--- MITgcm_contrib/articles/ceaice/ceaice.tex	2008/02/25 19:30:56	1.11
+++ MITgcm_contrib/articles/ceaice/ceaice.tex	2008/02/26 00:13:20	1.14
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.11 2008/02/25 19:30:56 mlosch Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/MITgcm_contrib/articles/ceaice/ceaice.tex,v 1.14 2008/02/26 00:13:20 dimitri Exp $
 % $Name:  $
 \documentclass[12pt]{article}
 
@@ -52,31 +52,58 @@
 \maketitle
 
 \begin{abstract}
-
 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.
+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}
+The availability of an adjoint model as a powerful research
+tool complementary to an ocean model was a major design
+requirement early on in the development of the MIT general
+circulation model (MITgcm) [Marshall et al. 1997a,
+Marotzke et al. 1999, Adcroft et al. 2002]. It was recognized
+that the adjoint permitted very efficient computation
+of gradients of various scalar-valued model diagnostics,
+norms or, generally, objective functions with respect
+to external or independent parameters. Such gradients
+arise in at least two major contexts. If the objective function
+is the sum of squared model vs. obervation differences
+weighted by e.g. the inverse error covariances, the gradient
+of the objective function can be used to optimize this measure
+of model vs. data misfit in a least-squares sense. One
+is then solving a problem of statistical state estimation.
+If the objective function is a key oceanographic quantity
+such as meridional heat or volume transport, ocean heat
+content or mean surface temperature index, the gradient
+provides a complete set of sensitivities of this quantity
+with respect to all independent variables simultaneously.
+
+References to existing sea-ice adjoint models, explaining that they are either
+for simplified configurations, for ice-only studies, or for short-duration
+studies to motivate the present work.
 
 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,
-  kreyscher00, zhang98, hunke97}. From the perspective of coupling a
+kreyscher00, zhang98, hunke97}. From the perspective of coupling a
 sea ice-model to a C-grid ocean model, the exchange of fluxes of heat
 and fresh-water pose no difficulty for a B-grid sea-ice model
 \citep[e.g.,][]{timmermann02a}. However, surface stress is defined at
@@ -84,7 +111,7 @@
 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.
 
@@ -92,39 +119,58 @@
 straits. In the limit of only one grid cell between coasts there is no
 flux allowed for a B-grid (with no-slip lateral boundary counditions),
 whereas the C-grid formulation allows a flux of sea-ice through this
-passage for all types of lateral boundary conditions. We (will)
+passage for all types of lateral boundary conditions. We
 demonstrate this effect in the Candian archipelago.
 
+Talk about problems that make the sea-ice-ocean code very sensitive and
+changes in the code that reduce these sensitivities.
+
+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.
+
+\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)
@@ -168,7 +214,7 @@
 $\Delta_{\min}=10^{-11}\text{\,s}^{-1}$ (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.
 
@@ -192,9 +238,9 @@
 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}
@@ -220,7 +266,7 @@
 \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}
@@ -250,8 +296,8 @@
 $P$ 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
@@ -260,7 +306,7 @@
 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}.
 
@@ -367,13 +413,18 @@
 \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
