ceaice_split_version/ceaice_part1/ceaice_arctic.tex

\section{Arctic Ocean Sensitivity Experiments}
\label{sec:arcticmodel}

This section presents results from regional coupled ocean and sea
ice simulations of the Arctic Ocean that exercise various capabilities of the
MITgcm sea ice model.  The objective is to
compare the old B-grid LSOR dynamic solver with the new C-grid LSOR and
EVP solvers. Additional experiments are carried out to illustrate
the differences between different lateral boundary conditions, ice advection
schemes, ocean-ice stress formulations, and alternate sea ice
thermodynamics.

The Arctic Ocean domain has 420 by 384 grid
boxes and is illustrated in \reffig{arctic_topog}.
\begin{figure}
\centering
\includegraphics*[width=\stdfigwidth]{\fpath/topography}
\caption{Bathymetry and domain boundaries of Arctic
  Domain, cut-out from the global solution.
  The white
  line encloses what is loosely referred to as the Canadian Arctic
  Archipelago in the text.
  %; the dashed line marks the boundaries of the inset on the right hand side.
  The letters label sections in the
  Canadian Archipelago, where ice transport is evaluated: 
  A: Nares Strait; %
  B: Peary Channel; %
  C: Prince Gustaf Adolf Sea; %
  D: Ballantyne Strait; %
  E: M'Clure Strait; %
  F: Amundsen Gulf; %
  G: Lancaster Sound; %
  H: Barrow Strait W.; %
  I: Barrow Strait E.; %
  J: Barrow Strait N.; %
  K: Fram Strait. %
  The sections A through F comprise the total Arctic inflow into the Canadian
  Archipelago. The white labels denote Ellesmere Island of the Queen
  Elizabeth Islands (QEI), Svalbard (SB), Franz Joseph Land (FJL),
  Severnaya Zemlya (SZ), and the New Siberian Islands (NSI). 
  \label{fig:arctic_topog}}
\end{figure}
For each sensitivity experiment, the model is integrated from
January~1, 1992 to March~31, 2000. \ml{[Reviewer 2: Why this period? I
  have no idea how to explain this] This period was chosen to be long
  enough to observe systematic differences due to details of the model
  configuration and short enough to allow many sensitivity
  experiments.}

\begin{table}
  \caption{Overview of forward model sensitivity experiments in a regional
    Arctic Ocean domain.
    \label{tab:experiments}}
  \centering
  \begin{tabular}{p{.15\linewidth}p{.76\linewidth}}
    {\em Experiment}& {\em Description} \\ \hline
    C-LSR-ns  &  The LSOR solver discretized on a C~grid with no-slip
    lateral boundary conditions (implemented via ghost-points). \\
    B-LSR-ns  &  The original LSOR solver of \citet{zhang97} on an
    Arakawa~B grid, implying no-slip lateral boundary conditions
    ($\vek{u}=0$ exactly), advection of ice variables with a 2nd-order
    central difference scheme plus explicit diffusion for stability. \\
    C-EVP-ns  &  The EVP solver of \citet{hunke01} on a C~grid with
    no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
    150\text{\,s}$. \\
    C-EVP-10  &  The EVP solver of \citet{hunke01} on a C~grid with
    no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
    10\text{\,s}$. \\
    C-LSR-fs  &  The LSOR solver on a C~grid with free-slip lateral
    boundary conditions (no lateral stress on coast lines). \\
    DST3FL    & C-LSR-ns with a third-order flux limited
    direct-space-time advection scheme for thermodynamic variables
    \citep{hundsdorfer94}. \\
    TEM       &  C-LSR-ns with a truncated ellipse method (TEM)
    rheology \citep{hibler97}. \\
    HB87      &  C-LSR-ns with ocean-ice stress coupling according
    to \citet{hibler87}.\\
    WTD       &   C-LSR-ns with 3-layer thermodynamics following
    \citet{winton00}.
  \end{tabular}
\end{table}

\reftab{experiments} gives an overview of all the experiments discussed in
this section. %
In all experiments except for DST3FL ice is advected
with the original second order central differences scheme that
requires small extra diffusion for stability reasons. %
The differences between integrations 
B-LSR-ns and C-LSR-ns can be interpreted as being
caused by model finite dimensional numerical truncation.  %

Both the LSOR and the EVP solvers aim to solve for the same
viscous-plastic rheology; while the LSOR solver is an iterative scheme
with a convergence criterion the EVP solution relaxes towards the VP
solution in the limit of infinite intergration time. \ml{The
  differences between integrations C-LSR-ns, C-EVP-ns, and C-EVP-10
  are caused by fundamentally different approaches to regularize large
  bulk and shear viscosities; LSOR and other iterative techniques need
  to clip large viscosities, while EVP introduces elastic waves that
  damp out with-in one sub-cycling sequence. Both LSOR and EVP
  solutions represent approximations to true viscous-plastic rheology
  and neither will be considered ``truth'' in the paper: On the one
  hand, LSOR (and other implicit solvers) requires many so-called
  pseudo time steps to converge in a non-linear sense
  \citep{lemieux09}, which makes this type of solver very
  expensive. We use only 2~(customary) pseudo time steps. On the other
  hand, the elastic wave energy in EVP damps out completely only after
  an infinite time compared to the damping time scale, so that in
  practice the rheology is not completely viscous-plastic.}

\ml{For the EVP solver we use two different damping time
  scales and sub-cycling time steps. In the C-EVP-10 experiment, the
  damping time scale is one third of the ocean model times step; the
  EVP model is sub-cycled 120 times within each 1200\,s ocean model
  time step resulting in $\Delta{t}_\mathrm{evp}=10\text{\,s}$. In the
  C-EVP-ns experiment, the damping time scale is one third of the
  forcing period of 6\,hrs; the EVP model is sub-cycled
%144 times within each 6\,hr forcing period 
  144 times within each 6\,hr forcing period resulting in
  $\Delta{t}_\mathrm{evp}=150\text{\,s}$. This choice requires a long
  ice dynamics time step of 6\,hrs; sea ice and ocean model are
  coupled after each ice dynamics time step. [NOT CLEAR YET:
  Alternatively one could still step the sea ice model with 1200\,s
  and couple every ocean time step, because the ocean state changes
  only slowly over 6\,hrs. This experiment give results that are
  different \ldots What should we do about that?] \reftab{timings}
  shows timings for the these three cases. Note that in our
  configuration on 36~CPUs of a SGI~Altix~3700 the EVP technique
  is faster than LSOR even for the short time step (C-EVP-10).}
%
For comparison purposes,
\citet{hunke01} used a sub-cycling time step of 30\,s for an ocean model time
step of 3600\,s and a damping time scale of 1296\,s.  
\begin{table}
  % timing with -fp-model precise for 2232 time steps
  % a) C-EVP-ns150relax7200 SEAICE_DYNSOLVER 1735 FORWARD_STEP 4306
  
  % timings with -mp -ftz for 2232 time steps
  % a) C-EVP-ns150relax7200 SEAICE_DYNSOLVER 20.5 FORWARD_STEP 2301
  % b) C-LSR-ns             SEAICE_DYNSOLVER 600  FORWARD_STEP 2887
  % c) C-EVP-10             SEAICE_DYNSOLVER 262  FORWARD_STEP 2541
  \caption{Integration throughput on 36 CPUs of a SGI
    Altix~3700. \label{tab:timings}}
  \centering
  \begin{tabular}{p{.2\linewidth}p{.25\linewidth}p{.25\linewidth}}
    & \multicolumn{2}{l}{\em Wall clock per integration month (2232 time
    steps)} \\
    {\em Experiment}& {\em ice dynamics}& {\em entire model} \\ \hline
    C-LSR-ns      & 600 sec & 2887 sec\\
    C-EVP-ns      & 20.5 sec & 2301 sec \\
    C-EVP-10      & 262 sec & 2541 sec
  \end{tabular}
\end{table}

Lateral boundary conditions on a coarse grid (coarse
compared to the roughness of the true coast line) are ill-defined so that
comparing a no-slip solution (C-LSR-ns) to a free-slip solution (C-LSR-fs)
gives another measure of uncertainty in the sea ice model.  The sensitivity
experiments also explore the response of the coupled ocean and sea ice model
to different numerics and physics, that is, to changes in advection
and diffusion properties (DST3FL), in rheology (TEM), in stress coupling
(HB87), and in thermodynamics (WTD).

Comparing the solutions obtained with different realizations of the
model dynamics is difficult because of the non-linear feedback of the
ice dynamics and thermodynamics. Already after a few months the
model trajectories have diverged far enough so that 
velocity differences are easier to interpret within the first 3~months 
of the integration while the ice distributions are still comparable.
The effect on ice-thickness of different numerics tends to accumulate 
along the time integration, resulting in larger differences - also 
easier to interpret - at the end of the integration.
\ml{We choose C-LSR-ns as the reference run for all comparisons
  bearing in mind that any other choice is equally valid.}
%Already after a few months the
%solutions have diverged so far from each other that comparing
%velocities only makes sense within the first 3~months of the
%integration while the ice distribution is still close to the initial
%conditions. At the end of the integration, the differences between the
%model solutions can be interpreted as accumulated model errors.

\reftab{differences} and \reftab{rmsdiff} summarizes the differences
in drift speed and effective ice thickness for all experiments
discussed in the following.
\begin{table}
  \caption{Overview over drift speed differences (JFM of first year of
    integration) and effective ice thickness differences (JFM of last year of
    integration) relative to C-LSR-ns. For reference the corresponding
    values for C-LSR-ns are given in the first line. 
    \label{tab:differences}} 
  \centering
  \begin{tabular}{lr@{\hspace{3ex}}r@{\hspace{3ex}}r@{\hspace{3ex}}r}
%  \begin{tabular}{p{.25\linewidth}p{.15\linewidth}p{.15\linewidth}p{.15\linewidth}p{.15\linewidth}}
    speed (cm/s) & & & & \\
                &   mean   & rms       & median    &      max \\ \hline
 C-LSR-ns (ref) &    3.295 &     4.711 &     2.502 &   28.599 \\ \hline
       B-LSR-ns &   -0.236 &     0.714 &    -0.071 &   14.355 \\
       C-EVP-ns &    0.887 &     1.366 &     0.523 &   11.061 \\
%       C-EVP-ns &    1.548 &     2.295 &     1.106 &   14.392 \\
       C-EVP-10 &    0.266 &     0.513 &     0.213 &   10.506 \\
       C-LSR-fs &    0.160 &     0.472 &     0.084 &    9.921 \\
         DST3FL &    0.035 &     0.301 &     0.008 &   10.251 \\
            TEM &    0.027 &     0.168 &     0.014 &    8.922 \\
           HB87 &    0.184 &     0.316 &     0.169 &    9.175 \\
            WTD &    0.354 &     1.418 &     0.039 &   26.298 \\
    & & & & \\
    thickness (m) & & & & \\
                &  mean    & rms      & median   & max      \\ \hline
 C-LSR-ns (ref) &    1.599 &    1.941 &    1.542 &   10.000 \\ \hline
       B-LSR-ns &    0.065 &    0.175 &    0.049 &    2.423 \\
       C-EVP-ns &   -0.096 &    0.467 &   -0.023 &    5.458 \\
%      C-EVP-ns &    0.216 &    0.601 &    0.169 &    5.652 \\
       C-EVP-10 &   -0.082 &    0.399 &   -0.020 &    5.993 \\
       C-LSR-fs &   -0.037 &    0.289 &   -0.005 &    3.947 \\
         DST3FL &    0.014 &    0.338 &   -0.018 &    9.246 \\
            TEM &   -0.020 &    0.138 &   -0.001 &    2.541 \\
           HB87 &   -0.052 &    0.114 &   -0.029 &    2.520 \\
            WTD &    0.518 &    0.667 &    0.528 &    4.144
  \end{tabular}
\end{table}
\begin{table}
  \caption{Root-mean-square differences for drift speed  (JFM of first year of
    integration) and effective thickness (JFM of last year of
    integration) for the ``Candian Arctic Archipelago'' defined in
    \reffig{arctic_topog} and the remaining domain (``rest''). 
    For reference the corresponding values for C-LSR-ns are given in
    the first line.  
%    \ml{[This table can be removed in the submitted version,
%      it just gives use number to work with in the
%      text.]} 
    \label{tab:rmsdiff}}
  \centering
  \begin{tabular}{lr@{\hspace{3ex}}r@{\hspace{3ex}}r@{\hspace{3ex}}
                   r@{\hspace{3ex}}r@{\hspace{3ex}}r}
           & \multicolumn{3}{c}{rms(speed) (cm/s)}
           & \multicolumn{3}{c}{rms(thickness) (m)} \\ 
           &  total &  CAA   & rest   & total  &  CAA   &  rest  \\ \hline
  C-LSR-ns (ref) &  4.711 &  1.425 &  5.037 &  1.941 &  3.304 &  1.625 \\ \hline
  B-LSR-ns &  0.714 &  0.445 &  0.747 &  0.175 &  0.369 &  0.117 \\
  C-EVP-ns &  1.366 &  0.915 &  1.424 &  0.467 &  1.207 &  0.150 \\
% C-EVP-ns &  2.295 &  2.184 &  2.312 &  0.601 &  1.213 &  0.431 \\
  C-EVP-10 &  0.513 &  0.259 &  0.543 &  0.399 &  1.044 &  0.105 \\
  C-LSR-fs &  0.472 &  0.266 &  0.497 &  0.289 &  0.741 &  0.099 \\
    DST3FL &  0.301 &  0.063 &  0.323 &  0.338 &  0.763 &  0.201 \\
       TEM &  0.168 &  0.066 &  0.179 &  0.138 &  0.359 &  0.040 \\
      HB87 &  0.316 &  0.114 &  0.337 &  0.114 &  0.236 &  0.079 \\
       WTD &  1.418 &  1.496 &  1.406 &  0.667 &  1.110 &  0.566 
  \end{tabular}
\end{table}

\subsection{Ice velocities in JFM 1992}

\newcommand{\subplotwidth}{0.47\textwidth}
\begin{figure*}[tp]
%\newcommand{\subplotwidth}{0.3\textwidth}
%\begin{figure*}[tp]
  \centering
  \subfigure[{\footnotesize C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-LSR-ns}}
  \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_B-LSR-ns-C-LSR-ns}}
  \\
  \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-EVP-ns150relax7200-C-LSR-ns}}
%  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-EVP-ns150-C-LSR-ns}}
  \subfigure[{\footnotesize C-EVP-10 $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-EVP-ns-C-LSR-ns}}
  \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged over the
    first 3~months of integration (cm/s); (b)-(h) difference between the
    C-LSR-ns reference solution and solutions with, respectively, the B-grid
    solver, the EVP-solver with $\Delta{t}_\mathrm{evp}=150\text{\,s}$, the
    EVP-solver with $\Delta{t}_\mathrm{evp}=10\text{\,s}$, free lateral slip,
    a different advection scheme (DST3FL) for thermodynamic variables, the
    truncated ellipse method (TEM), and a different ice-ocean stress
    formulation (HB87). %
    Color indicates speed or differences of speed and vectors indicate
    direction only. The direction vectors represent block averages
    over eight by eight grid points at every eighth velocity point. %
    Note that color scale varies from panel to panel.}
  \label{fig:iceveloc}
\end{figure*}
\addtocounter{figure}{-1}
\setcounter{subfigure}{4}
\begin{figure*}[tp]
  \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-LSR-fs-C-LSR-ns}}
  \subfigure[{\footnotesize DST3FL $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_adv33-C-LSR-ns}}
  \\
  \subfigure[{\footnotesize TEM $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_TEM-C-LSR-ns}}
  \subfigure[{\footnotesize HB87 $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_HB87-C-LSR-ns}}
  \caption{Continued.}
\end{figure*}

\reffig{iceveloc} shows ice velocities averaged over January,
February, and March (JFM) of 1992 for the C-LSR-ns solution; also
shown are the differences between this reference solution and various
sensitivity experiments. The velocity field of the C-LSR-ns
solution (\reffig{iceveloc}a) roughly resembles the drift velocities
of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
models in a cyclonic circulation regime \citep[their
Figure\,6]{martin07} with a Beaufort Gyre and a Transpolar Drift
shifted eastwards towards Alaska.

The difference between experiments C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
is most pronounced ($\sim 2$\,cm/s) along the coastlines, where the
discretization differs most between B and C~grids.  On a B~grid the tangential
velocity lies on the boundary, and is thus zero through the no-slip boundary
conditions, whereas on the C~grid it is half a cell width away from the
boundary, thus allowing more flow. The B-LSR-ns solution has less ice drift
through the Fram Strait and along Greenland's East Coast; also, the flow
through Baffin Bay and Davis Strait into the Labrador Sea is reduced with
respect to the C-LSR-ns solution.

\ml{The C-EVP-ns solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$
allows for increased drift by
order 4\,cm/s in the Beaufort Gyre and in the Transpolar Drift. In
general, drift velocities tend towards higher values in
the EVP solution with a root-mean-square (rms) difference of nearly
0.9\,cm/s.} As the number of sub-cycling time steps increases \ml{(and
the coupling interval decreases to each time step)}, the EVP
approximation converges towards VP dynamics: the C-EVP-10 solution
with $\Delta{t}_\mathrm{evp}=10\text{\,s}$ (\reffig{iceveloc}d) is
substantially closer to the C-LSR-ns solution ($\sim 2$\,cm/s from
\reffig{iceveloc}d and a root-mean-square of 0.5\,cm/s and only
0.26\,cm/s in the CAA). \ml{Both EVP solutions have a stronger
  Beaufort Gyre as in \citet{hunke99}.}

% As expected the differences between C-LSR-fs and C-LSR-ns
% (\reffig{iceveloc}e) are also largest ($\sim 2$\,cm/s) along the
% coastlines. In constrast to B-LSR-ns, the ice drift for C-LSR-fs is on
% average faster than for C-LSR-ns while for B-LSR-ns it is on average
% slower than for C-LSR-ns.  This is because the free-slip boundary
% condition of C-LSR-fs allows the flow to be faster than C-LSR-ns, for
% example, along the East Coast of Greenland, the North Coast of Alaska,
% and the East Coast of Baffin Island.
As expected the differences between C-LSR-fs and C-LSR-ns
(\reffig{iceveloc}e) are also largest ($\sim 2$\,cm/s) along the
coastlines. \ml{The free-slip boundary
condition of C-LSR-fs allows the flow to be faster, for
example, along the East Coast of Greenland, the North Coast of Alaska,
and the East Coast of Baffin Island, so that the ice drift for C-LSR-fs is on
average faster than for C-LSR-ns where for B-LSR-ns it is on average
slower.} % than for C-LSR-ns.

The more sophisticated advection scheme of \mbox{DST3FL}
(\reffig{iceveloc}f) has \ml{the largest effect along the ice edge
  \citep[see also][]{merryfield03}, where the gradients of thickness
  and concentration are largest and differences in velocity can reach
  5\,cm/s (maximum differences are 10\,cm/s at individual grid
  points)}.  Everywhere else the effect is very small (rms of
0.3\,cm/s) and can mostly be attributed to smaller numerical diffusion
(and to the absence of explicit diffusion that is required for
numerical stability in a simple second order central differences
scheme). %
Note, that the advection scheme has an indirect effect on the ice
drift, but a direct effect on the ice transport, and hence the ice
thickness distribution and ice strength; a modified ice strength then
leads to a modified drift field.

Compared to the other parameters, the ice rheology TEM
(\reffig{iceveloc}g) also has a very small (mostly $<0.5$\,cm/s and
the smallest rms-difference of all solutions)
effect on the solution. In general the ice drift tends to increase
because there is no tensile stress and ice can drift apart at
no cost.  Consequently, the largest effect on drift velocity can be
observed near the ice edge in the Labrador Sea. Note in experiments
\mbox{DST3FL} and TEM the drift pattern is slightly changed as opposed
to all other C-grid experiments, although this change is small.

By way of contrast, the ice-ocean stress formulation of
\citet{hibler87} results in stronger drift by up to 2\,cm/s almost
everywhere in the computational domain (\reffig{iceveloc}h).  The
increase is mostly aligned with the general direction of the flow,
implying that the \citet{hibler87} stress formulation reduces the
deceleration of drift by the ocean.

\subsection{Integrated effect on ice volume during JFM 2000}

\begin{figure*}[tp]
  \centering
  \subfigure[{\footnotesize C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-ns}}
  \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_B-LSR-ns-C-LSR-ns}}
  \\
  \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-EVP-ns150relax7200-C-LSR-ns}}
%  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-EVP-ns150-C-LSR-ns}}
  \subfigure[{\footnotesize C-EVP-10 $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-EVP-ns-C-LSR-ns}}
  \caption{(a) Effective thickness (volume per unit area) of the
    C-LSR-ns solution, averaged over the months January through March
    2000 (m); (b)-(h) difference between the
    C-LSR-ns reference solution and solutions with, respectively, the B-grid
    solver, the EVP-solver with $\Delta{t}_\mathrm{evp}=150\text{\,s}$, the
    EVP-solver with $\Delta{t}_\mathrm{evp}=10\text{\,s}$, free lateral slip,
    a different advection scheme (DST3FL) for thermodynamic variables, the
    truncated ellipse method (TEM), and a different ice-ocean stress
    formulation (m).}
  \label{fig:icethick}
\end{figure*}
\addtocounter{figure}{-1}
\setcounter{subfigure}{4}
\begin{figure*}[tp]
  \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-fs-C-LSR-ns}}
  \subfigure[{\footnotesize DST3FL $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_adv33-C-LSR-ns}}
  \\
  \subfigure[{\footnotesize TEM $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_TEM-C-LSR-ns}}
  \subfigure[{\footnotesize HB87 $-$ C-LSR-ns}]
  {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_HB87-C-LSR-ns}}
  \caption{Continued.}
\end{figure*}

\reffig{icethick}a shows the effective thickness (volume per unit area) of the
C-LSR-ns solution, averaged over January, February, and March of year 2000,
that is, eight years after the start of the simulation. By this time of the
integration, the differences in ice drift velocities have led to the evolution
of very different ice thickness distributions (as shown in
Figs.~\ref{fig:icethick}b--h) and concentrations (not shown) for each
sensitivity experiment.  The mean ice volume for the January--March 2000
period is also reported in \reftab{icevolume}.

\begin{table}
  \caption{Arctic ice volume averaged over Jan--Mar 2000, in
    km$^3$. Mean ice transport (and standard deviation in parenthesis)
    for the period Jan 1992 -- Dec 1999 through the Fram Strait (FS), the
    total northern inflow into the Canadian Arctic Archipelago (CAA), and the
    export through Lancaster Sound (LS), in $\text{km$^{3}$\,y$^{-1}$}$.
    \label{tab:icevolume}}
  \centering
  \begin{tabular}{lllll}
    & {\em Volume\;\;} &
    \multicolumn{3}{l}{{\em Sea ice transport} (km$^3$\,yr$^{-1}$)} \\
    {\em Experiment\;\;} & (km$^3$) & FS & CAA & LS \\ \hline
    C-LSR-ns   & 24,769 & 2196\,(1253) &  70\,(224) &  77\,(110) \\
    B-LSR-ns   & 23,824 & 2126\,(1278) &  34\,(122) &  43\,(76)  \\
    C-EVP-ns   & 22,426 & 2415\,(1394) & 209\,(561) & 159\,(133) \\
%    C-EVP-ns   & 27,056 & 2415\,(1394)\;\; & 352\,(735)\;\; & 256\,(151) \\
    C-EVP-10   & 22,633 & 2174\,(1260) & 186\,(496) & 133\,(128) \\
    C-LSR-fs   & 23,286 & 2236\,(1289) &  80\,(276) &  91\,(85)  \\
    DST3FL     & 24,023 & 2191\,(1261) &  88\,(251) &  84\,(129) \\
    TEM        & 23,529 & 2222\,(1258) &  60\,(242) &  87\,(112) \\
    HB87       & 23,060 & 2256\,(1327) &  64\,(230) &  77\,(114)
\\ WTD        & 31,634 & 2761\,(1563) &  23\,(140) &  94\,(63)
  \end{tabular}
\end{table}

The generally weaker ice drift velocities in the B-LSR-ns solution,
when compared to the C-LSR-ns solution, in particular through the
narrow passages in the Canadian Arctic Archipelago, lead to a larger build-up
of ice (2\,m or more) north of Greenland and north of the Archipelago in the
B-grid solution (\reffig{icethick}b).
The ice volume, however, is not larger everywhere.  Further west there are
patches of smaller ice volume in the B-grid solution, most likely
because the Beaufort Gyre is weaker and hence not as effective in
transporting ice westwards. There are also dipoles of ice volume
differences with more ice on the upstream side and less ice on the downstream
side of island groups, for example, of Franz Josef Land, of Severnaya Zemlya,
of the New Siberian Islands, and of the Queen Elizabeth Islands
(see \reffig{arctic_topog} for their geographical locations).  This is 
because ice tends to flow less easily along coastlines, around islands, and
through narrow channels in the B-LSR-ns solution than in the C-LSR-ns solution.

\ml{The C-EVP-ns solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$
  has thinner ($\sim 50$\,cm) ice in the central Arctic Ocean than the C-LSR-ns
solution (\reffig{icethick}c), and the rms-difference is with
47\,cm larger than in the B-LSR-ns case.}
%DM Is there a simple explanation we can give for this thicker ice?
Within the Canadian Arctic Archipelago, more
drift leads to faster ice export and reduced effective ice thickness. With a
shorter time step ($\Delta{t}_\mathrm{evp}=10\text{\,s}$) the EVP solution
converges towards the LSOR solution in the central Arctic
(\reffig{icethick}d). In the narrow straits in the Archipelago, however, the
ice thickness is not affected by the shorter time step and the ice is still
thinner by 2\,m or more, as it is in the EVP solution with
$\Delta{t}_\mathrm{evp}=150\text{\,s}$.

%DM C-EVP-10 is incredibly similar to C-LSR-fs - why is that?
%ML Ultimately I do not know, but the mechanism is described: weaker
%ML ice in EVP and less horizontal friction in C-LSR-fs along coasts
%ML basically have a similar effect. The velocities are not that similar.

Imposing a free-slip boundary condition in C-LSR-fs leads to much
smaller differences to C-LSR-ns (\reffig{icethick}e)
than the transition from the B~grid to the C~grid, except
in the Canadian Arctic Archipelago, where the free-slip solution
allows more flow (see \reftab{rmsdiff}). There, it reduces the effective ice
thickness by 2\,m or more where the ice is thick and the straits are
narrow (leading to an overall larger rms-difference than the B-LSR-ns
solution, see \reftab{rmsdiff}). Dipoles of ice thickness differences can also be observed 
around islands because the free-slip solution allows more flow around
islands than the no-slip solution. %
%ML Everywhere else the ice thickness is
%ML affected only slightly by the different boundary condition.
The differences in the Central Arctic are much smaller in absolute
value than the differences in the Canadian Arctic Archipelago
although there are also interesting changes in the ice-distribution
in the interior: Less ice in the Central Arctic is most likely
caused by more export (see \reftab{icevolume}).

The remaining sensitivity experiments, DST3FL, TEM, and HB87, have the
largest differences in effective ice thickness along the north coasts
of Greenland and Ellesmere Island in the Canadian Arctic Archipelago.
Although \ml{using the TEM rheology and the \citet{hibler87} ice-ocean
  stress formulation has different effects on the initial ice
  velocities (\reffig{iceveloc}g and~h)}, both experiments have
similarly reduced ice thicknesses in this area. The 3rd-order
advection scheme (DST3FL) has an opposite effect of similar magnitude,
pointing towards more implicit lateral stress with this numerical
scheme. %
The HB87 experiment shows ice thickness reduction in the entire Arctic
basin greater than in any other experiment, possibly because more
drift leads to faster export of ice.
%%ML then let's remove this statement
%In the Central Arctic all three sensitivity experiments are similar to
%the reference C-LSR-ns.
%% Hmmm - looking at figs it looks like 4(h) HB87 - C-LSR-ns is not so similar 
%% in the central Arctic.

% \begin{figure}[t]
%   \centering
%   \includegraphics[width=\stdfigwidth]{\fpath/rangehist}
%   \caption{Histogram of ranges ice thickness and drift
%     velocity differences between all model solutions (excluding WTD).}
%   \label{fig:rangehist}
% \end{figure}
% \reffig{rangehist} summarizes Figures~\ref{fig:iceveloc}
% and~\ref{fig:icethick} by showing histograms of maximum sea ice thickness and
% drift velocity differences between the various sensitivity experiments,
% excluding the \citet{winton00} thermodynamics (WTD) experiment, which is
% discussed separately in \refsec{TED}.  These histograms are obtained by
% computing the range of
% values between all model solutions (excluding WTD) at each grid
% point. The
% mean (median) range for ice thickness is 52 (37)\,cm and for drift speed
% 2.1 (1.7)\,cm/s; the maximal values are 9.2\,m and
% 18\,cm/s, respectively.
\begin{figure}[t]
  \centering
  \includegraphics[width=\stdfigwidth]{\fpath/diffhist}
  \caption{Histograms of ice thickness and drift velocity differences
    relative to C-LSR-ns. The black line is the cumulative number grid
    points in percent of all grid points. The colors indicate the
    distribution of these grid points between the various experiments
    in percent of the black line.}
  \label{fig:diffhist}
\end{figure}
\reffig{diffhist} summarizes Figures~\ref{fig:iceveloc}
and~\ref{fig:icethick} by showing histograms of sea ice thickness and
drift velocity differences to the reference C-LSR-ns. The black line
is the cumulative number grid points in percent of all grid points
\ml{of all models where differences up to the value on the abscissa
  are found. For example, ice thickness differences up to 50\,cm are
  found in 90\% of all grid points, or equally differences above
  50\,cm are only found in 10\% of all grid points. The colors
  indicate the distribution of these grid points between the various
  experiments. For example, more than 50\% of ice thickness
  differences between 40\,cm and 1\,m are found in the run WTD. The
  runs B-LSR-ns and C-EVP-ns only have a fairly larger number grid
  points with differences below 40\,cm. C-EVP-ns dominates nearly all
  velocity differences below 4.5\,cm/s. The remaining contributions
  are small except for small differences. Only very few points
  contribute to very large differences in thickness (above 1\,m) and
  velocity (above 4\,cm/s) indicated by the small slope of the
  cumlative number of grid point (black line).}

\begin{figure}[t]
  \centering
  \includegraphics[width=\stdfigwidth]{\fpath/nsidc_hist}
  \caption{Histograms of sea ice drift speeds for the interior Arctic
    for JFM 2000 in comparison to observations from NSIDC. \ml{[This
      figure needs to be remade with the proper data and the caption
      and text needs to be adjusted.]}
    \label{fig:nsidc_hist}}
\end{figure}
\ml{[This text is a stubb and will be extended if we decide to include
  such a comparison:] \reffig{nsidc_hist} compares histograms of
  modeled sea ice drift speeds in JFM (January, February, March) 2000
  with a data product by the National Snow and Ice Data Center (NSIDC)
  \citep{fowler03}. Drift speeds below 0.5\,cm/s have been discarded
  and the domain is restricted to the ``interior Arctic'' as in
  \citet{martin07}. Compared to the data product, all runs tend
  towards too high velocities with modes between 5 and 8\,cm/s.  Drift
  speeds below 4\,cm/s are generally less frequent than in the
  observations. The histograms are similar for the individual
  experiments with the run B-LSR-ns and the EVP runs being differnt in
  the low drift speed range. B-LSR-ns procudes too few low velocities,
  while C-EVP-10 reproduces the observations between 2 and 4\,cm/s.}

\subsection{Ice transports}
\label{sec:icetransports}

\begin{figure*}[tp]
%\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
%\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
%\centerline{{\includegraphics[width=\linewidth]{\fpath/ice_export}}}
\centerline{{\includegraphics[width=\mediumfigwidth]{\fpath/ice_export1996}}}
\caption{Transports of sea ice during 1996 for model sensitivity experiments
  listed in \reftab{experiments}.  Top panel shows flow through the northern
  edge of the Canadian Arctic Archipelago (Sections A--F in
  \reffig{arctic_topog}), middle panel shows flow through Lancaster Sound
  (Section G), and bottom panel shows flow through Fram Strait (Section K).
  Positive values indicate sea ice flux out of the Arctic Ocean. The time
  series are smoothed using a monthly running mean. The mean range, i.e., the
  time-mean difference between the model solution with maximum flux and that
  with minimum flux, is computed over the period January 1992 to December
  1999.
  \label{fig:archipelago}}
\end{figure*}
%DM Could we change order to be consistent with figs 3 and
%DM 4, i.e., C-LSR-ns, B-LSR-ns, ...

The difference in ice volume and in ice drift velocity between the various
sensitivity experiments has consequences for sea ice export from the Arctic
Ocean.  \ml{As an illustration (other years are similar)}, \reffig{archipelago} shows the 1996 time series of sea
ice transports through the northern edge of the Canadian Arctic Archipelago,
through Lancaster Sound, and through Fram Strait for each model sensitivity
experiment. The mean and standard deviation of these ice transports, over the
period January 1992 to December 1999, are reported in \reftab{icevolume}.  In
addition to sea ice dynamics, there are many factors, e.g., atmospheric and
oceanic forcing, drag coefficients, and ice strength, that control sea ice
export.  Although calibrating these various factors is beyond the scope of
this manuscript, it is nevertheless instructive to compare the values in
\reftab{icevolume} with published estimates, as is done next.  This is a
necessary step towards constraining this model with data, a key motivation for
developing the MITgcm sea ice model and its adjoint.

The export through Fram Strait for all the sensitivity experiments
is consistent with the value of $2300\pm610\text{\,km$^3$\,yr$^{-1}$}$
reported by \citet[and references therein]{serreze06}. 
%
Although Arctic sea ice is exported to the Atlantic Ocean principally through
the Fram Strait, \citet{serreze06} estimate that a considerable amount of sea
ice ($\sim 160\text{\,km$^3$\,yr$^{-1}$}$) is also exported through the
Canadian Arctic Archipelago.  This estimate, however, is associated with large
uncertainties.  For example, \citet{dey81} estimates an inflow into Baffin Bay
of $370$ to $537\text{\,km$^3$\,yr$^{-1}$}$ but a flow of only $102$ to
$137\text{\,km$^3$\,yr$^{-1}$}$ further upstream in Barrow Strait in the
1970's from satellite images; \citet{aagaard89} give approximately
$155\text{\,km$^3$\,yr$^{-1}$}$ for the export through the CAA.
The recent estimates of \citet{agnew08} for
Lancaster Sound are lower: $102\text{\,km$^3$\,yr$^{-1}$}$.  The model results
suggest annually averaged ice transports through Lancaster Sound ranging from
$43$ to $159\text{\,km$^3$\,yr$^{-1}$}$ and total northern inflow of
$34$ to $209\text{\,km$^3$\,yr$^{-1}$}$ (\reftab{icevolume}). These model
estimates and their standard deviations cannot be rejected based on the
observational estimates.

Generally, the EVP solutions have the highest maximum (export out of
the Arctic) and lowest minimum (import into the Arctic) fluxes as the
drift velocities are largest in these solutions.  In the extreme of
the Nares Strait, which is only a few grid points wide in our
configuration, both B- and C-grid LSOR solvers lead to practically no
ice transport, while the C-EVP solutions allow up to
$350\text{--}400\text{\,km$^3$\,yr$^{-1}$}$ in summer (not shown). 
\citet{tang04} report $300$ to $350\text{\,km$^3$\,yr$^{-1}$}$ and
\citet{kwok05:_nares_strait} $130\pm65\text{\,km$^3$\,yr$^{-1}$}$.  As
as consequence, the import into the Canadian Arctic Archipelago is
larger in all EVP solutions
%(range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$)
than in the LSOR solutions.
%get the order of magnitude right (range: $132$ to
%$165\text{\,km$^3$\,y$^{-1}$}$); 
The B-LSR-ns solution is even smaller by another factor of two than the
C-LSR solutions.
%underestimates the ice transport with $34\text{\,km$^3$\,y$^{-1}$}$.

\subsection{Thermodynamics}
\label{sec:TED}

The last sensitivity experiment (WTD) listed in \reftab{experiments}
is carried out using the 3-layer thermodynamics model of
\citet{winton00}.  This experiment has different albedo and basal heat
exchange formulations from all the other experiments. %
\ml{For example, the values for the albedos for dry ice, dry and wet
  snow are the same as for the zero-layer model, but ice albedos in
  WTD are computed following \citet{hansen83} and can become much
  smaller with a minimum value 
  $0.2\exp(-h/0.44\text{\,m})$ as a function of thickness $h$. Further the
  snow age is taken into account when computing the 
  snow albedo. This results in albedos that range from 0.22 to 0.95.
  Similarly, large differences can be found in the basal heat exchange
  parameterizations.}
%
For this reason, the resulting ice velocities, volume, and transports
have not been included in the earlier comparisons. \ml{The key
  difference with the ``zero-layer'' thermodynamic model is a delay in
  the seaice cycle of approximately one month in the maximum sea-ice
  thickness and two months in the minimum sea-ice thickness. This is
  shown in \reffig{seasonalcycle}, which compares the mean sea-ice
  thickness seasonal cycle of experiments with the zero-heat-capacity
  (C-LSR-ns) and three-layer (WTD) thermodynamic model. The mean ice
  thickness is computed for a sector in the western Arctic (75\degN\
  to 85\degN\ and 180\degW\ to 140\degW) in order to avoid confounding
  thickness and extent differences. Similar to \citet{semtner76}, the
  seasonal cycle for the ``zero-layer'' model (gray dashed line) is
  almost twice as large as for the three-layer thermodynamic model.}
\begin{figure}[t]
  \centering
  \includegraphics[width=\stdfigwidth]{\fpath/SeasonalCycleWest}
  \caption{Seasonal cycle of mean sea-ice thickness (cm) in a sector in
    the western Arctic (75\degN\ to 85\degN\ and 180\degW\ to
    140\degW) averaged over 1992--2000 of experiments C-LSR-ns and WTD.}
%   \includegraphics[width=\stdfigwidth]{\fpath/SeasonalCycle}
%   \caption{Seasonal cycle of sea-ice volume (km$^3$) averaged over
%     1992--2000 of experiments C-LSR-ns and WTD.}
  \label{fig:seasonalcycle}
\end{figure}

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