ceaice_split_version/ceaice_part1/ceaice_model.tex

\section{Sea ice 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}.
Many aspects of the original codes have been
adapted; these are the most important ones:
\begin{itemize}
\item the model has been rewritten for an Arakawa~C grid, both B- and
  C-grid variants are available; the finite-volume 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
%ph(
% This is Ian's work, and will be reported elsewhere, I insist;
% or Ian should be co-author
%\item growth and melt parameterizations have been refined and extended
%  in order to allow for automatic differentiation of the code.
%ph)
\end{itemize}
The sea ice model is tightly coupled to the ocean component of the
MITgcm \citep{marshall97:_hydros_quasi_hydros_nonhy,marshall97:_finit_volum_incom_navier_stokes}.
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}

%\ml{[Sergey says: a kind of repetition; need to revise previous section]}
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, the evolution equations for the
% internal stress tensor components are solved by sub-cycling the sea ice
% momentum solver within one ocean model time step.
Alternatively, the elastic-viscous-plastic (EVP) technique following
\citet{hunke01} relaxed the ice state towards the VP rheology by
sub-cycling the evolution equations for the internal stress tensor
components and 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 within 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}. 
In an alternative to the elliptic yield curve, 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
allow 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 coupled ocean 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 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 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
their effects on the sea-ice solution.

The finite-volume discretization of the momentum equation on the
Arakawa C~grid is straightforward. The stress tensor divergence, 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, see
\refapp{discretization} for details. 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}

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 \citep{semtner76}. 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.
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.
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 
%ph: it doesn't improve adjoint code *generation*, it improves the smoothness
\ml{improves the smoothness of the differentiated model}
%(see companion, part 2, paper).
\ml{(see \cite{fent:09} for details).}
%\ml{[ONCE IT IS SUBMITTED, otherwise pers. communcations:]}\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 $z=0$
at sea-level \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. 
%\ml{[Sergey says: any FV scheme will do this]}

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 scheme 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.
To avoid unphysical (negative) values for ice thickness and
concentration, a positive 2nd-order advection scheme with a SuperBee
flux limiter \citep{roe85}
is 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.
In the currently implemented solution, the sea-ice mass flux is used
to advect the enthalpy in order to ensure conservation of enthalpy
and to prevent false enthalpy extrema.

In \refsec{globalmodel} and \ref{sec:arcticmodel}
we exercise and compare several
of the options, which have been discussed above; we intercompare
the impact of the different formulations (all of which are widely
used in sea ice modeling today) on Arctic sea ice simulation 
\citep{proshutinsky07:_aomip}.
%% Got to here..... more later
%% Add reference to JGR special issue here.....
%\citep{prosh07:_aomipspecial}.


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "ceaice_part1"
%%% End: 
1	cnh	1.16	\section{Sea ice model formulation}
2	heimbach	1.1	\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	mlosch	1.15	%discretized on Arakawa~B grids \citep[e.g.,][]{hibler79, harder99,
7	mlosch	1.2	% kreyscher00, zhang98, hunke97}, although there are sea ice models
8	mlosch	1.15	%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	cnh	1.16	\citet{zhang97} first introduced by \citet{hibler79, hibler80}.
36			Many aspects of the original codes have been
37	mlosch	1.22	adapted; these are the most important ones:
38	heimbach	1.1	\begin{itemize}
39	mlosch	1.15	\item the model has been rewritten for an Arakawa~C grid, both B- and
40	mlosch	1.22	C-grid variants are available; the finite-volume C-grid code allows
41			for no-slip and free-slip lateral boundary conditions,
42	heimbach	1.1	\item two different solution methods for solving the nonlinear
43	dimitri	1.11	momentum equations, LSOR \citep{zhang97} and EVP
44			\citep{hunke97, hunke01}, have been adopted,
45			\item ice-ocean stress can be formulated as in \citet{hibler87},
46			\item ice concentration and thickness, snow, and ice salinity or enthalpy can
47			be advected by sophisticated, conservative
48			advection schemes with flux limiters, and
49	heimbach	1.23	%ph(
50			% This is Ian's work, and will be reported elsewhere, I insist;
51			% or Ian should be co-author
52			%\item growth and melt parameterizations have been refined and extended
53			% in order to allow for automatic differentiation of the code.
54			%ph)
55	heimbach	1.1	\end{itemize}
56	jmc	1.9	The sea ice model is tightly coupled to the ocean component of the
57	cnh	1.16	MITgcm \citep{marshall97:_hydros_quasi_hydros_nonhy,marshall97:_finit_volum_incom_navier_stokes}.
58	heimbach	1.1	Heat, fresh water fluxes and surface stresses are computed from the
59			atmospheric state and modified by the ice model at every time step.
60	mlosch	1.2	The remainder of this section describes the model equations and
61			details of their numerical realization. Further documentation and
62			model code can be found at \url{http://mitgcm.org}.
63
64			\subsection{Dynamics}
65			\label{sec:dynamics}
66
67	mlosch	1.22	%\ml{[Sergey says: a kind of repetition; need to revise previous section]}
68	mlosch	1.2	Sea-ice motion is driven by ice-atmosphere, ice-ocean and internal
69			stresses. The internal stresses are evaluated following a
70			viscous-plastic (VP) constitutive law with an elliptic yield curve as
71			in \citet{hibler79}. The full momentum equations for the sea-ice model
72			and the solution by line successive over-relaxation (LSOR) are
73	mlosch	1.22	described in \citet{zhang97}. %Alternatively, the momentum equation
74			% can be solved with an elastic-viscous-plastic (EVP) solver following
75			% \citet{hunke01}. In this technique, the evolution equations for the
76			% internal stress tensor components are solved by sub-cycling the sea ice
77			% momentum solver within one ocean model time step.
78			Alternatively, the elastic-viscous-plastic (EVP) technique following
79			\citet{hunke01} relaxed the ice state towards the VP rheology by
80			sub-cycling the evolution equations for the internal stress tensor
81			components and the sea ice momentum solver within one ocean model time
82			step.
83	mlosch	1.2
84			In both cases, the bulk viscosities can be bounded from above (if
85			required for numerical reasons). For stress tensor computations the
86			replacement pressure \citep{hibler95} is used so that the stress state
87	mlosch	1.22	always lies within the elliptic yield curve by definition.
88			%Alternatively, in the so-called truncated ellipse method (TEM) the
89			%shear viscosity is capped to suppress any tensile stress
90			%\citep{hibler97, geiger98}.
91			In an alternative to the elliptic yield curve, the so-called truncated
92			ellipse method (TEM), the shear viscosity is capped to suppress any
93			tensile stress \citep{hibler97, geiger98}.
94	mlosch	1.2
95			The horizontal gradient of the ocean's surface is estimated directly
96			from ocean sea surface height and pressure loading from atmosphere,
97	dimitri	1.11	ice and snow \citep{campin08}. Ice does not float on top of the
98			ocean, instead it depresses the ocean surface according to its thickness and
99			buoyancy.
100	mlosch	1.2
101	mlosch	1.15	Lateral boundary conditions are naturally ``no-slip'' for B~grids, as
102			the tangential velocities points lie on the boundary. For C~grids, the
103	cnh	1.16	lateral boundary condition for tangential velocities
104			allow alternatively no-slip or free-slip
105	mlosch	1.2	conditions. In ocean models free-slip boundary conditions in
106			conjunction with piecewise-constant (``castellated'') coastlines have
107			been shown to reduce to no-slip boundary conditions
108	cnh	1.16	\citep{adcroft98:_slippery_coast}; for coupled ocean sea-ice models the effects of
109	mlosch	1.2	lateral boundary conditions have not yet been studied (as far as we
110			know).
111
112			Moving sea ice exerts a surface stress on the ocean. In coupling the
113	jmc	1.9	sea-ice model to the ocean model, this stress is applied directly to
114	mlosch	1.2	the surface layer of the ocean model. An alternative ocean stress
115			formulation is given by \citet{hibler87}. Rather than applying the
116			interfacial stress directly, the stress is derived from integrating
117			over the ice thickness to the bottom of the oceanic surface layer. In
118	mlosch	1.15	the resulting equation for the combined ocean-ice momentum, the
119	mlosch	1.2	interfacial stress cancels and the total stress appears as the sum of
120	dimitri	1.6	wind stress and divergence of internal ice stresses \citep[see also
121	mlosch	1.2	Eq.\,2 of][]{hibler87}. While this formulation tightly embeds the
122	mlosch	1.15	sea ice into the surface layer of the ocean, its disadvantage is that
123	dimitri	1.11	the velocity in the surface layer of the ocean that is used to
124	mlosch	1.22	advect ocean tracers is an average over the ocean surface
125	dimitri	1.11	velocity and the ice velocity, leading to an inconsistency as the ice
126	mlosch	1.2	temperature and salinity are different from the oceanic variables.
127			Both stress coupling options are available for a direct comparison of
128	mlosch	1.18	their effects on the sea-ice solution.
129	mlosch	1.2
130	mlosch	1.21	The finite-volume discretization of the momentum equation on the
131			Arakawa C~grid is straightforward. The stress tensor divergence, in
132	mlosch	1.15	particular, is discretized naturally on the C~grid with the diagonal
133	mlosch	1.2	components of the stress tensor on the center points and the
134			off-diagonal term on the corner (or vorticity) points of the grid.
135			With this choice all derivatives are discretized as central
136	mlosch	1.21	differences and very little averaging is involved, see
137			\refapp{discretization} for details. Apart from the standard C-grid
138			implementation, the original B-grid implementation of \citet{zhang97}
139			is also available as an option in the code.
140	mlosch	1.2
141			\subsection{Thermodynamics}
142			\label{sec:thermodynamics}
143
144	cnh	1.16	Upward conductive heat flux
145	mlosch	1.2	is parameterized assuming a linear temperature profile and a constant
146	jmc	1.9	ice conductivity. This type of model is often referred to as a
147	mlosch	1.22	``zero-layer'' model \citep{semtner76}. The surface heat flux is
148			computed in a similar
149	mlosch	1.2	way to that of \citet{parkinson79} and \citet{manabe79}.
150
151			The conductive heat flux depends strongly on the ice thickness $h$.
152			However, the ice thickness in the model represents a mean over a
153			potentially very heterogeneous thickness distribution. In order to
154			parameterize a sub-grid scale distribution for heat flux computations,
155			the mean ice thickness $h$ is split into seven thickness categories
156			$H_{n}$ that are equally distributed between $2h$ and a minimum
157			imposed ice thickness of $5\text{\,cm}$ by $H_n= \frac{2n-1}{7}\,h$
158			for $n\in[1,7]$. The heat fluxes computed for each thickness category
159			is area-averaged to give the total heat flux \citep{hibler84}.
160
161	mlosch	1.12	The atmospheric heat flux is balanced by an oceanic heat flux.
162			The oceanic flux is proportional to the difference between
163	mlosch	1.3	ocean surface temperature and the freezing point temperature of sea
164	cnh	1.16	water, which is a function of salinity.
165			This flux is not assumed to instantaneously melt
166	mlosch	1.3	or create ice, but a time scale of three days is used to relax the
167	mlosch	1.8	ocean temperature to the freezing point. While this
168	dimitri	1.11	parameterization is not new \citep[it follows the ideas of,
169	mlosch	1.3	e.g.,][]{mellor86, mcphee92, lohmann98, notz03}, it avoids a
170			discontinuity in the functional relationship between model variables,
171	heimbach	1.23	which
172			%is crucial for making the code differentiable for adjoint code generation
173			%ph: it doesn't improve adjoint code generation, it improves the smoothness
174			\ml{improves the smoothness of the differentiated model}
175			%(see companion, part 2, paper).
176			\ml{(see \cite{fent:09} for details).}
177	dimitri	1.13	%\ml{[ONCE IT IS SUBMITTED, otherwise pers. communcations:]}\citep{fen09}
178	mlosch	1.2	The parameterization of lateral and vertical growth of sea ice follows
179			that of \citet{hibler79, hibler80}.
180
181			On top of the ice there is a layer of snow that modifies the heat flux
182			and the albedo as in \citet{zhang98}. If enough snow accumulates so
183			that its weight submerges the ice and the snow is flooded, a simple
184	dimitri	1.11	mass conserving parameterization of snow ice formation (a flood-freeze
185	mlosch	1.2	algorithm following Archimedes' principle) turns snow into ice until
186	jmc	1.19	the ice surface is back %at $z=0$
187	mlosch	1.20	at sea-level \citep{leppaeranta83}.
188	mlosch	1.2
189			The concentration $c$, effective ice thickness (ice volume per unit
190	dimitri	1.11	area, $c\cdot{h}$), effective snow thickness ($c\cdot{h}_{s}$), and effective
191			ice salinity (in g\,m$^{-2}$) are advected by ice velocities.
192	mlosch	1.2	%
193			From the various advection scheme that are available in the MITgcm
194	dimitri	1.11	\citep{mitgcm02}, we choose flux-limited schemes, i.e., multidimensional 2nd
195			and 3rd-order advection schemes with flux limiters
196			\citep{roe85, hundsdorfer94}, to preserve sharp gradients and
197	mlosch	1.2	edges that are typical of sea ice distributions and to rule out
198			unphysical over- and undershoots (negative thickness or
199	mlosch	1.18	concentration). These schemes, conserve volume and horizontal area and
200			are unconditionally stable, so that no extra diffusion is required.
201			%\ml{[Sergey says: any FV scheme will do this]}
202	mlosch	1.2
203	cnh	1.16	There is considerable doubt about the reliability of a
204	dimitri	1.11	``zero-layer'' thermodynamic model --- \citet{semtner84} found
205	mlosch	1.2	significant errors in phase (one month lead) and amplitude
206	dimitri	1.11	($\approx$50\%\,overestimate) in such models --- so that today many
207			sea ice models employ more complex thermodynamics. The MITgcm
208			sea ice model provides the option to use the thermodynamics model of
209			\citet{winton00}, which in turn
210			is based on the 3-layer model of \citet{semtner76} and which treats brine
211	cnh	1.16	content by means of enthalpy conservation. This scheme requires
212	jmc	1.9	additional state variables, namely the enthalpy of the two ice
213	dimitri	1.11	layers (instead of effective ice salinity), to be advected by ice velocities.
214	jmc	1.9	% \ml{[Jean-Michel, your
215			% turf: ]Care must be taken in advecting these quantities in order to
216			% ensure conservation of enthalpy. Currently this can only be
217			% accomplished with a 2nd-order advection scheme with flux limiter
218			% \citep{roe85}.}
219	dimitri	1.11	The internal sea ice temperature is inferred from ice enthalpy.
220	cnh	1.16	To avoid unphysical (negative) values for ice thickness and
221	mlosch	1.12	concentration, a positive 2nd-order advection scheme with a SuperBee
222			flux limiter \citep{roe85}
223			is used in this study to advect all sea-ice-related
224	dimitri	1.11	quantities of the \citet{winton00} thermodynamic model.
225	jmc	1.9	Because of the non-linearity of the advection scheme,
226			care must be taken in advecting these quantities: when simply using
227			ice velocity to advect enthalpy, the total energy (i.e., the volume
228			integral of enthalpy) is not conserved. Alternatively, one can advect
229			the energy content (i.e., product of ice-volume and enthalpy)
230			but then false enthalpy extrema can occur,
231			which then leads to unrealistic ice temperature.
232	mlosch	1.15	In the currently implemented solution, the sea-ice mass flux is used
233	cnh	1.17	to advect the enthalpy in order to ensure conservation of enthalpy
234	mlosch	1.15	and to prevent false enthalpy extrema.
235	heimbach	1.1
236	cnh	1.16	In \refsec{globalmodel} and \ref{sec:arcticmodel}
237			we exercise and compare several
238	mlosch	1.22	of the options, which have been discussed above; we intercompare
239	cnh	1.16	the impact of the different formulations (all of which are widely
240	cnh	1.17	used in sea ice modeling today) on Arctic sea ice simulation
241	mlosch	1.18	\citep{proshutinsky07:_aomip}.
242	cnh	1.16	%% Got to here..... more later
243			%% Add reference to JGR special issue here.....
244	mlosch	1.18	%\citep{prosh07:_aomipspecial}.
245
246
247	dimitri	1.11
248	heimbach	1.1	%%% Local Variables:
249			%%% mode: latex
250	mlosch	1.2	%%% TeX-master: "ceaice_part1"
251	heimbach	1.1	%%% End: