On the formulation of sea-ice models. Part 2: Lessons from multi-year adjoint sea-ice export sensitivities through the Canadian Arctic Archipelago
Introduction
This is the second part of a two-part paper (see Losch et al., 2010, for Part 1) describing the development of a sea-ice model for use in adjoint-based regional and global coupled ocean/sea-ice state estimation and sensitivity studies. It has been shown (e.g., Marotzke et al., 1999, Galanti et al., 2002, Galanti and Tziperman, 2003, Köhl, 2005, Bugnion et al., 2006a, Bugnion et al., 2006b, Losch and Heimbach, 2007, Moore et al., 2009, Veneziani et al., 2009) that adjoints are very valuable research tools to investigate sensitivities of key model diagnostics with respect to a wide variety of model inputs. Furthermore, increasing sophistication of global-scale as well as regional, polar state estimation systems, which attempt to synthesize observations and models (e.g., Miller et al., 2006, Duliere and Fichefet, 2007, Lisaæter et al., 2007, Stark et al., 2008, Stoessel, 2008, Panteleev et al., 2010) call for adequate representation of sea-ice in the model so as to represent relevant processes and to incorporate sea-ice observations in constraining the coupled system. The estimation system developed within the Estimating the Circulation and Climate of the Ocean (ECCO) consortium is based on the adjoint or Lagrange multiplier method (LMM) (e.g., Wunsch, 2006). It thus relies heavily on the availability of an adjoint model of the underlying general circulation model (Stammer et al., 2002a, Wunsch and Heimbach, 2007, Wunsch et al., 2008, Heimbach, 2008, and references therein).
Collectively, the lack, until recently, of an interactive sea-ice component in the ECCO approach, the experience gained (and the success) with the ocean-only problem, the importance of representing polar–subpolar interactions in ECCO-type calculations, and the need to incorporate sea-ice observations, make a compelling case for the development of a new sea-ice model. While many of its features are “conventional” (yet for the most part state-of-the-art), the ability to generate efficient adjoint code for coupled ocean/sea-ice simulations by means of automatic (or algorithmic) differentiation (AD: Griewank and Walther, 2008) sets this model apart from existing models. Whereas a few existing models (Kim et al., 2006a, Kim et al., 2006b) allow for the generation of tangent linear code for sea-ice-only model configurations by means of the so-called forward-mode AD, until very recently none of these were capable of producing efficient adjoint code by means of reverse-mode AD, let alone in a coupled ocean/sea-ice configuration, which can propagate sensitivities back and forth between the two components. Such coupled sensitivity propagation is highly desirable as it permits sea-ice and ocean observations to be used as simultaneous constraints on each other, yielding a truly coupled estimation problem.
In addition to the coupled ocean and sea-ice system described here, one other coupled adjoint system has recently become available for an Arctic configuration and was used to isolate dominant mechanisms responsible for the 2007 Arctic sea-ice minimum (Kauker et al., 2009). The availability of two adjoint modeling systems holds the prospect (for the first time) to compare adjoint calculations for a specific regional setup using different models. This is a proposed future objective within the Arctic Ocean Model Intercomparison Project (AOMIP).
The MITgcm sea-ice model was described in detail in Part 1. It borrows many components from current-generation sea-ice models, but these components were reformulated on an Arakawa C grid in order to match the MITgcm oceanic grid, and they were modified in many ways to permit efficient and accurate automatic differentiation. Part 1 provided a detailed discussion of the effect on the solution of various choices in the numerical implementation, in particular related to sea-ice dynamics. Such sensitivities are structural or configuration-based, rather than exploring a continuous space of control variables, and are best assessed in separate forward calculations. Special emphasis was put on aspects of the sea-ice dynamics, such as the use of different solvers for sea-ice rheology, the formulation of these solvers on an Arakawa B vs C grid, and the use of free-slip vs no-slip lateral boundary conditions. These scenarios provide important baseline trajectories for the adjoint calculations presented here, as they underscore the importance of the underlying state, around which the model is linearized.
Part 2 focusses on the adjoint component, its generation by means of AD, its reliability, and on the interpretability of adjoint variables. We investigate sensitivities of sea-ice transport through narrow straits, for which rheology configurations become crucial, and the dependence of adjoint sensitivities on the choices of configuration elements described in Part 1. The power of the adjoint is demonstrated through a case study of sea-ice transport through the Canadian Arctic Archipelago (CAA) measured in terms of its export through Lancaster Sound. Thereby we complement a recent study by Lietaer et al. (2008) that focused on the role of narrow straits in this region in setting the sea-ice mass balance in the Arctic. While Part 1 of the present paper showed that different grids, different rheologies, and different lateral boundary conditions lead to considerable differences in the computed sea-ice state, here we show that adjoint sensitivities may differ substantially depending on the baseline trajectory, around which the model is linearized. The present analysis provides important complementary information to the configuration sensitivities of Part 1: it enables us to extend analysis to continuous parameters, it demonstrates the degree of detail the adjoint variables contain, and it exposes causal relationships.
The remainder of Part 2 is organized as follows: Section 2 provides some details of the adjoint code generation by means of AD. Multi-year transient sensitivities of sea-ice export through the Canadian Arctic Archipelago are presented in Section 3. Extending the analysis of Part 1, we assess the consequences of the choices of lateral boundary conditions on the ensuing model sensitivities for various control variables. Discussion and conclusions are in Section 4.
Section snippets
MITgcm adjoint code generation
There is now a growing body of literature on adjoint applications in oceanography and adjoint code generation via AD. We therefore limit the description of the method to a brief summary. For discrete problems as considered here, the adjoint model operator (ADM) is the transpose of the Jacobian or tangent linear model operator (TLM) of the full (in general nonlinear) forward model (NLM), in this case, the MITgcm coupled ocean and sea-ice model. Consider a scalar-valued model diagnostics,
A case study: sensitivities of sea-ice export through Lancaster Sound
We demonstrate the power of the adjoint method in the context of investigating sea-ice export sensitivities through Lancaster Sound (LS). The rationale for this choice is to complement the analysis of sea-ice dynamics in the presence of narrow straits of Part 1. LS is one of the main paths of sea-ice export through the Canadian Arctic Archipelago (CAA) (Melling, 2002, Prinsenberg and Hamilton, 2005, Michel et al., 2006, Münchow et al., 2006, Kwok, 2006). Fig. 2 shows the intricate local
Discussion and conclusion
In this study, we have extended the MITgcm adjoint modeling capabilities to a coupled ocean and sea-ice configuration. The key development is a dynamic and thermodynamic sea-ice model akin to most state-of-the-art models but that is amenable to efficient, exact, parallel adjoint code generation via automatic differentiation. At least two natural lines of applications are made possible by the availability of the adjoint model: (i) use of the coupled adjoint modeling capabilities for
Acknowledgements
This work is a contribution to the ECCO2 project sponsored by the NASA Modeling Analysis and Prediction (MAP) program and to the ECCO-GODAE project sponsored by the National Oceanographic Partnership Program (NOPP). DM carried out this work at JPL/Caltech under contract with NASA. Computing resources were provided by NASA/ARC, NCAR/CSL, and JPL/SVF. Careful reviews by the two anonymous reviewers significantly improved the readability of the paper and are gratefully acknowledged.
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