ceaice_split_version/ceaice_part1/ceaice_model.tex

\section{Model Formulation}
\label{sec:model}

Traditionally, probably for historical reasons and the ease of
treating the Coriolis term, most standard sea-ice models are
discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
  kreyscher00, zhang98, hunke97}, although there are sea ice models
diretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,
  lemieux09}. %
\ml{[there is also MI-IM, but I only found this as a reference:
  \url{http://retro.met.no/english/r_and_d_activities/method/num_mod/MI-IM-Documentation.pdf}]}
From the perspective of coupling a sea ice-model to a C-grid ocean
model, the exchange of fluxes of heat and fresh-water pose no
difficulty for a B-grid sea-ice model \citep[e.g.,][]{timmermann02a}.
However, surface stress is defined at velocities points and thus needs
to be interpolated between a B-grid sea-ice model and a C-grid ocean
model. Smoothing implicitly associated with this interpolation may
mask grid scale noise and may contribute to stabilizing the solution.
On the other hand, by smoothing the stress signals are damped which
could lead to reduced variability of the system. By choosing a C-grid
for the sea-ice model, we circumvent this difficulty altogether and
render the stress coupling as consistent as the buoyancy coupling.

A further advantage of the C-grid formulation is apparent in narrow
straits. In the limit of only one grid cell between coasts there is no
flux allowed for a B-grid (with no-slip lateral boundary counditions),
and models have used topographies with artificially widened straits to
avoid this problem \citep{holloway07}. The C-grid formulation on the
other hand allows a flux of sea-ice through narrow passages if
free-slip along the boundaries is allowed. We demonstrate this effect
in the Candian Arctic Archipelago (CAA).

The MITgcm sea ice model (MITsim) is based on a variant of the
viscous-plastic (VP) dynamic-thermodynamic sea ice model
\citep{zhang97} first introduced by \citet{hibler79, hibler80}. In
order to adapt this model to the requirements of coupled
ice-ocean simulations, many important aspects of the original code have
been modified and improved:
\begin{itemize}
\item the code has been rewritten for an Arakawa C-grid, both B- and
  C-grid variants are available; the C-grid code allows for no-slip
  and free-slip lateral boundary conditions;
\item two different solution methods for solving the nonlinear
  momentum equations have been adopted: LSOR \citep{zhang97}, EVP
  \citep{hunke97};
\item ice-ocean stress can be formulated as in \citet{hibler87};
\item ice variables \ml{can be} advected by sophisticated, \ml{conservative}
  advection schemes \ml{with flux limiting};
\item growth and melt parameterizations have been refined and extended
  in order to allow for automatic differentiation of the code.
\end{itemize}

The sea ice model is tightly coupled to the ocean compontent of the
MITgcm \citep{marshall97:_finit_volum_incom_navier_stokes, mitgcm02}.
Heat, fresh water fluxes and surface stresses are computed from the
atmospheric state and modified by the ice model at every time step.
The model equations and details of their numerical realization are summarized
in the appendix. Further documentation and model code can be found at
\url{http://mitgcm.org}. 

%\subsection{C-grid}
%\begin{itemize}
%\item no-slip vs. free-slip for both lsr and evp; 
%  "diagnostics" to look at and use for comparison
%  \begin{itemize}
%  \item ice transport through Fram Strait/Denmark Strait/Davis
%    Strait/Bering strait (these are general) 
%  \item ice transport through narrow passages, e.g.\ Nares-Strait
%  \end{itemize}
%\item compare different advection schemes (if lsr turns out to be more
%  effective, then with lsr otherwise I prefer evp), eg. 
%  \begin{itemize}
%  \item default 2nd-order with diff1=0.002
%  \item 1st-order upwind with diff1=0.
%  \item DST3FL (SEAICEadvScheme=33 with diff1=0., should work, works for me)
%  \item 2nd-order wit flux limiter (SEAICEadvScheme=77 with diff1=0.)
%  \end{itemize}
%  That should be enough. Here, total ice mass and location of ice edge
%  is interesting. However, this comparison can be done in an idealized
%  domain, may not require full Arctic Domain?
%\item
%Do a little study on the parameters of LSR and EVP
%\begin{enumerate}
%\item convergence of LSR, how many iterations do you need to get a
%  true elliptic yield curve 
%\item vary deltaTevp and the relaxation parameter for EVP and see when
%  the EVP solution breaks down (relative to the forcing time scale).
%  For this, it is essential that the evp solver gives use "stripeless"
%  solutions, that is your dtevp = 1sec solutions/or 10sec solutions
%  with SEAICE\_evpDampC = 615.
%\end{enumerate}

%\end{itemize}

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1	heimbach	1.1	\section{Model Formulation}
2			\label{sec:model}
3
4			Traditionally, probably for historical reasons and the ease of
5			treating the Coriolis term, most standard sea-ice models are
6			discretized on Arakawa-B-grids \citep[e.g.,][]{hibler79, harder99,
7			kreyscher00, zhang98, hunke97}, although there are sea ice models
8			diretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,
9			lemieux09}. %
10			\ml{[there is also MI-IM, but I only found this as a reference:
11			\url{http://retro.met.no/english/r_and_d_activities/method/num_mod/MI-IM-Documentation.pdf}]}
12			From the perspective of coupling a sea ice-model to a C-grid ocean
13			model, the exchange of fluxes of heat and fresh-water pose no
14			difficulty for a B-grid sea-ice model \citep[e.g.,][]{timmermann02a}.
15			However, surface stress is defined at velocities points and thus needs
16			to be interpolated between a B-grid sea-ice model and a C-grid ocean
17			model. Smoothing implicitly associated with this interpolation may
18			mask grid scale noise and may contribute to stabilizing the solution.
19			On the other hand, by smoothing the stress signals are damped which
20			could lead to reduced variability of the system. By choosing a C-grid
21			for the sea-ice model, we circumvent this difficulty altogether and
22			render the stress coupling as consistent as the buoyancy coupling.
23
24			A further advantage of the C-grid formulation is apparent in narrow
25			straits. In the limit of only one grid cell between coasts there is no
26			flux allowed for a B-grid (with no-slip lateral boundary counditions),
27			and models have used topographies with artificially widened straits to
28			avoid this problem \citep{holloway07}. The C-grid formulation on the
29			other hand allows a flux of sea-ice through narrow passages if
30			free-slip along the boundaries is allowed. We demonstrate this effect
31			in the Candian Arctic Archipelago (CAA).
32
33			The MITgcm sea ice model (MITsim) is based on a variant of the
34			viscous-plastic (VP) dynamic-thermodynamic sea ice model
35			\citep{zhang97} first introduced by \citet{hibler79, hibler80}. In
36			order to adapt this model to the requirements of coupled
37			ice-ocean simulations, many important aspects of the original code have
38			been modified and improved:
39			\begin{itemize}
40			\item the code has been rewritten for an Arakawa C-grid, both B- and
41			C-grid variants are available; the C-grid code allows for no-slip
42			and free-slip lateral boundary conditions;
43			\item two different solution methods for solving the nonlinear
44			momentum equations have been adopted: LSOR \citep{zhang97}, EVP
45			\citep{hunke97};
46			\item ice-ocean stress can be formulated as in \citet{hibler87};
47			\item ice variables \ml{can be} advected by sophisticated, \ml{conservative}
48			advection schemes \ml{with flux limiting};
49			\item growth and melt parameterizations have been refined and extended
50			in order to allow for automatic differentiation of the code.
51			\end{itemize}
52
53			The sea ice model is tightly coupled to the ocean compontent of the
54			MITgcm \citep{marshall97:_finit_volum_incom_navier_stokes, mitgcm02}.
55			Heat, fresh water fluxes and surface stresses are computed from the
56			atmospheric state and modified by the ice model at every time step.
57			The model equations and details of their numerical realization are summarized
58			in the appendix. Further documentation and model code can be found at
59			\url{http://mitgcm.org}.
60
61			%\subsection{C-grid}
62			%\begin{itemize}
63			%\item no-slip vs. free-slip for both lsr and evp;
64			% "diagnostics" to look at and use for comparison
65			% \begin{itemize}
66			% \item ice transport through Fram Strait/Denmark Strait/Davis
67			% Strait/Bering strait (these are general)
68			% \item ice transport through narrow passages, e.g.\ Nares-Strait
69			% \end{itemize}
70			%\item compare different advection schemes (if lsr turns out to be more
71			% effective, then with lsr otherwise I prefer evp), eg.
72			% \begin{itemize}
73			% \item default 2nd-order with diff1=0.002
74			% \item 1st-order upwind with diff1=0.
75			% \item DST3FL (SEAICEadvScheme=33 with diff1=0., should work, works for me)
76			% \item 2nd-order wit flux limiter (SEAICEadvScheme=77 with diff1=0.)
77			% \end{itemize}
78			% That should be enough. Here, total ice mass and location of ice edge
79			% is interesting. However, this comparison can be done in an idealized
80			% domain, may not require full Arctic Domain?
81			%\item
82			%Do a little study on the parameters of LSR and EVP
83			%\begin{enumerate}
84			%\item convergence of LSR, how many iterations do you need to get a
85			% true elliptic yield curve
86			%\item vary deltaTevp and the relaxation parameter for EVP and see when
87			% the EVP solution breaks down (relative to the forcing time scale).
88			% For this, it is essential that the evp solver gives use "stripeless"
89			% solutions, that is your dtevp = 1sec solutions/or 10sec solutions
90			% with SEAICE\_evpDampC = 615.
91			%\end{enumerate}
92
93			%\end{itemize}
94
95			%%% Local Variables:
96			%%% mode: latex
97			%%% TeX-master: "ceaice"
98			%%% End: