articles/ceaice/ceaice_adjoint.tex

\section{Adjoint sensiivities of the MITsim}
\label{sec:adjoint}

\subsection{The adjoint of MITsim}

The adjoint model of the MITgcm has become an invaluable
tool for sensitivity analysis as well as state estimation \citep[for a
recent summary, see][]{heim:08}. The code has been developed and
tailored to be readily used with automatic differentiation tools for
adjoint code generation. This route was also taken in developing and
adapting the sea-ice compontent MITsim, so that tangent linear and
adjoint components can be obtained and kept up to date without
excessive effort.

The adjoint model operator (ADM) is the transpose of the tangent
linear model operator (TLM) of the full (in general nonlinear) forward
model, in this case the MITsim. This operator computes the gradients
of scalar-valued model diagnostics (so-called cost function or
objective function) with respect to many model inputs (so-called
independent or control variables).  These inputs can be two- or
three-dimensional fields of initial conditions of the ocean or sea-ice
state, model parameters such as mixing coefficients, or time-varying
surface or lateral (open) boundary conditions.  When combined, these
variables span a potentially high-dimensional (e.g.  O(10$^8$))
so-called control space. At this problem dimension, perturbing
individual parameters to assess model sensitivities quickly becomes
prohibitive. By contrast, transient sensitivities of the objective
function to any element of the control and model state space can be
computed very efficiently in one single adjoint model integration,
provided an adjoint model is available.

In anology to the TLM and ADM components of the MITgcm we rely on the
autmomatic differentiation (AD) tool ``Transformation of Algorithms in
Fortran'' (TAF) developed by Fastopt \citep{gier-kami:98} to generate
TLM and ADM code of the MITsim \citep[for details see][]{maro-etal:99,
  heim-etal:05}.  In short, the AD tool uses the nonlinear parent
model code to generate derivative code for the specified control space
and objective function. Advantages of this approach have been pointed
out, for example by \cite{gier-kami:98}.

Many issues of generating efficient exact adjoint sea-ice code are
similar to those for the ocean model's adjoint.  Linearizing the model
around the exact nonlinear model trajectory is a crucial aspect in the
presence of different regimes (e.g., is the thermodynamic growth term
for sea-ice evaluated near or far away from the freezing point of the
ocean surface?). Adapting the (parent) model code to support the AD
tool in providing exact and efficient adjoint code represents the main
work load initially. For legacy code, this task may become
substantial, but it is fairly straightforward when writing new code
with an AD tool in mind. Once this initial task is completed,
generating the adjoint code of a new model configuration takes about
10 minutes.

[HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]

\subsection{Special considerations}

* growth term(?)

* small active denominators

* dynamic solver (implicit function theorem)

* approximate adjoints


\subsection{An example: sensitivities of sea-ice export through
the Lancaster and Jones Sound}

We demonstrate the power of the adjoint method
in the context of investigating sea-ice export sensitivities through 
Lancaster and Jones Sound. The rationale for doing so is to complement
the analysis of sea-ice dynamics in the presence of narrow straits.
Lancaster Sound is one of the main outflow paths of sea-ice flowing
through the Canadian Arctic Archipelago (CAA). 
Export sensitivities reflect dominant
pathways through the CAA as resolved by the model.
Sensitivity maps can shed a very detailed light on various quantities
affecting the sea-ice export (and thus the underlying pathways).
Note that while the dominant circulation through Lancaster Sound is
toward the East, there is a small Westward flow to the North,
hugging the coast of Devon Island [ARE WE RESOLVING THIS?],
see e.g. \cite{mell:02, mich-etal:06,muen-etal:06}.

The model domain is a coarsened version of the Arctic face of the
high-resolution cubed-sphere configuration of the ECCO2 project
\citep[see][]{menemenlis05}. It covers the entire Arctic,
extends into the North Pacific such as to cover the entire
ice-covered regions, and comprises parts of the North Atlantic
down to XXN to enable analysis of remote influences of the 
North Atlantic current to sea-ice variability and export.
The horizontal resolution varies between XX and YY km
with 50 unevenly spaced vertical levels.
The adjoint models run efficiently on 80 processors
(benchmarks have been performed both on an SGI Altix as well as an 
IBM SP5 at NASA/ARC).

Following a 3-year spinup, the model has been integrated for four
years and five months between January 1989 and May 1993.
It is forced using realistic 6-hourly
NCEP/NCAR atmospheric state variables. Over the open ocean these are
converted into air-sea fluxes via the bulk formulae of
\citet{large04}.  Derivation of air-sea fluxes in the presence of
sea-ice is handled by the ice model as described in \refsec{model}.
The objective function is chosen $J$ as the
sea-ice export through 
Lancaster Sound at XX$^{\circ}$W
averaged over an 8-month period between October 1992 and May 1993.  

The adjoint model computes sensitivities
to sea-ice export back in time from 1993 to 1989 along this
trajectory.  In principle all adjoint model variable (i.e., Lagrange
multipliers) of the coupled ocean/sea-ice model
as well as the surface atmospheric state are available to
analyze the transient sensitivity behaviour.  
Over the open ocean, the adjoint of the bulk formula scheme
computes sensitivities to the time-varying atmospheric state.  Over
ice-covered parts, the sea-ice adjoint converts surface ocean
sensitivities to atmospheric sensitivities.

DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT

\subsection{Sensitivities to the sea-ice state}

\paragraph{Sensitivities to the sea-ice thickness}

The most readily interpretable ice-export sensitivity is that
to effective ice thickness, $\partial{J} / \partial{h}$.
Fig. XXX depcits transient $\partial{J} / \partial{h}$ using free-slip
(left column) and no-slip (right column) boundary conditions.
Sensitivity snapshots are depicted for (from top to bottom) 
12, 24, 36, and 48 months prior to May 2003.
The dominant features are\ml{ in accordance with expectations/as expected}:

(*)
Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.
a unit increase in sea-ice thickness in most places upstream
of Lancaster Sound will increase sea-ice export through Lancaster Sound.
The dominant pathway follows (backward in time) through Barrow Strait
into Viscount Melville Sound, and from there trough M'Clure Strait
into the Arctic Ocean (the "Northwest Passage"). 
Secondary paths are Northward from
Viscount Melville Sound through Byam Martin Channel into
Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.

(*)
As expected, at any given time the
region of influence is larger for the free-slip than no-slip simulation.
For the no-slip run, the region of influence is confined, after four years,
to just West of Barrow Strait (North of Prince of Wales Island),
and to the South of Penny Strait.
In contrast, sensitivities of the free-slip run extend
all the way to the Arctic interior both to the West
(M'Clure St.) and to the North (Ballantyne St., Prince Gustav Adolf Sea,
Massey Sound).

(*)
sensitivities seem to spread out in "pulses" (seasonal cycle)
[PLOT A TIME SERIES OF ADJheff in Barrow Strait)

(*)
The sensitivity in Baffin Bay are more complex.
The pattern evolves along the Western boundary, connecting
the Lancaster Sound Polynya, the Coburg Island Polynya, and the
North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
The sign of sensitivities has an oscillatory character
[AT FREQUENCY OF SEASONAL CYCLE?].
First, we need to establish whether forward perturbation runs
corroborate the oscillatory behaviour.
Then, several possible explanations:
(i) connection established through Nares Strait throughflow
which extends into Western boundary current in Northern Baffin Bay.
(ii) sea-ice concentration there is seasonal, i.e. partly
ice-free during the year. Seasonal cycle in sensitivity likely
connected to ice-free vs. ice-covered parts of the year.
Negative sensitivities can potentially be attributed
to blocking of Lancaster Sound ice export by Western boundary ice
in Baffin Bay.
(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
(in reality there's a Northern counter current hugging the coast of
Devon Island which we probably don't resolve).

Remote control of Kennedy Channel on Lancaster Sound ice export
seems a nice test for appropriateness of free-slip vs. no-slip BCs.

\paragraph{Sensitivities to the sea-ice area}

Fig. XXX depcits transient sea-ice export sensitivities
to changes in sea-ice concentration
 $\partial J / \partial area$ using free-slip
(left column) and no-slip (right column) boundary conditions.
Sensitivity snapshots are depicted for (from top to bottom) 
12, 24, 36, and 48 months prior to May 2003.
Contrary to the steady patterns seen for thickness sensitivities,
the ice-concentration sensitivities exhibit a strong seasonal cycle
in large parts of the domain (but synchronized on large scale).
The following discussion is w.r.t. free-slip run.

(*)
Months, during which sensitivities are negative:
\\
0 to 5   Db=N/A, Dr=5 (May-Jan) \\
10 to 17 Db=7, Dr=5 (Jul-Jan) \\
22 to 29 Db=7, Dr=5 (Jul-Jan) \\
34 to 41 Db=7, Dr=5 (Jul-Jan) \\
46 to 49 D=N/A \\
%
These negative sensitivities seem to be connected to months
during which main parts of the CAA are essentially entirely ice-covered.
This means that increase in ice concentration during this period
will likely reduce ice export due to blocking
[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
Only during periods where substantial parts of the CAA are
ice free (i.e. sea-ice concentration is less than one in larger parts of
the CAA) will an increase in ice-concentration increase ice export.

(*)
Sensitivities peak about 2-3 months before sign reversal, i.e.
max. negative sensitivities are expected end of July
[DOUBLE CHECK THIS].

(*) 
Peaks/bursts of sensitivities for months
14-17, 19-21, 27-29, 30-33, 38-40, 42-45

(*) 
Spatial "anti-correlation" (in sign) between main sensitivity branch
(essentially Northwest Passage and immediate connecting channels),
and remote places.
For example: month 20, 28, 31.5, 40, 43.
The timings of max. sensitivity extent are similar between
free-slip and no-slip run; and patterns are similar within CAA,
but differ in the Arctic Ocean interior.

(*) 
Interesting (but real?) patterns in Arctic Ocean interior.

\paragraph{Sensitivities to the sea-ice velocity}

(*)
Patterns of ADJuice at almost any point in time are rather complicated
(in particular with respect to spatial structure of signs).
Might warrant perturbation tests.
Patterns of ADJvice, on the other hand, are more spatially coherent,
but still hard to interpret (or even counter-intuitive
in many places).

(*)
"Growth in extent of sensitivities" goes in clear pulses:
almost no change between months: 0-5, 10-20, 24-32, 36-44
These essentially correspond to months of


\subsection{Sensitivities to the oceanic state}

\paragraph{Sensitivities to theta}

\textit{Sensitivities at the surface (z = 5 m)}

(*)
mabye redo with caxmax=0.02 or even 0.05

(*)
Core of negative sensitivities spreading through the CAA as 
one might expect [TEST]:
Increase in SST will decrease ice thickness and therefore ice export.

(*)
What's maybe unexpected is patterns of positive sensitivities
at the fringes of the "core", e.g. in the Southern channels
(Bellot St., Peel Sound, M'Clintock Channel), and to the North
(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
then shifting Northward into the Arctic interior).

(*)
Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
propagating into Lincoln Sea, then
entering Nares Strait and Smith Sound, periodically
warming or cooling[???] the Lancaster Sound exit.

\textit{Sensitivities at depth (z = 200 m)}

(*)
Negative sensitivities almost everywhere, as might be expected.

(*)
Sensitivity patterns between free-slip and no-slip BCs
are quite similar, except in Lincoln Sea (North of Nares St),
where the sign is reversed (but pattern remains similar).

\paragraph{Sensitivities to salt}

T.B.D.

\paragraph{Sensitivities to velocity}

T.B.D.

\subsection{Sensitivities to the atmospheric state}

\begin{itemize}
%
\item
plot of ATEMP for 12, 24, 36, 48 months
%
\item
plot of HEFF for 12, 24, 36, 48 months
%
\end{itemize}


\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
through Fram Strait in December 1995 to changes in sea-ice thickness
12, 24, 36, 48 months back in time. Corresponding sensitivities to
ocean surface temperature are depicted in
\reffig{4yradjthetalev1}(a--d).  The main characteristics is
consistency with expected advection of sea-ice over the relevant time
scales considered.  The general positive pattern means that an
increase in sea-ice thickness at location $(x,y)$ and time $t$ will
increase sea-ice export through Fram Strait at time $T_e$.  Largest
distances from Fram Strait indicate fastest sea-ice advection over the
time span considered.  The ice thickness sensitivities are in close
correspondence to ocean surface sentivitites, but of opposite sign.
An increase in temperature will incur ice melting, decrease in ice
thickness, and therefore decrease in sea-ice export at time $T_e$.

The picture is fundamentally different and much more complex
for sensitivities to ocean temperatures away from the surface.
\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
temperatures at roughly 400 m depth.
Primary features are the effect of the heat transport of the North
Atlantic current which feeds into the West Spitsbergen current,
the circulation around Svalbard, and ...


\ml{[based on the movie series
  zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
export through the Canadian Archipelag is highly sensitive to the
previous state of the ocean-ice system in the Archipelago and the
Western Arctic. According to the \ml{(adjoint)} senstivities of the
eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
cross-section G) with respect to ice volume (effective thickness), ocean
surface temperature, and vertical diffusivity near the surface
(\reffig{fouryearadj}) after 4 years of integration the following
mechanisms can be identified: near the ``observation'' (cross-section
G), smaller vertical diffusivities lead to lower surface temperatures
and hence to more ice that is available for export. Further away from
cross-section G, the sensitivity to vertical diffusivity has the
opposite sign, but temperature and ice volume sensitivities have the
same sign as close to the observation.

\begin{figure}[t!]
\centerline{
\subfigure[{\footnotesize -12 months}]
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
%
\subfigure[{\footnotesize -24 months}]
{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
}
%
\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
sea-ice thickness at various prior times.
\label{fig:4yradjheff}}
\end{figure}


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "ceaice"
%%% End: 
1	\section{Adjoint sensiivities of the MITsim}
2	\label{sec:adjoint}
3
4	\subsection{The adjoint of MITsim}
5
6	The adjoint model of the MITgcm has become an invaluable
7	tool for sensitivity analysis as well as state estimation \citep[for a
8	recent summary, see][]{heim:08}. The code has been developed and
9	tailored to be readily used with automatic differentiation tools for
10	adjoint code generation. This route was also taken in developing and
11	adapting the sea-ice compontent MITsim, so that tangent linear and
12	adjoint components can be obtained and kept up to date without
13	excessive effort.
14
15	The adjoint model operator (ADM) is the transpose of the tangent
16	linear model operator (TLM) of the full (in general nonlinear) forward
17	model, in this case the MITsim. This operator computes the gradients
18	of scalar-valued model diagnostics (so-called cost function or
19	objective function) with respect to many model inputs (so-called
20	independent or control variables). These inputs can be two- or
21	three-dimensional fields of initial conditions of the ocean or sea-ice
22	state, model parameters such as mixing coefficients, or time-varying
23	surface or lateral (open) boundary conditions. When combined, these
24	variables span a potentially high-dimensional (e.g. O(10$^8$))
25	so-called control space. At this problem dimension, perturbing
26	individual parameters to assess model sensitivities quickly becomes
27	prohibitive. By contrast, transient sensitivities of the objective
28	function to any element of the control and model state space can be
29	computed very efficiently in one single adjoint model integration,
30	provided an adjoint model is available.
31
32	In anology to the TLM and ADM components of the MITgcm we rely on the
33	autmomatic differentiation (AD) tool ``Transformation of Algorithms in
34	Fortran'' (TAF) developed by Fastopt \citep{gier-kami:98} to generate
35	TLM and ADM code of the MITsim \citep[for details see][]{maro-etal:99,
36	heim-etal:05}. In short, the AD tool uses the nonlinear parent
37	model code to generate derivative code for the specified control space
38	and objective function. Advantages of this approach have been pointed
39	out, for example by \cite{gier-kami:98}.
40
41	Many issues of generating efficient exact adjoint sea-ice code are
42	similar to those for the ocean model's adjoint. Linearizing the model
43	around the exact nonlinear model trajectory is a crucial aspect in the
44	presence of different regimes (e.g., is the thermodynamic growth term
45	for sea-ice evaluated near or far away from the freezing point of the
46	ocean surface?). Adapting the (parent) model code to support the AD
47	tool in providing exact and efficient adjoint code represents the main
48	work load initially. For legacy code, this task may become
49	substantial, but it is fairly straightforward when writing new code
50	with an AD tool in mind. Once this initial task is completed,
51	generating the adjoint code of a new model configuration takes about
52	10 minutes.
53
54	[HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]
55
56	\subsection{Special considerations}
57
58	* growth term(?)
59
60	* small active denominators
61
62	* dynamic solver (implicit function theorem)
63
64	* approximate adjoints
65
66
67	\subsection{An example: sensitivities of sea-ice export through
68	the Lancaster and Jones Sound}
69
70	We demonstrate the power of the adjoint method
71	in the context of investigating sea-ice export sensitivities through
72	Lancaster and Jones Sound. The rationale for doing so is to complement
73	the analysis of sea-ice dynamics in the presence of narrow straits.
74	Lancaster Sound is one of the main outflow paths of sea-ice flowing
75	through the Canadian Arctic Archipelago (CAA).
76	Export sensitivities reflect dominant
77	pathways through the CAA as resolved by the model.
78	Sensitivity maps can shed a very detailed light on various quantities
79	affecting the sea-ice export (and thus the underlying pathways).
80	Note that while the dominant circulation through Lancaster Sound is
81	toward the East, there is a small Westward flow to the North,
82	hugging the coast of Devon Island [ARE WE RESOLVING THIS?],
83	see e.g. \cite{mell:02, mich-etal:06,muen-etal:06}.
84
85	The model domain is a coarsened version of the Arctic face of the
86	high-resolution cubed-sphere configuration of the ECCO2 project
87	\citep[see][]{menemenlis05}. It covers the entire Arctic,
88	extends into the North Pacific such as to cover the entire
89	ice-covered regions, and comprises parts of the North Atlantic
90	down to XXN to enable analysis of remote influences of the
91	North Atlantic current to sea-ice variability and export.
92	The horizontal resolution varies between XX and YY km
93	with 50 unevenly spaced vertical levels.
94	The adjoint models run efficiently on 80 processors
95	(benchmarks have been performed both on an SGI Altix as well as an
96	IBM SP5 at NASA/ARC).
97
98	Following a 3-year spinup, the model has been integrated for four
99	years and five months between January 1989 and May 1993.
100	It is forced using realistic 6-hourly
101	NCEP/NCAR atmospheric state variables. Over the open ocean these are
102	converted into air-sea fluxes via the bulk formulae of
103	\citet{large04}. Derivation of air-sea fluxes in the presence of
104	sea-ice is handled by the ice model as described in \refsec{model}.
105	The objective function is chosen $J$ as the
106	sea-ice export through
107	Lancaster Sound at XX$^{\circ}$W
108	averaged over an 8-month period between October 1992 and May 1993.
109
110	The adjoint model computes sensitivities
111	to sea-ice export back in time from 1993 to 1989 along this
112	trajectory. In principle all adjoint model variable (i.e., Lagrange
113	multipliers) of the coupled ocean/sea-ice model
114	as well as the surface atmospheric state are available to
115	analyze the transient sensitivity behaviour.
116	Over the open ocean, the adjoint of the bulk formula scheme
117	computes sensitivities to the time-varying atmospheric state. Over
118	ice-covered parts, the sea-ice adjoint converts surface ocean
119	sensitivities to atmospheric sensitivities.
120
121	DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT
122
123	\subsection{Sensitivities to the sea-ice state}
124
125	\paragraph{Sensitivities to the sea-ice thickness}
126
127	The most readily interpretable ice-export sensitivity is that
128	to effective ice thickness, $\partial{J} / \partial{h}$.
129	Fig. XXX depcits transient $\partial{J} / \partial{h}$ using free-slip
130	(left column) and no-slip (right column) boundary conditions.
131	Sensitivity snapshots are depicted for (from top to bottom)
132	12, 24, 36, and 48 months prior to May 2003.
133	The dominant features are\ml{ in accordance with expectations/as expected}:
134
135	(*)
136	Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.
137	a unit increase in sea-ice thickness in most places upstream
138	of Lancaster Sound will increase sea-ice export through Lancaster Sound.
139	The dominant pathway follows (backward in time) through Barrow Strait
140	into Viscount Melville Sound, and from there trough M'Clure Strait
141	into the Arctic Ocean (the "Northwest Passage").
142	Secondary paths are Northward from
143	Viscount Melville Sound through Byam Martin Channel into
144	Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.
145
146	(*)
147	As expected, at any given time the
148	region of influence is larger for the free-slip than no-slip simulation.
149	For the no-slip run, the region of influence is confined, after four years,
150	to just West of Barrow Strait (North of Prince of Wales Island),
151	and to the South of Penny Strait.
152	In contrast, sensitivities of the free-slip run extend
153	all the way to the Arctic interior both to the West
154	(M'Clure St.) and to the North (Ballantyne St., Prince Gustav Adolf Sea,
155	Massey Sound).
156
157	(*)
158	sensitivities seem to spread out in "pulses" (seasonal cycle)
159	[PLOT A TIME SERIES OF ADJheff in Barrow Strait)
160
161	(*)
162	The sensitivity in Baffin Bay are more complex.
163	The pattern evolves along the Western boundary, connecting
164	the Lancaster Sound Polynya, the Coburg Island Polynya, and the
165	North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
166	The sign of sensitivities has an oscillatory character
167	[AT FREQUENCY OF SEASONAL CYCLE?].
168	First, we need to establish whether forward perturbation runs
169	corroborate the oscillatory behaviour.
170	Then, several possible explanations:
171	(i) connection established through Nares Strait throughflow
172	which extends into Western boundary current in Northern Baffin Bay.
173	(ii) sea-ice concentration there is seasonal, i.e. partly
174	ice-free during the year. Seasonal cycle in sensitivity likely
175	connected to ice-free vs. ice-covered parts of the year.
176	Negative sensitivities can potentially be attributed
177	to blocking of Lancaster Sound ice export by Western boundary ice
178	in Baffin Bay.
179	(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
180	(in reality there's a Northern counter current hugging the coast of
181	Devon Island which we probably don't resolve).
182
183	Remote control of Kennedy Channel on Lancaster Sound ice export
184	seems a nice test for appropriateness of free-slip vs. no-slip BCs.
185
186	\paragraph{Sensitivities to the sea-ice area}
187
188	Fig. XXX depcits transient sea-ice export sensitivities
189	to changes in sea-ice concentration
190	$\partial J / \partial area$ using free-slip
191	(left column) and no-slip (right column) boundary conditions.
192	Sensitivity snapshots are depicted for (from top to bottom)
193	12, 24, 36, and 48 months prior to May 2003.
194	Contrary to the steady patterns seen for thickness sensitivities,
195	the ice-concentration sensitivities exhibit a strong seasonal cycle
196	in large parts of the domain (but synchronized on large scale).
197	The following discussion is w.r.t. free-slip run.
198
199	(*)
200	Months, during which sensitivities are negative:
201	\\
202	0 to 5 Db=N/A, Dr=5 (May-Jan) \\
203	10 to 17 Db=7, Dr=5 (Jul-Jan) \\
204	22 to 29 Db=7, Dr=5 (Jul-Jan) \\
205	34 to 41 Db=7, Dr=5 (Jul-Jan) \\
206	46 to 49 D=N/A \\
207	%
208	These negative sensitivities seem to be connected to months
209	during which main parts of the CAA are essentially entirely ice-covered.
210	This means that increase in ice concentration during this period
211	will likely reduce ice export due to blocking
212	[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
213	Only during periods where substantial parts of the CAA are
214	ice free (i.e. sea-ice concentration is less than one in larger parts of
215	the CAA) will an increase in ice-concentration increase ice export.
216
217	(*)
218	Sensitivities peak about 2-3 months before sign reversal, i.e.
219	max. negative sensitivities are expected end of July
220	[DOUBLE CHECK THIS].
221
222	(*)
223	Peaks/bursts of sensitivities for months
224	14-17, 19-21, 27-29, 30-33, 38-40, 42-45
225
226	(*)
227	Spatial "anti-correlation" (in sign) between main sensitivity branch
228	(essentially Northwest Passage and immediate connecting channels),
229	and remote places.
230	For example: month 20, 28, 31.5, 40, 43.
231	The timings of max. sensitivity extent are similar between
232	free-slip and no-slip run; and patterns are similar within CAA,
233	but differ in the Arctic Ocean interior.
234
235	(*)
236	Interesting (but real?) patterns in Arctic Ocean interior.
237
238	\paragraph{Sensitivities to the sea-ice velocity}
239
240	(*)
241	Patterns of ADJuice at almost any point in time are rather complicated
242	(in particular with respect to spatial structure of signs).
243	Might warrant perturbation tests.
244	Patterns of ADJvice, on the other hand, are more spatially coherent,
245	but still hard to interpret (or even counter-intuitive
246	in many places).
247
248	(*)
249	"Growth in extent of sensitivities" goes in clear pulses:
250	almost no change between months: 0-5, 10-20, 24-32, 36-44
251	These essentially correspond to months of
252
253
254	\subsection{Sensitivities to the oceanic state}
255
256	\paragraph{Sensitivities to theta}
257
258	\textit{Sensitivities at the surface (z = 5 m)}
259
260	(*)
261	mabye redo with caxmax=0.02 or even 0.05
262
263	(*)
264	Core of negative sensitivities spreading through the CAA as
265	one might expect [TEST]:
266	Increase in SST will decrease ice thickness and therefore ice export.
267
268	(*)
269	What's maybe unexpected is patterns of positive sensitivities
270	at the fringes of the "core", e.g. in the Southern channels
271	(Bellot St., Peel Sound, M'Clintock Channel), and to the North
272	(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
273	then shifting Northward into the Arctic interior).
274
275	(*)
276	Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
277	propagating into Lincoln Sea, then
278	entering Nares Strait and Smith Sound, periodically
279	warming or cooling[???] the Lancaster Sound exit.
280
281	\textit{Sensitivities at depth (z = 200 m)}
282
283	(*)
284	Negative sensitivities almost everywhere, as might be expected.
285
286	(*)
287	Sensitivity patterns between free-slip and no-slip BCs
288	are quite similar, except in Lincoln Sea (North of Nares St),
289	where the sign is reversed (but pattern remains similar).
290
291	\paragraph{Sensitivities to salt}
292
293	T.B.D.
294
295	\paragraph{Sensitivities to velocity}
296
297	T.B.D.
298
299	\subsection{Sensitivities to the atmospheric state}
300
301	\begin{itemize}
302	%
303	\item
304	plot of ATEMP for 12, 24, 36, 48 months
305	%
306	\item
307	plot of HEFF for 12, 24, 36, 48 months
308	%
309	\end{itemize}
310
311
312
313	\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
314	through Fram Strait in December 1995 to changes in sea-ice thickness
315	12, 24, 36, 48 months back in time. Corresponding sensitivities to
316	ocean surface temperature are depicted in
317	\reffig{4yradjthetalev1}(a--d). The main characteristics is
318	consistency with expected advection of sea-ice over the relevant time
319	scales considered. The general positive pattern means that an
320	increase in sea-ice thickness at location $(x,y)$ and time $t$ will
321	increase sea-ice export through Fram Strait at time $T_e$. Largest
322	distances from Fram Strait indicate fastest sea-ice advection over the
323	time span considered. The ice thickness sensitivities are in close
324	correspondence to ocean surface sentivitites, but of opposite sign.
325	An increase in temperature will incur ice melting, decrease in ice
326	thickness, and therefore decrease in sea-ice export at time $T_e$.
327
328	The picture is fundamentally different and much more complex
329	for sensitivities to ocean temperatures away from the surface.
330	\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
331	temperatures at roughly 400 m depth.
332	Primary features are the effect of the heat transport of the North
333	Atlantic current which feeds into the West Spitsbergen current,
334	the circulation around Svalbard, and ...
335
336
337	\ml{[based on the movie series
338	zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
339	export through the Canadian Archipelag is highly sensitive to the
340	previous state of the ocean-ice system in the Archipelago and the
341	Western Arctic. According to the \ml{(adjoint)} senstivities of the
342	eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
343	cross-section G) with respect to ice volume (effective thickness), ocean
344	surface temperature, and vertical diffusivity near the surface
345	(\reffig{fouryearadj}) after 4 years of integration the following
346	mechanisms can be identified: near the ``observation'' (cross-section
347	G), smaller vertical diffusivities lead to lower surface temperatures
348	and hence to more ice that is available for export. Further away from
349	cross-section G, the sensitivity to vertical diffusivity has the
350	opposite sign, but temperature and ice volume sensitivities have the
351	same sign as close to the observation.
352
353	\begin{figure}[t!]
354	\centerline{
355	\subfigure[{\footnotesize -12 months}]
356	{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
357	%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
358	%
359	\subfigure[{\footnotesize -24 months}]
360	{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
361	}
362	%
363	\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
364	sea-ice thickness at various prior times.
365	\label{fig:4yradjheff}}
366	\end{figure}
367
368
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