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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)$,

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

next up previous
Next: Discussions Up: Results Previous: Nonlinearly active modes and
Dr Yuri V Lvov 2007-01-16