Exercise 4.

For the spectral energy density we have the Wiener-Khinchin formula (Frisch, 4.58),

$$E(k)=\frac{1}{\pi}\int_0^\infty kx\Gamma(x)\sin (kx) dx$$ (1)

where

$$\Gamma(x)=\langle \overline{v}(\overline{r})\cdot \overline{v... ...verline{r'})\rangle, \, x=\vert\overline{r}-\overline{r'}\vert$$ (2)

is the correlation function. Assume the scaling relation, $\Gamma(x) \sim x^n$ and derive a scaling relation for E(k). Eksperimentally it is foound that n=2/3.

For the advanced student:

Derive (1) from the 3 dimensional spectral energy density.