commit 683e1f92b790dddd6f2a82af2ef43aff81057289
parent baa656aae85634b19704b94f14803e76206b5729
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 3 Dec 2019 16:53:11 +0100
Better spacing before references
Diffstat:
1 file changed, 4 insertions(+), 4 deletions(-)
diff --git a/si.tex b/si.tex
@@ -240,7 +240,7 @@ where $\mu_\text{top}$ is constant for stress-controlled experiments and dynamic
We assign depth coordinates $z_i$, granular fluidity $g_i$, and fluid pressure $p_{\text{f},i}$ to a regular grid with ghost nodes and cell spacing $\Delta z$.
The fluidity field $g$ is solved for a set of mechanical forcings ($\mu$, $\sigma_\text{n}'$, boundary conditions for $g$), and material parameters ($A$, $b$, $d$).
-We rearrange Eq.~2 in the main text and split the Laplace operator ($\nabla^2$) into a 1D central finite difference 3-point stencil.
+We rearrange Eq.\ 2 in the main text and split the Laplace operator ($\nabla^2$) into a 1D central finite difference 3-point stencil.
An iterative scheme is applied to relax the following equation at each grid node $i$:
\begin{linenomath*}
\begin{equation}
@@ -262,9 +262,9 @@ We apply fixed-value (Dirichlet) boundary conditions for the fluidity field ($g(
This condition causes the velocity field transition towards a constant value at the domain edges.
Neumann boundary conditions, which are not used here, create a velocity profile resembling a free surface flow.
-The pore-pressure solution (Eq.~5 in the main text) is constrained by a hydrostatic pressure gradient at the bottom ($dp_\text{f}/dz (z=0) = \rho_\text{f}G$), and a pressure forcing at the top, for example sinusoidal: $p_\text{f}(z = L_z) = A_\text{f} \sin(2\pi f t) + p_{\text{f},0}$.
+The pore-pressure solution (Eq.\ 5 in the main text) is constrained by a hydrostatic pressure gradient at the bottom ($dp_\text{f}/dz (z=0) = \rho_\text{f}G$), and a pressure forcing at the top, for example sinusoidal: $p_\text{f}(z = L_z) = A_\text{f} \sin(2\pi f t) + p_{\text{f},0}$.
Here, $A_\text{f}$ is the forcing amplitude [Pa], $f$ is the forcing frequency [1/s], and $p_{\text{f},0}$ is the mean pore pressure over time [Pa].
-As for the granular flow solution, we also use operator splitting and finite differences to solve the equation for pore-pressure diffusion (Eq.~5 in the main text):
+As for the granular flow solution, we also use operator splitting and finite differences to solve the equation for pore-pressure diffusion (Eq.\ 5 in the main text):
\begin{linenomath*}
\begin{equation}
\Delta p_{\text{f},i} = \frac{1}{\phi_i \eta_\text{f} \beta_\text{f}}
@@ -453,7 +453,7 @@ In rate-\emph{limited} experiments, the iterative procedure is only performed fo
\begin{center}
\includegraphics[width=7.5cm]{experiments/fig-skin_depth.pdf}
\caption{\label{fig:skin_depth}%
- Skin depth of pore-pressure fluctuations (Eq.~6 in the main text) with forcing frequencies ranging from yearly to hourly periods.
+ Skin depth of pore-pressure fluctuations (Eq.\ 6 in the main text) with forcing frequencies ranging from yearly to hourly periods.
The permeability spans values common for tills \cite{Schwartz2003}.
}
\end{center}
