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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.  The third set of
10    results is from a yet smaller regional domain, which is used to illustrate
11    treatment of sea ice open boundary condition sin the MITgcm.
12    
13    \subsection{Global Ocean and Sea Ice Simulation}
14    \label{sec:global}
15    
16    The global ocean and sea ice results presented below were carried out as part
17    of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2)
18    project.  ECCO2 aims to produce increasingly accurate syntheses of all
19    available global-scale ocean and sea-ice data at resolutions that start to
20    resolve ocean eddies and other narrow current systems, which transport heat,
21    carbon, and other properties within the ocean \citep{menemenlis05}.  The
22    particular ECCO2 simulation discussed next is a baseline 28-year (1979-2006)
23    integration, labeled cube76, which has not yet been constrained by oceanic and
24    by sea ice data.  A cube-sphere grid projection is employed, which permits
25    relatively even grid spacing throughout the domain and which avoids polar
26    singularities \citep{adcroft04:_cubed_sphere}. Each face of the cube comprises
27    510 by 510 grid cells for a mean horizontal grid spacing of 18 km. There are
28    50 vertical levels ranging in thickness from 10 m near the surface to
29    approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from the
30    National Geophysical Data Center (NGDC) 2-minute gridded global relief data
31    (ETOPO2) and the model employs the partial-cell formulation of
32    \citet{adcroft97:_shaved_cells}, which permits accurate representation of the
33    bathymetry. 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    The ocean model is coupled to the sea-ice model discussed in
39    \refsec{model} using the following specific options.  The
40    zero-heat-capacity thermodynamics formulation of \citet{hibler80} is used to
41    compute sea ice thickness and concentration.  Snow cover and sea ice salinity
42    are prognostic.  Open water, dry ice, wet ice, dry snow, and wet snow albedo
43    are, respectively, 0.15, 0.88, 0.79, 0.97, and 0.83. Ice mechanics follow the
44    viscous plastic rheology of \citet{hibler79} and the ice momentum equation is
45    solved numerically using the C-grid implementation of the \citet{zhang97} LSR
46    dynamics model discussed hereinabove.  The ice is coupled to the ocean using
47    the rescaled vertical coordinate system, z$^\ast$, of
48    \citet{cam08}, that is, sea ice does not float above the ocean model but
49    rather deforms the ocean's model surface level.
50    
51    This particular ECCO2 simulation is initialized from temperature and salinity
52    fields derived from the Polar science center Hydrographic Climatology (PHC)
53    3.0 \citep{ste01a}. Surface boundary conditions for the period January 1979 to
54    July 2002 are derived from the European Centre for Medium-Range Weather
55    Forecasts (ECMWF) 40 year re-analysis (ERA-40) \citep{upp05}.  Surface
56    boundary conditions after September 2002 are derived from the ECMWF
57    operational analysis.  There is a one month transition period, August 2002,
58    during which the ERA-40 contribution decreases linearly from 1 to 0 and the
59    ECMWF analysis contribution increases linearly from 0 to 1.  Six-hourly
60    surface winds, temperature, humidity, downward short- and long-wave
61    radiations, and precipitation are converted to heat, freshwater, and wind
62    stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
63    radiation decays exponentially as per \citet{pau77}.  Low frequency
64    precipitation has been adjusted using the pentad (5-day) data from the Global
65    Precipitation Climatology Project (GPCP) \citep{huf01}.  The time-mean river
66    run-off from \citet{lar01} is applied globally, except in the Arctic Ocean
67    where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)
68    and prepared by P. Winsor (personnal communication, 2007) is specificied.
69    Additionally, there is a relaxation to the monthly-mean climatological sea
70    surface salinity values from PHC 3.0, a relaxation time scale of 101 days.
71    
72    Vertical mixing follows \citet{lar94} but with meridionally and vertically
73    varying background vertical diffusivity; at the surface, vertical diffusivity
74    is $4.4\times 10^{-6}$~m$^2$~s$^{-1}$ at the Equator, $3.6\times
75    10^{-6}$~m$^2$~s$^{-1}$ north of 70$^\circ$N, and $1.9\times
76    10^{-5}$~m$^2$~s$^{-1}$ south of 30$^\circ$S and between 30$^\circ$N and
77    60$^\circ$N , with sinusoidally varying values in between these latitudes;
78    vertically, diffusivity increases to $1.1\times 10^{-4}$~m$^2$~s$^{-1}$ at a a
79    depth of 6150 m as per \citet{bry79}.  A high order monotonicity-preserving
80    advection scheme \citep{dar04} is employed and there is no explicit horizontal
81    diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
82    the divergent flow as per \citet{kem08}.
83    
84  \subsection{Arctic Domain with Open Boundaries}  \subsection{Arctic Domain with Open Boundaries}
85  \label{sec:arctic}  \label{sec:arctic}
86    
87  The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It  A series of forward sensitivity experiments have been carried out on an
88  is carved out from, and obtains open boundary conditions from, the  Arctic Ocean domain with open boundaries.  The objective is to compare the old
89  global cubed-sphere configuration of the Estimating the Circulation  B-grid LSR dynamic solver with the new C-grid LSR and EVP solvers.  One
90  and Climate of the Ocean, Phase II (ECCO2) project  additional experiment is carried out to illustrate the differences between the
91  \citet{menemenlis05}.  The domain size is 420 by 384 grid boxes  two main options for sea ice thermodynamics in the MITgcm.
92  horizontally with mean horizontal grid spacing of 18 km.  
93    The Arctic domain of integration is illustrated in \reffig{arctic1}.  It
94    is carved out from, and obtains open boundary conditions from, the global
95    cubed-sphere configuration described above.  The horizontal domain size is
96    420 by 384 grid boxes.
97    
98  \begin{figure}  \begin{figure}
99  %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1.eps}}}  \centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/topography}}}
100  \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}  \caption{Bathymetry and domain boudaries of Arctic
101      Domain.\label{fig:arctic1}}
102  \end{figure}  \end{figure}
103    
104  There are 50 vertical levels ranging in thickness from 10 m near the surface  The main dynamic difference from cube sphere is that it does not use
105  to approximately 450 m at a maximum model depth of 6150 m. Bathymetry is from  rescaled vertical coordinates (z$^\ast$) and the surface boundary
106  the National Geophysical Data Center (NGDC) 2-minute gridded global relief  conditions for freshwater input are different, because those features
107  data (ETOPO2) and the model employs the partial-cell formulation of  are not supported by the open boundary code.
 \citet{adcroft97:_shaved_cells}, which permits accurate representation of the bathymetry. The  
 model is integrated in a volume-conserving configuration using a finite volume  
 discretization with C-grid staggering of the prognostic variables. In the  
 ocean, the non-linear equation of state of \citet{jackett95}.  The ocean model is  
 coupled to a sea-ice model described hereinabove.    
   
 This particular ECCO2 simulation is initialized from rest using the  
 January temperature and salinity distribution from the World Ocean  
 Atlas 2001 (WOA01) [Conkright et al., 2002] and it is integrated for  
 32 years prior to the 1996--2001 period discussed in the study. Surface  
 boundary conditions are from the National Centers for Environmental  
 Prediction and the National Center for Atmospheric Research  
 (NCEP/NCAR) atmospheric reanalysis [Kistler et al., 2001]. Six-hourly  
 surface winds, temperature, humidity, downward short- and long-wave  
 radiations, and precipitation are converted to heat, freshwater, and  
 wind stress fluxes using the \citet{large81, large82} bulk formulae.  
 Shortwave radiation decays exponentially as per Paulson and Simpson  
 [1977]. Additionally the time-mean river run-off from Large and Nurser  
 [2001] is applied and there is a relaxation to the monthly-mean  
 climatological sea surface salinity values from WOA01 with a  
 relaxation time scale of 3 months. Vertical mixing follows  
 \citet{large94} with background vertical diffusivity of  
 $1.5\times10^{-5}\text{\,m$^{2}$\,s$^{-1}$}$ and viscosity of  
 $10^{-3}\text{\,m$^{2}$\,s$^{-1}$}$. A third order, direct-space-time  
 advection scheme with flux limiter is employed \citep{hundsdorfer94}  
 and there is no explicit horizontal diffusivity. Horizontal viscosity  
 follows \citet{lei96} but  
 modified to sense the divergent flow as per Fox-Kemper and Menemenlis  
 [in press].  Shortwave radiation decays exponentially as per Paulson  
 and Simpson [1977].  Additionally, the time-mean runoff of Large and  
 Nurser [2001] is applied near the coastline and, where there is open  
 water, there is a relaxation to monthly-mean WOA01 sea surface  
 salinity with a time constant of 45 days.  
108    
109  Open water, dry  Open water, dry ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,
 ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,  
110  0.76, 0.94, and 0.8.  0.76, 0.94, and 0.8.
111    
112    The model is integrated from January, 1992 to March \ml{[???]}, 2000,
113    with five different dynamical solvers:
114    \begin{description}
115    \item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an Arakawa
116      B-grid, implying no-slip lateral boundary conditions;
117    \item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
118      boundary conditions;
119    \item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
120      conditions;
121    \item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
122      no-slip lateral boundary conditions; and
123    \item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
124      boundary conditions.
125    \end{description}
126    Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
127    that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
128    interpreted as pure model error. Lateral boundary conditions on a
129    coarse grid (compared to the roughness of the true coast line) are
130    unclear, so that comparing the no-slip solutions to the free-slip
131    solutions gives another measure of uncertainty in sea ice modeling.
132    
133    A principle difficulty in comparing the solutions obtained with
134    different variants of the dynamics solver lies in the non-linear
135    feedback of the ice dynamics and thermodynamics. Already after a few
136    months the solutions have diverged so far from each other that
137    comparing velocities only makes sense within the first 3~months of the
138    integration while the ice distribution is still close to the initial
139    conditions. At the end of the integration, the differences between the
140    model solutions can be interpreted as cumulated model uncertainties.
141    
142    \reffig{iceveloc} shows ice velocities averaged over Janunary,
143    February, and March (JFM) of 1992 for the C-LSR-ns solution; also
144    shown are the differences between B-grid and C-grid, LSR and EVP, and
145    no-slip and free-slip solution. The velocity field of the C-LSR-ns
146    solution (\reffig{iceveloc}a) roughly resembles the drift velocities
147    of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
148    models in an cyclonic circulation regime (CCR) \citep[their
149    Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
150    shifted eastwards towards Alaska.
151    
152    The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
153    is most pronounced
154    along the coastlines, where the discretization differs most between B
155    and C-grids: On a B-grid the tangential velocity is on the boundary
156    (and thus zero per the no-slip boundary conditions), whereas on the
157    C-grid the its half a cell width away from the boundary, thus allowing
158    more flow. The B-LSR-ns solution has less ice drift through the Fram
159    Strait and especially the along Greenland's east coast; also, the flow
160    through Baffin Bay and Davis Strait into the Labrador Sea is reduced
161    with respect the C-LSR-ns solution. \ml{[Do we expect this? Say
162      something about that]}
163    %
164    Compared to the differences between B and C-grid solutions the
165    C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
166    (\reffig{iceveloc}c).  As expected the differences are largest along
167    coastlines: because of the free-slip boundary conditions, flow is
168    faster in the C-LSR-fs solution, for example, along the east coast
169    of Greenland, the north coast of Alaska, and the east Coast of Baffin
170    Island.
171    \begin{figure}[htbp]
172      \centering
173      \subfigure[{\footnotesize C-LSR-ns}]
174      {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_lsr_noslip}}
175      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
176      {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_bgrid-lsr_noslip}}\\
177      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
178      {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_lsr_slip-lsr_noslip}}
179      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
180      {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_evp_noslip-lsr_noslip}}
181      \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
182        over the first 3 months of integration [cm/s]; (b)-(d) difference
183        between B-LSR-ns, C-LSR-fs, C-EVP-ns, and C-LSR-ns solutions
184        [cm/s]; color indicates speed (or differences of speed), vectors
185        indicate direction only.}
186      \label{fig:iceveloc}
187    \end{figure}
188    
189    The C-EVP-ns solution is very different from the C-LSR-ns solution
190    (\reffig{iceveloc}d). The EVP-approximation of the VP-dynamics allows
191    for increased drift by over 2\,cm/s in the Beaufort Gyre and the
192    transarctic drift. \ml{Also the Beaufort Gyre is moved towards Alaska
193      in the C-EVP-ns solution. [Really?]} In general, drift velocities are
194    biased towards higher values in the EVP solutions as can be seen from
195    a histogram of the differences in \reffig{drifthist}.
196    \begin{figure}[htbp]
197      \centering
198      \includegraphics[width=\textwidth]{\fpath/drifthist_evp_noslip-lsr_noslip}
199      \caption{Histogram of drift velocity differences for C-LSR-ns and
200        C-EVP-ns solution [cm/s].}
201      \label{fig:drifthist}
202    \end{figure}
203    
204    \reffig{icethick}a shows the effective thickness (volume per unit
205    area) of the C-LSR-ns solution, averaged over January, February, March
206    of year 2000. By this time of the integration, the differences in the
207    ice drift velocities have led to the evolution of very different ice
208    thickness distributions, which are shown in \reffig{icethick}b--d, and
209    area distributions (not shown).  \ml{Compared to other solutions, for
210      example, AOMIP the ice thickness distribution blablabal} \ml{[What
211      can I say about effective thickness?]}
212    \begin{figure}[htbp]
213      \centering
214      \subfigure[{\footnotesize C-LSR-ns}]
215      {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_lsr_noslip}}
216      \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
217      {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_bgrid-lsr_noslip}}\\
218      \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
219      {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_lsr_slip-lsr_noslip}}
220      \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
221      {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_evp_noslip-lsr_noslip}}
222      \caption{(a) Effective thickness (volume per unit area) of the
223        C-LSR-ns solution, averaged over the months Janurary through March
224        2000 [m]; (b)-(d) difference between B-LSR-ns, C-LSR-fs, C-EVP-ns,
225        and C-LSR-ns solutions [cm/s].}
226      \label{fig:icethick}
227    \end{figure}
228    
229    The generally weaker ice drift velocities in the B-LSR-ns solution,
230    when compared to the C-LSR-ns solution, in particular through the
231    narrow passages in the Canadian Archipelago, lead to a larger build-up
232    of ice north of Greenland and the Archipelago by 2\,m effective
233    thickness and more in the B-grid solution (\reffig{icethick}b). But
234    the ice volume in not larger everywhere: further west, there are
235    patches of smaller ice volume in the B-grid solution, most likely
236    because the Beaufort Gyre is weaker and hence not as effective in
237    transporting ice westwards. There are also dipoles of ice volume
238    differences on the \ml{luv [what is this in English?]} and the lee of
239    island groups, such as Franz-Josef-Land and \ml{IDONTKNOW}, which
240    \ml{\ldots [I find hard to interpret].}
241    
242    Imposing a free-slip boundary condition in C-LSR-fs leads to a much
243    smaller differences to C-LSR-ns than the transition from the B-grid to
244    the C-grid (\reffig{icethick}c), but in the Canadian Archipelago it
245    still reduces the effective ice thickness by up to 2\,m where the ice
246    is thick and the straits are narrow. Everywhere else the ice volume is
247    affected only slightly by the different boundary condition.
248    %
249    The C-EVP-ns solution has generally stronger drift velocities then the
250    C-LSR-ns solution. Consequently, more ice can be moved the eastern
251    part of the Arctic, where ice volumes are smaller, to the western
252    Arctic where ice piles up along the coast (\reffig{icethick}d). Within
253    the Canadian Archipelago, more drift leads to faster ice export and
254    reduced effective ice thickness.
255    
256    The difference in ice volume and ice drift velocities between the
257    different experiments has consequences for the ice transport out of
258    the Arctic. Although the main export of ice goes through the Fram
259    Strait, a considerable amoung of ice is exported through the Canadian
260    Archipelago \citep{???}. \reffig{archipelago} shows a time series of
261    daily averages ice transport through various straits in the Canadian
262    Archipelago and the Fram Strait for the different model solutions.
263    Generally, the C-EVP-ns solution has highest maxiumum (export out of
264    the Artic) and minimum (import into the Artic) fluxes as the drift
265    velocities area largest in this solution \ldots
266    \begin{figure}
267    \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
268    \caption{Transport through Canadian Archipelago for different solver flavors.
269    \label{fig:archipelago}}
270    \end{figure}
271    
272    \ml{[Transport to narrow straits, area?, more runs, TEM, advection
273      schemes, Winton TD, discussion about differences in terms of model
274      error? that's tricky as it means refering to Tremblay, thus our ice
275      models are all erroneous!]}
276    
277    In summary, we find that different dynamical solvers can yield very
278    different solutions. Compared to that the differences between
279    free-slip and no-slip solutions \emph{with the same solver} are
280    considerably smaller (the difference for the EVP solver is not shown,
281    but comparable to that for the LSOR solver)---albeit smaller, the
282    differences between free and no-slip solutions in ice drift can lead
283    to large differences in ice volume over integration time. At first,
284    this observation appears counterintuitive, as we expect that the
285    solution \emph{technique} should not affect the \emph{solution} to a
286    lower degree than actually modifying the equations. A more detailed
287    study on these differences is beyond the scope of this paper, but at
288    this point we may speculate, that the large difference between B-grid,
289    C-grid, LSOR, and EVP solutions stem from incomplete convergence of
290    the solvers due to linearization \citep[and Bruno Tremblay, personal
291    communication]{hunke01}: if the convergence of the non-linear momentum
292    equations is not complete for all linearized solvers, then one can
293    imagine that each solver stops at a different point in velocity-space
294    thus leading to different solutions for the ice drift velocities. If
295    this were true, this tantalizing circumstance had a dramatic impact on
296    sea-ice modeling in general, and we would need to improve the solution
297    technique of dynamic sea ice model, most likely at a very high
298    compuational cost (Bruno Tremblay, personal communication).
299    
300    
301    
302  \begin{itemize}  \begin{itemize}
303  \item Configuration  \item Configuration
304  \item OBCS from cube  \item OBCS from cube
# Line 82  ice, wet ice, dry snow, and wet snow alb Line 317  ice, wet ice, dry snow, and wet snow alb
317  \end{itemize}  \end{itemize}
318    
319  \begin{itemize}  \begin{itemize}
320  \item B-grid LSR no-slip  \item B-grid LSR no-slip: B-LSR-ns
321  \item C-grid LSR no-slip  \item C-grid LSR no-slip: C-LSR-ns
322  \item C-grid LSR slip  \item C-grid LSR slip:    C-LSR-fs
323  \item C-grid EVP no-slip  \item C-grid EVP no-slip: C-EVP-ns
324  \item C-grid EVP slip  \item C-grid EVP slip:    C-EVP-fs
325  \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)  \item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
326  \item C-grid LSR no-slip + Winton    new flag): C-LSR-ns+TEM
327    \item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
328    \item C-grid LSR no-slip + Winton:
329  \item  speed-performance-accuracy (small)  \item  speed-performance-accuracy (small)
330    ice transport through Canadian Archipelago differences    ice transport through Canadian Archipelago differences
331    thickness distribution differences    thickness distribution differences
# Line 104  We anticipate small differences between Line 341  We anticipate small differences between
341  \item VP vs.\ EVP: speed performance, accuracy?  \item VP vs.\ EVP: speed performance, accuracy?
342  \item ocean stress: different water mass properties beneath the ice  \item ocean stress: different water mass properties beneath the ice
343  \end{itemize}  \end{itemize}
344    
345    %\begin{figure}
346    %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM1992uvice}}}
347    %\caption{Surface sea ice velocity for different solver flavors.
348    %\label{fig:iceveloc}}
349    %\end{figure}
350    
351    %\begin{figure}
352    %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM2000heff}}}
353    %\caption{Sea ice thickness for different solver flavors.
354    %\label{fig:icethick}}
355    %\end{figure}
356    
357    %%% Local Variables:
358    %%% mode: latex
359    %%% TeX-master: "ceaice"
360    %%% End:
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