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-revision 1.1 by dimitri,
Wed Nov  7 14:35:09 2007 UTC
+revision 1.10 by dimitri,
Mon Feb 25 16:50:56 2008 UTC
 Line 1
+ % $Header$
+ % $Name$
  \documentclass[12pt]{article}
- \usepackage{epsfig}
- \usepackage{graphics}
+ \usepackage[]{graphicx}
  \usepackage{subfigure}
  \usepackage[round,comma]{natbib}
- \bibliographystyle{agu04}
+ \bibliographystyle{bib/agu04}
  \usepackage{amsmath,amssymb}
  \newcommand\bmmax{10} \newcommand\hmmax{10}
-Line 35
+Line 37
  \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}
-Line 47
+Line 52
  \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}
-Line 127 
 The ice strain rate is given by
+Line 142 
 The ice strain rate is given by
      \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
+ The maximum ice pressure $P_{\max}$, a measure of ice strength, depends on
- $h$ and compactness (concentration) $c$: \[P =
+ both thickness $h$ and compactness (concentration) $c$:
- P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},\] with the constants $P^{*}$ and
+ \begin{equation}
- $C^{*}$. The nonlinear bulk and shear viscosities $\eta$ and $\zeta$
+   P_{\max} = P^{*}c\,h\,e^{[C^{*}\cdot(1-c)]},
- are functions of ice strain rate invariants and ice strength such that
+ \label{eq:icestrength}
- the principal components of the stress lie on an elliptical yield
+ \end{equation}
- curve with the ratio of major to minor axis $e$ equal to $2$; they are
+ with the constants $P^{*}$ and $C^{*}$. The nonlinear bulk and shear
- given by:
+ 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} \\
+   \zeta =& \min\left(\frac{P_{\max}}{2\max(\Delta,\Delta_{\min})},
-   \eta =& \frac{P}{2\Delta{e}^2} \\
+    \zeta_{\max}\right) \\
+   \eta =& \frac{\zeta}{e^2} \\
    \intertext{with the abbreviation}
    \Delta = & \left[
      \left(\dot{\epsilon}_{11}^2+\dot{\epsilon}_{22}^2\right)
-Line 145 
 given by:
+Line 164 
 given by:
 \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 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,
-Line 287 
 the two ice layers and the thickness of
+Line 324 
 the two ice layers and the thickness of
  \section{Funnel Experiments}
  \label{sec:funnel}
- \begin{itemize}
+ For a first/detailed comparison between the different variants of the
- \item B-grid LSR no-slip
+ MIT sea ice model an idealized geometry of a periodic channel,
- \item C-grid LSR no-slip
+\,km long and 500\,m wide on a non-rotating plane, with converging
- \item C-grid LSR slip
+ walls forming a symmetric funnel and a narrow strait of 40\,km width
- \item C-grid EVP no-slip
+ is used. The horizontal resolution is 5\,km throughout the domain
- \item C-grid EVP slip
+ making the narrow strait 8 grid points wide. The ice model is
- \end{itemize}
+ initialized with a complete ice cover of 50\,cm uniform thickness. The
+ ice model is driven by a constant along channel eastward ocean current
- \subsection{B-grid vs.\ C-grid}
+ of 25\,cm/s that does not see the walls in the domain. All other
- Comparison between:
+ ice-ocean-atmosphere interactions are turned off, in particular there
- \begin{itemize}
+ is no feedback of ice dynamics on the ocean current. All thermodynamic
- \item B-grid, lsr, no-slip
+ processes are turned off so that ice thickness variations are only
- \item C-grid, lsr, no-slip
+ caused by convergent or divergent ice flow. Ice volume (effective
- \item C-grid, evp, no-slip
+ thickness) and concentration are advected with a third-order scheme
- \end{itemize}
+ with a flux limiter \citep{hundsdorfer94} to avoid undershoots. This
- all without ice-ocean stress, because ice-ocean stress does not work
+ scheme is unconditionally stable and does not require additional
- for B-grid.
+ 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
+\,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).
+ \ml{[Say something about performance? This is tricky, as the
+   perfomance depends strongly on the configuration. A run with slowly
+   changing forcing is favorable for LSR, because then only very few
+   iterations are required for convergences while EVP uses its fixed
+   number of internal timesteps. If the forcing in changing fast, LSR
+   needs far more iterations while EVP still uses the fixed number of
+   internal timesteps. I have produces runs where for slow forcing LSR
+   is much faster than EVP and for fast forcing, LSR is much slower
+   than EVP. EVP is certainly more efficient in terms of vectorization
+   and MFLOPS on our SX8, but is that a criterion?]}
  \subsection{C-grid}
  \begin{itemize}
-Line 350 
 differences between the two main options
+Line 503 
 differences between the two main options
  \subsection{Arctic Domain with Open Boundaries}
  \label{sec:arctic}
- The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It is
+ The Arctic domain of integration is illustrated in Fig.~\ref{???}.  It
- carved out from, and obtains open boundary conditions from, the global
+ is carved out from, and obtains open boundary conditions from, the
- cubed-sphere configuration of the Estimating the Circulation and Climate of
+ global cubed-sphere configuration of the Estimating the Circulation
- the Ocean, Phase II (ECCO2) project \cite{men05a}.  The domain size is 420 by
+ and Climate of the Ocean, Phase II (ECCO2) project
-grid boxes horizontally with mean horizontal grid spacing of 18 km.
+ \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes
+ horizontally with mean horizontal grid spacing of 18 km.
  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
+ This particular ECCO2 simulation is initialized from rest using the
- temperature and salinity distribution from the World Ocean Atlas 2001 (WOA01)
+ January temperature and salinity distribution from the World Ocean
- [Conkright et al., 2002] and it is integrated for 32 years prior to the
+ Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for
--2001 period discussed in the study. Surface boundary conditions are from
+years prior to the 1996--2001 period discussed in the study. Surface
- the National Centers for Environmental Prediction and the National Center for
+ boundary conditions are from the National Centers for Environmental
- Atmospheric Research (NCEP/NCAR) atmospheric reanalysis [Kistler et al.,
+ Prediction and the National Center for Atmospheric Research
-]. Six-hourly surface winds, temperature, humidity, downward short- and
+ (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly
- long-wave radiations, and precipitation are converted to heat, freshwater, and
+ surface winds, temperature, humidity, downward short- and long-wave
- wind stress fluxes using the Large and Pond [1981, 1982] bulk
+ radiations, and precipitation are converted to heat, freshwater, and
- formulae. Shortwave radiation decays exponentially as per Paulson and Simpson
+ wind stress fluxes using the \citet{large81, large82} bulk formulae.
- [1977]. Additionally the time-mean river run-off from Large and Nurser [2001]
+ Shortwave radiation decays exponentially as per Paulson and Simpson
- is applied and there is a relaxation to the monthly-mean climatological sea
+ [1977]. Additionally the time-mean river run-off from Large and Nurser
- surface salinity values from WOA01 with a relaxation time scale of 3
+ [2001] is applied and there is a relaxation to the monthly-mean
- months. Vertical mixing follows Large et al. [1994] with background vertical
+ climatological sea surface salinity values from WOA01 with a
- diffusivity of 1.5 � 10-5 m2 s-1 and viscosity of 10-3 m2 s-1. A third order,
+ relaxation time scale of 3 months. Vertical mixing follows
- direct-space-time advection scheme with flux limiter is employed and there is
+ \citet{large94} with background vertical diffusivity of
- no explicit horizontal diffusivity. Horizontal viscosity follows Leith [1996]
+ $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of
- but modified to sense the divergent flow as per Fox-Kemper and Menemenlis [in
+ $10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time
- press].  Shortwave radiation decays exponentially as per Paulson and Simpson
+ advection scheme with flux limiter is employed \citep{hundsdorfer94}
- [1977].  Additionally, the time-mean runoff of Large and Nurser [2001] is
+ and there is no explicit horizontal diffusivity. Horizontal viscosity
- applied near the coastline and, where there is open water, there is a
+ follows \citet{lei96} but
- relaxation to monthly-mean WOA01 sea surface salinity with a time constant of
+ modified to sense the divergent flow as per Fox-Kemper and Menemenlis
-days.
+ [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,
-Line 417 
 ice, wet ice, dry snow, and wet snow alb
+Line 575 
 ice, wet ice, dry snow, and wet snow alb
  \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
-Line 428 
 We anticipate small differences between
+Line 587 
 We anticipate small differences between
  \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
+ \item C-grid: less transport through narrow straits for no slip
-   conditons, less for free 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}
- \section{Adjoint sensitivity experiment}
- \label{sec:adjoint}
- Adjoint sensitivity experiment on 1/2-res setup
-  Sensitivity of sea ice volume flow through Fram Strait
  \section{Adjoint sensiivities of the MITsim}
  \subsection{The adjoint of MITsim}
-Line 516 
 storing vs. recomputation of the model s
+Line 669 
 storing vs. recomputation of the model s
  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.
-Line 539 
 may be derived in about 10 minutes.
+Line 692 
 may be derived in about 10 minutes.
  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
-Line 552 
 The adjoint models run efficiently on 80
+Line 706 
 The adjoint models run efficiently on 80
  (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
+ Following a 1-year spinup, the model has been integrated for four
- between 1992 and 1995.
+ years between 1992 and 1995. It is forced using realistic 6-hourly
- It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.
+ NCEP/NCAR atmospheric state variables. Over the open ocean these are
- Over the open ocean these are converted into
+ converted into air-sea fluxes via the bulk formulae of
- air-sea fluxes via the bulk formulae of Large and Yeager (2004).
+ \citet{large04}.  Derivation of air-sea fluxes in the presence of
- Derivation of air-sea fluxes in the presence of sea-ice is handled
+ sea-ice is handled by the ice model as described in \refsec{model}.
- by the ice model as described in Section XXX.
  The objective function chosen is sea-ice export through Fram Strait
- computed for December 1995
+ computed for December 1995.  The adjoint model computes sensitivities
- The adjoint model computes sensitivities to sea-ice export back in time
+ to sea-ice export back in time from 1995 to 1992 along this
- from 1995 to 1992 along this trajectory.
+ trajectory.  In principle all adjoint model variable (i.e., Lagrange
- In principle all adjoint model variable (i.e. Lagrange multipliers)
+ multipliers) of the coupled ocean/sea-ice model are available to
- of the coupled ocean/sea-ice model
+ analyze the transient sensitivity behaviour of the ocean and sea-ice
- are available to analyze the transient sensitivity behaviour
+ state.  Over the open ocean, the adjoint of the bulk formula scheme
- of the ocean and sea-ice state.
+ computes sensitivities to the time-varying atmospheric state.  Over
- Over the open ocean, the adjoint of the bulk formula scheme
+ ice-covered parts, the sea-ice adjoint converts surface ocean
- computes sensitivities to the time-varying atmospheric state.
+ sensitivities to atmospheric sensitivities.
- 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
- Fig. XXX(a--d) depict sensitivities of sea-ice export through Fram Strait
+, 24, 36, 48 months back in time. Corresponding sensitivities to
- in December 1995 to changes in sea-ice thickness
+ ocean surface temperature are depicted in
-, 24, 36, 48 months back in time.
+ \reffig{4yradjthetalev1}(a--d).  The main characteristics is
- Corresponding sensitivities to ocean surface temperature are
+ consistency with expected advection of sea-ice over the relevant time
- depicted in Fig. XXX(a--d).
+ scales considered.  The general positive pattern means that an
- The main characteristics is consistency with expected advection
+ increase in sea-ice thickness at location $(x,y)$ and time $t$ will
- of sea-ice over the relevant time scales considered.
+ increase sea-ice export through Fram Strait at time $T_e$.  Largest
- The general positive pattern means that an increase in
+ distances from Fram Strait indicate fastest sea-ice advection over the
- sea-ice thickness at location $(x,y)$ and time $t$ will increase
+ time span considered.  The ice thickness sensitivities are in close
- sea-ice export through Fram Strait at time $T_e$.
+ correspondence to ocean surface sentivitites, but of opposite sign.
- Largest distances from Fram Strait indicate fastest sea-ice advection
+ An increase in temperature will incur ice melting, decrease in ice
- over the time span considered.
+ thickness, and therefore decrease in sea-ice export at time $T_e$.
- 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,
-Line 600 
 the circulation around Svalbard, and ...
+Line 749 
 the circulation around Svalbard, and ...
  \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.
-Line 625 
 sea-ice thickness at various prior times
+Line 774 
 sea-ice thickness at various prior times
  \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
+ \caption{Same as \reffig{4yradjheff} but for sea surface temperature
  \label{fig:4yradjthetalev1}}
  \end{figure}
-Line 666 
 parameters that we use here. What about
+Line 815 
 parameters that we use here. What about
  \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}
-Line 676 
 helpful discussions.
+Line 825 
 helpful discussions.
  %%% 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.
+-layer Hibler salinity + snow
+-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
+/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

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+Added in v.1.10

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