Numerical solutions of the isotropic 3-wave kinetic equation

Abstract

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 analytic calculations of the Kolmogorov-Zakharov 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 Kolmogorov-Zakharov and equilibrium equipartition spectra.

Collective Oscillations in Irreversible Coagulation Driven by Monomer Inputs and Large-Cluster Outputs

“…In particular, the kinetic equation for isotropic 3-wave turbulence, which is closely analogous to Eq. (1), becomes nonlocal when j À j > 3 [23]. Furthermore, the oscillatory behavior discussed in this Letter may even have been already observed experimentally in measurements of nonequilibrium phase separation of binary mixtures with slowly ramped temperature [24,25].…”

The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

“…As a result of these very recent developments [5,6,8,9,[14][15][16][17] (see also Ref. [18] for a recent review), the last few years have seen a furthering in our understanding of the possible existence of equilibrium solutions in inviscid equations of hydrodynamics.…”

Anisotropic Wave Turbulence for Reduced Hydrodynamics with Rotationally Constrained Slow Inertial Waves

“…of Equation (43) for the concerned power law solution [7,30,36]. However, analyzing the convergence of the collision integral is a delicate task [52] and a rigorous commentary is not within the scope of this paper. However, in this section, we will argue on the convergence of the collision integral given by the r.h.s.…”

“…For a rigorous analysis of convergence of collision integrals in a three wave system, one may refer to the work by Connaughton [52]. Additionally, it is important to note that analysis of stability and evolutionary non-locality of concerned spectra subject to extra perturbative forcing in this anisotropic limit is beyond the scope of the current paper and the interested reader is referred to the work of Balk and Nazarenko [36] and Nazarenko [7] for details.…”