commit 559fa2b257fa7f084d5d7dd27748d14a3ac69259
parent 65fe096466dbcd95c90995f14c175d197df5ee66
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Thu, 12 Dec 2019 18:20:54 +0100
Update figures and si
Diffstat:
3 files changed, 21 insertions(+), 6 deletions(-)
diff --git a/experiments/fig-pulse_lag.pdf b/experiments/fig-pulse_lag.pdf
Binary files differ.
diff --git a/experiments/fig-skin_depth.pdf b/experiments/fig-skin_depth.pdf
Binary files differ.
diff --git a/si.tex b/si.tex
@@ -319,7 +319,7 @@ Here, $z'$ is depth below the ice-bed interface, i.e.\ $z' = L_z - z$.
\sigma_\mathrm{n}'(z',t)
=
\sigma_\mathrm{n}
- + (\rho_\mathrm{s} - \rho_\mathrm{f}) G z' % 1-phi here?
+ + (\rho_\mathrm{s} - \rho_\mathrm{f}) G z'
- 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)
@@ -330,7 +330,7 @@ Here, $z'$ is depth below the ice-bed interface, i.e.\ $z' = L_z - z$.
\begin{equation}
\frac{d\sigma_\mathrm{n}'}{dz'}(z',t)
=
- (\rho_\mathrm{s} - \rho_\mathrm{f}) G % 1-phi here?
+ (\rho_\mathrm{s} - \rho_\mathrm{f}) G
%- 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)
@@ -339,19 +339,34 @@ Here, $z'$ is depth below the ice-bed interface, i.e.\ $z' = L_z - z$.
\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{f}) G d_\mathrm{s}}{A_\mathrm{f}}
+% \exp \left( \frac{z'}{d_\mathrm{s}} \right)
+%\end{equation}
+%\end{linenomath*}
\begin{linenomath*}
\begin{equation}
+ 0 =
\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)
+ + \frac{(\rho_\mathrm{s} - \rho_\mathrm{f}) G d_\mathrm{s}}{A_\mathrm{f}}
+ \exp \left( \frac{z'}{d_\mathrm{s}} \right)
\end{equation}
\end{linenomath*}
+With sinusoidal water-pressure forcing, the above equation has no solution if $d\sigma_\mathrm{n}'/dz(z=0) > 0$.
+This can be the case if the pressure perturbation is too weak to reverse the effective normal stress curve at depth.
+As a result, shear deformation occurs at the top throughout the water-pressure cycle.
-\clearpage{}
+We use Brent's method \cite{Press2007} for numerically finding depth ($z'$) values that satisfy the above equation within $z' \in [0; 5d_s]$.
+Our implementation, the program \texttt{max\_depth\_simple\_shear}, takes command-line arguments of the same format as the main NGF program, \texttt{1d\_fd\_simple\_shear}, and prints the maximum deformation depth ($z'$) as the first column of output, and the skin depth ($d_\mathrm{s}$) as the second column.
+See "\texttt{max\_depth\_simple\_shear -h}" for usage details.
+\clearpage{}
%\noindent\textbf{Data Set S1.} %Type or paste caption here.
%upload your dataset(s) to AGU's journal submission site and select "Supporting Information (SI)" as the file type. Following naming %convention: ds01.
