--- MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex	2008/06/04 13:34:41	1.4
+++ MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex	2008/07/28 04:34:37	1.6
@@ -1,4 +1,4 @@
-\section{Adjoint sensiivities of the MITsim}
+\section{Adjoint sensitivities of the MITsim}
 \label{sec:adjoint}
 
 \subsection{The adjoint of MITsim}
@@ -65,305 +65,436 @@
 
 
 \subsection{An example: sensitivities of sea-ice export through
-the Lancaster and Jones Sound}
+the Lancaster Sound}
 
-We demonstrate the power of the adjoint method
-in the context of investigating sea-ice export sensitivities through 
-Lancaster and Jones Sound. The rationale for doing so is to complement
-the analysis of sea-ice dynamics in the presence of narrow straits.
-Lancaster Sound is one of the main outflow paths of sea-ice flowing
-through the Canadian Arctic Archipelago (CAA). 
-Export sensitivities reflect dominant
-pathways through the CAA as resolved by the model.
-Sensitivity maps can shed a very detailed light on various quantities
-affecting the sea-ice export (and thus the underlying pathways).
-Note that while the dominant circulation through Lancaster Sound is
-toward the East, there is a small Westward flow to the North,
-hugging the coast of Devon Island [ARE WE RESOLVING THIS?],
-see e.g. \cite{mell:02, mich-etal:06,muen-etal:06}.
-
-The model domain is a coarsened version of the Arctic face of the
-high-resolution cubed-sphere configuration of the ECCO2 project
-\citep[see][]{menemenlis05}. It covers the entire Arctic,
-extends into the North Pacific such as to cover the entire
-ice-covered regions, and comprises parts of the North Atlantic
-down to XXN to enable analysis of remote influences of the 
-North Atlantic current to sea-ice variability and export.
-The horizontal resolution varies between XX and YY km
-with 50 unevenly spaced vertical levels.
-The adjoint models run efficiently on 80 processors
-(benchmarks have been performed both on an SGI Altix as well as an 
-IBM SP5 at NASA/ARC).
-
-Following a 3-year spinup, the model has been integrated for four
-years and five months between January 1989 and May 1993.
-It is forced using realistic 6-hourly
-NCEP/NCAR atmospheric state variables. Over the open ocean these are
-converted into air-sea fluxes via the bulk formulae of
-\citet{large04}.  Derivation of air-sea fluxes in the presence of
-sea-ice is handled by the ice model as described in \refsec{model}.
-The objective function is chosen $J$ as the
-sea-ice export through 
-Lancaster Sound at XX$^{\circ}$W
-averaged over an 8-month period between October 1992 and May 1993.  
-
-The adjoint model computes sensitivities
-to sea-ice export back in time from 1993 to 1989 along this
-trajectory.  In principle all adjoint model variable (i.e., Lagrange
-multipliers) of the coupled ocean/sea-ice model
-as well as the surface atmospheric state are available to
-analyze the transient sensitivity behaviour.  
-Over the open ocean, the adjoint of the bulk formula scheme
-computes sensitivities to the time-varying atmospheric state.  Over
-ice-covered parts, the sea-ice adjoint converts surface ocean
-sensitivities to atmospheric sensitivities.
+We demonstrate the power of the adjoint method in the context of
+investigating sea-ice export sensitivities through Lancaster Sound.
+The rationale for doing so is to complement the analysis of sea-ice
+dynamics in the presence of narrow straits.  Lancaster Sound is one of
+the main paths of sea-ice flowing through the Canadian Arctic
+Archipelago (CAA).  Export sensitivities reflect dominant pathways
+through the CAA as resolved by the model.  Sensitivity maps can shed a
+very detailed light on various quantities affecting the sea-ice export
+(and thus the underlying pathways).  Note that while the dominant
+circulation through Lancaster Sound is toward the East, there is a
+small Westward flow to the North, hugging the coast of Devon Island
+\citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in
+our simulation.
+
+The model domain is the same as the one described in \refsec{forward},
+but with halved horizontal resolution.
+The adjoint models run efficiently on 80 processors (as validated
+by benchmarks on both an SGI Altix and an IBM SP5 at NASA/ARC).
+Following a 4-year spinup (1985 to 1988), the model is integrated for four
+years and nine months between January 1989 and September 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}.  The air-sea fluxes in the presence of
+%sea-ice are handled by the ice model as described in \refsec{model}.
+The objective function $J$ is chosen as the ``solid'' fresh water
+export, that is the export of ice and snow converted to units of fresh
+water,
+%
+\begin{equation}
+J \, = \, (\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u
+\end{equation} 
+%
+through Lancaster Sound at approximately 82\degW\ (cross-section G in
+\reffig{arctic_topog}) averaged \ml{PH: Maybe integrated quantity is
+more physical} over the final 12-month of the integration between October
+1992 and September 1993.
+
+The forward trajectory of the model integration resembles broadly that
+of the model in \refsec{forward}. Many details are different, owning
+to different resolution and integration period; for example, the solid
+fresh water transport through Lancaster Sound is
+%
+\ml{PH: Martin, where did you get these numbers from?}
+%
+$116\pm101\text{\,km$^{3}$\,y$^{-1}$}$ for a free slip simulation with
+the C-LSOR solver, but only $39\pm64\text{\,km$^{3}$\,y$^{-1}$}$ for a
+no slip simulation.
+
+The adjoint model is the transpose of the tangent linear (or Jacobian) model
+operator. It runs backward in time, from September 1993 to
+January 1989. Along its integration it accumulates the Lagrange multipliers
+of the model subject to the objective function (solid freshwater export),
+which can be interpreted as sensitivities of the objective function
+to each control variable and each element of the intermediate 
+coupled model state variables.
+Thus, all sensitivity elements of the coupled
+ocean/sea-ice model state as well as the surface atmospheric state are
+available for analysis of the transient sensitivity behavior.  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}
+\subsubsection{Adjoint sensitivities}
 
-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
+The most readily interpretable ice-export sensitivity is that to
+effective ice thickness, $\partial{J} / \partial{(hc)}$.
+\reffig{adjheff} shows transient $\partial{J} / \partial{(hc)}$ using
+free-slip (left column) and no-slip (right column) boundary
+conditions. Sensitivity snapshots are depicted for beginning of October 2002,
+i.e. 12 months back in time from September 1993 
+(the beginning of the averaging period for the objective
+function $J$, top),
+and for Jannuary 1989, the beginning of the forward integration (bottom).
+\begin{figure*}[t]
+  \includegraphics*[width=\textwidth]{\fpath/adjheff}
+  \caption{Sensitivity $\partial{J}/\partial{(hc)}$ in
+    m$^2$\,s$^{-1}$/m for two different times (rows) and two different
+    boundary conditions for sea ice drift. The color scale is chosen
+    to illustrate the patterns of the sensitivities; the maximum and
+    minimum values are given above the figures.
+    \label{fig:adjheff}}
+\end{figure*}
+
+As expected, the sensitivity patterns are predominantly positive,
+an increase in ice volume in most places ``upstream'' of
+Lancaster sound increases the solid fresh water export at the exit section.
+Also obvious is the transient nature of the sensitivity patterns
+(top panels vs. bottom panels),
+i.e. as time moves backward, an increasing area upstream of Lancaster Sound 
+contributes to the export sensitivity.
+The dominant pathway (free slip case) 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"). 
+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 \\
+The difference between the free slip and no slip solution is evident:
+by the end of the adjoint integration, in January 1989
+the free-slip sensitivities (bottom left) extend through most of the CAA
+and all the way into the Arctic interior, both to the West (M'Clure St.) 
+and to the North 
+(Ballantyne St., Prince Gustav Adolf Sea, Massey Sound), 
+whereas the no slip sensitivities (bottom right) are overall weaker 
+and remain mostly confined to Lancaster Sound and just West of Barrow Strait.
+In the free slip solution ice can drift more
+easily through narrow straits, and
+a positive ice volume anomaly further upstream in the CAA may increase
+ice export through the Lancaster Sound within a 4 year period.
+
+One peculiar feature in the October 1992 sensitivity maps (top panels)
+are the negative sensivities to the East and to the West.
+These can be explained by indirect effects: less ice to the East means
+less resistance to eastward drift and thus more export; similarly, less ice to
+the West means that more ice can be moved eastwards from the Barrow Strait
+into the Lancaster Sound leading to more ice export. 
+\ml{PH: The first explanation (East) I buy, the second (West) I don't}.
+
+The temporal evolution of several ice export sensitivities 
+(eqn. XX) along a zonal axis through
+Lancaster Sound, Barrow Strait,and  Melville Sound 
+(115\degW\ to 80\degW\ ), 
+are depicted as Hovmueller diagrams in \reffig{lancaster}.
+From top to bottom, sensitivities are with respect to effective 
+ice thickness ($hc$), 
+ocean surface temperature ($SST$) and precipitation ($p$) for free slip 
+(left column) and no slip (right column) ice drift boundary conditions.
 %
-These negative sensitivities seem to be connected to months
-during which main parts of the CAA are essentially entirely ice-covered.
-This means that increase in ice concentration during this period
-will likely reduce ice export due to blocking
-[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
-Only during periods where substantial parts of the CAA are
-ice free (i.e. sea-ice concentration is less than one in larger parts of
-the CAA) will an increase in ice-concentration increase ice export.
-
-(*)
-Sensitivities peak about 2-3 months before sign reversal, i.e.
-max. negative sensitivities are expected end of July
-[DOUBLE CHECK THIS].
-
-(*) 
-Peaks/bursts of sensitivities for months
-14-17, 19-21, 27-29, 30-33, 38-40, 42-45
-
-(*) 
-Spatial "anti-correlation" (in sign) between main sensitivity branch
-(essentially Northwest Passage and immediate connecting channels),
-and remote places.
-For example: month 20, 28, 31.5, 40, 43.
-The timings of max. sensitivity extent are similar between
-free-slip and no-slip run; and patterns are similar within CAA,
-but differ in the Arctic Ocean interior.
-
-(*) 
-Interesting (but real?) patterns in Arctic Ocean interior.
-
-\paragraph{Sensitivities to the sea-ice velocity}
-
-(*)
-Patterns of ADJuice at almost any point in time are rather complicated
-(in particular with respect to spatial structure of signs).
-Might warrant perturbation tests.
-Patterns of ADJvice, on the other hand, are more spatially coherent,
-but still hard to interpret (or even counter-intuitive
-in many places).
-
-(*)
-"Growth in extent of sensitivities" goes in clear pulses:
-almost no change between months: 0-5, 10-20, 24-32, 36-44
-These essentially correspond to months of
-
-
-\subsection{Sensitivities to the oceanic state}
-
-\paragraph{Sensitivities to theta}
-
-\textit{Sensitivities at the surface (z = 5 m)}
-
-(*)
-mabye redo with caxmax=0.02 or even 0.05
-
-(*)
-Core of negative sensitivities spreading through the CAA as 
-one might expect [TEST]:
-Increase in SST will decrease ice thickness and therefore ice export.
-
-(*)
-What's maybe unexpected is patterns of positive sensitivities
-at the fringes of the "core", e.g. in the Southern channels
-(Bellot St., Peel Sound, M'Clintock Channel), and to the North
-(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
-then shifting Northward into the Arctic interior).
-
-(*)
-Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
-propagating into Lincoln Sea, then
-entering Nares Strait and Smith Sound, periodically
-warming or cooling[???] the Lancaster Sound exit.
-
-\textit{Sensitivities at depth (z = 200 m)}
-
-(*)
-Negative sensitivities almost everywhere, as might be expected.
-
-(*)
-Sensitivity patterns between free-slip and no-slip BCs
-are quite similar, except in Lincoln Sea (North of Nares St),
-where the sign is reversed (but pattern remains similar).
-
-\paragraph{Sensitivities to salt}
-
-T.B.D.
-
-\paragraph{Sensitivities to velocity}
-
-T.B.D.
-
-\subsection{Sensitivities to the atmospheric state}
-
-\begin{itemize}
+\begin{figure*}
+  \includegraphics*[height=.8\textheight]{\fpath/lancaster_adj}
+  \caption{Hovermoeller diagrams of sensitivities (derivatives) of the
+    ``solid'' fresh water (i.e., ice and snow) export $J$ through Lancaster sound
+    (\reffig{arctic_topog}, cross-section G) with respect to effective
+    ice thickness ($hc$), ocean surface temperature (SST) and
+    precipitation ($p$) for two runs with free slip and no slip boundary
+    conditions for the sea ice drift. Also shown it the normalized ice
+    strengh $P/P^*=(hc)\,\exp[-C\,(1-c)]$ (bottom panel); each plot is
+    overlaid with the contours 1 and 3 of the normalized ice strength
+    for orientation.
+    \label{fig:lancaster}}
+\end{figure*}
 %
-\item
-plot of ATEMP for 12, 24, 36, 48 months
+
+The Hovmoeller diagrams of ice thickness (top row) and sea surface temperature
+(second row) sensitivities are coherent: 
+more ice in the Lancaster Sound leads
+to more export, and one way to get more ice is by colder surface
+temperatures (less melting from below). In the free slip case the
+sensitivities spread out in "pulses" following a seasonal cycle:
+can propagate westwards (backwards in time) when the ice
+strength is low in late summer, early autumn. 
+In contrast, during winter, the sensitivities show little to now
+westward propagation.
+In the no slip case the (normalized)
+ice strength does not fall below 1 during the winters of 1991 to 1993
+(mainly because the ice concentrations remain nearly 100\%, not
+shown). Ice is therefore blocked and cannot drift eastwards 
+(forward in time) through the 
+Melville Sound, Barrow Strait, Lancaster Sound channel system.
+Consequently, the sensitivities do not propagate westwards (backwards in
+time) and the export through Lancaster Sound is only affected by
+local ice formation and melting for the entire integration period.
+
+The sensitivities to precipitation exhibit an oscillatory behaviour:
+they are negative (more precipitation leads to less export) 
+before January (more precisely, late fall) and mostly positive after January
+(more precisely, January through July). 
+Times of positive sensitivities coincide with times of
+normalized ice strengths exceeding values of 3.
 %
-\item
-plot of HEFF for 12, 24, 36, 48 months
+\ml{PH: Problem is, that's not true for the first two years (backward),
+East of 95\degW\ , i.e. in Lancaster Sound.
+For example, at 90\degW\ the sensitivities are negative throughout 1992,
+and no clear correlation to ice strength is apparent there.}.
 %
-\end{itemize}
-
-
-
-\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
-through Fram Strait in December 1995 to changes in sea-ice thickness
-12, 24, 36, 48 months back in time. Corresponding sensitivities to
-ocean surface temperature are depicted in
-\reffig{4yradjthetalev1}(a--d).  The main characteristics is
-consistency with expected advection of sea-ice over the relevant time
-scales considered.  The general positive pattern means that an
-increase in sea-ice thickness at location $(x,y)$ and time $t$ will
-increase sea-ice export through Fram Strait at time $T_e$.  Largest
-distances from Fram Strait indicate fastest sea-ice advection over the
-time span considered.  The ice thickness sensitivities are in close
-correspondence to ocean surface sentivitites, but of opposite sign.
-An increase in temperature will incur ice melting, decrease in ice
-thickness, and therefore decrease in sea-ice export at time $T_e$.
-
-The picture is fundamentally different and much more complex
-for sensitivities to ocean temperatures away from the surface.
-\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
-temperatures at roughly 400 m depth.
-Primary features are the effect of the heat transport of the North
-Atlantic current which feeds into the West Spitsbergen current,
-the circulation around Svalbard, and ...
-
-
-\ml{[based on the movie series
-  zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
-export through the Canadian Archipelag is highly sensitive to the
-previous state of the ocean-ice system in the Archipelago and the
-Western Arctic. According to the \ml{(adjoint)} senstivities of the
-eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
-cross-section G) with respect to ice volume (effective thickness), ocean
-surface temperature, and vertical diffusivity near the surface
-(\reffig{fouryearadj}) after 4 years of integration the following
-mechanisms can be identified: near the ``observation'' (cross-section
-G), smaller vertical diffusivities lead to lower surface temperatures
-and hence to more ice that is available for export. Further away from
-cross-section G, the sensitivity to vertical diffusivity has the
-opposite sign, but temperature and ice volume sensitivities have the
-same sign as close to the observation.
-
-\begin{figure}[t!]
-\centerline{
-\subfigure[{\footnotesize -12 months}]
-{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
-%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
+Assuming that most precipation is snow in this area
 %
-\subfigure[{\footnotesize -24 months}]
-{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
-}
+\footnote{
+In the
+current implementation the model differentiates between snow and rain
+depending on the thermodynamic growth rate; when it is cold enough for
+ice to grow, all precipitation is assumed to be snow.}
 %
-\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
-sea-ice thickness at various prior times.
-\label{fig:4yradjheff}}
-\end{figure}
+the sensitivities can be interpreted in terms of the model physics.  Short
+wave radiation cannot penetrate the snow cover and has a higer albedo
+than ice (0.85 for dry snow and 0.75 for dry ice in our case); thus it
+protects the ice against melting in spring (after January).  
+\ml{PH: what about the direct effect of accumulation of precip. as snow
+which directly increases the volume.}.
+
+On the other hand, snow reduces the effective conductivity and thus the heat
+flux through the ice. This insulating effect slows down the cooling of
+the surface water underneath the ice and limits the ice growth from
+below, so that less snow in the ice-growing season leads to more new
+ice and thus more ice export.
+\ml{PH: Should probably discuss the effect of snow vs. rain.
+To me it seems that the "rain" effect doesn't really play
+because the neg. sensitivities are too late in the fall,
+probably mostly falling as snow.}.
+
+%Und jetzt weiss ich nicht mehr weiter, aber nun kann folgendes passiert sein:
+%1. snow insulates against melting from above during spring: more precip (snow) -> more export
+%2. less snow during fall -> more ice -> more export
+%3. precip is both snow and rain, depending on the sign of "FICE" (thermodynamic growth rate), with probably different implications
+
+
+\subsubsection{Forward sensitivities}
+
+\ml{[Here we need for integrations to show that the adjoint
+  sensitivites are not just academic. I suggest to perturb HEFF
+  and THETA initial conditions, and PRECIP somewhere in the Melville
+  Sound and then produce plots similar to reffig{lancaster}. For
+  PRECIP it would be great to have two perturbation experiments, one
+  where ADJprecip is posivite and one where ADJprecip is negative]}
+  
+
+%(*)
+%The sensitivity in Baffin Bay are more complex.
+%The pattern evolves along the Western boundary, connecting
+%the Lancaster Sound Polynya, the Coburg Island Polynya, and the
+%North Water Polynya, and reaches into Nares Strait and the Kennedy Channel.
+%The sign of sensitivities has an oscillatory character
+%[AT FREQUENCY OF SEASONAL CYCLE?].
+%First, we need to establish whether forward perturbation runs
+%corroborate the oscillatory behaviour.
+%Then, several possible explanations:
+%(i) connection established through Nares Strait throughflow
+%which extends into Western boundary current in Northern Baffin Bay.
+%(ii) sea-ice concentration there is seasonal, i.e. partly
+%ice-free during the year. Seasonal cycle in sensitivity likely
+%connected to ice-free vs. ice-covered parts of the year.
+%Negative sensitivities can potentially be attributed
+%to blocking of Lancaster Sound ice export by Western boundary ice
+%in Baffin Bay.
+%(iii) Alternatively to (ii), flow reversal in Lancaster Sound is a possibility
+%(in reality there's a Northern counter current hugging the coast of
+%Devon Island which we probably don't resolve).
+
+%Remote control of Kennedy Channel on Lancaster Sound ice export
+%seems a nice test for appropriateness of free-slip vs. no-slip BCs.
+
+%\paragraph{Sensitivities to the sea-ice area}
+
+%Fig. XXX depcits transient sea-ice export sensitivities
+%to changes in sea-ice concentration
+% $\partial J / \partial area$ using free-slip
+%(left column) and no-slip (right column) boundary conditions.
+%Sensitivity snapshots are depicted for (from top to bottom) 
+%12, 24, 36, and 48 months prior to May 2003.
+%Contrary to the steady patterns seen for thickness sensitivities,
+%the ice-concentration sensitivities exhibit a strong seasonal cycle
+%in large parts of the domain (but synchronized on large scale).
+%The following discussion is w.r.t. free-slip run.
+
+%(*)
+%Months, during which sensitivities are negative:
+%\\
+%0 to 5   Db=N/A, Dr=5 (May-Jan) \\
+%10 to 17 Db=7, Dr=5 (Jul-Jan) \\
+%22 to 29 Db=7, Dr=5 (Jul-Jan) \\
+%34 to 41 Db=7, Dr=5 (Jul-Jan) \\
+%46 to 49 D=N/A \\
+%%
+%These negative sensitivities seem to be connected to months
+%during which main parts of the CAA are essentially entirely ice-covered.
+%This means that increase in ice concentration during this period
+%will likely reduce ice export due to blocking
+%[NEED TO EXPLAIN WHY THIS IS NOT THE CASE FOR dJ/dHEFF].
+%Only during periods where substantial parts of the CAA are
+%ice free (i.e. sea-ice concentration is less than one in larger parts of
+%the CAA) will an increase in ice-concentration increase ice export.
+
+%(*)
+%Sensitivities peak about 2-3 months before sign reversal, i.e.
+%max. negative sensitivities are expected end of July
+%[DOUBLE CHECK THIS].
+
+%(*) 
+%Peaks/bursts of sensitivities for months
+%14-17, 19-21, 27-29, 30-33, 38-40, 42-45
+
+%(*) 
+%Spatial "anti-correlation" (in sign) between main sensitivity branch
+%(essentially Northwest Passage and immediate connecting channels),
+%and remote places.
+%For example: month 20, 28, 31.5, 40, 43.
+%The timings of max. sensitivity extent are similar between
+%free-slip and no-slip run; and patterns are similar within CAA,
+%but differ in the Arctic Ocean interior.
+
+%(*) 
+%Interesting (but real?) patterns in Arctic Ocean interior.
+
+%\paragraph{Sensitivities to the sea-ice velocity}
+
+%(*)
+%Patterns of ADJuice at almost any point in time are rather complicated
+%(in particular with respect to spatial structure of signs).
+%Might warrant perturbation tests.
+%Patterns of ADJvice, on the other hand, are more spatially coherent,
+%but still hard to interpret (or even counter-intuitive
+%in many places).
+
+%(*)
+%"Growth in extent of sensitivities" goes in clear pulses:
+%almost no change between months: 0-5, 10-20, 24-32, 36-44
+%These essentially correspond to months of
+
+
+%\subsection{Sensitivities to the oceanic state}
+
+%\paragraph{Sensitivities to theta}
+
+%\textit{Sensitivities at the surface (z = 5 m)}
+
+%(*)
+%mabye redo with caxmax=0.02 or even 0.05
+
+%(*)
+%Core of negative sensitivities spreading through the CAA as 
+%one might expect [TEST]:
+%Increase in SST will decrease ice thickness and therefore ice export.
+
+%(*)
+%What's maybe unexpected is patterns of positive sensitivities
+%at the fringes of the "core", e.g. in the Southern channels
+%(Bellot St., Peel Sound, M'Clintock Channel), and to the North
+%(initially MacLean St., Prince Gustav Adolf Sea, Hazen St.,
+%then shifting Northward into the Arctic interior).
+
+%(*)
+%Marked sensitivity from the Arctic interior roughly along 60$^{\circ}$W
+%propagating into Lincoln Sea, then
+%entering Nares Strait and Smith Sound, periodically
+%warming or cooling[???] the Lancaster Sound exit.
+
+%\textit{Sensitivities at depth (z = 200 m)}
+
+%(*)
+%Negative sensitivities almost everywhere, as might be expected.
+
+%(*)
+%Sensitivity patterns between free-slip and no-slip BCs
+%are quite similar, except in Lincoln Sea (North of Nares St),
+%where the sign is reversed (but pattern remains similar).
+
+%\paragraph{Sensitivities to salt}
+
+%T.B.D.
+
+%\paragraph{Sensitivities to velocity}
+
+%T.B.D.
+
+%\subsection{Sensitivities to the atmospheric state}
+
+%\begin{itemize}
+%%
+%\item
+%plot of ATEMP for 12, 24, 36, 48 months
+%%
+%\item
+%plot of HEFF for 12, 24, 36, 48 months
+%%
+%\end{itemize}
+
+
+
+%\reffig{4yradjheff}(a--d) depict sensitivities of sea-ice export
+%through Fram Strait in December 1995 to changes in sea-ice thickness
+%12, 24, 36, 48 months back in time. Corresponding sensitivities to
+%ocean surface temperature are depicted in
+%\reffig{4yradjthetalev1}(a--d).  The main characteristics is
+%consistency with expected advection of sea-ice over the relevant time
+%scales considered.  The general positive pattern means that an
+%increase in sea-ice thickness at location $(x,y)$ and time $t$ will
+%increase sea-ice export through Fram Strait at time $T_e$.  Largest
+%distances from Fram Strait indicate fastest sea-ice advection over the
+%time span considered.  The ice thickness sensitivities are in close
+%correspondence to ocean surface sentivitites, but of opposite sign.
+%An increase in temperature will incur ice melting, decrease in ice
+%thickness, and therefore decrease in sea-ice export at time $T_e$.
+
+%The picture is fundamentally different and much more complex
+%for sensitivities to ocean temperatures away from the surface.
+%\reffig{4yradjthetalev10??}(a--d) depicts ice export sensitivities to
+%temperatures at roughly 400 m depth.
+%Primary features are the effect of the heat transport of the North
+%Atlantic current which feeds into the West Spitsbergen current,
+%the circulation around Svalbard, and ...
+
+
+%%\begin{figure}[t!]
+%%\centerline{
+%%\subfigure[{\footnotesize -12 months}]
+%%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim072_cmax2.0E+02.eps}}
+%%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1.pdf}
+%%
+%%\subfigure[{\footnotesize -24 months}]
+%%{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim145_cmax2.0E+02.eps}}
+%%}
+%%
+%%\caption{Sensitivity of sea-ice export through Fram Strait in December 2005 to
+%%sea-ice thickness at various prior times.
+%%\label{fig:4yradjheff}}
+%%\end{figure}
+
+
+%\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.
 
 
 %%% Local Variables: 
