commit 18a948ba4b54d0352e4b4be097b57ffb5cebd5d4
parent ed98e8c2e898a7c08665bdb616513c5cf7c6090e
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Wed, 18 Dec 2019 13:43:55 +0100
Update text for simulation setup and results, swap fig 2 and 3
Diffstat:
1 file changed, 37 insertions(+), 41 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -182,16 +182,17 @@ Specifically designed laboratory experiments with various tills should inform th
\subsection{Simulation setup}
We apply the model in a one-dimensional setup with simple shear (Fig.~\ref{fig:validation}a).
-Parameter values and their references are listed in Table~S1.
+Parameter values 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 ($\sigma_\mathrm{n}(z=L_z)$) 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 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).
+For the experiments with variable water pressure, we apply a water-pressure forcing amplitude of $A_\mathrm{f}$ = 80 kPa that modulates effective normal stress at the top around 100 kPa.
+Unless noted, we use the value 0.94 for the dimensionless rate-dependence parameter $b$, which is impirically constrained from laboratory experiments on glass beads \cite{Forterre2003, Henann2016}.
+Under stress-controlled configurations, the model formulation produces a shear strain rate from sediment properties and applied stress.
+Rate-controlled conditions are implemented 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.
@@ -224,62 +225,57 @@ We first compare the modeled mechanical behavior to various tills tested in labo
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 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 modeled friction value can be linearly scaled by adjusting $\mu_\mathrm{s}$ in Eqs.~\ref{eq:cooperativity} and~\ref{eq:g_local}, and the model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:validation}c).
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 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.
+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 dynamical adjustment towards the critical state.
-Next we vary the top water pressure in a sinusoidal manner and observe the shear dynamics over a simulation time of seven days (Fig.~S1).
-The experiments are performed under both stress and rate-controlled configurations.
+Next, we vary the water pressure at the top in a sinusoidal manner and observe the resultant shear dynamics over a simulation time of seven days (Fig.~S1).
+The experiments are performed in both stress and rate-controlled configurations.
+Figure~\ref{fig:stick_slip_depth} shows depth variations in the rate-controlled configuration through a day of simulation time, which corresponds to a single wavelength of the water-pressure forcing.
+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, and create minima in effective normal stress away from the ice-bed interface.
+Due to diffusion, water pressure perturbations decay exponentially with depth and travel with a phase shift (Fig.~\ref{fig:stick_slip_depth}b).
+Maximum shear deformation moves downwards into the bed as effective normal stress at the top boundary increases (Fig.~\ref{fig:stick_slip_depth}c).
+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).
+
+Over the entire simulation period, the system shows stick-slip behavior under stress-controlled conditions where velocities range from 0 to $\sim$9 km/d with strong hysteresis (Fig.~\ref{fig:hysteresis}a).
The response during the first cycle ($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-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).
+Under rate-controlled conditions, the shear stress varies linearly as predicted for Mohr-Coulomb materials but with slight hysteresis at high effective normal stresses (Fig.~\ref{fig:hysteresis}d).
+Both driving modes produce deep deformation during periods where water-pressure at the ice-bed interface is at its lowest magnitude.
+Under stress-controlled conditions, the deep deformation produces an insignificant amount till flux (Fig.~\ref{fig:hysteresis}c), as the shear stress at this time as the shear stress is insufficient for slip (Fig.~\ref{fig:hysteresis}a).
+Instead, the majority of sediment transport occurs as shallow deformation during rapid slip events and high water pressures (Fig.~\ref{fig:hysteresis}b,c).
+On the contrary, the largest sediment transport under rate-controlled conditions occurs with low water pressures and high effective normal-stresses at the ice-bed interface (Fig.~\ref{fig:hysteresis}e,f).
+Pulse perturbations of various shape in water pressure are also able to cause deep deformation, and the maximum deformation depth increases with increasing perturbation amplitude (Fig.~S2).
-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.
+\begin{figure*}[htbp]
+ \begin{center}
+ \includegraphics[width=15.0cm]{experiments/fig-stick_slip_rate_depth.pdf}
+ \caption{\label{fig:stick_slip_depth}%
+ Pore-pressure diffusion and strain distribution with depth with a sinusoidal water-pressure forcing from the top.
+ The bed thickness is $L_z$ = 8 m, but here only the upper 4 m are shown.
+ The water-pressure forcing has a daily periodocity, and plot lines are one hour apart in simulation time.
+ The horizontal green line marks skin depth from Eq.~\ref{eq:skin_depth}.
+ }
+ \end{center}
+\end{figure*}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.49\textwidth]{experiments/fig-hysteresis_stress.pdf}
\includegraphics[width=0.49\textwidth]{experiments/fig-hysteresis_rate.pdf}
\caption{\label{fig:hysteresis}%
- Hysteresis of stress, velocity, and sediment flux under sinusoidal forcing in water pressure (same experiments as Fig.~\ref{fig:stick_slip}a,b).
- a) is under stress-controlled shear, b) is under rate-controlled shear.
- Black arrows denote temporal evolution of parameters.
+ Stress, velocity, and sediment flux under sinusoidal forcing in water pressure and effective normal stress at the ice-bed interface.
+ a-c) is under stress-controlled shear, d-f) is under rate-controlled shear.
+ Black arrows denote the temporal evolution.
}
\end{center}
\end{figure}
-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 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}
- \includegraphics[width=15.0cm]{experiments/fig-stick_slip_rate_depth.pdf}
- \caption{\label{fig:stick_slip_depth}%
- Pore-pressure diffusion and strain distribution with depth with a sinusoidal water-pressure forcing from the top (same experiment as Fig.~\ref{fig:stick_slip}b).
- The forcing has a daily periodocity, and plot lines are one hour in simulation time apart.
- The horizontal green line marks skin depth from Eq.~\ref{eq:skin_depth}.
- }
- \end{center}
-\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 (Fig.~\ref{fig:stick_slip_depth}a).
-However, water pressure variations at the ice-bed interface can reverse this depth trend.
-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}
