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.  The third set of
results is from a yet smaller regional domain, which is used to illustrate
treatment of sea ice open boundary condition sin 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} LSR
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 (GPCP) \citep{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}.

\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.  One
additional experiment is carried out to illustrate the differences between the
two main options for sea ice thermodynamics in the MITgcm.

The Arctic domain of integration is illustrated in \reffig{arctic1}.  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.\label{fig:arctic1}} 
\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 five different dynamical solvers:
\begin{description}
\item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an Arakawa
  B-grid, implying no-slip lateral boundary conditions;
\item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
  boundary conditions;
\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 is 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} \ml{[What
  can I say about effective thickness?]}
\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 on the \ml{luv [what is this in English?]} and the lee of
island groups, such as Franz-Josef-Land and \ml{IDONTKNOW}, which
\ml{\ldots [I find hard to interpret].}

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. Everywhere else the ice volume is
affected only slightly by the different boundary condition.
%
The C-EVP-ns solution has generally stronger drift velocities then the
C-LSR-ns solution. Consequently, more ice can be moved 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
daily averages 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 maxiumum (export out of
the Artic) and minimum (import into the Artic) fluxes as the drift
velocities area largest in this solution \ldots
\begin{figure}
\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
\caption{Transport through Canadian Archipelago for different solver flavors.
\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. Compared to that 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 comparable 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 integration time. At first,
this observation appears counterintuitive, as we expect that the
solution \emph{technique} should not affect the \emph{solution} to a
lower 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 \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}

%%% Local Variables: 
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1	dimitri	1.1	\section{Forward sensitivity experiments}
2			\label{sec:forward}
3
4	dimitri	1.2	This section presents results from global and regional coupled ocean and sea
5			ice simulations that exercise various capabilities of the MITgcm sea ice
6			model. The first set of results is from a global, eddy-permitting, ocean and
7			sea ice configuration. The second set of results is from a regional Arctic
8			configuration, which is used to compare the B-grid and C-grid dynamic solvers
9			and various other capabilities of the MITgcm sea ice model. The third set of
10			results is from a yet smaller regional domain, which is used to illustrate
11			treatment of sea ice open boundary condition sin the MITgcm.
12
13			\subsection{Global Ocean and Sea Ice Simulation}
14			\label{sec:global}
15
16			The global ocean and sea ice results presented below were carried out as part
17			of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2)
18			project. ECCO2 aims to produce increasingly accurate syntheses of all
19			available global-scale ocean and sea-ice data at resolutions that start to
20			resolve ocean eddies and other narrow current systems, which transport heat,
21			carbon, and other properties within the ocean \citep{menemenlis05}. The
22			particular ECCO2 simulation discussed next is a baseline 28-year (1979-2006)
23			integration, labeled cube76, which has not yet been constrained by oceanic and
24			by sea ice data. A cube-sphere grid projection is employed, which permits
25			relatively even grid spacing throughout the domain and which avoids polar
26			singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises
27			510 by 510 grid cells for a mean horizontal grid spacing of 18 km. There are
28			50 vertical levels ranging in thickness from 10 m near the surface to
29			approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from the
30			National Geophysical Data Center (NGDC) 2-minute gridded global relief data
31			(ETOPO2) and the model employs the partial-cell formulation of
32			\citet{adcroft97:_shaved_cells}, which permits accurate representation of the
33			bathymetry. The model is integrated in a volume-conserving configuration using
34			a finite volume discretization with C-grid staggering of the prognostic
35			variables. In the ocean, the non-linear equation of state of \citet{jac95} is
36			used.
37
38			The ocean model is coupled to the sea-ice model discussed in
39	mlosch	1.10	\refsec{model} using the following specific options. The
40			zero-heat-capacity thermodynamics formulation of \citet{hibler80} is used to
41	dimitri	1.2	compute sea ice thickness and concentration. Snow cover and sea ice salinity
42	dimitri	1.3	are prognostic. Open water, dry ice, wet ice, dry snow, and wet snow albedo
43			are, respectively, 0.15, 0.88, 0.79, 0.97, and 0.83. Ice mechanics follow the
44			viscous plastic rheology of \citet{hibler79} and the ice momentum equation is
45	mlosch	1.4	solved numerically using the C-grid implementation of the \citet{zhang97} LSR
46	dimitri	1.5	dynamics model discussed hereinabove. The ice is coupled to the ocean using
47			the rescaled vertical coordinate system, z$^\ast$, of
48			\citet{cam08}, that is, sea ice does not float above the ocean model but
49			rather deforms the ocean's model surface level.
50	dimitri	1.2
51	dimitri	1.3	This particular ECCO2 simulation is initialized from temperature and salinity
52	dimitri	1.5	fields derived from the Polar science center Hydrographic Climatology (PHC)
53			3.0 \citep{ste01a}. Surface boundary conditions for the period January 1979 to
54			July 2002 are derived from the European Centre for Medium-Range Weather
55			Forecasts (ECMWF) 40 year re-analysis (ERA-40) \citep{upp05}. Surface
56			boundary conditions after September 2002 are derived from the ECMWF
57			operational analysis. There is a one month transition period, August 2002,
58			during which the ERA-40 contribution decreases linearly from 1 to 0 and the
59			ECMWF analysis contribution increases linearly from 0 to 1. Six-hourly
60			surface winds, temperature, humidity, downward short- and long-wave
61			radiations, and precipitation are converted to heat, freshwater, and wind
62			stress fluxes using the \citet{large81,large82} bulk formulae. Shortwave
63			radiation decays exponentially as per \citet{pau77}. Low frequency
64			precipitation has been adjusted using the pentad (5-day) data from the Global
65			Precipitation Climatology Project (GPCP) \citep{huf01}. The time-mean river
66			run-off from \citet{lar01} is applied globally, except in the Arctic Ocean
67			where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)
68			and prepared by P. Winsor (personnal communication, 2007) is specificied.
69			Additionally, there is a relaxation to the monthly-mean climatological sea
70			surface salinity values from PHC 3.0, a relaxation time scale of 101 days.
71
72			Vertical mixing follows \citet{lar94} but with meridionally and vertically
73			varying background vertical diffusivity; at the surface, vertical diffusivity
74			is $4.4\times 10^{-6}$~m$^2$~s$^{-1}$ at the Equator, $3.6\times
75			10^{-6}$~m$^2$~s$^{-1}$ north of 70$^\circ$N, and $1.9\times
76			10^{-5}$~m$^2$~s$^{-1}$ south of 30$^\circ$S and between 30$^\circ$N and
77			60$^\circ$N , with sinusoidally varying values in between these latitudes;
78			vertically, diffusivity increases to $1.1\times 10^{-4}$~m$^2$~s$^{-1}$ at a a
79			depth of 6150 m as per \citet{bry79}. A high order monotonicity-preserving
80			advection scheme \citep{dar04} is employed and there is no explicit horizontal
81			diffusivity. Horizontal viscosity follows \citet{lei96} but modified to sense
82			the divergent flow as per \citet{kem08}.
83	dimitri	1.2
84			\subsection{Arctic Domain with Open Boundaries}
85			\label{sec:arctic}
86
87	dimitri	1.7	A series of forward sensitivity experiments have been carried out on an
88	dimitri	1.6	Arctic Ocean domain with open boundaries. The objective is to compare the old
89			B-grid LSR dynamic solver with the new C-grid LSR and EVP solvers. One
90			additional experiment is carried out to illustrate the differences between the
91			two main options for sea ice thermodynamics in the MITgcm.
92	dimitri	1.1
93	mlosch	1.10	The Arctic domain of integration is illustrated in \reffig{arctic1}. It
94	dimitri	1.6	is carved out from, and obtains open boundary conditions from, the global
95			cubed-sphere configuration described above. The horizontal domain size is
96			420 by 384 grid boxes.
97	dimitri	1.1
98			\begin{figure}
99	mlosch	1.10	\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/topography}}}
100			\caption{Bathymetry and domain boudaries of Arctic
101			Domain.\label{fig:arctic1}}
102	dimitri	1.1	\end{figure}
103
104	mlosch	1.10	The main dynamic difference from cube sphere is that it does not use
105			rescaled vertical coordinates (z$^\ast$) and the surface boundary
106			conditions for freshwater input are different, because those features
107			are not supported by the open boundary code.
108	dimitri	1.1
109	mlosch	1.10	Open water, dry ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,
110	dimitri	1.1	0.76, 0.94, and 0.8.
111
112	mlosch	1.10	The model is integrated from January, 1992 to March \ml{[???]}, 2000,
113			with five different dynamical solvers:
114			\begin{description}
115			\item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an Arakawa
116			B-grid, implying no-slip lateral boundary conditions;
117			\item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
118			boundary conditions;
119			\item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
120			conditions;
121			\item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
122			no-slip lateral boundary conditions; and
123			\item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
124			boundary conditions.
125			\end{description}
126			Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
127			that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
128			interpreted as pure model error. Lateral boundary conditions on a
129			coarse grid (compared to the roughness of the true coast line) are
130			unclear, so that comparing the no-slip solutions to the free-slip
131			solutions gives another measure of uncertainty in sea ice modeling.
132
133			A principle difficulty in comparing the solutions obtained with
134			different variants of the dynamics solver lies in the non-linear
135			feedback of the ice dynamics and thermodynamics. Already after a few
136			months the solutions have diverged so far from each other that
137			comparing velocities only makes sense within the first 3~months of the
138			integration while the ice distribution is still close to the initial
139			conditions. At the end of the integration, the differences between the
140			model solutions can be interpreted as cumulated model uncertainties.
141
142			\reffig{iceveloc} shows ice velocities averaged over Janunary,
143			February, and March (JFM) of 1992 for the C-LSR-ns solution; also
144			shown are the differences between B-grid and C-grid, LSR and EVP, and
145			no-slip and free-slip solution. The velocity field of the C-LSR-ns
146			solution (\reffig{iceveloc}a) roughly resembles the drift velocities
147			of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
148			models in an cyclonic circulation regime (CCR) \citep[their
149			Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
150			shifted eastwards towards Alaska.
151
152			The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
153			is most pronounced
154			along the coastlines, where the discretization differs most between B
155			and C-grids: On a B-grid the tangential velocity is on the boundary
156			(and thus zero per the no-slip boundary conditions), whereas on the
157			C-grid the its half a cell width away from the boundary, thus allowing
158			more flow. The B-LSR-ns solution has less ice drift through the Fram
159			Strait and especially the along Greenland's east coast; also, the flow
160			through Baffin Bay and Davis Strait into the Labrador Sea is reduced
161			with respect the C-LSR-ns solution. \ml{[Do we expect this? Say
162			something about that]}
163			%
164			Compared to the differences between B and C-grid solutions the
165			C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
166			(\reffig{iceveloc}c). As expected the differences are largest along
167			coastlines: because of the free-slip boundary conditions, flow is
168			faster in the C-LSR-fs solution, for example, along the east coast
169			of Greenland, the north coast of Alaska, and the east Coast of Baffin
170			Island.
171			\begin{figure}[htbp]
172			\centering
173			\subfigure[{\footnotesize C-LSR-ns}]
174			{\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_lsr_noslip}}
175			\subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
176			{\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_bgrid-lsr_noslip}}\\
177			\subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
178			{\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_lsr_slip-lsr_noslip}}
179			\subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
180			{\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_evp_noslip-lsr_noslip}}
181			\caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
182			over the first 3 months of integration [cm/s]; (b)-(d) difference
183			between B-LSR-ns, C-LSR-fs, C-EVP-ns, and C-LSR-ns solutions
184			[cm/s]; color indicates speed (or differences of speed), vectors
185			indicate direction only.}
186			\label{fig:iceveloc}
187			\end{figure}
188
189			The C-EVP-ns solution is very different from the C-LSR-ns solution
190			(\reffig{iceveloc}d). The EVP-approximation of the VP-dynamics allows
191			for increased drift by over 2\,cm/s in the Beaufort Gyre and the
192			transarctic drift. \ml{Also the Beaufort Gyre is moved towards Alaska
193			in the C-EVP-ns solution. [Really?]} In general, drift velocities are
194			biased towards higher values in the EVP solutions as can be seen from
195			a histogram of the differences in \reffig{drifthist}.
196			\begin{figure}[htbp]
197			\centering
198			\includegraphics[width=\textwidth]{\fpath/drifthist_evp_noslip-lsr_noslip}
199			\caption{Histogram of drift velocity differences for C-LSR-ns and
200			C-EVP-ns solution [cm/s].}
201			\label{fig:drifthist}
202			\end{figure}
203
204			\reffig{icethick}a shows the effective thickness (volume per unit
205			area) of the C-LSR-ns solution, averaged over January, February, March
206			of year 2000. By this time of the integration, the differences in the
207			ice drift velocities have led to the evolution of very different ice
208			thickness distributions, which are shown in \reffig{icethick}b--d, and
209			area distributions (not shown). \ml{Compared to other solutions, for
210			example, AOMIP the ice thickness distribution blablabal} \ml{[What
211			can I say about effective thickness?]}
212			\begin{figure}[htbp]
213			\centering
214			\subfigure[{\footnotesize C-LSR-ns}]
215			{\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_lsr_noslip}}
216			\subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
217			{\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_bgrid-lsr_noslip}}\\
218			\subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
219			{\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_lsr_slip-lsr_noslip}}
220			\subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
221			{\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_evp_noslip-lsr_noslip}}
222			\caption{(a) Effective thickness (volume per unit area) of the
223			C-LSR-ns solution, averaged over the months Janurary through March
224			2000 [m]; (b)-(d) difference between B-LSR-ns, C-LSR-fs, C-EVP-ns,
225			and C-LSR-ns solutions [cm/s].}
226			\label{fig:icethick}
227			\end{figure}
228
229			The generally weaker ice drift velocities in the B-LSR-ns solution,
230			when compared to the C-LSR-ns solution, in particular through the
231			narrow passages in the Canadian Archipelago, lead to a larger build-up
232			of ice north of Greenland and the Archipelago by 2\,m effective
233			thickness and more in the B-grid solution (\reffig{icethick}b). But
234			the ice volume in not larger everywhere: further west, there are
235			patches of smaller ice volume in the B-grid solution, most likely
236			because the Beaufort Gyre is weaker and hence not as effective in
237			transporting ice westwards. There are also dipoles of ice volume
238			differences on the \ml{luv [what is this in English?]} and the lee of
239			island groups, such as Franz-Josef-Land and \ml{IDONTKNOW}, which
240			\ml{\ldots [I find hard to interpret].}
241
242			Imposing a free-slip boundary condition in C-LSR-fs leads to a much
243			smaller differences to C-LSR-ns than the transition from the B-grid to
244			the C-grid (\reffig{icethick}c), but in the Canadian Archipelago it
245			still reduces the effective ice thickness by up to 2\,m where the ice
246			is thick and the straits are narrow. Everywhere else the ice volume is
247			affected only slightly by the different boundary condition.
248			%
249			The C-EVP-ns solution has generally stronger drift velocities then the
250			C-LSR-ns solution. Consequently, more ice can be moved the eastern
251			part of the Arctic, where ice volumes are smaller, to the western
252			Arctic where ice piles up along the coast (\reffig{icethick}d). Within
253			the Canadian Archipelago, more drift leads to faster ice export and
254			reduced effective ice thickness.
255
256			The difference in ice volume and ice drift velocities between the
257			different experiments has consequences for the ice transport out of
258			the Arctic. Although the main export of ice goes through the Fram
259			Strait, a considerable amoung of ice is exported through the Canadian
260			Archipelago \citep{???}. \reffig{archipelago} shows a time series of
261			daily averages ice transport through various straits in the Canadian
262			Archipelago and the Fram Strait for the different model solutions.
263			Generally, the C-EVP-ns solution has highest maxiumum (export out of
264			the Artic) and minimum (import into the Artic) fluxes as the drift
265			velocities area largest in this solution \ldots
266			\begin{figure}
267			\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
268			\caption{Transport through Canadian Archipelago for different solver flavors.
269			\label{fig:archipelago}}
270			\end{figure}
271
272			\ml{[Transport to narrow straits, area?, more runs, TEM, advection
273			schemes, Winton TD, discussion about differences in terms of model
274			error? that's tricky as it means refering to Tremblay, thus our ice
275			models are all erroneous!]}
276
277			In summary, we find that different dynamical solvers can yield very
278			different solutions. Compared to that the differences between
279			free-slip and no-slip solutions \emph{with the same solver} are
280			considerably smaller (the difference for the EVP solver is not shown,
281			but comparable to that for the LSOR solver)---albeit smaller, the
282			differences between free and no-slip solutions in ice drift can lead
283			to large differences in ice volume over integration time. At first,
284			this observation appears counterintuitive, as we expect that the
285			solution \emph{technique} should not affect the \emph{solution} to a
286			lower degree than actually modifying the equations. A more detailed
287			study on these differences is beyond the scope of this paper, but at
288			this point we may speculate, that the large difference between B-grid,
289			C-grid, LSOR, and EVP solutions stem from incomplete convergence of
290			the solvers due to linearization \citep[and Bruno Tremblay, personal
291			communication]{hunke01}: if the convergence of the non-linear momentum
292			equations is not complete for all linearized solvers, then one can
293			imagine that each solver stops at a different point in velocity-space
294			thus leading to different solutions for the ice drift velocities. If
295			this were true, this tantalizing circumstance had a dramatic impact on
296			sea-ice modeling in general, and we would need to improve the solution
297			technique of dynamic sea ice model, most likely at a very high
298			compuational cost (Bruno Tremblay, personal communication).
299
300
301
302	dimitri	1.1	\begin{itemize}
303			\item Configuration
304			\item OBCS from cube
305			\item forcing
306			\item 1/2 and full resolution
307			\item with a few JFM figs from C-grid LSR no slip
308			ice transport through Canadian Archipelago
309			thickness distribution
310			ice velocity and transport
311			\end{itemize}
312
313			\begin{itemize}
314			\item Arctic configuration
315			\item ice transport through straits and near boundaries
316			\item focus on narrow straits in the Canadian Archipelago
317			\end{itemize}
318
319			\begin{itemize}
320	mlosch	1.10	\item B-grid LSR no-slip: B-LSR-ns
321			\item C-grid LSR no-slip: C-LSR-ns
322			\item C-grid LSR slip: C-LSR-fs
323			\item C-grid EVP no-slip: C-EVP-ns
324			\item C-grid EVP slip: C-EVP-fs
325			\item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
326			new flag): C-LSR-ns+TEM
327			\item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
328			\item C-grid LSR no-slip + Winton:
329	dimitri	1.1	\item speed-performance-accuracy (small)
330			ice transport through Canadian Archipelago differences
331			thickness distribution differences
332			ice velocity and transport differences
333			\end{itemize}
334
335			We anticipate small differences between the different models due to:
336			\begin{itemize}
337			\item advection schemes: along the ice-edge and regions with large
338			gradients
339			\item C-grid: less transport through narrow straits for no slip
340			conditons, more for free slip
341			\item VP vs.\ EVP: speed performance, accuracy?
342			\item ocean stress: different water mass properties beneath the ice
343			\end{itemize}
344	dimitri	1.6
345	mlosch	1.10	%\begin{figure}
346			%\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM1992uvice}}}
347			%\caption{Surface sea ice velocity for different solver flavors.
348			%\label{fig:iceveloc}}
349			%\end{figure}
350
351			%\begin{figure}
352			%\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM2000heff}}}
353			%\caption{Sea ice thickness for different solver flavors.
354			%\label{fig:icethick}}
355			%\end{figure}
356	mlosch	1.9
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