Fluctuation theorem and extended thermodynamics of turbulence

Amilcare Porporato; Milad Hooshyar; Andrew D Bragg; Gabriel Katul

doi:10.1098/rspa.2020.0468

Fluctuation theorem and extended thermodynamics of turbulence

Proc Math Phys Eng Sci. 2020 Nov;476(2243):20200468. doi: 10.1098/rspa.2020.0468. Epub 2020 Nov 18.

Authors

Amilcare Porporato^{1

2}, Milad Hooshyar^{1

2

3}, Andrew D Bragg⁴, Gabriel Katul^{1

4

5}

Affiliations

¹ Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA.
² Princeton Environmental Institute and Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA.
³ Princeton Environmental Institute and Princeton Institute for International and Regional Studies, Princeton University, Princeton, NJ, USA.
⁴ Department of Civil and Environmental Engineering, Duke University, Durham, NC, USA.
⁵ Nicholas School of the Environment, Duke University, Durham, NC, USA.

Abstract

Turbulent flows are out-of-equilibrium because the energy supply at large scales and its dissipation by viscosity at small scales create a net transfer of energy among all scales. This energy cascade is modelled by approximating the spectral energy balance with a nonlinear Fokker-Planck equation consistent with accepted phenomenological theories of turbulence. The steady-state contributions of the drift and diffusion in the corresponding Langevin equation, combined with the killing term associated with the dissipation, induce a stochastic energy transfer across wavenumbers. The fluctuation theorem is shown to describe the scale-wise statistics of forward and backward energy transfer and their connection to irreversibility and entropy production. The ensuing turbulence entropy is used to formulate an extended turbulence thermodynamics.

Keywords: cascade; fluctuation theorem; turbulence.