The limit from an Euler-type system to the 2D Euler equations with Stratonovich transport noise is investigated. A weak convergence result for the vorticity field and a strong convergence result for the velocity field are proved. Our results aim to provide a stochastic reduction of fluid-dynamics models with three different time scales.
Additive noise in Partial Differential equations, in particular those of fluid mechanics, has relatively natural motivations. The aim of this work is showing that suitable multiscale arguments lead rigorously, from a model of fluid with additive noise, to transport type noise. The arguments apply both to small-scale random perturbations of the fluid acting on a large-scale passive scalar and to the action of the former on the large scales of the fluid itself. Our approach consists in studying the (stochastic) characteristics associated to small-scale random perturbations of the fluid, here modelled by stochastic 2D Euler equations with additive noise, and their convergence in the infinite scale separation limit.
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.
We study stochastic model reduction for evolution equations in infinite-dimensional Hilbert spaces and show the convergence to the reduced equations via abstract results of Wong–Zakai type for stochastic equations driven by a scaled Ornstein–Uhlenbeck process. Both weak and strong convergence are investigated, depending on the presence of quadratic interactions between reduced variables and driving noise. Finally, we are able to apply our results to a class of equations used in climate modeling.
We introduce a Gaussian measure formally preserved by the 2dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based on the techniques developed by Gubinelli and Jara in  for a hyperviscous version of the equations.
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