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: 
%%% mode: latex
%%% TeX-master: "ceaice"
%%% End: 
1	\section{Forward sensitivity experiments}
2	\label{sec:forward}
3
4	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	\refsec{model} using the following specific options. The
40	zero-heat-capacity thermodynamics formulation of \citet{hibler80} is used to
41	compute sea ice thickness and concentration. Snow cover and sea ice salinity
42	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	solved numerically using the C-grid implementation of the \citet{zhang97} LSR
46	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
51	This particular ECCO2 simulation is initialized from temperature and salinity
52	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
84	\subsection{Arctic Domain with Open Boundaries}
85	\label{sec:arctic}
86
87	A series of forward sensitivity experiments have been carried out on an
88	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
93	The Arctic domain of integration is illustrated in \reffig{arctic1}. It
94	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
98	\begin{figure}
99	\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/topography}}}
100	\caption{Bathymetry and domain boudaries of Arctic
101	Domain.\label{fig:arctic1}}
102	\end{figure}
103
104	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
109	Open water, dry ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,
110	0.76, 0.94, and 0.8.
111
112	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	\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	\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	\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
345	%\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
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