Energy spectra and passive tracer cascades in turbulent flows

TL;DR

This paper investigates how the energy spectrum influences the extent of passive tracer cascades in turbulent flows, revealing conditions under which wide cascades can coexist with inverse energy cascades in 2D and 3D turbulence.

Contribution

It provides new theoretical estimates on the extent of tracer cascades considering dual energy spectra and high Schmidt numbers in turbulent flows.

Findings

Tracer cascade range extends up to a logarithmic factor of the wavenumber beyond diffusion dominance.

Wide tracer cascades can coexist with inverse energy cascades at high Grashof numbers in 2D.

The extent of the tracer cascade depends on the Schmidt number and the energy spectrum.

Abstract

We study the influence of the energy spectrum on the extent of the cascade range of a passive tracer in turbulent flows. The interesting cases are when there are two different spectra over the potential range of the tracer cascade (in 2D when the tracer forcing is in the inverse energy cascade range, and in 3D when the Schmidt number Sc is large). The extent of the tracer cascade range is then limited by the width of the range for the shallower of the two energy spectra. Nevertheless, we show that in dimension $d=2,3$ the tracer cascade range extends (up to a logarithm) to $\kappa_{d\text{D}}^{p}$, where $\kappa_{d\text{D}}$ is the wavenumber beyond which diffusion should dominate and $p$ is arbitrarily close to 1, provided Sc is larger than a certain power (depending on $p$) of the Grashof number. We also derive estimates which suggest that in 2D, for Sc${}\sim1$ a wide tracer cascade can coexist with a significant inverse energy cascade at Grashof numbers large enough to produce a turbulent flow.