Scale-wise coherent vorticity extraction for conditional statistical modeling of homogeneous isotropic two-dimensional turbulence

Abstract

Classical statistical theories of turbulence have shown their limitations, in that they cannot predict much more than the energy spectrum in an idealized setting of statistical homogeneity and stationarity. We explore the applicability of a conditional statistical modeling approach: can we sort out what part of the information should be kept, and what part should be modeled statistically, or, in other words, “dissipated”? Our mathematical framework is the initial value problem for the two-dimensional (2D) Euler equations, which we approximate numerically by solving the 2D Navier–Stokes equations in the vanishing viscosity limit. In order to obtain a good approximation of the inviscid dynamics, we use a spectral method and a resolution going up to 8192 2. We introduce a macroscopic concept of dissipation, relying on a split of the flow between coherent and incoherent contributions: the coherent flow is constructed from the large wavelet coefficients of the vorticity field, and the incoherent flow from the small ones. In previous work, a unique threshold was applied to all wavelet coefficients, while here we also consider the effect of a scale by scale thresholding algorithm, called scale-wise coherent vorticity extraction. We study the statistical properties of the coherent and incoherent vorticity fields, and the transfers of enstrophy between them, and then use these results to propose, within a maximum entropy framework, a simple model for the incoherent vorticity. In the framework of this model, we show that the flow velocity can be predicted accurately in the L 2 norm for about 10 eddy turnover times.