commit 65fe096466dbcd95c90995f14c175d197df5ee66
parent ea8b41912f9acd9b30fcf0eef2a7b2907c0e7875
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 10 Dec 2019 16:05:22 +0100
Begin adding Liran's analytical solution to SI
Diffstat:
2 files changed, 43 insertions(+), 1 deletion(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -337,6 +337,7 @@ Due to higher hydraulic permeability, coarse tills are more susceptible to deep
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.
diff --git a/si.tex b/si.tex
@@ -176,7 +176,7 @@ Ben-Gurion University of the Negev}
\noindent\textbf{Contents of this file}
%%%Remove or add items as needed%%%
\begin{enumerate}
-\item Text S1
+\item Text S1 to S2
\item Figures S1 to S3
\item Table S1
%if Tables are larger than 1 page, upload as separate excel file
@@ -310,6 +310,47 @@ In rate-\emph{limited} experiments, the iterative procedure is only performed fo
\clearpage{}
+\noindent\textbf{Text S2. Analytical solution for maximum deformation depth}
+
+Here, $z'$ is depth below the ice-bed interface, i.e.\ $z' = L_z - z$.
+
+\begin{linenomath*}
+\begin{equation}
+ \sigma_\mathrm{n}'(z',t)
+ =
+ \sigma_\mathrm{n}
+ + (\rho_\mathrm{s} - \rho_\mathrm{f}) G z' % 1-phi here?
+ - p_\mathrm{f,top}
+ - A_\mathrm{f} \exp \left( - \frac{z'}{d_\mathrm{s}} \right)
+ \sin \left( \omega t - \frac{z'}{d_\mathrm{s}} \right)
+\end{equation}
+\end{linenomath*}
+
+\begin{linenomath*}
+\begin{equation}
+ \frac{d\sigma_\mathrm{n}'}{dz'}(z',t)
+ =
+ (\rho_\mathrm{s} - \rho_\mathrm{f}) G % 1-phi here?
+ %- p_\mathrm{f,top}
+ + \frac{A_\mathrm{f}}{d_\mathrm{s}} \exp \left( - \frac{z'}{d_\mathrm{s}} \right)
+ \left[ \sin \left( \omega t - \frac{z'}{d_\mathrm{s}} \right)
+ + \cos \left( \omega t - \frac{z'}{d_\mathrm{s}} \right) \right]
+\end{equation}
+\end{linenomath*}
+We would like to find the depth $z'$ where $d\sigma_\mathrm{n}'/dz' = 0$. At that depth the effective normal stress is at a minimum and deep deformation can occur.
+In our simulations we observe that the deepest deformation occurs when water pressure is at its minimum at the ice-bed interface, which means that $t=3\pi/2\omega$:
+\begin{linenomath*}
+\begin{equation}
+ \sin \left( \frac{3\pi}{2} - \frac{z'}{d_\mathrm{s}} \right)
+ + \cos \left( \frac{3\pi}{2} - \frac{z'}{d_\mathrm{s}} \right)
+ =
+ - \frac{(\rho_\mathrm{s} - \rho_\mathrm{s}) G d_\mathrm{s}}{A_\mathrm{f}}
+ - \exp \left( \frac{z'}{d_\mathrm{s}} \right)
+\end{equation}
+\end{linenomath*}
+
+\clearpage{}
+
%\noindent\textbf{Data Set S1.} %Type or paste caption here.
