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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}
# Line 65  generating the adjoint code of a new mod Line 65  generating the adjoint code of a new mod
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,
 Following a 3-year spinup, the model has been integrated for four  
 years and five months between January 1989 and May 1993.  
 It is forced using realistic 6-hourly  
 NCEP/NCAR atmospheric state variables. Over the open ocean these are  
 converted into air-sea fluxes via the bulk formulae of  
 \citet{large04}.  Derivation of air-sea fluxes in the presence of  
 sea-ice is handled by the ice model as described in \refsec{model}.  
 The objective function is chosen $J$ as the  
 sea-ice export through  
 Lancaster Sound at XX$^{\circ}$W  
 averaged over an 8-month period between October 1992 and May 1993.    
   
 The adjoint model computes sensitivities  
 to sea-ice export back in time from 1993 to 1989 along this  
 trajectory.  In principle all adjoint model variable (i.e., Lagrange  
 multipliers) of the coupled ocean/sea-ice model  
 as well as the surface atmospheric state are available to  
 analyze the transient sensitivity behaviour.    
 Over the open ocean, the adjoint of the bulk formula scheme  
 computes sensitivities to the time-varying atmospheric state.  Over  
 ice-covered parts, the sea-ice adjoint converts surface ocean  
 sensitivities to atmospheric sensitivities.  
   
 DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT  
   
 \subsection{Sensitivities to the sea-ice state}  
   
 \paragraph{Sensitivities to the sea-ice thickness}  
   
 The most readily interpretable ice-export sensitivity is that  
 to effective ice thickness, $\partial{J} / \partial{h}$.  
 Fig. XXX depcits transient $\partial{J} / \partial{h}$ using free-slip  
 (left column) and no-slip (right column) boundary conditions.  
 Sensitivity snapshots are depicted for (from top to bottom)  
 12, 24, 36, and 48 months prior to May 2003.  
 The dominant features are\ml{ in accordance with expectations/as expected}:  
   
 (*)  
 Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.  
 a unit increase in sea-ice thickness in most places upstream  
 of Lancaster Sound will increase sea-ice export through Lancaster Sound.  
 The dominant pathway follows (backward in time) through Barrow Strait  
 into Viscount Melville Sound, and from there trough M'Clure Strait  
 into the Arctic Ocean (the "Northwest Passage").  
 Secondary paths are Northward from  
 Viscount Melville Sound through Byam Martin Channel into  
 Prince Gustav Adolf Sea and through Penny Strait into MacLean Strait.  
   
 (*)  
 As expected, at any given time the  
 region of influence is larger for the free-slip than no-slip simulation.  
 For the no-slip run, the region of influence is confined, after four years,  
 to just West of Barrow Strait (North of Prince of Wales Island),  
 and to the South of Penny Strait.  
 In contrast, sensitivities of the free-slip run extend  
 all the way to the Arctic interior both to the West  
 (M'Clure St.) and to the North (Ballantyne St., Prince Gustav Adolf Sea,  
 Massey Sound).  
   
 (*)  
 sensitivities seem to spread out in "pulses" (seasonal cycle)  
 [PLOT A TIME SERIES OF ADJheff in Barrow Strait)  
   
 (*)  
 The sensitivity in Baffin Bay are more complex.  
 The pattern evolves along the Western boundary, connecting  
 the Lancaster Sound Polynya, the Coburg Island Polynya, and the  
 North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.  
 The sign of sensitivities has an oscillatory character  
 [AT FREQUENCY OF SEASONAL CYCLE?].  
 First, we need to establish whether forward perturbation runs  
 corroborate the oscillatory behaviour.  
 Then, several possible explanations:  
 (i) connection established through Nares Strait throughflow  
 which extends into Western boundary current in Northern Baffin Bay.  
 (ii) sea-ice concentration there is seasonal, i.e. partly  
 ice-free during the year. Seasonal cycle in sensitivity likely  
 connected to ice-free vs. ice-covered parts of the year.  
 Negative sensitivities can potentially be attributed  
 to blocking of Lancaster Sound ice export by Western boundary ice  
 in Baffin Bay.  
 (iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility  
 (in reality there's a Northern counter current hugging the coast of  
 Devon Island which we probably don't resolve).  
   
 Remote control of Kennedy Channel on Lancaster Sound ice export  
 seems a nice test for appropriateness of free-slip vs. no-slip BCs.  
   
 \paragraph{Sensitivities to the sea-ice area}  
   
 Fig. XXX depcits transient sea-ice export sensitivities  
 to changes in sea-ice concentration  
  $\partial J / \partial area$ using free-slip  
 (left column) and no-slip (right column) boundary conditions.  
 Sensitivity snapshots are depicted for (from top to bottom)  
 12, 24, 36, and 48 months prior to May 2003.  
 Contrary to the steady patterns seen for thickness sensitivities,  
 the ice-concentration sensitivities exhibit a strong seasonal cycle  
 in large parts of the domain (but synchronized on large scale).  
 The following discussion is w.r.t. free-slip run.  
   
 (*)  
 Months, during which sensitivities are negative:  
 \\  
 0 to 5   Db=N/A, Dr=5 (May-Jan) \\  
 10 to 17 Db=7, Dr=5 (Jul-Jan) \\  
 22 to 29 Db=7, Dr=5 (Jul-Jan) \\  
 34 to 41 Db=7, Dr=5 (Jul-Jan) \\  
 46 to 49 D=N/A \\  
98  %  %
99  These negative sensitivities seem to be connected to months  \begin{equation}
100  during which main parts of the CAA are essentially entirely ice-covered.  J \, = \, (\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u
101  This means that increase in ice concentration during this period  \end{equation}
102  will likely reduce ice export due to blocking  %
103  [NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].  through Lancaster Sound at approximately 82\degW\ (cross-section G in
104  Only during periods where substantial parts of the CAA are  \reffig{arctic_topog}) averaged \ml{PH: Maybe integrated quantity is
105  ice free (i.e. sea-ice concentration is less than one in larger parts of  more physical; ML: what did you actually compute? I did not scale
106  the CAA) will an increase in ice-concentration increase ice export.  anything, yet. Please insert what is actually done.} over the final
107    12-month of the integration between October 1992 and September 1993.
108  (*)  
109  Sensitivities peak about 2-3 months before sign reversal, i.e.  The forward trajectory of the model integration resembles broadly that
110  max. negative sensitivities are expected end of July  of the model in \refsec{forward}. Many details are different, owning
111  [DOUBLE CHECK THIS].  to different resolution and integration period; for example, the solid
112    fresh water transport through Lancaster Sound is
113  (*)  %
114  Peaks/bursts of sensitivities for months  \ml{PH: Martin, where did you get these numbers from?}
115  14-17, 19-21, 27-29, 30-33, 38-40, 42-45  \ml{[ML: I computed hu = -sum((SIheff+SIhsnow)*SIuice*area)/sum(area) at
116    $i=100,j=116:122$, and then took mean(hu) and std(hu). What are your numbers?]}
117  (*)  %
118  Spatial "anti-correlation" (in sign) between main sensitivity branch  $116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
119  (essentially Northwest Passage and immediate connecting channels),  the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a
120  and remote places.  no slip simulation. \ml{[Here we can say that the export through
121  For example: month 20, 28, 31.5, 40, 43.    Lancaster Sound is highly uncertain, making is a perfect candidate
122  The timings of max. sensitivity extent are similar between    for sensitivity, bla bla?]}
123  free-slip and no-slip run; and patterns are similar within CAA,  
124  but differ in the Arctic Ocean interior.  The adjoint model is the transpose of the tangent linear (or Jacobian) model
125    operator. It runs backwards in time, from September 1993 to
126  (*)  January 1989. During its integration it accumulates the Lagrange multipliers
127  Interesting (but real?) patterns in Arctic Ocean interior.  of the model subject to the objective function (solid freshwater export),
128    which can be interpreted as sensitivities of the objective function
129  \paragraph{Sensitivities to the sea-ice velocity}  to each control variable and each element of the intermediate
130    coupled model state variables.
131  (*)  Thus, all sensitivity elements of the coupled
132  Patterns of ADJuice at almost any point in time are rather complicated  ocean/sea-ice model state as well as the surface atmospheric state are
133  (in particular with respect to spatial structure of signs).  available for analysis of the transient sensitivity behavior.  Over the
134  Might warrant perturbation tests.  open ocean, the adjoint of the bulk formula scheme computes
135  Patterns of ADJvice, on the other hand, are more spatially coherent,  sensitivities to the time-varying atmospheric state.  Over ice-covered
136  but still hard to interpret (or even counter-intuitive  areas, the sea-ice adjoint converts surface ocean sensitivities to
137  in many places).  atmospheric sensitivities.
   
 (*)  
 "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}  
138    
139  T.B.D.  DISCUSS FORWARD STATE, INCLUDING SOME NUMBERS ON SEA-ICE EXPORT
140    
141  \subsection{Sensitivities to the atmospheric state}  \subsubsection{Adjoint sensitivities}
142    
143  \begin{itemize}  The most readily interpretable ice-export sensitivity is that to
144    effective ice thickness, $\partial{J} / \partial{(hc)}$.
145    \reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
146    free-slip (left column) and no-slip (right column) boundary
147    conditions. Sensitivity snapshots are depicted for beginning of October 1992,
148    that is 12 months before September 1993
149    (the beginning of the averaging period for the objective
150    function $J$, top),
151    and for Jannuary 1989, the beginning of the forward integration (bottom).
152    \begin{figure*}[t]
153      \includegraphics*[width=\textwidth]{\fpath/adjheff}
154      \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
155        m$^2$\,s$^{-1}$/m for two different times (rows) and two different
156        boundary conditions for sea ice drift. The color scale is chosen
157        to illustrate the patterns of the sensitivities; the maximum and
158        minimum values are given above the figures.
159        \label{fig:adjheff}}
160    \end{figure*}
161    
162    The sensitivity patterns for effective ice thickness are predominantly positive.
163    An increase in ice volume in most places ``upstream'' of
164    Lancaster sound increases the solid fresh water export at the exit section.
165    The transient nature of the sensitivity patterns
166    (top panels vs. bottom panels) is also obvious:
167    the area upstream of the Lancaster Sound that
168    contributes to the export sensitivity is larger in the earlier snapshot.
169    In the free slip case, the sensivity follows (backwards in time) the dominant pathway
170    through the Barrow Strait
171    into the Viscount Melville Sound, and from there trough the M'Clure Strait
172    into the Arctic Ocean (the ``Northwest Passage''). \ml{[Is that really
173      the Northwest Passage? I thought it would turn south in Barrow
174      Strait, but I am easily convinced because it makes a nicer story.]}
175    Secondary paths are northward from the
176    Viscount Melville Sound through the Byam Martin Channel into
177    the Prince Gustav Adolf Sea and through the Penny Strait into the
178    MacLean Strait. \ml{[Patrick, all these names, if mentioned in the
179      text need to be included somewhere in a figure (i.e. fig1). Can you
180      either do this in fig1 (based on martins\_figs.m) or send me a map
181      where these names are visible so I can do this unambiguously. I
182      don't know where  Byam
183      Martin Channel, Prince Gustav Adolf Sea, Penny Strait, MacLean
184      Strait, Ballantyne St., Massey Sound are.]}
185    
186    There are large differences between the free slip and no slip
187    solution.  By the end of the adjoint integration in January 1989, the
188    no slip sensitivities (bottom right) are generally weaker than the
189    free slip sensitivities and hardly reach beyond the western end of the
190    Barrow Strait. In contrast, the free-slip sensitivities (bottom left)
191    extend through most of the CAA and into the Arctic interior, both to
192    the West (M'Clure St.)  and to the North (Ballantyne St., Prince
193    Gustav Adolf Sea, Massey Sound), because in this case the ice can
194    drift more easily through narrow straits, so that a positive ice
195    volume anomaly anywhere upstream in the CAA increases ice export
196    through the Lancaster Sound within the simulated 4 year period.
197    
198    One peculiar feature in the October 1992 sensitivity maps (top panels)
199    are the negative sensivities to the East and to the West of the
200    Lancaster Sound.
201    These can be explained by indirect effects: less ice to the East means
202    less resistance to eastward drift and thus more export; similarly, less ice to
203    the West means that more ice can be moved eastwards from the Barrow Strait
204    into the Lancaster Sound leading to more ice export.
205    \ml{PH: The first explanation (East) I buy, the second (West) I
206      don't.} \ml{[ML: unfortunately, I don't have anything better to
207      offer, do you? Keep in mind that these sensitivites are very small
208      and only show up, because of the colorscale. In Fig6, they are
209      hardly visible.]}
210    
211    The temporal evolution of several ice export sensitivities (eqn. XX,
212    \ml{[which equation do you mean?]}) along a zonal axis through
213    Lancaster Sound, Barrow Strait, and Melville Sound (115\degW\ to
214    80\degW, averaged across the passages) are depicted as Hovmueller
215    diagrams in \reffig{lancasteradj}. These are, from top to bottom, the
216    sensitivities with respect to effective ice thickness ($hc$), ocean
217    surface temperature ($SST$) and precipitation ($p$) for free slip
218    (left column) and no slip (right column) ice drift boundary
219    conditions.
220  %  %
221  \item  \begin{figure*}
222  plot of ATEMP for 12, 24, 36, 48 months    \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
223      \caption{Hovermoeller diagrams along the axis Viscount Melville
224        Sound/Barrow Strait/Lancaster Sound. The diagrams show the
225        sensitivities (derivatives) of the ``solid'' fresh water (i.e.,
226        ice and snow) export $J$ through Lancaster sound
227        (\reffig{arctic_topog}, cross-section G) with respect to effective
228        ice thickness ($hc$), ocean surface temperature (SST) and
229        precipitation ($p$) for two runs with free slip and no slip
230        boundary conditions for the sea ice drift. Each plot is overlaid
231        with the contours 1 and 3 of the normalized ice strengh
232        $P/P^*=(hc)\,\exp[-C\,(1-c)]$ for orientation.
233        \label{fig:lancasteradj}}
234    \end{figure*}
235  %  %
236  \item  \begin{figure*}
237  plot of HEFF for 12, 24, 36, 48 months    \includegraphics*[height=.8\textheight]{\fpath/lancaster_fwd}
238      \caption{Hovermoeller diagrams along the axis Viscount Melville
239        Sound/Barrow Strait/Lancaster Sound of effective ice thickness
240        ($hc$), effective snow thickness ($h_{s}c$) and normalized ice
241        strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ for two runs with free slip
242        and no slip boundary conditions for the sea ice drift. Each plot
243        is overlaid with the contours 1 and 3 of the normalized ice
244        strength for orientation.
245        \label{fig:lancasterfwd}}
246    \end{figure*}
247  %  %
 \end{itemize}  
   
248    
249    The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature
250    (second row) sensitivities are coherent:
251    more ice in the Lancaster Sound leads
252    to more export, and one way to get more ice is by colder surface
253    temperatures (less melting from below). In the free slip case the
254    sensitivities spread out in "pulses" following a seasonal cycle:
255    ice can propagate eastwards (forward in time and thus sensitivites can
256    propagate westwards (backwards in time) when the ice strength is low
257    in late summer to early autumn.  
258    In contrast, during winter, the sensitivities show little to now
259    westward propagation, as the ice is frozen solid and does not move.
260    In the no slip case the (normalized)
261    ice strength does not fall below 1 during the winters of 1991 to 1993
262    (mainly because the ice concentrations remain near 100\%, not
263    shown). Ice is therefore blocked and cannot drift eastwards
264    (forward in time) through the Viscount
265    Melville Sound, Barrow Strait, Lancaster Sound channel system.
266    Consequently, the sensitivities do not propagate westwards (backwards in
267    time) and the export through Lancaster Sound is only affected by
268    local ice formation and melting for the entire integration period.
269    
270    The sensitivities to precipitation exhibit an oscillatory behaviour:
271    they are negative (more precipitation leads to less export)
272    before January (more precisely, late fall) and mostly positive after January
273    (more precisely, January through July).
274    Times of positive sensitivities coincide with times of
275    normalized ice strengths exceeding values of 3
276    %
277    \ml{PH: Problem is, that's not true for the first two years (backward),
278    east of 95\degW, that is, in the Lancaster Sound.
279    For example, at 90\degW\ the sensitivities are negative throughout 1992,
280    and no clear correlation to ice strength is apparent there.}
281    except between 95\degW\ and 85\degW, which is an area of
282    increased snow cover in spring. \ml{[ML: and no, I cannot explain
283      that. Can you?]}
284    
 \reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export  
 through Fram Strait in December 1995 to changes in sea-ice thickness  
 12, 24, 36, 48 months back in time. Corresponding sensitivities to  
 ocean surface temperature are depicted in  
 \reffig{4yradjthetalev1}(a--d).  The main characteristics is  
 consistency with expected advection of sea-ice over the relevant time  
 scales considered.  The general positive pattern means that an  
 increase in sea-ice thickness at location $(x,y)$ and time $t$ will  
 increase sea-ice export through Fram Strait at time $T_e$.  Largest  
 distances from Fram Strait indicate fastest sea-ice advection over the  
 time span considered.  The ice thickness sensitivities are in close  
 correspondence to ocean surface sentivitites, but of opposite sign.  
 An increase in temperature will incur ice melting, decrease in ice  
 thickness, and therefore decrease in sea-ice export at time $T_e$.  
   
 The picture is fundamentally different and much more complex  
 for sensitivities to ocean temperatures away from the surface.  
 \reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to  
 temperatures at roughly 400 m depth.  
 Primary features are the effect of the heat transport of the North  
 Atlantic current which feeds into the West Spitsbergen current,  
 the circulation around Svalbard, and ...  
   
   
 \ml{[based on the movie series  
   zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice  
 export through the Canadian Archipelag is highly sensitive to the  
 previous state of the ocean-ice system in the Archipelago and the  
 Western Arctic. According to the \ml{(adjoint)} senstivities of the  
 eastward ice transport through Lancaster Sound (\reffig{arctic_topog},  
 cross-section G) with respect to ice volume (effective thickness), ocean  
 surface temperature, and vertical diffusivity near the surface  
 (\reffig{fouryearadj}) after 4 years of integration the following  
 mechanisms can be identified: near the ``observation'' (cross-section  
 G), smaller vertical diffusivities lead to lower surface temperatures  
 and hence to more ice that is available for export. Further away from  
 cross-section G, the sensitivity to vertical diffusivity has the  
 opposite sign, but temperature and ice volume sensitivities have the  
 same sign as close to the observation.  
   
 \begin{figure}[t!]  
 \centerline{  
 \subfigure[{\footnotesize -12 months}]  
 {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}  
 %\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}  
285  %  %
286  \subfigure[{\footnotesize -24 months}]  Assuming that most precipation is snow in this area\footnote{
287  {\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}  In the
288  }  current implementation the model differentiates between snow and rain
289    depending on the thermodynamic growth rate; when it is cold enough for
290    ice to grow, all precipitation is assumed to be snow.}
291  %  %
292  \caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to  the sensitivities can be interpreted in terms of the model physics.
293  sea-ice thickness at various prior times.  The accumulation of snow directly increases the exported volume.
294  \label{fig:4yradjheff}}  Further, short wave radiation cannot penetrate the snow cover and has
295  \end{figure}  a higer albedo than ice (0.85 for dry snow and 0.75 for dry ice in our
296    case); thus it protects the ice against melting in spring (after
297    January).
298    
299    On the other hand, snow reduces the effective conductivity and thus the heat
300    flux through the ice. This insulating effect slows down the cooling of
301    the surface water underneath the ice and limits the ice growth from
302    below, so that less snow in the ice-growing season leads to more new
303    ice and thus more ice export.
304    \ml{PH: Should probably discuss the effect of snow vs. rain.
305    To me it seems that the "rain" effect doesn't really play a role
306    because the neg. sensitivities are too late in the fall,
307    probably mostly falling as snow.} \ml{[ML: correct, I looked at
308    NCEP/CORE air temperatures, and they are hardly above freezing in
309    Jul/Aug, but otherwise below freezing, that why I can assume that most
310    precip is snow. ]} \ml{[this is not very good but do you have anything
311    better?:]}
312    The negative sensitivities to precipitation between 95\degW\ and
313    85\degW\ in spring 1992 may be explained by a similar mechanism: in an
314    area of thick snow (almost 50\,cm), ice cannot melt and tends to block
315    the channel so that ice coming in from the West cannot pass thus
316    leading to less ice export in the next season.
317    
318    \subsubsection{Forward sensitivities}
319    
320    \ml{[Here we need for integrations to show that the adjoint
321      sensitivites are not just academic. I suggest to perturb HEFF
322      and THETA initial conditions, and PRECIP somewhere in the Melville
323      Sound and then produce plots similar to reffig{lancasteradj}. For
324      PRECIP it would be great to have two perturbation experiments, one
325      where ADJprecip is posivite and one where ADJprecip is negative]}
326      
327    
328    %(*)
329    %The sensitivity in Baffin Bay are more complex.
330    %The pattern evolves along the Western boundary, connecting
331    %the Lancaster Sound Polynya, the Coburg Island Polynya, and the
332    %North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
333    %The sign of sensitivities has an oscillatory character
334    %[AT FREQUENCY OF SEASONAL CYCLE?].
335    %First, we need to establish whether forward perturbation runs
336    %corroborate the oscillatory behaviour.
337    %Then, several possible explanations:
338    %(i) connection established through Nares Strait throughflow
339    %which extends into Western boundary current in Northern Baffin Bay.
340    %(ii) sea-ice concentration there is seasonal, i.e. partly
341    %ice-free during the year. Seasonal cycle in sensitivity likely
342    %connected to ice-free vs. ice-covered parts of the year.
343    %Negative sensitivities can potentially be attributed
344    %to blocking of Lancaster Sound ice export by Western boundary ice
345    %in Baffin Bay.
346    %(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
347    %(in reality there's a Northern counter current hugging the coast of
348    %Devon Island which we probably don't resolve).
349    
350    %Remote control of Kennedy Channel on Lancaster Sound ice export
351    %seems a nice test for appropriateness of free-slip vs. no-slip BCs.
352    
353    %\paragraph{Sensitivities to the sea-ice area}
354    
355    %Fig. XXX depcits transient sea-ice export sensitivities
356    %to changes in sea-ice concentration
357    % $\partial J / \partial area$ using free-slip
358    %(left column) and no-slip (right column) boundary conditions.
359    %Sensitivity snapshots are depicted for (from top to bottom)
360    %12, 24, 36, and 48 months prior to May 2003.
361    %Contrary to the steady patterns seen for thickness sensitivities,
362    %the ice-concentration sensitivities exhibit a strong seasonal cycle
363    %in large parts of the domain (but synchronized on large scale).
364    %The following discussion is w.r.t. free-slip run.
365    
366    %(*)
367    %Months, during which sensitivities are negative:
368    %\\
369    %0 to 5   Db=N/A, Dr=5 (May-Jan) \\
370    %10 to 17 Db=7, Dr=5 (Jul-Jan) \\
371    %22 to 29 Db=7, Dr=5 (Jul-Jan) \\
372    %34 to 41 Db=7, Dr=5 (Jul-Jan) \\
373    %46 to 49 D=N/A \\
374    %%
375    %These negative sensitivities seem to be connected to months
376    %during which main parts of the CAA are essentially entirely ice-covered.
377    %This means that increase in ice concentration during this period
378    %will likely reduce ice export due to blocking
379    %[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
380    %Only during periods where substantial parts of the CAA are
381    %ice free (i.e. sea-ice concentration is less than one in larger parts of
382    %the CAA) will an increase in ice-concentration increase ice export.
383    
384    %(*)
385    %Sensitivities peak about 2-3 months before sign reversal, i.e.
386    %max. negative sensitivities are expected end of July
387    %[DOUBLE CHECK THIS].
388    
389    %(*)
390    %Peaks/bursts of sensitivities for months
391    %14-17, 19-21, 27-29, 30-33, 38-40, 42-45
392    
393    %(*)
394    %Spatial "anti-correlation" (in sign) between main sensitivity branch
395    %(essentially Northwest Passage and immediate connecting channels),
396    %and remote places.
397    %For example: month 20, 28, 31.5, 40, 43.
398    %The timings of max. sensitivity extent are similar between
399    %free-slip and no-slip run; and patterns are similar within CAA,
400    %but differ in the Arctic Ocean interior.
401    
402    %(*)
403    %Interesting (but real?) patterns in Arctic Ocean interior.
404    
405    %\paragraph{Sensitivities to the sea-ice velocity}
406    
407    %(*)
408    %Patterns of ADJuice at almost any point in time are rather complicated
409    %(in particular with respect to spatial structure of signs).
410    %Might warrant perturbation tests.
411    %Patterns of ADJvice, on the other hand, are more spatially coherent,
412    %but still hard to interpret (or even counter-intuitive
413    %in many places).
414    
415    %(*)
416    %"Growth in extent of sensitivities" goes in clear pulses:
417    %almost no change between months: 0-5, 10-20, 24-32, 36-44
418    %These essentially correspond to months of
419    
420    
421    %\subsection{Sensitivities to the oceanic state}
422    
423    %\paragraph{Sensitivities to theta}
424    
425    %\textit{Sensitivities at the surface (z = 5 m)}
426    
427    %(*)
428    %mabye redo with caxmax=0.02 or even 0.05
429    
430    %(*)
431    %Core of negative sensitivities spreading through the CAA as
432    %one might expect [TEST]:
433    %Increase in SST will decrease ice thickness and therefore ice export.
434    
435    %(*)
436    %What's maybe unexpected is patterns of positive sensitivities
437    %at the fringes of the "core", e.g. in the Southern channels
438    %(Bellot St., Peel Sound, M'Clintock Channel), and to the North
439    %(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
440    %then shifting Northward into the Arctic interior).
441    
442    %(*)
443    %Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
444    %propagating into Lincoln Sea, then
445    %entering Nares Strait and Smith Sound, periodically
446    %warming or cooling[???] the Lancaster Sound exit.
447    
448    %\textit{Sensitivities at depth (z = 200 m)}
449    
450    %(*)
451    %Negative sensitivities almost everywhere, as might be expected.
452    
453    %(*)
454    %Sensitivity patterns between free-slip and no-slip BCs
455    %are quite similar, except in Lincoln Sea (North of Nares St),
456    %where the sign is reversed (but pattern remains similar).
457    
458    %\paragraph{Sensitivities to salt}
459    
460    %T.B.D.
461    
462    %\paragraph{Sensitivities to velocity}
463    
464    %T.B.D.
465    
466    %\subsection{Sensitivities to the atmospheric state}
467    
468    %\begin{itemize}
469    %%
470    %\item
471    %plot of ATEMP for 12, 24, 36, 48 months
472    %%
473    %\item
474    %plot of HEFF for 12, 24, 36, 48 months
475    %%
476    %\end{itemize}
477    
478    
479    
480    %\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
481    %through Fram Strait in December 1995 to changes in sea-ice thickness
482    %12, 24, 36, 48 months back in time. Corresponding sensitivities to
483    %ocean surface temperature are depicted in
484    %\reffig{4yradjthetalev1}(a--d).  The main characteristics is
485    %consistency with expected advection of sea-ice over the relevant time
486    %scales considered.  The general positive pattern means that an
487    %increase in sea-ice thickness at location $(x,y)$ and time $t$ will
488    %increase sea-ice export through Fram Strait at time $T_e$.  Largest
489    %distances from Fram Strait indicate fastest sea-ice advection over the
490    %time span considered.  The ice thickness sensitivities are in close
491    %correspondence to ocean surface sentivitites, but of opposite sign.
492    %An increase in temperature will incur ice melting, decrease in ice
493    %thickness, and therefore decrease in sea-ice export at time $T_e$.
494    
495    %The picture is fundamentally different and much more complex
496    %for sensitivities to ocean temperatures away from the surface.
497    %\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
498    %temperatures at roughly 400 m depth.
499    %Primary features are the effect of the heat transport of the North
500    %Atlantic current which feeds into the West Spitsbergen current,
501    %the circulation around Svalbard, and ...
502    
503    
504    %%\begin{figure}[t!]
505    %%\centerline{
506    %%\subfigure[{\footnotesize -12 months}]
507    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
508    %%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
509    %%
510    %%\subfigure[{\footnotesize -24 months}]
511    %%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
512    %%}
513    %%
514    %%\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
515    %%sea-ice thickness at various prior times.
516    %%\label{fig:4yradjheff}}
517    %%\end{figure}
518    
519    
520    %\ml{[based on the movie series
521    %  zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
522    %export through the Canadian Archipelag is highly sensitive to the
523    %previous state of the ocean-ice system in the Archipelago and the
524    %Western Arctic. According to the \ml{(adjoint)} senstivities of the
525    %eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
526    %cross-section G) with respect to ice volume (effective thickness), ocean
527    %surface temperature, and vertical diffusivity near the surface
528    %(\reffig{fouryearadj}) after 4 years of integration the following
529    %mechanisms can be identified: near the ``observation'' (cross-section
530    %G), smaller vertical diffusivities lead to lower surface temperatures
531    %and hence to more ice that is available for export. Further away from
532    %cross-section G, the sensitivity to vertical diffusivity has the
533    %opposite sign, but temperature and ice volume sensitivities have the
534    %same sign as close to the observation.
535    
536    
537  %%% Local Variables:  %%% Local Variables:
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