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revision 1.8 by dimitri, Fri Feb 29 01:28:05 2008 UTC revision 1.22 by dimitri, Thu Aug 14 16:12:41 2008 UTC
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1  \section{Forward sensitivity experiments}  \section{Forward Sensitivity Experiments in an Arctic Domain with Open
2    Boundaries}
3  \label{sec:forward}  \label{sec:forward}
4    
5  This section presents results from global and regional coupled ocean and sea  This section presents results from regional coupled ocean and sea
6  ice simulations that exercise various capabilities of the MITgcm sea ice  ice simulations of the Arctic Ocean that exercise various capabilities of the MITgcm sea ice
7  model.  The first set of results is from a global, eddy-permitting, ocean and  model.
8  sea ice configuration.  The second set of results is from a regional Arctic  The objective is to
9  configuration, which is used to compare the B-grid and C-grid dynamic solvers  compare the old B-grid LSOR dynamic solver with the new C-grid LSOR and
10  and various other capabilities of the MITgcm sea ice model.  The third set of  EVP solvers. Additional experiments are carried out to illustrate
11  results is from a yet smaller regional domain, which is used to illustrate  the differences between different ice advection schemes, ocean-ice
12  treatment of sea ice open boundary condition sin the MITgcm.  stress formulations and the two main options for sea ice
13    thermodynamics in the MITgcm.
14  \subsection{Global Ocean and Sea Ice Simulation}  
15  \label{sec:global}  \subsection{Model configuration and experiments}
16    \label{sec:arcticmodel}
17  The global ocean and sea ice results presented below were carried out as part  The Arctic model domain is illustrated in \reffig{arctic_topog}.
18  of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2)  \begin{figure*}
19  project.  ECCO2 aims to produce increasingly accurate syntheses of all  %\includegraphics*[width=0.44\linewidth,viewport=139 210 496 606,clip]{\fpath/topography}
20  available global-scale ocean and sea-ice data at resolutions that start to  %\includegraphics*[width=0.44\linewidth,viewport=0 0 496 606,clip]{\fpath/topography}
21  resolve ocean eddies and other narrow current systems, which transport heat,  %\includegraphics*[width=0.44\linewidth]{\fpath/topography}
22  carbon, and other properties within the ocean \citep{menemenlis05}.  The  %\includegraphics*[width=0.46\linewidth]{\fpath/archipelago}
23  particular ECCO2 simulation discussed next is a baseline 28-year (1979-2006)  \includegraphics*[width=\linewidth]{\fpath/topography}
24  integration, labeled cube76, which has not yet been constrained by oceanic and  \caption{Left: Bathymetry and domain boundaries of Arctic
25  by sea ice data.  A cube-sphere grid projection is employed, which permits    Domain.
26  relatively even grid spacing throughout the domain and which avoids polar    %; the dashed line marks the boundaries of the inset on the right hand side.
27  singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises    The letters in the inset label sections in the
28  510 by 510 grid cells for a mean horizontal grid spacing of 18 km. There are    Canadian Archipelago, where ice transport is evaluated:
29  50 vertical levels ranging in thickness from 10 m near the surface to    A: Nares Strait; %
30  approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from the    B: \ml{Meighen Island}; %
31  National Geophysical Data Center (NGDC) 2-minute gridded global relief data    C: Prince Gustaf Adolf Sea; %
32  (ETOPO2) and the model employs the partial-cell formulation of    D: \ml{Brock Island}; %
33  \citet{adcroft97:_shaved_cells}, which permits accurate representation of the    E: M'Clure Strait; %
34  bathymetry. The model is integrated in a volume-conserving configuration using    F: Amundsen Gulf; %
35  a finite volume discretization with C-grid staggering of the prognostic    G: Lancaster Sound; %
36  variables. In the ocean, the non-linear equation of state of \citet{jac95} is    H: Barrow Strait \ml{W.}; %
37  used.    I: Barrow Strait \ml{E.}; %
38      J: Barrow Strait \ml{N.}; %
39  The ocean model is coupled to the sea-ice model discussed in    K: Fram Strait. %
40  Section~\ref{sec:model} using the following specific options.  The    The sections A through F comprise the total inflow into the Canadian
41  zero-heat-capacity thermodynamics formulation of \citet{hib80} is used to    Archipelago. \ml{[May still need to check the geography.]}
42  compute sea ice thickness and concentration.  Snow cover and sea ice salinity    \label{fig:arctic_topog}}
43  are prognostic.  Open water, dry ice, wet ice, dry snow, and wet snow albedo  \end{figure*}
44  are, respectively, 0.15, 0.88, 0.79, 0.97, and 0.83. Ice mechanics follow the  It has 420 by 384 grid boxes and is carved out, and obtains open boundary
45  viscous plastic rheology of \citet{hibler79} and the ice momentum equation is  conditions from, a global cubed-sphere \citep{adcroft04:_cubed_sphere}
46  solved numerically using the C-grid implementation of the \citet{zhang97} LSR  configuration similar to that described in \citet{menemenlis05}. The
47  dynamics model discussed hereinabove.  The ice is coupled to the ocean using  particular simulation from which we obtain boundary conditions is a baseline
48  the rescaled vertical coordinate system, z$^\ast$, of  integration, labeled {\em ``cube76''}. Each face of the cube comprises 510 by
49  \citet{cam08}, that is, sea ice does not float above the ocean model but  510 grid cells for a mean horizontal grid spacing of 18\,km. There are 50
50  rather deforms the ocean's model surface level.  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  This particular ECCO2 simulation is initialized from temperature and salinity  The {\em cube76} simulation is initialized from temperature and salinity
63  fields derived from the Polar science center Hydrographic Climatology (PHC)  fields derived from the Polar science center Hydrographic Climatology (PHC)
64  3.0 \citep{ste01a}. Surface boundary conditions for the period January 1979 to  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  July 2002 are derived from the European Centre for Medium-Range Weather
66  Forecasts (ECMWF) 40 year re-analysis (ERA-40) \citep{upp05}.  Surface  Forecasts (ECMWF) 40 year re-analysis (ERA-40) \citep{upp05}.  Six-hourly
 boundary conditions after September 2002 are derived from the ECMWF  
 operational analysis.  There is a one month transition period, August 2002,  
 during which the ERA-40 contribution decreases linearly from 1 to 0 and the  
 ECMWF analysis contribution increases linearly from 0 to 1.  Six-hourly  
67  surface winds, temperature, humidity, downward short- and long-wave  surface winds, temperature, humidity, downward short- and long-wave
68  radiations, and precipitation are converted to heat, freshwater, and wind  radiations, and precipitation are converted to heat, freshwater, and wind
69  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
70  radiation decays exponentially as per \citet{pau77}.  Low frequency  radiation decays exponentially as per \citet{pau77}.  Low frequency
71  precipitation has been adjusted using the pentad (5-day) data from the Global  precipitation has been adjusted using the pentad (5-day) data from the Global
72  Precipitation Climatology Project (GPCP) \citep{huf01}.  The time-mean river  Precipitation Climatology Project \citep[GPCP,][]{huf01}.  The time-mean river
73  run-off from \citet{lar01} is applied globally, except in the Arctic Ocean  run-off from \citet{lar01} is applied globally, except in the Arctic Ocean
74  where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)  where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)
75  and prepared by P. Winsor (personnal communication, 2007) is specificied.  and prepared by P. Winsor (personnal communication, 2007) is specificied.
76  Additionally, there is a relaxation to the monthly-mean climatological sea  Additionally, there is a relaxation to the monthly-mean climatological sea
77  surface salinity values from PHC 3.0, a relaxation time scale of 101 days.  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  Vertical mixing follows \citet{lar94} but with meridionally and vertically
80  varying background vertical diffusivity; at the surface, vertical diffusivity  varying background vertical diffusivity; at the surface, vertical diffusivity
# Line 81  advection scheme \citep{dar04} is employ Line 88  advection scheme \citep{dar04} is employ
88  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
89  the divergent flow as per \citet{kem08}.  the divergent flow as per \citet{kem08}.
90    
91  \subsection{Arctic Domain with Open Boundaries}  The model configuration of {\em cube76} carries over to the Arctic domain
92  \label{sec:arctic}  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  A series of forward sensitivity experiments have been carried out on an  albedos of open water, dry ice, wet ice, dry snow, and wet snow, which
95  Arctic Ocean domain with open boundaries.  The objective is to compare the old  are now, respectively, 0.15, 0.85, 0.76, 0.94, and 0.8.  The Arctic Ocean
96  B-grid LSR dynamic solver with the new C-grid LSR and EVP solvers.  One  model is integrated from Jan~01, 1992 to Mar~31, 2000.
97  additional experiment is carried out to illustrate the differences between the  \reftab{experiments} gives an overview over the experiments discussed
98  two main options for sea ice thermodynamics in the MITgcm.  in \refsec{arcticresults}.
99    \begin{table}
100  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It    \caption{Overview over model simulations in \refsec{arcticresults}.
101  is carved out from, and obtains open boundary conditions from, the global      \label{tab:experiments}}
102  cubed-sphere configuration described above.  The horizontal domain size is    \begin{tabular}{p{.3\linewidth}p{.65\linewidth}}
103  420 by 384 grid boxes.      experiment name & description \\ \hline
104        B-LSR-ns       &  the original LSOR solver of \citet{zhang97} on an
105  \begin{figure}    Arakawa B-grid, implying no-slip lateral boundary conditions
106  %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1}}}    ($\vek{u}=0$ exactly), advection of ice variables with a 2nd-order
107  \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}    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}  \end{figure}
229    \addtocounter{figure}{-1}
230  Difference from cube sphere is that it does not use z* coordinates nor  \setcounter{subfigure}{4}
231  realfreshwater fluxes because it is not supported by open boundary code.  \begin{figure}[tp]
232      \subfigure[{\footnotesize C-LSR-ns TEM $-$ C-LSR-ns}]
233  Open water, dry    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_TEM-C-LSR-ns}}
234  ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,    \subfigure[{\footnotesize C-LSR-ns HB87 $-$ C-LSR-ns}]
235  0.76, 0.94, and 0.8.    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_HB87-C-LSR-ns}}
236      \\
237  \begin{itemize}    \subfigure[{\footnotesize  C-LSR-ns WTD $-$ C-LSR-ns}]
238  \item Configuration    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_ThSIce-C-LSR-ns}}
239  \item OBCS from cube    \subfigure[{\footnotesize C-LSR-ns DST3FL $-$ C-LSR-ns}]
240  \item forcing    {\includegraphics[width=\subplotwidth]{\fpath/JFMuv1992_adv33-C-LSR-ns}}
241  \item 1/2 and full resolution    \caption{continued}
 \item with a few JFM figs from C-grid LSR no slip  
   ice transport through Canadian Archipelago  
   thickness distribution  
   ice velocity and transport  
 \end{itemize}  
   
 \begin{itemize}  
 \item Arctic configuration  
 \item ice transport through straits and near boundaries  
 \item focus on narrow straits in the Canadian Archipelago  
 \end{itemize}  
   
 \begin{itemize}  
 \item B-grid LSR no-slip  
 \item C-grid LSR no-slip  
 \item C-grid LSR slip  
 \item C-grid EVP no-slip  
 \item C-grid EVP slip  
 \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)  
 \item C-grid LSR no-slip + Winton  
 \item  speed-performance-accuracy (small)  
   ice transport through Canadian Archipelago differences  
   thickness distribution differences  
   ice velocity and transport differences  
 \end{itemize}  
   
 We anticipate small differences between the different models due to:  
 \begin{itemize}  
 \item advection schemes: along the ice-edge and regions with large  
   gradients  
 \item C-grid: less transport through narrow straits for no slip  
   conditons, more for free slip  
 \item VP vs.\ EVP: speed performance, accuracy?  
 \item ocean stress: different water mass properties beneath the ice  
 \end{itemize}  
   
 \begin{figure}  
 \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM1992uvice}}}  
 \caption{Surface sea ice velocity for different solver flavors.  
 \label{fig:iceveloc}}  
242  \end{figure}  \end{figure}
243    
244  \begin{figure}  The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
245  \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}  is most pronounced along the coastlines, where the discretization
246  \caption{Transport through Canadian Archipelago for different solver flavors.  differs most between B and C-grids: On a B-grid the tangential
247  \label{fig:archipelago}}  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}  \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=0.6\linewidth]{\fpath/JFM2000heff}}}  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
448  \caption{Sea ice thickness for different solver flavors.  %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
449  \label{fig:icethick}}  %\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    model solutions (annual averages range from $2110$ to
461    $2300\text{\,km$^3$\,y$^{-1}$}$, except for \mbox{C-LSR-ns~WTD} with
462    $2760\text{\,km$^3$\,y$^{-1}$}$ and the EVP solution with the long
463    time step of 150\,s with nearly $3000\text{\,km$^3$\,y$^{-1}$}$),
464    while the export through the Candian Arctic Archipelago is smaller than
465    generally thought. For example, the ice transport through Lancaster
466    Sound is lower (annual averages are $43$ to
467    $256\text{\,km$^3$\,y$^{-1}$}$) than in \citet{dey81} who estimates an
468    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    upstream in Barrow Strait in the 1970's from satellite images.
471    Generally, the EVP solutions have the highest maximum (export out of
472    the Artic) and lowest minimum (import into the Artic) fluxes as the
473    drift velocities are largest in these solutions.  In the extreme of
474    the Nares Strait, which is only a few grid points wide in our
475    configuration, both B- and C-grid LSOR solvers lead to practically no
476    ice transport, while the C-EVP solutions allow up to
477    $600\text{\,km$^3$\,y$^{-1}$}$ in summer (not shown); \citet{tang04}
478    report $300$ to $350\text{\,km$^3$\,y$^{-1}$}$.  As as consequence,
479    the import into the Candian Arctic Archipelago is larger in all EVP solutions
480    %(range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$)
481    than in the LSOR solutions.
482    %get the order of magnitude right (range: $132$ to
483    %$165\text{\,km$^3$\,y$^{-1}$}$);
484    The B-LSR-ns solution is even smaller by another factor of two than the
485    C-LSR solutions (an exception is the WTD solution, where larger ice thickness
486    tends to block the transport).
487    %underestimates the ice transport with $34\text{\,km$^3$\,y$^{-1}$}$.
488    
489    %\ml{[Transport to narrow straits, area?, more runs, TEM, advection
490    %  schemes, Winton TD, discussion about differences in terms of model
491    %  error? that's tricky as it means refering to Tremblay, thus our ice
492    %  models are all erroneous!]}
493    
494    \subsubsection{Discussion}
495    
496    In summary, we find that different dynamical solvers can yield very
497    different solutions. In constrast to that, the differences between
498    free-slip and no-slip solutions \emph{with the same solver} are
499    considerably smaller (the difference for the EVP solver is not shown,
500    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    to equally large differences in ice volume, especially in the Canadian
503    Arctic Archipelago over the integration time. At first, this observation
504    seems counterintuitive, as we expect that the solution
505    \emph{technique} should not affect the \emph{solution} to a higher
506    degree than actually modifying the equations. A more detailed study on
507    these differences is beyond the scope of this paper, but at this point
508    we may speculate, that the large difference between B-grid, C-grid,
509    LSOR, and EVP solutions stem from incomplete convergence of the
510    solvers due to linearization and due to different methods of
511    linearization \citep[and Bruno Tremblay, personal
512    communication]{hunke01}: if the convergence of the non-linear momentum
513    equations is not complete for all linearized solvers, then one can
514    imagine that each solver stops at a different point in velocity-space
515    thus leading to different solutions for the ice drift velocities. If
516    this were true, this tantalizing circumstance would have a dramatic
517    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    high compuational cost (Bruno Tremblay, personal communication).
520    
521    Further, we observe that the EVP solutions tends to produce
522    effectively ``weaker'' ice that yields more easily to stress. This was
523    also observed by \citet{hunke99} in a fast response to changing winds,
524    their Figures\,10--12, where the EVP model adjusts quickly to a
525    cyclonic wind pattern, while the LSOR solution lags in time. This
526    property of the EVP solutions allows larger ice transports through
527    narrow straits, where the implicit solver LSOR forms rigid ice. The
528    underlying reasons for this striking difference need further
529    exploration.
530    
531    % THIS is now almost all in the text:
532    %\begin{itemize}
533    %\item Configuration
534    %\item OBCS from cube
535    %\item forcing
536    %\item 1/2 and full resolution
537    %\item with a few JFM figs from C-grid LSR no slip
538    %  ice transport through Canadian Archipelago
539    %  thickness distribution
540    %  ice velocity and transport
541    %\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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