--- MITgcm_contrib/articles/ceaice/ceaice.tex	2008/01/18 17:33:56	1.8
+++ MITgcm_contrib/articles/ceaice/ceaice.tex	2008/02/25 19:30:56	1.11
@@ -1,3 +1,5 @@
+% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.11 2008/02/25 19:30:56 mlosch Exp $
+% $Name:  $
 \documentclass[12pt]{article}
 
 \usepackage[]{graphicx}
@@ -50,7 +52,17 @@
 \maketitle
 
 \begin{abstract}
-  Some blabla
+
+As part of ongoing efforts to obtain a best possible synthesis of most
+available, global-scale, ocean and sea ice data, dynamic and thermodynamic
+sea-ice model components have been incorporated in the Massachusetts Institute
+of Technology general circulation model (MITgcm).  Sea-ice dynamics use either
+a visco-plastic rheology solved with a line successive relaxation (LSR)
+technique, reformulated on an Arakawa C-grid in order to match the oceanic and
+atmospheric grids of the MITgcm, and modified to permit efficient and accurate
+automatic differentiation of the coupled ocean and sea-ice model
+configurations.
+
 \end{abstract}
 
 \section{Introduction}
@@ -134,7 +146,7 @@
 both thickness $h$ and compactness (concentration) $c$:
 \begin{equation}
   P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
-\label{icestrength}
+\label{eq:icestrength}
 \end{equation}
 with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
 viscosities $\eta$ and $\zeta$ are functions of ice strain rate
@@ -309,131 +321,6 @@
 state variables to be advected by ice velocities, namely enthalphy of
 the two ice layers and the thickness of the overlying snow layer.
 
-\section{Funnel Experiments}
-\label{sec:funnel}
-
-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 have 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 EVP velocity
-fields 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 by variable
-atmospheric wind whose mean direction is diagnonal to the north-east
-corner of the domain; ice volume and concentration are held constant
-(no thermodynamics and 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{Ice flow, divergence and bulk viscosities of three
-    experiments with \citet{hunke01}'s test case: C-LSRns (top),
-    C-EVPns (middle), and C-EVPns with damping described in
-    \citet{hunke01} (bottom).}
-  \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 confined 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 $\nabla\cdot\vek{u}$ and
-$\log_{10}\zeta$ 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).
 
 \subsection{C-grid}
 \begin{itemize}
