Numerical models of the climate system play an important role in efforts to understand past climate variability and predict future climate changes. In many studies, climate models are driven by forcing fields that are either time-independent or that vary periodically (seasonally) and it is often highly desirable to obtain equilibrium solutions of the model. Existing methods, based on the simple expedient of integrating the model until the transients have died out, are too expensive to use routinely because the deep ocean takes several thousand years to equilibrate. The principal objective of this project is to develop a practical and efficient method for computing equilibrium solutions of periodically forced ocean general circulation models (OGCMs). The general approach will be to formulate the problem as a large system of nonlinear algebraic equations to be solved with a class of methods known as matrix-free Newton-Krylov, a combination of Newton-type methods for superlinearly convergent solution of nonlinear equations, and Krylov subspace methods for solving the Newton correction equations. To render this approach practical for global models with order (107) degrees of freedom, novel matrix free preconditioning strategies will be developed. The "matrix-free" nature of the proposed approach makes it extremely flexible, allowing its use with any ocean or climate model. The method can be applied to models forced at any period, including those driven by time-independent forcing, although the main focus here is the seasonal cycle. Preliminary results suggest that this scheme can accelerate the spin up of seasonally forced OGCMs by over two orders of magnitude over current practice. The convergence properties of this technique will be analyzed, and its efficiency assessed against traditional "acceleration" methods. While the primary target is ocean climate models with a nominal resolution of one , the method will also be applied to the next generation of higher resolution models, including eddy permitting ones. The technique will be applied to obtain equilibrium solutions for various forcing estimates for both present day climate from ocean reanalysis products, and that of the Last Glacial Maximum.
Intellectual merit: The slow dynamical adjustment timescale of the deep ocean is one of the principal obstacles to our ability to make more effective use of climate models. The proposed study will address this fundamental problem in climate simulation by developing practical algorithms for efficiently computing equilibrium solutions of seasonally forced OGCMs. A direct outcome of this research will be improved estimates of the circulation of both the modern ocean, and that of the Last Glacial Maximum.
Broader Impacts: By greatly reducing the computational cost of obtaining equilibrium solutions of climate models, this research will allow scientists to address questions of scientific and societal relevance that are currently unfeasible. These questions include systematic parameter sensitivity studies and simulations of paleoclimate, areas that are especially important for characterizing uncertainties in climate change simulations. A key advantage of the proposed approach is that it makes few assumptions about the underlying ocean or climate model code thus ensuring that the results of this research can be used by the widest possible group of researchers. This work is directly relevant to ongoing work in the areas of ocean circulation, paleoceanography, and ocean biogeochemistry. More broadly, while the specific objective is to address the ocean spin up problem, the computation of periodic solutions and limit cycles of systems modeled by partial differential equations is a very general one, and the proposed method is likely to have broad applicability in other disciplines. This research will contribute to the training and education of a graduate student. Numerical code developed as part of this research will be made freely available to the research community. Findings of this study will be published in journal articles and presented at conference meetings.
