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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  A second series of forward sensitivity experiments have been carried out on an  This section presents results from regional coupled ocean and sea
6  Arctic Ocean domain with open boundaries.  Once again the objective is to  ice simulations of the Arctic Ocean that exercise various capabilities of the MITgcm sea ice
7  compare the old B-grid LSR dynamic solver with the new C-grid LSR and EVP  model.
8  solvers.  One additional experiment is carried out to illustrate the  The objective is to
9  differences between the two main options for sea ice thermodynamics in the MITgcm.  compare the old B-grid LSOR dynamic solver with the new C-grid LSOR and
10    EVP solvers. Additional experiments are carried out to illustrate
11  \subsection{Arctic Domain with Open Boundaries}  the differences between different ice advection schemes, ocean-ice
12  \label{sec:arctic}  stress formulations and the two main options for sea ice
13    thermodynamics in the MITgcm.
14  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It  
15  is carved out from, and obtains open boundary conditions from, the  \subsection{Model configuration and experiments}
16  global cubed-sphere configuration of the Estimating the Circulation  \label{sec:arcticmodel}
17  and Climate of the Ocean, Phase II (ECCO2) project  The Arctic model domain is illustrated in \reffig{arctic_topog}.
18  \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes  \begin{figure*}
19  horizontally with mean horizontal grid spacing of 18 km.  %\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    The global ocean and sea ice results presented in \citet{menemenlis05}
49    were carried out as part
50    of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2)
51    project.  ECCO2 aims to produce increasingly accurate syntheses of all
52    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,
54    carbon, and other properties within the ocean \citep{menemenlis05}.  The
55    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
58    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
60    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
62    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. The model employs the
64    partial-cell formulation of
65    \citet{adcroft97:_shaved_cells}, which permits accurate representation of the
66    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
71    variables. In the ocean, the non-linear equation of state of \citet{jac95} is
72    used.
73    %
74    The global ocean model is coupled to a sea ice model in a
75    configuration similar to the case C-LSR-ns (see \reftab{experiments}),
76    with open water, dry ice, wet ice, dry snow, and wet snow albedos of,
77    respectively, 0.15, 0.88, 0.79, 0.97, and 0.83.
78    
79    This particular ECCO2 simulation is initialized from temperature and salinity
80    fields derived from the Polar science center Hydrographic Climatology (PHC)
81    3.0 \citep{ste01a}. Surface boundary conditions for the period January 1979 to
82    July 2002 are derived from the European Centre for Medium-Range Weather
83    Forecasts (ECMWF) 40 year re-analysis (ERA-40) \citep{upp05}.  Surface
84    boundary conditions after September 2002 are derived from the ECMWF
85    operational analysis.  There is a one month transition period, August 2002,
86    during which the ERA-40 contribution decreases linearly from 1 to 0 and the
87    ECMWF analysis contribution increases linearly from 0 to 1.  Six-hourly
88    surface winds, temperature, humidity, downward short- and long-wave
89    radiations, and precipitation are converted to heat, freshwater, and wind
90    stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
91    radiation decays exponentially as per \citet{pau77}.  Low frequency
92    precipitation has been adjusted using the pentad (5-day) data from the Global
93    Precipitation Climatology Project \citep[GPCP,][]{huf01}.  The time-mean river
94    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)
96    and prepared by P. Winsor (personnal communication, 2007) is specificied.
97    Additionally, there is a relaxation to the monthly-mean climatological sea
98    surface salinity values from PHC 3.0, a relaxation time scale of 101 days.
99    
100    Vertical mixing follows \citet{lar94} but with meridionally and vertically
101    varying background vertical diffusivity; at the surface, vertical diffusivity
102    is $4.4\times 10^{-6}$~m$^2$~s$^{-1}$ at the Equator, $3.6\times
103    10^{-6}$~m$^2$~s$^{-1}$ north of 70$^\circ$N, and $1.9\times
104    10^{-5}$~m$^2$~s$^{-1}$ south of 30$^\circ$S and between 30$^\circ$N and
105    60$^\circ$N , with sinusoidally varying values in between these latitudes;
106    vertically, diffusivity increases to $1.1\times 10^{-4}$~m$^2$~s$^{-1}$ at a a
107    depth of 6150 m as per \citet{bry79}.  A high order monotonicity-preserving
108    advection scheme \citep{dar04} is employed and there is no explicit horizontal
109    diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
110    the divergent flow as per \citet{kem08}.
111    
112    The model configuration of cube76 carries over to the Arctic domain
113    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    albedos of open water, dry ice, wet ice, dry snow, and wet snow, which
116    are now, respectively, 0.15, 0.85, 0.76, 0.94, and 0.8.
117    
118    The model is integrated from Jan~01, 1992 to Mar~31, 2000
119    \reftab{experiments} gives an overview over the experiments discussed
120    in \refsec{arcticresults}.
121    \begin{table}
122      \caption{Overview over model simulations in \refsec{arcticresults}.
123        \label{tab:experiments}}
124      \begin{tabular}{p{.3\linewidth}p{.65\linewidth}}
125        experiment name & description \\ \hline
126        B-LSR-ns       &  the original LSOR solver of \citet{zhang97} on an
127      Arakawa B-grid, implying no-slip lateral boundary conditions
128      ($\vek{u}=0$ 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}
251    \addtocounter{figure}{-1}
252    \setcounter{subfigure}{4}
253    \begin{figure}[t]
254      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
255      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_TEM-C-LSR-ns}}
256      \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
257      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_HB87-C-LSR-ns}}
258      \\
259      \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
260      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_ThSIce-C-LSR-ns}}
261      \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
262      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_adv33-C-LSR-ns}}
263      \caption{continued}
264    \end{figure}
265    The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
266    is most pronounced along the coastlines, where the discretization
267    differs most between B and C-grids: On a B-grid the tangential
268    velocity lies on the boundary (and is thus zero through the no-slip
269    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}
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=0.44\linewidth]{\fpath/arctic1.eps}}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
481  \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
482    %\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  There are 50 vertical levels ranging in thickness from 10 m near the surface  model solutions (annual averages range from $2110$ to
494  to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from  $2300\text{\,km$^3$\,y$^{-1}$}$, except for \mbox{C-LSR-ns~WTD} with
495  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  $2760\text{\,km$^3$\,y$^{-1}$}$ and the EVP solution with the long
496  data (ETOPO2) and the model employs the partial-cell formulation of  time step of 150\,s with nearly $3000\text{\,km$^3$\,y$^{-1}$}$),
497  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The  while the export through the Candian Archipelago is smaller than
498  model is integrated in a volume-conserving configuration using a finite volume  generally thought. For example, the ice transport through Lancaster
499  discretization with C-grid staggering of the prognostic variables. In the  Sound is lower (annual averages are $43$ to
500  ocean, the non-linear equation of state of \citet{jackett95}.  The ocean model is  $256\text{\,km$^3$\,y$^{-1}$}$) than in \citet{dey81} who estimates an
501  coupled to a sea-ice model described hereinabove.    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  This particular ECCO2 simulation is initialized from rest using the  upstream in Barrow Strait in the 1970ies from satellite images.
504  January temperature and salinity distribution from the World Ocean  Generally, the EVP solutions have the highest maximum (export out of
505  Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for  the Artic) and lowest minimum (import into the Artic) fluxes as the
506  32 years prior to the 1996--2001 period discussed in the study. Surface  drift velocities are largest in these solutions.  In the extreme of
507  boundary conditions are from the National Centers for Environmental  the Nares Strait, which is only a few grid points wide in our
508  Prediction and the National Center for Atmospheric Research  configuration, both B- and C-grid LSOR solvers lead to practically no
509  (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly  ice transport, while the C-EVP solutions allow up to
510  surface winds, temperature, humidity, downward short- and long-wave  $600\text{\,km$^3$\,y$^{-1}$}$ in summer (not shown); \citet{tang04}
511  radiations, and precipitation are converted to heat, freshwater, and  report $300$ to $350\text{\,km$^3$\,y$^{-1}$}$.  As as consequence,
512  wind stress fluxes using the \citet{large81, large82} bulk formulae.  the import into the Candian Archipelago is larger in all EVP solutions
513  Shortwave radiation decays exponentially as per Paulson and Simpson  %(range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$)
514  [1977]. Additionally the time-mean river run-off from Large and Nurser  than in the LSOR solutions.
515  [2001] is applied and there is a relaxation to the monthly-mean  %get the order of magnitude right (range: $132$ to
516  climatological sea surface salinity values from WOA01 with a  %$165\text{\,km$^3$\,y$^{-1}$}$);
517  relaxation time scale of 3 months. Vertical mixing follows  The B-LSR-ns solution is even smaller by another factor of two than the
518  \citet{large94} with background vertical diffusivity of  C-LSR solutions (an exception is the WTD solution, where larger ice thickness
519  $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of  tends to block the transport).
520  $10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time  %underestimates the ice transport with $34\text{\,km$^3$\,y$^{-1}$}$.
521  advection scheme with flux limiter is employed \citep{hundsdorfer94}  
522  and there is no explicit horizontal diffusivity. Horizontal viscosity  %\ml{[Transport to narrow straits, area?, more runs, TEM, advection
523  follows \citet{lei96} but  %  schemes, Winton TD, discussion about differences in terms of model
524  modified to sense the divergent flow as per Fox-Kemper and Menemenlis  %  error? that's tricky as it means refering to Tremblay, thus our ice
525  [in press].  Shortwave radiation decays exponentially as per Paulson  %  models are all erroneous!]}
526  and Simpson [1977].  Additionally, the time-mean runoff of Large and  
527  Nurser [2001] is applied near the coastline and, where there is open  \subsubsection{Discussion}
528  water, there is a relaxation to monthly-mean WOA01 sea surface  
529  salinity with a time constant of 45 days.  In summary, we find that different dynamical solvers can yield very
530    different solutions. In constrast to that, the differences between
531  Open water, dry  free-slip and no-slip solutions \emph{with the same solver} are
532  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,  considerably smaller (the difference for the EVP solver is not shown,
533  0.76, 0.94, and 0.8.  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  \begin{itemize}  to equally large differences in ice volume, especially in the Canadian
536  \item Configuration  Archipelago over the integration time. At first, this observation
537  \item OBCS from cube  seems counterintuitive, as we expect that the solution
538  \item forcing  \emph{technique} should not affect the \emph{solution} to a higher
539  \item 1/2 and full resolution  degree than actually modifying the equations. A more detailed study on
540  \item with a few JFM figs from C-grid LSR no slip  these differences is beyond the scope of this paper, but at this point
541    ice transport through Canadian Archipelago  we may speculate, that the large difference between B-grid, C-grid,
542    thickness distribution  LSOR, and EVP solutions stem from incomplete convergence of the
543    ice velocity and transport  solvers due to linearization and due to different methods of
544  \end{itemize}  linearization \citep[and Bruno Tremblay, personal
545    communication]{hunke01}: if the convergence of the non-linear momentum
546  \begin{itemize}  equations is not complete for all linearized solvers, then one can
547  \item Arctic configuration  imagine that each solver stops at a different point in velocity-space
548  \item ice transport through straits and near boundaries  thus leading to different solutions for the ice drift velocities. If
549  \item focus on narrow straits in the Canadian Archipelago  this were true, this tantalizing circumstance would have a dramatic
550  \end{itemize}  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  \begin{itemize}  high compuational cost (Bruno Tremblay, personal communication). Further,
553  \item B-grid LSR no-slip  we observe that the EVP solutions tends to produce effectively
554  \item C-grid LSR no-slip  ``weaker'' ice that yields more easily to stress. The fast response to
555  \item C-grid LSR slip  changing wind was also observed by \citet{hunke99}, their Fig.\,10--12,
556  \item C-grid EVP no-slip  where the EVP model adjusts quickly to a cyclonic wind pattern, while
557  \item C-grid EVP slip  the LSOR solution does not. This property of the EVP solutions allows
558  \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)  larger ice transports through narrow straits, where the implicit
559  \item C-grid LSR no-slip + Winton  solver LSOR forms rigid ice. The underlying reasons for this striking
560  \item  speed-performance-accuracy (small)  difference need further exploration.
561    ice transport through Canadian Archipelago differences  
562    thickness distribution differences  % THIS is now almost all in the text:
563    ice velocity and transport differences  %\begin{itemize}
564  \end{itemize}  %\item Configuration
565    %\item OBCS from cube
566  We anticipate small differences between the different models due to:  %\item forcing
567  \begin{itemize}  %\item 1/2 and full resolution
568  \item advection schemes: along the ice-edge and regions with large  %\item with a few JFM figs from C-grid LSR no slip
569    gradients  %  ice transport through Canadian Archipelago
570  \item C-grid: less transport through narrow straits for no slip  %  thickness distribution
571    conditons, more for free slip  %  ice velocity and transport
572  \item VP vs.\ EVP: speed performance, accuracy?  %\end{itemize}
573  \item ocean stress: different water mass properties beneath the ice  
574  \end{itemize}  %\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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