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1  \section{Adjoint sensiivities of the MITsim}  \section{Adjoint sensitivities of the MITsim}
2  \label{sec:adjoint}  \label{sec:adjoint}
3    
4  \subsection{The adjoint of MITsim}  \subsection{The adjoint of MITsim}
5    
6    The adjoint model of the MITgcm has become an invaluable
7  The ability to generate tangent linear and adjoint components  tool for sensitivity analysis as well as state estimation \citep[for a
8  of a coupled ocean sea-ice system was one of the main drivers  recent summary, see][]{heim:08}. The code has been developed and
9  behind the MITsim development.  tailored to be readily used with automatic differentiation tools for
10  For the ocean the adjoint capability has proven to be an  adjoint code generation. This route was also taken in developing and
11  invaluable tool for sensitivity analysis as well as state estimation,  adapting the sea-ice compontent MITsim, so that tangent linear and
12  as evidenced by various adjoint-based studies  adjoint components can be obtained and kept up to date without
13  (for a recent summary, see \cite{heim:08}).  excessive effort.
14    
15  The adjoint model operator (ADM) is the transpose of the tangent linear  The adjoint model operator (ADM) is the transpose of the tangent
16  model operator (TLM)  linear model operator (TLM) of the full (in general nonlinear) forward
17  of the full (in general nonlinear) forward model, i.e. the MITsim.  model, in this case the MITsim. This operator computes the gradients
18  It enables very efficient computation of gradients  of scalar-valued model diagnostics (so-called cost function or
19  of scalar-valued model diagnostics  objective function) with respect to many model inputs (so-called
20  (so-called cost function or objective function)  independent or control variables).  These inputs can be two- or
21  with respect to many model inputs (so-called independent or control variables).  three-dimensional fields of initial conditions of the ocean or sea-ice
22  These inputs can be two- or three-dimensional fields of initial  state, model parameters such as mixing coefficients, or time-varying
23  conditions of the ocean or sea-ice state, model parameters such as  surface or lateral (open) boundary conditions.  When combined, these
24  mixing coefficients, or time-varying surface or lateral (open) boundary conditions.  variables span a potentially high-dimensional (e.g.  O(10$^8$))
25  When combined, these variables span a potentially high-dimensional  so-called control space. At this problem dimension, perturbing
26  (e.g. O(10$^8$)) so-called control space. Performing parameter perturbations  individual parameters to assess model sensitivities quickly becomes
27  to assess model sensitivities quickly becomes prohibitive at these scales.  prohibitive. By contrast, transient sensitivities of the objective
28  Alternatively, transient sensitivities of the objective function  function to any element of the control and model state space can be
29  to any element of the  control and model state space can be computed  computed very efficiently in one single adjoint model integration,
30  very efficiently in  one single adjoint  provided an adjoint model is available.
31  model integration, provided an efficient adjoint model is available.  
32    In anology to the TLM and ADM components of the MITgcm we rely on the
33  Following closely the development and maintenance of the  autmomatic differentiation (AD) tool ``Transformation of Algorithms in
34  TLM and ADM components of the MITgcm we have relied heavily on the  Fortran'' (TAF) developed by Fastopt \citep{gier-kami:98} to generate
35  autmomatic differentiation (AD) tool  TLM and ADM code of the MITsim \citep[for details see][]{maro-etal:99,
36  "Transformation of Algorithms in Fortran" (TAF)    heim-etal:05}.  In short, the AD tool uses the nonlinear parent
37  developed by Fastopt \citep{gier-kami:98}.  model code to generate derivative code for the specified control space
38  to derive TLM and ADM code of the MITsim  and objective function. Advantages of this approach have been pointed
39  (for details see \cite{maro-etal:99}, \cite{heim-etal:05}).  out, for example by \cite{gier-kami:98}.
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.  Many issues of generating efficient exact adjoint sea-ice code are
42  Apart from its evident success, advantages of this approach have been  similar to those for the ocean model's adjoint.  Linearizing the model
43  pointed out, e.g. by \cite{gier-kami:98}.  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  Many issues underlying the efficient exact adjoint sea-ice code generation  for sea-ice evaluated near or far away from the freezing point of the
46  are similar to those arising for the ocean model's adjoint.  ocean surface?). Adapting the (parent) model code to support the AD
47  Linearizing the model around the exact nonlinear model trajectory,  tool in providing exact and efficient adjoint code represents the main
48  as we do, is a crucial aspect in the presence of different  work load initially. For legacy code, this task may become
49  regimes (e.g. effect of the seaice growth term at or away from the  substantial, but it is fairly straightforward when writing new code
50  freezing point of the ocean surface).  with an AD tool in mind. Once this initial task is completed,
51  Adjusting the (parent) model code to support the AD tool in  generating the adjoint code of a new model configuration takes about
52  providing exact and efficient adjoint code is the main initial work.  10 minutes.
 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.  
53    
54  [HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]  [HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]
55    
# Line 69  may be derived in about 10 minutes. Line 65  may be derived in about 10 minutes.
65    
66    
67  \subsection{An example: sensitivities of sea-ice export through  \subsection{An example: sensitivities of sea-ice export through
68  the Lancaster and Jones Sound}  the Lancaster Sound}
69    
70  We demonstrate the power of the adjoint method  We demonstrate the power of the adjoint method in the context of
71  in the context of investigating sea-ice export sensitivities through  investigating sea-ice export sensitivities through Lancaster Sound.
72  Lancaster and Jones Sound. The rationale for doing so is to complement  The rationale for doing so is to complement the analysis of sea-ice
73  the analysis of sea-ice dynamics in the presence of narrow straits.  dynamics in the presence of narrow straits.  Lancaster Sound is one of
74  Lancaster Sound is one of the main outflow paths of sea-ice flowing  the main outflow paths of sea-ice flowing through the Canadian Arctic
75  through the Canadian Arctic Archipelago (CAA).  Archipelago (CAA).  Export sensitivities reflect dominant pathways
76  Export sensitivities reflect dominant  through the CAA as resolved by the model.  Sensitivity maps can shed a
77  pathways through the CAA as resolved by the model.  very detailed light on various quantities affecting the sea-ice export
78  Sensitivity maps can shed a very detailed light on various quantities  (and thus the underlying pathways).  Note that while the dominant
79  affecting the sea-ice export (and thus the underlying pathways).  circulation through Lancaster Sound is toward the East, there is a
80  Note that while the dominant circulation through Lancaster Sound is  small Westward flow to the North, hugging the coast of Devon Island
81  toward the East, there is a small Westward flow to the North,  \citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in
82  hugging the coast of Devon Island [ARE WE RESOLVING THIS?],  our simulation.
 see e.g. \cite{mell:02, mich-etal:06,muen-etal:06}.  
83    
84  The model domain is a coarsened version of the Arctic face of the  The model domain is a coarsened version of the Arctic face of the
85  high-resolution cubed-sphere configuration of the ECCO2 project  high-resolution cubed-sphere configuration of the ECCO2 project
86  \citep[see][]{menemenlis05}. It covers the entire Arctic,  \citep{menemenlis05} as described in \refsec{forward}.  The horizontal
87  extends into the North Pacific such as to cover the entire  resolution is half of that in \refsec{forward} while the vertical grid
88  ice-covered regions, and comprises parts of the North Atlantic  is the same. \ml{[Is this important? Do we need to be more specific?:
89  down to XXN to enable analysis of remote influences of the    ]} The adjoint models run efficiently on 80 processors (as validated
90  North Atlantic current to sea-ice variability and export.  by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).
91  The horizontal resolution varies between XX and YY km  
92  with 50 unevenly spaced vertical levels.  Following a 3-year spinup, the model is integrated for four
93  The adjoint models run efficiently on 80 processors  years and five months between January 1989 and September 1993.
94  (benchmarks have been performed both on an SGI Altix as well as an  \ml{[Patrick: to what extent is this different from section 3?]}
95  IBM SP5 at NASA/ARC).  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.
96    %Over the open ocean these are
97  Following a 3-year spinup, the model has been integrated for four  %converted into air-sea fluxes via the bulk formulae of
98  years and five months between January 1989 and May 1993.  %\citet{large04}.  The air-sea fluxes in the presence of
99  It is forced using realistic 6-hourly  %sea-ice are handled by the ice model as described in \refsec{model}.
100  NCEP/NCAR atmospheric state variables. Over the open ocean these are  The objective function $J$ is chosen as the ``solid'' fresh water
101  converted into air-sea fluxes via the bulk formulae of  export, that is the export of ice and snow converted to units of fresh
102  \citet{large04}.  Derivation of air-sea fluxes in the presence of  water $(\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u$, through Lancaster
103  sea-ice is handled by the ice model as described in \refsec{model}.  Sound at approximately 82\degW\ (cross-section G in
104  The objective function chosen is  \reffig{arctic_topog}) averaged over a 12-month period between October
105  sea-ice export through  1992 and September 1993.
106  Lancaster Sound at XX$^{\circ}$W  
107  averaged over an 8-month period between October 1992 and May 1993.    The forward trajectory of the model integration resembles broadly that
108    of the model in \refsec{forward}. Many details are different, owning
109  The adjoint model computes sensitivities  to different resolution and integration period; for example, the solid
110  to sea-ice export back in time from 1993 to 1989 along this  fresh water transport through Lancaster Sound is
111  trajectory.  In principle all adjoint model variable (i.e., Lagrange  $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
112  multipliers) of the coupled ocean/sea-ice model  the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a
113  as well as the surface atmospheric state are available to  no slip simulation.
114  analyze the transient sensitivity behaviour.    
115  Over the open ocean, the adjoint of the bulk formula scheme  The adjoint model computes sensitivities of this export back in time
116  computes sensitivities to the time-varying atmospheric state.  Over  from 1993 to 1989 along this trajectory.  In principle all adjoint
117  ice-covered parts, the sea-ice adjoint converts surface ocean  model variable (i.e., Lagrange multipliers) of the coupled
118  sensitivities to atmospheric sensitivities.  ocean/sea-ice model as well as the surface atmospheric state are
119    available to analyze the transient sensitivity behavior.  Over the
120    open ocean, the adjoint of the bulk formula scheme computes
121    sensitivities to the time-varying atmospheric state.  Over ice-covered
122    parts, the sea-ice adjoint converts surface ocean sensitivities to
123    atmospheric sensitivities.
124    
125  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT
126    
127  \subsection{Sensitivities to the sea-ice state}  \subsubsection{Adjoint sensitivities}
   
 \paragraph{Sensitivities to the sea-ice thickness}  
128    
129  The most readily interpretable ice-export sensitivity is that  The most readily interpretable ice-export sensitivity is that to
130  to ice thickness, $\partial J / \partial heff$.  effective ice thickness, $\partial{J} / \partial{(hc)}$.
131  Fig. XXX depcits transient $\partial J / \partial heff$ using free-slip  \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
132  (left column) and no-slip (right column) boundary conditions.  free-slip (left column) and no-slip (right column) boundary
133  Sensitivity snapshots are depicted for (from top to bottom)  conditions. Sensitivity snapshots are depicted for 12 months prior to
134  12, 24, 36, and 48 months prior to May 2003.  September 1993 (at the beginning of the averaging period for the objective
135  The dominant features are in accordance with expectations:  function $J$, top) and at the beginning of the integration in January
136    1989 (bottom).
137  (*)  \begin{figure*}[t]
138  Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.    \includegraphics*[width=\textwidth]{\fpath/adjheff}
139  a unit increase in sea-ice thickness in most places upstream    \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
140  of Lancaster Sound will increase sea-ice export through Lancaster Sound.      m$^2$\,s$^{-1}$/m for two different times (rows) and two different
141  The dominant pathway follows (backward in time) through Barrow Strait      boundary conditions for sea ice drift. The color scale is chosen
142  into Viscount Melville Sound, and from there trough M'Clure Strait      to illustrate the patterns of the sensitivities; the maximum and
143  into the Arctic Ocean (the "Northwest Passage").      minimum values are given above the figures.
144  Secondary paths are Northward from      \label{fig:adjheff}}
145  Viscount Melville Sound through Byam Martin Channel into  \end{figure*}
146  Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.  
147    At the beginning of October 1992, the positive sensitivities in
148  (*)  the Lancaster Sound mean that an increase of ice volume increase the
149  As expected, at any given time the  solid fresh water export. The negative sensivities to the East and to the
150  region of influence is larger for the free-slip than no-slip simulation.  West can be explained by indirect effects: less ice to the East means
151  For the no-slip run, the region of influence is confined, after four years,  less resistance to eastward drift and thus more export; similarly, less ice to
152  to just West of Barrow Strait (North of Prince of Wales Island),  the West means that more ice can be moved eastwards from the Barrow Strait
153  and to the South of Penny Strait.  into the Lancaster Sound leading to more ice export. The sensitivities
154  In contrast, sensitivities of the free-slip run extend  are similar for both no slip and free slip solutions with a slightly larger
155  all the way to the Arctic interior both to the West  area covered by non-zero sensitivities in the free slip solution. At
156  (M'Clure St.) and to the North (Ballantyne St., Prince Gustav Adolf Sea,  the beginning of the integration (the end of the backward adjoint
157  Massey Sound).  integration) the free and no slip solutions are very different. The
158    sensitivities of the free slip solution extend through the enitre
159  (*)  Canadian Archipelago and into the Arctic while in the no slip solution
160  sensitivities seem to spread out in "pulses" (seasonal cycle)  they still are confined to the Lancaster Sound and the Barrow
161  [PLOT A TIME SERIES OF ADJheff in Barrow Strait)  Strait. This implies that in the free slip solution ice can drift more
162    easily through the narrow straits of the Canadian Archipelago, so that
163  (*)  a positive ice volume anomaly anywhere in the Canadian Archipelago is
164  The sensitivity in Baffin Bay are more complex.  moved through the Lancaster Sound within 4 years thus increasing the
165  The pattern evolves along the Western boundary, connecting  ice export.
166  the Lancaster Sound Polynya, the Coburg Island Polynya, and the  
167  North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.  The temporal evolution of several sensitivities along the zonal axis
168  The sign of sensitivities has an oscillatory character  Lancaster Sound-Barrow Strait-Melville Sound are shown in
169  [AT FREQUENCY OF SEASONAL CYCLE?].  \reffig{lancaster}.
170  First, we need to establish whether forward perturbation runs  \begin{figure*}
171  corroborate the oscillatory behaviour.    \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
172  Then, several possible explanations:    \caption{Hovermoeller diagrams of sensitivities (derivatives) of the
173  (i) connection established through Nares Strait throughflow      ``solid'' fresh water (i.e., ice and snow) export $J$ through Lancaster sound
174  which extends into Western boundary current in Northern Baffin Bay.      (\reffig{arctic_topog}, cross-section G) with respect to effective
175  (ii) sea-ice concentration there is seasonal, i.e. partly      ice thickness ($hc$), ocean surface temperature (SST) and
176  ice-free during the year. Seasonal cycle in sensitivity likely      precipitation ($p$) for two runs with free slip and no slip boundary
177  connected to ice-free vs. ice-covered parts of the year.      conditions for the sea ice drift. Also shown it the normalized ice
178  Negative sensitivities can potentially be attributed      strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ (bottom panel); each plot is
179  to blocking of Lancaster Sound ice export by Western boundary ice      overlaid with the contours 1 and 3 of the normalized ice strength
180  in Baffin Bay.      for orientation.
181  (iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility      \label{fig:lancaster}}
182  (in reality there's a Northern counter current hugging the coast of  \end{figure*}
183  Devon Island which we probably don't resolve).  \reffig{lancaster} shows the sensitivities of ``solid'' fresh water
184    export, that is ice and snow, through Lancaster sound (cross-section G
185  Remote control of Kennedy Channel on Lancaster Sound ice export  in \reffig{arctic_topog}) with respect to effective ice thickness
186  seems a nice test for appropriateness of free-slip vs. no-slip BCs.  ($hc$), ocean surface temperature (SST) and precipitation ($p$) for
187    two runs with free slip and no slip boundary conditions for the sea
188  \paragraph{Sensitivities to the sea-ice area}  ice drift. The Hovmoeller diagrams of sensitivities (derivatives) with
189    respect to effective ice thickness (top) and ocean surface temperature
190  Fig. XXX depcits transient sea-ice export sensitivities  (second from top) are coherent: more ice in the Lancaster Sound leads
191  to changes in sea-ice concentration  to more export and one way to get more ice is by colder surface
192   $\partial J / \partial area$ using free-slip  temperatures (less melting from below). In the free slip case the
193  (left column) and no-slip (right column) boundary conditions.  sensitivities can propagate westwards (backwards in time) when the ice
194  Sensitivity snapshots are depicted for (from top to bottom)  strength is low in late summer. In the no slip case the (normalized)
195  12, 24, 36, and 48 months prior to May 2003.  ice strength does not fall below 1 during the winters of 1991 to 1993
196  Contrary to the steady patterns seen for thickness sensitivities,  (mainly because the ice concentrations remain nearly 100\%, not
197  the ice-concentration sensitivities exhibit a strong seasonal cycle  shown), so that ice is blocked and cannot drift eastwards (forward in
198  in large parts of the domain (but synchronized on large scale).  time) in the Melville Sound-Barrow Strait-Lancaster Sound channel.
199  The following discussion is w.r.t. free-slip run.  Consequently the sensitivies do not propagate westwards (backwards in
200    time) and the export through Lancaster Sound is only affected by
201  (*)  local ice formation and melting.
202  Months, during which sensitivities are negative:  
203  \\  The sensitivities to precipitation are negative (more precipitation
204  0 to 5   Db=N/A, Dr=5 (May-Jan) \\  leads to less export) before January and mostly positive after
205  10 to 17 Db=7, Dr=5 (Jul-Jan) \\  January. Further they are mostly positive for normalized ice strengths
206  22 to 29 Db=7, Dr=5 (Jul-Jan) \\  over 3. Assuming that most precipation is snow in this area---in the
207  34 to 41 Db=7, Dr=5 (Jul-Jan) \\  current implementation the model differentiates between snow and rain
208  46 to 49 D=N/A \\  depending on the thermodynamic growth rate; when it is cold enough for
209  %  ice to grow, all precipitation is assumed to be snow---the
210  These negative sensitivities seem to be connected to months  sensitivities can be interpreted in terms of the model physics.  Short
211  during which main parts of the CAA are essentially entirely ice-covered.  wave radiation cannot penetrate a snow cover and has a higer albedo
212  This means that increase in ice concentration during this period  than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it
213  will likely reduce ice export due to blocking  protects the ice against melting in spring (after January).  On the
214  [NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].  other hand, snow reduces the effective conductivity and thus the heat
215  Only during periods where substantial parts of the CAA are  flux through the ice. This insulating effect slows down the cooling of
216  ice free (i.e. sea-ice concentration is less than one in larger parts of  the surface water underneath the ice and limits the ice growth from
217  the CAA) will an increase in ice-concentration increase ice export.  below, so that less snow in the ice-growing season leads to more new
218    ice and thus more ice export.
219  (*)  
220  Sensitivities peak about 2-3 months before sign reversal, i.e.  %Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein:
221  max. negative sensitivities are expected end of July  %1. snow insulates against melting from above during spring: more precip (snow) -> more export
222  [DOUBLE CHECK THIS].  %2. less snow during fall -> more ice -> more export
223    %3. precip is both snow and rain, depending on the sign of "FICE" (thermodynamic growth rate), with probably different implications
224  (*)  
225  Peaks/bursts of sensitivities for months  
226  14-17, 19-21, 27-29, 30-33, 38-40, 42-45  \subsubsection{Forward sensitivities}
227    
228  (*)  \ml{[Here we need for integrations to show that the adjoint
229  Spatial "anti-correlation" (in sign) between main sensitivity branch    sensitivites are not just academic. I suggest to perturb HEFF
230  (essentially Northwest Passage and immediate connecting channels),    and THETA initial conditions, and PRECIP somewhere in the Melville
231  and remote places.    Sound and then produce plots similar to reffig{lancaster}. For
232  For example: month 20, 28, 31.5, 40, 43.    PRECIP it would be great to have two perturbation experiments, one
233  The timings of max. sensitivity extent are similar between    where ADJprecip is posivite and one where ADJprecip is negative]}
234  free-slip and no-slip run; and patterns are similar within CAA,  %The dominant features are\ml{ in accordance with expectations/as expected}:
235  but differ in the Arctic Ocean interior.  
236    %(*)
237  (*)  %Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.
238  Interesting (but real?) patterns in Arctic Ocean interior.  %a unit increase in sea-ice thickness in most places upstream
239    %of Lancaster Sound will increase sea-ice export through Lancaster Sound.
240  \paragraph{Sensitivities to the sea-ice velocity}  %The dominant pathway follows (backward in time) through Barrow Strait
241    %into Viscount Melville Sound, and from there trough M'Clure Strait
242  (*)  %into the Arctic Ocean (the "Northwest Passage").
243  Patterns of ADJuice at almost any point in time are rather complicated  %Secondary paths are Northward from
244  (in particular with respect to spatial structure of signs).  %Viscount Melville Sound through Byam Martin Channel into
245  Might warrant perturbation tests.  %Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.
246  Patterns of ADJvice, on the other hand, are more spatially coherent,  
247  but still hard to interpret (or even counter-intuitive  %(*)
248  in many places).  %As expected, at any given time the
249    %region of influence is larger for the free-slip than no-slip simulation.
250  (*)  %For the no-slip run, the region of influence is confined, after four years,
251  "Growth in extent of sensitivities" goes in clear pulses:  %to just West of Barrow Strait (North of Prince of Wales Island),
252  almost no change between months: 0-5, 10-20, 24-32, 36-44  %and to the South of Penny Strait.
253  These essentially correspond to months of  %In contrast, sensitivities of the free-slip run extend
254    %all the way to the Arctic interior both to the West
255    %(M'Clure St.) and to the North (Ballantyne St., Prince Gustav Adolf Sea,
256  \subsection{Sensitivities to the oceanic state}  %Massey Sound).
257    
258  \paragraph{Sensitivities to theta}  %(*)
259    %sensitivities seem to spread out in "pulses" (seasonal cycle)
260  \textit{Sensitivities at the surface (z = 5 m)}  %[PLOT A TIME SERIES OF ADJheff in Barrow Strait)
261    
262  (*)  %(*)
263  mabye redo with caxmax=0.02 or even 0.05  %The sensitivity in Baffin Bay are more complex.
264    %The pattern evolves along the Western boundary, connecting
265  (*)  %the Lancaster Sound Polynya, the Coburg Island Polynya, and the
266  Core of negative sensitivities spreading through the CAA as  %North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
267  one might expect [TEST]:  %The sign of sensitivities has an oscillatory character
268  Increase in SST will decrease ice thickness and therefore ice export.  %[AT FREQUENCY OF SEASONAL CYCLE?].
269    %First, we need to establish whether forward perturbation runs
270  (*)  %corroborate the oscillatory behaviour.
271  What's maybe unexpected is patterns of positive sensitivities  %Then, several possible explanations:
272  at the fringes of the "core", e.g. in the Southern channels  %(i) connection established through Nares Strait throughflow
273  (Bellot St., Peel Sound, M'Clintock Channel), and to the North  %which extends into Western boundary current in Northern Baffin Bay.
274  (initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,  %(ii) sea-ice concentration there is seasonal, i.e. partly
275  then shifting Northward into the Arctic interior).  %ice-free during the year. Seasonal cycle in sensitivity likely
276    %connected to ice-free vs. ice-covered parts of the year.
277  (*)  %Negative sensitivities can potentially be attributed
278  Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W  %to blocking of Lancaster Sound ice export by Western boundary ice
279  propagating into Lincoln Sea, then  %in Baffin Bay.
280  entering Nares Strait and Smith Sound, periodically  %(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
281  warming or cooling[???] the Lancaster Sound exit.  %(in reality there's a Northern counter current hugging the coast of
282    %Devon Island which we probably don't resolve).
283  \textit{Sensitivities at depth (z = 200 m)}  
284    %Remote control of Kennedy Channel on Lancaster Sound ice export
285  (*)  %seems a nice test for appropriateness of free-slip vs. no-slip BCs.
286  Negative sensitivities almost everywhere, as might be expected.  
287    %\paragraph{Sensitivities to the sea-ice area}
288  (*)  
289  Sensitivity patterns between free-slip and no-slip BCs  %Fig. XXX depcits transient sea-ice export sensitivities
290  are quite similar, except in Lincoln Sea (North of Nares St),  %to changes in sea-ice concentration
291  where the sign is reversed (but pattern remains similar).  % $\partial J / \partial area$ using free-slip
292    %(left column) and no-slip (right column) boundary conditions.
293  \paragraph{Sensitivities to salt}  %Sensitivity snapshots are depicted for (from top to bottom)
294    %12, 24, 36, and 48 months prior to May 2003.
295  T.B.D.  %Contrary to the steady patterns seen for thickness sensitivities,
296    %the ice-concentration sensitivities exhibit a strong seasonal cycle
297  \paragraph{Sensitivities to velocity}  %in large parts of the domain (but synchronized on large scale).
298    %The following discussion is w.r.t. free-slip run.
299  T.B.D.  
300    %(*)
301  \subsection{Sensitivities to the atmospheric state}  %Months, during which sensitivities are negative:
302    %\\
303  \begin{itemize}  %0 to 5   Db=N/A, Dr=5 (May-Jan) \\
304  %  %10 to 17 Db=7, Dr=5 (Jul-Jan) \\
305  \item  %22 to 29 Db=7, Dr=5 (Jul-Jan) \\
306  plot of ATEMP for 12, 24, 36, 48 months  %34 to 41 Db=7, Dr=5 (Jul-Jan) \\
307  %  %46 to 49 D=N/A \\
308  \item  %%
309  plot of HEFF for 12, 24, 36, 48 months  %These negative sensitivities seem to be connected to months
310  %  %during which main parts of the CAA are essentially entirely ice-covered.
311  \end{itemize}  %This means that increase in ice concentration during this period
312    %will likely reduce ice export due to blocking
313    %[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
314    %Only during periods where substantial parts of the CAA are
315  \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export  %ice free (i.e. sea-ice concentration is less than one in larger parts of
316  through Fram Strait in December 1995 to changes in sea-ice thickness  %the CAA) will an increase in ice-concentration increase ice export.
317  12, 24, 36, 48 months back in time. Corresponding sensitivities to  
318  ocean surface temperature are depicted in  %(*)
319  \reffig{4yradjthetalev1}(a--d).  The main characteristics is  %Sensitivities peak about 2-3 months before sign reversal, i.e.
320  consistency with expected advection of sea-ice over the relevant time  %max. negative sensitivities are expected end of July
321  scales considered.  The general positive pattern means that an  %[DOUBLE CHECK THIS].
322  increase in sea-ice thickness at location $(x,y)$ and time $t$ will  
323  increase sea-ice export through Fram Strait at time $T_e$.  Largest  %(*)
324  distances from Fram Strait indicate fastest sea-ice advection over the  %Peaks/bursts of sensitivities for months
325  time span considered.  The ice thickness sensitivities are in close  %14-17, 19-21, 27-29, 30-33, 38-40, 42-45
326  correspondence to ocean surface sentivitites, but of opposite sign.  
327  An increase in temperature will incur ice melting, decrease in ice  %(*)
328  thickness, and therefore decrease in sea-ice export at time $T_e$.  %Spatial "anti-correlation" (in sign) between main sensitivity branch
329    %(essentially Northwest Passage and immediate connecting channels),
330  The picture is fundamentally different and much more complex  %and remote places.
331  for sensitivities to ocean temperatures away from the surface.  %For example: month 20, 28, 31.5, 40, 43.
332  \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to  %The timings of max. sensitivity extent are similar between
333  temperatures at roughly 400 m depth.  %free-slip and no-slip run; and patterns are similar within CAA,
334  Primary features are the effect of the heat transport of the North  %but differ in the Arctic Ocean interior.
335  Atlantic current which feeds into the West Spitsbergen current,  
336  the circulation around Svalbard, and ...  %(*)
337    %Interesting (but real?) patterns in Arctic Ocean interior.
338  \begin{figure}[t!]  
339  \centerline{  %\paragraph{Sensitivities to the sea-ice velocity}
340  \subfigure[{\footnotesize -12 months}]  
341  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}  %(*)
342  %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}  %Patterns of ADJuice at almost any point in time are rather complicated
343  %  %(in particular with respect to spatial structure of signs).
344  \subfigure[{\footnotesize -24 months}]  %Might warrant perturbation tests.
345  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}  %Patterns of ADJvice, on the other hand, are more spatially coherent,
346  }  %but still hard to interpret (or even counter-intuitive
347  %  %in many places).
348  \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to  
349  sea-ice thickness at various prior times.  %(*)
350  \label{fig:4yradjheff}}  %"Growth in extent of sensitivities" goes in clear pulses:
351  \end{figure}  %almost no change between months: 0-5, 10-20, 24-32, 36-44
352    %These essentially correspond to months of
353    
354    
355    %\subsection{Sensitivities to the oceanic state}
356    
357    %\paragraph{Sensitivities to theta}
358    
359    %\textit{Sensitivities at the surface (z = 5 m)}
360    
361    %(*)
362    %mabye redo with caxmax=0.02 or even 0.05
363    
364    %(*)
365    %Core of negative sensitivities spreading through the CAA as
366    %one might expect [TEST]:
367    %Increase in SST will decrease ice thickness and therefore ice export.
368    
369    %(*)
370    %What's maybe unexpected is patterns of positive sensitivities
371    %at the fringes of the "core", e.g. in the Southern channels
372    %(Bellot St., Peel Sound, M'Clintock Channel), and to the North
373    %(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
374    %then shifting Northward into the Arctic interior).
375    
376    %(*)
377    %Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
378    %propagating into Lincoln Sea, then
379    %entering Nares Strait and Smith Sound, periodically
380    %warming or cooling[???] the Lancaster Sound exit.
381    
382    %\textit{Sensitivities at depth (z = 200 m)}
383    
384    %(*)
385    %Negative sensitivities almost everywhere, as might be expected.
386    
387    %(*)
388    %Sensitivity patterns between free-slip and no-slip BCs
389    %are quite similar, except in Lincoln Sea (North of Nares St),
390    %where the sign is reversed (but pattern remains similar).
391    
392    %\paragraph{Sensitivities to salt}
393    
394    %T.B.D.
395    
396    %\paragraph{Sensitivities to velocity}
397    
398    %T.B.D.
399    
400    %\subsection{Sensitivities to the atmospheric state}
401    
402    %\begin{itemize}
403    %%
404    %\item
405    %plot of ATEMP for 12, 24, 36, 48 months
406    %%
407    %\item
408    %plot of HEFF for 12, 24, 36, 48 months
409    %%
410    %\end{itemize}
411    
412    
413    
414    %\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
415    %through Fram Strait in December 1995 to changes in sea-ice thickness
416    %12, 24, 36, 48 months back in time. Corresponding sensitivities to
417    %ocean surface temperature are depicted in
418    %\reffig{4yradjthetalev1}(a--d).  The main characteristics is
419    %consistency with expected advection of sea-ice over the relevant time
420    %scales considered.  The general positive pattern means that an
421    %increase in sea-ice thickness at location $(x,y)$ and time $t$ will
422    %increase sea-ice export through Fram Strait at time $T_e$.  Largest
423    %distances from Fram Strait indicate fastest sea-ice advection over the
424    %time span considered.  The ice thickness sensitivities are in close
425    %correspondence to ocean surface sentivitites, but of opposite sign.
426    %An increase in temperature will incur ice melting, decrease in ice
427    %thickness, and therefore decrease in sea-ice export at time $T_e$.
428    
429    %The picture is fundamentally different and much more complex
430    %for sensitivities to ocean temperatures away from the surface.
431    %\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
432    %temperatures at roughly 400 m depth.
433    %Primary features are the effect of the heat transport of the North
434    %Atlantic current which feeds into the West Spitsbergen current,
435    %the circulation around Svalbard, and ...
436    
437    
438    %%\begin{figure}[t!]
439    %%\centerline{
440    %%\subfigure[{\footnotesize -12 months}]
441    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
442    %%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
443    %%
444    %%\subfigure[{\footnotesize -24 months}]
445    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
446    %%}
447    %%
448    %%\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
449    %%sea-ice thickness at various prior times.
450    %%\label{fig:4yradjheff}}
451    %%\end{figure}
452    
453    
454    %\ml{[based on the movie series
455    %  zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
456    %export through the Canadian Archipelag is highly sensitive to the
457    %previous state of the ocean-ice system in the Archipelago and the
458    %Western Arctic. According to the \ml{(adjoint)} senstivities of the
459    %eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
460    %cross-section G) with respect to ice volume (effective thickness), ocean
461    %surface temperature, and vertical diffusivity near the surface
462    %(\reffig{fouryearadj}) after 4 years of integration the following
463    %mechanisms can be identified: near the ``observation'' (cross-section
464    %G), smaller vertical diffusivities lead to lower surface temperatures
465    %and hence to more ice that is available for export. Further away from
466    %cross-section G, the sensitivity to vertical diffusivity has the
467    %opposite sign, but temperature and ice volume sensitivities have the
468    %same sign as close to the observation.
469    
470    
471  %%% Local Variables:  %%% Local Variables:
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