--- MITgcm_contrib/articles/ceaice/ceaice.tex	2007/11/07 17:26:13	1.3
+++ MITgcm_contrib/articles/ceaice/ceaice.tex	2008/01/14 15:46:54	1.5
@@ -1,6 +1,6 @@
 \documentclass[12pt]{article}
-\usepackage{epsfig}
-\usepackage{graphics}
+
+\usepackage[]{graphicx}
 \usepackage{subfigure}
 
 \usepackage[round,comma]{natbib}
@@ -35,7 +35,10 @@
 \newlength{\mediumfigwidth}\setlength{\mediumfigwidth}{39pc}
 %\newlength{\widefigwidth}\setlength{\widefigwidth}{39pc}
 \newlength{\widefigwidth}\setlength{\widefigwidth}{\textwidth}
-\newcommand{\fpath}{.}
+\newcommand{\fpath}{figs}
+
+% commenting scheme
+\newcommand{\ml}[1]{\textsf{\slshape #1}}
 
 \title{A Dynamic-Thermodynamic Sea ice Model for Ocean Climate
   Estimation on an Arakawa C-Grid}
@@ -127,17 +130,21 @@
     \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) $c$: \[P =
-P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},\] with the constants $P^{*}$ and
-$C^{*}$. The nonlinear bulk and shear viscosities $\eta$ and $\zeta$
-are functions of ice strain rate invariants and ice strength such that
-the principal components of the stress lie on an elliptical yield
-curve with the ratio of major to minor axis $e$ equal to $2$; they are
-given by:
+The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on
+both thickness $h$ and compactness (concentration) $c$:
+\begin{equation}
+  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
+viscosities $\eta$ and $\zeta$ are functions of ice strain rate
+invariants and ice strength such that the principal components of the
+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)
@@ -145,6 +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,
@@ -287,23 +302,143 @@
 \section{Funnel Experiments}
 \label{sec:funnel}
 
-\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.
+For a first/detailed comparison between the different variants of the
+MIT sea ice model an idealized geometry of a periodic channel,
+1000\,km long and 500\,m wide on a non-rotating plane, with converging
+walls forming a symmetric funnel and a narrow strait of 40\,km width
+is used. The horizontal resolution is 5\,km throughout the domain
+making the narrow strait 8 grid points wide. The ice model is
+initialized with a complete ice cover of 50\,cm uniform thickness. The
+ice model is driven by a constant along channel eastward ocean current
+of 25\,cm/s that does not see the walls in the domain. All other
+ice-ocean-atmosphere interactions are turned off, in particular there
+is no feedback of ice dynamics on the ocean current. All thermodynamic
+processes are turned off so that ice thickness variations are only
+caused by convergent or divergent ice flow. Ice volume (effective
+thickness) and concentration are advected with a third-order scheme
+with a flux limiter \citep{hundsdorfer94} to avoid undershoots. This
+scheme is unconditionally stable and does not require additional
+diffusion. The time step used here is 1\,h.
+
+\reffig{funnelf0} compares the dynamic fields ice concentration $c$,
+effective thickness $h_{eff} = h\cdot{c}$, and velocities $(u,v)$ for
+five different cases at steady state (after 10\,years of integration): 
+\begin{description}
+\item[B-LSRns:] LSR solver with no-slip boundary conditions on a B-grid, 
+\item[C-LSRns:] LSR solver with no-slip boundary conditions on a C-grid,
+\item[C-LSRfs:] LSR solver with free-slip boundary conditions on a C-grid,
+\item[C-EVPns:] EVP solver with no-slip boundary conditions on a C-grid,
+\item[C-EVPfs:] EVP solver with free-slip boundary conditions on a C-grid,
+\end{description}
+\ml{[We have not implemented the EVP solver on a B-grid.]}
+\begin{figure*}[htbp]
+  \includegraphics[width=\widefigwidth]{\fpath/all_086280}
+  \caption{Ice concentration, effective thickness [m], and ice
+    velocities [m/s]
+    for 5 different numerical solutions.} 
+  \label{fig:funnelf0}
+\end{figure*}
+At a first glance, the solutions look similar. This is encouraging as
+the details of discretization and numerics should not affect the
+solutions to first order. In all cases the ice-ocean stress pushes the
+ice cover eastwards, where it converges in the funnel. In the narrow
+channel the ice moves quickly (nearly free drift) and leaves the
+channel as narrow band.
+
+A close look reveals interesting differences between the B- and C-grid
+results. The zonal velocity in the narrow channel is nearly the free
+drift velocity ( = ocean velocity) of 25\,cm/s for the C-grid
+solutions, regardless of the boundary conditions, while it is just
+above 20\,cm/s for the B-grid solution. The ice accelerates to
+25\,cm/s after it exits the channel. Concentrating on the solutions
+B-LSRns and C-LSRns, the ice volume (effective thickness) along the
+boundaries in the narrow channel is larger in the B-grid case although
+the ice concentration is reduces in the C-grid case. The combined
+effect leads to a larger actual ice thickness at smaller
+concentrations in the C-grid case. However, since the effective
+thickness determines the ice strength $P$ in Eq\refeq{icestrength},
+the ice strength and thus the bulk and shear viscosities are larger in
+the B-grid case leading to more horizontal friction. This circumstance
+might explain why the no-slip boundary conditions in the B-grid case
+appear to be more effective in reducing the flow within the narrow
+channel, than in the C-grid case. Further, the viscosities are also
+sensitive to details of the velocity gradients. Via $\Delta$, these
+gradients enter the viscosities in the denominator so that large
+gradients tend to reduce the viscosities. This again favors more flow
+along the boundaries in the C-grid case: larger velocities
+(\reffig{funnelf0}) on grid points that are closer to the boundary by
+a factor $\frac{1}{2}$ than in the B-grid case because of the stagger
+nature of the C-grid lead numerically to larger tangential gradients
+across the boundary; these in turn make the viscosities smaller for
+less tangential friction and allow more tangential flow along the
+boundaries.
+
+The above argument can also be invoked to explain the small
+differences between the free-slip and no-slip solutions on the C-grid.
+Because of the non-linearities in the ice viscosities, in particular
+along the boundaries, the no-slip boundary conditions has only a small
+impact on the solution.
+
+The difference between LSR and EVP solutions is largest in the
+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 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*}
+
+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}
@@ -594,21 +729,21 @@
 \begin{figure}[t!]
 \centerline{
 \subfigure[{\footnotesize -12 months}]
-{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
+{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
 %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
 %
 \subfigure[{\footnotesize -24 months}]
-{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
+{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
 }
 
 \centerline{
 \subfigure[{\footnotesize
 -36 months}]
-{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}
+{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}
 %
 \subfigure[{\footnotesize
 -48 months}]
-{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJheff_arc_lev1_tim292_cmax2.0E+02.eps}}
+{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim292_cmax2.0E+02.eps}}
 }
 \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
 sea-ice thickness at various prior times.
@@ -619,21 +754,21 @@
 \begin{figure}[t!]
 \centerline{
 \subfigure[{\footnotesize -12 months}]
-{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}
+{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}
 %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
 %
 \subfigure[{\footnotesize -24 months}]
-{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}
+{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}
 }
 
 \centerline{
 \subfigure[{\footnotesize
 -36 months}] 
-{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}
+{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}
 %
 \subfigure[{\footnotesize
 -48 months}]
-{\includegraphics*[width=0.44\linewidth]{figs/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}
+{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}
 }
 \caption{Same as Fig. XXX but for sea surface temperature
 \label{fig:4yradjthetalev1}}
@@ -662,7 +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}
 
@@ -670,3 +806,54 @@
 %%% mode: latex
 %%% TeX-master: t
 %%% End: 
+
+
+A Dynamic-Thermodynamic Sea ice Model for Ocean Climate
+  Estimation on an Arakawa C-Grid
+
+Introduction
+
+Ice Model:
+ Dynamics formulation.
+  B-C, LSR, EVP, no-slip, slip
+  parallellization
+ Thermodynamics formulation.
+  0-layer Hibler salinity + snow
+  3-layer Winton
+
+Idealized tests
+ Funnel Experiments
+ Downstream Island tests
+  B-grid LSR no-slip
+  C-grid LSR no-slip
+  C-grid LSR slip
+  C-grid EVP no-slip
+  C-grid EVP slip
+
+Arctic Setup
+ Configuration
+ OBCS from cube
+ forcing
+ 1/2 and full resolution
+ with a few JFM figs from C-grid LSR no slip
+  ice transport through Canadian Archipelago
+  thickness distribution
+  ice velocity and transport
+
+Arctic forward sensitivity experiments
+ B-grid LSR no-slip
+ C-grid LSR no-slip
+ C-grid LSR slip
+ C-grid EVP no-slip
+ C-grid EVP slip
+ C-grid LSR no-slip + Winton
+  speed-performance-accuracy (small)
+  ice transport through Canadian Archipelago differences
+  thickness distribution differences
+  ice velocity and transport differences
+
+Adjoint sensitivity experiment on 1/2-res setup
+ Sensitivity of sea ice volume flow through Fram Strait
+*** Sensitivity of sea ice volume flow through Canadian Archipelago
+
+Summary and conluding remarks
