--- MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex 2008/03/25 22:04:31 1.3 +++ MITgcm_contrib/articles/ceaice/ceaice_adjoint.tex 2008/06/04 13:34:41 1.4 @@ -3,57 +3,53 @@ \subsection{The adjoint of MITsim} - -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, -as evidenced by various adjoint-based studies -(for a recent summary, see \cite{heim:08}). - -The adjoint model operator (ADM) is the transpose of the tangent linear -model operator (TLM) -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, 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 efficient adjoint model is available. - -Following closely the development and maintenance of the -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 \citep{gier-kami:98}. -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. -Apart from its evident success, advantages of this approach have been -pointed out, e.g. by \cite{gier-kami:98}. - -Many issues underlying the efficient exact adjoint sea-ice code generation -are similar to those arising for the ocean model's adjoint. -Linearizing the model around the exact nonlinear model trajectory, -as we do, is a crucial aspect in the presence of different -regimes (e.g. effect of the seaice growth term at or away from the -freezing point of the ocean surface). -Adjusting the (parent) model code to support the AD tool in -providing exact and efficient adjoint code is the main initial work. -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. +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!] @@ -106,7 +102,7 @@ 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 +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. @@ -129,12 +125,12 @@ \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 +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 in accordance with expectations: +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. @@ -337,6 +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}]