Weak turbulence for inhomogeneous systems

Now, by analogy with homogeneous weak turbulence, we define the waveaction spectrum as

$$n_{\vec k,\vec x} = \langle \vert a(\vec k,\vec x) \vert^2 \rangle /F(0),$$

where averaging is performed over the random initial phases. Note that this definition is slightly different to the usual definition of the turbulence spectrum in homogeneous turbulence, i.e. the definition constructed from Fourier transforms, $n_{\vec k} \, \delta(\vec k- \vec k') = \langle a(\vec k) a(\vec k') \rangle$. Indeed, a Gabor transform can be viewed as a finite-box Fourier transform, where $\vec k= \vec k'$ in the definition of the spectrum and one replaces $\delta(\vec k- \vec k')$ with the box volume $F(0)$.

Multiplying () by $a^*(\vec k,\vec x)$ and combining the resulting equation with its complex conjugate, we get a generalization of ():

$$D_t n_{\vec k,\vec x} = -2 \Im \int F (\vec k+\vec k_1-\vec k_2-\vec k_3)$$

$$\times\langle a^*(\vec k,\vec x) a^*(\vec k_1,\vec x) a(\vec k_2,\vec x) a(\vec k_3,\vec x)\rangle \, d {\bf k_1} d {\bf k_2} d {\bf k_3},$$

with $D_{t}\equiv \partial_{t} + \dot{\vec{x}}\cdot\nabla +\dot{\vec{k}}\cdot\partial_{k}$. Note, that in the case of homogeneous turbulence, using the random phase assumption, in the above equation, would lead to the RHS becoming zero. This means that the nontrivial kinetic equation appears only in higher orders of the nonlinearity. For the inhomogeneous case, the nontrivial effect of the nonlinearity appears even at this (second) order. This can be seen via a frequency correction which, in turn, modifies the wave trajectories. This effect was considered by Zakharov et al [31] and it is especially important in systems where such frequency corrections result in modulational instabilities followed by collapsing events. In our case the nonlinearity is ``defocusing'' and, therefore, such an effect is less important. Indeed, in what follows we will neglect this effect as, at sufficiently small ratios of the inhomogeneity and turbulence intensity parameters, $\epsilon \ll \phi^2$, wave collision events are a far more dominant process.

Let us introduce notations

$$I^{k k_1}_{k_2 k_3} \equiv \langle a^*(\vec k,\vec x) a^*(\vec k_1,\vec x) a(\vec k_2,\vec x) a(\vec k_3,\vec x)\rangle,$$

and

$$I^{k k_1 k_2}_{k_3 k_4 k_5} \equiv \langle a^*(\vec k,\vec x) a^... ...c k_2,\vec x) a(\vec k_3,\vec x) a(\vec k_4,\vec x) a(\vec k_5,\vec x)\rangle.$$

Then, we have the following equation for the 4th-order moment,

$$D_t I^{k_1' k_2'}_{k_3' k_4'} = i(\tilde\omega_{k'_1} + \tilde\omega_{k_2'} - \tilde\omega_{k_3'} - \tilde\omega_{k_4'}) I^{k'_1 k_2'}_{k_3' k_4'}$$

$$+ \int \Big(I^{k_2 k_3 k_2'}_{k_1 k_3' k_4'} F(\vec k_1'+\vec k_1-\vec k_2 -\vec k_3)$$

$$+ I^{k_1' k_2 k_3 }_{k_1 k_3' k_4'} F(\vec k_2'+\vec k_1 -\vec k_2-\vec k_3)$$

$$- I^{k_1' k_2' k_1}_{k_4' k_2 k_3} F(\vec k_3' +\vec k_1-\vec k_2-\vec k_3)$$

$$- I^{k_1' k_2' k_1}_{k_3' k_2 k_3} F(\vec k_4'+\vec k_1-\vec k_2-\vec k_3) \, \Big)d k_1 d k_2 d k_3,$$ (34)

where we denote $\tilde\omega_{k}=k^2 + (\vec k\cdot \nabla_x U)$. Note that the first two terms on the RHS of this equation can be obtained one from another by exchanging $\vec k_1'$ and $\vec k_2'$, whereas the last two terms - by exchanging $\vec k_3'$ and $\vec k_4'$. To solve this equation, one can use the random phase assumption which is standard for the derivation of a weak homogeneous turbulence theory and which allows one to express the 6th-order moment in terms of the 2nd-order correlators. For homogeneous turbulence, the validity of this assumption was examined by Newell et al [18,33] who showed that initially Gaussian turbulence (characterized by random independent phases) remains Gaussian for the energy cascade range whereas in the particle cascade range deviations from Gaussianity grow toward low $k$ values. However, these deviations remain small over a large range of $k$ for small initial amplitudes and the random phase assumption can be used for these scales. Note that the deviations from Gaussianity at low $k$ correspond to the physical process of building a coherent condensate state. The results of [18,33] obtained for homogeneous GP turbulence will hold for trapped turbulence too because inhomogeneity has a neutral effect on the phase correlations. Indeed, according to the linear WKB equations the phases propagate unchanged along the rays. Thus we write

$$I^{123}_{456} \approx n_1 n_2 n_3 \Big( F^{3}_{4}(F^{2}_{5}F^{1}_{6} + F^{1}_{5}F^{2}_{6}) + F^{3}_{5} (F^{2}_{6}F^{1}_{4} + F^{2}_{4}F^{1}_{6})$$

$$+ F^{3}_{6} (F^{1}_{5}F^{2}_{4} + F^{1}_{4}F^{2}_{5}) \Big),$$

here we have used the shorthand notations, $F^{1}_{2}\equiv F(0) \, \delta(\vec k_1-\vec k_2)$ and $I^{123}_{456}=I^{k_1 k_2 k_3}_{k_4 k_5 k_6}$. Using this expression in () we have

$$\frac{D}{D t} I^{k_1 k_2}_{k_3 k_4} = i (\omega_{k_1}+\omega_{k_2} - \omega_{k_3} - \omega_{k_4}) I^{k_1 k_2}_{k_3 k_4}$$

$$+ 2 \left(n_{k_3} n_{k_4}(n_{k_1} + n_{k_2}) - n_{k_1} n_{k_2}(n_{k_3} + n_{k_4})\right).$$

Notice that the $\tilde \omega_k$ terms get replaced by $\omega_k$, since the $(\vec k\cdot \nabla_x U)$ terms drop out on the resonant manifold. Let us integrate this equation over the period $T$ which is less than both the slow WKB time $1/\epsilon$ and the nonlinear time $1/\sigma^4$. Then, one can ignore the time dependence in $n_k$ on the RHS of the above equation and we can take $\dot{k} = - \nabla U = const$ on the LHS.

The resulting equation can be easily integrated along the characteristics (rays) which in the limit $\omega T \to \infty$ gives

$$I^{k_1 k_2}_{k_3 k_4} = - 2 [ n_{k_3} n_{k_4}(n_{k_1} n_{k_2}) - n_{k_1} n_{k_2}(n_{k_3} +n_{k_4}) ]$$

$$\delta(\omega_{k_1}+\omega_{k_2} - \omega_{k_3} - \omega_{k_4}).$$ (35)

Note that to derive a similar expression in the theory of homogeneous weak turbulence one usually introduces an artificial ``dissipation'' to circumvent the pole and to get the correct sign in front of the delta function (see e.g. [14]). The roots of this problem can be found even at the level of the linear dynamics, where the use of Laplace (rather than Fourier) transforms provides a mathematical justification for the introduction of such a dissipation. However, in our case there is no need for us to introduce such a dissipation because inhomogeneity removes the degeneracy in the system. Substituting () into () we get the main equation describing weak turbulence, the four-wave kinetic equation

$$D_{t}n_{k} = \frac{1}{\pi} \int n_k n_1 n_2 n_3 \left( \frac{1}{n_{k}} + \frac{1}{n_{1}} - \frac{1}{n... ...}{n_{3}} \right) \delta \left(\vec{k} +\vec{k}_1 - \vec{k}_2 - \vec{k}_3\right)$$

$$\delta \left(\omega_k({\bf x}) + \omega_1({\bf x}) - \omega_2({\bf x}) - \omega_3({\bf x})\right) \, d \vec{k}_1 d \vec{k}_2 d \vec{k}_3,$$

where,

$$D_{t}\equiv \partial_{t} + \dot{\vec{x}}\cdot\nabla +\dot{\vec{k}}\cdot\partial_{k}, \dot{\vec{x}} = \partial_{k}\omega, \,\,\,\,\,\,\, \dot{\vec{k}} = -\nabla\omega.$$

We can see that the main difference between the kinetic equation for inhomogeneous media and homogeneous turbulence [11,12,13,22] is that the partial time derivative on the LHS is replaced by the full time derivative along the rays. Further, the frequency $\omega$ and spectrum $n$ are now functions not only of the wavenumber but also of the coordinate.

The same is true for the case when the ground state condensate is important for the wave dynamics [13]. The main interaction mechanism now become three wave interactions, with the kinetic equation

$$D_t n = \pi \int \vert V_{k k_1 k_2}\vert^2 \, f_{k12} \, \delta_{{{\bf k... ...bf k}} -\omega_{{\bf {k_1}}}-\omega_{{\bf {k_2}}}} d {\bf k}_{1} d {\bf k}_2 \,$$

$$- 2\pi\int \, \vert V_{k_1 k k_2}\vert^2\, f_{1k2}\, \delta_{{{\bf ... ...}} -\omega_{{\bf {k}}}-\omega_{{\bf {k_2}}}}} \, d {\bf k}_{1} d {\bf k}_2 \, ,$$ (36)

where $f_{k12} = n_{{\bf k_1}}n_{{\bf k_2}} - n_{{\bf k}}(n_{{\bf k_1}}+n_{{\bf k_2}}) \,$. Here, $n_k$, $D_t$ and $\omega$ are given by expressions (), () and () respectively and the expression for the interaction coefficient $V_{k k_1 k_2}$ can be found in [13]. Three-wave interactions always dominate over the four-wave process when $\rho \sim k^2$ (because $k \sim 1$ and $n \ll 1)$. In the case $\rho \ll k^2$, the relative importance of the three-wave and the four-wave processes can be established by comparing the characteristic times associated with these processes. The characteristic time of the three wave interactions for $\rho \ll k^2$ is

$$\tau_{3w} = k^{2-d}/\varrho n.$$

Thus, the 3-wave process will dominate the 4-wave one if the condensate is stronger than the waves, i.e. if $\varrho > n k^d \sim \phi^2$.