A Differential Approximation Model For Passive Scalar Turbulence

TL;DR

This paper introduces a simple differential approximation model for 2D passive scalar turbulence, successfully reproducing known spectral behaviors and validated through numerical simulations across various injection scenarios.

Contribution

It presents a novel third order differential approximation model that captures key spectral features of 2D passive scalar turbulence and aligns with established phenomenology.

Findings

Steady state solutions recover Kraichnan-Kolmogorov dual cascade.

Passive scalar spectra like Batchelor and Obukhov-Corssin are reproduced.

Numerical solutions confirm the analytical spectral behaviors.

Abstract

Two dimensional passive scalar turbulence is studied by means of a k-space diffusion model based on a third order differential approximation. This simple description of local nonlinear interactions in Fourier space is shown to present a general expression, in line with previous seminal works, and appears to be suitable for various 2D turbulence problems. Steady state solutions for the spectral energy density of the flow is shown to recover the Kraichnan-Kolmogorov phenomenology of the dual cascade, while various passive scalar spectra, such as Batchelor or Obukhov-Corssin spectra are recovered as steady state solutions of the spectral energy density of the passive scalar. These analytical results are then corroborated by numerical solutions of the time evolving problem with energy and passive scalar injection and dissipation on a logarithmic wavenumber space grid over a large range of scales. The particular power law spectra that is obtained is found to depend mainly on the location of the kinetic and passive scalar energy injections.