| Abstract: |
| Transport-type stochastic perturbations are a common way of representing turbulent effects in fluid dynamics models. A recent research thread on the topic is the so-called \emph{It\^o-Stratonovich diffusion limit}. By selecting Stratonovich transport noise with carefully arranged coefficients, one can show that the solution of certain SPDEs are close, in an appropriate topology, to a deterministic equation with an effective second order elliptic operator, linked to the Ito-Stratonovich corrector. In this talk, we deal with viscous a passive scalar model. Starting from [Flandoli \emph{et al.}, 2022, \emph{Philos. Trans. Roy. Soc. A}, 380(2219)], we consider a transport noise made by a sum of independent and compactly supported vector fields.
Due to the anisotropic nature of the noise, the identification of the limit equation is not straightforward as in other examples, as the Ito-Stratonovich corrector is a generic second order elliptic operator with non-constant coefficients. Using tools from Homogenisation theory, we obtain a representation for the limiting effective diffusivity matrix.
Exploiting this representation, we study both analytically and numerically asymptotics, in the zero-viscosity regime, of the effective diffusivity across a number of vector field regimes parametrised by the radius of their support. |
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