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  \subsection{The adjoint of MITsim}
- The ability to generate tangent linear and adjoint model components
- of the MITsim has been a main design task.
+ The ability to generate tangent linear and adjoint components
+ of a coupled ocean sea-ice system was one of the main drivers
+ behind the MITsim development.
  For the ocean the adjoint capability has proven to be an
- invaluable tool for sensitivity analysis as well as state estimation.
+ invaluable tool for sensitivity analysis as well as state estimation,
- In short, the adjoint enables very efficient computation of the gradient
+ as evidenced by various adjoint-based studies
- of scalar-valued model diagnostics (called cost function or objective function)
+ (for a recent summary, see \cite{heim:08}).
- with respect to many model "variables".
- These variables can be two- or three-dimensional fields of initial
+ The adjoint model operator (ADM) is the transpose of the tangent linear
- conditions, model parameters such as mixing coefficients, or
+ model operator (TLM)
- time-varying surface or lateral (open) boundary conditions.
+ of the full (in general nonlinear) forward model, i.e. the MITsim.
+ It enables very efficient computation of 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. Performing parameter perturbations
  to assess model sensitivities quickly becomes prohibitive at these scales.
- Alternatively, (time-varying) sensitivities of the objective function
+ Alternatively, transient sensitivities of the objective function
- to any element of the  control space can be computed very efficiently in
+ to any element of the  control and model state space can be computed
- one single adjoint
+ very efficiently in  one single adjoint
  model integration, provided an efficient adjoint model is available.
- [REFERENCES]
+ Following closely the development and maintenance of the
+ TLM and ADM components of the MITgcm we have relied heavily on the
- 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)
+ developed by Fastopt \citep{gier-kami:98}.
- to derive TLM and ADM code of the MITsim.
+ to derive TLM and ADM code of the MITsim
+ (for details see \cite{maro-etal:99}, \cite{heim-etal:05}).
  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)
+ Apart from its evident success, advantages of this approach have been
- several advantages:
+ pointed out, e.g. by \cite{gier-kami:98}.
- (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
+ Many issues underlying the efficient exact adjoint sea-ice code generation
- development of the parent model, and adjustments to the parent model
+ are similar to those arising for the ocean model's adjoint.
- to extend the automatically generated adjoint are incremental changes
+ Linearizing the model around the exact nonlinear model trajectory,
- only, rather than extensive re-developments,
+ as we do, is a crucial aspect in the presence of different
- (3) the parallel structure of the parent model is preserved
+ regimes (e.g. effect of the seaice growth term at or away from the
- by the adjoint model, ensuring efficient use in high performance
+ freezing point of the ocean surface).
- computing environments.
+ Adjusting the (parent) model code to support the AD tool in
+ providing exact and efficient adjoint code is the main initial work.
- Some initial code adjustments are required to support dependency analysis
+ This may be substantial for legacy code, but fairly straightforward
- of the flow reversal and certain language limitations which may lead
+ when coding with "AD application in mind".
- 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.
-Line 93 
 may be derived in about 10 minutes.
+Line 68 
 may be derived in about 10 minutes.
  * 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
+ in the context of investigating sea-ice export sensitivities through
- (for details of this study see Heimbach et al., 2007).
+ Lancaster and Jones Sound. The rationale for doing so is to complement
- %\citep[for details of this study see][]{heimbach07}. %Heimbach et al., 2007).
+ the analysis of sea-ice dynamics in the presence of narrow straits.
- The domain chosen is a coarsened version of the Arctic face of the
+ 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
-Line 112 
 The adjoint models run efficiently on 80
+Line 99 
 The adjoint models run efficiently on 80
  (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
+ Following a 3-year spinup, the model has been integrated for four
- years between 1992 and 1995. It is forced using realistic 6-hourly
+ 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
+ The objective function chosen is
- computed for December 1995.  The adjoint model computes sensitivities
+ sea-ice export through
- to sea-ice export back in time from 1995 to 1992 along this
+ 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
+ multipliers) of the coupled ocean/sea-ice model
- analyze the transient sensitivity behaviour of the ocean and sea-ice
+ as well as the surface atmospheric state are available to
- state.  Over the open ocean, the adjoint of the bulk formula scheme
+ 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 ice thickness, $\partial J / \partial heff$.
+ Fig. XXX depcits transient $\partial J / \partial heff$ using free-slip
+ (left column) and no-slip (right column) boundary conditions.
+ Sensitivity snapshots are depicted for (from top to bottom)
+, 24, 36, and 48 months prior to May 2003.
+ The dominant features are in accordance with expectations:
+ (*)
+ 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)
+, 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:
+ \\
+to 5   Db=N/A, Dr=5 (May-Jan) \\
+to 17 Db=7, Dr=5 (Jul-Jan) \\
+to 29 Db=7, Dr=5 (Jul-Jan) \\
+to 41 Db=7, Dr=5 (Jul-Jan) \\
+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
+-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
 , 24, 36, 48 months back in time. Corresponding sensitivities to
-Line 161 
 the circulation around Svalbard, and ...
+Line 346 
 the circulation around Svalbard, and ...
  \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}
  %%% Local Variables:
  %%% mode: latex
  %%% TeX-master: "ceaice"

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