with constant flux can be found. Besides this assumption, it is also assumed that wave fields are homogeneous, and that free waves are uncorrelated.

Weak turbulence theory has been applied to capillary and gravito-capillary waves [@zakharov_kolmogorov_1992], vibrations on a plate [@during_weak_2006], rotating flows [@galtier_weak_2003], and magnetohydrodynamic waves [@galtier_weak_2000; @nazarenko_wave_2011; @schekochihin_weak_2012]. For some of these systems, the predictions of the theory are compatible with results obtained from experiments or from numerical simulations. For example, see Refs. [@deike_decay_2012; @falcon_capillary_2009; @kolmakov_quasiadiabatic_2004] for capillary waves, [@cobelli_different_2011] for gravitocapillary waves, [@mordant_are_2008; @boudaoud_observation_2008; @cobelli_space-time_2009] for vibrations on a plate, and [@leerink_multiscale_2012; @mininni_energy_2007] for magnetohydrodynamic waves. Although agreement has been found between theory, numerical simulations and experiments, there are also discrepancies. In some of these cases the compatibility is limited to the spectrum of certain fields (see, e.g., [@mininni_energy_2007]), or to specific configurations used to generate the excitations. Moreover, for many systems it is also not clear whether the weak turbulence approximation holds for all times, as the solutions are not homogeneous in wavenumber space and at sufficiently small scales eddies may become faster than waves resulting in strong turbulence [@chen_resonant_2005].

One of the most important applications of weak turbulence is in surface gravity waves. In oceanography, 