Correlations of phases, phase factors and amplitudes.

WT closure relies on the RPA properties of the wave fields, i.e. that the amplitudes $A_k$ and the phase factors $\psi_k$ are statistically independent variables. On the other hand, WT calculation for the phases $\phi_k$ shows that these quantities get correlated. In order to check these properties and predictions numerically, let us introduce a function that measures the degree of statistical dependence (or independence) of some Fourier-space variables $X({\bf k}_1)$ and $Y({\bf k}_2)$,

$${\cal C}_{X,Y} ({\bf k}_1, {\bf k}_2) = \frac{ \langle X({\bf k}_1) Y({\bf k}_2) \rangle - \langle X({\bf... ...2 } \sqrt{ \langle Y^2({\bf k}_2) \rangle - \langle Y({\bf k}_2) \rangle^2 } }.$$ (31)

For example, we can examine to what degree amplitudes $A$ and independent of the phase factors $\psi$ by looking at the function ${\cal C}_{A,\psi} ({\bf k}_1, {\bf k}_2)$ for different values of ${\bf k}_1$ and ${\bf k}_2$. Independence of the amplitudes at different wavenumbers can be examined by the auto-correlation function ${\cal C}_{A,A} ({\bf k}_1, {\bf k}_2)$, and similar for the phase factors and the phases. We restrict ourselves with choosing ${\bf k}_1 =(15,0)$ and ${\bf k}_2 =(k,0)$ with $k \in (10,64)$. Figure 15 shows the values of correlators ${\cal C}_{\phi,\phi} ({\bf k}_1, {\bf k}_2)$ and ${\cal C}_{\psi,\psi} ({\bf k}_1, {\bf k}_2)$ as functions of $k$. In agreement with WT predictions, auto-correlations of $\psi_k$'s are very small whereas the ones of $\phi_k$'s are significant (except, of course, for $k=15$ where by definition these correlators are equal to one). Correlators ${\cal C}_{A,A} ({\bf k}_1, {\bf k}_2)$ and ${\cal C}_{A,\psi} ({\bf k}_1, {\bf k}_2)$ are shown in figure 16. Again, we see a good agreement with the WT prediction: these correlations are very small (except, again, ${\cal C}_{A,A}(15,15)=1$).

Figure 15: Two-point auto-correlations for the phases ${\cal C}_{\phi,\phi}({\bf k_1}, {\bf k})$ (dashed line) and for the phase factors ${\cal C}_{\psi,\psi}({\bf k_1}, {\bf k})$ (solid line) with one point fixed at ${\bf k} = (15,0)$.

[width=10cm]Psi_psi_corr.eps

Figure 16: Two-point auto-correlation of amplitudes ${\cal C}_{A,A}({\bf k_1}, {\bf k})$ (thin curve) and two-point correlation between the amplitudes and phase factors ${\cal C}_{A,\psi}({\bf k_1}, {\bf k})$ (thick curve) with one point fixed at ${\bf k} = (15,0)$.

[width=10cm]A_psi_corr.eps