--- MITgcm_contrib/articles/ceaice/ceaice.tex	2008/01/10 15:47:32	1.4
+++ MITgcm_contrib/articles/ceaice/ceaice.tex	2008/01/14 15:46:54	1.5
@@ -1,6 +1,7 @@
 \documentclass[12pt]{article}
 
-\usepackage{graphicx,subfigure}
+\usepackage[]{graphicx}
+\usepackage{subfigure}
 
 \usepackage[round,comma]{natbib}
 \bibliographystyle{bib/agu04}
@@ -129,10 +130,10 @@
     \frac{\partial{u_{i}}}{\partial{x_{j}}} +
     \frac{\partial{u_{j}}}{\partial{x_{i}}}\right).
 \end{equation*}
-The pressure $P$, a measure of ice strength, depends on both thickness
-$h$ and compactness (concentration) $c$: 
+The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on
+both thickness $h$ and compactness (concentration) $c$:
 \begin{equation}
-  P = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
+  P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
 \label{icestrength}
 \end{equation}
 with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
@@ -141,8 +142,9 @@
 stress lie on an elliptical yield curve with the ratio of major to
 minor axis $e$ equal to $2$; they are given by:
 \begin{align*}
-  \zeta =& \frac{P}{2\Delta} \\
-  \eta =& \frac{P}{2\Delta{e}^2} \\
+  \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})},
+   \zeta_{\max}\right) \\
+  \eta =& \frac{\zeta}{e^2} \\
   \intertext{with the abbreviation}
   \Delta = & \left[
     \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
@@ -150,6 +152,14 @@
     2\dot{\epsilon}_{11}\dot{\epsilon}_{22} (1-e^{-2})
   \right]^{-\frac{1}{2}}
 \end{align*}
+The bulk viscosities are bounded above by imposing both a minimum
+$\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (for numerical reasons) and a
+maximum $\zeta_{\max} = P_{\max}/\Delta^*$, where
+$\Delta^*=(5\times10^{12}/2\times10^4)\text{\,s}^{-1}$. For stress
+tensor compuation the replacement pressure $P = 2\,\Delta\zeta$
+\citep{hibler95} is used so that the stress state always lies on the
+elliptic yield curve by definition.
+
 In the current implementation, the VP-model is integrated with the
 semi-implicit line successive over relaxation (LSOR)-solver of
 \citet{zhang98}, which allows for long time steps that, in our case,
@@ -373,32 +383,62 @@
 effective thickness and meridional velocity fields. The velocity field
 appears to be a little noisy. This noise has been address by
 \citet{hunke01}. It can be dealt with by reducing EVP's internal time
-step (increasing the number of iterations) or by regularizing the bulk
-and shear viscosities. We revisit the latter option by reproducing the
-results of \citet{hunke01} for the C-grid no-slip cases.
+step (increasing the number of iterations along with the computational
+cost) or by regularizing the bulk and shear viscosities. We revisit
+the latter option by reproducing some of the results of
+\citet{hunke01}, namely the experiment described in her section~4, for
+our C-grid no-slip cases: in a square domain with a few islands the
+ice model is initialized with constant ice thickness and linearly
+increasing ice concentration to the east. The model dynamics are
+forced with a constant anticyclonic ocean gyre and variable
+atmospheric wind whose mean directed diagnonally to the north-east
+corner of the domain; ice volume and concentration are held constant
+(no advection by ice velocity).  \reffig{hunke01} shows the ice
+velocity field, its divergence, and the bulk viscosity $\zeta$ for the
+cases C-LRSns and C-EVPns, and for a C-EVPns case, where
+\citet{hunke01}'s regularization has been implemented; compare to
+Fig.\,4 in \citet{hunke01}. The regularization contraint limits ice
+strength and viscosities as a function of damping time scale,
+resolution and EVP-time step, effectively allowing the elastic waves to
+damp out more quickly \citep{hunke01}.
 \begin{figure*}[htbp]
   \includegraphics[width=\widefigwidth]{\fpath/hun12days}
   \caption{Hunke's test case.}
   \label{fig:hunke01}
 \end{figure*}
 
-\begin{itemize}
-\item B-grid LSR no-slip
-\item C-grid LSR no-slip
-\item C-grid LSR slip
-\item C-grid EVP no-slip
-\item C-grid EVP slip
-\end{itemize}
-
-\subsection{B-grid vs.\ C-grid}
-Comparison between:
-\begin{itemize}
-\item B-grid, lsr, no-slip
-\item C-grid, lsr, no-slip 
-\item C-grid, evp, no-slip
-\end{itemize}
-all without ice-ocean stress, because ice-ocean stress does not work
-for B-grid.
+In the far right (``east'') side of the domain the ice concentration
+is close to one and the ice should be nearly rigid. The applied wind
+tends to push ice toward the upper right corner. Because the highly
+compact ice is confinded by the boundary, it resists any further
+compression and exhibits little motion in the rigid region on the
+right hand side. The C-LSRns solution (top row) allows high
+viscosities in the rigid region suppressing nearly all flow.
+\citet{hunke01}'s regularization for the C-EVPns solution (bottom row)
+clearly suppresses the noise present in $\nabla\cdot\vek{u}$ in the
+unregularized case (middle row), at the cost of reduced viscosities
+These reduced viscosities lead to small but finite ice velocities
+which in turn can have a strong effect on solutions in the limit of
+nearly rigid regimes (arching and blocking, not shown).
+
+
+%\begin{itemize}
+%\item B-grid LSR no-slip
+%\item C-grid LSR no-slip
+%\item C-grid LSR slip
+%\item C-grid EVP no-slip
+%\item C-grid EVP slip
+%\end{itemize}
+
+%\subsection{B-grid vs.\ C-grid}
+%Comparison between:
+%\begin{itemize}
+%\item B-grid, lsr, no-slip
+%\item C-grid, lsr, no-slip 
+%\item C-grid, evp, no-slip
+%\end{itemize}
+%all without ice-ocean stress, because ice-ocean stress does not work
+%for B-grid.
 
 \subsection{C-grid}
 \begin{itemize}
@@ -757,7 +797,8 @@
 We thank Jinlun Zhang for providing the original B-grid code and many
 helpful discussions.
 
-\bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}
+%\bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}
+\bibliography{journal_abrvs,seaice,genocean,maths,mitgcmuv,bib/fram}
 
 \end{document}
 
