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revision 1.8 by dimitri, Fri Feb 29 01:28:05 2008 UTC revision 1.20 by mlosch, Sat Jul 5 10:39:16 2008 UTC
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1  \section{Forward sensitivity experiments}  \section{Forward Sensitivity Experiments in an Arctic Domain with Open
2    Boundaries}
3  \label{sec:forward}  \label{sec:forward}
4    
5  This section presents results from global and regional coupled ocean and sea  This section presents results from regional coupled ocean and sea
6  ice simulations that exercise various capabilities of the MITgcm sea ice  ice simulations of the Arctic Ocean that exercise various capabilities of the MITgcm sea ice
7  model.  The first set of results is from a global, eddy-permitting, ocean and  model.
8  sea ice configuration.  The second set of results is from a regional Arctic  The objective is to
9  configuration, which is used to compare the B-grid and C-grid dynamic solvers  compare the old B-grid LSOR dynamic solver with the new C-grid LSOR and
10  and various other capabilities of the MITgcm sea ice model.  The third set of  EVP solvers. Additional experiments are carried out to illustrate
11  results is from a yet smaller regional domain, which is used to illustrate  the differences between different ice advection schemes, ocean-ice
12  treatment of sea ice open boundary condition sin the MITgcm.  stress formulations and the two main options for sea ice
13    thermodynamics in the MITgcm.
14  \subsection{Global Ocean and Sea Ice Simulation}  
15  \label{sec:global}  \subsection{Model configuration and experiments}
16    \label{sec:arcticmodel}
17  The global ocean and sea ice results presented below were carried out as part  The Arctic model domain is illustrated in \reffig{arctic_topog}.
18    \begin{figure*}
19    %\includegraphics*[width=0.44\linewidth,viewport=139 210 496 606,clip]{\fpath/topography}
20    %\includegraphics*[width=0.44\linewidth,viewport=0 0 496 606,clip]{\fpath/topography}
21    %\includegraphics*[width=0.44\linewidth]{\fpath/topography}
22    %\includegraphics*[width=0.46\linewidth]{\fpath/archipelago}
23    \includegraphics*[width=\linewidth]{\fpath/topography}
24    \caption{Left: Bathymetry and domain boundaries of Arctic
25      Domain.
26      %; the dashed line marks the boundaries of the inset on the right hand side.
27      The letters in the inset label sections in the
28      Canadian Archipelago, where ice transport is evaluated:
29      A: Nares Strait; %
30      B: \ml{Meighen Island}; %
31      C: Prince Gustaf Adolf Sea; %
32      D: \ml{Brock Island}; %
33      E: M'Clure Strait; %
34      F: Amundsen Gulf; %
35      G: Lancaster Sound; %
36      H: Barrow Strait \ml{W.}; %
37      I: Barrow Strait \ml{E.}; %
38      J: Barrow Strait \ml{N.}; %
39      K: Fram Strait. %
40      The sections A through F comprise the total inflow into the Canadian
41      Archipelago. \ml{[May still need to check the geography.]}
42      \label{fig:arctic_topog}}
43    \end{figure*}
44    It has 420 by 384 grid boxes and is carved out, and obtains open
45    boundary conditions from, a global cubed-sphere configuration
46    similar to that described in \citet{menemenlis05}.
47    
48    \ml{[Some of this could be part of the introduction?]}%
49    The global ocean and sea ice results presented in \citet{menemenlis05}
50    were carried out as part
51  of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2)  of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2)
52  project.  ECCO2 aims to produce increasingly accurate syntheses of all  project.  ECCO2 aims to produce increasingly accurate syntheses of all
53  available global-scale ocean and sea-ice data at resolutions that start to  available global-scale ocean and sea-ice data at resolutions that start to
54  resolve ocean eddies and other narrow current systems, which transport heat,  resolve ocean eddies and other narrow current systems, which transport heat,
55  carbon, and other properties within the ocean \citep{menemenlis05}.  The  carbon, and other properties within the ocean \citep{menemenlis05}.  The
56  particular ECCO2 simulation discussed next is a baseline 28-year (1979-2006)  particular ECCO2 simulation from which we obtain the boundary
57    conditions is a baseline 28-year (1979-2006)
58  integration, labeled cube76, which has not yet been constrained by oceanic and  integration, labeled cube76, which has not yet been constrained by oceanic and
59  by sea ice data.  A cube-sphere grid projection is employed, which permits  by sea ice data. A cube-sphere grid projection is employed, which permits
60  relatively even grid spacing throughout the domain and which avoids polar  relatively even grid spacing throughout the domain and which avoids polar
61  singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises  singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises
62  510 by 510 grid cells for a mean horizontal grid spacing of 18 km. There are  510 by 510 grid cells for a mean horizontal grid spacing of 18\,km. There are
63  50 vertical levels ranging in thickness from 10 m near the surface to  50 vertical levels ranging in thickness from 10 m near the surface to
64  approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from the  approximately 450 m at a maximum model depth of 6150 m. The model employs the
65  National Geophysical Data Center (NGDC) 2-minute gridded global relief data  partial-cell formulation of
 (ETOPO2) and the model employs the partial-cell formulation of  
66  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the
67  bathymetry. The model is integrated in a volume-conserving configuration using  bathymetry. Bathymetry is from the S2004 (W.~Smith, unpublished) blend of the
68    \citet{smi97} and the General Bathymetric Charts of the Oceans (GEBCO) one
69    arc-minute bathymetric grid. % (see Fig.~\ref{fig:CubeBathymetry}).
70    The model is integrated in a volume-conserving configuration using
71  a finite volume discretization with C-grid staggering of the prognostic  a finite volume discretization with C-grid staggering of the prognostic
72  variables. In the ocean, the non-linear equation of state of \citet{jac95} is  variables. In the ocean, the non-linear equation of state of \citet{jac95} is
73  used.  used.
74    %
75  The ocean model is coupled to the sea-ice model discussed in  The global ocean model is coupled to a sea ice model in a
76  Section~\ref{sec:model} using the following specific options.  The  configuration similar to the case C-LSR-ns (see \reftab{experiments}),
77  zero-heat-capacity thermodynamics formulation of \citet{hib80} is used to  with open water, dry ice, wet ice, dry snow, and wet snow albedos of,
78  compute sea ice thickness and concentration.  Snow cover and sea ice salinity  respectively, 0.15, 0.88, 0.79, 0.97, and 0.83.
 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.  
79    
80  This particular ECCO2 simulation is initialized from temperature and salinity  This particular ECCO2 simulation is initialized from temperature and salinity
81  fields derived from the Polar science center Hydrographic Climatology (PHC)  fields derived from the Polar science center Hydrographic Climatology (PHC)
# Line 62  radiations, and precipitation are conver Line 91  radiations, and precipitation are conver
91  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
92  radiation decays exponentially as per \citet{pau77}.  Low frequency  radiation decays exponentially as per \citet{pau77}.  Low frequency
93  precipitation has been adjusted using the pentad (5-day) data from the Global  precipitation has been adjusted using the pentad (5-day) data from the Global
94  Precipitation Climatology Project (GPCP) \citep{huf01}.  The time-mean river  Precipitation Climatology Project \citep[GPCP,][]{huf01}.  The time-mean river
95  run-off from \citet{lar01} is applied globally, except in the Arctic Ocean  run-off from \citet{lar01} is applied globally, except in the Arctic Ocean
96  where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)  where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)
97  and prepared by P. Winsor (personnal communication, 2007) is specificied.  and prepared by P. Winsor (personnal communication, 2007) is specificied.
# Line 81  advection scheme \citep{dar04} is employ Line 110  advection scheme \citep{dar04} is employ
110  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
111  the divergent flow as per \citet{kem08}.  the divergent flow as per \citet{kem08}.
112    
113  \subsection{Arctic Domain with Open Boundaries}  The model configuration of cube76 carries over to the Arctic domain
114  \label{sec:arctic}  configuration except for numerical details related to the non-linear
115    free surface that are not supported by the open boundary code, and the
116  A series of forward sensitivity experiments have been carried out on an  albedos of open water, dry ice, wet ice, dry snow, and wet snow, which
117  Arctic Ocean domain with open boundaries.  The objective is to compare the old  are now, respectively, 0.15, 0.85, 0.76, 0.94, and 0.8.
118  B-grid LSR dynamic solver with the new C-grid LSR and EVP solvers.  One  
119  additional experiment is carried out to illustrate the differences between the  The model is integrated from Jan~01, 1992 to Mar~31, 2000.
120  two main options for sea ice thermodynamics in the MITgcm.  \reftab{experiments} gives an overview over the experiments discussed
121    in \refsec{arcticresults}.
122  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It  \begin{table}
123  is carved out from, and obtains open boundary conditions from, the global    \caption{Overview over model simulations in \refsec{arcticresults}.
124  cubed-sphere configuration described above.  The horizontal domain size is      \label{tab:experiments}}
125  420 by 384 grid boxes.    \begin{tabular}{p{.3\linewidth}p{.65\linewidth}}
126        experiment name & description \\ \hline
127  \begin{figure}      B-LSR-ns       &  the original LSOR solver of \citet{zhang97} on an
128  %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1}}}    Arakawa B-grid, implying no-slip lateral boundary conditions
129  \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}    ($\vek{u}=0$ exactly), advection of ice variables with a 2nd-order
130      central difference scheme plus explicit diffusion for stability \\
131        C-LSR-ns       &  the LSOR solver discretized on a C-grid with no-slip lateral
132      boundary conditions (implemented via ghost-points) \\
133        C-LSR-fs       &  the LSOR solver on a C-grid with free-slip lateral boundary
134      conditions \\
135        C-EVP-ns       &  the EVP solver of \citet{hunke01} on a C-grid with
136      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
137      150\text{\,s}$ \\
138        C-EVP-ns10     &  the EVP solver of \citet{hunke01} on a C-grid with
139      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
140      10\text{\,s}$ \\
141        C-LSR-ns HB87  &  C-LSR-ns with ocean-ice stress coupling according
142      to \citet{hibler87}\\
143        C-LSR-ns TEM   &  C-LSR-ns with a truncated ellispe method (TEM)
144      rheology \citep{hibler97} \\
145        C-LSR-ns WTD   &   C-LSR-ns with 3-layer thermodynamics following
146      \citet{winton00} \\
147        C-LSR-ns DST3FL& C-LSR-ns with a third-order flux limited
148      direct-space-time advection scheme for thermodynamic variables
149      \citep{hundsdorfer94}
150      \end{tabular}
151    \end{table}
152    %\begin{description}
153    %\item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an
154    %  Arakawa B-grid, implying no-slip lateral boundary conditions
155    %  ($\vek{u}=0$ exactly);
156    %\item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
157    %  boundary conditions (implemented via ghost-points);
158    %\item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
159    %  conditions;
160    %\item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
161    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
162    %  150\text{\,s}$;
163    %\item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
164    %  boundary conditions  and $\Delta{t}_\mathrm{evp} = 150\text{\,s}$;
165    %\item[C-LSR-ns DST3FL:] C-LSR-ns with a third-order flux limited
166    %  direct-space-time advection scheme \citep{hundsdorfer94};
167    %\item[C-LSR-ns TEM:] C-LSR-ns with a truncated ellispe method (TEM)
168    %  rheology \citep{hibler97};
169    %\item[C-LSR-ns HB87:] C-LSR-ns with ocean-ice stress coupling according
170    %  to \citet{hibler87};
171    %\item[C-LSR-ns WTD:] C-LSR-ns with 3-layer thermodynamics following
172    %  \citet{winton00};
173    %%\item[C-EVP-ns damp:] C-EVP-ns with additional damping to reduce small
174    %%  scale noise \citep{hunke01};
175    %\item[C-EVP-ns10:] the EVP solver of \citet{hunke01} on a C-grid with
176    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
177    %  10\text{\,s}$.
178    %\end{description}
179    Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
180    that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
181    interpreted as pure model error. Lateral boundary conditions on a
182    coarse grid (coarse compared to the roughness of the true coast line) are
183    unclear, so that comparing the no-slip solutions to the free-slip
184    solutions gives another measure of uncertainty in sea ice modeling.
185    The remaining experiments explore further sensitivities of the system
186    to different physics (change in rheology, advection and diffusion
187    properties, stress coupling, and thermodynamics) and different time
188    steps for the EVP solutions: \citet{hunke01} uses 120 subcycling steps
189    for the EVP solution. We use two interpretations of this choice where
190    the EVP model is subcycled 120 times within a (short) model timestep
191    of 1200\,s resulting in a very long and expensive integration
192    ($\Delta{t}_\mathrm{evp}=10\text{\,s}$) and 144 times within the
193    forcing timescale of 6\,h ($\Delta{t}_\mathrm{evp}=150\text{\,s}$).
194    
195    \subsection{Results}
196    \label{sec:arcticresults}
197    
198    Comparing the solutions obtained with different realizations of the
199    model dynamics is difficult because of the non-linear feedback of the
200    ice dynamics and thermodynamics. Already after a few months the
201    solutions have diverged so far from each other that comparing
202    velocities only makes sense within the first 3~months of the
203    integration while the ice distribution is still close to the initial
204    conditions. At the end of the integration, the differences between the
205    model solutions can be interpreted as cumulated model uncertainties.
206    
207    \subsubsection{Ice velocities in JFM 1992}
208    
209    \reffig{iceveloc} shows ice velocities averaged over Janunary,
210    February, and March (JFM) of 1992 for the C-LSR-ns solution; also
211    shown are the differences between this reference solution and various
212    sensitivity experiments. The velocity field of the C-LSR-ns
213    solution (\reffig{iceveloc}a) roughly resembles the drift velocities
214    of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
215    models in a cyclonic circulation regime (CCR) \citep[their
216    Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
217    shifted eastwards towards Alaska.
218    %
219    \newcommand{\subplotwidth}{0.47\textwidth}
220    %\newcommand{\subplotwidth}{0.3\textwidth}
221    \begin{figure}[tp]
222      \centering
223      \subfigure[{\footnotesize C-LSR-ns}]
224      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-LSR-ns}}
225      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
226      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_B-LSR-ns-C-LSR-ns}}
227      \\
228      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
229      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-LSR-fs-C-LSR-ns}}
230      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
231      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-EVP-ns150-C-LSR-ns}}
232    %  \\
233    %  \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
234    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_TEM-C-LSR-ns}}
235    %  \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
236    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_HB87-C-LSR-ns}}
237    %  \\
238    %  \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
239    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_ThSIce-C-LSR-ns}}
240    %  \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
241    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_adv33-C-LSR-ns}}
242      \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
243        over the first 3 months of integration [cm/s]; (b)-(h) difference
244        between solutions with B-grid, free lateral slip, EVP-solver,
245        truncated ellipse method (TEM), different ice-ocean stress
246        formulation (HB87), different thermodynamics (WTD), different
247        advection for thermodynamic variables (DST3FL) and the C-LSR-ns
248        reference solution [cm/s]; color indicates speed (or differences
249        of speed), vectors indicate direction only.}
250      \label{fig:iceveloc}
251  \end{figure}  \end{figure}
252    \addtocounter{figure}{-1}
253  Difference from cube sphere is that it does not use z* coordinates nor  \setcounter{subfigure}{4}
254  realfreshwater fluxes because it is not supported by open boundary code.  \begin{figure}[tp]
255      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
256  Open water, dry    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_TEM-C-LSR-ns}}
257  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,    \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
258  0.76, 0.94, and 0.8.    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_HB87-C-LSR-ns}}
259      \\
260  \begin{itemize}    \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
261  \item Configuration    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_ThSIce-C-LSR-ns}}
262  \item OBCS from cube    \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
263  \item forcing    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_adv33-C-LSR-ns}}
264  \item 1/2 and full resolution    \caption{continued}
 \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  
 \item C-grid LSR no-slip  
 \item C-grid LSR slip  
 \item C-grid EVP no-slip  
 \item C-grid EVP slip  
 \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)  
 \item C-grid LSR no-slip + Winton  
 \item  speed-performance-accuracy (small)  
   ice transport through Canadian Archipelago differences  
   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}}  
265  \end{figure}  \end{figure}
266    
267  \begin{figure}  The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
268  \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}  is most pronounced along the coastlines, where the discretization
269  \caption{Transport through Canadian Archipelago for different solver flavors.  differs most between B and C-grids: On a B-grid the tangential
270  \label{fig:archipelago}}  velocity lies on the boundary (and is thus zero through the no-slip
271    boundary conditions), whereas on the C-grid it is half a cell width
272    away from the boundary, thus allowing more flow. The B-LSR-ns solution
273    has less ice drift through the Fram Strait and along
274    Greenland's east coast; also, the flow through Baffin Bay and Davis
275    Strait into the Labrador Sea is reduced with respect the C-LSR-ns
276    solution.  \ml{[Do we expect this? Say something about that]}
277    %
278    Compared to the differences between B and C-grid solutions, the
279    C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
280    (\reffig{iceveloc}c).  As expected the differences are largest along
281    coastlines: because of the free-slip boundary conditions, flow is
282    faster in the C-LSR-fs solution, for example, along the east coast
283    of Greenland, the north coast of Alaska, and the east Coast of Baffin
284    Island.
285    
286    The C-EVP-ns solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$ is
287    very different from the C-LSR-ns solution (\reffig{iceveloc}d). The
288    EVP-approximation of the VP-dynamics allows for increased drift by up
289    to 8\,cm/s in the Beaufort Gyre and the transarctic drift.  In
290    general, drift velocities are strongly biased towards higher values in
291    the EVP solutions.
292    
293    Compared to the other parameters, the ice rheology TEM
294    (\reffig{iceveloc}e) has a very small effect on the solution. In
295    general the ice drift tends to be increased, because there is no
296    tensile stress and ice can be ``pulled appart'' at no cost.
297    Consequently, the largest effect on drift velocity can be observed
298    near the ice edge in the Labrador Sea. In contrast, the drift is
299    stronger almost everywhere in the computational domain in the run with
300    the ice-ocean stress formulation of \citet{hibler87}
301    (\reffig{iceveloc}f). The increase is mostly aligned with the general
302    direction of the flow, implying that the different stress formulation
303    reduces the deceleration of drift by the ocean.
304    
305    The 3-layer thermodynamics following \citet{winton00} requires
306    additional information on initial conditions for enthalphy. These
307    different initial conditions make a comparison of the first months
308    difficult to interpret. The drift in the Beaufort Gyre is slightly
309    reduced relative to the reference run C-LSR-ns, but the drift through
310    the Fram Strait is increased. The drift velocities near the ice edge
311    are very different, because the ice extend is already larger in
312    \mbox{C-LSR-ns~WTD}; inward from the ice egde, this run has smaller
313    drift velocities, because the ice motion is more contrained by a
314    larger ice extent than in \mbox{C-LSR-ns}, where the ice at the same
315    geographical position is nearly in free drift.
316    
317    A more sophisticated advection scheme (\mbox{C-LSR-ns DST3FL},
318    \reffig{iceveloc}h) has some effect along the ice edge, where
319    the gradients of thickness and concentration are largest. Everywhere
320    else the effect is very small and can mostly be attributed to smaller
321    numerical diffusion (and to the absense of explicit diffusion that is
322    requird for numerical stability in a simple second order central
323    differences scheme).
324    
325    \subsubsection{Ice volume during JFM 2000}
326    
327    \reffig{icethick}a shows the effective thickness (volume per unit
328    area) of the C-LSR-ns solution, averaged over January, February, March
329    of year 2000. By this time of the integration, the differences in the
330    ice drift velocities have led to the evolution of very different ice
331    thickness distributions, which are shown in \reffig{icethick}b--h, and
332    concentrations (not shown).
333    \begin{figure}[tp]
334      \centering
335      \subfigure[{\footnotesize C-LSR-ns}]
336      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-ns}}
337      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
338      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_B-LSR-ns-C-LSR-ns}}
339      \\
340      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
341      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-fs-C-LSR-ns}}
342      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
343      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-EVP-ns150-C-LSR-ns}}
344      \caption{(a) Effective thickness (volume per unit area) of the
345        C-LSR-ns solution, averaged over the months Janurary through March
346        2000 [m]; (b)-(h) difference between solutions with B-grid, free
347        lateral slip, EVP-solver, truncated ellipse method (TEM),
348        different ice-ocean stress formulation (HB87), different
349        thermodynamics (WTD), different advection for thermodynamic
350        variables (DST3FL) and the C-LSR-ns reference solution [m].}
351      \label{fig:icethick}
352  \end{figure}  \end{figure}
353    \addtocounter{figure}{-1}
354    \setcounter{subfigure}{4}
355    \begin{figure}[tp]
356      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
357      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_TEM-C-LSR-ns}}
358      \subfigure[{\footnotesize C-EVP-ns HB87 $-$ C-LSR-ns}]
359      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_HB87-C-LSR-ns}}
360      \\
361      \subfigure[{\footnotesize C-LSR-ns WTD $-$ C-LSR-ns}]
362      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_ThSIce-C-LSR-ns}}
363      \subfigure[{\footnotesize C-EVP-ns DST3FL $-$ C-LSR-ns}]
364      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_adv33-C-LSR-ns}}
365      \caption{continued}
366    \end{figure}
367    The generally weaker ice drift velocities in the B-LSR-ns solution,
368    when compared to the C-LSR-ns solution, in particular through the
369    narrow passages in the Canadian Archipelago, lead to a larger build-up
370    of ice north of Greenland and the Archipelago by 2\,m effective
371    thickness and more in the B-grid solution (\reffig{icethick}b). But
372    the ice volume in not larger everywhere: further west, there are
373    patches of smaller ice volume in the B-grid solution, most likely
374    because the Beaufort Gyre is weaker and hence not as effective in
375    transporting ice westwards. There are also dipoles of ice volume
376    differences with more ice on the upstream side of island groups and
377    less ice in their lee, such as Franz-Josef-Land and
378    Severnaya Semlya\ml{/or Nordland?},
379    because ice tends to flow along coasts less easily in the B-LSR-ns
380    solution.
381    
382    Imposing a free-slip boundary condition in C-LSR-fs leads to much
383    smaller differences to C-LSR-ns in the central Arctic than the
384    transition from the B-grid to the C-grid (\reffig{icethick}c), except
385    in the Canadian Archipelago. There it reduces the effective ice
386    thickness by 2\,m and more where the ice is thick and the straits are
387    narrow.  Dipoles of ice thickness differences can also be observed
388    around islands, because the free-slip solution allows more flow around
389    islands than the no-slip solution. Everywhere else the ice volume is
390    affected only slightly by the different boundary condition.
391    %
392    The C-EVP-ns solution has much thicker ice in the central Arctic Ocean
393    than the C-LSR-ns solution (\reffig{icethick}d, note the color scale).
394    Within the Canadian Archipelago, more drift leads to faster ice export
395    and reduced effective ice thickness. With a shorter time step of
396    $\Delta{t}_\mathrm{evp}=10\text{\,s}$ the EVP solution converges to
397    the LSOR solution (not shown). Only in the narrow straits in the
398    Archipelago the ice thickness is not affected by the shorter time step
399    and the ice is still thinner by 2\,m and more, as in the EVP solution
400    with $\Delta{t}_\mathrm{evp}=150\text{\,s}$.
401    
402    In year 2000, there is more ice everywhere in the domain in
403    C-LSR-ns~WTD (\reffig{icethick}g, note the color scale).
404    This difference, which is even more pronounced in summer (not shown),
405    can be attributed to direct effects of the different thermodynamics in
406    this run. The remaining runs have the largest differences in effective
407    ice thickness long the north coasts of Greenland and Ellesmere Island.
408    Although the effects of TEM and \citet{hibler87}'s ice-ocean stress
409    formulation are so different on the initial ice velocities, both runs
410    have similarly reduced ice thicknesses in this area. The 3rd-order
411    advection scheme has an opposite effect of similar magnitude, pointing
412    towards more implicit lateral stress with this numerical scheme.
413    
414    The observed difference of order 2\,m and less are smaller than the
415    differences that were observed between different hindcast models and climate
416    models in \citet{gerdes07}. There the range of sea ice volume of
417    different sea ice-ocean models (which shared very similar forcing
418    fields) was on the order of $10,000\text{km$^{3}$}$; this range was
419    even larger for coupled climate models. Here, the range (and the
420    averaging period) is smaller than $4,000\text{km$^{3}$}$ except for
421    the run \mbox{C-LSR-ns~WTD} where the more complicated thermodynamics
422    leads to generally thicker ice (\reffig{icethick} and
423    \reftab{icevolume}).
424    \begin{table}[t]
425      \begin{tabular}{lr@{\hspace{5ex}}r@{$\pm$}rr@{$\pm$}rr@{$\pm$}r}
426        model run & ice volume
427        & \multicolumn{6}{c}{ice transport [$\text{flux$\pm$ std.,
428            km$^{3}$\,y$^{-1}$}$]}\\
429        & [$\text{km$^{3}$}$]
430        & \multicolumn{2}{c}{FS}
431        & \multicolumn{2}{c}{NI}
432        & \multicolumn{2}{c}{LS} \\ \hline
433        B-LSR-ns       & 23,824 & 2126 & 1278 &   34 &  122 &   43 &   76 \\
434        C-LSR-ns       & 24,769 & 2196 & 1253 &   70 &  224 &   77 &  110 \\
435        C-LSR-fs       & 23,286 & 2236 & 1289 &   80 &  276 &   91 &   85 \\
436        C-EVP-ns       & 27,056 & 3050 & 1652 &  352 &  735 &  256 &  151 \\
437        C-EVP-ns10     & 22,633 & 2174 & 1260 &  186 &  496 &  133 &  128 \\
438        C-LSR-ns HB87  & 23,060 & 2256 & 1327 &   64 &  230 &   77 &  114 \\
439        C-LSR-ns TEM   & 23,529 & 2222 & 1258 &   60 &  242 &   87 &  112 \\
440        C-LSR-ns WTD   & 31,634 & 2761 & 1563 &   23 &  140 &   94 &   63 \\
441        C-LSR-ns DST3FL& 24,023 & 2191 & 1261 &   88 &  251 &   84 &  129
442      \end{tabular}
443      \caption{Arctic ice volume averaged over Jan--Mar 2000, in
444        $\text{km$^{3}$}$. Mean ice transport and standard deviation for the
445        period Jan 1992 -- Dec 1999 through the Fram Strait (FS), the
446        total northern inflow into the Canadian Archipelago (NI), and the
447        export through Lancaster Sound (LS), in $\text{km$^{3}$\,y$^{-1}$}$.
448      \label{tab:icevolume}}
449    \end{table}
450    
451    \subsubsection{Ice transports}
452    
453    The difference in ice volume and ice drift velocities between the
454    different experiments has consequences for the ice transport out of
455    the Arctic. Although by far the most exported ice drifts through the
456    Fram Strait (approximately $2300\pm610\text{\,km$^3$\,y$^{-1}$}$), a
457    considerable amount (order $160\text{\,km$^3$\,y$^{-1}$}$) ice is
458    exported through the Canadian Archipelago \citep[and references
459    therein]{serreze06}. Note, that ice transport estimates are associated
460    with large uncertainties; also note that tuning an Arctic sea
461    ice-ocean model to reproduce observations is not our goal, but we use
462    the published numbers as an orientation.
463    
464    \reffig{archipelago} shows an excerpt of a time series of daily
465    averaged ice transports, smoothed with a monthly running mean, through
466    various straits in the Canadian Archipelago and the Fram Strait for
467    the different model solutions; \reftab{icevolume} summarizes the
468    time series.
469  \begin{figure}  \begin{figure}
470  \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM2000heff}}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
471  \caption{Sea ice thickness for different solver flavors.  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
472  \label{fig:icethick}}  %\centerline{{\includegraphics[width=\linewidth]{\fpath/ice_export}}}
473    \centerline{{\includegraphics[width=\linewidth]{\fpath/ice_export1996}}}
474    \caption{Transport through Canadian Archipelago for different solver
475      flavors. The letters refer to the labels of the sections in
476      \reffig{arctic_topog}; positive values are flux out of the Arctic;
477      legend abbreviations are explained in \reftab{experiments}. The mean
478      range of the different model solution is taken over the period Jan
479      1992 to Dec 1999.
480    \label{fig:archipelago}}
481  \end{figure}  \end{figure}
482    The export through Fram Strait agrees with the observations in all
483    model solutions (annual averages range from $2110$ to
484    $2300\text{\,km$^3$\,y$^{-1}$}$, except for \mbox{C-LSR-ns~WTD} with
485    $2760\text{\,km$^3$\,y$^{-1}$}$ and the EVP solution with the long
486    time step of 150\,s with nearly $3000\text{\,km$^3$\,y$^{-1}$}$),
487    while the export through the Candian Archipelago is smaller than
488    generally thought. For example, the ice transport through Lancaster
489    Sound is lower (annual averages are $43$ to
490    $256\text{\,km$^3$\,y$^{-1}$}$) than in \citet{dey81} who estimates an
491    inflow into Baffin Bay of $370$ to $537\text{\,km$^3$\,y$^{-1}$}$, but
492    a flow of only $102$ to $137\text{\,km$^3$\,y$^{-1}$}$ further
493    upstream in Barrow Strait in the 1970ies from satellite images.
494    Generally, the EVP solutions have the highest maximum (export out of
495    the Artic) and lowest minimum (import into the Artic) fluxes as the
496    drift velocities are largest in these solutions.  In the extreme of
497    the Nares Strait, which is only a few grid points wide in our
498    configuration, both B- and C-grid LSOR solvers lead to practically no
499    ice transport, while the C-EVP solutions allow up to
500    $600\text{\,km$^3$\,y$^{-1}$}$ in summer (not shown); \citet{tang04}
501    report $300$ to $350\text{\,km$^3$\,y$^{-1}$}$.  As as consequence,
502    the import into the Candian Archipelago is larger in all EVP solutions
503    %(range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$)
504    than in the LSOR solutions.
505    %get the order of magnitude right (range: $132$ to
506    %$165\text{\,km$^3$\,y$^{-1}$}$);
507    The B-LSR-ns solution is even smaller by another factor of two than the
508    C-LSR solutions (an exception is the WTD solution, where larger ice thickness
509    tends to block the transport).
510    %underestimates the ice transport with $34\text{\,km$^3$\,y$^{-1}$}$.
511    
512    %\ml{[Transport to narrow straits, area?, more runs, TEM, advection
513    %  schemes, Winton TD, discussion about differences in terms of model
514    %  error? that's tricky as it means refering to Tremblay, thus our ice
515    %  models are all erroneous!]}
516    
517    \subsubsection{Discussion}
518    
519    In summary, we find that different dynamical solvers can yield very
520    different solutions. In constrast to that, the differences between
521    free-slip and no-slip solutions \emph{with the same solver} are
522    considerably smaller (the difference for the EVP solver is not shown,
523    but similar to that for the LSOR solver). Albeit smaller, the
524    differences between free and no-slip solutions in ice drift can lead
525    to equally large differences in ice volume, especially in the Canadian
526    Archipelago over the integration time. At first, this observation
527    seems counterintuitive, as we expect that the solution
528    \emph{technique} should not affect the \emph{solution} to a higher
529    degree than actually modifying the equations. A more detailed study on
530    these differences is beyond the scope of this paper, but at this point
531    we may speculate, that the large difference between B-grid, C-grid,
532    LSOR, and EVP solutions stem from incomplete convergence of the
533    solvers due to linearization and due to different methods of
534    linearization \citep[and Bruno Tremblay, personal
535    communication]{hunke01}: if the convergence of the non-linear momentum
536    equations is not complete for all linearized solvers, then one can
537    imagine that each solver stops at a different point in velocity-space
538    thus leading to different solutions for the ice drift velocities. If
539    this were true, this tantalizing circumstance would have a dramatic
540    impact on sea-ice modeling in general, and we would need to improve
541    the solution techniques for dynamic sea ice models, most likely at a very
542    high compuational cost (Bruno Tremblay, personal communication).
543    
544    Further, we observe that the EVP solutions tends to produce
545    effectively ``weaker'' ice that yields more easily to stress. This was
546    also observed by \citet{hunke99} in a fast response to changing winds,
547    their Figures\,10--12, where the EVP model adjusts quickly to a
548    cyclonic wind pattern, while the LSOR solution lags in time. This
549    property of the EVP solutions allows larger ice transports through
550    narrow straits, where the implicit solver LSOR forms rigid ice. The
551    underlying reasons for this striking difference need further
552    exploration.
553    
554    % THIS is now almost all in the text:
555    %\begin{itemize}
556    %\item Configuration
557    %\item OBCS from cube
558    %\item forcing
559    %\item 1/2 and full resolution
560    %\item with a few JFM figs from C-grid LSR no slip
561    %  ice transport through Canadian Archipelago
562    %  thickness distribution
563    %  ice velocity and transport
564    %\end{itemize}
565    
566    %\begin{itemize}
567    %\item Arctic configuration
568    %\item ice transport through straits and near boundaries
569    %\item focus on narrow straits in the Canadian Archipelago
570    %\end{itemize}
571    
572    %\begin{itemize}
573    %\item B-grid LSR no-slip: B-LSR-ns
574    %\item C-grid LSR no-slip: C-LSR-ns
575    %\item C-grid LSR slip:    C-LSR-fs
576    %\item C-grid EVP no-slip: C-EVP-ns
577    %\item C-grid EVP slip:    C-EVP-fs
578    %\item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
579    %  new flag): C-LSR-ns+TEM
580    %\item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
581    %\item C-grid LSR no-slip + Winton:
582    %\item  speed-performance-accuracy (small)
583    %  ice transport through Canadian Archipelago differences
584    %  thickness distribution differences
585    %  ice velocity and transport differences
586    %\end{itemize}
587    
588    %We anticipate small differences between the different models due to:
589    %\begin{itemize}
590    %\item advection schemes: along the ice-edge and regions with large
591    %  gradients
592    %\item C-grid: less transport through narrow straits for no slip
593    %  conditons, more for free slip
594    %\item VP vs.\ EVP: speed performance, accuracy?
595    %\item ocean stress: different water mass properties beneath the ice
596    %\end{itemize}
597    
598    %%% Local Variables:
599    %%% mode: latex
600    %%% TeX-master: "ceaice"
601    %%% End:
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