liran's feedback
Diffstat:
2 files changed, 26 insertions(+), 26 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -272,12 +272,12 @@ The method is unconditionally stable and second-order accurate in time and space
Our implementation of grain and fluid dynamics is highly efficient, and for the presented experiments each time step completes in less than 1 ms on a single CPU core.
\subsubsection{Rate-controlled experiments} % quick edit, needs rewrite. perhaps also move somewhere else
-The continuum model is in its presented form suited for resolving strain rate and shear velocity from a given stress forcing, i.e. in a stress-controlled setup.
+The continuum model in its presented form is suited for resolving strain rate and shear velocity from a given stress forcing, i.e. in a stress-controlled setup.
However, certain experiments are best approached in a rate-controlled manner where a specified shear rate results in strain-rate distribution and shear stress.
-For our system of equations this is an inverse problem that can be tackled by adjusting the applied stress until the resultant shear velocity matches the desired value.
+For our system of equations, this is an inverse problem that can be tackled by adjusting the applied friction at the top until the resultant shear velocity matches the desired value.
We implement an automatic iterative procedure that can be set to match a shear velocity, or limit the velocity beneath a given value.
First, an initial top velocity value $v_x^*$ is calculated in a forward manner on the base of an arbitrary value for friction $\mu^*$.
-The difference between $v_x^*$ and the desired velocity $v_x^\text{d}$ is calculated as a normalized residual $r$:
+The difference between $v_x^*$ and the desired velocity $v_x^\text{d}$ is used in the calculation of a normalized residual $r$:
\begin{linenomath*}
\begin{equation}
r = \frac{v_x^\text{d} - v_x^*}{v_x^* + 10^{-12}}.
@@ -297,8 +297,8 @@ where $\theta = 10^{-2}$ is a chosen relaxation factor.
The computations are then rerun with the new applied stress until the tolerance criteria is met.
In rate-\emph{limited} experiments, the iterative procedure is only performed for negative residual values.
-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.
+%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}.
@@ -309,7 +309,7 @@ For the first experiment with variable water pressure, we apply a water-pressure
\includegraphics{experimental-setup.pdf}
\caption{\label{fig:experimental-setup}%
Experimental setup for 1d shear experiments.
- The upper boundary conditions for the granular solver are fixed shear stress $\tau(z=L_z)$ (stress controlled), or fixed shear velocity $v_x(z=L_z)$ (rate controlled).
+ The upper boundary conditions for the granular solver are fixed applied friction $\mu(z=L_z)$ (stress controlled), or fixed shear velocity $v_x(z=L_z)$ (rate controlled).
The upper normal stress ($\sigma_\text{n}$) is constant, but effective normal stress ($\sigma'_\text{n}$) varies if $p_\text{f}$ changes.
The granular phase lower boundary condition is no slip.
For the fluid solver, the top is fixed fluid pressure $p_\text{f}(z=L_z)$, which can be constant or vary in time.
@@ -337,7 +337,7 @@ For the first experiment with variable water pressure, we apply a water-pressure
\end{tabular}
}
\caption{\label{tab:params}%
- Material parameters for experiments emulating discrete element method (DEM) particles \citep[e.g.,][]{Damsgaard2013},
+ Material parameters for model simulations emulating discrete element method (DEM) particles \citep[e.g.,][]{Damsgaard2013},
the subglacial till under the mountain glacier Storgl\"aciaren, Sweden. \citep[e.g.,][]{Iverson1995, Hooke1997, Iverson1998},
the West Antarctic Ice Sheet (WIS) till at the Upstream B site \citep[e.g.,][]{Engelhardt1998, Tulaczyk2000, Leeman2016},
and the Two Rivers till of the Lake Michigan Lobe of the Laurentide palaeo-ice sheet \citep[e.g.,][]{Clark1994, Iverson1998}.
@@ -364,7 +364,7 @@ For the first experiment with variable water pressure, we apply a water-pressure
\textbf{a:}
Rate dependence in till friction from laboratory experiments \citep[after][]{Iverson2010}.
\textbf{b:}
- Influence of rate-dependence factor $b$ in Eq.~\ref{eq:g_local} on post-failure friction in continuum model.
+ Influence of rate-dependence factor $b$ in Eq.~\ref{eq:g_local} on post-failure friction in the current continuum model.
Here, $\mu_\text{s} = 0.5$ and $\sigma_\text{n}' = 100$ kPa.
\textbf{c:}
Mohr-Coulomb analysis of till samples in laboratory experiments \citep[after][]{Iverson2010}.
@@ -377,9 +377,9 @@ For the first experiment with variable water pressure, we apply a water-pressure
We first compare the modeled mechanical behavior to various tills tested in laboratory settings (Fig.~\ref{fig:rate_dependence}, after \citet{Iverson2010}).
Over five orders of strain-rate magnitude, some tills show slight rate weakening and others are slightly rate strengthening (Fig.~\ref{fig:rate_dependence}a).
Shear-strain rates up to 5.000 a$^{-1}$ are realistic for natural glacier systems \citep{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:rate_dependence}b).
+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:rate_dependence}b), making the model under these conditions rate strengthening.
The modeled friction value can be linearly scaled by adjusting $\mu_\text{s}$ 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:rate_dependence}d).
+Our model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:rate_dependence}d), which is useful as these parameters are often constrained from Mohr-Coulomb tests on till samples.
\begin{figure*}[htbp]
\begin{center}
@@ -393,9 +393,9 @@ Our model can simulate any combination of effective friction (or friction angle
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 \citet{Damsgaard2013}.
+Instead, we compare the modeled strain distribution with discrete-element results from \citet{Damsgaard2013} which allow us to calibrate $A$.
By inserting relevant material parameters for grain size, friction, stress, and shear velocity (DEM, Table~\ref{tab:params}), we almost exactly replicate the strain distribution with the NGF model (Fig.~\ref{fig:strain_distribution}).
-Sediment advection is pressure dependent, with low effective normal stresses producing shallow deformation, and high effective normal stresses deepening the material mobilization.
+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.
\begin{figure}[htbp]
@@ -410,10 +410,11 @@ The DEM results took more than two months of computational time, whereas the con
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 $\sigma_\text{n}'$ = 100 kPa.
-The grain size $d$ has a major influence on the strain distribution, where finer materials show deeper deformation.
+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 ($\mu_\text{s}$), as the material is less willing to fail.
-Our implementation of cohesion does not influence strain.
+Our implementation of cohesion does not influence strain after yielding.
Static friction and cohesion both linearly scale the bulk friction, as expected with Mohr-Coulomb materials (see also Fig.~\ref{fig:rate_dependence}).
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.
@@ -454,9 +455,9 @@ The depth of maximum shear-strain rate corresponds to the depth of minimum in ef
\begin{center}
\includegraphics[width=15.0cm]{experiments/fig-stick_slip_rate_depth.pdf}
\caption{\label{fig:stick_slip_depth}%
- Pore-pressure diffusion and strain distribution with depth with a sinusoidal water-pressure forcing from the top (same experiment as Fig.~\ref{fig:stick_slip}a).
+ Pore-pressure diffusion and strain distribution with depth with a sinusoidal water-pressure forcing from the top (same experiment as Fig.~\ref{fig:stick_slip}b).
The forcing has a daily periodocity, and plot lines are one hour in simulation time apart.
- The horizontal magenta line marks skin depth from Eq.~\ref{eq:skin_depth}.
+ The horizontal green line marks skin depth from Eq.~\ref{eq:skin_depth}.
}
\end{center}
\end{figure*}
@@ -464,7 +465,7 @@ The depth of maximum shear-strain rate corresponds to the depth of minimum in ef
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.% \citep[p.\ 271 in][]{Turcotte2002}.
-Deep deformation occurs when the effective normal stress is less at depth than at the top.
+Deep deformation occurs when the effective normal stress is smaller at depth than at the top.
\begin{figure}[htbp]
\begin{center}
@@ -472,7 +473,7 @@ Deep deformation occurs when the effective normal stress is less at depth than a
\includegraphics[width=0.49\textwidth]{experiments/fig-pulse_square.pdf}
\caption{\label{fig:pulse}%
Pulse perturbations in top water pressure with linearly increasing amplitude over time.
- The perturbations are triangular (left) or square (right).
+ The experiments are rate controlled, and are forced with triangular (left) or square (right) perturbations in top pore pressure.
The fluid is reset to hydrostatic pressure distribution before each new pressure pulse.
}
\end{center}
@@ -480,15 +481,14 @@ Deep deformation occurs when the effective normal stress is less at depth than a
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.
-The response in maximum deformation depth is non-linear for triangular perturbations, and linear with square perturbations.
\section{Discussion}%
\label{sec:discussion}
-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}.
+The stress-dependent sediment advection without variations in the pore pressure 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, 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.,][]{Sergienko2013, McCracken2016, Iverson2017b, Hermanowski2019b}.
+Topography of non-planar ice-bed interfaces (proto-drumlins, ribbed moraines, etc.) may be transported and modulated through this variable transport capacity, unless stress differences are overprinted by spatial variations in water pressure \citep[e.g.,][]{Sergienko2013, McCracken2016, Iverson2017b, Hermanowski2019b}.
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.
@@ -500,7 +500,7 @@ Our numerical solution to pore-pressure diffusion can be forced with any water-p
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 of smooth forcings 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},
+As long as fluid and diffusion properties are constant and the layer is sufficiently thick, 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}
d_\text{s}
@@ -527,19 +527,19 @@ Practically all of the shear strain through a perturbation cycle occurs above th
\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.
+It is worth noting that to induce deep deformation the water pressure deviations need to exceed the initial effective stress gradient.
This means that minima in effective normal stress are increasingly difficult to create at larger depths through pure diffusion from the ice-bed interface.
-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}).
+Due to higher hydraulic permeability, coarse tills are more susceptible to deep deformation, but deep strain requires longer-lasting perturbations in water pressure (Fig.~\ref{fig:skin_depth}).
Contrarily, fine-grained tills are unlikely to cause deep deformation.
\citet{Truffer2000} and \citet{Kjaer2006} both observed deep deformation in glacier settings with relatively coarse subglacial tills.
Lateral water input at depth may be a viable alternate mechanism for creating occasional episodes of deep slip, in particular when horizontal bedding decreases vertical permeability \citep[e.g.,][]{Kjaer2006}.
-
+TODO: LAKE DRAINAGE
\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.
+The mechanics adhere to Mohr-Coulomb plasticity, with a weak and 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.
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.
diff --git a/experiments/fig-rate_dependence.pdf b/experiments/fig-rate_dependence.pdf
Binary files differ.
