commit d3524ed214e5dd9c254e375c738c7413e1d37f74
parent 2155b2767f7cc48c73887962b690c097dba5b7a2
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 7 Oct 2019 16:14:26 +0200
Work on intro
Diffstat:
1 file changed, 31 insertions(+), 25 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -51,35 +51,42 @@ We show that past pulses in water pressure can transfer shear away from the ice-
\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}.
-Modeling of till erosion, transport, and deposition requires quantification of strain distribution in the sediment.
-With analytical models or numerical continuum models
-
-
-Early on, till was assumed to be viscous \citep{Boulton1987} which allowed the formulation of analytical solutions to the coupled ice-till problem \citep[e.g.][]{Alley1987, Walder1994, Hindmarsh1999, Fowler2000}.
-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 hard-bed sliding laws without cavitation \citep[e.g.][]{Budd1979}, acting as a negative feedback on perturbations in glacier flow rate.
+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{equation}
+ \dot{\gamma} = a \frac{\tau^n}{(\sigma_\text{n}')^m},
+ \label{eq:rheology}
+\end{equation}
+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.
+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}.
+%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}.
+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.
-
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 be contributed by adjustment of sediment-internal stresses, giving a highly-nonlinear rate dependence \citep[e.g.][]{Kamb1991, Damsgaard2016, Hart2019}.
+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 basal sliding problem is investigated by inverting observations of glacier-surface flow velocities to subglacial sliding physics.
-Ice streams primarily move by sliding and subglacial sediment deformation, and velocities can vary by three orders of magnitude during a tidal cycle \citep[e.g.][]{Bindschadler2003, Tulaczyk2006}.
-The nonlinearity manifests itself in stress exponent values from 2 to over 10 \citep[e.g.][]{Tulaczyk2006, Gudmundsson2006, King2011, Gudmundsson2011, Walker2012, Rosier2014, Goldberg2014, Thompson2014, Rosier2015, Gillet-Chaulet2016, Minchew2016}.
-The degree of non-linearity may pose drastically different contributions to global mean sea-level rise in future scenarios \citep[e.g.][]{Parizek2013, Ritz2015}.
+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).
-% Damsgaard2013
-
-% Tsai, Schoof
-
-The \emph{undrained plastic-bed} model by \citet{Tulaczyk2000b} uses the relationship between sediment void-ratio, effective normal stress, and shear stress to link hydraulics with till strength.
-This parameterization has been included in ice sheet models, successfully reproducing transient flow dynamics \citep{Bougamont2011, vanDerWel2013, Gillet-Chaulet2016, Bougamont2019}.
+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}.
-\citet{Schoof2010} produced a reguralized form of the Coulomb friction sliding law, making it feasible to simulate sliding over plastic tills in ice-sheet models.
-However, the parameterization does not describe how sediment is advected during shear, but exclusively describes the friction at the ice-bed interface.
%\citet{Iverson2001} demonstrated that perturbations in effective stress at depth can distribute deformation away from the ice-bed interface.
@@ -87,7 +94,6 @@ However, the parameterization does not describe how sediment is advected during
-% Minchew2019
% mu(I)
% GDR-MiDi2004
@@ -146,7 +152,7 @@ In the above mathematical framework, the material slightly strengthens when the
\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, Damsgaard2017}:
+We prescribe the transient evolution of pore-fluid pressure ($p_\text{f}$) by Darcian pressure diffusion \citep[e.g.][]{Goren2010, Goren2011, Damsgaard2017b}:
\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}),
\label{eq:p_f}
@@ -424,8 +430,8 @@ Practically all of the shear strain through a perturbation cycle occurs above th
\section*{Appendix}%
\label{sec:appendix}
-The source code for the grain-water model is available at \url{https://gitlab.com/admesg/1d_fd_simple_shear}.
-All results and figures can be reproduced by following the instructions in the experiment repository for this publication, available at \url{https://gitlab.com/admesg/continuum_granular_exp_manus1}.
+The source code for the grain-water model is available at \url{https://src.adamsgaard.dk/1d_fd_simple_shear}.
+All results and figures can be reproduced by following the instructions in the experiment repository for this publication, available at \url{https://src.adamsgaard.dk/continuum_granular_exp_manus1}.
%% Bibliography
\printbibliography{}
