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revision 1.8 by dimitri, Fri Feb 29 01:28:05 2008 UTC revision 1.18 by mlosch, Thu Jul 3 18:16:22 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    
15    \subsection{Model configuration and experiments}
16    \label{sec:arcticmodel}
17    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  \subsection{Global Ocean and Sea Ice Simulation}  The global ocean and sea ice results presented in \citet{menemenlis05}
49  \label{sec:global}  were carried out as part
   
 The global ocean and sea ice results presented below were carried out as part  
50  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)
51  project.  ECCO2 aims to produce increasingly accurate syntheses of all  project.  ECCO2 aims to produce increasingly accurate syntheses of all
52  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
53  resolve ocean eddies and other narrow current systems, which transport heat,  resolve ocean eddies and other narrow current systems, which transport heat,
54  carbon, and other properties within the ocean \citep{menemenlis05}.  The  carbon, and other properties within the ocean \citep{menemenlis05}.  The
55  particular ECCO2 simulation discussed next is a baseline 28-year (1979-2006)  particular ECCO2 simulation from which we obtain the boundary
56    conditions is a baseline 28-year (1979-2006)
57  integration, labeled cube76, which has not yet been constrained by oceanic and  integration, labeled cube76, which has not yet been constrained by oceanic and
58  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
59  relatively even grid spacing throughout the domain and which avoids polar  relatively even grid spacing throughout the domain and which avoids polar
60  singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises  singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises
61  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
62  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
63  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
64  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  
65  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the
66  bathymetry. The model is integrated in a volume-conserving configuration using  bathymetry. Bathymetry is from the S2004 (W.~Smith, unpublished) blend of the
67    \citet{smi97} and the General Bathymetric Charts of the Oceans (GEBCO) one
68    arc-minute bathymetric grid. % (see Fig.~\ref{fig:CubeBathymetry}).
69    The model is integrated in a volume-conserving configuration using
70  a finite volume discretization with C-grid staggering of the prognostic  a finite volume discretization with C-grid staggering of the prognostic
71  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
72  used.  used.
73    %
74  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
75  Section~\ref{sec:model} using the following specific options.  The  configuration similar to the case C-LSR-ns (see \reftab{experiments}),
76  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,
77  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.  
78    
79  This particular ECCO2 simulation is initialized from temperature and salinity  This particular ECCO2 simulation is initialized from temperature and salinity
80  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 90  radiations, and precipitation are conver
90  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
91  radiation decays exponentially as per \citet{pau77}.  Low frequency  radiation decays exponentially as per \citet{pau77}.  Low frequency
92  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
93  Precipitation Climatology Project (GPCP) \citep{huf01}.  The time-mean river  Precipitation Climatology Project \citep[GPCP,][]{huf01}.  The time-mean river
94  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
95  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)
96  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 109  advection scheme \citep{dar04} is employ
109  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
110  the divergent flow as per \citet{kem08}.  the divergent flow as per \citet{kem08}.
111    
112  \subsection{Arctic Domain with Open Boundaries}  The model configuration of cube76 carries over to the Arctic domain
113  \label{sec:arctic}  configuration except for numerical details related to the non-linear
114    free surface that are not supported by the open boundary code, and the
115  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
116  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.
117  B-grid LSR dynamic solver with the new C-grid LSR and EVP solvers.  One  
118  additional experiment is carried out to illustrate the differences between the  The model is integrated from Jan~01, 1992 to Mar~31, 2000
119  two main options for sea ice thermodynamics in the MITgcm.  \reftab{experiments} gives an overview over the experiments discussed
120    in \refsec{arcticresults}.
121  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It  \begin{table}
122  is carved out from, and obtains open boundary conditions from, the global    \caption{Overview over model simulations in \refsec{arcticresults}.
123  cubed-sphere configuration described above.  The horizontal domain size is      \label{tab:experiments}}
124  420 by 384 grid boxes.    \begin{tabular}{p{.3\linewidth}p{.65\linewidth}}
125        experiment name & description \\ \hline
126  \begin{figure}      B-LSR-ns       &  the original LSOR solver of \citet{zhang97} on an
127  %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1}}}    Arakawa B-grid, implying no-slip lateral boundary conditions
128  \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}    ($\vek{u}=0$ exactly), advection of ice variables with a 2nd-order
129      central difference scheme plus explicit diffusion for stability \\
130        C-LSR-ns       &  the LSOR solver discretized on a C-grid with no-slip lateral
131      boundary conditions (implemented via ghost-points) \\
132        C-LSR-fs       &  the LSOR solver on a C-grid with free-slip lateral boundary
133      conditions \\
134        C-EVP-ns       &  the EVP solver of \citet{hunke01} on a C-grid with
135      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
136      150\text{\,s}$ \\
137        C-EVP-ns10     &  the EVP solver of \citet{hunke01} on a C-grid with
138      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
139      10\text{\,s}$ \\
140        C-LSR-ns HB87  &  C-LSR-ns with ocean-ice stress coupling according
141      to \citet{hibler87}\\
142        C-LSR-ns TEM   &  C-LSR-ns with a truncated ellispe method (TEM)
143      rheology \citep{hibler97} \\
144        C-LSR-ns WTD   &   C-LSR-ns with 3-layer thermodynamics following
145      \citet{winton00} \\
146        C-LSR-ns DST3FL& C-LSR-ns with a third-order flux limited
147      direct-space-time advection scheme for thermodynamic variables
148      \citep{hundsdorfer94}
149      \end{tabular}
150    \end{table}
151    %\begin{description}
152    %\item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an
153    %  Arakawa B-grid, implying no-slip lateral boundary conditions
154    %  ($\vek{u}=0$ exactly);
155    %\item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
156    %  boundary conditions (implemented via ghost-points);
157    %\item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
158    %  conditions;
159    %\item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
160    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
161    %  150\text{\,s}$;
162    %\item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
163    %  boundary conditions  and $\Delta{t}_\mathrm{evp} = 150\text{\,s}$;
164    %\item[C-LSR-ns DST3FL:] C-LSR-ns with a third-order flux limited
165    %  direct-space-time advection scheme \citep{hundsdorfer94};
166    %\item[C-LSR-ns TEM:] C-LSR-ns with a truncated ellispe method (TEM)
167    %  rheology \citep{hibler97};
168    %\item[C-LSR-ns HB87:] C-LSR-ns with ocean-ice stress coupling according
169    %  to \citet{hibler87};
170    %\item[C-LSR-ns WTD:] C-LSR-ns with 3-layer thermodynamics following
171    %  \citet{winton00};
172    %%\item[C-EVP-ns damp:] C-EVP-ns with additional damping to reduce small
173    %%  scale noise \citep{hunke01};
174    %\item[C-EVP-ns10:] the EVP solver of \citet{hunke01} on a C-grid with
175    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
176    %  10\text{\,s}$.
177    %\end{description}
178    Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
179    that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
180    interpreted as pure model error. Lateral boundary conditions on a
181    coarse grid (coarse compared to the roughness of the true coast line) are
182    unclear, so that comparing the no-slip solutions to the free-slip
183    solutions gives another measure of uncertainty in sea ice modeling.
184    The remaining experiments explore further sensitivities of the system
185    to different physics (change in rheology, advection and diffusion
186    properties, stress coupling, and thermodynamics) and different time
187    steps for the EVP solutions: \citet{hunke01} uses 120 subcycling steps
188    for the EVP solution. We use two interpretations of this choice where
189    the EVP model is subcycled 120 times within a (short) model timestep
190    of 1200\,s resulting in a very long and expensive integration
191    ($\Delta{t}_\mathrm{evp}=10\text{\,s}$) and 120 times within the
192    forcing timescale of 6\,h ($\Delta{t}_\mathrm{evp}=150\text{\,s}$).
193    
194    \subsection{Results}
195    \label{sec:arcticresults}
196    
197    Comparing the solutions obtained with different realizations of the
198    model dynamics is difficult because of the non-linear feedback of the
199    ice dynamics and thermodynamics. Already after a few months the
200    solutions have diverged so far from each other that comparing
201    velocities only makes sense within the first 3~months of the
202    integration while the ice distribution is still close to the initial
203    conditions. At the end of the integration, the differences between the
204    model solutions can be interpreted as cumulated model uncertainties.
205    
206    \subsubsection{Ice velocities in JFM 1992}
207    
208    \reffig{iceveloc} shows ice velocities averaged over Janunary,
209    February, and March (JFM) of 1992 for the C-LSR-ns solution; also
210    shown are the differences between B-grid and C-grid, LSR and EVP, and
211    no-slip and free-slip solution. The velocity field of the C-LSR-ns
212    solution (\reffig{iceveloc}a) roughly resembles the drift velocities
213    of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
214    models in a cyclonic circulation regime (CCR) \citep[their
215    Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
216    shifted eastwards towards Alaska.
217    
218    \newcommand{\subplotwidth}{0.44\textwidth}
219    %\newcommand{\subplotwidth}{0.3\textwidth}
220    \begin{figure}[tp]
221      \centering
222      \subfigure[{\footnotesize C-LSR-ns}]
223      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-LSR-ns}}
224      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
225      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_B-LSR-ns-C-LSR-ns}}
226      \\
227      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
228      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-LSR-fs-C-LSR-ns}}
229      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
230      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-EVP-ns150-C-LSR-ns}}
231    %  \\
232    %  \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
233    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_TEM-C-LSR-ns}}
234    %  \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
235    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_HB87-C-LSR-ns}}
236    %  \\
237    %  \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
238    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_ThSIce-C-LSR-ns}}
239    %  \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
240    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_adv33-C-LSR-ns}}
241      \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
242        over the first 3 months of integration [cm/s]; (b)-(h) difference
243        between solutions with B-grid, free lateral slip, EVP-solver,
244        truncated ellipse method (TEM), different ice-ocean stress
245        formulation (HB87), different thermodynamics (WTD), different
246        advection for thermodynamic variables (DST3FL) and the C-LSR-ns
247        reference solution [cm/s]; color indicates speed (or differences
248        of speed), vectors indicate direction only.}
249      \label{fig:iceveloc}
250  \end{figure}  \end{figure}
251    \addtocounter{figure}{-1}
252  Difference from cube sphere is that it does not use z* coordinates nor  \setcounter{subfigure}{4}
253  realfreshwater fluxes because it is not supported by open boundary code.  \begin{figure}[t]
254      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
255  Open water, dry    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_TEM-C-LSR-ns}}
256  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,    \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
257  0.76, 0.94, and 0.8.    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_HB87-C-LSR-ns}}
258      \\
259  \begin{itemize}    \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
260  \item Configuration    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_ThSIce-C-LSR-ns}}
261  \item OBCS from cube    \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
262  \item forcing    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_adv33-C-LSR-ns}}
263  \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}}  
264  \end{figure}  \end{figure}
265    The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
266  \begin{figure}  is most pronounced along the coastlines, where the discretization
267  \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}  differs most between B and C-grids: On a B-grid the tangential
268  \caption{Transport through Canadian Archipelago for different solver flavors.  velocity lies on the boundary (and is thus zero through the no-slip
269  \label{fig:archipelago}}  boundary conditions), whereas on the C-grid it is half a cell width
270    away from the boundary, thus allowing more flow. The B-LSR-ns solution
271    has less ice drift through the Fram Strait and especially the along
272    Greenland's east coast; also, the flow through Baffin Bay and Davis
273    Strait into the Labrador Sea is reduced with respect the C-LSR-ns
274    solution.  \ml{[Do we expect this? Say something about that]}
275    %
276    Compared to the differences between B and C-grid solutions, the
277    C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
278    (\reffig{iceveloc}c).  As expected the differences are largest along
279    coastlines: because of the free-slip boundary conditions, flow is
280    faster in the C-LSR-fs solution, for example, along the east coast
281    of Greenland, the north coast of Alaska, and the east Coast of Baffin
282    Island.
283    
284    The C-EVP-ns solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$ is
285    very different from the C-LSR-ns solution (\reffig{iceveloc}d). The
286    EVP-approximation of the VP-dynamics allows for increased drift by
287    over 2\,cm/s in the Beaufort Gyre and the transarctic drift.
288    %\ml{Also the Beaufort Gyre is moved towards Alaska in the C-EVP-ns
289    %solution. [Really?, No]}
290    In general, drift velocities are biased towards higher values in the
291    EVP solutions.
292    % as can be seen from a histogram of the differences in
293    %\reffig{drifthist}.
294    %\begin{figure}[htbp]
295    %  \centering
296    %  \includegraphics[width=\textwidth]{\fpath/drifthist_C-EVP-ns-C-LSR-ns}
297    %  \caption{Histogram of drift velocity differences for C-LSR-ns and
298    %    C-EVP-ns solution [cm/s].}
299    %  \label{fig:drifthist}
300    %\end{figure}
301    
302    Compared to the other parameters, the ice rheology TEM
303    (\reffig{iceveloc}(e)) has a very small effect on the solution. In
304    general the ice drift tends to be increased, because there is no
305    tensile stress and ice can be ``pulled appart'' at no cost.
306    Consequently, the largest effect on drift velocity can be observed
307    near the ice edge in the Labrador Sea. In contrast, in the run with
308    the ice-ocean stress formulation of \citet{hibler87},
309    \reffig{iceveloc}(f) the drift is stronger almost everywhere in the
310    computational domain. The increase is mostly aligned with the general
311    direction of the flow, implying that the different stress formulation
312    reduces the deceleration of drift by the ocean.
313    
314    The 3-layer thermodynamics following \citet{winton00} requires
315    additional information on initial conditions for enthalphy. These
316    different initial conditions make a comparison of the first months
317    difficult to interpret. The drift in the Beaufort Gyre is slightly
318    reduced relative to the reference run C-LSR-ns, but the drift through
319    the Fram Strait is increased. The drift velocities near the ice edge
320    are very different, because the ice extend is already larger in
321    \mbox{C-LSR-ns~WTD}; inward from the ice egde, this run has smaller
322    drift velocities, because the ice motion is more contrained by a
323    larger ice extent than in \mbox{C-LSR-ns}, where the ice at the same
324    place is drifting nearly freely.
325    
326    A more sophisticated advection scheme (\mbox{C-LSR-ns DST3FL},
327    \reffig{iceveloc}(h)) has its largest effect along the ice edge, where
328    the gradients of thickness and concentration are largest. Everywhere
329    else the effect is very small and can mostly be attributed to smaller
330    numerical diffusion (and to the absense of explicitly diffusion for
331    numerical stability).
332    
333    \subsubsection{Ice volume during JFM 2000}
334    
335    \reffig{icethick}a shows the effective thickness (volume per unit
336    area) of the C-LSR-ns solution, averaged over January, February, March
337    of year 2000. By this time of the integration, the differences in the
338    ice drift velocities have led to the evolution of very different ice
339    thickness distributions, which are shown in \reffig{icethick}b--d, and
340    concentrations (not shown).
341    \begin{figure}[tp]
342      \centering
343      \subfigure[{\footnotesize C-LSR-ns}]
344      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-ns}}
345      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
346      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_B-LSR-ns-C-LSR-ns}}
347      \\
348      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
349      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-fs-C-LSR-ns}}
350      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
351      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-EVP-ns150-C-LSR-ns}}
352      \caption{(a) Effective thickness (volume per unit area) of the
353        C-LSR-ns solution, averaged over the months Janurary through March
354        2000 [m]; (b)-(h) difference between solutions with B-grid, free
355        lateral slip, EVP-solver, truncated ellipse method (TEM),
356        different ice-ocean stress formulation (HB87), different
357        thermodynamics (WTD), different advection for thermodynamic
358        variables (DST3FL) and the C-LSR-ns reference solution [m].}
359      \label{fig:icethick}
360  \end{figure}  \end{figure}
361    \addtocounter{figure}{-1}
362    \setcounter{subfigure}{4}
363    \begin{figure}[t]
364      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
365      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_TEM-C-LSR-ns}}
366      \subfigure[{\footnotesize C-EVP-ns HB87 $-$ C-LSR-ns}]
367      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_HB87-C-LSR-ns}}
368      \\
369      \subfigure[{\footnotesize C-LSR-ns WTD $-$ C-LSR-ns}]
370      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_ThSIce-C-LSR-ns}}
371      \subfigure[{\footnotesize C-EVP-ns DST3FL $-$ C-LSR-ns}]
372      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_adv33-C-LSR-ns}}
373      \caption{continued}
374    \end{figure}
375    The generally weaker ice drift velocities in the B-LSR-ns solution,
376    when compared to the C-LSR-ns solution, in particular through the
377    narrow passages in the Canadian Archipelago, lead to a larger build-up
378    of ice north of Greenland and the Archipelago by 2\,m effective
379    thickness and more in the B-grid solution (\reffig{icethick}b). But
380    the ice volume in not larger everywhere: further west, there are
381    patches of smaller ice volume in the B-grid solution, most likely
382    because the Beaufort Gyre is weaker and hence not as effective in
383    transporting ice westwards. There are also dipoles of ice volume
384    differences with more ice on the upstream side of island groups and
385    less ice in their lee, such as Franz-Josef-Land and
386    Severnaya Semlya\ml{/or Nordland?},
387    because ice tends to flow along coasts less easily in the B-LSR-ns
388    solution.
389    
390    Imposing a free-slip boundary condition in C-LSR-fs leads to a much
391    smaller differences to C-LSR-ns in the central Arctic than the
392    transition from the B-grid to the C-grid (\reffig{icethick}c), except
393    in the Canadian Archipelago. There it reduces the effective ice
394    thickness by 2\,m and more where the ice is thick and the straits are
395    narrow.  Dipoles of ice thickness differences can also be observed
396    around islands, because the free-slip solution allows more flow around
397    islands than the no-slip solution. Everywhere else the ice volume is
398    affected only slightly by the different boundary condition.
399    %
400    The C-EVP-ns solution has generally stronger drift velocities than the
401    C-LSR-ns solution. Consequently, more ice can be moved from the
402    eastern part of the Arctic, where ice volumes are smaller, to the
403    western Arctic (\reffig{icethick}d). Within the Canadian Archipelago,
404    more drift leads to faster ice export and reduced effective ice
405    thickness. With a shorter time step of
406    $\Delta{t}_\mathrm{evp}=10\text{\,s}$ the EVP solution seems to
407    converge to the LSOR solution (not shown). Only in the narrow straits
408    in the Archipelago the ice thickness is not affected by the shorter
409    time step and the ice is still thinner by 2\,m and more, as in the EVP
410    solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$.
411    
412    In year 2000, there is more ice everywhere in the domain in
413    \mbox{C-LSR-ns WTD}, \reffig{icethick}(g). This difference, which is
414    even more pronounced in summer (not shown), can be attributed to
415    direct effects of the different thermodynamics in this run. The
416    remaining runs have the largest differences in effective ice thickness
417    long the north coasts of Greenland and Ellesmere Island. Although the
418    effects of TEM and \citet{hibler87}'s ice-ocean stress formulation are
419    so different on the initial ice velocities, both runs have similarly
420    reduced ice thicknesses in this area. The 3rd-order advection scheme
421    has an opposite effect of similar magnitude, point towards more
422    implicit lateral stress with this numerical scheme.
423    
424    The observed difference of order 2\,m and less are smaller than the
425    differences that were observed between different hindcast models and climate
426    models in \citet{gerdes07}. There the range of sea ice volume of
427    different sea ice-ocean models (which shared very similar forcing
428    fields) was on the order of $10,000\text{km$^{3}$}$; this range was
429    even larger for coupled climate models. Here, the range (and the
430    averaging period) is smaller than $4,000\text{km$^{3}$}$ except for
431    the run \mbox{C-LSR-ns~WTD} where the more complicated thermodynamics
432    leads to generally thicker ice (\reffig{icethick} and
433    \reftab{icevolume}).
434    \begin{table}[htbp]
435      \begin{tabular}{lr@{\hspace{5ex}}r@{$\pm$}rr@{$\pm$}rr@{$\pm$}r}
436        model run & ice volume
437        & \multicolumn{6}{c}{ice transport [$\text{flux$\pm$ std.,
438            km$^{3}$\,y$^{-1}$}$]}\\
439        & [$\text{km$^{3}$}$]
440        & \multicolumn{2}{c}{FS}
441        & \multicolumn{2}{c}{NI}
442        & \multicolumn{2}{c}{LS} \\ \hline
443        B-LSR-ns       & 23,824 & 2126 & 1278 &   34 &  122 &   43 &   76 \\
444        C-LSR-ns       & 24,769 & 2196 & 1253 &   70 &  224 &   77 &  110 \\
445        C-LSR-fs       & 23,286 & 2236 & 1289 &   80 &  276 &   91 &   85 \\
446        C-EVP-ns       & 27,056 & 3050 & 1652 &  352 &  735 &  256 &  151 \\
447        C-EVP-ns10     & 22,633 & 2174 & 1260 &  186 &  496 &  133 &  128 \\
448        C-LSR-ns HB87  & 23,060 & 2256 & 1327 &   64 &  230 &   77 &  114 \\
449        C-LSR-ns TEM   & 23,529 & 2222 & 1258 &   60 &  242 &   87 &  112 \\
450        C-LSR-ns WTD   & 31,634 & 2761 & 1563 &   23 &  140 &   94 &   63 \\
451        C-LSR-ns DST3FL& 24,023 & 2191 & 1261 &   88 &  251 &   84 &  129
452      \end{tabular}
453      \caption{Arctic ice volume averaged over Jan--Mar 2000, in
454        $\text{km$^{3}$}$. Mean ice transport and standard deviation for the
455        period Jan 1992 -- Dec 1999 through the Fram Strait (FS), the
456        total northern inflow into the Canadian Archipelago (NI), and the
457        export through Lancaster Sound (LS), in $\text{km$^{3}$\,y$^{-1}$}$.}
458      \label{tab:icevolume}
459    \end{table}
460    
461    The difference in ice volume and ice drift velocities between the
462    different experiments has consequences for the ice transport out of
463    the Arctic. Although by far the most exported ice drifts through the
464    Fram Strait (approximately $2300\pm610\text{\,km$^3$\,y$^{-1}$}$), a
465    considerable amount (order $160\text{\,km$^3$\,y$^{-1}$}$) ice is
466    exported through the Canadian Archipelago \citep[and references
467    therein]{serreze06}. Note, that ice transport estimates are associated
468    with large uncertainties; also note that tuning an Arctic sea
469    ice-ocean model to reproduce observations is not our goal, but we use
470    the published numbers as an orientation.
471    
472    \subsubsection{Ice transports}
473    
474    \reffig{archipelago} shows an excerpt of a time series of daily
475    averaged, smoothed with monthly running means, ice transports through
476    various straits in the Canadian Archipelago and the Fram Strait for
477    the different model solutions and \reftab{icevolume} summarizes the
478    time series.
479  \begin{figure}  \begin{figure}
480  \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM2000heff}}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
481  \caption{Sea ice thickness for different solver flavors.  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
482  \label{fig:icethick}}  %\centerline{{\includegraphics[width=\linewidth]{\fpath/ice_export}}}
483    \centerline{{\includegraphics[width=\linewidth]{\fpath/ice_export1996}}}
484    \caption{Transport through Canadian Archipelago for different solver
485      flavors. The letters refer to the labels of the sections in
486      \reffig{arctic_topog}; positive values are flux out of the Arctic;
487      legend abbreviations are explained in \reftab{experiments}. The mean
488      range of the different model solution is taken over the period Jan
489      1992 to Dec 1999.
490    \label{fig:archipelago}}
491  \end{figure}  \end{figure}
492    The export through Fram Strait agrees with the observations in all
493    model solutions (annual averages range from $2110$ to
494    $2300\text{\,km$^3$\,y$^{-1}$}$, except for \mbox{C-LSR-ns~WTD} with
495    $2760\text{\,km$^3$\,y$^{-1}$}$ and the EVP solution with the long
496    time step of 150\,s with nearly $3000\text{\,km$^3$\,y$^{-1}$}$),
497    while the export through the Candian Archipelago is smaller than
498    generally thought. For example, the ice transport through Lancaster
499    Sound is lower (annual averages are $43$ to
500    $256\text{\,km$^3$\,y$^{-1}$}$) than in \citet{dey81} who estimates an
501    inflow into Baffin Bay of $370$ to $537\text{\,km$^3$\,y$^{-1}$}$, but
502    a flow of only $102$ to $137\text{\,km$^3$\,y$^{-1}$}$ further
503    upstream in Barrow Strait in the 1970ies from satellite images.
504    Generally, the EVP solutions have the highest maximum (export out of
505    the Artic) and lowest minimum (import into the Artic) fluxes as the
506    drift velocities are largest in these solutions.  In the extreme of
507    the Nares Strait, which is only a few grid points wide in our
508    configuration, both B- and C-grid LSOR solvers lead to practically no
509    ice transport, while the C-EVP solutions allow up to
510    $600\text{\,km$^3$\,y$^{-1}$}$ in summer (not shown); \citet{tang04}
511    report $300$ to $350\text{\,km$^3$\,y$^{-1}$}$.  As as consequence,
512    the import into the Candian Archipelago is larger in all EVP solutions
513    %(range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$)
514    than in the LSOR solutions.
515    %get the order of magnitude right (range: $132$ to
516    %$165\text{\,km$^3$\,y$^{-1}$}$);
517    The B-LSR-ns solution is even smaller by another factor of two than the
518    C-LSR solutions (an exception is the WTD solution, where larger ice thickness
519    tends to block the transport).
520    %underestimates the ice transport with $34\text{\,km$^3$\,y$^{-1}$}$.
521    
522    %\ml{[Transport to narrow straits, area?, more runs, TEM, advection
523    %  schemes, Winton TD, discussion about differences in terms of model
524    %  error? that's tricky as it means refering to Tremblay, thus our ice
525    %  models are all erroneous!]}
526    
527    \subsubsection{Discussion}
528    
529    In summary, we find that different dynamical solvers can yield very
530    different solutions. In constrast to that, the differences between
531    free-slip and no-slip solutions \emph{with the same solver} are
532    considerably smaller (the difference for the EVP solver is not shown,
533    but similar to that for the LSOR solver). Albeit smaller, the
534    differences between free and no-slip solutions in ice drift can lead
535    to equally large differences in ice volume, especially in the Canadian
536    Archipelago over the integration time. At first, this observation
537    seems counterintuitive, as we expect that the solution
538    \emph{technique} should not affect the \emph{solution} to a higher
539    degree than actually modifying the equations. A more detailed study on
540    these differences is beyond the scope of this paper, but at this point
541    we may speculate, that the large difference between B-grid, C-grid,
542    LSOR, and EVP solutions stem from incomplete convergence of the
543    solvers due to linearization and due to different methods of
544    linearization \citep[and Bruno Tremblay, personal
545    communication]{hunke01}: if the convergence of the non-linear momentum
546    equations is not complete for all linearized solvers, then one can
547    imagine that each solver stops at a different point in velocity-space
548    thus leading to different solutions for the ice drift velocities. If
549    this were true, this tantalizing circumstance would have a dramatic
550    impact on sea-ice modeling in general, and we would need to improve
551    the solution techniques for dynamic sea ice models, most likely at a very
552    high compuational cost (Bruno Tremblay, personal communication). Further,
553    we observe that the EVP solutions tends to produce effectively
554    ``weaker'' ice that yields more easily to stress. The fast response to
555    changing wind was also observed by \citet{hunke99}, their Fig.\,10--12,
556    where the EVP model adjusts quickly to a cyclonic wind pattern, while
557    the LSOR solution does not. This property of the EVP solutions allows
558    larger ice transports through narrow straits, where the implicit
559    solver LSOR forms rigid ice. The underlying reasons for this striking
560    difference need further exploration.
561    
562    % THIS is now almost all in the text:
563    %\begin{itemize}
564    %\item Configuration
565    %\item OBCS from cube
566    %\item forcing
567    %\item 1/2 and full resolution
568    %\item with a few JFM figs from C-grid LSR no slip
569    %  ice transport through Canadian Archipelago
570    %  thickness distribution
571    %  ice velocity and transport
572    %\end{itemize}
573    
574    %\begin{itemize}
575    %\item Arctic configuration
576    %\item ice transport through straits and near boundaries
577    %\item focus on narrow straits in the Canadian Archipelago
578    %\end{itemize}
579    
580    %\begin{itemize}
581    %\item B-grid LSR no-slip: B-LSR-ns
582    %\item C-grid LSR no-slip: C-LSR-ns
583    %\item C-grid LSR slip:    C-LSR-fs
584    %\item C-grid EVP no-slip: C-EVP-ns
585    %\item C-grid EVP slip:    C-EVP-fs
586    %\item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
587    %  new flag): C-LSR-ns+TEM
588    %\item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
589    %\item C-grid LSR no-slip + Winton:
590    %\item  speed-performance-accuracy (small)
591    %  ice transport through Canadian Archipelago differences
592    %  thickness distribution differences
593    %  ice velocity and transport differences
594    %\end{itemize}
595    
596    %We anticipate small differences between the different models due to:
597    %\begin{itemize}
598    %\item advection schemes: along the ice-edge and regions with large
599    %  gradients
600    %\item C-grid: less transport through narrow straits for no slip
601    %  conditons, more for free slip
602    %\item VP vs.\ EVP: speed performance, accuracy?
603    %\item ocean stress: different water mass properties beneath the ice
604    %\end{itemize}
605    
606    %%% Local Variables:
607    %%% mode: latex
608    %%% TeX-master: "ceaice"
609    %%% End:
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