Jeroen Wouters, Valerio Lucarini
In this paper we consider the problem of disentangling multi-level systems by connecting the seemingly unrelated approaches of the Mori- Zwanzig projection operator technique and of the Ruelle response theory, for which we propose a new derivation. In a previous paper we have shown that by using the Ruelle response theory on a weakly coupled system it is possible to construct a surrogate dynamics for the slow variables, such that the expectation value of any observable agrees, up to second order in the coupling strength, to its expectation evaluated on the full dynamics, where both slow and fast variables are involved. We show here that such surrogate dynamics agrees up to second order to the effective dynamics one can derive by expanding perturbatively the Mori-Zwanzing projection operator, which creates, instead, an accurate representation of the trajec- tories of the slow variables. In the case of e.g. geophysical fluid dynamics, this implies that the parametrizations of unresolved processes suited for prediction (numerical weather forecast) and those suited for the represen- tation of long term statistical properties (climate) are closely related, if one takes into account, in addition to the widely adopted stochastic forc- ing, the usually neglected memory effects. This bears relevance for the current trend of aiming at seamless prediction.
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http://arxiv.org/abs/1208.3080
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