Abstract
In this paper we propose a stochastic model reduction procedure for deterministic equations from geophysical fluid dynamics. Once large-scale and small-scale components of the dynamics have been identified, our method consists in modelling stochastically the small scales and, as a result, we obtain that a transport-type Stratonovich noise is sufficient to model the influence of the small scale structures on the large scales ones. This work aims to contribute to motivate the use of stochastic models in fluid mechanics and identifies examples of noise of interest for the reduction of complexity of the interaction between scales. The ideas are presented in full generality and applied to specific examples in the last section.
Highlights
This work deals with stochastic models in fluid mechanics
In this paper we propose a stochastic model reduction procedure for deterministic equations from geophysical fluid dynamics
It claims that stochastic models may reduce the complexity of interaction between scales and the noise arising from such a reduction is not the classical additive noise added to the equations in most of the literature, but a multiplicative one of transport type, described here and in related works
Summary
This work deals with stochastic models in fluid mechanics. The literature on the subject is very large, but it is mostly of theoretical nature. From the numerical point of view, especially when one is interested in the simulations of complex turbulent flows like weather forecast, one necessarily has to deal with the fact that limited computational power often implies an under-representation of the real physical processes with spatial or temporal scale smaller than a certain threshold, typically the length of the grid parametrisation and the time discretisation step These small scale processes may have a non-trivial impact on the large scales ones, and it is important to take this impact into the account in order to obtain accurate description of the evolution of the simulated process, see in [1] and the references therein. The special form of the limiting equation (stochastic PDE with Stratonovich transport noise) gives access to a vast range of results and techniques from stochastic analysis to study some properties of a geophysical system like, for instance, the existence of invariant measures, ergodicity, Large Deviations estimates for small intensity of the noise, and others
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