Appendix: Wave Turbulence with sources and sinks

One of the central discoveries in Wave Turbulence was the power-law Kolmogorov-Zakharov (KZ) spectrum, $n^{kz} \sim k^\nu$, which realise themselves in presence of the energy sources and sinks separated by a large inertial range of scales. The exponent $\nu$ depends on the scaling properties of the interaction coefficient and the frequency. Most of the previous WT literature is devoted to study of KZ spectra and a good review of these works can be found in [1]. We are not going review these studies here, but instead we are going to find out how adding such energy sources and sinks will this modify the evolution equations for the statistics. Instead of the Hamiltonian equation (1) let us consider

$$i\dot c_l = \frac{\partial {\cal H}}{\partial \bar c_l} + i \tilde \gamma_l c_l,$$ (92)

where $\tilde \gamma_l$ describes sources and sinks of the energy, e.g. due to instability and viscosity respectively. Easy to see that this linear term will not change the structure of the $N$-mode PDF equation (65) but it will lead to re-definition of the flux:

$${\cal F}_j \to {\cal F}_j - \tilde \gamma_j s_j P.$$ (93)

In the one-mode equations, this simply means renormalisation

$$\gamma \to \gamma - \tilde \gamma.$$ (94)

Thus we arrive at a simple message that the energy sources and sinks do not produce any ``sources'' or ``sinks'' for the flux of probability.

In the inertial range, there is no flux modification and one can easily find $F=0$ solution of (90) for the one-mode PDF,

$$P^{(a)}_{j} = (1/n_j^{kz}) \exp(-s_j /n_j^{kz}),$$ (95)

where $n_j^{kz}$ is the KZ spectrum (solving the kinetic equation in the inertial range). However, it is easy to check by substitution that the product of such one-mode PDF's, ${\cal P} = \Pi_j (1/n_j^{kz}) \exp(-s_j /n_j^{kz})$, is not an exact solution to the multi-mode equation (65). Thus, there have to be corrections to this expression related either to a finite flux, an amplitude correlation or both.