manus_continuum_granular1

manuscript files for first continuum-till paper
git clone git://src.adamsgaard.dk/manus_continuum_granular1
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commit cb25136d1a2087b21154c01d5c7576d2432ce30e
parent 701b42111317a0a4a863e663d0965a020b6889dc
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date:   Tue, 12 Nov 2019 11:44:26 +0100

Begin incorporating Liran's comments

Diffstat:
Mcontinuum-granular-manuscript1.tex | 93++++++++++++++++++++++++++++++++++++++++++-------------------------------------
1 file changed, 50 insertions(+), 43 deletions(-)

diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex @@ -51,10 +51,12 @@ maxcitenames=2, backend=bibtex8]{biblatex} The dynamic interplay between fast ice flow, meltwater drainage, and till deformation is crucial for understanding glacier and ice-sheet behavior. Subglacial sediment transport constructs landforms that influence glacier stress balance and post-glaciation geomorphology. Till yield strength is highly dependent on water pressure and follows the Mohr-Coulomb rheology. -However, the physical transport of till during subglacial shear is not well understood, and is for that reason not included in prognostic ice-sheet models. -We present a water-saturated continuum model that is consistent with Mohr-Coulomb mechanics and is suitable for coupled glacier-sediment-hydrology modeling. -We show that pulses in water pressure can shift deformation away from the ice-bed interface and far into the bed, resulting in significant till advection. -Deep deformation is most likely in tills with relatively high hydraulic permeability, forced by slow and large water-pressure variations. +However, the physical transport of till during subglacial shear is not well understood. +Prognostic ice-sheet models assume that fast flow is produced by shear on the ice-bed interface, while geological observations indicate that internal deformation within till layers is common. +We present a water-saturated continuum model that is both inspired by the granular nature of till layers and is consistent with Mohr-Coulomb mechanics, allowing +modeling of the critical aspects of glacier-sediment-hydrology system. +Model results indicate that pulses in water pressure can shift deformation away from the ice-bed interface and far into the bed, resulting in significant till advection. +Deep deformation is most likely in tills with relatively high hydraulic permeability, forced by long-lasting and large water-pressure variations. \end{abstract} @@ -65,12 +67,15 @@ Fast glacier and ice-sheet flow often ocurrs over weak sedimentary deposits, whe 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 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}. +Variations in subglacial water pressure are common and can be caused by internal and external factors. +For example, \citet{Kavanaugh2009} showed from field observations that hydraulic flow paths and sediment deformation can constantly rearrange. +Large pressure variations can occur through episodic drainage of ice-surface lakes \citep[e.g.,][]{Christoffersen2018} or subglacial lakes \citep[e.g.,][]{Palmer2015}. +The pore pressure relieves some of the overburden ice weight and reduces the compressive solid stress on the granular skeleton lowering Terzaghi's effective stress \citep{Terzaghi1943}. +Pore-pressure induced stress modifications control the kinematics of the ice and sub-ice sediments, which in turn may feed back into the dynamics of the system. +The interplay of ice, water, and sediment is therefore 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. +This idea 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. A viscous rheology implies that the 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. @@ -78,29 +83,29 @@ 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, regardless of strain rate. \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 under glaciers. +In spite of a limited 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}. +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}, mountain glaciers \citep[e.g.,][]{Kavanaugh2006}, 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 methods allowing plastic beds in ice-flow models offer no treatment of sediment erosion, transport, and deposition as there exists no appropriate formulation for modeling strain in subglacial till and associated advective transport. +The deadlock was partially resolved when \citet{Schoof2006} and \citet{Bueler2009} showed that Mohr-Coulomb friction can be included in ice-sheet models through mathematical reguralization. +While the methods describe the mechanical effect of the bed on the flowing ice, they offer no treatment of sediment erosion, transport, and deposition as strain in the sedimentary bed is not included. +Yet, \citet{Ritz2015} demonstrated that the description of basal friction is highly influential for future Antarctic ice-sheet contributions to global-mean sea level rise, where sliding over plastic beds is likely to increase future contributions to sea-level rise. \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 pore-pressure diffusion and add strength contributions from cohesion. +We rely on the original model by \citet{Henann2013} that was developed for critical state deformation of dry and cohesionless granular materials. +To correctly account for subglacial tills that 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 the notion of pore-pressure controlled effective stress, pore-pressure diffusion and add strength contributions from cohesion. 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. +In the following, we present the \citet{Henann2013} model and our modifications for modeling water-saturated and cohesive 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. -Remnants of pressure deviations within the glacier bed provide hysteresis to stress and strain histories. +The model produces rich dynamics that is consistent and could explain previously poorly-understood field observations. 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 produces stick-slip dynamics under variable water pressures, as observed in mountain glaciers \citep[e.g.,][]{Fischer1997} and ice streams \citep[e.g.,][]{Bindschadler2003}. +Remnants of pressure deviations within the glacier bed cause hysteresis in stress and strain. 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. @@ -108,8 +113,8 @@ The model source code is constructed with minimal external dependencies, is free \label{sec:methods} 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. +\citet{GDR-MiDi2004} introduced a non-dimensional inertia number that summarizes the mechanical behavior of dry and dense granular deformation. +This finding evolved into an empirical continuum rheology in \citet{daCruz2005} and \citet{Jop2006}, where stress and porosity depend 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}. @@ -117,49 +122,51 @@ Granular shear zones are an example of the non-locality as they have a minimum w \subsection{Non-local granular fluidity (NGF) model}% \label{sub:granular_flow} \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. +The model builds on the previous continuum rheology for granular materials by \citet{daCruz2005} and \citet{Jop2006}, and with non-local effects accurately describes strain distribution in a variety of experimental settings. +In the \citet{Henann2013} model, 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}. +The modeled sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit \citep[e.g.,][]{Henann2013, Henann2016}, but unlike classical plastic models it includes a closed form relation that predicts the stress-strain rate relation beyond yield. 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, + \dot{\gamma} \approx \dot{\gamma}^\text{p} = g(\mu, \sigma_\text{n}') \mu, \label{eq:shear_strain_rate} \end{equation} \end{linenomath*} where $\mu = \tau/\sigma_\text{n}'$ is the dimensionless ratio between shear stress ($\tau$ [Pa]) and effective normal stress ($\sigma_\text{n}' = \sigma_\text{n} - p_f$ [Pa]). Water pressure is $p_\text{f}$ [Pa] and $g$ [s$^{-1}$] is the granular fluidity. -The fluidity is a kinematic variable governed by grain velocity fluctuations and packing fraction \citep{Zhang2017}, and consists of local and non-local components. -The local contribution to fluidity is defined as: +The fluidity $g$ is a kinematic variable governed by grain velocity fluctuations and packing fraction \citep{Zhang2017}, and consists of local and non-local components: \begin{linenomath*} \begin{equation} - g_\text{local}(\mu, \sigma_\text{n}') = - \begin{cases} - \sqrt{d^2 \sigma_\text{n}' / \rho_\text{s}} ((\mu - C/\sigma_\text{n}') - \mu_\text{s})/(b\mu) &\text{if } \mu - C/\sigma_\text{n}' > \mu_\text{s} \text{, and}\\ - 0 &\text{if } \mu - C/\sigma_\text{n}' \leq \mu_\text{s}. - \end{cases} - \label{eq:g_local} + \nabla^2 g = \frac{1}{\xi^2(\mu)} (g - g_\text{local}), + \label{eq:g} \end{equation} \end{linenomath*} -where $d$ [m] is the representative grain diameter, $\mu_\text{s}$ [-] is the static Coulomb yield coefficient, $C$ [Pa] is the material cohesion, and $b$ [-] is the non-linear rate dependence beyond yield. -The failure point is determined by the Mohr-Coulomb constituent relation. -Beyond failure, the flow is governed by a Poisson-type equation that distributes strain in space according to material properties and stress state. -The degree of non-locality is scaled by the cooperativity length $\xi$: +The above Poisson-type equation acts to distribute strain in space according to material properties and non-local stress state. +The degree of non-locality is scaled by the cooperativity length $\xi$, wich, in turn, scales with nonlocal amplitude $A$ [-]: \begin{linenomath*} \begin{equation} - \nabla^2 g = \frac{1}{\xi^2(\mu)} (g - g_\text{local}), - \label{eq:g} + \xi(\mu) = \frac{Ad}{\sqrt{|(\mu - C/\sigma_\text{n}') - \mu_\text{s}|}}, + \label{eq:cooperativity} \end{equation} \end{linenomath*} -where +where $d$ [m] is the representative grain diameter, $\mu_\text{s}$ [-] is the static Coulomb yield coefficient, and $C$ [Pa] is the material cohesion. +The local contribution to fluidity is defined as: \begin{linenomath*} \begin{equation} - \xi(\mu) = \frac{Ad}{\sqrt{|(\mu - C/\sigma_\text{n}') - \mu_\text{s}|}}. - \label{eq:cooperativity} + g_\text{local}(\mu, \sigma_\text{n}') = + \begin{cases} + \sqrt{d^2 \sigma_\text{n}' / \rho_\text{s}} ((\mu - C/\sigma_\text{n}') - \mu_\text{s})/(b\mu) &\text{if } \mu - C/\sigma_\text{n}' > \mu_\text{s} \text{, and}\\ + 0 &\text{if } \mu - C/\sigma_\text{n}' \leq \mu_\text{s}. + \end{cases} + \label{eq:g_local} \end{equation} \end{linenomath*} -The non-locality scales with nonlocal amplitude $A$ [-] and grain size $d$. +where $\rho_\text{s}$ is grain mineral density and $b$ [-] controls the non-linear rate dependence beyond yield. +The failure point is principally determined by the Mohr-Coulomb constituent relation in the conditional of Eq.~\ref{eq:g_local}. +However, the non-locality in Eq.~\ref{eq:g} infers that deformation can occur in places that otherwise would not fail. + + In the above framework, the material strengthens when the shear zone size is restricted by thickness of the granular bed. \subsection{Fluid-pressure evolution}%