Numerical solutions of the isotropic 3-wave kinetic equation

We show that the isotropic 3-wave kinetic equation is equivalent to the mean-field rate equations for an aggregation–fragmentation problem with an unusual fragmentation mechanism. This analogy is used to write the theory of 3-wave turbulence almost entirely in terms of a single scaling parameter. A new numerical method for solving the kinetic equation over a large range of frequencies is developed by extending Lee’s method for solving aggregation equations. The new algorithm is validated against some analytical calculations of the Kolmogorov–Zakharov (KZ) constant for some families of model interaction coefficients. The algorithm is then applied to study some wave turbulence problems in which the finiteness of the dissipation scale is an essential feature. Firstly, it is shown that for finite capacity cascades, the dissipation of energy becomes independent of the cut-off frequency as this cut-off is taken to infinity. This is an explicit indication of the presence of a dissipative anomaly. Secondly, a preliminary numerical study is presented of the so-called bottleneck effect in a wave turbulence context. It is found that the structure of the bottleneck depends non-trivially on the interaction coefficient. Finally, some results are presented on the complementary phenomenon of thermalisation in closed wave systems which demonstrates explicitly for the first time the existence of so-called mixed solutions of the kinetic equation which exhibit aspects of both KZ and equilibrium equipartition spectra.