Emergence of long-range correlations and thermal spectra in forced turbulence

TL;DR

This paper explains how long-range correlations and thermal spectra emerge in forced turbulence, linking them to momentum diffusion and invariants, and discusses conditions affecting the spectral shape.

Contribution

It demonstrates that long-range correlations and a thermal spectrum result from turbulent momentum diffusion, extending up to a slowly growing maximum scale, and clarifies the role of forcing in spectral development.

Findings

Thermal spectra arise from turbulent diffusion of linear momentum.

Long-range correlations extend up to a maximum scale growing with time.

Saffman's decay laws apply during decay after forcing stops.

Abstract

Recent numerical studies have shown that forced, statistically isotropic turbulence develops a `thermal equilibrium' spectrum, $\mathcal{E}(k) \propto k^2$, at large scales. This behaviour presents a puzzle, as it appears to imply the growth of a non-zero Saffman integral, which would require the longitudinal velocity correlation function, $\chi(r)$, to satisfy $\chi(r\to \infty)\propto r^{-3}$. As is well known, the Saffman integral is an invariant of decaying turbulence, precisely because non-local interactions (i.e., interactions via exchange of pressure waves) are too weak to generate such correlations. Subject to certain restrictions on the nature of the forcing, we argue that the same should be true for forced turbulence. We show that long-range correlations and a $k^2$ spectrum arise as a result of the turbulent diffusion of linear momentum, and extend only up to a maximum scale that grows slowly with time. This picture has a number of interesting consequences. First, if the forcing generates eddies with significant linear momentum (as in so-called Saffman turbulence), a thermal spectrum is not reached - instead, a shallower spectrum develops. Secondly, the energy of turbulence that is forced for a while and then allowed to decay obeys Saffman's decay laws for a period that is much longer than the duration of the forcing stage.