commit 76eb1fda78781efe4bc83141688d0834d6ebae2d
parent ab60478166212d3b6621705e8410a795ad289d4e
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 21 Oct 2019 10:25:18 +0200
Add comma after e.g.
Diffstat:
1 file changed, 54 insertions(+), 29 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -48,6 +48,8 @@ maxcitenames=2, backend=bibtex8]{biblatex}
\maketitle
\begin{abstract}
+The dynamic interplay between fast ice flow, meltwater drainage, and till deformation.
+
Subglacial sediment mechanics are of primary importance to glacier and ice-sheet flow patterns at high basal water pressures.
Till yield strength follows the non-linear Mohr–Coulomb rheology, but that does not by itself describe the spatial distribution of strain.
Here we present a water-saturated granular continuum model that is consistent with laboratory experiments on till and follows Mohr–Coulomb behavior.
@@ -59,8 +61,31 @@ We show that past pulses in water pressure can transfer shear away from the ice-
\section{Introduction}%
\label{sec:introduction}
-Subglacial sediment deformation is in many settings of primary importance to glacier flow \citep[e.g.][]{Boulton1974, Engelhardt1990, Fischer1994, Truffer2006}.
-Sediment mechanics influence glacier stability, sediment transport, and bedform genesis, which is why till rheology is long debated \citep[e.g.][]{Alley1986, Boulton1987, Kamb1991, Iverson1995, Hindmarsh1997, Hooke1997, Fowler2003, Kavanaugh2006, Iverson2010, Hart2011, Fowler2018}.
+
+% Never include expressions such as 'is discussed' or 'is described'
+
+%% The review
+% Pick out 3-10 papers providing background to my research and say something about each of them.
+% For example, paraphrase a sentance or two from each abstract.
+% Organize the review so that it leads up to something, namely, my claim.
+
+
+%% The claim
+% Why the paper's agenda is a worthwile extension of the historical review.
+% Personal pronouns should be used in the claim and anywhere else the author expresses judgement, opinion, or choice.
+
+
+%% The agenda
+% Summarize what we will show the reader as the paper progresses.
+% Tell how the paper works to fulfill our claim.
+% In this way, the agenda should clarify the claim.
+% The agenda is not as important as the review and the claim: keep it short.
+% If some of the conclusions can be made in simple statements, state them right after the agenda.
+
+
+
+Subglacial sediment deformation is in many settings of primary importance to glacier flow \citep[e.g.,][]{Boulton1974, Engelhardt1990, Fischer1994, Truffer2006}.
+Sediment mechanics influence glacier stability, sediment transport, and bedform genesis, which is why till rheology is long debated \citep[e.g.,][]{Alley1986, Boulton1987, Kamb1991, Iverson1995, Hindmarsh1997, Hooke1997, Fowler2003, Kavanaugh2006, Iverson2010, Hart2011, Fowler2018}.
Modeling of till transport requires that the strain distribution in the soft bed can be described by the stress field and material properties.
The simplest invoked equation is of the form:
\begin{linenomath*}
@@ -73,22 +98,22 @@ where $\dot{\gamma}$ is the shear-strain rate [s$^{-1}$], and $a$ is a material
The shear- and effective normal stress [Pa] is denoted by $\tau$ and $\sigma_\text{n}'$, respectively.
The stress exponent ($n$ and $m$) values characterize the mechanical non-linearity.
Exponent values of 1 produce linear viscous behavior, and a material is perfectly plastic when $n$ and $m$ values go to infinity.
-The degree of non-linearity of subglacial till may pose drastically different ice-stream behavior \citep[e.g.][]{Bougamont2011, Tsai2015} and contributions to global mean sea-level rise \citep[e.g.][]{Parizek2013, Ritz2015}.
+The degree of non-linearity of subglacial till may pose drastically different ice-stream behavior \citep[e.g.,][]{Bougamont2011, Tsai2015} and contributions to global mean sea-level rise \citep[e.g.,][]{Parizek2013, Ritz2015}.
Early on, till was assumed to be mildly non-linear viscous with a constant rate dependence \citep[$a=3.99$, $n=1.33$ and $m=1.8$ in][]{Boulton1987}.
-This rheology allowed the formulation of analytical solutions to the coupled ice-till problem \citep[e.g.][]{Alley1987, Walder1994, Hindmarsh1999, Fowler2000, Schoof2007}.
+This rheology allowed the formulation of analytical solutions to the coupled ice-till problem \citep[e.g.,][]{Alley1987, Walder1994, Hindmarsh1999, Fowler2000, Schoof2007}.
%The viscous rheology implies that the till looses all strength if deformation rates approach zero, and the till strength is without an upper bound as strain rate increases.
-Resultant glacier sliding laws are similar to empirical soft-bed sliding laws without cavitation \citep[e.g.][]{Budd1979}.
+Resultant glacier sliding laws are similar to empirical soft-bed sliding laws without cavitation \citep[e.g.,][]{Budd1979}.
Increasing shear stress acts as a negative feedback on perturbations in glacier flow rate.
-However, laboratory experiments on tills \citep[e.g.][]{Kamb1991, Iverson1998, Tulaczyk2000, Rathbun2008, Iverson2010, Iverson2015} and field observations \citep[e.g.][]{Iverson1995, Hooke1997, Tulaczyk2006} have concluded that till strength is nearly independent of deformation rate, and behaves according to Mohr-Coulomb plasticity.
+However, laboratory experiments on tills \citep[e.g.,][]{Kamb1991, Iverson1998, Tulaczyk2000, Rathbun2008, Iverson2010, Iverson2015} and field observations \citep[e.g.,][]{Iverson1995, Hooke1997, Tulaczyk2006} have concluded that till strength is nearly independent of deformation rate, and behaves according to Mohr-Coulomb plasticity.
In some cases, it has been observed that till strength slightly decreases at faster shear rates \citep{Iverson1998, Iverson2015}, which could potentially amplify changes in glacier velocities.
-The presence of water can add a transient rate dependence due to volumetric adjustmend during early shear \citep[e.g.][]{Iverson1997, Moore2002, Damsgaard2015}, but this rate dependence is generally shortlived.
-Furthermore, pre-failure creep can occur as sediment reacts to stress change with a highly-nonlinear rate dependence \citep[e.g.][]{Kamb1991, Damsgaard2016, Hart2019}.
+The presence of water can add a transient rate dependence due to volumetric adjustmend during early shear \citep[e.g.,][]{Iverson1997, Moore2002, Damsgaard2015}, but this rate dependence is generally shortlived.
+Furthermore, pre-failure creep can occur as sediment reacts to stress change with a highly-nonlinear rate dependence \citep[e.g.,][]{Kamb1991, Damsgaard2016, Hart2019}.
Besides smaller scale laboratory and field investigations, the larger scale effects of basal friction are investigated by inverting observations of glacier-surface flow velocities to sliding physics.
Ice streams move primarily by sliding and subglacial sediment deformation.
-In places, such as Whillans Ice Plain, small perturbations in stress and water pressure cause a dramatic response in ice flow \citep[e.g.][]{Bindschadler2003, Tulaczyk2006}.
-The inferred stress exponent values characterize the collective behavior of the basal ice and subglacial sediment, and range from 2 to over 10 \citep[e.g.][]{Tulaczyk2006, Gudmundsson2006, King2011, Gudmundsson2011, Walker2012, Rosier2014, Goldberg2014, Thompson2014, Rosier2015, Gillet-Chaulet2016, Minchew2016}.
+In places, such as Whillans Ice Plain, small perturbations in stress and water pressure cause a dramatic response in ice flow \citep[e.g.,][]{Bindschadler2003, Tulaczyk2006}.
+The inferred stress exponent values characterize the collective behavior of the basal ice and subglacial sediment, and range from 2 to over 10 \citep[e.g.,][]{Tulaczyk2006, Gudmundsson2006, King2011, Gudmundsson2011, Walker2012, Rosier2014, Goldberg2014, Thompson2014, Rosier2015, Gillet-Chaulet2016, Minchew2016}.
% Till continuity (Alley and Cuffey).
@@ -97,14 +122,14 @@ The model utilizes the relationship between porosity and effective normal stress
A common implementation is a regularized form where shear stress vanishes when shear-strain rates go to zero \citep{Schoof2006, Bueler2009, Schoof2010}.
In these forms the shear stress is still limited to the Mohr-Coulomb value at higher rates \citep{Schoof2006, Bueler2009, Schoof2010}.
However, the Coulomb-frictional parameterizations do not describe the actual sediment deformation, but describe the basal friction felt by the flowing ice.
-Shear deformation is known to deepen under increasing effective normal stress \citep{Fischer1997, Iverson1999, Boulton2001, Damsgaard2013}, and this may be a primary ingredient for growth of subglacial topography \citep[e.g.][]{Schoof2007}.
+Shear deformation is known to deepen under increasing effective normal stress \citep{Fischer1997, Iverson1999, Boulton2001, Damsgaard2013}, and this may be a primary ingredient for growth of subglacial topography \citep[e.g.,][]{Schoof2007}.
In order to model soft-bed sliding and till continuity, a model is necessary that accurately describes subglacial shear strain while being in accordance to Mohr-Coulomb friction and sediment near-plastcitiy.
The discrete element model for sediment deformation by \citet{Damsgaard2013} includes Coulomb-frictional physics and sediment strain distribution, but is far too detailed and costly for coupled ice-till computations.
\subsection{Insights from granular physics}%
Soils, tills, and other sediments are granular materials, consisting of discrete grains interacting with frictional losses.
-A key characteristic of granular materials is the ability to change mechanical phase \citep[jammed, flowing, in suspension, e.g.][]{Jaeger1996, deGennes1999, Forterre2008}.
+A key characteristic of granular materials is the ability to change mechanical phase \citep[jammed, flowing, in suspension, e.g.,][]{Jaeger1996, deGennes1999, Forterre2008}.
The holy grail within the field of granular physics is to find a unifying mathematical framework describing behavior across all phases.
The fundamental understanding of the strength of granular materials goes all the way back to the 18th century.
The Mohr-Coulomb constitutive relation postulates a linear relationship betweeen effective normal stress on a shear zone and the maximum shear stress it can support.
@@ -133,16 +158,16 @@ The relationships for strength and porosity act as constitutive relations, makin
However, these models are \emph{local}, meaning that the spatially local stress state determines the local strain-rate response alone.
Compared to glacier models, this formulation corresponds to the shallow-ice approximation where local ice-surface slope is the sole factor for local shear-strain rate.
But, like many glacial settings, granular deformation is often \emph{non-local}.
-For example, granular shear zones have a minimum width, dependent on grain characteristics \citep[e.g.][]{Nedderman1980, Kamrin2018}.
+For example, granular shear zones have a minimum width, dependent on grain characteristics \citep[e.g.,][]{Nedderman1980, Kamrin2018}.
Furthermore, $\mu(I)$ rheology does not work for slow flows as the thickness of shear bands depends on the shear velocity and vanishes in the quasi-static limit \citep{Forterre2008}.
\citet{Henann2013} presented the non-local granular fluidity (NGF) model where a \emph{fluidity} field accounts for the non-local effects on deformation.
The NGF model builds on the previous $\mu(I)$ rheology, but accurately describes strain distribution in a variety of experimental settings.
However, the NGF model assumes all the material to be in the critical state and a uniform porosity throughout the domain.
-The fluidity acts as a state variable, describing the phase transition between jammed and flowing parts \citep[e.g.][]{Zhang2017}.
+The fluidity acts as a state variable, describing the phase transition between jammed and flowing parts \citep[e.g.,][]{Zhang2017}.
The resultant granular rheology is based on observations, non-dimensionalized on the base of material properties.
The \citet{Henann2013} is fundamentally different than viscous rheologies proposed for glacial tills, but is more akin to statistical mechanics.
-The NGF model allows upscaling of the discrete element method \citep[e.g.][]{Cundall1979, Damsgaard2013}, while remaining true to the physics.
+The NGF model allows upscaling of the discrete element method \citep[e.g.,][]{Cundall1979, Damsgaard2013}, while remaining true to the physics.
However, the NGF model is dry, and in the context of subglacial mechanics, dry models are generally not useful.
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.
@@ -153,7 +178,7 @@ In this paper we expand the steady-state NGF continuum model for granular flow b
\subsection{Granular flow}%
\label{sub:granular_flow}
-In the NGF model, the sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit \citep[e.g.][]{Henann2013, Henann2016}.
+In the NGF model, the sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit \citep[e.g.,][]{Henann2013, Henann2016}.
We assume that elasticity is negligible and set the total shear rate $\dot{\gamma}$ to consist of a plastic contribution $\dot{\gamma}^\text{p}$:
\begin{linenomath*}
\begin{equation}
@@ -197,7 +222,7 @@ In the above framework, the material strengthens when the shear zone size is res
\subsection{Fluid-pressure evolution}%
\label{sub:fluid_pressure_evolution}
-We prescribe the transient evolution of pore-fluid pressure ($p_\text{f}$) by Darcian pressure diffusion \citep[e.g.][]{Goren2010, Goren2011, Damsgaard2017b}:
+We prescribe the transient evolution of pore-fluid pressure ($p_\text{f}$) by Darcian pressure diffusion \citep[e.g.,][]{Goren2010, Goren2011, Damsgaard2017b}:
\begin{linenomath*}
\begin{equation}
\frac{\partial p_\text{f}}{\partial t} = \frac{1}{\phi\mu_\text{f}\beta_\text{f}} \nabla \cdot (k \nabla p_\text{f}),
@@ -212,12 +237,12 @@ The fluid pressure is used to determine the effective normal stress used in the
\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 volume changes \citep[e.g.][]{Iverson2000, Iverson2010-2, Damsgaard2015} or compaction \citep[e.g.][]{Dewhurst1996}.
+For that reason the model is not able to simulate uniaxial compaction or shear-induced volume changes \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 representative grain size $d$ scales the non-locality and strain distribution.
-However, it is awkward to describe grain size distributions of diamictons with a single value \citep[e.g.][]{Hooke1995}.
-We expect that a volumetrically dominant grain size dominates the strain distribution, outside of effects of ploughing by large clasts \citep[e.g.][]{Tulaczyk1999}.
+However, it is awkward to describe grain size distributions of diamictons with a single value \citep[e.g.,][]{Hooke1995}.
+We expect that a volumetrically dominant grain size dominates the strain distribution, outside of effects of ploughing by large clasts \citep[e.g.,][]{Tulaczyk1999}.
%Future research will investigate how wide grain-size distributions affect strain distribution.
\subsection{Numerical solution procedure}%
@@ -292,7 +317,7 @@ As for the granular flow solution, we also use operator splitting and finite dif
\end{linenomath*}
For each time step $\Delta t$, a solution to Eq.~\ref{eq:p_f_solution} is first found by explicit temporal integration.
We then use Jacobian iterations to find an implicit solution to the same equation using underrelaxation.
-For the final pressure field at $t + \Delta t$ we mix the explicit and implicit solutions with equal weight, which is known as the Crank-Nicholson method \citep[e.g.][]{Patankar1980, Ferziger2002, Press2007}.
+For the final pressure field at $t + \Delta t$ we mix the explicit and implicit solutions with equal weight, which is known as the Crank-Nicholson method \citep[e.g.,][]{Patankar1980, Ferziger2002, Press2007}.
The method is unconditionally stable and second-order accurate in time and space.
\subsubsection{Rate-controlled experiments} % quick edit, needs rewrite. perhaps also move somewhere else
@@ -361,10 +386,10 @@ 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},
- 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}.
+ Material parameters for experiments 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}.
The parameter values originate from the following references:
a: \citet{Damsgaard2013},
b: \citet{Iverson1998},
@@ -515,7 +540,7 @@ The response in maximum deformation depth is non-linear for triangular perturbat
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, 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 the 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.
@@ -524,7 +549,7 @@ We couple the water-pressure diffusion with a more complex sediment rheology tha
The slight rate dependence (Fig.~\ref{fig:rate_dependence}) makes it relatively trivial to couple to ice-flow models, while retaining realistic sediment physics.
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 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}
@@ -554,10 +579,10 @@ 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}.
+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}.
+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}%
