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1  \section{Forward sensitivity experiments}  \section{Forward sensitivity experiments}
2  \label{sec:forward}  \label{sec:forward}
3    
4  A second series of forward sensitivity experiments have been carried out on an  This section presents results from global and regional coupled ocean and sea
5  Arctic Ocean domain with open boundaries.  Once again the objective is to  ice simulations that exercise various capabilities of the MITgcm sea ice
6  compare the old B-grid LSR dynamic solver with the new C-grid LSR and EVP  model.  The first set of results is from a global, eddy-permitting, ocean and
7  solvers.  One additional experiment is carried out to illustrate the  sea ice configuration.  The second set of results is from a regional Arctic
8  differences between the two main options for sea ice thermodynamics in the MITgcm.  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.
10    
11    \subsection{Global Ocean and Sea Ice Simulation}
12    \label{sec:global}
13    
14    The global ocean and sea ice results presented below were carried out as part
15    of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2)
16    project.  ECCO2 aims to produce increasingly accurate syntheses of all
17    available global-scale ocean and sea-ice data at resolutions that start to
18    resolve ocean eddies and other narrow current systems, which transport heat,
19    carbon, and other properties within the ocean \citep{menemenlis05}.  The
20    particular ECCO2 simulation discussed next is a baseline 28-year (1979-2006)
21    integration, labeled cube76, which has not yet been constrained by oceanic and
22    by sea ice data.  A cube-sphere grid projection is employed, which permits
23    relatively even grid spacing throughout the domain and which avoids polar
24    singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises
25    510 by 510 grid cells for a mean horizontal grid spacing of 18 km. There are
26    50 vertical levels ranging in thickness from 10 m near the surface to
27    approximately 450 m at a maximum model depth of 6150 m. The model employs the
28    partial-cell formulation of
29    \citet{adcroft97:_shaved_cells}, which permits accurate representation of the
30    bathymetry. Bathymetry is from the S2004 (Smith, unpublished) blend of the
31    \citet{smi97} and the General Bathymetric Charts of the Oceans (GEBCO) one
32    arc-minute bathymetric grid (see Fig.~\ref{fig:CubeBathymetry}).
33    The model is integrated in a volume-conserving configuration using
34    a finite volume discretization with C-grid staggering of the prognostic
35    variables. In the ocean, the non-linear equation of state of \citet{jac95} is
36    used.
37    
38    \begin{figure}[h]
39      \centering
40      \includegraphics[width=\textwidth]{\fpath/CubeBathymetry}
41      \caption{Bathymetry of the global cubed sphere model configuration.  The
42        solid lines indicate domain boundaries for the regional Arctic
43        configuration discussed in Section~\ref{sec:arctic}.}
44      \label{fig:CubeBathymetry}
45    \end{figure}
46    
47    The ocean model is coupled to the sea-ice model discussed in
48    \refsec{model} using the following specific options.  The
49    zero-heat-capacity thermodynamics formulation of \citet{hibler80} is
50    used to compute sea ice thickness and concentration.  Snow cover and
51    sea ice salinity are prognostic.  Open water, dry ice, wet ice, dry
52    snow, and wet snow albedo are, respectively, 0.15, 0.88, 0.79, 0.97,
53    and 0.83. Ice mechanics follow the viscous plastic rheology of
54    \citet{hibler79} and the ice momentum equation is solved numerically
55    using the C-grid implementation of the \citet{zhang97}'s LSOR dynamics
56    model discussed hereinabove.  The ice is coupled to the ocean using
57    the rescaled vertical coordinate system, z$^\ast$, of \citet{cam08},
58    that is, sea ice does not float above the ocean model but rather
59    deforms the ocean's model surface level.
60    
61    This particular ECCO2 simulation is initialized from temperature and salinity
62    fields derived from the Polar science center Hydrographic Climatology (PHC)
63    3.0 \citep{ste01a}. Surface boundary conditions for the period January 1979 to
64    July 2002 are derived from the European Centre for Medium-Range Weather
65    Forecasts (ECMWF) 40 year re-analysis (ERA-40) \citep{upp05}.  Surface
66    boundary conditions after September 2002 are derived from the ECMWF
67    operational analysis.  There is a one month transition period, August 2002,
68    during which the ERA-40 contribution decreases linearly from 1 to 0 and the
69    ECMWF analysis contribution increases linearly from 0 to 1.  Six-hourly
70    surface winds, temperature, humidity, downward short- and long-wave
71    radiations, and precipitation are converted to heat, freshwater, and wind
72    stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
73    radiation decays exponentially as per \citet{pau77}.  Low frequency
74    precipitation has been adjusted using the pentad (5-day) data from the Global
75    Precipitation Climatology Project \citep[GPCP][]{huf01}.  The time-mean river
76    run-off from \citet{lar01} is applied globally, except in the Arctic Ocean
77    where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)
78    and prepared by P. Winsor (personnal communication, 2007) is specificied.
79    Additionally, there is a relaxation to the monthly-mean climatological sea
80    surface salinity values from PHC 3.0, a relaxation time scale of 101 days.
81    
82    Vertical mixing follows \citet{lar94} but with meridionally and vertically
83    varying background vertical diffusivity; at the surface, vertical diffusivity
84    is $4.4\times 10^{-6}$~m$^2$~s$^{-1}$ at the Equator, $3.6\times
85    10^{-6}$~m$^2$~s$^{-1}$ north of 70$^\circ$N, and $1.9\times
86    10^{-5}$~m$^2$~s$^{-1}$ south of 30$^\circ$S and between 30$^\circ$N and
87    60$^\circ$N , with sinusoidally varying values in between these latitudes;
88    vertically, diffusivity increases to $1.1\times 10^{-4}$~m$^2$~s$^{-1}$ at a a
89    depth of 6150 m as per \citet{bry79}.  A high order monotonicity-preserving
90    advection scheme \citep{dar04} is employed and there is no explicit horizontal
91    diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
92    the divergent flow as per \citet{kem08}.
93    
94    \ml{[Dimitris, here you need to either provide figures, so that I can
95      write text, or you can provide both figures and text. I guess, one
96      figure, showing the northern and southern hemisphere in summer and
97      winter is fine (four panels), as we are showing so many figures in
98      the next section.]}
99    
100    
101  \subsection{Arctic Domain with Open Boundaries}  \subsection{Arctic Domain with Open Boundaries}
102  \label{sec:arctic}  \label{sec:arctic}
103    
104  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It  A series of forward sensitivity experiments have been carried out on
105  is carved out from, and obtains open boundary conditions from, the  an Arctic Ocean domain with open boundaries.  The objective is to
106  global cubed-sphere configuration of the Estimating the Circulation  compare the old B-grid LSR dynamic solver with the new C-grid LSR and
107  and Climate of the Ocean, Phase II (ECCO2) project  EVP solvers.  Additional experiments are carried out to illustrate
108  \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes  the differences between different ice advection schemes, ocean-ice
109  horizontally with mean horizontal grid spacing of 18 km.  stress formulations and the two main options for sea ice
110    thermodynamics in the MITgcm.
111    
112    The Arctic domain of integration is illustrated in
113    \reffig{arctic_topog}.  It is carved out from, and obtains open
114    boundary conditions from, the global cubed-sphere configuration
115    described above.  The horizontal domain size is 420 by 384 grid boxes.
116    \begin{figure*}
117    %\includegraphics*[width=0.44\linewidth,viewport=139 210 496 606,clip]{\fpath/topography}
118    %\includegraphics*[width=0.44\linewidth,viewport=0 0 496 606,clip]{\fpath/topography}
119    %\includegraphics*[width=0.44\linewidth]{\fpath/topography}
120    %\includegraphics*[width=0.46\linewidth]{\fpath/archipelago}
121    \includegraphics*[width=\linewidth]{\fpath/topography}
122    \caption{Left: Bathymetry and domain boudaries of Arctic
123      Domain; the dashed line marks the boundaries of the inset on the
124      right hand side. The letters in the inset label sections in the
125      Canadian Archipelago, where ice transport is evaluated:
126      A: Nares Strait; %
127      B: \ml{Meighen Island}; %
128      C: Prince Gustaf Adolf Sea; %
129      D: \ml{Brock Island}; %
130      E: M'Clure Strait; %
131      F: Amundsen Gulf; %
132      G: Lancaster Sound; %
133      H: Barrow Strait \ml{W.}; %
134      I: Barrow Strait \ml{E.}; %
135      J: Barrow Strait \ml{N.}; %
136      K: Fram Strait. %
137      The sections A through F comprise the total inflow into the Canadian
138      Archipelago. \ml{[May still need to check the geography.]}
139      \label{fig:arctic_topog}}
140    \end{figure*}
141    
142    The main dynamic difference from cube sphere is that the Arctic domain
143    configuration does not use rescaled vertical coordinates (z$^\ast$)
144    and the surface boundary conditions for freshwater input are
145    different, because those features are not supported by the open
146    boundary code.
147    %
148    Open water, dry ice, wet ice, dry snow, and wet snow albedo are,
149    respectively, 0.15, 0.85, 0.76, 0.94, and 0.8.
150    
151    The model is integrated from Jan~01, 1992 to Mar~31, 2000,
152    with three different dynamical solvers, two different boundary
153    conditions, different stress coupling, rheology, and advection
154    schemes. \reftab{experiments} gives an overview over the experiments
155    discussed in this section.
156    \begin{table}[t]
157      \caption{Overview over model simulations in \refsec{arctic}.
158        \label{tab:experiments}}
159      \begin{tabular}{p{.3\linewidth}p{.65\linewidth}}
160        experiment name & description \\ \hline
161        B-LSR-ns       &  the original LSOR solver of \citet{zhang97} on an
162      Arakawa B-grid, implying no-slip lateral boundary conditions
163      ($\vek{u}=0$ exactly) \\
164        C-LSR-ns       &  the LSOR solver discretized on a C-grid with no-slip lateral
165      boundary conditions (implemented via ghost-points) \\
166        C-LSR-fs       &  the LSOR solver on a C-grid with free-slip lateral boundary
167      conditions \\
168        C-EVP-ns       &  the EVP solver of \citet{hunke01} on a C-grid with
169      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
170      150\text{\,s}$ \\
171        C-EVP-ns10     &  the EVP solver of \citet{hunke01} on a C-grid with
172      no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
173      10\text{\,s}$ \\
174        C-LSR-ns HB87  &  C-LSR-ns with ocean-ice stress coupling according
175      to \citet{hibler87}\\
176        C-LSR-ns TEM   &  C-LSR-ns with a truncated ellispe method (TEM)
177      rheology \citep{hibler97} \\
178        C-LSR-ns WTD   &   C-LSR-ns with 3-layer thermodynamics following
179      \citet{winton00} \\
180        C-LSR-ns DST3FL& C-LSR-ns with a third-order flux limited
181      direct-space-time advection scheme for thermodynamic variables
182      \citep{hundsdorfer94}
183      \end{tabular}
184    \end{table}
185    %\begin{description}
186    %\item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an
187    %  Arakawa B-grid, implying no-slip lateral boundary conditions
188    %  ($\vek{u}=0$ exactly);
189    %\item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
190    %  boundary conditions (implemented via ghost-points);
191    %\item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
192    %  conditions;
193    %\item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
194    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
195    %  150\text{\,s}$;
196    %\item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
197    %  boundary conditions  and $\Delta{t}_\mathrm{evp} = 150\text{\,s}$;
198    %\item[C-LSR-ns DST3FL:] C-LSR-ns with a third-order flux limited
199    %  direct-space-time advection scheme \citep{hundsdorfer94};
200    %\item[C-LSR-ns TEM:] C-LSR-ns with a truncated ellispe method (TEM)
201    %  rheology \citep{hibler97};
202    %\item[C-LSR-ns HB87:] C-LSR-ns with ocean-ice stress coupling according
203    %  to \citet{hibler87};
204    %\item[C-LSR-ns WTD:] C-LSR-ns with 3-layer thermodynamics following
205    %  \citet{winton00};
206    %%\item[C-EVP-ns damp:] C-EVP-ns with additional damping to reduce small
207    %%  scale noise \citep{hunke01};
208    %\item[C-EVP-ns10:] the EVP solver of \citet{hunke01} on a C-grid with
209    %  no-slip lateral boundary conditions and $\Delta{t}_\mathrm{evp} =
210    %  10\text{\,s}$.
211    %\end{description}
212    Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
213    that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
214    interpreted as pure model error. Lateral boundary conditions on a
215    coarse grid (coarse compared to the roughness of the true coast line) are
216    unclear, so that comparing the no-slip solutions to the free-slip
217    solutions gives another measure of uncertainty in sea ice modeling.
218    The remaining experiments explore further sensitivities of the system
219    to different physics (change in rheology, advection and diffusion
220    properties, stress coupling, and thermodynamics) and different time
221    steps for the EVP solutions: \citet{hunke01} uses 120 subcycling steps
222    for the EVP solution. We use two interpretations of this choice where
223    the EVP model is subcycled 120 times within a (short) model timestep
224    of 1200\,s resulting in a very long and expensive integration
225    ($\Delta{t}_\mathrm{evp}=10\text{\,s}$) and 120 times within the
226    forcing timescale of 6\,h ($\Delta{t}_\mathrm{evp}=150\text{\,s}$).
227    
228    A principle difficulty in comparing the solutions obtained with
229    different realizations of the model dynamics lies in the non-linear
230    feedback of the ice dynamics and thermodynamics. Already after a few
231    months the solutions have diverged so far from each other that
232    comparing velocities only makes sense within the first 3~months of the
233    integration while the ice distribution is still close to the initial
234    conditions. At the end of the integration, the differences between the
235    model solutions can be interpreted as cumulated model uncertainties.
236    
237    \reffig{iceveloc} shows ice velocities averaged over Janunary,
238    February, and March (JFM) of 1992 for the C-LSR-ns solution; also
239    shown are the differences between B-grid and C-grid, LSR and EVP, and
240    no-slip and free-slip solution. The velocity field of the C-LSR-ns
241    solution (\reffig{iceveloc}a) roughly resembles the drift velocities
242    of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
243    models in a cyclonic circulation regime (CCR) \citep[their
244    Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
245    shifted eastwards towards Alaska.
246    
247    \newcommand{\subplotwidth}{0.44\textwidth}
248    %\newcommand{\subplotwidth}{0.3\textwidth}
249    \begin{figure}[tp]
250      \centering
251      \subfigure[{\footnotesize C-LSR-ns}]
252      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-LSR-ns}}
253      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
254      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_B-LSR-ns-C-LSR-ns}}
255      \\
256      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
257      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-LSR-fs-C-LSR-ns}}
258      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
259      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_C-EVP-ns150-C-LSR-ns}}
260    %  \\
261    %  \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
262    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_TEM-C-LSR-ns}}
263    %  \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
264    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_HB87-C-LSR-ns}}
265    %  \\
266    %  \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
267    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_ThSIce-C-LSR-ns}}
268    %  \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
269    %  {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_adv33-C-LSR-ns}}
270      \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
271        over the first 3 months of integration [cm/s]; (b)-(h) difference
272        between solutions with B-grid, free lateral slip, EVP-solver,
273        truncated ellipse method (TEM), different ice-ocean stress
274        formulation (HB87), different thermodynamics (WTD), different
275        advection for thermodynamic variables (DST3FL) and the C-LSR-ns
276        reference solution [cm/s]; color indicates speed (or differences
277        of speed), vectors indicate direction only.}
278      \label{fig:iceveloc}
279    \end{figure}
280    \addtocounter{figure}{-1}
281    \setcounter{subfigure}{4}
282    \begin{figure}[t]
283      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
284      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_TEM-C-LSR-ns}}
285      \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
286      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_HB87-C-LSR-ns}}
287      \\
288      \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
289      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_ThSIce-C-LSR-ns}}
290      \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
291      {\includegraphics[width=\subplotwidth]{\fpath/JFMuv_adv33-C-LSR-ns}}
292      \caption{continued}
293    \end{figure}
294    The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
295    is most pronounced along the coastlines, where the discretization
296    differs most between B and C-grids: On a B-grid the tangential
297    velocity lies on the boundary (and is thus zero through the no-slip
298    boundary conditions), whereas on the C-grid it is half a cell width
299    away from the boundary, thus allowing more flow. The B-LSR-ns solution
300    has less ice drift through the Fram Strait and especially the along
301    Greenland's east coast; also, the flow through Baffin Bay and Davis
302    Strait into the Labrador Sea is reduced with respect the C-LSR-ns
303    solution.  \ml{[Do we expect this? Say something about that]}
304    %
305    Compared to the differences between B and C-grid solutions, the
306    C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
307    (\reffig{iceveloc}c).  As expected the differences are largest along
308    coastlines: because of the free-slip boundary conditions, flow is
309    faster in the C-LSR-fs solution, for example, along the east coast
310    of Greenland, the north coast of Alaska, and the east Coast of Baffin
311    Island.
312    
313    The C-EVP-ns solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$ is
314    very different from the C-LSR-ns solution (\reffig{iceveloc}d). The
315    EVP-approximation of the VP-dynamics allows for increased drift by
316    over 2\,cm/s in the Beaufort Gyre and the transarctic drift.
317    %\ml{Also the Beaufort Gyre is moved towards Alaska in the C-EVP-ns
318    %solution. [Really?, No]}
319    In general, drift velocities are biased towards higher values in the
320    EVP solutions.
321    % as can be seen from a histogram of the differences in
322    %\reffig{drifthist}.
323    %\begin{figure}[htbp]
324    %  \centering
325    %  \includegraphics[width=\textwidth]{\fpath/drifthist_C-EVP-ns-C-LSR-ns}
326    %  \caption{Histogram of drift velocity differences for C-LSR-ns and
327    %    C-EVP-ns solution [cm/s].}
328    %  \label{fig:drifthist}
329    %\end{figure}
330    
331    Compared to the other parameters, the ice rheology TEM
332    (\reffig{iceveloc}(e)) has a very small effect on the solution. In
333    general the ice drift tends to be increased, because there is no
334    tensile stress and ice can be ``pulled appart'' at no cost.
335    Consequently, the largest effect on drift velocity can be observed
336    near the ice edge in the Labrador Sea. In contrast, in the run with
337    the ice-ocean stress formulation of \citet{hibler87},
338    \reffig{iceveloc}(f) the drift is stronger almost everywhere in the
339    computational domain. The increase is mostly aligned with the general
340    direction of the flow, implying that the different stress formulation
341    reduces the deceleration of drift by the ocean.
342    
343    The 3-layer thermodynamics following \citet{winton00} requires
344    additional information on initial conditions for enthalphy. These
345    different initial conditions make a comparison of the first months
346    difficult to interpret. The drift in the Beaufort Gyre is slightly
347    reduced relative to the reference run C-LSR-ns, but the drift through
348    the Fram Strait is increased. The drift velocities near the ice edge
349    are very different, because the ice extend is already larger in
350    \mbox{C-LSR-ns~WTD}; inward from the ice egde, this run has smaller
351    drift velocities, because the ice motion is more contrained by a
352    larger ice extent than in \mbox{C-LSR-ns}, where the ice at the same
353    place is drifting nearly freely.
354    
355    A more sophisticated advection scheme (\mbox{C-LSR-ns DST3FL},
356    \reffig{iceveloc}(h)) has its largest effect along the ice edge, where
357    the gradients of thickness and concentration are largest. Everywhere
358    else the effect is very small and can mostly be attributed to smaller
359    numerical diffusion (and to the absense of explicitly diffusion for
360    numerical stability).
361    
362    \reffig{icethick}a shows the effective thickness (volume per unit
363    area) of the C-LSR-ns solution, averaged over January, February, March
364    of year 2000. By this time of the integration, the differences in the
365    ice drift velocities have led to the evolution of very different ice
366    thickness distributions, which are shown in \reffig{icethick}b--d, and
367    concentrations (not shown).
368    \begin{figure}[tp]
369      \centering
370      \subfigure[{\footnotesize C-LSR-ns}]
371      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-ns}}
372      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
373      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_B-LSR-ns-C-LSR-ns}}
374      \\
375      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
376      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-LSR-fs-C-LSR-ns}}
377      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
378      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_C-EVP-ns150-C-LSR-ns}}
379      \caption{(a) Effective thickness (volume per unit area) of the
380        C-LSR-ns solution, averaged over the months Janurary through March
381        2000 [m]; (b)-(h) difference between solutions with B-grid, free
382        lateral slip, EVP-solver, truncated ellipse method (TEM),
383        different ice-ocean stress formulation (HB87), different
384        thermodynamics (WTD), different advection for thermodynamic
385        variables (DST3FL) and the C-LSR-ns reference solution [m].}
386      \label{fig:icethick}
387    \end{figure}
388    \addtocounter{figure}{-1}
389    \setcounter{subfigure}{4}
390    \begin{figure}[t]
391      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
392      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_TEM-C-LSR-ns}}
393      \subfigure[{\footnotesize C-EVP-ns HB87 $-$ C-LSR-ns}]
394      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_HB87-C-LSR-ns}}
395      \\
396      \subfigure[{\footnotesize C-LSR-ns WTD $-$ C-LSR-ns}]
397      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_ThSIce-C-LSR-ns}}
398      \subfigure[{\footnotesize C-EVP-ns DST3FL $-$ C-LSR-ns}]
399      {\includegraphics[width=\subplotwidth]{\fpath/JFMheff2000_adv33-C-LSR-ns}}
400      \caption{continued}
401    \end{figure}
402    %
403    The generally weaker ice drift velocities in the B-LSR-ns solution,
404    when compared to the C-LSR-ns solution, in particular through the
405    narrow passages in the Canadian Archipelago, lead to a larger build-up
406    of ice north of Greenland and the Archipelago by 2\,m effective
407    thickness and more in the B-grid solution (\reffig{icethick}b). But
408    the ice volume in not larger everywhere: further west, there are
409    patches of smaller ice volume in the B-grid solution, most likely
410    because the Beaufort Gyre is weaker and hence not as effective in
411    transporting ice westwards. There are also dipoles of ice volume
412    differences with more ice on the upstream side of island groups and
413    less ice in their lee, such as Franz-Josef-Land and
414    Severnaya Semlya\ml{/or Nordland?},
415    because ice tends to flow along coasts less easily in the B-LSR-ns
416    solution.
417    
418    Imposing a free-slip boundary condition in C-LSR-fs leads to a much
419    smaller differences to C-LSR-ns in the central Arctic than the
420    transition from the B-grid to the C-grid (\reffig{icethick}c), except
421    in the Canadian Archipelago. There it reduces the effective ice
422    thickness by 2\,m and more where the ice is thick and the straits are
423    narrow.  Dipoles of ice thickness differences can also be observed
424    around islands, because the free-slip solution allows more flow around
425    islands than the no-slip solution. Everywhere else the ice volume is
426    affected only slightly by the different boundary condition.
427    %
428    The C-EVP-ns solution has generally stronger drift velocities than the
429    C-LSR-ns solution. Consequently, more ice can be moved from the
430    eastern part of the Arctic, where ice volumes are smaller, to the
431    western Arctic (\reffig{icethick}d). Within the Canadian Archipelago,
432    more drift leads to faster ice export and reduced effective ice
433    thickness. With a shorter time step of
434    $\Delta{t}_\mathrm{evp}=10\text{\,s}$ the EVP solution seems to
435    converge to the LSOR solution (not shown). Only in the narrow straits
436    in the Archipelago the ice thickness is not affected by the shorter
437    time step and the ice is still thinner by 2\,m and more, as in the EVP
438    solution with $\Delta{t}_\mathrm{evp}=150\text{\,s}$.
439    
440    In year 2000, there more ice everywhere in the domain in
441    \mbox{C-LSR-ns WTD}, \reffig{icethick}(g). This difference, which is
442    even more pronounced in summer (not shown), can be attributed to
443    direct effects of the different thermodynamics in this run. The
444    remaining runs have the largest differences in effective ice thickness
445    long the north coasts of Greenland and Ellesmere Island. Although the
446    effects of TEM and \citet{hibler87}'s ice-ocean stress formulation are
447    so different on the initial ice velocities, both runs have similarly
448    reduced ice thicknesses in this area. The 3rd-order advection scheme
449    has an opposite effect of similar magnitude, point towards more
450    implicit lateral stress with this numerical scheme.
451    
452    The observed difference of order 2\,m and less are smaller than the
453    differences that were observed between different hindcast models and climate
454    models in \citet{gerdes07}. There the range of sea ice volume of
455    different sea ice-ocean models (which shared very similar forcing
456    fields) was on the order of $10,000\text{km$^{3}$}$; this range was
457    even larger for coupled climate models. Here, the range (and the
458    averaging period) is smaller than $4,000\text{km$^{3}$}$ except for
459    the run \mbox{C-LSR-ns~WTD} where the more complicated thermodynamics
460    leads to generally thicker ice (\reffig{icethick} and
461    \reftab{icevolume}).
462    \begin{table}[htbp]
463      \begin{tabular}{lr@{\hspace{5ex}}r@{$\pm$}rr@{$\pm$}rr@{$\pm$}r}
464        model run & ice volume
465        & \multicolumn{6}{c}{ice transport [$\text{flux$\pm$ std.,
466            km$^{3}$\,y$^{-1}$}$]}\\
467        & [$\text{km$^{3}$}$]
468        & \multicolumn{2}{c}{FS}
469        & \multicolumn{2}{c}{NI}
470        & \multicolumn{2}{c}{LS} \\ \hline
471        B-LSR-ns       & 23,824 & 2126 & 1278 &   34 &  122 &   43 &   76 \\
472        C-LSR-ns       & 24,769 & 2196 & 1253 &   70 &  224 &   77 &  110 \\
473        C-LSR-fs       & 23,286 & 2236 & 1289 &   80 &  276 &   91 &   85 \\
474        C-EVP-ns       & 27,056 & 3050 & 1652 &  352 &  735 &  256 &  151 \\
475        C-EVP-ns10     & 22,633 & 2174 & 1260 &  186 &  496 &  133 &  128 \\
476        C-LSR-ns HB87  & 23,060 & 2256 & 1327 &   64 &  230 &   77 &  114 \\
477        C-LSR-ns TEM   & 23,529 & 2222 & 1258 &   60 &  242 &   87 &  112 \\
478        C-LSR-ns WTD   & 31,634 & 2761 & 1563 &   23 &  140 &   94 &   63 \\
479        C-LSR-ns DST3FL& 24,023 & 2191 & 1261 &   88 &  251 &   84 &  129
480      \end{tabular}
481      \caption{Arctic ice volume averaged over Jan--Mar 2000, in
482        $\text{km$^{3}$}$. Mean ice transport and standard deviation for the
483        period Jan 1992 -- Dec 1999 through the Fram Strait (FS), the
484        total northern inflow into the Canadian Archipelago (NI), and the
485        export through Lancaster Sound (LS), in $\text{km$^{3}$\,y$^{-1}$}$.}
486      \label{tab:icevolume}
487    \end{table}
488    
489    The difference in ice volume and ice drift velocities between the
490    different experiments has consequences for the ice transport out of
491    the Arctic. Although by far the most exported ice drifts through the
492    Fram Strait (approximately $2300\pm610\text{\,km$^3$\,y$^{-1}$}$), a
493    considerable amount (order $160\text{\,km$^3$\,y$^{-1}$}$) ice is
494    exported through the Canadian Archipelago \citep[and references
495    therein]{serreze06}. Note, that ice transport estimates are associated
496    with large uncertainties; also note that tuning an Arctic sea
497    ice-ocean model to reproduce observations is not our goal, but we use
498    the published numbers as an orientation.
499    
500    \reffig{archipelago} shows a time series of daily averaged, smoothed
501    with monthly running means, ice transports through various straits in
502    the Canadian Archipelago and the Fram Strait for the different model
503    solutions and \reftab{icevolume} summarizes the time series. The
504    export through Fram Strait agrees with the observations in all model
505    solutions (annual averages range from $2110$ to
506    $2300\text{\,km$^3$\,y$^{-1}$}$, except for \mbox{C-LSR-ns~WTD} with
507    $2760\text{\,km$^3$\,y$^{-1}$}$ and the EVP solution with the long
508    time step of 150\,s with nearly $3000\text{\,km$^3$\,y$^{-1}$}$),
509    while the export through the Candian Archipelago is smaller than
510    generally thought. For example, the ice transport through Lancaster
511    Sound is lower (annual averages are $43$ to
512    $256\text{\,km$^3$\,y$^{-1}$}$) than in \citet{dey81} who estimates an
513    inflow into Baffin Bay of $370$ to $537\text{\,km$^3$\,y$^{-1}$}$, but
514    a flow of only $102$ to $137\text{\,km$^3$\,y$^{-1}$}$ further
515    upstream in Barrow Strait in the 1970ies from satellite images.
516    Generally, the EVP solutions have the highest maximum (export out of
517    the Artic) and lowest minimum (import into the Artic) fluxes as the
518    drift velocities are largest in these solutions.  In the extreme of
519    the Nares Strait, which is only a few grid points wide in our
520    configuration, both B- and C-grid LSOR solvers lead to practically no
521    ice transport, while the C-EVP solutions allow up to
522    $600\text{\,km$^3$\,y$^{-1}$}$ in summer; \citet{tang04} report $300$
523    to $350\text{\,km$^3$\,y$^{-1}$}$.  As as consequence, the import into
524    the Candian Archipelago is larger in all EVP solutions
525    %(range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$)
526    than in the LSOR solutions.
527    %get the order of magnitude right (range: $132$ to
528    %$165\text{\,km$^3$\,y$^{-1}$}$);
529    The B-LSR-ns solution is even smaller by another factor of two than the
530    C-LSR solutions (an exception is the WTD solution, where larger ice thickness
531    tends to block the transport).
532    %underestimates the ice transport with $34\text{\,km$^3$\,y$^{-1}$}$.
533  \begin{figure}  \begin{figure}
534  %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1.eps}}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
535  \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
536    \centerline{{\includegraphics[width=\linewidth]{\fpath/ice_export}}}
537    \caption{Transport through Canadian Archipelago for different solver
538      flavors. The letters refer to the labels of the sections in
539      \reffig{arctic_topog}; positive values are flux out of the Arctic;
540      legend abbreviations are explained in \reftab{experiments}.
541    \label{fig:archipelago}}
542  \end{figure}  \end{figure}
543    
544  There are 50 vertical levels ranging in thickness from 10 m near the surface  %\ml{[Transport to narrow straits, area?, more runs, TEM, advection
545  to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from  %  schemes, Winton TD, discussion about differences in terms of model
546  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  %  error? that's tricky as it means refering to Tremblay, thus our ice
547  data (ETOPO2) and the model employs the partial-cell formulation of  %  models are all erroneous!]}
548  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The  
549  model is integrated in a volume-conserving configuration using a finite volume  In summary, we find that different dynamical solvers can yield very
550  discretization with C-grid staggering of the prognostic variables. In the  different solutions. In constrast to that, the differences between
551  ocean, the non-linear equation of state of \citet{jackett95}.  The ocean model is  free-slip and no-slip solutions \emph{with the same solver} are
552  coupled to a sea-ice model described hereinabove.    considerably smaller (the difference for the EVP solver is not shown,
553    but similar to that for the LSOR solver). Albeit smaller, the
554  This particular ECCO2 simulation is initialized from rest using the  differences between free and no-slip solutions in ice drift can lead
555  January temperature and salinity distribution from the World Ocean  to equally large differences in ice volume, especially in the Canadian
556  Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for  Archipelago over the integration time. At first, this observation
557  32 years prior to the 1996--2001 period discussed in the study. Surface  seems counterintuitive, as we expect that the solution
558  boundary conditions are from the National Centers for Environmental  \emph{technique} should not affect the \emph{solution} to a higher
559  Prediction and the National Center for Atmospheric Research  degree than actually modifying the equations. A more detailed study on
560  (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly  these differences is beyond the scope of this paper, but at this point
561  surface winds, temperature, humidity, downward short- and long-wave  we may speculate, that the large difference between B-grid, C-grid,
562  radiations, and precipitation are converted to heat, freshwater, and  LSOR, and EVP solutions stem from incomplete convergence of the
563  wind stress fluxes using the \citet{large81, large82} bulk formulae.  solvers due to linearization and due to different methods of
564  Shortwave radiation decays exponentially as per Paulson and Simpson  linearization \citep[and Bruno Tremblay, personal
565  [1977]. Additionally the time-mean river run-off from Large and Nurser  communication]{hunke01}: if the convergence of the non-linear momentum
566  [2001] is applied and there is a relaxation to the monthly-mean  equations is not complete for all linearized solvers, then one can
567  climatological sea surface salinity values from WOA01 with a  imagine that each solver stops at a different point in velocity-space
568  relaxation time scale of 3 months. Vertical mixing follows  thus leading to different solutions for the ice drift velocities. If
569  \citet{large94} with background vertical diffusivity of  this were true, this tantalizing circumstance would have a dramatic
570  $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of  impact on sea-ice modeling in general, and we would need to improve
571  $10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time  the solution techniques for dynamic sea ice models, most likely at a very
572  advection scheme with flux limiter is employed \citep{hundsdorfer94}  high compuational cost (Bruno Tremblay, personal communication). Further,
573  and there is no explicit horizontal diffusivity. Horizontal viscosity  we observe that the EVP solutions tends to produce effectively
574  follows \citet{lei96} but  ``weaker'' ice that yields more easily to stress. The fast response to
575  modified to sense the divergent flow as per Fox-Kemper and Menemenlis  changing wind was also observed by \citet{hunke99}, their Fig.\,10--12,
576  [in press].  Shortwave radiation decays exponentially as per Paulson  where the EVP model adjusts quickly to a cyclonic wind pattern, while
577  and Simpson [1977].  Additionally, the time-mean runoff of Large and  the LSOR solution does not. This property of the EVP solutions allows
578  Nurser [2001] is applied near the coastline and, where there is open  larger ice transports through narrow straits, where the implicit
579  water, there is a relaxation to monthly-mean WOA01 sea surface  solver LSOR forms rigid ice. The underlying reasons for this striking
580  salinity with a time constant of 45 days.  difference need further exploration.
581    
582  Open water, dry  % THIS is now almost all in the text:
583  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,  %\begin{itemize}
584  0.76, 0.94, and 0.8.  %\item Configuration
585    %\item OBCS from cube
586  \begin{itemize}  %\item forcing
587  \item Configuration  %\item 1/2 and full resolution
588  \item OBCS from cube  %\item with a few JFM figs from C-grid LSR no slip
589  \item forcing  %  ice transport through Canadian Archipelago
590  \item 1/2 and full resolution  %  thickness distribution
591  \item with a few JFM figs from C-grid LSR no slip  %  ice velocity and transport
592    ice transport through Canadian Archipelago  %\end{itemize}
593    thickness distribution  
594    ice velocity and transport  %\begin{itemize}
595  \end{itemize}  %\item Arctic configuration
596    %\item ice transport through straits and near boundaries
597  \begin{itemize}  %\item focus on narrow straits in the Canadian Archipelago
598  \item Arctic configuration  %\end{itemize}
599  \item ice transport through straits and near boundaries  
600  \item focus on narrow straits in the Canadian Archipelago  %\begin{itemize}
601  \end{itemize}  %\item B-grid LSR no-slip: B-LSR-ns
602    %\item C-grid LSR no-slip: C-LSR-ns
603  \begin{itemize}  %\item C-grid LSR slip:    C-LSR-fs
604  \item B-grid LSR no-slip  %\item C-grid EVP no-slip: C-EVP-ns
605  \item C-grid LSR no-slip  %\item C-grid EVP slip:    C-EVP-fs
606  \item C-grid LSR slip  %\item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
607  \item C-grid EVP no-slip  %  new flag): C-LSR-ns+TEM
608  \item C-grid EVP slip  %\item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
609  \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)  %\item C-grid LSR no-slip + Winton:
610  \item C-grid LSR no-slip + Winton  %\item  speed-performance-accuracy (small)
611  \item  speed-performance-accuracy (small)  %  ice transport through Canadian Archipelago differences
612    ice transport through Canadian Archipelago differences  %  thickness distribution differences
613    thickness distribution differences  %  ice velocity and transport differences
614    ice velocity and transport differences  %\end{itemize}
615  \end{itemize}  
616    %We anticipate small differences between the different models due to:
617  We anticipate small differences between the different models due to:  %\begin{itemize}
618  \begin{itemize}  %\item advection schemes: along the ice-edge and regions with large
619  \item advection schemes: along the ice-edge and regions with large  %  gradients
620    gradients  %\item C-grid: less transport through narrow straits for no slip
621  \item C-grid: less transport through narrow straits for no slip  %  conditons, more for free slip
622    conditons, more for free slip  %\item VP vs.\ EVP: speed performance, accuracy?
623  \item VP vs.\ EVP: speed performance, accuracy?  %\item ocean stress: different water mass properties beneath the ice
624  \item ocean stress: different water mass properties beneath the ice  %\end{itemize}
625  \end{itemize}  
626    %%% Local Variables:
627    %%% mode: latex
628    %%% TeX-master: "ceaice"
629    %%% End:
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