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
%discretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,
%  lemieux08}. %
%\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 velocity 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 conditions),
%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 Canadian Arctic Archipelago (CAA).

The MITgcm sea ice model is based on a variant of the
viscous-plastic (VP) dynamic-thermodynamic sea-ice model of
\citet{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, LSOR \citep{zhang97} and EVP
  \citep{hunke97, hunke01}, have been adopted,
\item ice-ocean stress can be formulated as in \citet{hibler87},
\item ice concentration and thickness, snow, and ice salinity or enthalpy can
  be advected by sophisticated, conservative
  advection schemes with flux limiters, and
\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 component 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 remainder of this section describes the model equations and
details of their numerical realization. Further documentation and
model code can be found at \url{http://mitgcm.org}.

\subsection{Dynamics}
\label{sec:dynamics}

Sea-ice motion is driven by ice-atmosphere, ice-ocean and internal
stresses. The internal stresses are evaluated following a
viscous-plastic (VP) constitutive law with an elliptic yield curve as
in \citet{hibler79}. The full momentum equations for the sea-ice model
and the solution by line successive over-relaxation (LSOR) are
described in \citet{zhang97}.  Alternatively, the momentum equation
can be solved with an elastic-viscous-plastic (EVP) solver following
\citet{hunke01}. In this technique, evolution equations for the
internal stress tensor components are solved by sub-cycling the sea ice
momentum solver within one ocean model time step.

In both cases, the bulk viscosities can be bounded from above (if
required for numerical reasons). For stress tensor computations the
replacement pressure \citep{hibler95} is used so that the stress state
always lies on the elliptic yield curve by definition. Alternatively,
in the so-called truncated ellipse method (TEM) the shear viscosity is
capped to suppress any tensile stress \citep{hibler97, geiger98}.

The horizontal gradient of the ocean's surface is estimated directly
from ocean sea surface height and pressure loading from atmosphere,
ice and snow \citep{campin08}.  Ice does not float on top of the
ocean, instead it depresses the ocean surface according to its thickness and
buoyancy.

Lateral boundary conditions are naturally ``no-slip'' for B-grids, as
the tangential velocities points lie on the boundary. For C-grids, the
lateral boundary condition for tangential velocities is realized via
``ghost points'', allowing alternatively no-slip or free-slip
conditions. In ocean models free-slip boundary conditions in
conjunction with piecewise-constant (``castellated'') coastlines have
been shown to reduce to no-slip boundary conditions
\citep{adcroft98:_slippery_coast}; for sea-ice models the effects of
lateral boundary conditions have not yet been studied (as far as we
know).

Moving sea ice exerts a surface stress on the ocean. In coupling the
sea-ice model to the ocean model, this stress is applied directly to
the surface layer of the ocean model. An alternative ocean stress
formulation is given by \citet{hibler87}. Rather than applying the
interfacial stress directly, the stress is derived from integrating
over the ice thickness to the bottom of the oceanic surface layer. In
the resulting equation for the \emph{combined} ocean-ice momentum, the
interfacial stress cancels and the total stress appears as the sum of
wind stress and divergence of internal ice stresses \citep[see also
Eq.\,2 of][]{hibler87}. While this formulation tightly embeds the
sea-ice into the surface layer of the ocean, its disadvantage is that
the velocity in the surface layer of the ocean that is used to
advect ocean tracers is really an average over the ocean surface
velocity and the ice velocity, leading to an inconsistency as the ice
temperature and salinity are different from the oceanic variables.
Both stress coupling options are available for a direct comparison of
the their effects on the sea-ice solution.

The discretization of the momentum equation is straightforward. It is
similar to that of \citet{zhang98, zhang03}, but differs fundamentally
in the underlying grid, namely the Arakawa C-grid. The EVP model, in
particular, is discretized naturally on the C-grid with the diagonal
components of the stress tensor on the center points and the
off-diagonal term on the corner (or vorticity) points of the grid.
With this choice all derivatives are discretized as central
differences and very little averaging is involved. Apart from the
standard C-grid implementation, the original B-grid implementation of
\citet{zhang97} is also available as an option in the code.

\subsection{Thermodynamics}
\label{sec:thermodynamics}

In its original formulation the sea ice model \citep{menemenlis05}
uses simple thermodynamics following the appendix of
\citet{semtner76}. This formulation does not allow storage of heat,
that is, the heat capacity of ice is zero. Upward conductive heat flux
is parameterized assuming a linear temperature profile and a constant
ice conductivity. This type of model is often referred to as a
``zero-layer'' model. The surface heat flux is computed in a similar
way to that of \citet{parkinson79} and \citet{manabe79}.

The conductive heat flux depends strongly on the ice thickness $h$.
However, the ice thickness in the model represents a mean over a
potentially very heterogeneous thickness distribution.  In order to
parameterize a sub-grid scale distribution for heat flux computations,
the mean ice thickness $h$ is split into seven thickness categories
$H_{n}$ that are equally distributed between $2h$ and a minimum
imposed ice thickness of $5\text{\,cm}$ by $H_n= \frac{2n-1}{7}\,h$
for $n\in[1,7]$. The heat fluxes computed for each thickness category
is area-averaged to give the total heat flux \citep{hibler84}.

The atmospheric heat flux is balanced by an oceanic heat flux from
below.  The oceanic flux is proportional to the difference between
ocean surface temperature and the freezing point temperature of sea
water, which is a function of salinity. Contrary to
\citet{menemenlis05}, this flux is not assumed to instantaneously melt
or create ice, but a time scale of three days is used to relax the
ocean temperature to the freezing point. While this
parameterization is not new \citep[it follows the ideas of,
e.g.,][]{mellor86, mcphee92, lohmann98, notz03}, it avoids a
discontinuity in the functional relationship between model variables,
which is crucial for making the code differentiable for adjoint code
generation \citep{fen09}.
%
The parameterization of lateral and vertical growth of sea ice follows
that of \citet{hibler79, hibler80}.

On top of the ice there is a layer of snow that modifies the heat flux
and the albedo as in \citet{zhang98}.  If enough snow accumulates so
that its weight submerges the ice and the snow is flooded, a simple
mass conserving parameterization of snow ice formation (a flood-freeze
algorithm following Archimedes' principle) turns snow into ice until
the ice surface is back at $z=0$ \citep{leppaeranta83}.

The concentration $c$, effective ice thickness (ice volume per unit
area, $c\cdot{h}$), effective snow thickness ($c\cdot{h}_{s}$), and effective
ice salinity (in g\,m$^{-2}$) are advected by ice velocities.
%
From the various advection scheme that are available in the MITgcm
\citep{mitgcm02}, we choose flux-limited schemes, i.e., multidimensional 2nd
and 3rd-order advection schemes with flux limiters
\citep{roe85, hundsdorfer94}, to preserve sharp gradients and
edges that are typical of sea ice distributions and to rule out
unphysical over- and undershoots (negative thickness or
concentration). These schemes conserve volume and horizontal area and
are unconditionally stable, so that no extra diffusion is required.
The original 2nd order central differences scheme, which requires additional
diffusion, has been retained for legacy tests and for comparison
with the newer schemes.

Finally, there is considerable doubt about the reliability of a
``zero-layer'' thermodynamic model --- \citet{semtner84} found
significant errors in phase (one month lead) and amplitude
($\approx$50\%\,overestimate) in such models --- so that today many
sea ice models employ more complex thermodynamics. The MITgcm
sea ice model provides the option to use the thermodynamics model of
\citet{winton00}, which in turn
is based on the 3-layer model of \citet{semtner76} and which treats brine
content by means of enthalpy conservation. This model requires
additional state variables, namely the enthalpy of the two ice
layers (instead of effective ice salinity), to be advected by ice velocities.
%  \ml{[Jean-Michel, your
%  turf: ]Care must be taken in advecting these quantities in order to
%  ensure conservation of enthalpy. Currently this can only be
%  accomplished with a 2nd-order advection scheme with flux limiter
%  \citep{roe85}.}
The internal sea ice temperature is inferred from ice enthalpy.
Since negative ice concentration or negative ice thickness is
not allowed, only positive advection schemes are suitable;
the 2nd order advection scheme with SuperBee limiter falls into this
category and has been used in this study to advect all sea-ice-related
quantities of the \citet{winton00} thermodynamic model.
Because of the non-linearity of the advection scheme,
care must be taken in advecting these quantities: when simply using 
ice velocity to advect enthalpy, the total energy (i.e., the volume
integral of enthalpy) is not conserved. Alternatively, one can advect 
the energy content (i.e., product of ice-volume and enthalpy)
but then false enthalpy extrema can occur,
which then leads to unrealistic ice temperature.
The solution currently implemented consists in
using the sea ice mass flux to advect the enthalpy: this
ensures conservation of enthalphy and prevents
the occurrence of false enthalpy extrema.

In \refsec{globalmodel} and \ref{sec:arcticmodel} we provide example
applications of the MITgcm sea ice model and we exercise and compare several
of the options, which have been discussed hereiabove.

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "ceaice_part1"
%%% End: 
1	heimbach	1.1	\section{Model Formulation}
2			\label{sec:model}
3
4	mlosch	1.2	%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	jmc	1.9	%discretized on a C-grid \citep[e.g.,][]{ip91, tremblay97,
9	mlosch	1.5	% lemieux08}. %
10	mlosch	1.2	%\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 velocity 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	jmc	1.9	%flux allowed for a B-grid (with no-slip lateral boundary conditions),
27	mlosch	1.2	%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	jmc	1.9	%in the Canadian Arctic Archipelago (CAA).
32	heimbach	1.1
33	dimitri	1.10	The MITgcm sea ice model is based on a variant of the
34	mlosch	1.2	viscous-plastic (VP) dynamic-thermodynamic sea-ice model of
35			\citet{zhang97} first introduced by \citet{hibler79, hibler80}. In
36			order to adapt this model to the requirements of coupled ice-ocean
37			simulations, many important aspects of the original code have been
38			modified and improved:
39	heimbach	1.1	\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	dimitri	1.11	and free-slip lateral boundary conditions,
43	heimbach	1.1	\item two different solution methods for solving the nonlinear
44	dimitri	1.11	momentum equations, LSOR \citep{zhang97} and EVP
45			\citep{hunke97, hunke01}, have been adopted,
46			\item ice-ocean stress can be formulated as in \citet{hibler87},
47			\item ice concentration and thickness, snow, and ice salinity or enthalpy can
48			be advected by sophisticated, conservative
49			advection schemes with flux limiters, and
50	heimbach	1.1	\item growth and melt parameterizations have been refined and extended
51			in order to allow for automatic differentiation of the code.
52			\end{itemize}
53
54	jmc	1.9	The sea ice model is tightly coupled to the ocean component of the
55	heimbach	1.1	MITgcm \citep{marshall97:_finit_volum_incom_navier_stokes, mitgcm02}.
56			Heat, fresh water fluxes and surface stresses are computed from the
57			atmospheric state and modified by the ice model at every time step.
58	mlosch	1.2	The remainder of this section describes the model equations and
59			details of their numerical realization. Further documentation and
60			model code can be found at \url{http://mitgcm.org}.
61
62			\subsection{Dynamics}
63			\label{sec:dynamics}
64
65			Sea-ice motion is driven by ice-atmosphere, ice-ocean and internal
66			stresses. The internal stresses are evaluated following a
67			viscous-plastic (VP) constitutive law with an elliptic yield curve as
68			in \citet{hibler79}. The full momentum equations for the sea-ice model
69			and the solution by line successive over-relaxation (LSOR) are
70			described in \citet{zhang97}. Alternatively, the momentum equation
71			can be solved with an elastic-viscous-plastic (EVP) solver following
72			\citet{hunke01}. In this technique, evolution equations for the
73	dimitri	1.6	internal stress tensor components are solved by sub-cycling the sea ice
74			momentum solver within one ocean model time step.
75	mlosch	1.2
76			In both cases, the bulk viscosities can be bounded from above (if
77			required for numerical reasons). For stress tensor computations the
78			replacement pressure \citep{hibler95} is used so that the stress state
79			always lies on the elliptic yield curve by definition. Alternatively,
80			in the so-called truncated ellipse method (TEM) the shear viscosity is
81			capped to suppress any tensile stress \citep{hibler97, geiger98}.
82
83			The horizontal gradient of the ocean's surface is estimated directly
84			from ocean sea surface height and pressure loading from atmosphere,
85	dimitri	1.11	ice and snow \citep{campin08}. Ice does not float on top of the
86			ocean, instead it depresses the ocean surface according to its thickness and
87			buoyancy.
88	mlosch	1.2
89			Lateral boundary conditions are naturally ``no-slip'' for B-grids, as
90			the tangential velocities points lie on the boundary. For C-grids, the
91			lateral boundary condition for tangential velocities is realized via
92			``ghost points'', allowing alternatively no-slip or free-slip
93			conditions. In ocean models free-slip boundary conditions in
94			conjunction with piecewise-constant (``castellated'') coastlines have
95			been shown to reduce to no-slip boundary conditions
96			\citep{adcroft98:_slippery_coast}; for sea-ice models the effects of
97			lateral boundary conditions have not yet been studied (as far as we
98			know).
99
100			Moving sea ice exerts a surface stress on the ocean. In coupling the
101	jmc	1.9	sea-ice model to the ocean model, this stress is applied directly to
102	mlosch	1.2	the surface layer of the ocean model. An alternative ocean stress
103			formulation is given by \citet{hibler87}. Rather than applying the
104			interfacial stress directly, the stress is derived from integrating
105			over the ice thickness to the bottom of the oceanic surface layer. In
106			the resulting equation for the \emph{combined} ocean-ice momentum, the
107			interfacial stress cancels and the total stress appears as the sum of
108	dimitri	1.6	wind stress and divergence of internal ice stresses \citep[see also
109	mlosch	1.2	Eq.\,2 of][]{hibler87}. While this formulation tightly embeds the
110			sea-ice into the surface layer of the ocean, its disadvantage is that
111	dimitri	1.11	the velocity in the surface layer of the ocean that is used to
112			advect ocean tracers is really an average over the ocean surface
113			velocity and the ice velocity, leading to an inconsistency as the ice
114	mlosch	1.2	temperature and salinity are different from the oceanic variables.
115			Both stress coupling options are available for a direct comparison of
116			the their effects on the sea-ice solution.
117
118	dimitri	1.6	The discretization of the momentum equation is straightforward. It is
119	mlosch	1.2	similar to that of \citet{zhang98, zhang03}, but differs fundamentally
120			in the underlying grid, namely the Arakawa C-grid. The EVP model, in
121			particular, is discretized naturally on the C-grid with the diagonal
122			components of the stress tensor on the center points and the
123			off-diagonal term on the corner (or vorticity) points of the grid.
124			With this choice all derivatives are discretized as central
125			differences and very little averaging is involved. Apart from the
126			standard C-grid implementation, the original B-grid implementation of
127			\citet{zhang97} is also available as an option in the code.
128
129			\subsection{Thermodynamics}
130			\label{sec:thermodynamics}
131
132			In its original formulation the sea ice model \citep{menemenlis05}
133			uses simple thermodynamics following the appendix of
134			\citet{semtner76}. This formulation does not allow storage of heat,
135			that is, the heat capacity of ice is zero. Upward conductive heat flux
136			is parameterized assuming a linear temperature profile and a constant
137	jmc	1.9	ice conductivity. This type of model is often referred to as a
138	mlosch	1.2	``zero-layer'' model. The surface heat flux is computed in a similar
139			way to that of \citet{parkinson79} and \citet{manabe79}.
140
141			The conductive heat flux depends strongly on the ice thickness $h$.
142			However, the ice thickness in the model represents a mean over a
143			potentially very heterogeneous thickness distribution. In order to
144			parameterize a sub-grid scale distribution for heat flux computations,
145			the mean ice thickness $h$ is split into seven thickness categories
146			$H_{n}$ that are equally distributed between $2h$ and a minimum
147			imposed ice thickness of $5\text{\,cm}$ by $H_n= \frac{2n-1}{7}\,h$
148			for $n\in[1,7]$. The heat fluxes computed for each thickness category
149			is area-averaged to give the total heat flux \citep{hibler84}.
150
151			The atmospheric heat flux is balanced by an oceanic heat flux from
152			below. The oceanic flux is proportional to the difference between
153	mlosch	1.3	ocean surface temperature and the freezing point temperature of sea
154			water, which is a function of salinity. Contrary to
155	mlosch	1.2	\citet{menemenlis05}, this flux is not assumed to instantaneously melt
156	mlosch	1.3	or create ice, but a time scale of three days is used to relax the
157	mlosch	1.8	ocean temperature to the freezing point. While this
158	dimitri	1.11	parameterization is not new \citep[it follows the ideas of,
159	mlosch	1.3	e.g.,][]{mellor86, mcphee92, lohmann98, notz03}, it avoids a
160			discontinuity in the functional relationship between model variables,
161	dimitri	1.6	which is crucial for making the code differentiable for adjoint code
162	dimitri	1.7	generation \citep{fen09}.
163	mlosch	1.2	%
164			The parameterization of lateral and vertical growth of sea ice follows
165			that of \citet{hibler79, hibler80}.
166
167			On top of the ice there is a layer of snow that modifies the heat flux
168			and the albedo as in \citet{zhang98}. If enough snow accumulates so
169			that its weight submerges the ice and the snow is flooded, a simple
170	dimitri	1.11	mass conserving parameterization of snow ice formation (a flood-freeze
171	mlosch	1.2	algorithm following Archimedes' principle) turns snow into ice until
172			the ice surface is back at $z=0$ \citep{leppaeranta83}.
173
174			The concentration $c$, effective ice thickness (ice volume per unit
175	dimitri	1.11	area, $c\cdot{h}$), effective snow thickness ($c\cdot{h}_{s}$), and effective
176			ice salinity (in g\,m$^{-2}$) are advected by ice velocities.
177	mlosch	1.2	%
178			From the various advection scheme that are available in the MITgcm
179	dimitri	1.11	\citep{mitgcm02}, we choose flux-limited schemes, i.e., multidimensional 2nd
180			and 3rd-order advection schemes with flux limiters
181			\citep{roe85, hundsdorfer94}, to preserve sharp gradients and
182	mlosch	1.2	edges that are typical of sea ice distributions and to rule out
183			unphysical over- and undershoots (negative thickness or
184	dimitri	1.6	concentration). These schemes conserve volume and horizontal area and
185	mlosch	1.2	are unconditionally stable, so that no extra diffusion is required.
186	dimitri	1.11	The original 2nd order central differences scheme, which requires additional
187			diffusion, has been retained for legacy tests and for comparison
188			with the newer schemes.
189	mlosch	1.2
190			Finally, there is considerable doubt about the reliability of a
191	dimitri	1.11	``zero-layer'' thermodynamic model --- \citet{semtner84} found
192	mlosch	1.2	significant errors in phase (one month lead) and amplitude
193	dimitri	1.11	($\approx$50\%\,overestimate) in such models --- so that today many
194			sea ice models employ more complex thermodynamics. The MITgcm
195			sea ice model provides the option to use the thermodynamics model of
196			\citet{winton00}, which in turn
197			is based on the 3-layer model of \citet{semtner76} and which treats brine
198	jmc	1.9	content by means of enthalpy conservation. This model requires
199			additional state variables, namely the enthalpy of the two ice
200	dimitri	1.11	layers (instead of effective ice salinity), to be advected by ice velocities.
201	jmc	1.9	% \ml{[Jean-Michel, your
202			% turf: ]Care must be taken in advecting these quantities in order to
203			% ensure conservation of enthalpy. Currently this can only be
204			% accomplished with a 2nd-order advection scheme with flux limiter
205			% \citep{roe85}.}
206	dimitri	1.11	The internal sea ice temperature is inferred from ice enthalpy.
207	jmc	1.9	Since negative ice concentration or negative ice thickness is
208			not allowed, only positive advection schemes are suitable;
209			the 2nd order advection scheme with SuperBee limiter falls into this
210	dimitri	1.11	category and has been used in this study to advect all sea-ice-related
211			quantities of the \citet{winton00} thermodynamic model.
212	jmc	1.9	Because of the non-linearity of the advection scheme,
213			care must be taken in advecting these quantities: when simply using
214			ice velocity to advect enthalpy, the total energy (i.e., the volume
215			integral of enthalpy) is not conserved. Alternatively, one can advect
216			the energy content (i.e., product of ice-volume and enthalpy)
217			but then false enthalpy extrema can occur,
218			which then leads to unrealistic ice temperature.
219			The solution currently implemented consists in
220	dimitri	1.11	using the sea ice mass flux to advect the enthalpy: this
221	jmc	1.9	ensures conservation of enthalphy and prevents
222			the occurrence of false enthalpy extrema.
223	heimbach	1.1
224	dimitri	1.11	In \refsec{globalmodel} and \ref{sec:arcticmodel} we provide example
225			applications of the MITgcm sea ice model and we exercise and compare several
226			of the options, which have been discussed hereiabove.
227
228	heimbach	1.1	%%% Local Variables:
229			%%% mode: latex
230	mlosch	1.2	%%% TeX-master: "ceaice_part1"
231	heimbach	1.1	%%% End: