articles/ceaice/ceaice_forward.tex

\section{Forward sensitivity experiments}
\label{sec:forward}

This section presents results from global and regional coupled ocean and sea
ice simulations that exercise various capabilities of the MITgcm sea ice
model.  The first set of results is from a global, eddy-permitting, ocean and
sea ice configuration.  The second set of results is from a regional Arctic
configuration, which is used to compare the B-grid and C-grid dynamic solvers
and various other capabilities of the MITgcm sea ice model.  
%
\ml{[do we really want to do this?:] The third set of
results is from a yet smaller regional domain, which is used to illustrate
treatment of sea ice open boundary condition in the MITgcm.}

\subsection{Global Ocean and Sea Ice Simulation}
\label{sec:global}

The global ocean and sea ice results presented below were carried out as part
of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2)
project.  ECCO2 aims to produce increasingly accurate syntheses of all
available global-scale ocean and sea-ice data at resolutions that start to
resolve ocean eddies and other narrow current systems, which transport heat,
carbon, and other properties within the ocean \citep{menemenlis05}.  The
particular ECCO2 simulation discussed next is a baseline 28-year (1979-2006)
integration, labeled cube76, which has not yet been constrained by oceanic and
by sea ice data.  A cube-sphere grid projection is employed, which permits
relatively even grid spacing throughout the domain and which avoids polar
singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises
510 by 510 grid cells for a 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
\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 \citet{jac95} is
used.

The ocean model is coupled to the sea-ice model discussed in
\refsec{model} using the following specific options.  The
zero-heat-capacity thermodynamics formulation of \citet{hibler80} is
used to compute sea ice thickness and concentration.  Snow cover and
sea ice salinity are prognostic.  Open water, dry ice, wet ice, dry
snow, and wet snow albedo are, respectively, 0.15, 0.88, 0.79, 0.97,
and 0.83. Ice mechanics follow the viscous plastic rheology of
\citet{hibler79} and the ice momentum equation is solved numerically
using the C-grid implementation of the \citet{zhang97}'s LSOR dynamics
model discussed hereinabove.  The ice is coupled to the ocean using
the rescaled vertical coordinate system, z$^\ast$, of \citet{cam08},
that is, sea ice does not float above the ocean model but rather
deforms the ocean's model surface level.

This particular ECCO2 simulation is initialized from temperature and salinity
fields derived from the Polar science center Hydrographic Climatology (PHC)
3.0 \citep{ste01a}. Surface boundary conditions for the period January 1979 to
July 2002 are derived from the European Centre for Medium-Range Weather
Forecasts (ECMWF) 40 year re-analysis (ERA-40) \citep{upp05}.  Surface
boundary conditions after September 2002 are derived from the ECMWF
operational analysis.  There is a one month transition period, August 2002,
during which the ERA-40 contribution decreases linearly from 1 to 0 and the
ECMWF analysis contribution increases linearly from 0 to 1.  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 \citet{pau77}.  Low frequency
precipitation has been adjusted using the pentad (5-day) data from the Global
Precipitation Climatology Project \citep[GPCP][]{huf01}.  The time-mean river
run-off from \citet{lar01} is applied globally, except in the Arctic Ocean
where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)
and prepared by P. Winsor (personnal communication, 2007) is specificied.
Additionally, there is a relaxation to the monthly-mean climatological sea
surface salinity values from PHC 3.0, a relaxation time scale of 101 days.

Vertical mixing follows \citet{lar94} but with meridionally and vertically
varying background vertical diffusivity; at the surface, vertical diffusivity
is $4.4\times 10^{-6}$~m$^2$~s$^{-1}$ at the Equator, $3.6\times
10^{-6}$~m$^2$~s$^{-1}$ north of 70$^\circ$N, and $1.9\times
10^{-5}$~m$^2$~s$^{-1}$ south of 30$^\circ$S and between 30$^\circ$N and
60$^\circ$N , with sinusoidally varying values in between these latitudes;
vertically, diffusivity increases to $1.1\times 10^{-4}$~m$^2$~s$^{-1}$ at a a
depth of 6150 m as per \citet{bry79}.  A high order monotonicity-preserving
advection scheme \citep{dar04} is employed and there is no explicit horizontal
diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
the divergent flow as per \citet{kem08}.

\ml{[Dimitris, here you need to either provide figures, so that I can
  write text, or you can provide both figures and text. I guess, one
  figure, showing the northern and southern hemisphere in summer and
  winter is fine (four panels), as we are showing so many figures in
  the next section.]}


\subsection{Arctic Domain with Open Boundaries}
\label{sec:arctic}

A series of forward sensitivity experiments have been carried out on
an Arctic Ocean domain with open boundaries.  The objective is to
compare the old B-grid LSR dynamic solver with the new C-grid LSR and
EVP solvers.  Additional experiments are is carried out to illustrate
the differences between different ice advection schemes, ocean-ice
stress formulations and the two main options for sea ice
thermodynamics in the MITgcm.

The Arctic domain of integration is illustrated in
\reffig{arctic_topog}.  It is carved out from, and obtains open
boundary conditions from, the global cubed-sphere configuration
described above.  The horizontal domain size is 420 by 384 grid boxes.
\begin{figure}
\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/topography}}}
\caption{Bathymetry and domain boudaries of Arctic
  Domain. The letters label sections in the Canadian Archipelago,
  where ice transport is evaluated.
  \label{fig:arctic_topog}}
\end{figure}

The main dynamic difference from cube sphere is that it does not use
rescaled vertical coordinates (z$^\ast$) and the surface boundary
conditions for freshwater input are different, because those features
are not supported by the open boundary code.

Open water, dry ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,
0.76, 0.94, and 0.8.

The model is integrated from January, 1992 to March \ml{[???]}, 2000,
with three different dynamical solvers and two different boundary
conditions:
\begin{description}
\item[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);
\item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
  boundary conditions (implemented via ghost-points);
\item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
  conditions;
\item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
  no-slip lateral boundary conditions; and
\item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
  boundary conditions.
\end{description}
Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
interpreted as pure model error. Lateral boundary conditions on a
coarse grid (compared to the roughness of the true coast line) are
unclear, so that comparing the no-slip solutions to the free-slip
solutions gives another measure of uncertainty in sea ice modeling.

A principle difficulty in comparing the solutions obtained with
different variants of the dynamics solver lies in the non-linear
feedback of the ice dynamics and thermodynamics. 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 cumulated model uncertainties. 

\reffig{iceveloc} shows ice velocities averaged over Janunary,
February, and March (JFM) of 1992 for the C-LSR-ns solution; also
shown are the differences between B-grid and C-grid, LSR and EVP, and
no-slip and free-slip solution. 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 an cyclonic circulation regime (CCR) \citep[their
Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
shifted eastwards towards Alaska.

The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
is most pronounced 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 thus zero per the no-slip boundary
conditions), whereas on the C-grid the its 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 especially the along Greenland's
east coast; also, the flow through Baffin Bay and Davis Strait into
the Labrador Sea is reduced with respect the C-LSR-ns solution.
\ml{[Do we expect this? Say something about that]}
%
Compared to the differences between B and C-grid solutions,the
C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
(\reffig{iceveloc}c).  As expected the differences are largest along
coastlines: because of the free-slip boundary conditions, flow is
faster in the C-LSR-fs solution, for example, along the east coast
of Greenland, the north coast of Alaska, and the east Coast of Baffin
Island.
\begin{figure}[htbp]
  \centering
  \subfigure[{\footnotesize C-LSR-ns}]
  {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_lsr_noslip}}
  \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
  {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_bgrid-lsr_noslip}}\\
  \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
  {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_lsr_slip-lsr_noslip}}
  \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
  {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_evp_noslip-lsr_noslip}}
  \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
    over the first 3 months of integration [cm/s]; (b)-(d) difference
    between B-LSR-ns, C-LSR-fs, C-EVP-ns, and C-LSR-ns solutions
    [cm/s]; color indicates speed (or differences of speed), vectors
    indicate direction only.}
  \label{fig:iceveloc}
\end{figure}

The C-EVP-ns solution is very different from the C-LSR-ns solution
(\reffig{iceveloc}d). The EVP-approximation of the VP-dynamics allows
for increased drift by over 2\,cm/s in the Beaufort Gyre and the
transarctic drift. \ml{Also the Beaufort Gyre is moved towards Alaska
  in the C-EVP-ns solution. [Really?]} In general, drift velocities are
biased towards higher values in the EVP solutions as can be seen from
a histogram of the differences in \reffig{drifthist}.
\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{\fpath/drifthist_evp_noslip-lsr_noslip}
  \caption{Histogram of drift velocity differences for C-LSR-ns and
    C-EVP-ns solution [cm/s].} 
  \label{fig:drifthist}
\end{figure}

\reffig{icethick}a shows the effective thickness (volume per unit
area) of the C-LSR-ns solution, averaged over January, February, March
of year 2000. By this time of the integration, the differences in the
ice drift velocities have led to the evolution of very different ice
thickness distributions, which are shown in \reffig{icethick}b--d, and
area distributions (not shown).  \ml{Compared to other solutions, for
  example, AOMIP the ice thickness distribution blablabal}
\begin{figure}[htbp]
  \centering
  \subfigure[{\footnotesize C-LSR-ns}]
  {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_lsr_noslip}}
  \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
  {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_bgrid-lsr_noslip}}\\
  \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
  {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_lsr_slip-lsr_noslip}}
  \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
  {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_evp_noslip-lsr_noslip}}
  \caption{(a) Effective thickness (volume per unit area) of the
    C-LSR-ns solution, averaged over the months Janurary through March
    2000 [m]; (b)-(d) difference between B-LSR-ns, C-LSR-fs, C-EVP-ns,
    and C-LSR-ns solutions [cm/s].}
  \label{fig:icethick}
\end{figure}
%
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 Archipelago, lead to a larger build-up
of ice north of Greenland and the Archipelago by 2\,m effective
thickness and more in the B-grid solution (\reffig{icethick}b). But
the ice volume in 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 \ml{luv [what is this in English?,
  upstream]} and less ice in the the lee of island groups, such as
Franz-Josef-Land and \ml{IDONTKNOW}, because ice tends to flow along
coasts less easily in the B-LSR-ns solution.

Imposing a free-slip boundary condition in C-LSR-fs leads to a much
smaller differences to C-LSR-ns than the transition from the B-grid to
the C-grid (\reffig{icethick}c), but in the Canadian Archipelago it
still reduces the effective ice thickness by up to 2\,m where the ice
is thick and the straits are narrow. 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.
Everywhere else the ice volume is affected only slightly by the
different boundary condition.
%
The C-EVP-ns solution has generally stronger drift velocities than the
C-LSR-ns solution. Consequently, more ice can be moved from the eastern
part of the Arctic, where ice volumes are smaller, to the western
Arctic where ice piles up along the coast (\reffig{icethick}d). Within
the Canadian Archipelago, more drift leads to faster ice export and
reduced effective ice thickness.

The difference in ice volume and ice drift velocities between the
different experiments has consequences for the ice transport out of
the Arctic. Although the main export of ice goes through the Fram
Strait, a considerable amoung of ice is exported through the Canadian
Archipelago \citep{???}. \reffig{archipelago} shows a time series of
\ml{[maybe smooth to longer time scales:] daily averaged} ice
transport through various straits in the Canadian Archipelago and the
Fram Strait for the different model solutions.  Generally, the
C-EVP-ns solution has highest maximum (export out of the Artic) and
minimum (import into the Artic) fluxes as the drift velocities are
largest in this solution \ldots
\begin{figure}
\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
\caption{Transport through Canadian Archipelago for different solver
  flavors. The letters refer to the labels of the sections in
  \reffig{arctic_topog}.
\label{fig:archipelago}}
\end{figure}

\ml{[Transport to narrow straits, area?, more runs, TEM, advection
  schemes, Winton TD, discussion about differences in terms of model
  error? that's tricky as it means refering to Tremblay, thus our ice
  models are all erroneous!]}

In summary, we find that different dynamical solvers can yield very
different solutions. In contrast, the differences between free-slip
and no-slip solutions \emph{with the same solver} are considerably
smaller (the difference for the EVP solver is not shown, but similar
to that for the LSOR solver). Albeit smaller, the differences between
free and no-slip solutions in ice drift can lead to large differences
in ice volume over the integration time. At first, this observation
seems counterintuitive, as we expect that the solution
\emph{technique} should not affect the \emph{solution} to a higher
degree than actually modifying the equations. A more detailed study on
these differences is beyond the scope of this paper, but at this point
we may speculate, that the large difference between B-grid, C-grid,
LSOR, and EVP solutions stem from incomplete convergence of the
solvers due to linearization and due to different methods of
linearization \citep[and Bruno Tremblay, personal
communication]{hunke01}: if the convergence of the non-linear momentum
equations is not complete for all linearized solvers, then one can
imagine that each solver stops at a different point in velocity-space
thus leading to different solutions for the ice drift velocities. If
this were true, this tantalizing circumstance had a dramatic impact on
sea-ice modeling in general, and we would need to improve the solution
technique of dynamic sea ice model, most likely at a very high
compuational cost (Bruno Tremblay, personal communication).


\begin{itemize}
\item Configuration
\item OBCS from cube
\item forcing
\item 1/2 and full resolution
\item with a few JFM figs from C-grid LSR no slip
  ice transport through Canadian Archipelago
  thickness distribution
  ice velocity and transport
\end{itemize}

\begin{itemize}
\item Arctic configuration
\item ice transport through straits and near boundaries
\item focus on narrow straits in the Canadian Archipelago
\end{itemize}

\begin{itemize}
\item B-grid LSR no-slip: B-LSR-ns
\item C-grid LSR no-slip: C-LSR-ns
\item C-grid LSR slip:    C-LSR-fs
\item C-grid EVP no-slip: C-EVP-ns
\item C-grid EVP slip:    C-EVP-fs
\item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
  new flag): C-LSR-ns+TEM
\item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
\item C-grid LSR no-slip + Winton:
\item  speed-performance-accuracy (small)
  ice transport through Canadian Archipelago differences
  thickness distribution differences
  ice velocity and transport differences
\end{itemize}

We anticipate small differences between the different models due to:
\begin{itemize}
\item advection schemes: along the ice-edge and regions with large
  gradients
\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}

%\begin{figure}
%\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM1992uvice}}}
%\caption{Surface sea ice velocity for different solver flavors.
%\label{fig:iceveloc}}
%\end{figure}

%\begin{figure}
%\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM2000heff}}}
%\caption{Sea ice thickness for different solver flavors.
%\label{fig:icethick}}
%\end{figure}

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