commit b3eb177bad02fa955c93db8374504da9bccd5455
parent bfacabe59c857e9789be11a2bc62fc2b7b10a6ef
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 16 Dec 2019 14:37:42 +0100
Update with more on analytical solution
Diffstat:
2 files changed, 22 insertions(+), 18 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -321,8 +321,20 @@ As long as fluid and diffusion properties are constant and the layer is sufficie
\end{linenomath*}
where $D$ is the hydraulic diffusivity [m$^2$/s] and $P$ [s] is the period of the oscillations.
The remaining terms were previously defined.
-Figure~\ref{fig:skin_depth}a shows the skin depth for water at 0$^\circ$C under a range of permeabilities and forcing frequencies.
-We find that skin depth calculations can be a useful starting point for determining scenarios where deep deformation is possible.
+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):
+\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{f}) G d_\mathrm{s}}{A_\mathrm{f}}
+ \exp \left( \frac{z'}{d_\mathrm{s}} \right)
+ \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.
diff --git a/si.tex b/si.tex
@@ -311,9 +311,9 @@ 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.\ $z' = L_z - z$.
-
+The depth profile and transient behavior of effective normal stress $\sigma_\mathrm{n}'$ can be found by extending a solution for dispersion of a sinusoidal forcing through a diffusive medium \cite<Eq.~4.88> {Turcotte2002}.
+It is assumed that the bed is a semi-infinite halfspace.
+Here, $z'$ is depth below the ice-bed interface, i.e.\ $z' = L_z - z$:
\begin{linenomath*}
\begin{equation}
\sigma_\mathrm{n}'(z',t)
@@ -322,10 +322,10 @@ Here, $z'$ is depth below the ice-bed interface, i.e.\ $z' = L_z - z$.
+ (\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)
+ \sin \left( \omega t - \frac{z'}{d_\mathrm{s}} \right).
\end{equation}
\end{linenomath*}
-
+The vertical gradient of the effective normal stress is found by,
\begin{linenomath*}
\begin{equation}
\frac{d\sigma_\mathrm{n}'}{dz'}(z',t)
@@ -334,20 +334,12 @@ Here, $z'$ is depth below the ice-bed interface, i.e.\ $z' = L_z - z$.
%- 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]
+ + \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.
+For this study, we want to find the depth $z'$ where $d\sigma_\mathrm{n}'/dz' = 0$.
+At this 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 =
