There has been a recent burst of activity in the atmosphere-ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction. Here a systematic mathematical strategy for stochastic climate modeling is developed, and some of the new phenomena in the resulting equations for the climate variables alone are explored. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. All of these phenomena are derived from a systematic self-consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit. The theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated. The feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography. Explicit concrete examples with the new phenomena are presented for the stochastically forced three-mode interaction equations. The conjecture of Smith and Waleffe [Phys. Fluids 11 (1999), 1608-1622] for stochastically forced three-wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach. Examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self-consistency of the mathematical approach with the predictions of equilibrium statistical mechanics. In particular, for these examples, the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables.
There has been a recent burst of activity in the atmosphere/ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction. Here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory. This analysis demonstrates the potential need for more sophisticated models beyond stable linear Langevin equations. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. The strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta-plane with topography and a mean flow. In this example, the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom.A n area with great importance for future developments in climate prediction involves simplified stochastic modeling of nonlinear features of the coupled atmosphere/ocean system. The practical reasons for such needs are easy to understand. In the foreseeable future, it will be impossible to resolve the effects of the coupled atmosphere/ocean system through computer models with detailed resolution of the atmosphere on decadal time scales. However, the questions of interest also change. For example, for climate prediction, one is not interested in whether there is a significant deflection of the storm track northward in the Atlantic during a specific week in January of a given year, but rather, whether the mean and variance of the storm track are large during several years of winter seasons and what is the impact of this trend on the overall pole-ward transport of heat in both the atmosphere and ocean. The idea of simplified stochastic modeling for unresolved space-time scales in climate modeling is over 20 years old and emerged from fundamental papers by Hasselman (1) and Leith (2). In the atmosphere/ocean community, there is a recent flourishing of ideas utilizing simple stable linear Langevin stochastic equations to model and predict short-term and decadal climate changes such as El Niño (3, 4), the North Atlantic Oscillation (5, 6), and mid-latitude storm tracks (7-9) with notable positive results but also failure of this simplified stochastic model in some circumstances (10).Here we introduce and analyze several idealized models for stochastic climate modeling and utilize unambiguous mathematical theory to demonstrate the potential need for more sophisticated stochastic models beyond those developed in earlier works (3-10). In particular, explicit examples demonstrate that simple Langevin models can emerge in stochastic climate modeling with an unstable climate mean and, in appropriate circumstances, stochastic models need to incorporate both suitable nonli...
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