Spectral scaling of the Leray-$\alpha$ model for two-dimensional turbulence

TL;DR

This paper investigates the spectral scaling behavior of the Leray-$eta$ model in two-dimensional turbulence, confirming that the energy spectrum scaling depends on the dominant cascading quantity's characteristic time scale.

Contribution

It demonstrates that the spectral scaling of the Leray-$eta$ model aligns with theoretical predictions based on the dominant cascading quantity's time scale, extending understanding from previous models.

Findings

Leray-$eta$ model exhibits a $k^{-5}$ energy spectrum scaling for $keta o ext{large}$.

Spectral scaling depends on the characteristic time scale of the dominant cascade.

Results support the hypothesis that the cascading quantity determines spectral scaling.

Abstract

We present data from high-resolution numerical simulations of the Navier-Stokes-$\alpha$ and the Leray-$\alpha$ models for two-dimensional turbulence. It was shown previously (Lunasin et al., J. Turbulence, 8, (2007), 751-778), that for wavenumbers $k$ such that $k\alpha\gg 1$, the energy spectrum of the smoothed velocity field for the two-dimensional Navier-Stokes-$\alpha$ (NS-$\alpha$) model scales as $k^{-7}$. This result is in agreement with the scaling deduced by dimensional analysis of the flux of the conserved enstrophy using its characteristic time scale. We therefore hypothesize that the spectral scaling of any $\alpha$-model in the sub-$\alpha$ spatial scales must depend only on the characteristic time scale and dynamics of the dominant cascading quantity in that regime of scales. The data presented here, from simulations of the two-dimensional Leray-$\alpha$ model, confirm our hypothesis. We show that for $k\alpha\gg 1$, the energy spectrum for the two-dimensional Leray-$\alpha$ scales as $k^{-5}$, as expected by the characteristic time scale for the flux of the conserved enstrophy of the Leray-$\alpha$ model. These results lead to our conclusion that the dominant directly cascading quantity of the model equations must determine the scaling of the energy spectrum.