commit a7f0be01be04a19c219c757e98ea06eb7adb9809
parent c302d2c9abe63ca642d8c82fecf931973746e05a
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 3 Dec 2019 16:14:08 +0100
Improve discussion
Diffstat:
1 file changed, 22 insertions(+), 28 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -297,25 +297,23 @@ The representative grain size $d$ has a major influence on the strain distributi
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:mohr_coulomb}).
+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.
\section{Discussion}%
\label{sec:discussion}
-
-Granular deformation occurs where the effective normal stress is at its lowest value.
-Due to sediment non-locality, the stress minima needs to be of a sufficient thickness in order for a shear zone establishment.
-
-
-In this study it is assumed that there is a strong coupling between ice and bed.
-However, overpressurization and slip at the ice-bed interface may cause episodic decoupling at the interface and reduce bed deformation, as observed under Whillans Ice Stream, West Antarctica \cite<e.g.,>[] {Engelhardt1998}, and in deposits from Pleistocene glaciations \cite<e.g.,>[] {Piotrowski2001}.
-We see the presented framework as a significant improvement of treating sediment advection in ice-flow models, but acknowledge that a complete understanding of the sediment mass budget requires improved models of ice-bed interface physics.
-
-The stress-dependent sediment advection without variations in the pore pressure observed in Fig.~\ref{fig:strain_distribution} is relevant for instability theories of subglacial landform development \cite{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 this variable transport capacity, unless stress differences are overprinted by spatial variations in water pressure \cite<e.g.,>[] {Sergienko2013, McCracken2016, Iverson2017b, Hermanowski2019b}.
+This study has the specific aim of quantifying advective sediment transport under shear.
+As granular deformation is associated with finite length scales, it is crucial to include non-local terms in the granular model equations instead of applying earlier \emph{local} sediment models \cite<e.g.,>[] {daCruz2005, Jop2006, Forterre2008}.
+However, the modeled sediment flux may present an upper bound since we assume that there is a strong mechanical coupling between ice and bed.
+Overpressurization and slip at the ice-bed interface may cause episodic decoupling and reduce bed deformation.
+Interface slip is observed both under contemporary ice streams \cite<e.g.,>[] {Engelhardt1998}, and in deposits from past glaciations \cite<e.g.,>[] {Piotrowski2001}.
+Still, we see the presented framework as a significant improvement of treating sediment advection in ice-flow models, but acknowledge that a complete understanding of the sediment mass budget requires improved models of ice-bed interface physics.
+
+Without water-pressure variations, the sediment advection is stress dependent (Fig.~\ref{fig:validation}d), which is a prerequisite for instability theories of subglacial landform development \cite{Hindmarsh1999, Fowler2000, Schoof2007, Fowler2018}.
+From geometrical considerations, it is likely that up-ice sloping stoss sides of subglacial topography experience higher bed-normal stress than down-ice sloping lee sides.
+With all other physical conditions being equal, our results indicate that shear-driven sediment advection would be larger on the stoss side than on the lee side.
+Topography of non-planar ice-bed interfaces (proto-drumlins, ribbed moraines, etc.) may be transported and modulated through this spatially variable transport capacity, unless stress differences are overprinted by variations in water pressure \cite<e.g.,>[] {Sergienko2013, McCracken2016, Iverson2017b, Hermanowski2019b}.
At depth, the water pressure variations display exponential decay in amplitude and increasing lag.
As long as fluid and diffusion properties are constant and the layer is sufficiently thick, an analytical solution to skin depth $d_\mathrm{s}$ [m] in our system follows the form (after Eq.~4.90 in \citeA{Turcotte2002}),
@@ -329,27 +327,23 @@ 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.
-Unrelated to skin depth, the forcing amplitude determines if pressure anomalies at depth are sufficiently large to facilitate shear (Fig.~\ref{fig:pulse}).
-Figure~\ref{fig:skin_depth} shows the skin depth for water under a range of permeabilities and forcing frequencies.
-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 (magenta line in Fig.~\ref{fig:stick_slip_depth}).
-
+Figure~S3 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.
-It is worth noting that to induce deep deformation the water pressure deviations need to exceed the initial effective stress gradient.
-This means that 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.~\ref{fig:skin_depth}).
-Contrarily, fine-grained tills are unlikely to cause deep deformation.
-Lateral water input 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}.
-%TODO: LAKE DRAINAGE
+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).
+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}.
\section{Conclusion}%
\label{sec:conclusion}
We present a new model for coupled computation of subglacial till and water.
The model is based on the concept of non-local granular fluidity \cite{Henann2013}, but is extended with cohesion and pore-pressure diffusion.
The mechanics adhere to Mohr-Coulomb plasticity, with a weak and non-linear rate dependence governed by stress and sediment properties.
-In agreement with laboratory results, the material is effectively rate-independent at glacial shear velocities.
-A simple shear experimental setup is adapted for analyzing the mechanical response under different stresses and water-pressure variations.
-With cyclical water-pressure variations at the ice-bed interface, deep deformation occurs when remnant high water pressures at depth overcome the lithostatic gradient.
+The material is effectively rate-independent at glacial shear velocities, generally in agreement with laboratory results.
+We adapt a simple shear experimental setup for analyzing the mechanical response under different stresses and water-pressure variations.
+With cyclical water-pressure variations at the ice-bed interface, deep deformation occurs when remnant high water pressures at depth overcome the effective lithostatic gradient.
Deep deformation may be common in coarse-grained subglacial tills with strong annual water-pressure differences.
Similarly, sudden water-pressure pulses are powerful drivers for single events of deep deformation.
