Incontinuum mechanics,wave turbulence is a set ofnonlinearwaves deviated far fromthermal equilibrium. Such a state is usually accompanied bydissipation. It is eitherdecaying turbulence or requires an external source ofenergy to sustain it. Examples are waves on afluid surface excited bywinds orships, and waves inplasma excited byelectromagnetic waves etc.
External sources by some resonant mechanism usually excite waves withfrequencies andwavelengths in some narrow interval. For example, shaking a container with frequency ω excites surface waveswith frequency ω/2 (parametric resonance, discovered byMichael Faraday). When waveamplitudes are small – which usually means that the wave is far frombreaking – only those waves exist that are directly excited by an external source.
When, however, wave amplitudes are not very small (for surface waves: when the fluid surface is inclined by more than few degrees) waves with different frequencies start tointeract. That leads to an excitation of waves with frequencies and wavelengths in wide intervals, not necessarily in resonance with an external source. In experiments with high shaking amplitudes one initially observes waves that are inresonance with one another. Thereafter, both longer and shorter waves appear as a result of wave interaction. The appearance of shorter waves is referred to as a direct cascade while longer waves are part of aninverse cascade of wave turbulence.
Two generic types of wave turbulence should be distinguished:statistical wave turbulence (SWT) anddiscrete wave turbulence (DWT).
In SWT theoryexact 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 byVladimir E. Zakharov. These solutions are calledKolmogorov–Zakharov (KZ) energy spectra and have the formk−α, withk thewavenumber and α a positive constant depending on the specific wave system.[1] The form of KZ-spectradoes not depend on the details of initial energy distribution over the wave field 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 inKartashova (2006), are exact and quasi-resonances. Previous to the two-layer model of wave turbulence, the standard counterpart of SWT werelow-dimensioned systems characterized bya small number of modes included. However, DWT is characterized byresonance clustering,[2] 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).[3]
In some wave turbulent systems both discrete and statistical layers of turbulence are observedsimultaneously, this wave turbulent regime have been described inZakharov et al. (2005) and is calledmesoscopic. Accordingly, three wave turbulent regimes can be singled out—kinetic, discrete and mesoscopic described by KZ-spectra, resonance clustering and their coexistence correspondingly.[4]Energetic behavior of kinetic wave turbulent regime is usually described byFeynman-typediagrams (i.e.Wyld's diagrams), while NR-diagrams are suitable for representing finite resonance clusters in discrete regime and energy cascades in mesoscopic regimes.
