commit 701b42111317a0a4a863e663d0965a020b6889dc
parent 90ed5d47b7711f1eb3d55ba5d5d1cc187e90a7fe
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Thu, 7 Nov 2019 12:48:12 +0100
Improve writing throughout
Diffstat:
1 file changed, 18 insertions(+), 20 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -367,13 +367,12 @@ For the first experiment with variable water pressure, we apply a water-pressure
\end{center}
\end{figure*}
-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 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}.
+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).
-The modeled friction value can be shifted up and down by adjusting $\mu_\text{s}$ in Eqs.~\ref{eq:g_local} and~\ref{eq:cooperativity}.
-Tills are Mohr-Coulomb materials, where the shear stress linearly depends on effective normal stress (Fig.~\ref{fig:rate_dependence}c).
-Our modified NGF model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:rate_dependence}d).
+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).
\begin{figure*}[htbp]
\begin{center}
@@ -388,9 +387,9 @@ Our modified NGF model can simulate any combination of effective friction (or fr
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}.
-By inserting relevant material parameters for grain size, friction, stress, and shear velocity (Table~\ref{tab:params}), we almost exactly replicate the strain distribution with the NGF model (Fig.~\ref{fig:strain_distribution}).
+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.
-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.
+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]
\begin{center}
@@ -439,12 +438,10 @@ The response during the first cycle ($t<1$ d) is slightly different from later c
\end{center}
\end{figure}
-Under both stress and rate-controlled conditions, the grain/fluid system displays strong hysteresis in shear velocity and strain distribution (Fig.~\ref{fig:stick_slip_depth}).
+Under both stress and rate-controlled conditions, the grain/fluid system displays strong hysteresis in shear velocity and strain distribution (Fig.~\ref{fig:hysteresis}).
The granular deformation primarily occurs where the effective normal stress is the lowest value.
-When water pressure drops at the ice-bed interface, there is a remnant of high fluid-pressure diffusing downwards.
-The depth of maximum shear-strain rate follows the minimum in effective normal stress into the bed, until the effective normal stress at the top exceeds the minimum at depth.
-It is worth noting that the minima in $\sigma'_\text{n}$ need a width comparable to the shear zone thickness, due to granular non-locality.
-Otherwise the material rather deforms in a zone where the depth integral of effective normal stress is lower.
+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 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}
@@ -459,8 +456,8 @@ Otherwise the material rather deforms in a zone where the depth integral of effe
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 top water pressure is at a minimum, and the effective normal stress is less at depth than at the top.
+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.
\begin{figure}[htbp]
\begin{center}
@@ -488,12 +485,14 @@ Topography of non-planar ice-bed interfaces (proto-drumlins, ribbed moraines, et
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.
+\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.
+However, these studies did not include non-local granular effects assosiated with granular deformation.
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.
+Our numerical solution to pore-pressure diffusion can be forced with any water-pressure signal from the ice-bed interface.
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}.
+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},
\begin{linenomath*}
\begin{equation}
@@ -523,10 +522,10 @@ Practically all of the shear strain through a perturbation cycle occurs above th
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}.
+\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}.
\section{Conclusion}%
@@ -535,7 +534,6 @@ 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.
