Stochastic Modelling of Small-Scale Perturbation

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.

Stochastic modelling in fluid dynamics: Itô versus Stratonovich

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations.