2.8 The phenomenology of fully developed turbulence: Fluid-like case

Here we present the phenomenology of fully developed turbulence, as far as the scaling properties are concerned. In this way we are able to recover a universal form for the spectral pseudo-energy in the stationary case. In real space a common tool to investigate statistical properties of turbulence is represented by field increments $± ± ± dzl (r) = [z (r + l) - z (r)] .e$ , being $e$ the longitudinal direction. These stochastic quantities represent fluctuations⁶ across eddies at the scale $l$ . The scaling invariance of MHD equations (cf. Section 2.3), from a phenomenological point of view, implies that we expect solutions where $dz±l ~ lh$ . All the statistical properties of the field depend only on the scale $l$ , on the mean energy dissipation rate $e±$ , and on the viscosity $n$ . Also, $± e$ is supposed to be the common value of the injection, transfer and dissipation rates. Moreover, the dependence on the viscosity only arises at small scales, near the bottom of the inertial range. Under these assumptions the typical energy dissipation rate per unit mass scales as $( ) e± ~ dz± 2/t± l l$ . The time $t± l$ associated with the scale $l$ is the typical time needed for the energy to be transferred on a smaller scale, say the eddy turnover time $± ± tl ~ l/dzl$ , so that

± ( ±)2 ± e ~ dzl dz /l.

When we conjecture that both $dz±$ fluctuations have the same scaling laws, namely $dz± ~ lh$ , we recover the Kolmogorov scaling for the field increments

Usually, we refer to this scaling as the K41 model (Kolmogorov, 1941

; Frisch, 1995

). Note that, since from dimensional considerations the scaling of the energy transfer rate should be $e± ~ l1-3h$ , $h = 1/3$ is the choice to guarantee the absence of scaling for $e±$ .

In the real space turbulence properties can be described using either the probability distribution functions (PDFs hereafter) of increments, or the longitudinal structure functions, which represents nothing but the higher order moments of the field. Disregarding the magnetic field, in a purely fully developed fluid turbulence, this is defined as $S(p)= <dup > l l$ . These quantities, in the inertial range, behave as a power law $(p) qp S l ~ l$ , so that it is interesting to compute the set of scaling exponent $qp$ . Using, from a phenomenological point of view, the scaling for field increments (see Equation (22 )), it is straightforward to compute the scaling laws $S(pl) ~ lp/3$ . Then $qp = p/3$ results to be a linear function of the order $p$ .

When we assume the scaling law $± h dzl ~ l$ , we can compute the high-order moments of the structure functions for increments of the Elsässer variables, namely $< > (dz±l )p ~ lqp$ , thus obtaining a linear scaling $q = p/3 p$ , similar to usual fluid flows. For Gaussianly distributed fields, a particular role is played by the second-order moment, because all moments can be computed from $(2) S l$ . It is straightforward to translate the dimensional analysis results to Fourier spectra. The spectral property of the field can be recovered from $S(l2)$ , say in the homogeneous and isotropic case

( ) (2) integral oo sin-kl- Sl = 4 0 E(k) 1- kl dk,

where $k ~ 1/l$ is the wave vector, so that in the inertial range where Equation (26 ) is verified

The Kolmogorov spectrum (see Equation (23

)) is largely observed in all experimental investigations of turbulence, and is considered as the main result of the K41 phenomenology of turbulence (Frisch, 1995

). However, spectral analysis does not provide a complete description of the statistical properties of the field, unless this has Gaussian properties. The same considerations can be made for the spectral pseudo-energies $E ±(k)$ , which are related to the $2nd$ order structure functions $<[ ±]2> dzl$ .