The turbulent stress spectrum in the inertial and subinertial ranges

TL;DR

This paper derives the relationship between turbulent energy spectra and stress spectra for velocity and magnetic fields, revealing how spectral slopes behave across different turbulence regimes and discussing astrophysical implications.

Contribution

It provides a numerical solution linking energy spectra to stress spectra in turbulence, including effects of helicity and different spectral slopes, with applications to astrophysics and cosmology.

Findings

Stress spectrum follows Kolmogorov slope for Kolmogorov energy spectra.

Shallower spectra lead to stress spectra approaching white noise.

Helical fields increase stress spectrum in the subinertial range.

Abstract

For velocity and magnetic fields, the turbulent pressure and, more generally, the squared fields such as the components of the turbulent stress tensor, play important roles in astrophysics. For both one and three dimensions, we derive the equations relating the energy spectra of the fields to the spectra of their squares. We solve the resulting integrals numerically and show that for turbulent energy spectra of Kolmogorov type, the spectral slope of the stress spectrum is also of Kolmogorov type. For shallower turbulence spectra, the slope of the stress spectrum quickly approaches that of white noise, regardless of how blue the spectrum of the field is. For fully helical fields, the stress spectrum is elevated by about a factor of two in the subinertial range, while that in the inertial range remains unchanged. We discuss possible implications for understanding the spectrum of primordial gravitational waves from causally generated magnetic fields during cosmological phase transitions in the early universe. We also discuss potential diagnostic applications to the interstellar medium, where polarization and scintillation measurements characterize the square of the magnetic field.