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Evolution equations for the PDF's, moments and spectrum.

In [5] the following equation for for the $ N$-mode PDF was obtained for the four-wave systems,

$\displaystyle \dot {\cal P} = { \pi {\epsilon}^2 } \int \vert W^{jl}_{nm}\vert^...
...a s} \right]_4 {\cal P} \right) \, d{\bf k}_j d{\bf k}_l d{\bf k}_m d{\bf k}_n,$ (16)

where

$\displaystyle \left[{\delta \over \delta s} \right]_4 = {\delta \over \delta s_...
...delta \over \delta s_l} - {\delta \over \delta s_m} -{\delta \over \delta s_n}.$ (17)

Here $ N \to \infty
$ limit has already been taken and $ {\delta
\over \delta s_j}$ means the variational derivative. Using this equation, one can prove that RPA property holds over the nonlinear time, i.e. the $ N$-mode PDF remains of the product factorized form with accuracy sufficient for the WT closures to work [5]. Using RPA, we get for the one-point marginals [5],

$\displaystyle {\partial P_a \over \partial t}+ {\partial F \over \partial s_j} =0,$ (18)

with $ F$ is a probability flux in the s-space,

$\displaystyle F=-s_j (\gamma P_a +\eta_j {\delta P_a \over \delta s_j}),$ (19)

where
$\displaystyle \eta_j$ $\displaystyle =$ $\displaystyle 4 \pi \epsilon^2 \int
\vert W^{jl}_{nm}\vert^2 \delta^{jl}_{nm}
\...
...(\omega^{jl}_{nm}) n_l n_m n_n \, d { {\bf k}_l} d { {\bf k}_m} d { {\bf k}_n,}$ (20)
$\displaystyle \gamma_j$ $\displaystyle =$ $\displaystyle 4 \pi \epsilon^2 \int \vert W^{jl}_{nm}\vert^2 \delta^{jl}_{nm}
\...
...n_l (n_m + n_n) - n_m n_n\Big] \, d { {\bf k}_l} d { {\bf k}_m} d { {\bf k}_n.}$ (21)

Here we introduced the wave-action spectrum,

$\displaystyle n_j = \langle A_j^2 \rangle.$ (22)

From (14) we get the following equation for the moments $ M^{(p)}_j = \langle A_j^{2p} \rangle $:

$\displaystyle \dot M^{(p)}_j = -p \gamma_j M^{(p)}_j + p^2 \eta_j M^{(p-1)}_j.$ (23)

which, for $ p=1$ gives the standard wave kinetic equation (WKE),

$\displaystyle \dot n_j = - \gamma_j n_j + \eta_j .$ (24)


next up previous
Next: Preservation of the RPA Up: Theoretical WT predictions Previous: Theoretical WT predictions
Dr Yuri V Lvov 2007-01-16