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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 boundary
45    conditions from, a global cubed-sphere \citep{adcroft04:_cubed_sphere}
46    configuration similar to that described in \citet{menemenlis05}. The
47    particular simulation from which we obtain boundary conditions is a baseline
48    integration, labeled {\em ``cube76''}. Each face of the cube comprises 510 by
49    510 grid cells for a mean horizontal grid spacing of 18\,km. There are 50
50    vertical levels ranging in thickness from 10 m near the surface to
51    approximately 450 m at a maximum model depth of 6150 m. The model employs the
52    partial-cell formulation of \citet{adcroft97:_shaved_cells}, which permits
53    accurate representation of the bathymetry. Bathymetry is from the S2004
54    (W.~Smith, unpublished) blend of the \citet{smi97} and the General Bathymetric
55    Charts of the Oceans (GEBCO) one arc-minute bathymetric grid.  The model is
56    integrated in a volume-conserving configuration using a finite volume
57    discretization with C-grid staggering of the prognostic variables. In the
58    ocean, the non-linear equation of state of \citet{jac95} is used.  The global
59    ocean model is coupled to a sea ice model in a configuration similar to the
60    case C-LSR-ns (see \reftab{experiments}).
61    
62    The {\em cube76} simulation is initialized from temperature and salinity
63    fields derived from the Polar science center Hydrographic Climatology (PHC)
64    3.0 \citep{ste01a}. Surface boundary conditions for the period January 1979 to
65    July 2002 are derived from the European Centre for Medium-Range Weather
66    Forecasts (ECMWF) 40 year re-analysis (ERA-40) \citep{upp05}.  Six-hourly
67    surface winds, temperature, humidity, downward short- and long-wave
68    radiations, and precipitation are converted to heat, freshwater, and wind
69    stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
70    radiation decays exponentially as per \citet{pau77}.  Low frequency
71    precipitation has been adjusted using the pentad (5-day) data from the Global
72    Precipitation Climatology Project \citep[GPCP,][]{huf01}.  The time-mean river
73    run-off from \citet{lar01} is applied globally, except in the Arctic Ocean
74    where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)
75    and prepared by P. Winsor (personnal communication, 2007) is specificied.
76    Additionally, there is a relaxation to the monthly-mean climatological sea
77    surface salinity values from PHC 3.0, with a relaxation time scale of 101 days.
78    
79    Vertical mixing follows \citet{lar94} but with meridionally and vertically
80    varying background vertical diffusivity; at the surface, vertical diffusivity
81    is $4.4\times 10^{-6}$~m$^2$~s$^{-1}$ at the Equator, $3.6\times
82    10^{-6}$~m$^2$~s$^{-1}$ north of 70$^\circ$N, and $1.9\times
83    10^{-5}$~m$^2$~s$^{-1}$ south of 30$^\circ$S and between 30$^\circ$N and
84    60$^\circ$N , with sinusoidally varying values in between these latitudes;
85    vertically, diffusivity increases to $1.1\times 10^{-4}$~m$^2$~s$^{-1}$ at a a
86    depth of 6150 m as per \citet{bry79}.  A high order monotonicity-preserving
87    advection scheme \citep{dar04} is employed and there is no explicit horizontal
88    diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
89    the divergent flow as per \citet{kem08}.
90    
91    The model configuration of {\em cube76} carries over to the Arctic domain
92    configuration except for numerical details related to the non-linear
93    free surface that are not supported by the open boundary code, and the
94    albedos of open water, dry ice, wet ice, dry snow, and wet snow, which
95    are now, respectively, 0.15, 0.85, 0.76, 0.94, and 0.8.  The Arctic Ocean
96    model is integrated from Jan~01, 1992 to Mar~31, 2000.
97    \reftab{experiments} gives an overview over the experiments discussed
98    in \refsec{arcticresults}.
99    \begin{table}
100      \caption{Overview over model simulations in \refsec{arcticresults}.
101        \label{tab:experiments}}
102      \begin{tabular}{p{.3\linewidth}p{.65\linewidth}}
103        experiment name & description \\ \hline
104        B-LSR-ns       &  the original LSOR solver of \citet{zhang97} on an
105      Arakawa B-grid, implying no-slip lateral boundary conditions
106      ($\vek{u}=0$ exactly), advection of ice variables with a 2nd-order
107      central difference scheme plus explicit diffusion for stability \\
108        C-LSR-ns       &  the LSOR solver discretized on a C-grid with no-slip lateral
109      boundary conditions (implemented via ghost-points) \\
110        C-LSR-fs       &  the LSOR solver on a C-grid with free-slip lateral boundary
111      conditions \\
112        C-EVP-ns       &  the EVP solver of \citet{hunke01} on a C-grid with
113      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
114      150\text{\,s}$ \\
115        C-EVP-ns10     &  the EVP solver of \citet{hunke01} on a C-grid with
116      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
117      10\text{\,s}$ \\
118        C-LSR-ns HB87  &  C-LSR-ns with ocean-ice stress coupling according
119      to \citet{hibler87}\\
120        C-LSR-ns TEM   &  C-LSR-ns with a truncated ellispe method (TEM)
121      rheology \citep{hibler97} \\
122        C-LSR-ns WTD   &   C-LSR-ns with 3-layer thermodynamics following
123      \citet{winton00} \\
124        C-LSR-ns DST3FL& C-LSR-ns with a third-order flux limited
125      direct-space-time advection scheme for thermodynamic variables
126      \citep{hundsdorfer94}
127      \end{tabular}
128    \end{table}
129    %\begin{description}
130    %\item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an
131    %  Arakawa B-grid, implying no-slip lateral boundary conditions
132    %  ($\vek{u}=0$ exactly);
133    %\item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
134    %  boundary conditions (implemented via ghost-points);
135    %\item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
136    %  conditions;
137    %\item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
138    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
139    %  150\text{\,s}$;
140    %\item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
141    %  boundary conditions  and $\Delta{t}_\mathrm{evp} = 150\text{\,s}$;
142    %\item[C-LSR-ns DST3FL:] C-LSR-ns with a third-order flux limited
143    %  direct-space-time advection scheme \citep{hundsdorfer94};
144    %\item[C-LSR-ns TEM:] C-LSR-ns with a truncated ellispe method (TEM)
145    %  rheology \citep{hibler97};
146    %\item[C-LSR-ns HB87:] C-LSR-ns with ocean-ice stress coupling according
147    %  to \citet{hibler87};
148    %\item[C-LSR-ns WTD:] C-LSR-ns with 3-layer thermodynamics following
149    %  \citet{winton00};
150    %%\item[C-EVP-ns damp:] C-EVP-ns with additional damping to reduce small
151    %%  scale noise \citep{hunke01};
152    %\item[C-EVP-ns10:] the EVP solver of \citet{hunke01} on a C-grid with
153    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
154    %  10\text{\,s}$.
155    %\end{description}
156    Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
157    that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
158    interpreted as pure model error. Lateral boundary conditions on a
159    coarse grid (coarse compared to the roughness of the true coast line) are
160    unclear, so that comparing the no-slip solutions to the free-slip
161    solutions gives another measure of uncertainty in sea ice modeling.
162    The remaining experiments explore further sensitivities of the system
163    to different physics (change in rheology, advection and diffusion
164    properties, stress coupling, and thermodynamics) and different time
165    steps for the EVP solutions: \citet{hunke01} uses 120 subcycling steps
166    for the EVP solution. We use two interpretations of this choice where
167    the EVP model is subcycled 120 times within a (short) model timestep
168    of 1200\,s resulting in a very long and expensive integration
169    ($\Delta{t}_\mathrm{evp}=10\text{\,s}$) and 144 times within the
170    forcing timescale of 6\,h ($\Delta{t}_\mathrm{evp}=150\text{\,s}$).
171    
172    \subsection{Results}
173    \label{sec:arcticresults}
174    
175    Comparing the solutions obtained with different realizations of the
176    model dynamics is difficult because of the non-linear feedback of the
177    ice dynamics and thermodynamics. Already after a few months the
178    solutions have diverged so far from each other that comparing
179    velocities only makes sense within the first 3~months of the
180    integration while the ice distribution is still close to the initial
181    conditions. At the end of the integration, the differences between the
182    model solutions can be interpreted as cumulated model uncertainties.
183    
184    \subsubsection{Ice velocities in JFM 1992}
185    
186    \reffig{iceveloc} shows ice velocities averaged over January,
187    February, and March (JFM) of 1992 for the C-LSR-ns solution; also
188    shown are the differences between this reference solution and various
189    sensitivity experiments. The velocity field of the C-LSR-ns
190    solution (\reffig{iceveloc}a) roughly resembles the drift velocities
191    of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
192    models in a cyclonic circulation regime (CCR) \citep[their
193    Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
194    shifted eastwards towards Alaska.
195    %
196    \newcommand{\subplotwidth}{0.47\textwidth}
197    %\newcommand{\subplotwidth}{0.3\textwidth}
198    \begin{figure}[tp]
199      \centering
200      \subfigure[{\footnotesize C-LSR-ns}]
201      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-LSR-ns}}
202      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
203      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_B-LSR-ns-C-LSR-ns}}
204      \\
205      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
206      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-LSR-fs-C-LSR-ns}}
207      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
208      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_C-EVP-ns150-C-LSR-ns}}
209    %  \\
210    %  \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
211    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_TEM-C-LSR-ns}}
212    %  \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
213    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_HB87-C-LSR-ns}}
214    %  \\
215    %  \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
216    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_ThSIce-C-LSR-ns}}
217    %  \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
218    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_adv33-C-LSR-ns}}
219      \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
220        over the first 3 months of integration [cm/s]; (b)-(h) difference
221        between solutions with B-grid, free lateral slip, EVP-solver,
222        truncated ellipse method (TEM), different ice-ocean stress
223        formulation (HB87), different thermodynamics (WTD), different
224        advection for thermodynamic variables (DST3FL) and the C-LSR-ns
225        reference solution [cm/s]; color indicates speed (or differences
226        of speed), vectors indicate direction only.}
227      \label{fig:iceveloc}
228    \end{figure}
229    \addtocounter{figure}{-1}
230    \setcounter{subfigure}{4}
231    \begin{figure}[tp]
232      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
233      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_TEM-C-LSR-ns}}
234      \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
235      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_HB87-C-LSR-ns}}
236      \\
237      \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
238      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_ThSIce-C-LSR-ns}}
239      \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
240      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_adv33-C-LSR-ns}}
241      \caption{continued}
242    \end{figure}
243    
244    The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
245    is most pronounced along the coastlines, where the discretization
246    differs most between B and C-grids: On a B-grid the tangential
247    velocity lies on the boundary (and is thus zero through the no-slip
248    boundary conditions), whereas on the C-grid it is half a cell width
249    away from the boundary, thus allowing more flow. The B-LSR-ns solution
250    has less ice drift through the Fram Strait and along
251    Greenland's east coast; also, the flow through Baffin Bay and Davis
252    Strait into the Labrador Sea is reduced with respect to the C-LSR-ns
253    solution.  \ml{[Do we expect this? Say something about that]}
254    %
255    Compared to the differences between B and C-grid solutions, the
256    C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
257    (\reffig{iceveloc}c).  As expected the differences are largest along
258    coastlines: because of the free-slip boundary conditions, flow is
259    faster in the C-LSR-fs solution, for example, along the east coast
260    of Greenland, the north coast of Alaska, and the east Coast of Baffin
261    Island.
262    
263    The C-EVP-ns solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$ is
264    very different from the C-LSR-ns solution (\reffig{iceveloc}d). The
265    EVP-approximation of the VP-dynamics allows for increased drift by up
266    to 8\,cm/s in the Beaufort Gyre and the transarctic drift.  In
267    general, drift velocities are strongly biased towards higher values in
268    the EVP solutions.
269    
270    Compared to the other parameters, the ice rheology TEM
271    (\reffig{iceveloc}e) has a very small effect on the solution. In
272    general the ice drift tends to be increased, because there is no
273    tensile stress and ice can be ``pulled appart'' at no cost.
274    Consequently, the largest effect on drift velocity can be observed
275    near the ice edge in the Labrador Sea. In contrast, the drift is
276    stronger almost everywhere in the computational domain in the run with
277    the ice-ocean stress formulation of \citet{hibler87}
278    (\reffig{iceveloc}f). The increase is mostly aligned with the general
279    direction of the flow, implying that the different stress formulation
280    reduces the deceleration of drift by the ocean.
281    
282    The 3-layer thermodynamics following \citet{winton00} requires
283    additional information on initial conditions for enthalphy. These
284    different initial conditions make a comparison of the first months
285    difficult to interpret. The drift in the Beaufort Gyre is slightly
286    reduced relative to the reference run C-LSR-ns, but the drift through
287    the Fram Strait is increased. The drift velocities near the ice edge
288    are very different, because the ice extent is already larger in
289    \mbox{C-LSR-ns~WTD}; inward from the ice egde, this run has smaller
290    drift velocities, because the ice motion is more contrained by a
291    larger ice extent than in \mbox{C-LSR-ns}, where the ice at the same
292    geographical position is nearly in free drift.
293    
294    A more sophisticated advection scheme (\mbox{C-LSR-ns DST3FL},
295    \reffig{iceveloc}h) has some effect along the ice edge, where
296    the gradients of thickness and concentration are largest. Everywhere
297    else the effect is very small and can mostly be attributed to smaller
298    numerical diffusion (and to the absense of explicit diffusion that is
299    required for numerical stability in a simple second order central
300    differences scheme).
301    
302    \subsubsection{Ice volume during JFM 2000}
303    
304    \reffig{icethick}a shows the effective thickness (volume per unit
305    area) of the C-LSR-ns solution, averaged over January, February, March
306    of year 2000. By this time of the integration, the differences in the
307    ice drift velocities have led to the evolution of very different ice
308    thickness distributions, which are shown in \reffig{icethick}b--h, and
309    concentrations (not shown).
310    \begin{figure}[tp]
311      \centering
312      \subfigure[{\footnotesize C-LSR-ns}]
313      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-ns}}
314      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
315      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_B-LSR-ns-C-LSR-ns}}
316      \\
317      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
318      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-fs-C-LSR-ns}}
319      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
320      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-EVP-ns150-C-LSR-ns}}
321      \caption{(a) Effective thickness (volume per unit area) of the
322        C-LSR-ns solution, averaged over the months Janurary through March
323        2000 [m]; (b)-(h) difference between solutions with B-grid, free
324        lateral slip, EVP-solver, truncated ellipse method (TEM),
325        different ice-ocean stress formulation (HB87), different
326        thermodynamics (WTD), different advection for thermodynamic
327        variables (DST3FL) and the C-LSR-ns reference solution [m].}
328      \label{fig:icethick}
329    \end{figure}
330    \addtocounter{figure}{-1}
331    \setcounter{subfigure}{4}
332    \begin{figure}[tp]
333      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
334      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_TEM-C-LSR-ns}}
335      \subfigure[{\footnotesize C-EVP-ns HB87 $-$ C-LSR-ns}]
336      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_HB87-C-LSR-ns}}
337      \\
338      \subfigure[{\footnotesize C-LSR-ns WTD $-$ C-LSR-ns}]
339      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_ThSIce-C-LSR-ns}}
340      \subfigure[{\footnotesize C-EVP-ns DST3FL $-$ C-LSR-ns}]
341      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_adv33-C-LSR-ns}}
342      \caption{continued}
343    \end{figure}
344    The generally weaker ice drift velocities in the B-LSR-ns solution,
345    when compared to the C-LSR-ns solution, in particular through the
346    narrow passages in the Canadian Arctic Archipelago, lead to a larger build-up
347    of ice north of Greenland and the Archipelago by 2\,m effective
348    thickness and more in the B-grid solution (\reffig{icethick}b). But
349    the ice volume in not larger everywhere: further west, there are
350    patches of smaller ice volume in the B-grid solution, most likely
351    because the Beaufort Gyre is weaker and hence not as effective in
352    transporting ice westwards. There are also dipoles of ice volume
353    differences with more ice on the upstream side of island groups and
354    less ice in their lee, such as Franz-Josef-Land and
355    Severnaya Semlya\ml{/or Nordland?},
356    because ice tends to flow along coasts less easily in the B-LSR-ns
357    solution.
358    
359    Imposing a free-slip boundary condition in C-LSR-fs leads to much
360    smaller differences to C-LSR-ns in the central Arctic than the
361    transition from the B-grid to the C-grid (\reffig{icethick}c), except
362    in the Canadian Arctic Archipelago. There it reduces the effective ice
363    thickness by 2\,m and more where the ice is thick and the straits are
364    narrow.  Dipoles of ice thickness differences can also be observed
365    around islands, because the free-slip solution allows more flow around
366    islands than the no-slip solution. Everywhere else the ice volume is
367    affected only slightly by the different boundary condition.
368    %
369    The C-EVP-ns solution has much thicker ice in the central Arctic Ocean
370    than the C-LSR-ns solution (\reffig{icethick}d, note the color scale).
371    Within the Canadian Arctic Archipelago, more drift leads to faster ice export
372    and reduced effective ice thickness. With a shorter time step of
373    $\Delta{t}_\mathrm{evp}=10\text{\,s}$ the EVP solution converges to
374    the LSOR solution (not shown). Only in the narrow straits in the
375    Archipelago the ice thickness is not affected by the shorter time step
376    and the ice is still thinner by 2\,m and more, as in the EVP solution
377    with $\Delta{t}_\mathrm{evp}=150\text{\,s}$.
378    
379    In year 2000, there is more ice everywhere in the domain in
380    C-LSR-ns~WTD (\reffig{icethick}g, note the color scale).
381    This difference, which is even more pronounced in summer (not shown),
382    can be attributed to direct effects of the different thermodynamics in
383    this run. The remaining runs have the largest differences in effective
384    ice thickness along the north coasts of Greenland and Ellesmere Island.
385    Although the effects of TEM and \citet{hibler87}'s ice-ocean stress
386    formulation are so different on the initial ice velocities, both runs
387    have similarly reduced ice thicknesses in this area. The 3rd-order
388    advection scheme has an opposite effect of similar magnitude, pointing
389    towards more implicit lateral stress with this numerical scheme.
390    
391    The observed difference of order 2\,m and less are smaller than the
392    differences that were observed between different hindcast models and climate
393    models in \citet{gerdes07}. There the range of sea ice volume of
394    different sea ice-ocean models (which shared very similar forcing
395    fields) was on the order of $10,000\text{km$^{3}$}$; this range was
396    even larger for coupled climate models. Here, the range (and the
397    averaging period) is smaller than $4,000\text{km$^{3}$}$ except for
398    the run \mbox{C-LSR-ns~WTD} where the more complete thermodynamics
399    lead to generally thicker ice (\reffig{icethick} and
400    \reftab{icevolume}).
401    \begin{table}[t]
402      \begin{tabular}{lr@{\hspace{5ex}}r@{$\pm$}rr@{$\pm$}rr@{$\pm$}r}
403        model run & ice volume
404        & \multicolumn{6}{c}{ice transport [$\text{flux$\pm$ std.,
405            km$^{3}$\,y$^{-1}$}$]}\\
406        & [$\text{km$^{3}$}$]
407        & \multicolumn{2}{c}{FS}
408        & \multicolumn{2}{c}{NI}
409        & \multicolumn{2}{c}{LS} \\ \hline
410        B-LSR-ns       & 23,824 & 2126 & 1278 &   34 &  122 &   43 &   76 \\
411        C-LSR-ns       & 24,769 & 2196 & 1253 &   70 &  224 &   77 &  110 \\
412        C-LSR-fs       & 23,286 & 2236 & 1289 &   80 &  276 &   91 &   85 \\
413        C-EVP-ns       & 27,056 & 3050 & 1652 &  352 &  735 &  256 &  151 \\
414        C-EVP-ns10     & 22,633 & 2174 & 1260 &  186 &  496 &  133 &  128 \\
415        C-LSR-ns HB87  & 23,060 & 2256 & 1327 &   64 &  230 &   77 &  114 \\
416        C-LSR-ns TEM   & 23,529 & 2222 & 1258 &   60 &  242 &   87 &  112 \\
417        C-LSR-ns WTD   & 31,634 & 2761 & 1563 &   23 &  140 &   94 &   63 \\
418        C-LSR-ns DST3FL& 24,023 & 2191 & 1261 &   88 &  251 &   84 &  129
419      \end{tabular}
420      \caption{Arctic ice volume averaged over Jan--Mar 2000, in
421        $\text{km$^{3}$}$. Mean ice transport and standard deviation for the
422        period Jan 1992 -- Dec 1999 through the Fram Strait (FS), the
423        total northern inflow into the Canadian Arctic Archipelago (NI), and the
424        export through Lancaster Sound (LS), in $\text{km$^{3}$\,y$^{-1}$}$.
425      \label{tab:icevolume}}
426    \end{table}
427    
428    \subsubsection{Ice transports}
429    
430    The difference in ice volume and ice drift velocities between the
431    different experiments has consequences for the ice transport out of
432    the Arctic. Although by far the most exported ice drifts through the
433    Fram Strait (approximately $2300\pm610\text{\,km$^3$\,y$^{-1}$}$), a
434    considerable amount (order $160\text{\,km$^3$\,y$^{-1}$}$) of ice is
435    exported through the Canadian Arctic Archipelago \citep[and references
436    therein]{serreze06}. Note, that ice transport estimates are associated
437    with large uncertainties and that the results presented herein have not
438    yet been constrained by observations; we use
439    the published numbers as an orientation.
440    
441    \reffig{archipelago} shows an excerpt of a time series of daily
442    averaged ice transports, smoothed with a monthly running mean, through
443    various straits in the Canadian Arctic Archipelago and the Fram Strait for
444    the different model solutions; \reftab{icevolume} summarizes the
445    time series.
446  \begin{figure}  \begin{figure}
447  %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1.eps}}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
448  \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
449    %\centerline{{\includegraphics[width=\linewidth]{\fpath/ice_export}}}
450    \centerline{{\includegraphics[width=\linewidth]{\fpath/ice_export1996}}}
451    \caption{Transport through Canadian Arctic Archipelago for different solver
452      flavors. The letters refer to the labels of the sections in
453      \reffig{arctic_topog}; positive values are flux out of the Arctic;
454      legend abbreviations are explained in \reftab{experiments}. The mean
455      range of the different model solution is taken over the period Jan
456      1992 to Dec 1999.
457    \label{fig:archipelago}}
458  \end{figure}  \end{figure}
459    The export through Fram Strait agrees with the observations in all
460  There are 50 vertical levels ranging in thickness from 10 m near the surface  model solutions (annual averages range from $2110$ to
461  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
462  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  $2760\text{\,km$^3$\,y$^{-1}$}$ and the EVP solution with the long
463  data (ETOPO2) and the model employs the partial-cell formulation of  time step of 150\,s with nearly $3000\text{\,km$^3$\,y$^{-1}$}$),
464  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The  while the export through the Candian Arctic Archipelago is smaller than
465  model is integrated in a volume-conserving configuration using a finite volume  generally thought. For example, the ice transport through Lancaster
466  discretization with C-grid staggering of the prognostic variables. In the  Sound is lower (annual averages are $43$ to
467  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
468  coupled to a sea-ice model described hereinabove.    inflow into Baffin Bay of $370$ to $537\text{\,km$^3$\,y$^{-1}$}$, but
469    a flow of only $102$ to $137\text{\,km$^3$\,y$^{-1}$}$ further
470  This particular ECCO2 simulation is initialized from rest using the  upstream in Barrow Strait in the 1970's from satellite images.
471  January temperature and salinity distribution from the World Ocean  Generally, the EVP solutions have the highest maximum (export out of
472  Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for  the Artic) and lowest minimum (import into the Artic) fluxes as the
473  32 years prior to the 1996--2001 period discussed in the study. Surface  drift velocities are largest in these solutions.  In the extreme of
474  boundary conditions are from the National Centers for Environmental  the Nares Strait, which is only a few grid points wide in our
475  Prediction and the National Center for Atmospheric Research  configuration, both B- and C-grid LSOR solvers lead to practically no
476  (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly  ice transport, while the C-EVP solutions allow up to
477  surface winds, temperature, humidity, downward short- and long-wave  $600\text{\,km$^3$\,y$^{-1}$}$ in summer (not shown); \citet{tang04}
478  radiations, and precipitation are converted to heat, freshwater, and  report $300$ to $350\text{\,km$^3$\,y$^{-1}$}$.  As as consequence,
479  wind stress fluxes using the \citet{large81, large82} bulk formulae.  the import into the Candian Arctic Archipelago is larger in all EVP solutions
480  Shortwave radiation decays exponentially as per Paulson and Simpson  %(range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$)
481  [1977]. Additionally the time-mean river run-off from Large and Nurser  than in the LSOR solutions.
482  [2001] is applied and there is a relaxation to the monthly-mean  %get the order of magnitude right (range: $132$ to
483  climatological sea surface salinity values from WOA01 with a  %$165\text{\,km$^3$\,y$^{-1}$}$);
484  relaxation time scale of 3 months. Vertical mixing follows  The B-LSR-ns solution is even smaller by another factor of two than the
485  \citet{large94} with background vertical diffusivity of  C-LSR solutions (an exception is the WTD solution, where larger ice thickness
486  $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of  tends to block the transport).
487  $10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time  %underestimates the ice transport with $34\text{\,km$^3$\,y$^{-1}$}$.
488  advection scheme with flux limiter is employed \citep{hundsdorfer94}  
489  and there is no explicit horizontal diffusivity. Horizontal viscosity  %\ml{[Transport to narrow straits, area?, more runs, TEM, advection
490  follows \citet{lei96} but  %  schemes, Winton TD, discussion about differences in terms of model
491  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
492  [in press].  Shortwave radiation decays exponentially as per Paulson  %  models are all erroneous!]}
493  and Simpson [1977].  Additionally, the time-mean runoff of Large and  
494  Nurser [2001] is applied near the coastline and, where there is open  \subsubsection{Discussion}
495  water, there is a relaxation to monthly-mean WOA01 sea surface  
496  salinity with a time constant of 45 days.  In summary, we find that different dynamical solvers can yield very
497    different solutions. In constrast to that, the differences between
498  Open water, dry  free-slip and no-slip solutions \emph{with the same solver} are
499  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,
500  0.76, 0.94, and 0.8.  but similar to that for the LSOR solver). Albeit smaller, the
501    differences between free and no-slip solutions in ice drift can lead
502  \begin{itemize}  to equally large differences in ice volume, especially in the Canadian
503  \item Configuration  Arctic Archipelago over the integration time. At first, this observation
504  \item OBCS from cube  seems counterintuitive, as we expect that the solution
505  \item forcing  \emph{technique} should not affect the \emph{solution} to a higher
506  \item 1/2 and full resolution  degree than actually modifying the equations. A more detailed study on
507  \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
508    ice transport through Canadian Archipelago  we may speculate, that the large difference between B-grid, C-grid,
509    thickness distribution  LSOR, and EVP solutions stem from incomplete convergence of the
510    ice velocity and transport  solvers due to linearization and due to different methods of
511  \end{itemize}  linearization \citep[and Bruno Tremblay, personal
512    communication]{hunke01}: if the convergence of the non-linear momentum
513  \begin{itemize}  equations is not complete for all linearized solvers, then one can
514  \item Arctic configuration  imagine that each solver stops at a different point in velocity-space
515  \item ice transport through straits and near boundaries  thus leading to different solutions for the ice drift velocities. If
516  \item focus on narrow straits in the Canadian Archipelago  this were true, this tantalizing circumstance would have a dramatic
517  \end{itemize}  impact on sea-ice modeling in general, and we would need to improve
518    the solution techniques for dynamic sea ice models, most likely at a very
519  \begin{itemize}  high compuational cost (Bruno Tremblay, personal communication).
520  \item B-grid LSR no-slip  
521  \item C-grid LSR no-slip  Further, we observe that the EVP solutions tends to produce
522  \item C-grid LSR slip  effectively ``weaker'' ice that yields more easily to stress. This was
523  \item C-grid EVP no-slip  also observed by \citet{hunke99} in a fast response to changing winds,
524  \item C-grid EVP slip  their Figures\,10--12, where the EVP model adjusts quickly to a
525  \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)  cyclonic wind pattern, while the LSOR solution lags in time. This
526  \item C-grid LSR no-slip + Winton  property of the EVP solutions allows larger ice transports through
527  \item  speed-performance-accuracy (small)  narrow straits, where the implicit solver LSOR forms rigid ice. The
528    ice transport through Canadian Archipelago differences  underlying reasons for this striking difference need further
529    thickness distribution differences  exploration.
530    ice velocity and transport differences  
531  \end{itemize}  % THIS is now almost all in the text:
532    %\begin{itemize}
533  We anticipate small differences between the different models due to:  %\item Configuration
534  \begin{itemize}  %\item OBCS from cube
535  \item advection schemes: along the ice-edge and regions with large  %\item forcing
536    gradients  %\item 1/2 and full resolution
537  \item C-grid: less transport through narrow straits for no slip  %\item with a few JFM figs from C-grid LSR no slip
538    conditons, more for free slip  %  ice transport through Canadian Archipelago
539  \item VP vs.\ EVP: speed performance, accuracy?  %  thickness distribution
540  \item ocean stress: different water mass properties beneath the ice  %  ice velocity and transport
541  \end{itemize}  %\end{itemize}
542    
543    %\begin{itemize}
544    %\item Arctic configuration
545    %\item ice transport through straits and near boundaries
546    %\item focus on narrow straits in the Canadian Archipelago
547    %\end{itemize}
548    
549    %\begin{itemize}
550    %\item B-grid LSR no-slip: B-LSR-ns
551    %\item C-grid LSR no-slip: C-LSR-ns
552    %\item C-grid LSR slip:    C-LSR-fs
553    %\item C-grid EVP no-slip: C-EVP-ns
554    %\item C-grid EVP slip:    C-EVP-fs
555    %\item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
556    %  new flag): C-LSR-ns+TEM
557    %\item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
558    %\item C-grid LSR no-slip + Winton:
559    %\item  speed-performance-accuracy (small)
560    %  ice transport through Canadian Archipelago differences
561    %  thickness distribution differences
562    %  ice velocity and transport differences
563    %\end{itemize}
564    
565    %We anticipate small differences between the different models due to:
566    %\begin{itemize}
567    %\item advection schemes: along the ice-edge and regions with large
568    %  gradients
569    %\item C-grid: less transport through narrow straits for no slip
570    %  conditons, more for free slip
571    %\item VP vs.\ EVP: speed performance, accuracy?
572    %\item ocean stress: different water mass properties beneath the ice
573    %\end{itemize}
574    
575    %%% Local Variables:
576    %%% mode: latex
577    %%% TeX-master: "ceaice"
578    %%% End:
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