commit af47cfa8b0d1094514e5e81af7daeb048862f947
parent 432afb24da68958100e786b3e1694779886dc53e
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 3 Dec 2019 13:50:36 +0100
Add BC description to methods section
Diffstat:
1 file changed, 12 insertions(+), 7 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -181,13 +181,19 @@ The issue is left for future research and here we assume that $d$ corresponds to
Specifically designed laboratory experiments with various tills should inform the treatment of length scale, outside of ploughing effects by large clasts protruding from the basal ice \cite<e.g.,>[] {Tulaczyk1999}.
\subsection{Simulation setup}
-Parameter values and their references are listed in Table~S1.
-For the first experiment with variable water pressure, we apply a water-pressure forcing amplitude of 50 kPa that modulates effective stress at the top around 100 kPa (Fig.~\ref{fig:stick_slip}).
+We apply the model in a one-dimensional setup with simple shear (Fig.~\ref{fig:validation}a).
+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.
+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.
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.
Whillans Ice Plain, West Antarctica is an example of this setting, where a low surface slope and low driving stress results in stick-slip movement \cite<e.g.,>[] {Bindschadler2003}.
A rate-controlled setup is the opposite end member, where changes in bed friction do not influence ice flow velocity.
+Parameter values and their references are listed in Table~S1.
\begin{figure*}[htbp]
\begin{center}
@@ -211,11 +217,10 @@ A rate-controlled setup is the opposite end member, where changes in bed frictio
\section{Results}%
\label{sec:results}
-
-We first compare the modeled mechanical behavior to various tills tested in laboratory settings (Fig.~\ref{fig:rate_dependence} and~\ref{fig:mohr_coulomb}, after \citeA{Iverson2010}).
-Over five orders of strain-rate magnitude, some tills show slight rate weakening and others are slightly rate strengthening (Fig.~\ref{fig:rate_dependence}a).
-Shear-strain rates up to 5.000 a$^{-1}$ are realistic for natural glacier systems \cite{Cuffey2010}.
-Our 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:rate_dependence}b), making the model under these conditions rate strengthening.
+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}.
+Our 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 modeled friction value can be linearly scaled by adjusting $\mu_\mathrm{s}$ in Eqs.~\ref{eq:g_local} and~\ref{eq:cooperativity}.
Our model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:mohr_coulomb}), which is useful as these parameters are often constrained from Mohr-Coulomb tests on till samples.
