ceaice_split_version/ceaice_part1/ceaice_model.tex

\section{Sea ice model formulation}
\label{sec:model}

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, freshwater 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
seawater, 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
improves the smoothness of the differentiated model
%(see companion, part 2, paper).
\citep[see][for details]{fent:09}.
%\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 schemes 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.
%
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	dimitri	1.10	The MITgcm sea ice model is based on a variant of the
5	mlosch	1.2	viscous-plastic (VP) dynamic-thermodynamic sea-ice model of
6	cnh	1.16	\citet{zhang97} first introduced by \citet{hibler79, hibler80}.
7			Many aspects of the original codes have been
8	mlosch	1.22	adapted; these are the most important ones:
9	heimbach	1.1	\begin{itemize}
10	mlosch	1.15	\item the model has been rewritten for an Arakawa~C grid, both B- and
11	mlosch	1.22	C-grid variants are available; the finite-volume C-grid code allows
12			for no-slip and free-slip lateral boundary conditions,
13	heimbach	1.1	\item two different solution methods for solving the nonlinear
14	dimitri	1.11	momentum equations, LSOR \citep{zhang97} and EVP
15			\citep{hunke97, hunke01}, have been adopted,
16			\item ice-ocean stress can be formulated as in \citet{hibler87},
17			\item ice concentration and thickness, snow, and ice salinity or enthalpy can
18			be advected by sophisticated, conservative
19			advection schemes with flux limiters, and
20	heimbach	1.23	%ph(
21			% This is Ian's work, and will be reported elsewhere, I insist;
22			% or Ian should be co-author
23			%\item growth and melt parameterizations have been refined and extended
24			% in order to allow for automatic differentiation of the code.
25			%ph)
26	heimbach	1.1	\end{itemize}
27	jmc	1.9	The sea ice model is tightly coupled to the ocean component of the
28	cnh	1.16	MITgcm \citep{marshall97:_hydros_quasi_hydros_nonhy,marshall97:_finit_volum_incom_navier_stokes}.
29	mlosch	1.25	Heat, freshwater fluxes and surface stresses are computed from the
30	heimbach	1.1	atmospheric state and modified by the ice model at every time step.
31	mlosch	1.2	The remainder of this section describes the model equations and
32			details of their numerical realization. Further documentation and
33			model code can be found at \url{http://mitgcm.org}.
34
35			\subsection{Dynamics}
36			\label{sec:dynamics}
37
38	mlosch	1.22	%\ml{[Sergey says: a kind of repetition; need to revise previous section]}
39	mlosch	1.2	Sea-ice motion is driven by ice-atmosphere, ice-ocean and internal
40			stresses. The internal stresses are evaluated following a
41			viscous-plastic (VP) constitutive law with an elliptic yield curve as
42			in \citet{hibler79}. The full momentum equations for the sea-ice model
43			and the solution by line successive over-relaxation (LSOR) are
44	mlosch	1.22	described in \citet{zhang97}. %Alternatively, the momentum equation
45			% can be solved with an elastic-viscous-plastic (EVP) solver following
46			% \citet{hunke01}. In this technique, the evolution equations for the
47			% internal stress tensor components are solved by sub-cycling the sea ice
48			% momentum solver within one ocean model time step.
49			Alternatively, the elastic-viscous-plastic (EVP) technique following
50			\citet{hunke01} relaxed the ice state towards the VP rheology by
51			sub-cycling the evolution equations for the internal stress tensor
52			components and the sea ice momentum solver within one ocean model time
53			step.
54	mlosch	1.2
55			In both cases, the bulk viscosities can be bounded from above (if
56			required for numerical reasons). For stress tensor computations the
57			replacement pressure \citep{hibler95} is used so that the stress state
58	mlosch	1.22	always lies within the elliptic yield curve by definition.
59			%Alternatively, in the so-called truncated ellipse method (TEM) the
60			%shear viscosity is capped to suppress any tensile stress
61			%\citep{hibler97, geiger98}.
62			In an alternative to the elliptic yield curve, the so-called truncated
63			ellipse method (TEM), the shear viscosity is capped to suppress any
64			tensile stress \citep{hibler97, geiger98}.
65	mlosch	1.2
66			The horizontal gradient of the ocean's surface is estimated directly
67			from ocean sea surface height and pressure loading from atmosphere,
68	dimitri	1.11	ice and snow \citep{campin08}. Ice does not float on top of the
69			ocean, instead it depresses the ocean surface according to its thickness and
70			buoyancy.
71	mlosch	1.2
72	mlosch	1.15	Lateral boundary conditions are naturally ``no-slip'' for B~grids, as
73			the tangential velocities points lie on the boundary. For C~grids, the
74	cnh	1.16	lateral boundary condition for tangential velocities
75			allow alternatively no-slip or free-slip
76	mlosch	1.2	conditions. In ocean models free-slip boundary conditions in
77			conjunction with piecewise-constant (``castellated'') coastlines have
78			been shown to reduce to no-slip boundary conditions
79	cnh	1.16	\citep{adcroft98:_slippery_coast}; for coupled ocean sea-ice models the effects of
80	mlosch	1.2	lateral boundary conditions have not yet been studied (as far as we
81			know).
82
83			Moving sea ice exerts a surface stress on the ocean. In coupling the
84	jmc	1.9	sea-ice model to the ocean model, this stress is applied directly to
85	mlosch	1.2	the surface layer of the ocean model. An alternative ocean stress
86			formulation is given by \citet{hibler87}. Rather than applying the
87			interfacial stress directly, the stress is derived from integrating
88			over the ice thickness to the bottom of the oceanic surface layer. In
89	mlosch	1.15	the resulting equation for the combined ocean-ice momentum, the
90	mlosch	1.2	interfacial stress cancels and the total stress appears as the sum of
91	dimitri	1.6	wind stress and divergence of internal ice stresses \citep[see also
92	mlosch	1.2	Eq.\,2 of][]{hibler87}. While this formulation tightly embeds the
93	mlosch	1.15	sea ice into the surface layer of the ocean, its disadvantage is that
94	dimitri	1.11	the velocity in the surface layer of the ocean that is used to
95	mlosch	1.22	advect ocean tracers is an average over the ocean surface
96	dimitri	1.11	velocity and the ice velocity, leading to an inconsistency as the ice
97	mlosch	1.2	temperature and salinity are different from the oceanic variables.
98			Both stress coupling options are available for a direct comparison of
99	mlosch	1.18	their effects on the sea-ice solution.
100	mlosch	1.2
101	mlosch	1.21	The finite-volume discretization of the momentum equation on the
102			Arakawa C~grid is straightforward. The stress tensor divergence, in
103	mlosch	1.15	particular, is discretized naturally on the C~grid with the diagonal
104	mlosch	1.2	components of the stress tensor on the center points and the
105			off-diagonal term on the corner (or vorticity) points of the grid.
106			With this choice all derivatives are discretized as central
107	mlosch	1.21	differences and very little averaging is involved, see
108			\refapp{discretization} for details. Apart from the standard C-grid
109			implementation, the original B-grid implementation of \citet{zhang97}
110			is also available as an option in the code.
111	mlosch	1.2
112			\subsection{Thermodynamics}
113			\label{sec:thermodynamics}
114
115	cnh	1.16	Upward conductive heat flux
116	mlosch	1.2	is parameterized assuming a linear temperature profile and a constant
117	jmc	1.9	ice conductivity. This type of model is often referred to as a
118	mlosch	1.22	``zero-layer'' model \citep{semtner76}. The surface heat flux is
119			computed in a similar
120	mlosch	1.2	way to that of \citet{parkinson79} and \citet{manabe79}.
121
122			The conductive heat flux depends strongly on the ice thickness $h$.
123			However, the ice thickness in the model represents a mean over a
124			potentially very heterogeneous thickness distribution. In order to
125			parameterize a sub-grid scale distribution for heat flux computations,
126			the mean ice thickness $h$ is split into seven thickness categories
127			$H_{n}$ that are equally distributed between $2h$ and a minimum
128			imposed ice thickness of $5\text{\,cm}$ by $H_n= \frac{2n-1}{7}\,h$
129			for $n\in[1,7]$. The heat fluxes computed for each thickness category
130			is area-averaged to give the total heat flux \citep{hibler84}.
131
132	mlosch	1.12	The atmospheric heat flux is balanced by an oceanic heat flux.
133			The oceanic flux is proportional to the difference between
134	mlosch	1.25	ocean surface temperature and the freezing point temperature of
135			seawater, which is a function of salinity.
136	cnh	1.16	This flux is not assumed to instantaneously melt
137	mlosch	1.3	or create ice, but a time scale of three days is used to relax the
138	mlosch	1.8	ocean temperature to the freezing point. While this
139	dimitri	1.11	parameterization is not new \citep[it follows the ideas of,
140	mlosch	1.3	e.g.,][]{mellor86, mcphee92, lohmann98, notz03}, it avoids a
141			discontinuity in the functional relationship between model variables,
142	heimbach	1.23	which
143			%is crucial for making the code differentiable for adjoint code generation
144			%ph: it doesn't improve adjoint code generation, it improves the smoothness
145	mlosch	1.24	improves the smoothness of the differentiated model
146	heimbach	1.23	%(see companion, part 2, paper).
147	mlosch	1.25	\citep[see][for details]{fent:09}.
148	dimitri	1.13	%\ml{[ONCE IT IS SUBMITTED, otherwise pers. communcations:]}\citep{fen09}
149	mlosch	1.2	The parameterization of lateral and vertical growth of sea ice follows
150			that of \citet{hibler79, hibler80}.
151
152			On top of the ice there is a layer of snow that modifies the heat flux
153			and the albedo as in \citet{zhang98}. If enough snow accumulates so
154			that its weight submerges the ice and the snow is flooded, a simple
155	dimitri	1.11	mass conserving parameterization of snow ice formation (a flood-freeze
156	mlosch	1.2	algorithm following Archimedes' principle) turns snow into ice until
157	jmc	1.19	the ice surface is back %at $z=0$
158	mlosch	1.20	at sea-level \citep{leppaeranta83}.
159	mlosch	1.2
160			The concentration $c$, effective ice thickness (ice volume per unit
161	dimitri	1.11	area, $c\cdot{h}$), effective snow thickness ($c\cdot{h}_{s}$), and effective
162			ice salinity (in g\,m$^{-2}$) are advected by ice velocities.
163	mlosch	1.2	%
164	mlosch	1.25	From the various advection schemes that are available in the MITgcm
165	dimitri	1.11	\citep{mitgcm02}, we choose flux-limited schemes, i.e., multidimensional 2nd
166			and 3rd-order advection schemes with flux limiters
167			\citep{roe85, hundsdorfer94}, to preserve sharp gradients and
168	mlosch	1.2	edges that are typical of sea ice distributions and to rule out
169			unphysical over- and undershoots (negative thickness or
170	mlosch	1.18	concentration). These schemes, conserve volume and horizontal area and
171			are unconditionally stable, so that no extra diffusion is required.
172			%\ml{[Sergey says: any FV scheme will do this]}
173	mlosch	1.2
174	cnh	1.16	There is considerable doubt about the reliability of a
175	dimitri	1.11	``zero-layer'' thermodynamic model --- \citet{semtner84} found
176	mlosch	1.2	significant errors in phase (one month lead) and amplitude
177	dimitri	1.11	($\approx$50\%\,overestimate) in such models --- so that today many
178			sea ice models employ more complex thermodynamics. The MITgcm
179			sea ice model provides the option to use the thermodynamics model of
180			\citet{winton00}, which in turn
181			is based on the 3-layer model of \citet{semtner76} and which treats brine
182	cnh	1.16	content by means of enthalpy conservation. This scheme requires
183	jmc	1.9	additional state variables, namely the enthalpy of the two ice
184	dimitri	1.11	layers (instead of effective ice salinity), to be advected by ice velocities.
185	mlosch	1.24	%
186	dimitri	1.11	The internal sea ice temperature is inferred from ice enthalpy.
187	cnh	1.16	To avoid unphysical (negative) values for ice thickness and
188	mlosch	1.12	concentration, a positive 2nd-order advection scheme with a SuperBee
189			flux limiter \citep{roe85}
190			is used in this study to advect all sea-ice-related
191	dimitri	1.11	quantities of the \citet{winton00} thermodynamic model.
192	jmc	1.9	Because of the non-linearity of the advection scheme,
193			care must be taken in advecting these quantities: when simply using
194			ice velocity to advect enthalpy, the total energy (i.e., the volume
195			integral of enthalpy) is not conserved. Alternatively, one can advect
196			the energy content (i.e., product of ice-volume and enthalpy)
197			but then false enthalpy extrema can occur,
198			which then leads to unrealistic ice temperature.
199	mlosch	1.15	In the currently implemented solution, the sea-ice mass flux is used
200	cnh	1.17	to advect the enthalpy in order to ensure conservation of enthalpy
201	mlosch	1.15	and to prevent false enthalpy extrema.
202	heimbach	1.1
203	cnh	1.16	In \refsec{globalmodel} and \ref{sec:arcticmodel}
204			we exercise and compare several
205	mlosch	1.22	of the options, which have been discussed above; we intercompare
206	cnh	1.16	the impact of the different formulations (all of which are widely
207	cnh	1.17	used in sea ice modeling today) on Arctic sea ice simulation
208	mlosch	1.18	\citep{proshutinsky07:_aomip}.
209	cnh	1.16	%% Got to here..... more later
210			%% Add reference to JGR special issue here.....
211	mlosch	1.18	%\citep{prosh07:_aomipspecial}.
212
213
214	dimitri	1.11
215	heimbach	1.1	%%% Local Variables:
216			%%% mode: latex
217	mlosch	1.2	%%% TeX-master: "ceaice_part1"
218	heimbach	1.1	%%% End: