commit baa656aae85634b19704b94f14803e76206b5729
parent 6e2b194b7883bda23cc821870911dedd1d82d3a0
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 3 Dec 2019 16:52:14 +0100
Fix references in SI
Diffstat:
1 file changed, 6 insertions(+), 6 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.~\ref{eq:g} 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 ($g(
This condition causes the velocity field transition towards a constant value at the domain edges.
Neumann boundary conditions, which are not used here, create a velocity profile resembling a free surface flow.
-The pore-pressure solution (Eq.~\ref{eq:p_f}) is constrained by a hydrostatic pressure gradient at the bottom ($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.~\ref{eq:p_f}):
+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}}
@@ -282,8 +282,8 @@ The method is unconditionally stable and second-order accurate in time and space
Our implementation of grain and fluid dynamics is highly efficient, and for the presented experiments each time step completes in less than 1 ms on a single CPU core.
-\subsubsection{Rate-controlled experiments} % quick edit, needs rewrite. perhaps also move somewhere else
-The continuum model in its presented form is suited for resolving strain rate and shear velocity from a given stress forcing, i.e. in a stress-controlled setup.
+\textbf{S1.1 Rate-controlled experiments}
+The continuum model in its presented form is suited for resolving strain rate and shear velocity from a given stress forcing, i.e.\ in a stress-controlled setup.
However, certain experiments are best approached in a rate-controlled manner where a specified shear rate results in strain-rate distribution and shear stress.
For our system of equations, this is an inverse problem that can be tackled by adjusting the applied friction at the top until the resultant shear velocity matches the desired value.
We implement an automatic iterative procedure that can be set to match a shear velocity, or limit the velocity beneath a given value.
@@ -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.~\ref{eq:skin_depth}) 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}
