commit b03651da54d34dd531feb12b8abcdbb418c4961e
parent eab76b7dc633dd7daa5b6f45d5a87e4459683d99
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Fri, 13 Dec 2019 12:13:02 +0100
Update main text with Liran's feedback
Diffstat:
1 file changed, 39 insertions(+), 37 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -37,9 +37,9 @@
\correspondingauthor{Anders Damsgaard}{anders@adamsgaard.dk}
\begin{keypoints} % each keypoint max 140 chars
-\item Non-local continuum model is consistent with subglacial till rheology
-\item Sediment advection depends on the subglacial stress and sediment properties
-\item Water-pressure perturbations can cause deep shear mobilization of till beneath the ice-bed interface
+\item Non-local continuum model of deforming grains is consistent with subglacial till rheology
+\item Sediment transport depends on the subglacial stress and sediment properties
+\item Diffusion of water-pressure perturbations generates deep shear zones within the till
\end{keypoints}
\begin{abstract}
@@ -58,7 +58,8 @@ TODO
\section{Introduction}
-Fast glacier and ice-sheet flow often ocurrs over weak sedimentary deposits, where basal slip accounts for nearly all movement \cite <e.g.,>[]{Cuffey2010}.
+Fast glacier and ice-sheet flow often ocurrs over weak sedimentary deposits.
+While internal ice deformation is small, basal sediment deformation and slip at the ice-bed interface accounts for nearly all movement \cite <e.g.,>[]{Cuffey2010}.
Basal sediments, called subglacial till, are diamictons commonly consisting of reworked sediments and erosional products \cite <e.g.,>[] {Evans2006}.
Meltwater fully saturates the pore space, and variations in subglacial water pressure are common and can be caused by internal variability \cite<e.g.,>[] {Kavanaugh2009} or external water input \cite<e.g.,>[] {Andrews2014, Christoffersen2018}.
In-situ field observations demonstrate that deformation of this layer can contribute significantly to the glacier movement \cite<e.g.,>[] {Boulton1979, Humphrey1993, Truffer2000}.
@@ -69,7 +70,7 @@ Mohr-Coulomb plastic materials have a yield strength that linearly scales with e
\citeA{Iverson2010} reviewed possible viscous contributions during till-water deformation, but deemed them to be of minor importance.
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 \cite<e.g.,>[] {Jenson1995}, landform formation \cite<e.g.,>[] {Hindmarsh1999, Fowler2000}, localization of water drainage \cite<e.g.,>[] {Walder1994, Ng2000b}, and ice-sheet behavior in a changing climate \cite<e.g.,>[] {Pollard2009}.
-Meanwhile, the Mohr-Coulomb plastic model continued to gain further empirical support from laboratory testing \cite<e.g.,>[] {Rathbun2008, Iverson2015}, as well as field observations on mountain glaciers \cite<e.g.,>[] {Hooke1997, Truffer2006, Iverson2007}, mountain glaciers \cite<e.g.,>[] {Kavanaugh2006}, and ice sheets \cite<e.g.,>[] {Tulaczyk2006, Gillet-Chaulet2016, Minchew2016}.
+Meanwhile, the Mohr-Coulomb plastic model continued to gain further empirical support from laboratory testing \cite<e.g.,>[] {Rathbun2008, Iverson2015}, as well as field observations on mountain glaciers \cite<e.g.,>[] {Hooke1997, Truffer2006, Kavanaugh2006, Iverson2007}, and ice sheets \cite<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.
The deadlock was partially resolved when \citeA{Schoof2006} and \citeA{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.
@@ -89,7 +90,6 @@ We discuss its applicability and technical limitations before comparing the simu
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 \cite{Truffer2000, Kjaer2006}.
The model produces stick-slip dynamics under variable water pressures, as observed in mountain glaciers \cite<e.g.,>[] {Fischer1997} and ice streams \cite<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.
@@ -99,7 +99,8 @@ The model source code is constructed with minimal external dependencies, is free
Soils, tills, and other sediments are granular materials, consisting of discrete grains that interact with frictional losses.
\citeA{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 \citeA{daCruz2005} and \citeA{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.
+However, these continuum models are \emph{local}, meaning that local stresses determines the local strain-rate response alone.
+This means that material properties do not influence shear zone width.
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 \cite<e.g.,>[] {Nedderman1980, Forterre2008, Kamrin2018}.
@@ -127,7 +128,7 @@ The fluidity $g$ is a kinematic variable governed by grain velocity fluctuations
\end{equation}
\end{linenomath*}
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 $A$ [-]:
+The degree of non-locality is scaled by the cooperativity length $\xi$, which, in turn, scales with nonlocal amplitude $A$ [-]:
\begin{linenomath*}
\begin{equation}
\xi(\mu) = \frac{Ad}{\sqrt{|(\mu - C/\sigma_\mathrm{n}') - \mu_\mathrm{s}|}},
@@ -149,7 +150,6 @@ The local contribution to fluidity is defined as:
where $\rho_\mathrm{s}$ 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 cases where the surrounding areas have a high local fluidity.
-This characteristic also strengthens the material if the shear zone size is restricted by bed geometry.
\subsection{Fluid-pressure evolution}%
\label{sub:fluid_pressure_evolution}
@@ -185,10 +185,13 @@ We apply the model in a one-dimensional setup with simple shear (Fig.~\ref{fig:v
Parameter values and their references are listed in Table~S1.
The lower boundary condition for the granular phase is no slip ($v_x(L=0) = 0$).
The upper boundary condition for the granular phase is fixed shear friction $\mu(z=L_z)$ under stress controlled settings, or fixed shear velocity $v_x(z=L_z)$ for rate-controlled experiments.
-The upper normal stress ($\sigma_\mathrm{n}$) is constant, but effective normal stress ($\sigma_\mathrm{n}' = \sigma_\mathrm{n} - p_\mathrm{f}$) varies if water pressure $p_\mathrm{f}$ changes.
+The upper normal stress ($\sigma_\mathrm{n}(z=L_z)$) is constant, and normal stress linearly increases with depth due to material weight.
+Effective normal stress ($\sigma_\mathrm{n}' = \sigma_\mathrm{n} - p_\mathrm{f}$) varies if water pressure $p_\mathrm{f}$ changes.
For the water-pressure solver, the top pressure ($p_\mathrm{f}(z=L_z)$) is either constant or varied through time.
The water pressure is set to follow the hydrostatic gradient at the lower boundary ($dp_\mathrm{f}/dz(z=0) = \rho_\mathrm{f}G$).
-For the experiments with variable water pressure, we apply a water-pressure forcing amplitude of 50 kPa that modulates effective stress at the top around 100 kPa.
+For the experiments with variable water pressure, we apply a water-pressure forcing amplitude of 50 kPa that modulates effective normal stress at the top around 100 kPa.
+The presented model formulation produces a shear strain rate from sediment properties and applied stress.
+Inverse simulations are performed by iteratively adjusting the applied stress until the output shear velocity matches the desired value (full details in Text S1.1).
Many simulations are performed under both stress- and rate-controlled shear, which both idealize the driving glacier physics.
Real glacier settings fall somewhere in between, depending on how important basal friction is to the overall stress balance.
Stress-controlled conditions approximate a setting where ice flow directly responds to changes in subglacial strain rates.
@@ -206,7 +209,7 @@ A rate-controlled setup is the opposite end member, where changes in bed frictio
a) Experimental setup for the NGF model in one-dimensional shear.
The upper boundary is constant friction $\mu$ (stress controlled), or constant velocity $v_x$ (velocity controlled).
b) Rate dependence of critical-state friction in laboratory experiments on till (after \citeA{Iverson2010}), and the NGF model with material friction $\mu_\mathrm{s}$ = 0.5 and effective normal stress $\sigma_\mathrm{n}'$ = 100 kPa.
- The dimensionles parameter $b$ controls frictional rate dependence (Eq.~\ref{eq:g_local}).
+ The dimensionles parameter $b$ controls frictional rate dependence (Eq.~\ref{eq:g_local}), where $b$ = 0.94 is appropriate for glass beads \cite{Henann2016}.
c) Mohr-Coulomb analysis of till samples and NGF model.
d) Modeled strain distribution under varying effective normal stress ($\sigma_\mathrm{n}'$) with the discrete-element method (DEM, \citeA{Damsgaard2013}) and the NGF model.
}
@@ -220,14 +223,14 @@ A rate-controlled setup is the opposite end member, where changes in bed frictio
We first compare the modeled mechanical behavior to various tills tested in laboratory settings.
Over five orders of strain-rate magnitude, some tills show slight rate weakening and others are slightly rate strengthening (Fig.~\ref{fig:validation}b).
Shear-strain rates up to $\sim5 \times 10^3$ a$^{-1}$ are realistic for natural glacier systems \cite{Cuffey2010}.
-The 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:validation}b), making the model under these conditions rate strengthening.
+The model is effectively rate-independent over most of the range, but higher values of $b$, which determines the degree of rate dependence, provide larger frictional resistance at extreme shear-strain rates (Fig.~\ref{fig:validation}b).
The modeled friction value can be linearly scaled by adjusting $\mu_\mathrm{s}$ in Eqs.~\ref{eq:cooperativity} and~\ref{eq:g_local}.
The model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:validation}c), which is useful as these parameters are often constrained from Mohr-Coulomb tests on till samples.
The NGF model contains parameter $A$ for adjusting the degree of material non-locality (Eq.~\ref{eq:cooperativity}).
However, at present 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 \citeA{Damsgaard2013}.
-By inserting relevant material parameters for grain size, friction, stress, and shear velocity (DEM, Table~S1), the NGF model model approximates the strain distribution well (Fig.~\ref{fig:validation}d).
+By inserting relevant material parameters for grain size, friction, stress, and shear velocity (DEM, Table~S1), the NGF model approximates the strain distribution well (Fig.~\ref{fig:validation}d).
Both models show that 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 and adjustment towards the critical state.
@@ -247,15 +250,15 @@ The DEM results took more than two months of computational time, whereas the con
Next we vary the top water pressure and observe the shear dynamics over a simulation time of seven days (Fig.~\ref{fig:stick_slip}).
The experiments are performed under both stress and rate-controlled configurations.
The response during the first cycle ($t<1$ d) is slightly different from later cycles ($t>1$ d) as the model is initialized with a hydrostatic water-pressure distribution.
-Under stress-controlled conditions (Fig.~\ref{fig:stick_slip}a), the system shows stick-slip behavior where velocities range from 0 to $\sim$9 km/d.
-The depth of maximum deformation moves into the bed as effective normal stress at the top boundary increases.
-Under the stress-controlled conditions the till flux peaks during rapid slip as water pressure at the ice-bed interface is at its maximum.
+Under stress-controlled conditions (Fig.~\ref{fig:stick_slip}a-d), the system shows stick-slip behavior where velocities range from 0 to $\sim$9 km/d (Fig.~\ref{fig:stick_slip}b).
+The depth of maximum deformation moves into the bed as effective normal stress at the top boundary increases (Fig.~\ref{fig:stick_slip}c).
+Under the stress-controlled conditions the till flux peaks during rapid slip as water pressure at the ice-bed interface is at its maximum (Fig.~\ref{fig:stick_slip}d).
-In the rate-controlled configuration (Fig.~\ref{fig:stick_slip}b), the shear stress varies as effective normal stress oscillates, as expected from a Mohr-Coulomb material.
-As in the stress-controlled configuration, deformation propagates into the bed as effective normal stress increases at the top.
-Contrary to the stress-controlled setup, the till flux is under rate-controlled shear largest during deep deformation events, which occur when the ice-bed water pressure is at its minimum value.
+In the rate-controlled configuration (Fig.~\ref{fig:stick_slip}e-h), the shear stress varies as effective normal stress oscillates (Fig.~\ref{fig:stick_slip}f), as expected from a Mohr-Coulomb material.
+As in the stress-controlled configuration, deformation propagates into the bed as effective normal stress increases at the top (Fig.~\ref{fig:stick_slip}g).
+Contrary to the stress-controlled setup, the till flux is largest under rate-controlled shear during deep deformation events (Fig.~\ref{fig:stick_slip}h), which occur when the ice-bed water pressure is at its minimum value.
We find that pulse perturbations of various shape in water pressure are also able to cause deep deformation (Fig.~S1).
-The maximum deformation depth increases with increasing perturbation amplitude, with a temporal lag governed by pressure diffusion.
+The maximum deformation depth increases with increasing perturbation amplitude.%, with a temporal lag governed by pressure diffusion.
\begin{figure}[htbp]
\begin{center}
@@ -272,7 +275,7 @@ The maximum deformation depth increases with increasing perturbation amplitude,
Figure~\ref{fig:hysteresis} demonstrates that the grain/fluid system displays strong hysteresis in shear velocity, strain distribution, and till flux under both stress and rate-controlled conditions.
As noted from the previous figure, the sediment flux has different maxima between the driving modes.
Stress-controlled shear produces a large sediment flux in a thin deforming layer close to the boundary during slip events when effective pressure is at its lowest value (Fig.~\ref{fig:hysteresis}a).
-On the other hand, under rate-controlled conditions the majority of sediment flux occurs as a plug-like flow where deformation occurs deep in the bed during maxima in effective normal stress at the ice-bed interface (Fig.~\ref{fig:hysteresis}b).
+On the other hand, under rate-controlled conditions the deformation occurs deep in the bed during maxima in effective normal stress at the ice-bed interface (Fig.~\ref{fig:hysteresis}b).
\begin{figure*}[htbp]
\begin{center}
@@ -286,20 +289,10 @@ On the other hand, under rate-controlled conditions the majority of sediment flu
\end{figure*}
Figure~\ref{fig:stick_slip_depth} shows depth variations through a day of simulation time, which corresponds to a single wavelength of water-pressure forcing.
-The effective normal stress generally increases with depth according to the difference in grain and fluid density.
+The effective normal stress generally increases with depth according to the difference in grain and fluid density (Fig.~\ref{fig:stick_slip_depth}a).
However, water pressure variations at the ice-bed interface can reverse this depth trend.
-Granular failure generally occurs where effective normal stress is at its minimum, as long as there is enough space to accomodate shear zone size.
-Due to diffusion, water pressure perturbations decay exponentially with depth and travel with a phase shift.
-
-Figure~S2 contains a systematic analysis of parameter influence in the model equations.
-Several observations emerge from this parameter sensitivity analysis.
-The representative grain size $d$ has a major influence on the strain distribution, where finer materials show deeper deformation.
-The material is slightly weaker with larger grain sizes.
-The shear zone is more narrow with higher material static friction coefficients ($\mu_\mathrm{s}$), as the material is less willing to fail.
-Our implementation of cohesion does not influence strain after yield.
-Static friction and cohesion both linearly scale the bulk friction, as expected with Mohr-Coulomb materials (see also Fig.~\ref{fig:validation}c).
-The non-local amplitude $A$ slightly changes the curvature of the shear strain profile, but does not affect the overall friction.
-There is a significant strengthening when the bed thickness $L_z$ begins to constrict the shear zone thickness.
+Due to diffusion, water pressure perturbations decay exponentially with depth and travel with a phase shift (Fig.~\ref{fig:stick_slip_depth}b).
+Granular failure generally occurs where effective normal stress is at its minimum, resulting in a plug-like flow (Fig.~\ref{fig:stick_slip_depth}c,d) during reversal of the depth trend in effective normal stress (Fig.~\ref{fig:stick_slip_depth}b).
\section{Discussion}%
\label{sec:discussion}
@@ -328,15 +321,24 @@ As long as fluid and diffusion properties are constant and the layer is sufficie
\end{linenomath*}
where $D$ is the hydraulic diffusivity [m$^2$/s] and $P$ [s] is the period of the oscillations.
The remaining terms were previously defined.
-Figure~S3 shows the skin depth for water at 0$^\circ$C under a range of permeabilities and forcing frequencies.
+Figure~\ref{fig:skin_depth}a shows the skin depth for water at 0$^\circ$C under a range of permeabilities and forcing frequencies.
We find that skin depth calculations can be a useful starting point for determining scenarios where deep deformation is possible.
The stick-slip experiments (Fig.~\ref{fig:stick_slip}) correspond to a skin depth of 2.2 meter.
Practically all of the shear strain through a perturbation cycle occurs above the skin depth (green horizontal line in Fig.~\ref{fig:stick_slip_depth}).
However, minima in effective normal stress are increasingly difficult to create at larger depths through pure diffusion from the ice-bed interface.
-Due to higher hydraulic permeability, coarse tills are more susceptible to deep deformation, but deep strain requires longer-lasting perturbations in water pressure (Fig.~S3).
+Due to higher hydraulic permeability, coarse tills are more susceptible to deep deformation (Fig.~\ref{fig:skin_depth}a), but deep strain requires larger perturbations in water pressure (Fig.~\ref{fig:skin_depth}b,c).
On the contrary, fine-grained tills are unlikely to undergo deep deformation.
Instead, lateral water input from lake drainage or hydrological rerouting at depth may be a viable alternate mechanism for creating occasional episodes of deep slip, in particular when horizontal bedding decreases vertical permeability \cite<e.g.,>[] {Kjaer2006}.
+\begin{figure}[htbp]
+ \begin{center}
+ \includegraphics[width=15cm]{experiments/fig-skin_depth.pdf}
+ \caption{\label{fig:skin_depth}%
+ Skin depth of pore-pressure fluctuations (Eq.\ 6 in the main text) with forcing frequencies ranging from yearly to hourly periods.
+ The permeability spans values common for tills \cite{Schwartz2003}.
+ }
+ \end{center}
+\end{figure}
\section{Conclusion}%
\label{sec:conclusion}
