--- MITgcm_contrib/articles/ceaice/ceaice.tex	2008/01/15 16:04:50	1.6
+++ MITgcm_contrib/articles/ceaice/ceaice.tex	2008/02/25 23:45:46	1.13
@@ -1,3 +1,5 @@
+% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.13 2008/02/25 23:45:46 dimitri Exp $
+% $Name:  $
 \documentclass[12pt]{article}
 
 \usepackage[]{graphicx}
@@ -50,17 +52,30 @@
 \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, a dynamic and thermodynamic
+sea-ice model has been coupled to the Massachusetts Institute of Technology
+general circulation model (MITgcm).  Ice mechanics follow a viscous plastic
+rheology and the ice momentum equations are solved numerically using either
+line successive relaxation (LSR) or elastic-viscous-plastic (EVP) dynamic
+models.  Ice thermodynamics are represented using either a zero-heat-capacity
+formulation or a two-layer formulation that conserves enthalpy.  The model
+includes prognostic variables for snow and for sea-ice salinity.  The above
+sea ice model components were borrowed from current-generation climate models
+but they were reformulated on an Arakawa C-grid in order to match the MITgcm
+oceanic grid and they were modified in many ways to permit efficient and
+accurate automatic differentiation.  This paper describes the MITgcm sea ice
+model; it presents example Arctic and Antarctic results from a realistic,
+eddy-permitting, global ocean and sea-ice configuration; it compares B-grid
+and C-grid dynamic solvers in a regional Arctic configuration; and it presents
+example results from coupled ocean and sea-ice adjoint-model integrations.
+
 \end{abstract}
 
 \section{Introduction}
 \label{sec:intro}
 
-more blabla
-
-\section{Model}
-\label{sec:model}
-
 Traditionally, probably for historical reasons and the ease of
 treating the Coriolis term, most standard sea-ice models are
 discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
@@ -72,7 +87,7 @@
 sea-ice model and a C-grid ocean model. While the smoothing implicitly
 associated with this interpolation may mask grid scale noise, it may
 in two-way coupling lead to a computational mode as will be shown. By
-choosing a C-grid for the sea-ice model, we circumvene this difficulty
+choosing a C-grid for the sea-ice model, we circumvent this difficulty
 altogether and render the stress coupling as consistent as the
 buoyancy coupling.
 
@@ -83,36 +98,46 @@
 passage for all types of lateral boundary conditions. We (will)
 demonstrate this effect in the Candian archipelago.
 
+\section{Model}
+\label{sec:model}
+
 \subsection{Dynamics}
 \label{sec:dynamics}
 
-The momentum equations of the sea-ice model are standard with
+The momentum equation of the sea-ice model is
 \begin{equation}
   \label{eq:momseaice}
   m \frac{D\vek{u}}{Dt} = -mf\vek{k}\times\vek{u} + \vtau_{air} +
-  \vtau_{ocean} - m \nabla{\phi(0)} + \vek{F},
+  \vtau_{ocean} - mg \nabla{\phi(0)} + \vek{F},
 \end{equation}
-where $\vek{u} = u\vek{i}+v\vek{j}$ is the ice velocity vectory, $m$
-the ice mass per unit area, $f$ the Coriolis parameter, $g$ is the
-gravity accelation, $\nabla\phi$ is the gradient (tilt) of the sea
-surface height potential beneath the ice. $\phi$ is the sum of
-atmpheric pressure $p_{a}$ and loading due to ice and snow
-$(m_{i}+m_{s})g$. $\vtau_{air}$ and $\vtau_{ocean}$ are the wind and
-ice-ocean stresses, respectively.  $\vek{F}$ is the interaction force
-and $\vek{i}$, $\vek{j}$, and $\vek{k}$ are the unit vectors in the
-$x$, $y$, and $z$ directions.  Advection of sea-ice momentum is
-neglected. The wind and ice-ocean stress terms are given by
+where $m=m_{i}+m_{s}$ is the ice and snow mass per unit area;
+$\vek{u}=u\vek{i}+v\vek{j}$ is the ice velocity vector;
+$\vek{i}$, $\vek{j}$, and $\vek{k}$ are unit vectors in the $x$, $y$, and $z$
+directions, respectively;
+$f$ is the Coriolis parameter;
+$\vtau_{air}$ and $\vtau_{ocean}$ are the wind-ice and ocean-ice stresses,
+respectively;
+$g$ is the gravity accelation;
+$\nabla\phi(0)$ is the gradient (or tilt) of the sea surface height;
+$\phi(0)$ is the sea surface height potential in response to ocean dynamics
+and to atmospheric pressure loading;
+and $\vek{F}=\nabla\cdot\sigma$ is the divergence of the internal ice stress
+tensor $\sigma_{ij}$.
+When using the rescaled vertical coordinate system, z$^\ast$, of
+\citet{cam08}, $\phi(0)$ also includes a term due to snow and ice loading, $mg$.
+Advection of sea-ice momentum is neglected. The wind and ice-ocean stress
+terms are given by
 \begin{align*}
-  \vtau_{air} =& \rho_{air} |\vek{U}_{air}|R_{air}(\vek{U}_{air}) \\ 
-  \vtau_{ocean} =& \rho_{ocean} |\vek{U}_{ocean}-\vek{u}| 
+  \vtau_{air}   = & \rho_{air}  C_{air}   |\vek{U}_{air}  -\vek{u}|
+                   R_{air}  (\vek{U}_{air}  -\vek{u}), \\ 
+  \vtau_{ocean} = & \rho_{ocean}C_{ocean} |\vek{U}_{ocean}-\vek{u}| 
                    R_{ocean}(\vek{U}_{ocean}-\vek{u}), \\ 
 \end{align*}
 where $\vek{U}_{air/ocean}$ are the surface winds of the atmosphere
-and surface currents of the ocean, respectively. $C_{air/ocean}$ are
-air and ocean drag coefficients, $\rho_{air/ocean}$ reference
-densities, and $R_{air/ocean}$ rotation matrices that act on the
-wind/current vectors. $\vek{F} = \nabla\cdot\sigma$ is the divergence
-of the interal stress tensor $\sigma_{ij}$. 
+and surface currents of the ocean, respectively; $C_{air/ocean}$ are
+air and ocean drag coefficients; $\rho_{air/ocean}$ are reference
+densities; and $R_{air/ocean}$ are rotation matrices that act on the
+wind/current vectors.
 
 For an isotropic system this stress tensor can be related to the ice
 strain rate and strength by a nonlinear viscous-plastic (VP)
@@ -134,7 +159,7 @@
 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
@@ -156,7 +181,7 @@
 $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (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 $P = 2\,\Delta\zeta$
+tensor computation 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.
 
@@ -180,9 +205,9 @@
 treated explicitly.
 
 \citet{hunke97}'s introduced an elastic contribution to the strain
-rate elatic-viscous-plastic in order to regularize
+rate elastic-viscous-plastic in order to regularize
 Eq.\refeq{vpequation} in such a way that the resulting
-elatic-viscous-plastic (EVP) and VP models are identical at steady
+elastic-viscous-plastic (EVP) and VP models are identical at steady
 state,
 \begin{equation}
   \label{eq:evpequation}
@@ -208,7 +233,7 @@
 \dot{\epsilon}_{11}+\dot{\epsilon}_{22}$, and the horizontal tension
 and shearing strain rates, $D_T =
 \dot{\epsilon}_{11}-\dot{\epsilon}_{22}$ and $D_S =
-2\dot{\epsilon}_{12}$, respectively and using the above abbreviations,
+2\dot{\epsilon}_{12}$, respectively, and using the above abbreviations,
 the equations can be written as:
 \begin{align}
   \label{eq:evpstresstensor1}
@@ -238,8 +263,8 @@
 $P$ at vorticity points.
 
 For a general curvilinear grid, one needs in principle to take metric
-terms into account that arise in the transformation a curvilinear grid
-on the sphere. However, for now we can neglect these metric terms
+terms into account that arise in the transformation of a curvilinear grid
+on the sphere. For now, however, we can neglect these metric terms
 because they are very small on the cubed sphere grids used in this
 paper; in particular, only near the edges of the cubed sphere grid, we
 expect them to be non-zero, but these edges are at approximately
@@ -248,7 +273,7 @@
 cartesian.  However, for last-glacial-maximum or snowball-earth-like
 simulations the question of metric terms needs to be reconsidered.
 Either, one includes these terms as in \citet{zhang03}, or one finds a
-vector-invariant formulation fo the sea-ice internal stress term that
+vector-invariant formulation for the sea-ice internal stress term that
 does not require any metric terms, as it is done in the ocean dynamics
 of the MITgcm \citep{adcroft04:_cubed_sphere}.
 
@@ -309,131 +334,6 @@
 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}
@@ -480,13 +380,18 @@
 \subsection{Arctic Domain with Open Boundaries}
 \label{sec:arctic}
 
-The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It
+The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It
 is carved out from, and obtains open boundary conditions from, the
 global cubed-sphere configuration of the Estimating the Circulation
 and Climate of the Ocean, Phase II (ECCO2) project
 \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes
 horizontally with mean horizontal grid spacing of 18 km.
 
+\begin{figure}
+%\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1.eps}}}
+\caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}
+\end{figure}
+
 There are 50 vertical levels ranging in thickness from 10 m near the surface
 to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from
 the National Geophysical Data Center (NGDC) 2-minute gridded global relief
@@ -794,8 +699,7 @@
 We thank Jinlun Zhang for providing the original B-grid code and many
 helpful discussions. ML thanks Elizabeth Hunke for multiple explanations.
 
-%\bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}
-\bibliography{journal_abrvs,seaice,genocean,maths,mixing,mitgcmuv,bib/fram}
+\bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}
 
 \end{document}
 
