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-revision 1.7 by dimitri,
Fri Feb 29 01:25:55 2008 UTC
+revision 1.14 by mlosch,
Tue Apr 29 14:04:15 2008 UTC
 Line 6 
 ice simulations that exercise various ca
  model.  The first set of results is from a global, eddy-permitting, ocean and
  sea ice configuration.  The second set of results is from a regional Arctic
  configuration, which is used to compare the B-grid and C-grid dynamic solvers
- and various other capabilities of the MITgcm sea ice model.  The third set of
+ and various other capabilities of the MITgcm sea ice model.
+ %
+ \ml{[do we really want to do this?:] The third set of
  results is from a yet smaller regional domain, which is used to illustrate
- treatment of sea ice open boundary condition sin the MITgcm.
+ treatment of sea ice open boundary condition in the MITgcm.}
  \subsection{Global Ocean and Sea Ice Simulation}
  \label{sec:global}
-Line 36 
 variables. In the ocean, the non-linear
+Line 38 
 variables. In the ocean, the non-linear
  used.
  The ocean model is coupled to the sea-ice model discussed in
- Section~\ref{sec:model} using the following specific options.  The
+ \refsec{model} using the following specific options.  The
- zero-heat-capacity thermodynamics formulation of \citet{hib80} is used to
+ zero-heat-capacity thermodynamics formulation of \citet{hibler80} is
- compute sea ice thickness and concentration.  Snow cover and sea ice salinity
+ used to compute sea ice thickness and concentration.  Snow cover and
- are prognostic.  Open water, dry ice, wet ice, dry snow, and wet snow albedo
+ sea ice salinity are prognostic.  Open water, dry ice, wet ice, dry
- are, respectively, 0.15, 0.88, 0.79, 0.97, and 0.83. Ice mechanics follow the
+ snow, and wet snow albedo are, respectively, 0.15, 0.88, 0.79, 0.97,
- viscous plastic rheology of \citet{hibler79} and the ice momentum equation is
+ and 0.83. Ice mechanics follow the viscous plastic rheology of
- solved numerically using the C-grid implementation of the \citet{zhang97} LSR
+ \citet{hibler79} and the ice momentum equation is solved numerically
- dynamics model discussed hereinabove.  The ice is coupled to the ocean using
+ using the C-grid implementation of the \citet{zhang97}'s LSOR dynamics
- the rescaled vertical coordinate system, z$^\ast$, of
+ model discussed hereinabove.  The ice is coupled to the ocean using
- \citet{cam08}, that is, sea ice does not float above the ocean model but
+ the rescaled vertical coordinate system, z$^\ast$, of \citet{cam08},
- rather deforms the ocean's model surface level.
+ that is, sea ice does not float above the ocean model but rather
+ deforms the ocean's model surface level.
  This particular ECCO2 simulation is initialized from temperature and salinity
  fields derived from the Polar science center Hydrographic Climatology (PHC)
-Line 62 
 radiations, and precipitation are conver
+Line 65 
 radiations, and precipitation are conver
  stress fluxes using the \citet{large81,large82} bulk formulae.  Shortwave
  radiation decays exponentially as per \citet{pau77}.  Low frequency
  precipitation has been adjusted using the pentad (5-day) data from the Global
- Precipitation Climatology Project (GPCP) \citep{huf01}.  The time-mean river
+ Precipitation Climatology Project \citep[GPCP][]{huf01}.  The time-mean river
  run-off from \citet{lar01} is applied globally, except in the Arctic Ocean
  where monthly mean river runoff based on the Arctic Runoff Data Base (ARDB)
  and prepared by P. Winsor (personnal communication, 2007) is specificied.
-Line 81 
 advection scheme \citep{dar04} is employ
+Line 84 
 advection scheme \citep{dar04} is employ
  diffusivity.  Horizontal viscosity follows \citet{lei96} but modified to sense
  the divergent flow as per \citet{kem08}.
+ \ml{[Dimitris, here you need to either provide figures, so that I can
+   write text, or you can provide both figures and text. I guess, one
+   figure, showing the northern and southern hemisphere in summer and
+   winter is fine (four panels), as we are showing so many figures in
+   the next section.]}
  \subsection{Arctic Domain with Open Boundaries}
  \label{sec:arctic}
- A series of forward sensitivity experiments have been carried out on an
+ A series of forward sensitivity experiments have been carried out on
- Arctic Ocean domain with open boundaries.  The objective is to compare the old
+ an Arctic Ocean domain with open boundaries.  The objective is to
- B-grid LSR dynamic solver with the new C-grid LSR and EVP solvers.  One
+ compare the old B-grid LSR dynamic solver with the new C-grid LSR and
- additional experiment is carried out to illustrate the differences between the
+ EVP solvers.  Additional experiments are is carried out to illustrate
- two main options for sea ice thermodynamics in the MITgcm.
+ the differences between different ice advection schemes, ocean-ice
+ stress formulations and the two main options for sea ice
- The Arctic domain of integration is illustrated in Fig.~\ref{fig:arctic1}.  It
+ thermodynamics in the MITgcm.
- is carved out from, and obtains open boundary conditions from, the global
- cubed-sphere configuration described above.  The horizontal domain size is
+ The Arctic domain of integration is illustrated in
-by 384 grid boxes.
+ \reffig{arctic_topog}.  It is carved out from, and obtains open
+ boundary conditions from, the global cubed-sphere configuration
+ described above.  The horizontal domain size is 420 by 384 grid boxes.
+ \begin{figure*}
+ \includegraphics*[width=0.44\linewidth,viewport=139 210 496 606,clip]{\fpath/topography}
+ %\includegraphics*[width=0.44\linewidth,viewport=0 0 496 606,clip]{\fpath/topography}
+ \includegraphics*[width=0.46\linewidth]{\fpath/archipelago}
+ \caption{Left: Bathymetry and domain boudaries of Arctic
+   Domain; the dashed line marks the boundaries of the inset on the
+   right hand side. The letters in the inset label sections in the
+   Canadian Archipelago, where ice transport is evaluated:
+   A: Nares Strait; %
+   B: \ml{Meighen Island}; %
+   C: Prince Gustaf Adolf Sea; %
+   D: \ml{Brock Island}; %
+   E: McClure Strait; %
+   F: Amundsen Gulf; %
+   G: Lancaster Sound; %
+   H: Barrow Strait \ml{W.}; %
+   I: Barrow Strait \ml{E.}; %
+   J: Barrow Strait \ml{N.}. %
+   \label{fig:arctic_topog}}
+ \end{figure*}
+ The main dynamic difference from cube sphere is that it does not use
+ rescaled vertical coordinates (z$^\ast$) and the surface boundary
+ conditions for freshwater input are different, because those features
+ are not supported by the open boundary code.
+ Open water, dry ice, wet ice, dry snow, and wet snow albedo are,
+ respectively, 0.15, 0.85, 0.76, 0.94, and 0.8.
+ The model is integrated from January, 1992 to March \ml{[???]}, 2000,
+ with three different dynamical solvers and two different boundary
+ conditions:
+ \begin{description}
+ \item[B-LSR-ns:] the original LSOR solver of \citet{zhang97} on an
+   Arakawa B-grid, implying no-slip lateral boundary conditions
+   ($\vek{u}=0$ exactly);
+ \item[C-LSR-ns:] the LSOR solver discretized on a C-grid with no-slip lateral
+   boundary conditions (implemented via ghost-points);
+ \item[C-LSR-fs:] the LSOR solver on a C-grid with free-slip lateral boundary
+   conditions;
+ \item[C-EVP-ns:] the EVP solver of \citet{hunke01} on a C-grid with
+   no-slip lateral boundary conditions;
+ \item[C-EVP-fs:] the EVP solver on a C-grid with free-slip lateral
+   boundary conditions;
+ \item[C-LSR-ns adv33:] C-LSR-ns with a third-order flux limited
+   direct-space-time advection scheme \citep{hundsdorfer94};
+ \item[C-LSR-ns TEM:] C-LSR-ns with a truncated
+   ellispe method (TEM) rheology \citep{hibler97};
+ \item[C-LSR-ns HB87:] C-LSR-ns with ocean-ice stress coupling according
+   to \citet{hibler87};
+ \item[C-EVP-ns damp:] C-EVP-ns with additional damping to reduce small
+   scale noise \citep{hunke01}.
+ \end{description}
+ Both LSOR and EVP solvers solve the same viscous-plastic rheology, so
+ that differences between runs B-LSR-ns, C-LSR-ns, and C-EVP-ns can be
+ interpreted as pure model error. Lateral boundary conditions on a
+ coarse grid (compared to the roughness of the true coast line) are
+ unclear, so that comparing the no-slip solutions to the free-slip
+ solutions gives another measure of uncertainty in sea ice
+ modeling. The remaining experiments explore further
+ sensitivities of the system to different physics (change in rheology,
+ advection and diffusion properties and stress coupling) and numerics
+ (numerical method to damp noise in the EVP solutions).
+ A principle difficulty in comparing the solutions obtained with
+ different variants of the dynamics solver lies in the non-linear
+ feedback of the ice dynamics and thermodynamics. Already after a few
+ months the solutions have diverged so far from each other that
+ comparing velocities only makes sense within the first 3~months of the
+ integration while the ice distribution is still close to the initial
+ conditions. At the end of the integration, the differences between the
+ model solutions can be interpreted as cumulated model uncertainties.
+ \reffig{iceveloc} shows ice velocities averaged over Janunary,
+ February, and March (JFM) of 1992 for the C-LSR-ns solution; also
+ shown are the differences between B-grid and C-grid, LSR and EVP, and
+ no-slip and free-slip solution. The velocity field of the C-LSR-ns
+ solution (\reffig{iceveloc}a) roughly resembles the drift velocities
+ of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
+ models in an cyclonic circulation regime (CCR) \citep[their
+ Figure\,6]{martin07} with a Beaufort Gyre and a transpolar drift
+ shifted eastwards towards Alaska.
+ The difference beween runs C-LSR-ns and B-LSR-ns (\reffig{iceveloc}b)
+ is most pronounced along the coastlines, where the discretization
+ differs most between B and C-grids: On a B-grid the tangential
+ velocity lies on the boundary (and is thus zero through the no-slip
+ boundary conditions), whereas on the C-grid it is half a cell width
+ away from the boundary, thus allowing more flow. The B-LSR-ns solution
+ has less ice drift through the Fram Strait and especially the along
+ Greenland's east coast; also, the flow through Baffin Bay and Davis
+ Strait into the Labrador Sea is reduced with respect the C-LSR-ns
+ solution.  \ml{[Do we expect this? Say something about that]}
+ %
+ Compared to the differences between B and C-grid solutions,the
+ C-LSR-fs ice drift field differs much less from the C-LSR-ns solution
+ (\reffig{iceveloc}c).  As expected the differences are largest along
+ coastlines: because of the free-slip boundary conditions, flow is
+ faster in the C-LSR-fs solution, for example, along the east coast
+ of Greenland, the north coast of Alaska, and the east Coast of Baffin
+ Island.
+ \begin{figure}[htbp]
+   \centering
+   \subfigure[{\footnotesize C-LSR-ns}]
+ %  {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_lsr_noslip}}
+   {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_C-LSR-ns}}
+   \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
+ %  {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_bgrid-lsr_noslip}}\\
+   {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_B-LSR-ns-C-LSR-ns}}\\
+   \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
+ %  {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_lsr_slip-lsr_noslip}}
+   {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_C-LSR-fs-C-LSR-ns}}
+   \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
+ %  {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_evp_noslip-lsr_noslip}}
+   {\includegraphics[width=0.44\textwidth]{\fpath/JFMuv_C-EVP-ns-C-LSR-ns}}
+   \caption{(a) Ice drift velocity of the C-LSR-ns solution averaged
+     over the first 3 months of integration [cm/s]; (b)-(d) difference
+     between B-LSR-ns, C-LSR-fs, C-EVP-ns, and C-LSR-ns solutions
+     [cm/s]; color indicates speed (or differences of speed), vectors
+     indicate direction only.}
+   \label{fig:iceveloc}
+ \end{figure}
+ The C-EVP-ns solution is very different from the C-LSR-ns solution
+ (\reffig{iceveloc}d). The EVP-approximation of the VP-dynamics allows
+ for increased drift by over 2\,cm/s in the Beaufort Gyre and the
+ transarctic drift. \ml{Also the Beaufort Gyre is moved towards Alaska
+   in the C-EVP-ns solution. [Really?]} In general, drift velocities are
+ biased towards higher values in the EVP solutions as can be seen from
+ a histogram of the differences in \reffig{drifthist}.
+ \begin{figure}[htbp]
+   \centering
+   \includegraphics[width=\textwidth]{\fpath/drifthist_C-EVP-ns-C-LSR-ns}
+   \caption{Histogram of drift velocity differences for C-LSR-ns and
+     C-EVP-ns solution [cm/s].}
+   \label{fig:drifthist}
+ \end{figure}
+ \reffig{icethick}a shows the effective thickness (volume per unit
+ area) of the C-LSR-ns solution, averaged over January, February, March
+ of year 2000. By this time of the integration, the differences in the
+ ice drift velocities have led to the evolution of very different ice
+ thickness distributions, which are shown in \reffig{icethick}b--d, and
+ area distributions (not shown).  \ml{Compared to other solutions, for
+   example, AOMIP the ice thickness distribution blablabal}
+ \begin{figure}[htbp]
+   \centering
+   \subfigure[{\footnotesize C-LSR-ns}]
+   {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_lsr_noslip}}
+   \subfigure[{\footnotesize B-LSR-ns $-$ C-LSR-ns}]
+   {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_bgrid-lsr_noslip}}\\
+   \subfigure[{\footnotesize C-LSR-fs $-$ C-LSR-ns}]
+   {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_lsr_slip-lsr_noslip}}
+   \subfigure[{\footnotesize C-EVP-ns $-$ C-LSR-ns}]
+   {\includegraphics[width=0.44\textwidth]{\fpath/JFMheff2000_evp_noslip-lsr_noslip}}
+   \caption{(a) Effective thickness (volume per unit area) of the
+     C-LSR-ns solution, averaged over the months Janurary through March
+[m]; (b)-(d) difference between B-LSR-ns, C-LSR-fs, C-EVP-ns,
+     and C-LSR-ns solutions [cm/s].}
+   \label{fig:icethick}
+ \end{figure}
+ %
+ The generally weaker ice drift velocities in the B-LSR-ns solution,
+ when compared to the C-LSR-ns solution, in particular through the
+ narrow passages in the Canadian Archipelago, lead to a larger build-up
+ of ice north of Greenland and the Archipelago by 2\,m effective
+ thickness and more in the B-grid solution (\reffig{icethick}b). But
+ the ice volume in not larger everywhere: further west, there are
+ patches of smaller ice volume in the B-grid solution, most likely
+ because the Beaufort Gyre is weaker and hence not as effective in
+ transporting ice westwards. There are also dipoles of ice volume
+ differences with more ice on the upstream side of island groups and
+ less ice in their lee, such as Franz-Josef-Land and \ml{IDONTKNOW},
+ because ice tends to flow along coasts less easily in the B-LSR-ns
+ solution.
+ Imposing a free-slip boundary condition in C-LSR-fs leads to a much
+ smaller differences to C-LSR-ns than the transition from the B-grid to
+ the C-grid (\reffig{icethick}c), but in the Canadian Archipelago it
+ still reduces the effective ice thickness by up to 2\,m where the ice
+ is thick and the straits are narrow. Dipoles of ice thickness
+ differences can also be observed around islands, because the free-slip
+ solution allows more flow around islands than the no-slip solution.
+ Everywhere else the ice volume is affected only slightly by the
+ different boundary condition.
+ %
+ The C-EVP-ns solution has generally stronger drift velocities than the
+ C-LSR-ns solution. Consequently, more ice can be moved from the eastern
+ part of the Arctic, where ice volumes are smaller, to the western
+ Arctic where ice piles up along the coast (\reffig{icethick}d). Within
+ the Canadian Archipelago, more drift leads to faster ice export and
+ reduced effective ice thickness.
+ The difference in ice volume and ice drift velocities between the
+ different experiments has consequences for the ice transport out of
+ the Arctic. Although by far the most exported ice drifts through the
+ Fram Strait (approximately $2300\pm610\text{\,km$^3$\,y$^{-1}$}$), a
+ considerable amount (order $160\text{\,km$^3$\,y$^{-1}$}$) ice is
+ exported through the Canadian Archipelago \citep[and references
+ therein]{serreze06}.  \reffig{archipelago} shows a time series of
+ \ml{[maybe smooth to different time scales:] daily averaged, smoothed
+   with monthly running means,} ice transports through various straits
+ in the Canadian Archipelago and the Fram Strait for the different
+ model solutions. The export through Fram Strait agrees with the
+ observations in all model solutions (annual averages range from $2112$
+ to $2425\text{\,km$^3$\,y$^{-1}$}$), while the export through
+ Lancaster Sound is lower (annual averages are $66$ to
+ $256\text{\,km$^3$\,y$^{-1}$}$) than observed
+ \citep[???][]{lancaster}.  Generally, the C-EVP solutions have highest
+ maximum (export out of the Artic) and minimum (import into the Artic)
+ fluxes as the drift velocities are largest in this solution.  In the
+ extreme, both B- and C-grid LSOR solvers have practically no ice
+ transport through the Nares Strait, which is only a few grid points
+ wide, while the C-EVP solutions allow up to
+ $600\text{\,km$^3$\,y$^{-1}$}$ in summer. As as consequence, the
+ import into the Candian Archipelago is overestimated in all EVP
+ solutions (range: $539$ to $773\text{\,km$^3$\,y$^{-1}$}$), while the
+ C-LSR solutions get the order of magnitude right (range: $132$ to
+ $165\text{\,km$^3$\,y$^{-1}$}$); the B-LSR-ns solution grossly
+ underestimates the ice transport with $77\text{\,km$^3$\,y$^{-1}$}$.
  \begin{figure}
- %\centerline{{\includegraphics*[width=0.44\linewidth]{\fpath/arctic1}}}
+ %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
- \caption{Bathymetry of Arctic Domain.\label{fig:arctic1}}
+ \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/ice_export}}}
+ \caption{Transport through Canadian Archipelago for different solver
+   flavors. The letters refer to the labels of the sections in
+   \reffig{arctic_topog}; positive values are flux out of the Arctic;
+   legend abbreviations are explained in \reftab{experiments}.
+ \label{fig:archipelago}}
  \end{figure}
- Difference from cube sphere is that it does not use z* coordinates nor
+ \ml{[Transport to narrow straits, area?, more runs, TEM, advection
- realfreshwater fluxes because it is not supported by open boundary code.
+   schemes, Winton TD, discussion about differences in terms of model
+   error? that's tricky as it means refering to Tremblay, thus our ice
+   models are all erroneous!]}
+ In summary, we find that different dynamical solvers can yield very
+ different solutions. In contrast, the differences between free-slip
+ and no-slip solutions \emph{with the same solver} are considerably
+ smaller (the difference for the EVP solver is not shown, but similar
+ to that for the LSOR solver). Albeit smaller, the differences between
+ free and no-slip solutions in ice drift can lead to large differences
+ in ice volume over the integration time. At first, this observation
+ seems counterintuitive, as we expect that the solution
+ \emph{technique} should not affect the \emph{solution} to a higher
+ degree than actually modifying the equations. A more detailed study on
+ these differences is beyond the scope of this paper, but at this point
+ we may speculate, that the large difference between B-grid, C-grid,
+ LSOR, and EVP solutions stem from incomplete convergence of the
+ solvers due to linearization and due to different methods of
+ linearization \citep[and Bruno Tremblay, personal
+ communication]{hunke01}: if the convergence of the non-linear momentum
+ equations is not complete for all linearized solvers, then one can
+ imagine that each solver stops at a different point in velocity-space
+ thus leading to different solutions for the ice drift velocities. If
+ this were true, this tantalizing circumstance had a dramatic impact on
+ sea-ice modeling in general, and we would need to improve the solution
+ technique of dynamic sea ice model, most likely at a very high
+ compuational cost (Bruno Tremblay, personal communication).
- Open water, dry
- ice, wet ice, dry snow, and wet snow albedo are, respectively, 0.15, 0.85,
-.76, 0.94, and 0.8.
  \begin{itemize}
  \item Configuration
-Line 125 
 ice, wet ice, dry snow, and wet snow alb
+Line 385 
 ice, wet ice, dry snow, and wet snow alb
  \end{itemize}
  \begin{itemize}
- \item B-grid LSR no-slip
+ \item B-grid LSR no-slip: B-LSR-ns
- \item C-grid LSR no-slip
+ \item C-grid LSR no-slip: C-LSR-ns
- \item C-grid LSR slip
+ \item C-grid LSR slip:    C-LSR-fs
- \item C-grid EVP no-slip
+ \item C-grid EVP no-slip: C-EVP-ns
- \item C-grid EVP slip
+ \item C-grid EVP slip:    C-EVP-fs
- \item C-grid LSR + TEM (truncated ellipse method, no tensile stress, new flag)
+ \item C-grid LSR + TEM (truncated ellipse method, no tensile stress,
- \item C-grid LSR no-slip + Winton
+   new flag): C-LSR-ns+TEM
+ \item C-grid LSR with different advection scheme: 33 vs 77, vs. default?
+ \item C-grid LSR no-slip + Winton:
  \item  speed-performance-accuracy (small)
    ice transport through Canadian Archipelago differences
    thickness distribution differences
-Line 148 
 We anticipate small differences between
+Line 410 
 We anticipate small differences between
  \item ocean stress: different water mass properties beneath the ice
  \end{itemize}
- \begin{figure}
+ %\begin{figure}
- \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM1992uvice}}}
+ %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM1992uvice}}}
- \caption{Surface sea ice velocity for different solver flavors.
+ %\caption{Surface sea ice velocity for different solver flavors.
- \label{fig:iceveloc}}
+ %\label{fig:iceveloc}}
- \end{figure}
+ %\end{figure}
- \begin{figure}
+ %\begin{figure}
- \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/Jan1992xport}}}
+ %\centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM2000heff}}}
- \caption{Transport through Canadian Archipelago for different solver flavors.
+ %\caption{Sea ice thickness for different solver flavors.
- \label{fig:archipelago}}
+ %\label{fig:icethick}}
- \end{figure}
+ %\end{figure}
- \begin{figure}
+ %%% Local Variables:
- \centerline{{\includegraphics*[width=0.6\linewidth]{\fpath/JFM2000heff}}}
+ %%% mode: latex
- \caption{Sea ice thickness nsitivity for different solver flavors.
+ %%% TeX-master: "ceaice"
- \label{fig:icethick}}
+ %%% End:
- \end{figure}

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