# Energy and enstrophy spectra and fluxes for the inertial-dissipation range of two-dimensional turbulence

## TL;DR

This study derives extended spectra and flux expressions for 2D turbulence beyond the inertial range using Pao's conjecture, validated by numerical simulations showing good agreement except at small wavenumbers due to flow unsteadiness.

## Contribution

The paper introduces new spectral and flux models for 2D turbulence that incorporate exponential factors and validates them through numerical simulations.

## Key findings

- Model predictions agree with simulations for most wavenumbers.
- Flux fluctuations at small wavenumbers are due to flow unsteadiness.
- Shell-to-shell energy transfers show forward transfer among neighbors and backward transfer elsewhere.

## Abstract

In this paper, using Pao's conjecture [Y.-H. Pao, Phys. Fluids 11, 1371 (1968)], we derive expressions for the spectra and fluxes of kinetic energy and enstrophy for two-dimensional (2D) forced turbulence that extend beyond the inertial range. In these expressions, the fluxes and the spectra contain additional factors of the exponential form. To validate these model predictions, we perform numerical simulations of 2D turbulence with an external force applied at $k=k_f$ in the intermediate range. The numerical results match with the model predictions, except for the energy and enstrophy fluxes for $k<k_f$, where the fluxes exhibit significant fluctuations. We show that these fluctuations arise due to the unsteady nature of the flow at small wavenumbers. For the $k<k_f$, the shell-to-shell energy transfers computed using numerical data show forward energy transfers among the neighbouring shells, but backward energy transfers for other shells.