commit 01e950341255cd872455afcafa7bc3ae1c3e4a78
parent ac805ca950007b6fe4bc936e0eef7998bfa39049
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 16 Dec 2019 15:05:15 +0100
Add more discussion for new skin depth figure
Diffstat:
1 file changed, 11 insertions(+), 9 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -322,7 +322,7 @@ As long as fluid and diffusion properties are constant and the layer is sufficie
where $D$ is the hydraulic diffusivity [m$^2$/s] and $P$ [s] is the period of the oscillations.
The remaining terms were previously defined.
However, as the deformation pattern depends on both hydraulic properties and the forcing amplitude (Fig.~S3), the skin depth alone is insufficient to judge the occurence of deep deformation.
-Instead, we analytical solution for diffusive pressure perturbation can be used to find the largest depth $z'$ containing a minimum of effective normal stress over the cause of a pressure-perturbation cycle (see SI Text S1):
+We constrain an analytical solution for diffusive pressure perturbation to find the largest depth $z'$ containing a minimum of effective normal stress over the cause of a pressure-perturbation cycle (see SI Text S1):
\begin{linenomath*}
\begin{equation}
0 =
@@ -333,14 +333,13 @@ Instead, we analytical solution for diffusive pressure perturbation can be used
\label{eq:max_depth}
\end{equation}
\end{linenomath*}
-Figure~\ref{fig:skin_depth}a shows the skin depth for water at 0$^\circ$C under a range of permeabilities and forcing frequencies, while panels~\ref{fig:skin_depth}b and~c show solutions to Eq.~\ref{eq:max_depth}.
-
-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 (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}.
+Figure~\ref{fig:skin_depth}a shows the skin depth for water at 0$^\circ$C under a range of permeabilities and forcing frequencies, while panels~\ref{fig:skin_depth}b and~c show the maximum expected deformation depth from solutions to Eq.~\ref{eq:max_depth}.
+Minima in effective normal stress are increasingly difficult to create at larger depths through pure diffusion from the ice-bed interface.
+The deepest deformation occurs when the combination of forcing amplitude and skin depth is optimal.
+At higher skin depths, the lithostatic stress increase exceeds the pressure perturbation at depth.
+Conversely, deformation depth is restricted at lower skin depths because the pressure signal propagates too slowly through the bed relative to the forcing frequency.
+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, but deformation is still expected to occasionally occur away from the ice-bed interface.
\begin{figure}[htbp]
\begin{center}
@@ -352,6 +351,9 @@ Instead, lateral water input from lake drainage or hydrological rerouting at dep
\end{center}
\end{figure}
+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.
