--- MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex	2008/02/26 19:27:26	1.1
+++ MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex	2008/06/04 13:34:41	1.4
@@ -3,82 +3,53 @@
 
 \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!]
 
@@ -93,13 +64,25 @@
 * approximate adjoints
 
 
-\subsection{An example: sensitivities of sea-ice export through Fram Strait}
+\subsection{An example: sensitivities of sea-ice export through
+the Lancaster and Jones 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
+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
@@ -112,23 +95,221 @@
 (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
+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 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
+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 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
+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 \\
+%
+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
@@ -152,6 +333,23 @@
 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}]
@@ -161,41 +359,14 @@
 \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.pdf}
-%
-\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}
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: "ceaice"
+%%% End: 
