Wave Turbulence - Statistical Wave Turbulence and Discrete Wave Turbulence

Statistical Wave Turbulence and Discrete Wave Turbulence

Two generic types of wave turbulence should be distinguished: statistical wave turbulence (SWT) and discrete wave turbulence (DWT). In SWT theory exact and quasi-resonances are omitted, which allows using some statistical assumptions and describing the wave system by kinetic equations and their stationary solutions – the approach developed by Vladimir E. Zakharov. These solutions are called Kolmogorov–Zakharov (KZ) energy spectra and have the form k−α, with k the wavenumber and α a positive constant depending on the specific wave system. The form of KZ-spectra does not depend on the details of initial energy distribution over the wavefield or on the initial magnitude of the complete energy in a wave turbulent system. Only the fact the energy is conserved at some inertial interval is important.

The subject of DWT, first introduced in Kartashova (2006), are exact and quasi-resonances. Previous to the two-layer model of wave turbulence, the standard counterpart of SWT were low-dimensioned systems characterized by a small number of modes included. However, DWT is characterized by resonance clustering, and not by the number of modes in particular resonance clusters – which can be fairly big. As a result, while SWT is completely described by statistical methods, in DWT both integrable and chaotic dynamics are accounted for. A graphical representation of a resonant cluster of wave components is given by the corresponding NR-diagram (nonlinear resonance diagram).

In some wave turbulent systems both discrete and statistical layers of turbulence are observed simultaneously, this wave turbulent regime have been described in Zakharov et al. (2005) and is called mesoscopic. Accordingly, three wave turbulent regimes can be singled out—kinetic, discrete and mesoscopic described by KZ-spectra, resonance clustering and their coexistence correspondingly. Energetic behavior of kinetic wave turbulent regime is usually described by Feynmann-type diagrams (i.e. Wyld's diagrams), while NR-diagrams are suitable for representing finite resonance clusters in discrete regime and energy cascades in mesoscopic regimes.