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 numerical formulation (for
example B-grid versus C-grid or EVP versus LSR) has as much, or even
greater, influence on solution than physical parameterizations (for
example free-slip versus no-slip boundary conditions). This is
especially true in regions of
\begin{itemize}
\item convergence as seen in the north of Greenland ice thickness in figure 4.
\item along coasts, see the eastern coast of Greenland in figure 3
(where velocity differences
are apparent).
\item in the vicinity of straits, see the Canadian Arctic Archipelago
in figures 3 and 4.
\end{itemize}
One upshot of all these experiments is that sea-ice export from the
Arctic into both the Baffin Bay and 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 and so 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 is 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}.  However, we note that the EVP LSOR differences
decrease with descreasing sub-cycling time step but 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
\citep{lemieux09}. However, this effect is most likely reduced and
constrained to small areas as in \citet{lemieux09} by the small time
step that we used. Whether more pseudo time steps make the LSOR
solution generate weaker ice remains unclear. Preliminary tests showed
that the viscositites even increase especially in areas of thick ice,
when the number of pseudo time steps is increased (not shown).

\ml{[from here on reinserted text]}\\
Other numerical formulation
%possibilities
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.  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 %hypothesize
that the changes are
due to the effective ``slipperiness'' of the coast line 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 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
the ocean surface according to its thickness and buoyancy.
\\\ml{[end of reinserted text]}\\

%[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			We have shown that changes in discretization details, in boundary
5			conditions, and in sea-ice-dynamics formulation lead to considerable
6			differences in model results. Notably numerical formulation (for
7			example B-grid versus C-grid or EVP versus LSR) has as much, or even
8			greater, influence on solution than physical parameterizations (for
9			example free-slip versus no-slip boundary conditions). This is
10			especially true in regions of
11			\begin{itemize}
12			\item convergence as seen in the north of Greenland ice thickness in figure 4.
13			\item along coasts, see the eastern coast of Greenland in figure 3
14			(where velocity differences
15			are apparent).
16			\item in the vicinity of straits, see the Canadian Arctic Archipelago
17			in figures 3 and 4.
18			\end{itemize}
19			One upshot of all these experiments is that sea-ice export from the
20			Arctic into both the Baffin Bay and GIN (Greenland/Iceland/Norwegian) Sea
21			regions is highly sensitive to numerical formulation. Changes in
22			export in turn impact deep-water mass formation in the northern North
23			Atlantic and so uncertainties due to numerical formulation might
24			potentially have wide reaching impacts outside of the Arctic.
25
26			The relatively large differences between solutions with different
27			dynamical solvers is somewhat surprising. The expectation is that the
28			solution technique should not affect the solution to a higher degree
29			than actually modifying the equations. The EVP solutions tend to
30			produce effectively ``weaker'' ice that yields more easily to stress
31			than the LSOR solutions , similar to the findings in \citet{hunke99}.
32			The differences between LSOR and EVP can, in part, stem from
33			incomplete convergence of the solvers due to linearization and due to
34			different methods of linearization \citep[and B. Tremblay, pers.
35			comm. 2008]{hunke01}. However, we note that the EVP LSOR differences
36			decrease with descreasing sub-cycling time step but the difference
37			remains significant even at a 3 second sub-cycling period. For the
38			LSOR solutions we use 2 pseudo time steps, so that the convergence of
39			the non-linear momentum equations may not be complete
40			\citep{lemieux09}. However, this effect is most likely reduced and
41			constrained to small areas as in \citet{lemieux09} by the small time
42			step that we used. Whether more pseudo time steps make the LSOR
43			solution generate weaker ice remains unclear. Preliminary tests showed
44			that the viscositites even increase especially in areas of thick ice,
45			when the number of pseudo time steps is increased (not shown).
46
47	mlosch	1.2	\ml{[from here on reinserted text]}\\
48			Other numerical formulation
49			%possibilities
50			choices that were tested include switching from one horizontal grid
51			staggering (C-grid) to another (B-grid). This change significantly affects
52			narrow straits, for example, in the Canadian Arctic Archipelago, and subsequent
53			conditions upstream and downstream of the straits. It also affects flows of
54			ice along the West Greenland coast. Similar, but smaller, differences
55			between B-grid and C-grid sea ice solutions were noted in the
56			coarser-resolution study of \citet{bouillon09}.
57			The differences between the no-slip and free-slip lateral boundary
58			conditions are also most significant near the coast. The difference
59			in volume transports through the Fram Strait, the Canadian Arctic
60			Archipelago, and Lancaster Sound for the no-slip and free-slip
61			experiments and for the B-, C- grid experiments and EVP-10 and LSR are
62			similar. As in the case of oceanic boundary conditions
63			\citep{adcroft98:_slippery_coast}, we expect %hypothesize
64			that the changes are
65			due to the effective ``slipperiness'' of the coast line boundary
66			condition.
67
68			The flux-limited scheme without explicit diffusion (DST3FL) is
69			recommended. This is because the flux-limited scheme preserves sharp
70			gradients and edges that are typical of sea ice distributions and
71			because it avoids unphysical (negative) values for ice thickness and
72			concentration \citep[see also][]{merryfield03}. The flux limited
73			scheme conserves volume and horizontal area and is unconditionally
74			stable, so that no extra diffusion is required.
75
76			Changing the ice rheology to the truncated ellipse method (TEM) primarily
77			impacts the solution in the Canadian Arctic Archipelago and the West
78			Greenland coast as
79			does altering the stress formulation on the ice solution. We interpret
80			this result as
81			indicating that the CAA and West Greenland current are regions of
82			high-sensitivity.
83			Here, more ice leads to rigid structure that inhibits ice flow
84			and yields ice accumulation upstream.
85
86			Although the \citet{hibler87} stress
87			formulation appears more natural for advecting sea ice, the advection of
88			oceanic properties is problematic: Thermodynamic and passive tracers in the
89			top ocean model level are advected with a velocity that is the average over
90			ice drift and ocean currents rather than an average of surface oceanic
91			currents alone. For our purposes, the preferred ice-ocean coupling uses the
92			rescaled vertical coordinates of \citet{campin08}, which allows the ice to
93			the ocean surface according to its thickness and buoyancy.
94			\\\ml{[end of reinserted text]}\\
95
96	mlosch	1.1	%[Reviewer 2: statement about robustness, against forcing, choice
97			% of computational interval etc.:]
98			A few comments regarding the robustness of our results against choice
99			of forcing, integration period, and horizontal resolution follow.
100			Strictly speaking, our results refer to an 8-year integration with
101			18~km horizontal grid spacing. We find that the differences between
102			the solutions have an obvious trend after the first season but that
103			this trend flattens out after a few seasons. We do not expect the
104			differences to increase dramatically with additional integration time,
105			since the simulated multi-year sea ice has reached a quasi
106			equilibrium. Surface atmospheric conditions are specified every 6
107			hours. Models with weaker ice can react more quickly to a change in
108			wind forcing, therefore we speculate that the differences between EVP
109			and LSOR integrations would change with different forcing: less
110			variable wind forcing would lead to smaller differences, while larger
111			flucations in the forcing would increase them. In the same way, we
112			expect that with coarser grids, the ocean component is much less
113			variable so that in this case one will only find smaller differences
114			between ice models.
115
116			The MITgcm sea ice model
117			%makes possible,
118			enables, within the same code, the direct comparison of various widely
119			used dynamics and thermodynamics model components. What sets apart
120			the MITgcm sea ice model from other current-generation sea ice models
121			is the ability to derive an accurate, stable, and efficient adjoint
122			model using automatic differentiation source transformation tools.
123			This capability is the topic of a companion, second paper. The adjoint
124			model greatly facilitates and enhances exploration of the model's
125			parameter space. It lays the foundation for coupled ocean and sea ice
126			state estimation.
127
128
129
130
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