--- MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex	2008/03/04 20:30:40	1.2
+++ MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex	2008/07/25 15:01:19	1.5
@@ -1,84 +1,55 @@
-\section{Adjoint sensiivities of the MITsim}
+\section{Adjoint sensitivities of the MITsim}
 \label{sec:adjoint}
 
 \subsection{The adjoint of MITsim}
 
-The ability to generate tangent linear and adjoint model components
-of the MITsim has been a main design task.
-For the ocean the adjoint capability has proven to be an
-invaluable tool for sensitivity analysis as well as state estimation.
-In short, the adjoint enables very efficient computation of the gradient
-of scalar-valued model diagnostics (called cost function or objective function)
-with respect to many model "variables".
-These variables can be two- or three-dimensional fields of initial 
-conditions, model parameters such as mixing coefficients, or
-time-varying surface or lateral (open) boundary conditions.
-When combined, these variables span a potentially high-dimensional
-(e.g. O(10$^8$)) so-called control space. Performing parameter perturbations
-to assess model sensitivities quickly becomes prohibitive at these scales.
-Alternatively, (time-varying) sensitivities of the objective function 
-to any element of the  control space can be computed very efficiently in 
-one single adjoint 
-model integration, provided an efficient adjoint model is available.
-
-[REFERENCES]
-
-
-The adjoint operator (ADM) is the transpose of the tangent linear operator (TLM)
-of the full (in general nonlinear) forward model, i.e. the MITsim.
-The TLM maps perturbations of elements of the control space
-(e.g. initial ice thickness distribution)
-via the model Jacobian
-to a perturbation in the objective function 
-(e.g. sea-ice export at the end of the integration interval).
-\textit{Tangent} linearity ensures that the derivatives are evaluated
-with respect to the underlying model trajectory at each point in time.
-This is crucial for nonlinear trajectories and the presence of different
-regimes (e.g. effect of the seaice growth term at or away from the
-freezing point of the ocean surface).
-Ensuring tangent linearity can be easily achieved by integrating
-the full model in sync with the TLM to provide the underlying model state.
-Ensuring \textit{tangent} adjoints is equally crucial, but much more
-difficult to achieve because of the reverse nature of the integration:
-the adjoint accumulates sensitivities backward in time,
-starting from a unit perturbation of the objective function.
-The adjoint model requires the model state in reverse order.
-This presents one of the major complications in deriving an
-exact, i.e. \textit{tangent} adjoint model.
-
-Following closely the development and maintenance of TLM and ADM
-components of the MITgcm we have relied heavily on the
-autmomatic differentiation (AD) tool
-"Transformation of Algorithms in Fortran" (TAF)
-developed by Fastopt (Giering and Kaminski, 1998)
-to derive TLM and ADM code of the MITsim.
-Briefly, the nonlinear parent model is fed to the AD tool which produces 
-derivative code for the specified control space and objective function.
-Following this approach has (apart from its evident success)
-several advantages:
-(1) the adjoint model is the exact adjoint operator of the parent model,
-(2) the adjoint model can be kept up to date with respect to ongoing
-development of the parent model, and adjustments to the parent model
-to extend the automatically generated adjoint are incremental changes
-only, rather than extensive re-developments,
-(3) the parallel structure of the parent model is preserved 
-by the adjoint model, ensuring efficient use in high performance
-computing environments.
-
-Some initial code adjustments are required to support dependency analysis
-of the flow reversal and certain language limitations which may lead
-to irreducible flow graphs (e.g. GOTO statements).
-The problem of providing the required model state in reverse order
-at the time of evaluating nonlinear or conditional
-derivatives is solved via balancing
-storing vs. recomputation of the model state in a multi-level
-checkpointing loop.
-Again, an initial code adjustment is required to support TAFs 
-checkpointing capability.
-The code adjustments are sufficiently simple so as not to cause
-major limitations to the full nonlinear parent model.
-Once in place, an adjoint model of a new model configuration
-may be derived in about 10 minutes.
+The adjoint model of the MITgcm has become an invaluable
+tool for sensitivity analysis as well as state estimation \citep[for a
+recent summary, see][]{heim:08}. The code has been developed and
+tailored to be readily used with automatic differentiation tools for
+adjoint code generation. This route was also taken in developing and
+adapting the sea-ice compontent MITsim, so that tangent linear and
+adjoint components can be obtained and kept up to date without
+excessive effort.
+
+The adjoint model operator (ADM) is the transpose of the tangent
+linear model operator (TLM) of the full (in general nonlinear) forward
+model, in this case the MITsim. This operator computes the gradients
+of scalar-valued model diagnostics (so-called cost function or
+objective function) with respect to many model inputs (so-called
+independent or control variables).  These inputs can be two- or
+three-dimensional fields of initial conditions of the ocean or sea-ice
+state, model parameters such as mixing coefficients, or time-varying
+surface or lateral (open) boundary conditions.  When combined, these
+variables span a potentially high-dimensional (e.g.  O(10$^8$))
+so-called control space. At this problem dimension, perturbing
+individual parameters to assess model sensitivities quickly becomes
+prohibitive. By contrast, transient sensitivities of the objective
+function to any element of the control and model state space can be
+computed very efficiently in one single adjoint model integration,
+provided an adjoint model is available.
+
+In anology to the TLM and ADM components of the MITgcm we rely on the
+autmomatic differentiation (AD) tool ``Transformation of Algorithms in
+Fortran'' (TAF) developed by Fastopt \citep{gier-kami:98} to generate
+TLM and ADM code of the MITsim \citep[for details see][]{maro-etal:99,
+  heim-etal:05}.  In short, the AD tool uses the nonlinear parent
+model code to generate derivative code for the specified control space
+and objective function. Advantages of this approach have been pointed
+out, for example by \cite{gier-kami:98}.
+
+Many issues of generating efficient exact adjoint sea-ice code are
+similar to those for the ocean model's adjoint.  Linearizing the model
+around the exact nonlinear model trajectory is a crucial aspect in the
+presence of different regimes (e.g., is the thermodynamic growth term
+for sea-ice evaluated near or far away from the freezing point of the
+ocean surface?). Adapting the (parent) model code to support the AD
+tool in providing exact and efficient adjoint code represents the main
+work load initially. For legacy code, this task may become
+substantial, but it is fairly straightforward when writing new code
+with an AD tool in mind. Once this initial task is completed,
+generating the adjoint code of a new model configuration takes about
+10 minutes.
 
 [HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]
 
@@ -93,112 +64,409 @@
 * approximate adjoints
 
 
-\subsection{An example: sensitivities of sea-ice export through Fram Strait}
+\subsection{An example: sensitivities of sea-ice export through
+the Lancaster Sound}
 
-We demonstrate the power of the adjoint method
-in the context of investigating sea-ice export sensitivities through Fram Strait
-(for details of this study see Heimbach et al., 2007).
-%\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).
-The domain chosen is a coarsened version of the Arctic face of the
+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 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
+\citep{mell:02, mich-etal:06,muen-etal:06}, which is not resolved in
+our simulation.
+
+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 1-year spinup, the model has been integrated for four
-years between 1992 and 1995. It is forced using realistic 6-hourly
-NCEP/NCAR atmospheric state variables. Over the open ocean these are
-converted into air-sea fluxes via the bulk formulae of
-\citet{large04}.  Derivation of air-sea fluxes in the presence of
-sea-ice is handled by the ice model as described in \refsec{model}.
-The objective function chosen is sea-ice export through Fram Strait
-computed for December 1995.  The adjoint model computes sensitivities
-to sea-ice export back in time from 1995 to 1992 along this
-trajectory.  In principle all adjoint model variable (i.e., Lagrange
-multipliers) of the coupled ocean/sea-ice model are available to
-analyze the transient sensitivity behaviour of the ocean and sea-ice
-state.  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.
-
-\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}}
-}
-
-\centerline{
-\subfigure[{\footnotesize
--36 months}]
-{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim218_cmax2.0E+02.eps}}
-%
-\subfigure[{\footnotesize
--48 months}]
-{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJheff_arc_lev1_tim292_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}
-
-
-\begin{figure}[t!]
-\centerline{
-\subfigure[{\footnotesize -12 months}]
-{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim072_cmax5.0E+01.eps}}
-%\includegraphics*[width=.3\textwidth]{H_c.bin_res_100_lev1}
-%
-\subfigure[{\footnotesize -24 months}]
-{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim145_cmax5.0E+01.eps}}
-}
-
-\centerline{
-\subfigure[{\footnotesize
--36 months}] 
-{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim218_cmax5.0E+01.eps}}
-%
-\subfigure[{\footnotesize
--48 months}]
-{\includegraphics*[width=0.44\linewidth]{\fpath/run_4yr_ADJtheta_arc_lev1_tim292_cmax5.0E+01.eps}}
-}
-\caption{Same as \reffig{4yradjheff} but for sea surface temperature
-\label{fig:4yradjthetalev1}}
-\end{figure}
+\citep{menemenlis05} as described in \refsec{forward}.  The horizontal
+resolution is half of that in \refsec{forward} while the vertical grid
+is the same. \ml{[Is this important? Do we need to be more specific?:
+  ]} 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 3-year spinup, the model is integrated for four
+years and five months between January 1989 and September 1993.
+\ml{[Patrick: to what extent is this different from section 3?]}
+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 $(\rho_{i} h_{i}c + \rho_{s} h_{s}c)\,u$, through Lancaster
+Sound at approximately 82\degW\ (cross-section G in
+\reffig{arctic_topog}) averaged over a 12-month period 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
+$116\pm101\text{\,km$^{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 computes sensitivities of this 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 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
+
+\subsubsection{Adjoint sensitivities}
+
+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 12 months prior to
+September 1993 (at the beginning of the averaging period for the objective
+function $J$, top) and at the beginning of the integration in January
+1989 (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*}
+
+At the beginning of October 1992, the positive sensitivities in
+the Lancaster Sound mean that an increase of ice volume increase the
+solid fresh water export. The negative sensivities to the East and to the
+West 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. The sensitivities
+are similar for both no slip and free slip solutions with a slightly larger
+area covered by non-zero sensitivities in the free slip solution. At
+the beginning of the integration (the end of the backward adjoint
+integration) the free and no slip solutions are very different. The
+sensitivities of the free slip solution extend through the enitre
+Canadian Archipelago and into the Arctic while in the no slip solution
+they still are confined to the Lancaster Sound and the Barrow
+Strait. This implies that in the free slip solution ice can drift more
+easily through the narrow straits of the Canadian Archipelago, so that
+a positive ice volume anomaly anywhere in the Canadian Archipelago is
+moved through the Lancaster Sound within 4 years thus increasing the
+ice export.
+
+The temporal evolution of several sensitivities along the zonal axis
+Lancaster Sound-Barrow Strait-Melville Sound are shown in
+\reffig{lancaster}.
+\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*}
+\reffig{lancaster} shows the sensitivities of ``solid'' fresh water
+export, that is ice and snow, through Lancaster sound (cross-section G
+in \reffig{arctic_topog}) 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. The Hovmoeller diagrams of sensitivities (derivatives) with
+respect to effective ice thickness (top) and ocean surface temperature
+(second from top) 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 can propagate westwards (backwards in time) when the ice
+strength is low in late summer. 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), so that ice is blocked and cannot drift eastwards (forward in
+time) in the Melville Sound-Barrow Strait-Lancaster Sound channel.
+Consequently the sensitivies do not propagate westwards (backwards in
+time) and the export through Lancaster Sound is only affected by
+local ice formation and melting.
+
+The sensitivities to precipitation are negative (more precipitation
+leads to less export) before January and mostly positive after
+January. Further they are mostly positive for normalized ice strengths
+over 3. Assuming that most precipation is snow in this area---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---the
+sensitivities can be interpreted in terms of the model physics.  Short
+wave radiation cannot penetrate a 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).  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.
+
+%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 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 \\
+%%
+%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: 
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