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

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