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3    
4  \subsection{The adjoint of MITsim}  \subsection{The adjoint of MITsim}
5    
6    The adjoint model of the MITgcm has become an invaluable
7  The ability to generate tangent linear and adjoint components  tool for sensitivity analysis as well as state estimation \citep[for a
8  of a coupled ocean sea-ice system was one of the main drivers  recent summary, see][]{heim:08}. The code has been developed and
9  behind the MITsim development.  tailored to be readily used with automatic differentiation tools for
10  For the ocean the adjoint capability has proven to be an  adjoint code generation. This route was also taken in developing and
11  invaluable tool for sensitivity analysis as well as state estimation,  adapting the sea-ice compontent MITsim, so that tangent linear and
12  as evidenced by various adjoint-based studies  adjoint components can be obtained and kept up to date without
13  (for a recent summary, see \cite{heim:08}).  excessive effort.
14    
15  The adjoint model operator (ADM) is the transpose of the tangent linear  The adjoint model operator (ADM) is the transpose of the tangent
16  model operator (TLM)  linear model operator (TLM) of the full (in general nonlinear) forward
17  of the full (in general nonlinear) forward model, i.e. the MITsim.  model, in this case the MITsim. This operator computes the gradients
18  It enables very efficient computation of gradients  of scalar-valued model diagnostics (so-called cost function or
19  of scalar-valued model diagnostics  objective function) with respect to many model inputs (so-called
20  (so-called cost function or objective function)  independent or control variables).  These inputs can be two- or
21  with respect to many model inputs (so-called independent or control variables).  three-dimensional fields of initial conditions of the ocean or sea-ice
22  These inputs can be two- or three-dimensional fields of initial  state, model parameters such as mixing coefficients, or time-varying
23  conditions of the ocean or sea-ice state, model parameters such as  surface or lateral (open) boundary conditions.  When combined, these
24  mixing coefficients, or time-varying surface or lateral (open) boundary conditions.  variables span a potentially high-dimensional (e.g.  O(10$^8$))
25  When combined, these variables span a potentially high-dimensional  so-called control space. At this problem dimension, perturbing
26  (e.g. O(10$^8$)) so-called control space. Performing parameter perturbations  individual parameters to assess model sensitivities quickly becomes
27  to assess model sensitivities quickly becomes prohibitive at these scales.  prohibitive. By contrast, transient sensitivities of the objective
28  Alternatively, transient sensitivities of the objective function  function to any element of the control and model state space can be
29  to any element of the  control and model state space can be computed  computed very efficiently in one single adjoint model integration,
30  very efficiently in  one single adjoint  provided an adjoint model is available.
31  model integration, provided an efficient adjoint model is available.  
32    In anology to the TLM and ADM components of the MITgcm we rely on the
33  Following closely the development and maintenance of the  autmomatic differentiation (AD) tool ``Transformation of Algorithms in
34  TLM and ADM components of the MITgcm we have relied heavily on the  Fortran'' (TAF) developed by Fastopt \citep{gier-kami:98} to generate
35  autmomatic differentiation (AD) tool  TLM and ADM code of the MITsim \citep[for details see][]{maro-etal:99,
36  "Transformation of Algorithms in Fortran" (TAF)    heim-etal:05}.  In short, the AD tool uses the nonlinear parent
37  developed by Fastopt \citep{gier-kami:98}.  model code to generate derivative code for the specified control space
38  to derive TLM and ADM code of the MITsim  and objective function. Advantages of this approach have been pointed
39  (for details see \cite{maro-etal:99}, \cite{heim-etal:05}).  out, for example by \cite{gier-kami:98}.
40  Briefly, the nonlinear parent model is fed to the AD tool which produces  
41  derivative code for the specified control space and objective function.  Many issues of generating efficient exact adjoint sea-ice code are
42  Apart from its evident success, advantages of this approach have been  similar to those for the ocean model's adjoint.  Linearizing the model
43  pointed out, e.g. by \cite{gier-kami:98}.  around the exact nonlinear model trajectory is a crucial aspect in the
44    presence of different regimes (e.g., is the thermodynamic growth term
45  Many issues underlying the efficient exact adjoint sea-ice code generation  for sea-ice evaluated near or far away from the freezing point of the
46  are similar to those arising for the ocean model's adjoint.  ocean surface?). Adapting the (parent) model code to support the AD
47  Linearizing the model around the exact nonlinear model trajectory,  tool in providing exact and efficient adjoint code represents the main
48  as we do, is a crucial aspect in the presence of different  work load initially. For legacy code, this task may become
49  regimes (e.g. effect of the seaice growth term at or away from the  substantial, but it is fairly straightforward when writing new code
50  freezing point of the ocean surface).  with an AD tool in mind. Once this initial task is completed,
51  Adjusting the (parent) model code to support the AD tool in  generating the adjoint code of a new model configuration takes about
52  providing exact and efficient adjoint code is the main initial work.  10 minutes.
 This may be substantial for legacy code, but fairly straightforward  
 when coding with "AD application in mind".  
 Once in place, an adjoint model of a new model configuration  
 may be derived in about 10 minutes.  
53    
54  [HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]  [HIGHLIGHT COUPLED NATURE OF THE ADJOINT!]
55    
# Line 106  NCEP/NCAR atmospheric state variables. O Line 102  NCEP/NCAR atmospheric state variables. O
102  converted into air-sea fluxes via the bulk formulae of  converted into air-sea fluxes via the bulk formulae of
103  \citet{large04}.  Derivation of air-sea fluxes in the presence of  \citet{large04}.  Derivation of air-sea fluxes in the presence of
104  sea-ice is handled by the ice model as described in \refsec{model}.  sea-ice is handled by the ice model as described in \refsec{model}.
105  The objective function chosen is  The objective function is chosen $J$ as the
106  sea-ice export through  sea-ice export through
107  Lancaster Sound at XX$^{\circ}$W  Lancaster Sound at XX$^{\circ}$W
108  averaged over an 8-month period between October 1992 and May 1993.    averaged over an 8-month period between October 1992 and May 1993.  
# Line 129  DISCUSS FORWARD STATE, INCLUDING SOME NU Line 125  DISCUSS FORWARD STATE, INCLUDING SOME NU
125  \paragraph{Sensitivities to the sea-ice thickness}  \paragraph{Sensitivities to the sea-ice thickness}
126    
127  The most readily interpretable ice-export sensitivity is that  The most readily interpretable ice-export sensitivity is that
128  to ice thickness, $\partial J / \partial heff$.  to effective ice thickness, $\partial{J} / \partial{h}$.
129  Fig. XXX depcits transient $\partial J / \partial heff$ using free-slip  Fig. XXX depcits transient $\partial{J} / \partial{h}$ using free-slip
130  (left column) and no-slip (right column) boundary conditions.  (left column) and no-slip (right column) boundary conditions.
131  Sensitivity snapshots are depicted for (from top to bottom)  Sensitivity snapshots are depicted for (from top to bottom)
132  12, 24, 36, and 48 months prior to May 2003.  12, 24, 36, and 48 months prior to May 2003.
133  The dominant features are in accordance with expectations:  The dominant features are\ml{ in accordance with expectations/as expected}:
134    
135  (*)  (*)
136  Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.  Dominant pattern (for the free-slip run) is that of positive sensitivities, i.e.
# Line 337  Primary features are the effect of the h Line 333  Primary features are the effect of the h
333  Atlantic current which feeds into the West Spitsbergen current,  Atlantic current which feeds into the West Spitsbergen current,
334  the circulation around Svalbard, and ...  the circulation around Svalbard, and ...
335    
336    
337    \ml{[based on the movie series
338      zzz\_run\_export\_canarch\_freeslip\_4yr\_1989\_ADJ*:]} The ice
339    export through the Canadian Archipelag is highly sensitive to the
340    previous state of the ocean-ice system in the Archipelago and the
341    Western Arctic. According to the \ml{(adjoint)} senstivities of the
342    eastward ice transport through Lancaster Sound (\reffig{arctic_topog},
343    cross-section G) with respect to ice volume (effective thickness), ocean
344    surface temperature, and vertical diffusivity near the surface
345    (\reffig{fouryearadj}) after 4 years of integration the following
346    mechanisms can be identified: near the ``observation'' (cross-section
347    G), smaller vertical diffusivities lead to lower surface temperatures
348    and hence to more ice that is available for export. Further away from
349    cross-section G, the sensitivity to vertical diffusivity has the
350    opposite sign, but temperature and ice volume sensitivities have the
351    same sign as close to the observation.
352    
353  \begin{figure}[t!]  \begin{figure}[t!]
354  \centerline{  \centerline{
355  \subfigure[{\footnotesize -12 months}]  \subfigure[{\footnotesize -12 months}]
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