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revision 1.8 by dimitri, Fri Feb 29 01:28:05 2008 UTC revision 1.15 by mlosch, Mon Jun 2 13:25:40 2008 UTC
# Line 6  ice simulations that exercise various ca Line 6  ice simulations that exercise various ca
6  model.  The first set of results is from a global, eddy-permitting, ocean and  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  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  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  and various other capabilities of the MITgcm sea ice model.  
10    %
11    \ml{[do we really want to do this?:] The third set of
12  results is from a yet smaller regional domain, which is used to illustrate  results is from a yet smaller regional domain, which is used to illustrate
13  treatment of sea ice open boundary condition sin the MITgcm.  treatment of sea ice open boundary condition in the MITgcm.}
14    
15  \subsection{Global Ocean and Sea Ice Simulation}  \subsection{Global Ocean and Sea Ice Simulation}
16  \label{sec:global}  \label{sec:global}
# Line 36  variables. In the ocean, the non-linear Line 38  variables. In the ocean, the non-linear
38  used.  used.
39    
40  The ocean model is coupled to the sea-ice model discussed in  The ocean model is coupled to the sea-ice model discussed in
41  Section~\ref{sec:model} using the following specific options.  The  \refsec{model} using the following specific options.  The
42  zero-heat-capacity thermodynamics formulation of \citet{hib80} is used to  zero-heat-capacity thermodynamics formulation of \citet{hibler80} is
43  compute sea ice thickness and concentration.  Snow cover and sea ice salinity  used to compute sea ice thickness and concentration.  Snow cover and
44  are prognostic.  Open water, dry ice, wet ice, dry snow, and wet snow albedo  sea ice salinity are prognostic.  Open water, dry ice, wet ice, dry
45  are, respectively, 0.15, 0.88, 0.79, 0.97, and 0.83. Ice mechanics follow the  snow, and wet snow albedo are, respectively, 0.15, 0.88, 0.79, 0.97,
46  viscous plastic rheology of \citet{hibler79} and the ice momentum equation is  and 0.83. Ice mechanics follow the viscous plastic rheology of
47  solved numerically using the C-grid implementation of the \citet{zhang97} LSR  \citet{hibler79} and the ice momentum equation is solved numerically
48  dynamics model discussed hereinabove.  The ice is coupled to the ocean using  using the C-grid implementation of the \citet{zhang97}'s LSOR dynamics
49  the rescaled vertical coordinate system, z$^\ast$, of  model discussed hereinabove.  The ice is coupled to the ocean using
50  \citet{cam08}, that is, sea ice does not float above the ocean model but  the rescaled vertical coordinate system, z$^\ast$, of \citet{cam08},
51  rather deforms the ocean's model surface level.  that is, sea ice does not float above the ocean model but rather
52    deforms the ocean's model surface level.
53    
54  This particular ECCO2 simulation is initialized from temperature and salinity  This particular ECCO2 simulation is initialized from temperature and salinity
55  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 65  radiations, and precipitation are conver
65  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
66  radiation decays exponentially as per \citet{pau77}.  Low frequency  radiation decays exponentially as per \citet{pau77}.  Low frequency
67  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
68  Precipitation Climatology Project (GPCP) \citep{huf01}.  The time-mean river  Precipitation Climatology Project \citep[GPCP][]{huf01}.  The time-mean river
69  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
70  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)
71  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 84  advection scheme \citep{dar04} is employ
84  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
85  the divergent flow as per \citet{kem08}.  the divergent flow as per \citet{kem08}.
86    
87    \ml{[Dimitris, here you need to either provide figures, so that I can
88      write text, or you can provide both figures and text. I guess, one
89      figure, showing the northern and southern hemisphere in summer and
90      winter is fine (four panels), as we are showing so many figures in
91      the next section.]}
92    
93    
94  \subsection{Arctic Domain with Open Boundaries}  \subsection{Arctic Domain with Open Boundaries}
95  \label{sec:arctic}  \label{sec:arctic}
96    
97  A series of forward sensitivity experiments have been carried out on an  A series of forward sensitivity experiments have been carried out on
98  Arctic Ocean domain with open boundaries.  The objective is to compare the old  an Arctic Ocean domain with open boundaries.  The objective is to
99  B-grid LSR dynamic solver with the new C-grid LSR and EVP solvers.  One  compare the old B-grid LSR dynamic solver with the new C-grid LSR and
100  additional experiment is carried out to illustrate the differences between the  EVP solvers.  Additional experiments are carried out to illustrate
101  two main options for sea ice thermodynamics in the MITgcm.  the differences between different ice advection schemes, ocean-ice
102    stress formulations and the two main options for sea ice
103  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It  thermodynamics in the MITgcm.
104  is carved out from, and obtains open boundary conditions from, the global  
105  cubed-sphere configuration described above.  The horizontal domain size is  The Arctic domain of integration is illustrated in
106  420 by 384 grid boxes.  \reffig{arctic_topog}.  It is carved out from, and obtains open
107    boundary conditions from, the global cubed-sphere configuration
108  \begin{figure}  described above.  The horizontal domain size is 420 by 384 grid boxes.
109  %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1}}}  \begin{figure*}
110  \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}  %\includegraphics*[width=0.44\linewidth,viewport=139 210 496 606,clip]{\fpath/topography}
111    %\includegraphics*[width=0.44\linewidth,viewport=0 0 496 606,clip]{\fpath/topography}
112    \includegraphics*[width=0.44\linewidth]{\fpath/topography}
113    \includegraphics*[width=0.46\linewidth]{\fpath/archipelago}
114    \caption{Left: Bathymetry and domain boudaries of Arctic
115      Domain; the dashed line marks the boundaries of the inset on the
116      right hand side. The letters in the inset label sections in the
117      Canadian Archipelago, where ice transport is evaluated:
118      A: Nares Strait; %
119      B: \ml{Meighen Island}; %
120      C: Prince Gustaf Adolf Sea; %
121      D: \ml{Brock Island}; %
122      E: M'Clure Strait; %
123      F: Amundsen Gulf; %
124      G: Lancaster Sound; %
125      H: Barrow Strait \ml{W.}; %
126      I: Barrow Strait \ml{E.}; %
127      J: Barrow Strait \ml{N.}. %
128      The sections A through F comprise the total inflow into the Canadian
129      Archipelago. \ml{[May still need to check the geography.]}
130      \label{fig:arctic_topog}}
131    \end{figure*}
132    
133    The main dynamic difference from cube sphere is that the Arctic domain
134    configuration does not use rescaled vertical coordinates (z$^\ast$)
135    and the surface boundary conditions for freshwater input are
136    different, because those features are not supported by the open
137    boundary code.
138    %
139    Open water, dry ice, wet ice, dry snow, and wet snow albedo are,
140    respectively, 0.15, 0.85, 0.76, 0.94, and 0.8.
141    
142    The model is integrated from Jan~01, 1992 to Mar~31, 2000,
143    with three different dynamical solvers, two different boundary
144    conditions, different stress coupling, rheology, and advection
145    schemes. \reftab{experiments} gives an overview over the experiments
146    discussed in this section.
147    \begin{table}[htbp]
148      \begin{tabular}{p{.3\linewidth}p{.65\linewidth}}
149        experiment name & description \\ \hline
150        B-LSR-ns       &  the original LSOR solver of \citet{zhang97} on an
151      Arakawa B-grid, implying no-slip lateral boundary conditions
152      ($\vek{u}=0$ exactly) \\
153        C-LSR-ns       &  the LSOR solver discretized on a C-grid with no-slip lateral
154      boundary conditions (implemented via ghost-points) \\
155        C-LSR-fs       &  the LSOR solver on a C-grid with free-slip lateral boundary
156      conditions \\
157        C-EVP-ns       &  the EVP solver of \citet{hunke01} on a C-grid with
158      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
159      150\text{\,s}$ \\
160        C-EVP-ns10     &  the EVP solver of \citet{hunke01} on a C-grid with
161      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
162      10\text{\,s}$ \\
163        C-LSR-ns HB87  &  C-LSR-ns with ocean-ice stress coupling according
164      to \citet{hibler87}\\
165        C-LSR-ns TEM   &  C-LSR-ns with a truncated ellispe method (TEM)
166      rheology \citep{hibler97} \\
167        C-LSR-ns WTD   &   C-LSR-ns with 3-layer thermodynamics following
168      \citet{winton00} \\
169        C-LSR-ns DST3FL& C-LSR-ns with a third-order flux limited
170      direct-space-time advection scheme for thermodynamic variables
171      \citep{hundsdorfer94}
172      \end{tabular}
173      \caption{Overview over model simulations in \refsec{arctic}.
174        \label{tab:experiments}}
175    \end{table}
176    %\begin{description}
177    %\item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an
178    %  Arakawa B-grid, implying no-slip lateral boundary conditions
179    %  ($\vek{u}=0$ exactly);
180    %\item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
181    %  boundary conditions (implemented via ghost-points);
182    %\item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
183    %  conditions;
184    %\item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
185    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
186    %  150\text{\,s}$;
187    %\item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
188    %  boundary conditions  and $\Delta{t}_\mathrm{evp} = 150\text{\,s}$;
189    %\item[C-LSR-ns DST3FL:] C-LSR-ns with a third-order flux limited
190    %  direct-space-time advection scheme \citep{hundsdorfer94};
191    %\item[C-LSR-ns TEM:] C-LSR-ns with a truncated ellispe method (TEM)
192    %  rheology \citep{hibler97};
193    %\item[C-LSR-ns HB87:] C-LSR-ns with ocean-ice stress coupling according
194    %  to \citet{hibler87};
195    %\item[C-LSR-ns WTD:] C-LSR-ns with 3-layer thermodynamics following
196    %  \citet{winton00};
197    %%\item[C-EVP-ns damp:] C-EVP-ns with additional damping to reduce small
198    %%  scale noise \citep{hunke01};
199    %\item[C-EVP-ns10:] the EVP solver of \citet{hunke01} on a C-grid with
200    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
201    %  10\text{\,s}$.
202    %\end{description}
203    Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
204    that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
205    interpreted as pure model error. Lateral boundary conditions on a
206    coarse grid (coarse compared to the roughness of the true coast line) are
207    unclear, so that comparing the no-slip solutions to the free-slip
208    solutions gives another measure of uncertainty in sea ice modeling.
209    The remaining experiments explore further sensitivities of the system
210    to different physics (change in rheology, advection and diffusion
211    properties, stress coupling, and thermodynamics) and different time
212    steps for the EVP solutions: \citet{hunke01} uses 120 subcycling steps
213    for the EVP solution. We use two interpretations of this choice where
214    the EVP model is subcycled 120 times within a (short) model timestep
215    of 1200\,s resulting in a very long and expensive integration
216    ($\Delta{t}_\mathrm{evp}=10\text{\,s}$) and 120 times within the
217    forcing timescale of 6\,h ($\Delta{t}_\mathrm{evp}=150\text{\,s}$).
218    
219    A principle difficulty in comparing the solutions obtained with
220    different realizations of the model dynamics lies in the non-linear
221    feedback of the ice dynamics and thermodynamics. Already after a few
222    months the solutions have diverged so far from each other that
223    comparing velocities only makes sense within the first 3~months of the
224    integration while the ice distribution is still close to the initial
225    conditions. At the end of the integration, the differences between the
226    model solutions can be interpreted as cumulated model uncertainties.
227    
228    \reffig{iceveloc} shows ice velocities averaged over Janunary,
229    February, and March (JFM) of 1992 for the C-LSR-ns solution; also
230    shown are the differences between B-grid and C-grid, LSR and EVP, and
231    no-slip and free-slip solution. The velocity field of the C-LSR-ns
232    solution (\reffig{iceveloc}a) roughly resembles the drift velocities
233    of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
234    models in a cyclonic circulation regime (CCR) \citep[their
235    Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
236    shifted eastwards towards Alaska.
237    
238    The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
239    is most pronounced along the coastlines, where the discretization
240    differs most between B and C-grids: On a B-grid the tangential
241    velocity lies on the boundary (and is thus zero through the no-slip
242    boundary conditions), whereas on the C-grid it is half a cell width
243    away from the boundary, thus allowing more flow. The B-LSR-ns solution
244    has less ice drift through the Fram Strait and especially the along
245    Greenland's east coast; also, the flow through Baffin Bay and Davis
246    Strait into the Labrador Sea is reduced with respect the C-LSR-ns
247    solution.  \ml{[Do we expect this? Say something about that]}
248    %
249    Compared to the differences between B and C-grid solutions,the
250    C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
251    (\reffig{iceveloc}c).  As expected the differences are largest along
252    coastlines: because of the free-slip boundary conditions, flow is
253    faster in the C-LSR-fs solution, for example, along the east coast
254    of Greenland, the north coast of Alaska, and the east Coast of Baffin
255    Island.
256    %\newcommand{\subplotwidth}{0.44\textwidth}
257    \newcommand{\subplotwidth}{0.3\textwidth}
258    \begin{figure}[htbp]
259      \centering
260      \subfigure[{\footnotesize C-LSR-ns}]
261      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-LSR-ns}}
262      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
263      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_B-LSR-ns-C-LSR-ns}}
264      \\
265      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
266      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-LSR-fs-C-LSR-ns}}
267      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
268      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-EVP-ns150-C-LSR-ns}}
269      \\
270      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
271      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_TEM-C-LSR-ns}}
272      \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
273      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_HB87-C-LSR-ns}}
274      \\
275      \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
276      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_ThSIce-C-LSR-ns}}
277      \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
278      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_adv33-C-LSR-ns}}
279      \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
280        over the first 3 months of integration [cm/s]; (b)-(h) difference
281        between solutions with B-grid, free lateral slip, EVP-solver,
282        truncated ellipse method (TEM), different ice-ocean stress
283        formulation (HB87), different thermodynamics (WTD), different
284        advection for thermodynamic variables (DST3FL) and the C-LSR-ns
285        reference solution [cm/s]; color indicates speed (or differences
286        of speed), vectors indicate direction only.}
287      \label{fig:iceveloc}
288  \end{figure}  \end{figure}
289    
290  Difference from cube sphere is that it does not use z* coordinates nor  The C-EVP-ns solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$ is
291  realfreshwater fluxes because it is not supported by open boundary code.  very different from the C-LSR-ns solution (\reffig{iceveloc}d). The
292    EVP-approximation of the VP-dynamics allows for increased drift by
293  Open water, dry  over 2\,cm/s in the Beaufort Gyre and the transarctic drift.
294  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,  %\ml{Also the Beaufort Gyre is moved towards Alaska in the C-EVP-ns
295  0.76, 0.94, and 0.8.  %solution. [Really?, No]}
296    In general, drift velocities are biased towards higher values in the
297  \begin{itemize}  EVP solutions.
298  \item Configuration  % as can be seen from a histogram of the differences in
299  \item OBCS from cube  %\reffig{drifthist}.
300  \item forcing  %\begin{figure}[htbp]
301  \item 1/2 and full resolution  %  \centering
302  \item with a few JFM figs from C-grid LSR no slip  %  \includegraphics[width=\textwidth]{\fpath/drifthist_C-EVP-ns-C-LSR-ns}
303    ice transport through Canadian Archipelago  %  \caption{Histogram of drift velocity differences for C-LSR-ns and
304    thickness distribution  %    C-EVP-ns solution [cm/s].}
305    ice velocity and transport  %  \label{fig:drifthist}
306  \end{itemize}  %\end{figure}
307    
308  \begin{itemize}  \reffig{icethick}a shows the effective thickness (volume per unit
309  \item Arctic configuration  area) of the C-LSR-ns solution, averaged over January, February, March
310  \item ice transport through straits and near boundaries  of year 2000. By this time of the integration, the differences in the
311  \item focus on narrow straits in the Canadian Archipelago  ice drift velocities have led to the evolution of very different ice
312  \end{itemize}  thickness distributions, which are shown in \reffig{icethick}b--d, and
313    concentrations (not shown).
314  \begin{itemize}  \begin{figure}[htbp]
315  \item B-grid LSR no-slip    \centering
316  \item C-grid LSR no-slip    \subfigure[{\footnotesize C-LSR-ns}]
317  \item C-grid LSR slip    {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-ns}}
318  \item C-grid EVP no-slip    \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
319  \item C-grid EVP slip    {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_B-LSR-ns-C-LSR-ns}}
320  \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)    \\
321  \item C-grid LSR no-slip + Winton    \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
322  \item  speed-performance-accuracy (small)    {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-fs-C-LSR-ns}}
323    ice transport through Canadian Archipelago differences    \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
324    thickness distribution differences    {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-EVP-ns150-C-LSR-ns}}
325    ice velocity and transport differences    \\
326  \end{itemize}    \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
327      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_TEM-C-LSR-ns}}
328  We anticipate small differences between the different models due to:    \subfigure[{\footnotesize C-EVP-ns HB87 $-$ C-LSR-ns}]
329  \begin{itemize}    {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_HB87-C-LSR-ns}}
330  \item advection schemes: along the ice-edge and regions with large    \\
331    gradients    \subfigure[{\footnotesize C-LSR-ns WTD $-$ C-LSR-ns}]
332  \item C-grid: less transport through narrow straits for no slip    {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_ThSIce-C-LSR-ns}}
333    conditons, more for free slip    \subfigure[{\footnotesize C-EVP-ns DST3FL $-$ C-LSR-ns}]
334  \item VP vs.\ EVP: speed performance, accuracy?    {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_adv33-C-LSR-ns}}
335  \item ocean stress: different water mass properties beneath the ice    \caption{(a) Effective thickness (volume per unit area) of the
336  \end{itemize}      C-LSR-ns solution, averaged over the months Janurary through March
337        2000 [m]; (b)-(d) difference between solutions with B-grid, free
338  \begin{figure}      lateral slip, EVP-solver, truncated ellipse method (TEM),
339  \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM1992uvice}}}      different ice-ocean stress formulation (HB87), different
340  \caption{Surface sea ice velocity for different solver flavors.      thermodynamics (WTD), different advection for thermodynamic
341  \label{fig:iceveloc}}      variables (DST3FL) and the C-LSR-ns reference solution [m].}
342      \label{fig:icethick}
343  \end{figure}  \end{figure}
344    %
345    The generally weaker ice drift velocities in the B-LSR-ns solution,
346    when compared to the C-LSR-ns solution, in particular through the
347    narrow passages in the Canadian Archipelago, lead to a larger build-up
348    of ice north of Greenland and the Archipelago by 2\,m effective
349    thickness and more in the B-grid solution (\reffig{icethick}b). But
350    the ice volume in not larger everywhere: further west, there are
351    patches of smaller ice volume in the B-grid solution, most likely
352    because the Beaufort Gyre is weaker and hence not as effective in
353    transporting ice westwards. There are also dipoles of ice volume
354    differences with more ice on the upstream side of island groups and
355    less ice in their lee, such as Franz-Josef-Land and
356    Severnaya Semlya\ml{/or Nordland?},
357    because ice tends to flow along coasts less easily in the B-LSR-ns
358    solution.
359    
360    Imposing a free-slip boundary condition in C-LSR-fs leads to a much
361    smaller differences to C-LSR-ns in the central Arctic than the
362    transition from the B-grid to the C-grid (\reffig{icethick}c), except
363    in the Canadian Archipelago. There it reduces the effective ice
364    thickness by 2\,m and more where the ice is thick and the straits are
365    narrow.  Dipoles of ice thickness differences can also be observed
366    around islands, because the free-slip solution allows more flow around
367    islands than the no-slip solution.  Everywhere else the ice volume is
368    affected only slightly by the different boundary condition.
369    %
370    The C-EVP-ns solution has generally stronger drift velocities than the
371    C-LSR-ns solution. Consequently, more ice can be moved from the
372    eastern part of the Arctic, where ice volumes are smaller, to the
373    western Arctic (\reffig{icethick}d). Within the Canadian Archipelago,
374    more drift leads to faster ice export and reduced effective ice
375    thickness. With a shorter time step of
376    $\Delta{t}_\mathrm{evp}=10\text{\,s}$ the EVP solution seems to
377    converge to the LSOR solution (not shown). Only in the narrow straits
378    in the Archipelago the ice thickness is not affected by the shorter
379    time step and the ice is still thinner by 2\,m and more, as in the EVP
380    solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$.
381    
382    The observed difference of order 2\,m and less are smaller than the
383    differences that were observed between different hindcast and climate
384    models in \citet{gerdes07}. There the range of sea ice volume of
385    different sea ice-ocean models (which shared very similar forcing
386    fields) was on the order of $10,000\text{km$^{3}$}$; this range was
387    even larger for coupled climate models. Here, the range (and the
388    averaging period) is smaller than $4,000\text{km$^{3}$}$ except for
389    the run \mbox{C-LSR-ns~WTD} where the more complicated thermodynamics
390    leads to generally thicker ice (\reffig{icethick} and
391    \reftab{icevolume}).
392    \begin{table}[htbp]
393      \begin{tabular}{lr@{\hspace{5ex}}r@{$\pm$}rr@{$\pm$}rr@{$\pm$}r}
394        model run & ice volume
395        & \multicolumn{6}{c}{ice transport [$\text{flux$\pm$ std.,
396            km$^{3}$\,y$^{-1}$}$]}\\
397        & [$\text{km$^{3}$}$]
398        & \multicolumn{2}{c}{FS}
399        & \multicolumn{2}{c}{NI}
400        & \multicolumn{2}{c}{LS} \\ \hline
401        B-LSR-ns       & 23,824 & 2126 & 1278 &   34 &  122 &   43 &   76 \\
402        C-LSR-ns       & 24,769 & 2196 & 1253 &   70 &  224 &   77 &  110 \\
403        C-LSR-fs       & 23,286 & 2236 & 1289 &   80 &  276 &   91 &   85 \\
404        C-EVP-ns       & 27,056 & 3050 & 1652 &  352 &  735 &  256 &  151 \\
405        C-EVP-ns10     & 22,633 & 2174 & 1260 &  186 &  496 &  133 &  128 \\
406        C-LSR-ns HB87  & 23,060 & 2256 & 1327 &   64 &  230 &   77 &  114 \\
407        C-LSR-ns TEM   & 23,529 & 2222 & 1258 &   60 &  242 &   87 &  112 \\
408        C-LSR-ns WTD   & 31,634 & 2761 & 1563 &   23 &  140 &   94 &   63 \\
409        C-LSR-ns DST3FL& 24,023 & 2191 & 1261 &   88 &  251 &   84 &  129
410      \end{tabular}
411      \caption{Arctic ice volume averaged over Jan--Mar 2000, in
412        $\text{km$^{3}$}$. Mean ice transport and standard deviation for the
413        period Jan 1992 -- Dec 1999 through the Fram Strait (FS), the
414        total northern inflow into the Canadian Archipelago (NI), and the
415        export through Lancaster Sound (LS), in $\text{km$^{3}$\,y$^{-1}$}$.}
416      \label{tab:icevolume}
417    \end{table}
418    
419    The difference in ice volume and ice drift velocities between the
420    different experiments has consequences for the ice transport out of
421    the Arctic. Although by far the most exported ice drifts through the
422    Fram Strait (approximately $2300\pm610\text{\,km$^3$\,y$^{-1}$}$), a
423    considerable amount (order $160\text{\,km$^3$\,y$^{-1}$}$) ice is
424    exported through the Canadian Archipelago \citep[and references
425    therein]{serreze06}. Note, that ice transport estimates are associated
426    with large uncertainties; also note that tuning an Arctic sea
427    ice-ocean model to reproduce observations is not our goal, but we use
428    the published numbers as an orientation.
429    
430    \reffig{archipelago} shows a time series of daily averaged, smoothed
431    with monthly running means, ice transports through various straits in
432    the Canadian Archipelago and the Fram Strait for the different model
433    solutions and \reftab{icevolume} summarizes the time series. The
434    export through Fram Strait agrees with the observations in all model
435    solutions (annual averages range from $2110$ to
436    $2300\text{\,km$^3$\,y$^{-1}$}$, except for \mbox{C-LSR-ns~WTD} with
437    $2760\text{\,km$^3$\,y$^{-1}$}$ and the EVP solution with the long
438    time step of 150\,s with nearly $3000\text{\,km$^3$\,y$^{-1}$}$),
439    while the export through the Candian Archipelago is smaller than
440    generally thought. For example, the ice transport through Lancaster
441    Sound is lower (annual averages are $43$ to
442    $256\text{\,km$^3$\,y$^{-1}$}$) than in \citet{dey81} who estimates an
443    inflow into Baffin Bay of $370$ to $537\text{\,km$^3$\,y$^{-1}$}$, but
444    a flow of only $102$ to $137\text{\,km$^3$\,y$^{-1}$}$ further
445    upstream in Barrow Strait in the 1970ies from satellite images.
446    Generally, the EVP solutions have the highest maximum (export out of
447    the Artic) and lowest minimum (import into the Artic) fluxes as the
448    drift velocities are largest in these solutions.  In the extreme of
449    the Nares Strait, which is only a few grid points wide in our
450    configuration, both B- and C-grid LSOR solvers lead to practically no
451    ice transport, while the C-EVP solutions allow up to
452    $600\text{\,km$^3$\,y$^{-1}$}$ in summer; \citet{tang04} report $300$
453    to $350\text{\,km$^3$\,y$^{-1}$}$.  As as consequence, the import into
454    the Candian Archipelago is larger in all EVP solutions
455    %(range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$)
456    than in the LSOR solutions.
457    %get the order of magnitude right (range: $132$ to
458    %$165\text{\,km$^3$\,y$^{-1}$}$);
459    The B-LSR-ns solution is even smaller by another factor of two than the
460    C-LSR solutions (an exception is the WTD solution, where larger ice thickness
461    tends to block the transport).
462    %underestimates the ice transport with $34\text{\,km$^3$\,y$^{-1}$}$.
463  \begin{figure}  \begin{figure}
464  \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
465  \caption{Transport through Canadian Archipelago for different solver flavors.  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
466    \centerline{{\includegraphics*[width=\linewidth]{\fpath/ice_export}}}
467    \caption{Transport through Canadian Archipelago for different solver
468      flavors. The letters refer to the labels of the sections in
469      \reffig{arctic_topog}; positive values are flux out of the Arctic;
470      legend abbreviations are explained in \reftab{experiments}.
471  \label{fig:archipelago}}  \label{fig:archipelago}}
472  \end{figure}  \end{figure}
473    
474  \begin{figure}  %\ml{[Transport to narrow straits, area?, more runs, TEM, advection
475  \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM2000heff}}}  %  schemes, Winton TD, discussion about differences in terms of model
476  \caption{Sea ice thickness for different solver flavors.  %  error? that's tricky as it means refering to Tremblay, thus our ice
477  \label{fig:icethick}}  %  models are all erroneous!]}
478  \end{figure}  
479    In summary, we find that different dynamical solvers can yield very
480    different solutions. In constrast to that, the differences between
481    free-slip and no-slip solutions \emph{with the same solver} are
482    considerably smaller (the difference for the EVP solver is not shown,
483    but similar to that for the LSOR solver). Albeit smaller, the
484    differences between free and no-slip solutions in ice drift can lead
485    to equally large differences in ice volume, especially in the Canadian
486    Archipelago over the integration time. At first, this observation
487    seems counterintuitive, as we expect that the solution
488    \emph{technique} should not affect the \emph{solution} to a higher
489    degree than actually modifying the equations. A more detailed study on
490    these differences is beyond the scope of this paper, but at this point
491    we may speculate, that the large difference between B-grid, C-grid,
492    LSOR, and EVP solutions stem from incomplete convergence of the
493    solvers due to linearization and due to different methods of
494    linearization \citep[and Bruno Tremblay, personal
495    communication]{hunke01}: if the convergence of the non-linear momentum
496    equations is not complete for all linearized solvers, then one can
497    imagine that each solver stops at a different point in velocity-space
498    thus leading to different solutions for the ice drift velocities. If
499    this were true, this tantalizing circumstance would have a dramatic
500    impact on sea-ice modeling in general, and we would need to improve
501    the solution techniques for dynamic sea ice models, most likely at a very
502    high compuational cost (Bruno Tremblay, personal communication). Further,
503    we observe that the EVP solutions tends to produce effectively
504    ``weaker'' ice that yields more easily to stress. The fast response to
505    changing wind was also observed by \citet{hunke99}, their Fig.\,10--12,
506    where the EVP model adjusts quickly to a cyclonic wind pattern, while
507    the LSOR solution does not. This property of the EVP solutions allows
508    larger ice transports through narrow straits, where the implicit
509    solver LSOR forms rigid ice. The underlying reasons for this striking
510    difference need further exploration.
511    
512    % THIS is now almost all in the text:
513    %\begin{itemize}
514    %\item Configuration
515    %\item OBCS from cube
516    %\item forcing
517    %\item 1/2 and full resolution
518    %\item with a few JFM figs from C-grid LSR no slip
519    %  ice transport through Canadian Archipelago
520    %  thickness distribution
521    %  ice velocity and transport
522    %\end{itemize}
523    
524    %\begin{itemize}
525    %\item Arctic configuration
526    %\item ice transport through straits and near boundaries
527    %\item focus on narrow straits in the Canadian Archipelago
528    %\end{itemize}
529    
530    %\begin{itemize}
531    %\item B-grid LSR no-slip: B-LSR-ns
532    %\item C-grid LSR no-slip: C-LSR-ns
533    %\item C-grid LSR slip:    C-LSR-fs
534    %\item C-grid EVP no-slip: C-EVP-ns
535    %\item C-grid EVP slip:    C-EVP-fs
536    %\item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
537    %  new flag): C-LSR-ns+TEM
538    %\item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
539    %\item C-grid LSR no-slip + Winton:
540    %\item  speed-performance-accuracy (small)
541    %  ice transport through Canadian Archipelago differences
542    %  thickness distribution differences
543    %  ice velocity and transport differences
544    %\end{itemize}
545    
546    %We anticipate small differences between the different models due to:
547    %\begin{itemize}
548    %\item advection schemes: along the ice-edge and regions with large
549    %  gradients
550    %\item C-grid: less transport through narrow straits for no slip
551    %  conditons, more for free slip
552    %\item VP vs.\ EVP: speed performance, accuracy?
553    %\item ocean stress: different water mass properties beneath the ice
554    %\end{itemize}
555    
556    %%% Local Variables:
557    %%% mode: latex
558    %%% TeX-master: "ceaice"
559    %%% End:
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