commit 156ae779f40f1f2c32106bf78f3eb7d29f746b12
parent 5d35bcab717193c3b009c9ac8c525a33fafa8c6b
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Wed, 26 Jun 2019 13:15:00 +0200
Add skin depth solution
Diffstat:
2 files changed, 19 insertions(+), 0 deletions(-)
diff --git a/BIBnew.bib b/BIBnew.bib
@@ -8816,3 +8816,12 @@ Winton and A. T. Wittenberg and F. Zeng and R. Zhang and J. P. Dunne},
title = {A theory of glacial quarrying for landscape evolution models},
journal = {Geology}
}
+
+@book{Turcotte2002,
+ doi = {10.1017/cbo9780511807442.012},
+ year=2002,
+ publisher = {Cambridge University Press, Cambridge},
+ pages = {848},
+ author = {D. L. Turcotte and G. Schubert},
+ title = {Geodynamics},
+}
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -147,6 +147,16 @@ The method is unconditionally stable and second-order accurate in time and space
\section{Results}%
\label{sec:results}
+The water pressure variations vary with the same periodocity as the forcing, but with exponential decay in amplitude and increasing lag at depth.
+The skin depth is defined as the distance where the fluctuation amplitude decreases to $1/e$ of its surface value.
+As long as fluid and diffusion properties are constant,
+ an analytical solution to skin depth in our system follows the form \citep[after Eq. 4.90 in][]{Turcotte2002},
+\begin{equation}
+ d_\text{s} = \left( \frac{k}{\phi\mu_\text{f}\beta_\text{f}\pi f} \right)^{1/2}
+ \label{eq:skin_depth}
+\end{equation}
+The above relation implies that the amplitude in water-pressure forcing does not influence the maximum depth of slip.
+
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7.5cm]{experiments/fig1.pdf}
