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{hunke01, hunke02}, have been adopted,
\item ice-ocean stress can be formulated as in \citet{hibler87} as an
  alternative to the standard method of applying ice-ocean stress
  directly,
\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; and by the horizontal surface elevation gradient
of the ocean. 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}. %
\ml{Implicit solvers such as LSOR usually require bounding very high
  viscosities for numerical stability. Alternatively, the
  elastic-viscous-plastic (EVP) technique following \citet{hunke01}
  regularizes large viscosities by adding an extra term in the
  constitutive law that introduces  damped elastic waves. The
  EVP-solver relaxes 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. Neither solver requires limiting the viscosities from below.}

\ml{[Reviewer 2 asks for more explanation of VP/EVP/LSOR, do we want
  to give more?]}

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.  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).
\ml{Free-slip boundary conditions are not implemented for the B~grid.}

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 through the ice
is parameterized assuming a linear temperature profile and a constant
ice conductivity implying zero heat capacity for ice. 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, that is,
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	mlosch	1.27	\citep{hunke01, hunke02}, have been adopted,
16	mlosch	1.26	\item ice-ocean stress can be formulated as in \citet{hibler87} as an
17			alternative to the standard method of applying ice-ocean stress
18			directly,
19	dimitri	1.11	\item ice concentration and thickness, snow, and ice salinity or enthalpy can
20			be advected by sophisticated, conservative
21	mlosch	1.26	advection schemes with flux limiters.%, and
22	heimbach	1.23	%ph(
23			% This is Ian's work, and will be reported elsewhere, I insist;
24			% or Ian should be co-author
25			%\item growth and melt parameterizations have been refined and extended
26			% in order to allow for automatic differentiation of the code.
27			%ph)
28	heimbach	1.1	\end{itemize}
29	jmc	1.9	The sea ice model is tightly coupled to the ocean component of the
30	cnh	1.16	MITgcm \citep{marshall97:_hydros_quasi_hydros_nonhy,marshall97:_finit_volum_incom_navier_stokes}.
31	mlosch	1.25	Heat, freshwater fluxes and surface stresses are computed from the
32	heimbach	1.1	atmospheric state and modified by the ice model at every time step.
33	mlosch	1.2	The remainder of this section describes the model equations and
34			details of their numerical realization. Further documentation and
35			model code can be found at \url{http://mitgcm.org}.
36
37			\subsection{Dynamics}
38			\label{sec:dynamics}
39
40	mlosch	1.22	%\ml{[Sergey says: a kind of repetition; need to revise previous section]}
41	mlosch	1.2	Sea-ice motion is driven by ice-atmosphere, ice-ocean and internal
42	mlosch	1.26	stresses; and by the horizontal surface elevation gradient
43			of the ocean. The internal stresses are evaluated following a
44	mlosch	1.2	viscous-plastic (VP) constitutive law with an elliptic yield curve as
45			in \citet{hibler79}. The full momentum equations for the sea-ice model
46			and the solution by line successive over-relaxation (LSOR) are
47	mlosch	1.28	described in \citet{zhang97}. %
48			\ml{Implicit solvers such as LSOR usually require bounding very high
49			viscosities for numerical stability. Alternatively, the
50			elastic-viscous-plastic (EVP) technique following \citet{hunke01}
51			regularizes large viscosities by adding an extra term in the
52			constitutive law that introduces damped elastic waves. The
53			EVP-solver relaxes the ice state towards the VP rheology by
54			sub-cycling the evolution equations for the internal stress tensor
55			components and the sea ice momentum solver within one ocean model
56			time step. Neither solver requires limiting the viscosities from below.}
57
58			\ml{[Reviewer 2 asks for more explanation of VP/EVP/LSOR, do we want
59			to give more?]}
60
61			For stress tensor computations the replacement pressure
62			\citep{hibler95} is used so that the stress state always lies within
63			the elliptic yield curve by definition. In an alternative to the
64			elliptic yield curve, the so-called truncated ellipse method (TEM),
65			the shear viscosity is capped to suppress any tensile stress
66			\citep{hibler97, geiger98}.
67	mlosch	1.2
68			The horizontal gradient of the ocean's surface is estimated directly
69			from ocean sea surface height and pressure loading from atmosphere,
70	dimitri	1.11	ice and snow \citep{campin08}. Ice does not float on top of the
71			ocean, instead it depresses the ocean surface according to its thickness and
72			buoyancy.
73	mlosch	1.2
74	mlosch	1.15	Lateral boundary conditions are naturally ``no-slip'' for B~grids, as
75	mlosch	1.27	the tangential velocities points lie on the boundary. For C~grids,
76			the lateral boundary condition for tangential velocities allow
77			alternatively no-slip or free-slip conditions. In ocean models
78			free-slip boundary conditions in conjunction with piecewise-constant
79			(``castellated'') coastlines have been shown to reduce to no-slip
80			boundary conditions \citep{adcroft98:_slippery_coast}; for coupled
81			ocean sea-ice models the effects of lateral boundary conditions have
82			not yet been studied (as far as we know).
83			\ml{Free-slip boundary conditions are not implemented for the B~grid.}
84	mlosch	1.2
85			Moving sea ice exerts a surface stress on the ocean. In coupling the
86	jmc	1.9	sea-ice model to the ocean model, this stress is applied directly to
87	mlosch	1.2	the surface layer of the ocean model. An alternative ocean stress
88			formulation is given by \citet{hibler87}. Rather than applying the
89			interfacial stress directly, the stress is derived from integrating
90			over the ice thickness to the bottom of the oceanic surface layer. In
91	mlosch	1.15	the resulting equation for the combined ocean-ice momentum, the
92	mlosch	1.2	interfacial stress cancels and the total stress appears as the sum of
93	dimitri	1.6	wind stress and divergence of internal ice stresses \citep[see also
94	mlosch	1.2	Eq.\,2 of][]{hibler87}. While this formulation tightly embeds the
95	mlosch	1.15	sea ice into the surface layer of the ocean, its disadvantage is that
96	dimitri	1.11	the velocity in the surface layer of the ocean that is used to
97	mlosch	1.22	advect ocean tracers is an average over the ocean surface
98	dimitri	1.11	velocity and the ice velocity, leading to an inconsistency as the ice
99	mlosch	1.2	temperature and salinity are different from the oceanic variables.
100			Both stress coupling options are available for a direct comparison of
101	mlosch	1.18	their effects on the sea-ice solution.
102	mlosch	1.2
103	mlosch	1.21	The finite-volume discretization of the momentum equation on the
104			Arakawa C~grid is straightforward. The stress tensor divergence, in
105	mlosch	1.15	particular, is discretized naturally on the C~grid with the diagonal
106	mlosch	1.2	components of the stress tensor on the center points and the
107			off-diagonal term on the corner (or vorticity) points of the grid.
108			With this choice all derivatives are discretized as central
109	mlosch	1.21	differences and very little averaging is involved, see
110			\refapp{discretization} for details. Apart from the standard C-grid
111			implementation, the original B-grid implementation of \citet{zhang97}
112			is also available as an option in the code.
113	mlosch	1.2
114			\subsection{Thermodynamics}
115			\label{sec:thermodynamics}
116
117	mlosch	1.26	Upward conductive heat flux through the ice
118	mlosch	1.2	is parameterized assuming a linear temperature profile and a constant
119	mlosch	1.26	ice conductivity implying zero heat capacity for ice. This type of
120			model is often referred to as a ``zero-layer'' model
121			\citep{semtner76}. The surface heat flux is computed in a similar
122	mlosch	1.2	way to that of \citet{parkinson79} and \citet{manabe79}.
123
124			The conductive heat flux depends strongly on the ice thickness $h$.
125			However, the ice thickness in the model represents a mean over a
126			potentially very heterogeneous thickness distribution. In order to
127			parameterize a sub-grid scale distribution for heat flux computations,
128			the mean ice thickness $h$ is split into seven thickness categories
129			$H_{n}$ that are equally distributed between $2h$ and a minimum
130			imposed ice thickness of $5\text{\,cm}$ by $H_n= \frac{2n-1}{7}\,h$
131			for $n\in[1,7]$. The heat fluxes computed for each thickness category
132			is area-averaged to give the total heat flux \citep{hibler84}.
133
134	mlosch	1.12	The atmospheric heat flux is balanced by an oceanic heat flux.
135			The oceanic flux is proportional to the difference between
136	mlosch	1.25	ocean surface temperature and the freezing point temperature of
137			seawater, which is a function of salinity.
138	cnh	1.16	This flux is not assumed to instantaneously melt
139	mlosch	1.3	or create ice, but a time scale of three days is used to relax the
140	mlosch	1.8	ocean temperature to the freezing point. While this
141	dimitri	1.11	parameterization is not new \citep[it follows the ideas of,
142	mlosch	1.3	e.g.,][]{mellor86, mcphee92, lohmann98, notz03}, it avoids a
143			discontinuity in the functional relationship between model variables,
144	heimbach	1.23	which
145			%is crucial for making the code differentiable for adjoint code generation
146			%ph: it doesn't improve adjoint code generation, it improves the smoothness
147	mlosch	1.24	improves the smoothness of the differentiated model
148	heimbach	1.23	%(see companion, part 2, paper).
149	mlosch	1.25	\citep[see][for details]{fent:09}.
150	dimitri	1.13	%\ml{[ONCE IT IS SUBMITTED, otherwise pers. communcations:]}\citep{fen09}
151	mlosch	1.2	The parameterization of lateral and vertical growth of sea ice follows
152			that of \citet{hibler79, hibler80}.
153
154			On top of the ice there is a layer of snow that modifies the heat flux
155			and the albedo as in \citet{zhang98}. If enough snow accumulates so
156			that its weight submerges the ice and the snow is flooded, a simple
157	dimitri	1.11	mass conserving parameterization of snow ice formation (a flood-freeze
158	mlosch	1.2	algorithm following Archimedes' principle) turns snow into ice until
159	jmc	1.19	the ice surface is back %at $z=0$
160	mlosch	1.20	at sea-level \citep{leppaeranta83}.
161	mlosch	1.2
162			The concentration $c$, effective ice thickness (ice volume per unit
163	dimitri	1.11	area, $c\cdot{h}$), effective snow thickness ($c\cdot{h}_{s}$), and effective
164			ice salinity (in g\,m$^{-2}$) are advected by ice velocities.
165	mlosch	1.2	%
166	mlosch	1.25	From the various advection schemes that are available in the MITgcm
167	mlosch	1.28	\citep{mitgcm02}, we choose flux-limited schemes, that is,
168			multidimensional 2nd and 3rd-order advection schemes with flux
169			limiters \citep{roe85, hundsdorfer94}, to preserve sharp gradients and
170	mlosch	1.2	edges that are typical of sea ice distributions and to rule out
171			unphysical over- and undershoots (negative thickness or
172	mlosch	1.18	concentration). These schemes, conserve volume and horizontal area and
173	mlosch	1.28	are unconditionally stable, so that no extra diffusion is required.
174	mlosch	1.18	%\ml{[Sergey says: any FV scheme will do this]}
175	mlosch	1.2
176	cnh	1.16	There is considerable doubt about the reliability of a
177	dimitri	1.11	``zero-layer'' thermodynamic model --- \citet{semtner84} found
178	mlosch	1.2	significant errors in phase (one month lead) and amplitude
179	dimitri	1.11	($\approx$50\%\,overestimate) in such models --- so that today many
180			sea ice models employ more complex thermodynamics. The MITgcm
181			sea ice model provides the option to use the thermodynamics model of
182			\citet{winton00}, which in turn
183			is based on the 3-layer model of \citet{semtner76} and which treats brine
184	cnh	1.16	content by means of enthalpy conservation. This scheme requires
185	jmc	1.9	additional state variables, namely the enthalpy of the two ice
186	dimitri	1.11	layers (instead of effective ice salinity), to be advected by ice velocities.
187	mlosch	1.24	%
188	dimitri	1.11	The internal sea ice temperature is inferred from ice enthalpy.
189	cnh	1.16	To avoid unphysical (negative) values for ice thickness and
190	mlosch	1.12	concentration, a positive 2nd-order advection scheme with a SuperBee
191			flux limiter \citep{roe85}
192			is used in this study to advect all sea-ice-related
193	dimitri	1.11	quantities of the \citet{winton00} thermodynamic model.
194	jmc	1.9	Because of the non-linearity of the advection scheme,
195			care must be taken in advecting these quantities: when simply using
196			ice velocity to advect enthalpy, the total energy (i.e., the volume
197			integral of enthalpy) is not conserved. Alternatively, one can advect
198			the energy content (i.e., product of ice-volume and enthalpy)
199			but then false enthalpy extrema can occur,
200			which then leads to unrealistic ice temperature.
201	mlosch	1.15	In the currently implemented solution, the sea-ice mass flux is used
202	cnh	1.17	to advect the enthalpy in order to ensure conservation of enthalpy
203	mlosch	1.15	and to prevent false enthalpy extrema.
204	heimbach	1.1
205	cnh	1.16	In \refsec{globalmodel} and \ref{sec:arcticmodel}
206			we exercise and compare several
207	mlosch	1.22	of the options, which have been discussed above; we intercompare
208	cnh	1.16	the impact of the different formulations (all of which are widely
209	cnh	1.17	used in sea ice modeling today) on Arctic sea ice simulation
210	mlosch	1.18	\citep{proshutinsky07:_aomip}.
211	cnh	1.16	%% Got to here..... more later
212			%% Add reference to JGR special issue here.....
213	mlosch	1.18	%\citep{prosh07:_aomipspecial}.
214
215
216	dimitri	1.11
217	heimbach	1.1	%%% Local Variables:
218			%%% mode: latex
219	mlosch	1.2	%%% TeX-master: "ceaice_part1"
220	heimbach	1.1	%%% End: