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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 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.
83  see e.g. \cite{mell:02, mich-etal:06,muen-etal:06}.  
84    The model domain is the same as the one described in \refsec{forward},
85  The model domain is a coarsened version of the Arctic face of the  but with halved horizontal resolution.
86  high-resolution cubed-sphere configuration of the ECCO2 project  The adjoint models run efficiently on 80 processors (as validated
87  \citep[see][]{menemenlis05}. It covers the entire Arctic,  by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).
88  extends into the North Pacific such as to cover the entire  Following a 4-year spinup (1985 to 1988), the model is integrated for four
89  ice-covered regions, and comprises parts of the North Atlantic  years and nine months between January 1989 and September 1993.
90  down to XXN to enable analysis of remote influences of the  It is forced using realistic 6-hourly NCEP/NCAR atmospheric state variables.
91  North Atlantic current to sea-ice variability and export.  %Over the open ocean these are
92  The horizontal resolution varies between XX and YY km  %converted into air-sea fluxes via the bulk formulae of
93  with 50 unevenly spaced vertical levels.  %\citet{large04}.  The air-sea fluxes in the presence of
94  The adjoint models run efficiently on 80 processors  %sea-ice are handled by the ice model as described in \refsec{model}.
95  (benchmarks have been performed both on an SGI Altix as well as an  The objective function $J$ is chosen as the ``solid'' fresh water
96  IBM SP5 at NASA/ARC).  export, that is the export of ice and snow converted to units of fresh
97    water,
98  Following a 3-year spinup, the model has been integrated for four  %
99  years and five months between January 1989 and May 1993.  \begin{equation}
100  It is forced using realistic 6-hourly  J \, = \, (\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u
101  NCEP/NCAR atmospheric state variables. Over the open ocean these are  \end{equation}
102  converted into air-sea fluxes via the bulk formulae of  %
103  \citet{large04}.  Derivation of air-sea fluxes in the presence of  through Lancaster Sound at approximately 82\degW\ (cross-section G in
104  sea-ice is handled by the ice model as described in \refsec{model}.  \reffig{arctic_topog}) averaged \ml{PH: Maybe integrated quantity is
105  The objective function chosen is  more physical} over the final 12-month of the integration between October
106  sea-ice export through  1992 and September 1993.
107  Lancaster Sound at XX$^{\circ}$W  
108  averaged over an 8-month period between October 1992 and May 1993.    The forward trajectory of the model integration resembles broadly that
109    of the model in \refsec{forward}. Many details are different, owning
110  The adjoint model computes sensitivities  to different resolution and integration period; for example, the solid
111  to sea-ice export back in time from 1993 to 1989 along this  fresh water transport through Lancaster Sound is
112  trajectory.  In principle all adjoint model variable (i.e., Lagrange  %
113  multipliers) of the coupled ocean/sea-ice model  \ml{PH: Martin, where did you get these numbers from?}
114  as well as the surface atmospheric state are available to  %
115  analyze the transient sensitivity behaviour.    $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
116  Over the open ocean, the adjoint of the bulk formula scheme  the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a
117  computes sensitivities to the time-varying atmospheric state.  Over  no slip simulation.
118  ice-covered parts, the sea-ice adjoint converts surface ocean  
119  sensitivities to atmospheric sensitivities.  The adjoint model is the transpose of the tangent linear (or Jacobian) model
120    operator. It runs backward in time, from September 1993 to
121    January 1989. Along its integration it accumulates the Lagrange multipliers
122    of the model subject to the objective function (solid freshwater export),
123    which can be interpreted as sensitivities of the objective function
124    to each control variable and each element of the intermediate
125    coupled model state variables.
126    Thus, all sensitivity elements of the coupled
127    ocean/sea-ice model state as well as the surface atmospheric state are
128    available for analysis of the transient sensitivity behavior.  Over the
129    open ocean, the adjoint of the bulk formula scheme computes
130    sensitivities to the time-varying atmospheric state.  Over ice-covered
131    parts, the sea-ice adjoint converts surface ocean sensitivities to
132    atmospheric sensitivities.
133    
134  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT
135    
136  \subsection{Sensitivities to the sea-ice state}  \subsubsection{Adjoint sensitivities}
137    
138  \paragraph{Sensitivities to the sea-ice thickness}  The most readily interpretable ice-export sensitivity is that to
139    effective ice thickness, $\partial{J} / \partial{(hc)}$.
140  The most readily interpretable ice-export sensitivity is that  \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
141  to ice thickness, $\partial J / \partial heff$.  free-slip (left column) and no-slip (right column) boundary
142  Fig. XXX depcits transient $\partial J / \partial heff$ using free-slip  conditions. Sensitivity snapshots are depicted for beginning of October 2002,
143  (left column) and no-slip (right column) boundary conditions.  i.e. 12 months back in time from September 1993
144  Sensitivity snapshots are depicted for (from top to bottom)  (the beginning of the averaging period for the objective
145  12, 24, 36, and 48 months prior to May 2003.  function $J$, top),
146  The dominant features are in accordance with expectations:  and for Jannuary 1989, the beginning of the forward integration (bottom).
147    \begin{figure*}[t]
148  (*)    \includegraphics*[width=\textwidth]{\fpath/adjheff}
149  Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.    \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
150  a unit increase in sea-ice thickness in most places upstream      m$^2$\,s$^{-1}$/m for two different times (rows) and two different
151  of Lancaster Sound will increase sea-ice export through Lancaster Sound.      boundary conditions for sea ice drift. The color scale is chosen
152  The dominant pathway follows (backward in time) through Barrow Strait      to illustrate the patterns of the sensitivities; the maximum and
153        minimum values are given above the figures.
154        \label{fig:adjheff}}
155    \end{figure*}
156    
157    As expected, the sensitivity patterns are predominantly positive,
158    an increase in ice volume in most places ``upstream'' of
159    Lancaster sound increases the solid fresh water export at the exit section.
160    Also obvious is the transient nature of the sensitivity patterns
161    (top panels vs. bottom panels),
162    i.e. as time moves backward, an increasing area upstream of Lancaster Sound
163    contributes to the export sensitivity.
164    The dominant pathway (free slip case) follows (backward in time)
165    through Barrow Strait
166  into Viscount Melville Sound, and from there trough M'Clure Strait  into Viscount Melville Sound, and from there trough M'Clure Strait
167  into the Arctic Ocean (the "Northwest Passage").  into the Arctic Ocean (the ``Northwest Passage'').
168  Secondary paths are Northward from  Secondary paths are Northward from
169  Viscount Melville Sound through Byam Martin Channel into  Viscount Melville Sound through Byam Martin Channel into
170  Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.  Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.
171    
172  (*)  The difference between the free slip and no slip solution is evident:
173  As expected, at any given time the  by the end of the adjoint integration, in January 1989
174  region of influence is larger for the free-slip than no-slip simulation.  the free-slip sensitivities (bottom left) extend through most of the CAA
175  For the no-slip run, the region of influence is confined, after four years,  and all the way into the Arctic interior, both to the West (M'Clure St.)
176  to just West of Barrow Strait (North of Prince of Wales Island),  and to the North
177  and to the South of Penny Strait.  (Ballantyne St., Prince Gustav Adolf Sea, Massey Sound),
178  In contrast, sensitivities of the free-slip run extend  whereas the no slip sensitivities (bottom right) are overall weaker
179  all the way to the Arctic interior both to the West  and remain mostly confined to Lancaster Sound and just West of Barrow Strait.
180  (M'Clure St.) and to the North (Ballantyne St., Prince Gustav Adolf Sea,  In the free slip solution ice can drift more
181  Massey Sound).  easily through narrow straits, and
182    a positive ice volume anomaly further upstream in the CAA may increase
183  (*)  ice export through the Lancaster Sound within a 4 year period.
184  sensitivities seem to spread out in "pulses" (seasonal cycle)  
185  [PLOT A TIME SERIES OF ADJheff in Barrow Strait)  One peculiar feature in the October 1992 sensitivity maps (top panels)
186    are the negative sensivities to the East and to the West.
187  (*)  These can be explained by indirect effects: less ice to the East means
188  The sensitivity in Baffin Bay are more complex.  less resistance to eastward drift and thus more export; similarly, less ice to
189  The pattern evolves along the Western boundary, connecting  the West means that more ice can be moved eastwards from the Barrow Strait
190  the Lancaster Sound Polynya, the Coburg Island Polynya, and the  into the Lancaster Sound leading to more ice export.
191  North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.  \ml{PH: The first explanation (East) I buy, the second (West) I don't}.
192  The sign of sensitivities has an oscillatory character  
193  [AT FREQUENCY OF SEASONAL CYCLE?].  The temporal evolution of several ice export sensitivities
194  First, we need to establish whether forward perturbation runs  (eqn. XX) along a zonal axis through
195  corroborate the oscillatory behaviour.  Lancaster Sound, Barrow Strait,and  Melville Sound
196  Then, several possible explanations:  (115\degW\ to 80\degW\ ),
197  (i) connection established through Nares Strait throughflow  are depicted as Hovmueller diagrams in \reffig{lancaster}.
198  which extends into Western boundary current in Northern Baffin Bay.  From top to bottom, sensitivities are with respect to effective
199  (ii) sea-ice concentration there is seasonal, i.e. partly  ice thickness ($hc$),
200  ice-free during the year. Seasonal cycle in sensitivity likely  ocean surface temperature ($SST$) and precipitation ($p$) for free slip
201  connected to ice-free vs. ice-covered parts of the year.  (left column) and no slip (right column) ice drift boundary conditions.
 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 \\  
202  %  %
203  These negative sensitivities seem to be connected to months  \begin{figure*}
204  during which main parts of the CAA are essentially entirely ice-covered.    \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
205  This means that increase in ice concentration during this period    \caption{Hovermoeller diagrams of sensitivities (derivatives) of the
206  will likely reduce ice export due to blocking      ``solid'' fresh water (i.e., ice and snow) export $J$ through Lancaster sound
207  [NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].      (\reffig{arctic_topog}, cross-section G) with respect to effective
208  Only during periods where substantial parts of the CAA are      ice thickness ($hc$), ocean surface temperature (SST) and
209  ice free (i.e. sea-ice concentration is less than one in larger parts of      precipitation ($p$) for two runs with free slip and no slip boundary
210  the CAA) will an increase in ice-concentration increase ice export.      conditions for the sea ice drift. Also shown it the normalized ice
211        strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ (bottom panel); each plot is
212  (*)      overlaid with the contours 1 and 3 of the normalized ice strength
213  Sensitivities peak about 2-3 months before sign reversal, i.e.      for orientation.
214  max. negative sensitivities are expected end of July      \label{fig:lancaster}}
215  [DOUBLE CHECK THIS].  \end{figure*}
   
 (*)  
 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}  
216  %  %
217  \item  
218  plot of ATEMP for 12, 24, 36, 48 months  The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature
219    (second row) sensitivities are coherent:
220    more ice in the Lancaster Sound leads
221    to more export, and one way to get more ice is by colder surface
222    temperatures (less melting from below). In the free slip case the
223    sensitivities spread out in "pulses" following a seasonal cycle:
224    can propagate westwards (backwards in time) when the ice
225    strength is low in late summer, early autumn.
226    In contrast, during winter, the sensitivities show little to now
227    westward propagation.
228    In the no slip case the (normalized)
229    ice strength does not fall below 1 during the winters of 1991 to 1993
230    (mainly because the ice concentrations remain nearly 100\%, not
231    shown). Ice is therefore blocked and cannot drift eastwards
232    (forward in time) through the
233    Melville Sound, Barrow Strait, Lancaster Sound channel system.
234    Consequently, the sensitivities do not propagate westwards (backwards in
235    time) and the export through Lancaster Sound is only affected by
236    local ice formation and melting for the entire integration period.
237    
238    The sensitivities to precipitation exhibit an oscillatory behaviour:
239    they are negative (more precipitation leads to less export)
240    before January (more precisely, late fall) and mostly positive after January
241    (more precisely, January through July).
242    Times of positive sensitivities coincide with times of
243    normalized ice strengths exceeding values of 3.
244  %  %
245  \item  \ml{PH: Problem is, that's not true for the first two years (backward),
246  plot of HEFF for 12, 24, 36, 48 months  East of 95\degW\ , i.e. in Lancaster Sound.
247    For example, at 90\degW\ the sensitivities are negative throughout 1992,
248    and no clear correlation to ice strength is apparent there.}.
249  %  %
250  \end{itemize}  Assuming that most precipation is snow in this area
   
   
   
 \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}  
251  %  %
252  \subfigure[{\footnotesize -24 months}]  \footnote{
253  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}  In the
254  }  current implementation the model differentiates between snow and rain
255    depending on the thermodynamic growth rate; when it is cold enough for
256    ice to grow, all precipitation is assumed to be snow.}
257  %  %
258  \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to  the sensitivities can be interpreted in terms of the model physics.  Short
259  sea-ice thickness at various prior times.  wave radiation cannot penetrate the snow cover and has a higer albedo
260  \label{fig:4yradjheff}}  than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it
261  \end{figure}  protects the ice against melting in spring (after January).  
262    \ml{PH: what about the direct effect of accumulation of precip. as snow
263    which directly increases the volume.}.
264    
265    On the other hand, snow reduces the effective conductivity and thus the heat
266    flux through the ice. This insulating effect slows down the cooling of
267    the surface water underneath the ice and limits the ice growth from
268    below, so that less snow in the ice-growing season leads to more new
269    ice and thus more ice export.
270    \ml{PH: Should probably discuss the effect of snow vs. rain.
271    To me it seems that the "rain" effect doesn't really play
272    because the neg. sensitivities are too late in the fall,
273    probably mostly falling as snow.}.
274    
275    %Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein:
276    %1. snow insulates against melting from above during spring: more precip (snow) -> more export
277    %2. less snow during fall -> more ice -> more export
278    %3. precip is both snow and rain, depending on the sign of "FICE" (thermodynamic growth rate), with probably different implications
279    
280    
281    \subsubsection{Forward sensitivities}
282    
283    \ml{[Here we need for integrations to show that the adjoint
284      sensitivites are not just academic. I suggest to perturb HEFF
285      and THETA initial conditions, and PRECIP somewhere in the Melville
286      Sound and then produce plots similar to reffig{lancaster}. For
287      PRECIP it would be great to have two perturbation experiments, one
288      where ADJprecip is posivite and one where ADJprecip is negative]}
289      
290    
291    %(*)
292    %The sensitivity in Baffin Bay are more complex.
293    %The pattern evolves along the Western boundary, connecting
294    %the Lancaster Sound Polynya, the Coburg Island Polynya, and the
295    %North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
296    %The sign of sensitivities has an oscillatory character
297    %[AT FREQUENCY OF SEASONAL CYCLE?].
298    %First, we need to establish whether forward perturbation runs
299    %corroborate the oscillatory behaviour.
300    %Then, several possible explanations:
301    %(i) connection established through Nares Strait throughflow
302    %which extends into Western boundary current in Northern Baffin Bay.
303    %(ii) sea-ice concentration there is seasonal, i.e. partly
304    %ice-free during the year. Seasonal cycle in sensitivity likely
305    %connected to ice-free vs. ice-covered parts of the year.
306    %Negative sensitivities can potentially be attributed
307    %to blocking of Lancaster Sound ice export by Western boundary ice
308    %in Baffin Bay.
309    %(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
310    %(in reality there's a Northern counter current hugging the coast of
311    %Devon Island which we probably don't resolve).
312    
313    %Remote control of Kennedy Channel on Lancaster Sound ice export
314    %seems a nice test for appropriateness of free-slip vs. no-slip BCs.
315    
316    %\paragraph{Sensitivities to the sea-ice area}
317    
318    %Fig. XXX depcits transient sea-ice export sensitivities
319    %to changes in sea-ice concentration
320    % $\partial J / \partial area$ using free-slip
321    %(left column) and no-slip (right column) boundary conditions.
322    %Sensitivity snapshots are depicted for (from top to bottom)
323    %12, 24, 36, and 48 months prior to May 2003.
324    %Contrary to the steady patterns seen for thickness sensitivities,
325    %the ice-concentration sensitivities exhibit a strong seasonal cycle
326    %in large parts of the domain (but synchronized on large scale).
327    %The following discussion is w.r.t. free-slip run.
328    
329    %(*)
330    %Months, during which sensitivities are negative:
331    %\\
332    %0 to 5   Db=N/A, Dr=5 (May-Jan) \\
333    %10 to 17 Db=7, Dr=5 (Jul-Jan) \\
334    %22 to 29 Db=7, Dr=5 (Jul-Jan) \\
335    %34 to 41 Db=7, Dr=5 (Jul-Jan) \\
336    %46 to 49 D=N/A \\
337    %%
338    %These negative sensitivities seem to be connected to months
339    %during which main parts of the CAA are essentially entirely ice-covered.
340    %This means that increase in ice concentration during this period
341    %will likely reduce ice export due to blocking
342    %[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
343    %Only during periods where substantial parts of the CAA are
344    %ice free (i.e. sea-ice concentration is less than one in larger parts of
345    %the CAA) will an increase in ice-concentration increase ice export.
346    
347    %(*)
348    %Sensitivities peak about 2-3 months before sign reversal, i.e.
349    %max. negative sensitivities are expected end of July
350    %[DOUBLE CHECK THIS].
351    
352    %(*)
353    %Peaks/bursts of sensitivities for months
354    %14-17, 19-21, 27-29, 30-33, 38-40, 42-45
355    
356    %(*)
357    %Spatial "anti-correlation" (in sign) between main sensitivity branch
358    %(essentially Northwest Passage and immediate connecting channels),
359    %and remote places.
360    %For example: month 20, 28, 31.5, 40, 43.
361    %The timings of max. sensitivity extent are similar between
362    %free-slip and no-slip run; and patterns are similar within CAA,
363    %but differ in the Arctic Ocean interior.
364    
365    %(*)
366    %Interesting (but real?) patterns in Arctic Ocean interior.
367    
368    %\paragraph{Sensitivities to the sea-ice velocity}
369    
370    %(*)
371    %Patterns of ADJuice at almost any point in time are rather complicated
372    %(in particular with respect to spatial structure of signs).
373    %Might warrant perturbation tests.
374    %Patterns of ADJvice, on the other hand, are more spatially coherent,
375    %but still hard to interpret (or even counter-intuitive
376    %in many places).
377    
378    %(*)
379    %"Growth in extent of sensitivities" goes in clear pulses:
380    %almost no change between months: 0-5, 10-20, 24-32, 36-44
381    %These essentially correspond to months of
382    
383    
384    %\subsection{Sensitivities to the oceanic state}
385    
386    %\paragraph{Sensitivities to theta}
387    
388    %\textit{Sensitivities at the surface (z = 5 m)}
389    
390    %(*)
391    %mabye redo with caxmax=0.02 or even 0.05
392    
393    %(*)
394    %Core of negative sensitivities spreading through the CAA as
395    %one might expect [TEST]:
396    %Increase in SST will decrease ice thickness and therefore ice export.
397    
398    %(*)
399    %What's maybe unexpected is patterns of positive sensitivities
400    %at the fringes of the "core", e.g. in the Southern channels
401    %(Bellot St., Peel Sound, M'Clintock Channel), and to the North
402    %(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
403    %then shifting Northward into the Arctic interior).
404    
405    %(*)
406    %Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
407    %propagating into Lincoln Sea, then
408    %entering Nares Strait and Smith Sound, periodically
409    %warming or cooling[???] the Lancaster Sound exit.
410    
411    %\textit{Sensitivities at depth (z = 200 m)}
412    
413    %(*)
414    %Negative sensitivities almost everywhere, as might be expected.
415    
416    %(*)
417    %Sensitivity patterns between free-slip and no-slip BCs
418    %are quite similar, except in Lincoln Sea (North of Nares St),
419    %where the sign is reversed (but pattern remains similar).
420    
421    %\paragraph{Sensitivities to salt}
422    
423    %T.B.D.
424    
425    %\paragraph{Sensitivities to velocity}
426    
427    %T.B.D.
428    
429    %\subsection{Sensitivities to the atmospheric state}
430    
431    %\begin{itemize}
432    %%
433    %\item
434    %plot of ATEMP for 12, 24, 36, 48 months
435    %%
436    %\item
437    %plot of HEFF for 12, 24, 36, 48 months
438    %%
439    %\end{itemize}
440    
441    
442    
443    %\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
444    %through Fram Strait in December 1995 to changes in sea-ice thickness
445    %12, 24, 36, 48 months back in time. Corresponding sensitivities to
446    %ocean surface temperature are depicted in
447    %\reffig{4yradjthetalev1}(a--d).  The main characteristics is
448    %consistency with expected advection of sea-ice over the relevant time
449    %scales considered.  The general positive pattern means that an
450    %increase in sea-ice thickness at location $(x,y)$ and time $t$ will
451    %increase sea-ice export through Fram Strait at time $T_e$.  Largest
452    %distances from Fram Strait indicate fastest sea-ice advection over the
453    %time span considered.  The ice thickness sensitivities are in close
454    %correspondence to ocean surface sentivitites, but of opposite sign.
455    %An increase in temperature will incur ice melting, decrease in ice
456    %thickness, and therefore decrease in sea-ice export at time $T_e$.
457    
458    %The picture is fundamentally different and much more complex
459    %for sensitivities to ocean temperatures away from the surface.
460    %\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
461    %temperatures at roughly 400 m depth.
462    %Primary features are the effect of the heat transport of the North
463    %Atlantic current which feeds into the West Spitsbergen current,
464    %the circulation around Svalbard, and ...
465    
466    
467    %%\begin{figure}[t!]
468    %%\centerline{
469    %%\subfigure[{\footnotesize -12 months}]
470    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
471    %%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
472    %%
473    %%\subfigure[{\footnotesize -24 months}]
474    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
475    %%}
476    %%
477    %%\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
478    %%sea-ice thickness at various prior times.
479    %%\label{fig:4yradjheff}}
480    %%\end{figure}
481    
482    
483    %\ml{[based on the movie series
484    %  zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
485    %export through the Canadian Archipelag is highly sensitive to the
486    %previous state of the ocean-ice system in the Archipelago and the
487    %Western Arctic. According to the \ml{(adjoint)} senstivities of the
488    %eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
489    %cross-section G) with respect to ice volume (effective thickness), ocean
490    %surface temperature, and vertical diffusivity near the surface
491    %(\reffig{fouryearadj}) after 4 years of integration the following
492    %mechanisms can be identified: near the ``observation'' (cross-section
493    %G), smaller vertical diffusivities lead to lower surface temperatures
494    %and hence to more ice that is available for export. Further away from
495    %cross-section G, the sensitivity to vertical diffusivity has the
496    %opposite sign, but temperature and ice volume sensitivities have the
497    %same sign as close to the observation.
498    
499    
500  %%% Local Variables:  %%% Local Variables:
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