articles/ceaice/ceaice_adjoint.tex

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

\subsection{The adjoint of MITsim}


The ability to generate tangent linear and adjoint components
of a coupled ocean sea-ice system was one of the main drivers
behind the MITsim development.
For the ocean the adjoint capability has proven to be an
invaluable tool for sensitivity analysis as well as state estimation,
as evidenced by various adjoint-based studies
(for a recent summary, see \cite{heim:08}).

The adjoint model operator (ADM) is the transpose of the tangent linear 
model operator (TLM)
of the full (in general nonlinear) forward model, i.e. the MITsim.
It enables very efficient computation of 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. Performing parameter perturbations
to assess model sensitivities quickly becomes prohibitive at these scales.
Alternatively, 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 efficient adjoint model is available.

Following closely the development and maintenance of the
TLM and ADM components of the MITgcm we have relied heavily on the
autmomatic differentiation (AD) tool
"Transformation of Algorithms in Fortran" (TAF)
developed by Fastopt \citep{gier-kami:98}.
to derive TLM and ADM code of the MITsim
(for details see \cite{maro-etal:99}, \cite{heim-etal:05}).
Briefly, the nonlinear parent model is fed to the AD tool which produces 
derivative code for the specified control space and objective function.
Apart from its evident success, advantages of this approach have been 
pointed out, e.g. by \cite{gier-kami:98}.

Many issues underlying the efficient exact adjoint sea-ice code generation
are similar to those arising for the ocean model's adjoint.
Linearizing the model around the exact nonlinear model trajectory,
as we do, is a crucial aspect in the presence of different
regimes (e.g. effect of the seaice growth term at or away from the
freezing point of the ocean surface).
Adjusting the (parent) model code to support the AD tool in
providing exact and efficient adjoint code is the main initial work.
This may be substantial for legacy code, but fairly straightforward
when coding with "AD application in mind".
Once in place, an adjoint model of a new model configuration
may be derived in 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 chosen is 
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 ice thickness, $\partial J / \partial heff$.
Fig. XXX depcits transient $\partial J / \partial heff$ 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 in accordance with expectations:

(*)
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 ...

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