commit 26136d9adf3d6ece87d2b5e4d247f4fd53571ac2
parent 7e35bec83dd5d7e56331debc16cc0e713a141a61
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Wed, 30 Oct 2019 14:49:13 +0100
Rework introduction and the beginning of methods
Diffstat:
1 file changed, 59 insertions(+), 162 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -62,171 +62,65 @@ Deep deformation is most likely in tills with relatively high hydraulic permeabi
\label{sec:introduction}
Fast glacier and ice-sheet flow often ocurrs over weak sedimentary deposits, where basal slip accounts for nearly all movement \citep[e.g.,][]{Cuffey2010}.
-Basal sediments, called subglacial till, are diamictons commonly consisting of reworked sediments and erosional products \citep[e.g.][]{Evans2006}.
+Basal sediments, called subglacial till, are diamictons commonly consisting of reworked sediments and erosional products \citep[e.g.,][]{Evans2006}.
%\citet{Hooke1995} demonstrated that the grain-size distribution is fractal.
-Water is typically generated at the ice-bed interface because of frictional heating, and fully saturates the pore space.
-The pressure of the pore-water relieves some of the overburden ice weight, reducing the compressive stress on granular skeleton to Terzaghi's effective stress \citep{Terzaghi1943}.
-Water viscosity does not significantly contribute to critical-state shear stress.
-\citet{Iverson2010} reviewed the possible viscous contributions during sediment dilation and compaction but deemed them to be of minor importance.
-
-Early on, \citet{Boulton1987} presented observations of in-situ deformation of subglacial till and concluded that till rheology was mildly non-linear viscous.
-However, \citet{Kamb1991}, \citet{Iverson1998}, and \citet{Tulaczyk2000} demonstrated from laboratory shear tests that rate-independent Mohr-Coulomb plasticity is a far better rheological description.
-In spite of vaning observational basis, viscous rheologies did not fall out of favor as they allow for mathematical modeling of till deformation patterns.
-In particular, the rheology was used to explain subglacial landform formation \citep[e.g.,]{Hindmarsh1999, Fowler2000}, water drainage localization \citep[e.g.][]{Walder1994}, and ice-sheet mass loss \citep[e.g.,][]{Pollard2009}.
-\citet{Ritz2015} demonstrated that the assertion of basal behavior heavily influences future Antarctic contributions to global-mean sea level rise in ice-sheet models
-We know that the assumptions matter, but not what to replace them with.
-However, the dynamic interplay of ice, water, and sediment is still not well understood.
-Small-scale laboratory experiments have provided revolutionary insights into till mechanics \citep{Iverson2015}, but few of the discovered sediment processes have found their way into numerical ice-sheet models.
-In particular, sediment advection during shear is
-
-Here comes you!
-Shed new unique light on this matter with a model that is related to granular-scale mechanics, but can be applied in an ice-sheet kind of setting.
-For that we need the continuum formulation is really hard, many other people have tried but it has been questionable and contradicts laboratory experiments.
-Here comes our contribution which is consistent with laboratory experiments, but we can also connect to field data because of the continuum scale, which no laboratory experiment can ever do.
-Building on new modeling advances in granular mechanics.
-
-
-%% The review
-% Pick out 3-10 papers providing background to my research and say something about each of them.
-% For example, paraphrase a sentence or two from each abstract.
-% Organize the review so that it leads up to something, namely, my claim.
-
-Fast ice flow is often ocurring over weak sedimentary beds \citep[e.g.,][]{Kamb1991, Cuffey2010}.
-Early on, \citet{Boulton1987} concluded that subglacial till behaved mildly-non-linear viscous.
-
-
-The basal sediment mechanics
-\citet{Ritz2015}
-Early on, till was assumed to be mildly non-linear viscous with a constant rate dependence \citep{Boulton1987}.
-Increasing shear stress acts as a negative feedback on perturbations in glacier flow rate.
-
-
-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 basal sediment mechanics
-\citet{Ritz2015}
-Early on, till was assumed to be mildly non-linear viscous with a constant rate dependence \citep{Boulton1987}.
-Increasing shear stress acts as a negative feedback on perturbations in glacier flow rate.
-
-
-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 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.
-
-
-%% Contribution sentence
-We adapt a continuum model for dry granular flows by \citet{Henann2013} to the subglacial environment by adding pore-pressure dynamics and cohesion.
-Our model makes it possible to simulate subglacial till strength and sediment advection from the mechanical state and water pressure at the ice-bed interface.
-
-
-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*}
-\begin{equation}
- \dot{\gamma} = a \frac{\tau^n}{(\sigma_\text{n}')^m},
- \label{eq:rheology}
-\end{equation}
-\end{linenomath*}
-where $\dot{\gamma}$ is the shear-strain rate [s$^{-1}$], and $a$ is a material rate dependence [s$^{-1}$].
-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.
-
-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}.
-
-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}.
-
-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}.
-
-% Till continuity (Alley and Cuffey).
-
-The \emph{undrained plastic-bed} model by \citet{Tulaczyk2000b} was the first model to characterize subglacial till as a rate-independent material, where yield strength is solely dependent on compressive stress.
-The model utilizes the relationship between porosity and effective normal stress in order to couple till strength with subglacial hydraulics, and successfully reproduces transient flow dynamics in ice-flow models \citep{Bougamont2011, vanDerWel2013, Robel2013, Gillet-Chaulet2016, Bougamont2019}.
-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}.
-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-plasticity.
-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}.
-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.
-For most sedimentary materials cohesion C is close to zero.
-Importantly, the Mohr-Coulomb relationship only describes the yield point of inertialess deformation, and there is no length scale to shear zones.
-
-\citet{Bagnold1954} realized that granular flows show a complex rate dependence.
-It was later shown that a dimensionless inertia number $I$ summarizes the mechanical behavior of cohesionless, critical-state, granular flows \citep{GDR-MiDi2004}:
-\begin{linenomath*}
-\begin{equation}
- I = \frac{\dot{\gamma} d}{\sqrt{\sigma_\text{n}/\rho_\text{s}}},
- \label{eq:inertia_number}
-\end{equation}
-\end{linenomath*}
-where $d$ is the representative grain size, $\sigma_\text{n}$ is the normal stress, and $\rho_\text{s}$ is the density of the grain material.
-$I$ describes the relative importance of the microscopic and macroscopic time scales.
-For example, if a granular material is sheared quickly, $I$ goes up as inertia increases and grains spend less time in a locked arrangement.
-On the contrary, if normal stress increases $I$ goes down as grain mobility in the shear zone is restricted.
-For a given material, the critical-state ratio between shear stress and normal stress $\mu = \tau/\sigma_\text{n}$ depends non-linearly on $I$ \citep{daCruz2005, Jop2006}.
-The dependence allows for empirical relationships where $\mu(I)$ takes form of a highly non-linear Bingham rheology.
-Tuned to experiments on simple granular materials, the material is more rate dependent at large inertia numbers (i.e., ``landslide regime'').
-With smaller inertia numbers, the behavior smoothly transitions to a ``pseudo-static regime'' of near rate independence ($I < 10^{-3}$).
-
-Similarly, shear-zone porosity was in monodisperse materials found to linearly depend on $I$ \citep{Pouliquen2006}.
-The relationships for strength and porosity act as constitutive relations, making continuum modeling of granular flows possible.
-\citet{Pailha2008} and \citet{Pailha2009} demonstrated simple coupling of the continuum granular model to pore-pressure dynamics.
-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}.
-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 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.
-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.
+Water is typically generated at the ice-bed interface from frictional heating, and fully saturates the pore space.
+The pore pressure relieves some of the overburden ice weight and reduces the compressive stress on the granular skeleton to Terzaghi's effective stress \citep{Terzaghi1943}.
+%Water viscosity does not significantly contribute to critical-state shear stress.
+The dynamic interplay of ice, water, and sediment is important for glacier and ice-sheet dynamics, but remains poorly understood \citep[e.g.,][]{Clarke2005}.
+
+\citet{Boulton1979} presented the pioneering idea that sedimentary beds and hydrological processes can significantly influence glacier and ice-sheet flow and stability.
+This discovery sparked interest in understanding the associated sediment physics.
+\citet{Boulton1987} argued that a viscous rheological model with mild stress non-linearity appropriately described their in-situ observations of a deforming bed.
+The viscous rheology implies that stress required to deform the till is strongly dependent on how fast it is deformed.
+Secondly, viscous materials have no upper bound to their strength with increasing strain rates.
+However, this result was hard to reproduce.
+\citet{Kamb1991}, \citet{Iverson1998}, and \citet{Tulaczyk2000} demonstrated from laboratory shear tests that rate-independent Mohr-Coulomb plasticity, as common for sedimentary materials, is a far better rheological description for subglacial till.
+Mohr-Coulomb plastic materials have a yield strength that linearly scales with effective stress.
+\citet{Iverson2010} reviewed possible viscous contributions during till-water deformation, but deemed them to be of minor importance.
+In spite of a vaning observational basis, viscous rheologies continued to be applied as they allow for mathematical modeling of till advection.
+Tills with viscous rheology were used to explain coupled ice-bed processes including subglacial sediment transport \citep[e.g.,][]{Jenson1995}, landform formation \citep[e.g.,][]{Hindmarsh1999, Fowler2000}, localization of water drainage \citep[e.g.,][]{Walder1994, Ng2000b}, and ice-sheet behavior in a changing climate \citep[e.g.,][]{Pollard2009}.
+Meanwhile, the Mohr-Coulomb plastic model continued to gain mounting empirical support from further laboratory testing \citep[e.g.,][]{Rathbun2008, Iverson2015}, as well as field observations on mountain glaciers \citep[e.g.,][]{Hooke1997, Truffer2006, Iverson2007} and ice sheets \citep[e.g.,][]{Tulaczyk2006, Gillet-Chaulet2016, Minchew2016}.
+Inconveniently, the plastic rheology caused a deadlock for the typical continuum modeling of ice and till, as the Mohr-Coulomb constitutive model offers no direct relation between stress and strain rate.
+\citet{Schoof2006} and \citet{Bueler2009} showed that Mohr-Coulomb friction can be included in ice-sheet models through mathematical reguralization.
+\citet{Ritz2015} demonstrated that the type of basal friction is highly influential for future Antarctic contributions to global-mean sea level rise.
+However, the deformable bed is assumed to be devoid of reshaping through erosion, transport, and deposition as there exists no appropriate formulation for modeling strain in subglacial till and associated advective transport.
+
+\citet{Damsgaard2013} and \citet{Damsgaard2016} demonstrated that strain distribution and plasticity can be modeled in water-saturated tills by explicitly considering each sediment grain.
+Unfortunately, intense computational requirements associated with the grain-scale modeling entirely outrule model applicability for coupled ice-till simulations.
+Instead, simulation of landform to ice-sheet scale requires continuum models.
+In this study we build on continuum-modeling advances in granular mechanics and produce a model appropriate for water-saturated sediment deformation in the subglacial environment.
+The original model by \citet{Henann2013} is developed for critical state deformation of dry and cohesionless granular materials.
+However, subglacial tills are water saturated and often contain a certain amount of cohesion that generally increases with clay content \citep[e.g.,][]{Iverson1997}.
+We extend the \citet{Henann2013} model by including sediment-strength contributions from cohesion and effective stress variations from pore-pressure diffusion.
+The resultant model contributes the methodological basis required for understanding the coupled dynamics of ice flow over deformable beds.
+Different from previous continuum models for till, our model remains true to rheological properties observed in laboratory and field settings.
+
+In the following, we present the \citet{Henann2013} model and our modifications for modeling water-saturated subglacial till.
+We discuss its applicability and technical limitations before comparing the simulated sediment behavior to published results from laboratory experiments on tills.
+The model produces stick-slip dynamics under variable water pressures, and remnants of pressure deviations within the modeled glacier bed constribute hysteresis to stress and strain.
+In particular, the model demonstrates its ability to produce deformation deep away from the ice-bed interface, as occasionally observed in field settings \citep{Truffer2000, Kjaer2006}.
+The model source code is constructed with minimal external dependencies, is freely available, and is straight-forward to couple to models of ice-sheet dynamics and glacier hydrology.
\section{Methods}%
\label{sec:methods}
-\subsection{Granular flow}%
+Soils, tills, and other sediments are granular materials, consisting of discrete grains that interact with frictional losses.
+\citet{GDR-MiDi2004} demonstrated that inertia summarizes the mechanical behavior of dry granular deformation.
+This finding evolved into an empirical continuum rheology in \citet{daCruz2005} and \citet{Jop2006}, where stress and porosity depends on inertia in a non-linear manner.
+However, these continuum models are \emph{local}, meaning that the spatially local state determines the local strain-rate response alone.
+Granular deformation contains numerous non-local effects, where flow rates in neighboring areas influence the tendency of a sediment parcel to deform.
+Granular shear zones are an example of the non-locality as they have a minimum width dependent on grain characteristics \citep[e.g.,][]{Nedderman1980, Forterre2008, Kamrin2018}.
+
+\subsection{Non-local granular fluidity (NGF) model}%
\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}.
-We assume that elasticity is negligible and set the total shear rate $\dot{\gamma}$ to consist of a plastic contribution $\dot{\gamma}^\text{p}$:
+\citet{Henann2013} presented the non-local granular fluidity (NGF) model where a \emph{fluidity} field variable accounts for the non-local effects on deformation.
+The model builds on the previous continuum rheology for granular materials by \citet{daCruz2005} and \citet{Jop2006}, but accurately describes strain distribution in a variety of experimental settings.
+All material is assumed to have a uniform porosity and be in the critical state.
+Fluidity acts as a state variable, describing the phase transition between non-deforming (jammed) and actively deforming (flowing) parts of the sediment \citep[e.g.,][]{Zhang2017}.
+The modeled sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit \citep[e.g.,][]{Henann2013, Henann2016}.
+For the purposes in this paper, we assume that plastic shear strain ($\dot{\gamma}^\text{p}$) contributes all of the resultant deformation ($\dot{\gamma}$):
\begin{linenomath*}
\begin{equation}
\dot{\gamma} \approx \dot{\gamma}^\text{p} = g(\mu_\text{c}, \sigma_\text{n}') \mu,
@@ -285,12 +179,14 @@ The fluid pressure is used to determine the effective normal stress used in the
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}.
-A transient granular model with state-dependent porosity is currently under development.
+We currently have a transient granular continuum model with state-dependent porosity under development.
+However, \citet{Iverson2010} argued that the majority of actively deforming subglacial sediment may be in the critical state.
+For that reason, we see this contribution as a valuable first step.
-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}.
+In the NGF model, the representative grain size $d$ scales the non-locality and strain distribution.
+However, it is awkward to describe grain size distributions of diamictons with fractal grain size distribution with a single length scale \citep{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.
+Future research will investigate how wide grain-size distributions affect strain distribution, and will benchmark against specifically designed laboratory experiments on tills.
\subsection{Numerical solution procedure}%
\label{sub:numerical_solution_procedure}
@@ -649,6 +545,7 @@ Similarly, sudden water-pressure pulses are powerful drivers for single events o
\label{sec:appendix}
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 simulation parameters can be specified as command-line arguments.
+Run \texttt{./1d\_fd\_simple\_shear -h} after compiling for usage information.
The results and figures in this paper 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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