2.10 Some exact relationships

So far, we have been discussing about the inertial range of turbulence. What this means from a heuristic point of view is somewhat clear, but from an operational point of view this would be rather obscure. In this regard, a very important result on turbulence, due to Kolmogorov (1941

), is the so called “ $4/5$ -law” which, being obtained from the Navier-Stokes equations, is ”…one of the most important results in fully developed turbulence because it is both exact and nontrivial” (cf. Frisch, 1995

). Under the hypothesis of homogeneity, isotropy, and, in the limit of infinite Reynolds number, assuming that the turbulent flow has a finite nonzero mean dissipation energy rate $e$ (cf. Frisch, 1995

), the third-order velocity structure function behaves linearly with $l$ , namely

Following a similar approach developed by Yaglom (1949 ), Politano and Pouquet (1995 ) derived an exact relation, from MHD equations, for the third-order correlator involving Elsässer variables:

Both Equations (26

, 27

) can be used, or better, in a certain sense they might be used, as a formal definition of inertial range. Since they are exact relationships derived from Navier-Stokes and MHD equations under usual hypotheses, they represent a kind of “zeroth-order” conditions on experimental and theoretical analysis of the inertial range properties of turbulence. Using Equation (27

) for $± Yl$ , or better a little different form (Politano and Pouquet, 1995

), namely $< > [dz± ]2 |dz ±| l l$ , as a formal definition, these authors found that an inertial range is observed in numerical simulations (Politano et al., 1998a ). At odds with numerical simulations, Equation (27

) is not easily verified if applied to solar wind data since an extended range is not clearly defined. However, we will come back to this important point in the future version of this review, where we will show how $± Yl$ functions behave in the solar wind.

As far as the shell model is concerned, the existence of a cascade towards small scales is expressed by an exact relation, which is equivalent to Equation (27 ). Using Equations (20 ) the scale-by-scale pseudo-energy budget is given by

d sum ± 2 [ ±] sum 2 ± 2 sum [ ± ±*] -- |Z n| = knIm T n - 2nk n| Z n| + 2 R e Zn Fn . dt n n n

The second and third terms on the right hand side present, respectively, the rate of pseudo-energy dissipation and the rate of pseudo-energy injection. The first term represents the flux of pseudo-energy along the wave vectors, responsible for the redistribution of pseudo-energies on the wave vectors, and is given by

( ) ± ± ± ± 2---a---c ± ± ± Tn = (a + c)Z n Zn+1Z n+2 + c Z n-1Zn+1Z n + ± ± ± (c - a) ± ± ± + (2- a- c)Z n Z n+2Z n+1 + ----- Z)Z n Z n+1Z n-1. (28) c

Using the same assumptions as before, namely: i) the forcing terms act only on the largest scales, ii) the system can reach a statistically stationary state, and iii) in the limit of fully developed turbulence, $n --> 0$ , the mean pseudo-energy dissipation rates tend to finite positive limits $± e$ , it can be found that

This is an exact relation which is valid in the inertial range of turbulence. Even in this case it can be used as an operative definition of the inertial range in the shell model, that is, the inertial range of the energy cascade in the shell model is defined as the range of scales $kn$ , where the law from Equation (29

) is verified.