--- MITgcm_contrib/articles/ceaice/ceaice.tex	2008/01/10 15:47:32	1.4
+++ MITgcm_contrib/articles/ceaice/ceaice.tex	2008/02/25 23:45:46	1.13
@@ -1,6 +1,9 @@
+% $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,subfigure}
+\usepackage[]{graphicx}
+\usepackage{subfigure}
 
 \usepackage[round,comma]{natbib}
 \bibliographystyle{bib/agu04}
@@ -49,17 +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,
@@ -71,7 +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.
 
@@ -82,36 +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)
@@ -129,11 +155,11 @@
     \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)]},
-\label{icestrength}
+  P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
+\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
@@ -141,8 +167,9 @@
 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 +177,24 @@
     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 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.
+
+In the so-called truncated ellipse method the shear viscosity $\eta$
+is capped to suppress any tensile stress \citep{hibler97, geiger98}:
+\begin{equation}
+  \label{eq:etatem}
+  \eta = \min(\frac{\zeta}{e^2}
+  \frac{\frac{P}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})}
+  {\sqrt{(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})^2
+      +4\dot{\epsilon}_{12}^2}}
+\end{equation}
+
 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,
@@ -160,9 +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}
@@ -188,7 +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}
@@ -218,8 +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
@@ -228,7 +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}.
 
@@ -289,116 +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 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) 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.
-\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.
 
 \subsection{C-grid}
 \begin{itemize}
@@ -445,45 +380,55 @@
 \subsection{Arctic Domain with Open Boundaries}
 \label{sec:arctic}
 
-The Arctic domain of integration is illustrated in Fig.~\ref{???}.  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 \cite{men05a}.  The domain size is 420 by
-384 grid boxes horizontally with mean horizontal grid spacing of 18 km.  
+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
 data (ETOPO2) and the model employs the partial-cell formulation of
-\cite{adc97}, which permits accurate representation of the bathymetry. The
+\citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The
 model is integrated in a volume-conserving configuration using a finite volume
 discretization with C-grid staggering of the prognostic variables. In the
-ocean, the non-linear equation of state of \cite{jac95}.  The ocean model is
+ocean, the non-linear equation of state of \citet{jackett95}.  The ocean model is
 coupled to a sea-ice model described hereinabove.  
 
-This particular ECCO2 simulation is initialized from rest using the January
-temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01)
-[Conkright et al., 2002] and it is integrated for 32 years prior to the
-1996-2001 period discussed in the study. Surface boundary conditions are from
-the National Centers for Environmental Prediction and the National Center for
-Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al.,
-2001]. Six-hourly surface winds, temperature, humidity, downward short- and
-long-wave radiations, and precipitation are converted to heat, freshwater, and
-wind stress fluxes using the Large and Pond [1981, 1982] bulk
-formulae. Shortwave radiation decays exponentially as per Paulson and Simpson
-[1977]. Additionally the time-mean river run-off from Large and Nurser [2001]
-is applied and there is a relaxation to the monthly-mean climatological sea
-surface salinity values from WOA01 with a relaxation time scale of 3
-months. Vertical mixing follows Large et al. [1994] with background vertical
-diffusivity of 1.5 � 10-5 m2 s-1 and viscosity of 10-3 m2 s-1. A third order,
-direct-space-time advection scheme with flux limiter is employed and there is
-no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996]
-but modified to sense the divergent flow as per Fox-Kemper and Menemenlis [in
-press].  Shortwave radiation decays exponentially as per Paulson and Simpson
-[1977].  Additionally, the time-mean runoff of Large and Nurser [2001] is
-applied near the coastline and, where there is open water, there is a
-relaxation to monthly-mean WOA01 sea surface salinity with a time constant of
-45 days.
+This particular ECCO2 simulation is initialized from rest using the
+January temperature and salinity distribution from the World Ocean
+Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for
+32 years prior to the 1996--2001 period discussed in the study. Surface
+boundary conditions are from the National Centers for Environmental
+Prediction and the National Center for Atmospheric Research
+(NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly
+surface winds, temperature, humidity, downward short- and long-wave
+radiations, and precipitation are converted to heat, freshwater, and
+wind stress fluxes using the \citet{large81, large82} bulk formulae.
+Shortwave radiation decays exponentially as per Paulson and Simpson
+[1977]. Additionally the time-mean river run-off from Large and Nurser
+[2001] is applied and there is a relaxation to the monthly-mean
+climatological sea surface salinity values from WOA01 with a
+relaxation time scale of 3 months. Vertical mixing follows
+\citet{large94} with background vertical diffusivity of
+$1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
+$10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time
+advection scheme with flux limiter is employed \citep{hundsdorfer94}
+and there is no explicit horizontal diffusivity. Horizontal viscosity
+follows \citet{lei96} but
+modified to sense the divergent flow as per Fox-Kemper and Menemenlis
+[in press].  Shortwave radiation decays exponentially as per Paulson
+and Simpson [1977].  Additionally, the time-mean runoff of Large and
+Nurser [2001] is applied near the coastline and, where there is open
+water, there is a relaxation to monthly-mean WOA01 sea surface
+salinity with a time constant of 45 days.
 
 Open water, dry
 ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,
@@ -512,6 +457,7 @@
 \item C-grid LSR slip
 \item C-grid EVP no-slip
 \item C-grid EVP slip
+\item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)
 \item C-grid LSR no-slip + Winton
 \item  speed-performance-accuracy (small)
   ice transport through Canadian Archipelago differences
@@ -523,8 +469,8 @@
 \begin{itemize}
 \item advection schemes: along the ice-edge and regions with large
   gradients
-\item C-grid: more transport through narrow straits for no slip
-  conditons, less for free slip
+\item C-grid: less transport through narrow straits for no slip
+  conditons, more for free slip
 \item VP vs.\ EVP: speed performance, accuracy?
 \item ocean stress: different water mass properties beneath the ice
 \end{itemize}
@@ -605,7 +551,7 @@
 checkpointing loop.
 Again, an initial code adjustment is required to support TAFs 
 checkpointing capability.
-The code adjustments are sufficiently simply so as not to cause
+The code adjustments are sufficiently simple so as not to cause
 major limitations to the full nonlinear parent model.
 Once in place, an adjoint model of a new model configuration
 may be derived in about 10 minutes.
@@ -628,9 +574,10 @@
 We demonstrate the power of the adjoint method
 in the context of investigating sea-ice export sensitivities through Fram Strait
 (for details of this study see Heimbach et al., 2007).
+%\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).
 The domain chosen is a coarsened version of the Arctic face of the
 high-resolution cubed-sphere configuration of the ECCO2 project
-(see Menemenlis et al. 2005). It covers the entire Arctic,
+\citep[see][]{menemenlis05}. It covers the entire Arctic,
 extends into the North Pacific such as to cover the entire
 ice-covered regions, and comprises parts of the North Atlantic
 down to XXN to enable analysis of remote influences of the 
@@ -641,46 +588,41 @@
 (benchmarks have been performed both on an SGI Altix as well as an 
 IBM SP5 at NASA/ARC).
 
-Following a 1-year spinup, the model has been integrated for four years 
-between 1992 and 1995.
-It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.
-Over the open ocean these are converted into
-air-sea fluxes via the bulk formulae of Large and Yeager (2004).
-Derivation of air-sea fluxes in the presence of sea-ice is handled
-by the ice model as described in Section XXX.
+Following a 1-year spinup, the model has been integrated for four
+years between 1992 and 1995. It is forced using realistic 6-hourly
+NCEP/NCAR atmospheric state variables. Over the open ocean these are
+converted into air-sea fluxes via the bulk formulae of
+\citet{large04}.  Derivation of air-sea fluxes in the presence of
+sea-ice is handled by the ice model as described in \refsec{model}.
 The objective function chosen is sea-ice export through Fram Strait
-computed for December 1995
-The adjoint model computes sensitivities to sea-ice export back in time
-from 1995 to 1992 along this trajectory.
-In principle all adjoint model variable (i.e. Lagrange multipliers)
-of the coupled ocean/sea-ice model
-are available to analyze the transient sensitivity behaviour
-of the ocean and sea-ice state.
-Over the open ocean, the adjoint of the bulk formula scheme
-computes sensitivities to the time-varying atmospheric state.
-Over ice-covered parts, the sea-ice adjoint converts
-surface ocean sensitivities to atmospheric sensitivities.
-
-Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait
-in December 1995 to changes in sea-ice thickness 
-12, 24, 36, 48 months back in time.
-Corresponding sensitivities to ocean surface temperature are
-depicted in Fig. XXX(a--d).
-The main characteristics is consistency with expected advection
-of sea-ice over the relevant time scales considered.
-The general positive pattern means that an increase in
-sea-ice thickness at location $(x,y)$ and time $t$ will increase
-sea-ice export through Fram Strait at time $T_e$.
-Largest distances from Fram Strait indicate fastest sea-ice advection
-over the time span considered.
-The ice thickness sensitivities are in close correspondence to
-ocean surface sentivitites, but of opposite sign.
-An increase in temperature will incur ice melting, decrease in ice thickness,
-and therefore decrease in sea-ice export at time $T_e$.
+computed for December 1995.  The adjoint model computes sensitivities
+to sea-ice export back in time from 1995 to 1992 along this
+trajectory.  In principle all adjoint model variable (i.e., Lagrange
+multipliers) of the coupled ocean/sea-ice model are available to
+analyze the transient sensitivity behaviour of the ocean and sea-ice
+state.  Over the open ocean, the adjoint of the bulk formula scheme
+computes sensitivities to the time-varying atmospheric state.  Over
+ice-covered parts, the sea-ice adjoint converts surface ocean
+sensitivities to atmospheric sensitivities.
+
+\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
+through Fram Strait in December 1995 to changes in sea-ice thickness
+12, 24, 36, 48 months back in time. Corresponding sensitivities to
+ocean surface temperature are depicted in
+\reffig{4yradjthetalev1}(a--d).  The main characteristics is
+consistency with expected advection of sea-ice over the relevant time
+scales considered.  The general positive pattern means that an
+increase in sea-ice thickness at location $(x,y)$ and time $t$ will
+increase sea-ice export through Fram Strait at time $T_e$.  Largest
+distances from Fram Strait indicate fastest sea-ice advection over the
+time span considered.  The ice thickness sensitivities are in close
+correspondence to ocean surface sentivitites, but of opposite sign.
+An increase in temperature will incur ice melting, decrease in ice
+thickness, and therefore decrease in sea-ice export at time $T_e$.
 
 The picture is fundamentally different and much more complex
 for sensitivities to ocean temperatures away from the surface.
-Fig. XXX (a--d) depicts ice export sensitivities to
+\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
 temperatures at roughly 400 m depth.
 Primary features are the effect of the heat transport of the North
 Atlantic current which feeds into the West Spitsbergen current,
@@ -730,7 +672,7 @@
 -48 months}]
 {\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
+\caption{Same as \reffig{4yradjheff} but for sea surface temperature
 \label{fig:4yradjthetalev1}}
 \end{figure}
 
@@ -755,7 +697,7 @@
 
 \paragraph{Acknowledgements}
 We thank Jinlun Zhang for providing the original B-grid code and many
-helpful discussions.
+helpful discussions. ML thanks Elizabeth Hunke for multiple explanations.
 
 \bibliography{bib/journal_abrvs,bib/seaice,bib/genocean,bib/maths,bib/mitgcmuv,bib/fram}
 
