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Empirical Model Reduction and the Modeling Hierarchy in Climate Dynamics and the Geosciences

Modern climate dynamics uses a two-fisted approach in attacking and solving the problems of atmospheric and oceanic flows. The two fists are: (i) observational analyses; and (ii) simulations of the geofluids, including the coupled atmosphere–ocean system, using a hierarchy of dynamical models. These models represent interactions between many of processes that act on a broad range of spatial and time scales, from a few kilometers to tens of thousands, and from diurnal to multi-decadal, respectively. The evolution of virtual climates simulated by the most detailed and realistic models in the hierarchy is typically as difficult to interpret as that of the actual climate system, based on the available observations thereof. Highly simplified models of weather and climate, though, help gain a deeper understanding of a few isolated processes, as well as clues on how the interaction between these processes and the rest of the climate system may participate in shaping climate variability. Finally, models of intermediate complexity, which resolve well a subset of the climate system and parameterize the remainder of the processes or scales of motion, serve as a conduit between the models at the two ends of the hierarchy. We present here a methodology for constructing intermediate models based almost entirely on the observed evolution of selected climate fields, without reference to dynamical equations that may govern this evolution; these models parameterize unresolved processes as multivariate stochastic forcing. This methodology may be applied with equal success to actual observational data sets, as well as to data sets resulting from a high-end model simulation. We illustrate this methodology by its applications to: (i) observed and simulated low-frequency variability of atmospheric flows in the Northern Hemisphere; (ii) observed evolution of tropical sea-surface temperatures; and (iii) observed air–sea interaction in the Southern Ocean. Similar results have been obtained for (iv) radialdiffusion model simulations of Earth’s radiation belts, but are not included here because of space restrictions. In each case, the reduced stochastic model represents surprisingly well a variety of linear and nonlinear statistical properties of the resolved fields. Our methodology thus provides efficient means of constructing reduced, numerically inexpensive climate models. These models can be thought of as stochastic–dynamic prototypes of more complex deterministic models, as in examples (i) and (iv), but work just as well in the situation when the actual governing equations are poorly known, as in (ii) and (iii). These models can serve as competitive prediction tools, as in (ii), or be included as stochastic parameterizations of certain processes within more complex climate models, as 1 Dept. of Mathematical Sciences, Atmospheric Science Group, University of WisconsinMilwaukee, P. O. Box 413, Milwaukee, WI 53201. E-mail: kravtsov@uwm.edu . 2 Dept. of Atmospheric and Oceanic Sciences & Institute of Geophysics and Planetary Physics, University of California at Los Angeles, Los Angeles, CA 90095. 3 Geosciences Department & Laboratoire de Météorologie Dynamique (CNRS and IPSL), Ecole Normale Supérieure, F-75231 Paris Cedex 05, France