ceaice_split_version/ceaice_part1/ceaice_concl2.tex

\section{Conclusions}
\label{sec:concl}

We have shown that changes in discretization details, in boundary conditions,
and in sea-ice-dynamics formulation lead to considerable differences in model
results. Notably the sea-ice-dynamics formulation, e.g., B-grid versus C-grid
or EVP versus LSOR, has as much or even greater influence on the solution than
physical parameterizations, e.g., free-slip versus no-slip boundary
conditions. This is especially true
\begin{itemize}
\item in regions of convergence (see ice thickness north of Greenland in
  Fig.~4),
\item along coasts (see eastern coast of Greenland in Fig.~3 where velocity
  differences are apparent),
\item and in the vicinity of straits (see the Canadian Arctic Archipelago in
  Figs.~3 and 4).
\end{itemize}
These experiments demonstrate that sea-ice export from the Arctic into both
the Baffin Bay and the GIN (Greenland/Iceland/Norwegian) Sea regions is highly
sensitive to numerical formulation. Changes in export in turn impact
deep-water mass formation in the northern North Atlantic.  Therefore
uncertainties due to numerical formulation might potentially have wide
reaching impacts outside of the Arctic.

The relatively large differences between solutions with different dynamical
solvers is somewhat surprising. The expectation was that the solution
technique should not affect the solution to a higher degree than actually
modifying the equations.  The EVP solutions tend to produce effectively
``weaker'' ice that yields more easily to stress than the LSOR solutions,
similar to the findings in \citet{hunke99}.  The differences between LSOR and
EVP can, in part, stem from incomplete convergence of the solvers due to
linearization and due to different methods of linearization \citep[and B.\
Tremblay, pers.\ comm.\ 2008]{hunke01}.  We note that the EVP-to-LSOR
differences decrease with descreasing sub-cycling time step but that the
difference remains significant even at a 3-second sub-cycling period. For the
LSOR solutions we use 2 pseudo time steps so that the convergence of the
non-linear momentum equations may not be complete. This effect is most likely
reduced and constrained to small areas as in \citet{lemieux09} because of the
small time step that we used. Whether more pseudo time steps make the LSOR
solution generate weaker ice requires further investigation. Preliminary tests
indicate that the viscosity increases with increasing number of LSOR pseudo
time steps, especially in areas of thick ice (not shown).

Other numerical formulation
choices that were tested include switching from one horizontal grid
staggering (C-grid) to another (B-grid).  This change significantly affects
narrow straits, for example, in the Canadian Arctic Archipelago, and subsequent
conditions upstream and downstream of the straits.  It also affects flows of
ice along the West Greenland coast.  Similar, but smaller, differences
between B-grid and C-grid sea ice solutions were noted in the
coarser-resolution study of \citet{bouillon09}.
The differences between the no-slip and free-slip lateral boundary
conditions are also most significant near the coast.
%DM I DON'T UNDERSTAND THIS STATEMENT SO HAVE REMOVED IT
% The difference
% in volume transports through the Fram Strait, the Canadian Arctic
% Archipelago, and Lancaster Sound for the no-slip and free-slip
% experiments and for the B-, C- grid experiments and EVP-10 and LSR are
% similar.
As in the case of oceanic boundary conditions
\citep{adcroft98:_slippery_coast}, we expect
that the changes are
due to the effective ``slipperiness'' of the coastline boundary
condition.

The flux-limited scheme without explicit diffusion (DST3FL) is
recommended.  This is because the flux-limited scheme preserves sharp
gradients and edges that are typical of sea ice distributions and
because it avoids unphysical (negative) values for ice thickness and
concentration \citep[see also][]{merryfield03}. The flux limited
scheme conserves volume and horizontal area and is unconditionally
stable, so that no extra diffusion is required.

Changing the ice rheology to the truncated ellipse method (TEM) primarily
impacts the solution in the Canadian Arctic Archipelago and the West
Greenland coast as
does altering the stress formulation on the ice solution. We interpret
this result as
indicating that the CAA and West Greenland current are regions of
high-sensitivity.
Here, more ice leads to a rigid structure that inhibits ice flow
and yields ice accumulation upstream.

Although the \citet{hibler87} stress
formulation appears more natural for advecting sea ice, the advection of
oceanic properties is problematic:  Thermodynamic and passive tracers in the
top ocean model level are advected with a velocity that is the average over
ice drift and ocean currents rather than an average of surface oceanic
currents alone.  For our purposes, the preferred ice-ocean coupling uses the
rescaled vertical coordinates of \citet{campin08}, which allows the ice to
depress the ocean surface according to its thickness and buoyancy.

%[Reviewer 2: statement about robustness, against forcing, choice
%  of computational interval etc.:]
A few comments regarding the robustness of our results against choice
of forcing, integration period, and horizontal resolution follow.
Strictly speaking, our results refer to an 8-year integration with
18~km horizontal grid spacing.  We find that the differences between
the solutions have an obvious trend after the first season but that
this trend flattens out after a few seasons.  We do not expect the
differences to increase dramatically with additional integration time,
since the simulated multi-year sea ice has reached a quasi
equilibrium. Surface atmospheric conditions are specified every 6
hours. Models with weaker ice can react more quickly to a change in
wind forcing, therefore we speculate that the differences between EVP
and LSOR integrations would change with different forcing: less
variable wind forcing would lead to smaller differences, while larger
flucations in the forcing would increase them. In the same way, we
expect that with coarser grids, the ocean component is much less
variable so that in this case one will only find smaller differences
between ice models.

The MITgcm sea ice model
%makes possible,
enables, within the same code, the direct comparison of various widely
used dynamics and thermodynamics model components.  What sets apart
the MITgcm sea ice model from other current-generation sea ice models
is the ability to derive an accurate, stable, and efficient adjoint
model using automatic differentiation source transformation tools.
This capability is the topic of a companion, second paper. The adjoint
model greatly facilitates and enhances exploration of the model's
parameter space. It lays the foundation for coupled ocean and sea ice
state estimation.


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1	mlosch	1.1	\section{Conclusions}
2			\label{sec:concl}
3
4	dimitri	1.3	We have shown that changes in discretization details, in boundary conditions,
5			and in sea-ice-dynamics formulation lead to considerable differences in model
6			results. Notably the sea-ice-dynamics formulation, e.g., B-grid versus C-grid
7			or EVP versus LSOR, has as much or even greater influence on the solution than
8			physical parameterizations, e.g., free-slip versus no-slip boundary
9			conditions. This is especially true
10	mlosch	1.1	\begin{itemize}
11	dimitri	1.3	\item in regions of convergence (see ice thickness north of Greenland in
12			Fig.~4),
13			\item along coasts (see eastern coast of Greenland in Fig.~3 where velocity
14			differences are apparent),
15			\item and in the vicinity of straits (see the Canadian Arctic Archipelago in
16			Figs.~3 and 4).
17	mlosch	1.1	\end{itemize}
18	dimitri	1.3	These experiments demonstrate that sea-ice export from the Arctic into both
19			the Baffin Bay and the GIN (Greenland/Iceland/Norwegian) Sea regions is highly
20			sensitive to numerical formulation. Changes in export in turn impact
21			deep-water mass formation in the northern North Atlantic. Therefore
22			uncertainties due to numerical formulation might potentially have wide
23			reaching impacts outside of the Arctic.
24
25			The relatively large differences between solutions with different dynamical
26			solvers is somewhat surprising. The expectation was that the solution
27			technique should not affect the solution to a higher degree than actually
28			modifying the equations. The EVP solutions tend to produce effectively
29			``weaker'' ice that yields more easily to stress than the LSOR solutions,
30			similar to the findings in \citet{hunke99}. The differences between LSOR and
31			EVP can, in part, stem from incomplete convergence of the solvers due to
32			linearization and due to different methods of linearization \citep[and B.\
33			Tremblay, pers.\ comm.\ 2008]{hunke01}. We note that the EVP-to-LSOR
34			differences decrease with descreasing sub-cycling time step but that the
35			difference remains significant even at a 3-second sub-cycling period. For the
36			LSOR solutions we use 2 pseudo time steps so that the convergence of the
37			non-linear momentum equations may not be complete. This effect is most likely
38			reduced and constrained to small areas as in \citet{lemieux09} because of the
39			small time step that we used. Whether more pseudo time steps make the LSOR
40			solution generate weaker ice requires further investigation. Preliminary tests
41			indicate that the viscosity increases with increasing number of LSOR pseudo
42			time steps, especially in areas of thick ice (not shown).
43	mlosch	1.1
44	mlosch	1.2	Other numerical formulation
45			choices that were tested include switching from one horizontal grid
46			staggering (C-grid) to another (B-grid). This change significantly affects
47			narrow straits, for example, in the Canadian Arctic Archipelago, and subsequent
48			conditions upstream and downstream of the straits. It also affects flows of
49			ice along the West Greenland coast. Similar, but smaller, differences
50			between B-grid and C-grid sea ice solutions were noted in the
51			coarser-resolution study of \citet{bouillon09}.
52			The differences between the no-slip and free-slip lateral boundary
53	dimitri	1.3	conditions are also most significant near the coast.
54			%DM I DON'T UNDERSTAND THIS STATEMENT SO HAVE REMOVED IT
55			% The difference
56			% in volume transports through the Fram Strait, the Canadian Arctic
57			% Archipelago, and Lancaster Sound for the no-slip and free-slip
58			% experiments and for the B-, C- grid experiments and EVP-10 and LSR are
59			% similar.
60			As in the case of oceanic boundary conditions
61			\citep{adcroft98:_slippery_coast}, we expect
62	mlosch	1.2	that the changes are
63	dimitri	1.3	due to the effective ``slipperiness'' of the coastline boundary
64	mlosch	1.2	condition.
65
66			The flux-limited scheme without explicit diffusion (DST3FL) is
67			recommended. This is because the flux-limited scheme preserves sharp
68			gradients and edges that are typical of sea ice distributions and
69			because it avoids unphysical (negative) values for ice thickness and
70			concentration \citep[see also][]{merryfield03}. The flux limited
71			scheme conserves volume and horizontal area and is unconditionally
72			stable, so that no extra diffusion is required.
73
74			Changing the ice rheology to the truncated ellipse method (TEM) primarily
75			impacts the solution in the Canadian Arctic Archipelago and the West
76			Greenland coast as
77			does altering the stress formulation on the ice solution. We interpret
78			this result as
79			indicating that the CAA and West Greenland current are regions of
80			high-sensitivity.
81	dimitri	1.3	Here, more ice leads to a rigid structure that inhibits ice flow
82	mlosch	1.2	and yields ice accumulation upstream.
83
84			Although the \citet{hibler87} stress
85			formulation appears more natural for advecting sea ice, the advection of
86			oceanic properties is problematic: Thermodynamic and passive tracers in the
87			top ocean model level are advected with a velocity that is the average over
88			ice drift and ocean currents rather than an average of surface oceanic
89			currents alone. For our purposes, the preferred ice-ocean coupling uses the
90			rescaled vertical coordinates of \citet{campin08}, which allows the ice to
91	dimitri	1.3	depress the ocean surface according to its thickness and buoyancy.
92	mlosch	1.2
93	mlosch	1.1	%[Reviewer 2: statement about robustness, against forcing, choice
94			% of computational interval etc.:]
95			A few comments regarding the robustness of our results against choice
96			of forcing, integration period, and horizontal resolution follow.
97			Strictly speaking, our results refer to an 8-year integration with
98			18~km horizontal grid spacing. We find that the differences between
99			the solutions have an obvious trend after the first season but that
100			this trend flattens out after a few seasons. We do not expect the
101			differences to increase dramatically with additional integration time,
102			since the simulated multi-year sea ice has reached a quasi
103			equilibrium. Surface atmospheric conditions are specified every 6
104			hours. Models with weaker ice can react more quickly to a change in
105			wind forcing, therefore we speculate that the differences between EVP
106			and LSOR integrations would change with different forcing: less
107			variable wind forcing would lead to smaller differences, while larger
108			flucations in the forcing would increase them. In the same way, we
109			expect that with coarser grids, the ocean component is much less
110			variable so that in this case one will only find smaller differences
111			between ice models.
112
113			The MITgcm sea ice model
114			%makes possible,
115			enables, within the same code, the direct comparison of various widely
116			used dynamics and thermodynamics model components. What sets apart
117			the MITgcm sea ice model from other current-generation sea ice models
118			is the ability to derive an accurate, stable, and efficient adjoint
119			model using automatic differentiation source transformation tools.
120			This capability is the topic of a companion, second paper. The adjoint
121			model greatly facilitates and enhances exploration of the model's
122			parameter space. It lays the foundation for coupled ocean and sea ice
123			state estimation.
124
125
126
127
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