Wave turbulence is a set of
waves deviated far from
thermal equilibrium. Such state is accompanied by
dissipation. It is either
decaying turbulence or requires external source of
energy to sustain it. Examples are waves on a
fluid surface excited by
winds or
ships, and waves in
plasma excited by
electromagnetic waves etc.
External sources by some resonant mechanism usually excite waves with
frequencies and
wavelengths in some narrow interval. For example, shaking container with the frequency ω excites surface waves
with the frequency ω/2 (parametric resonance discovered by
Michael Faraday).
When wave amplitudes are small (which usually means that the wave is far from breaking) 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 to interact. That leads to an excitation of waves with frequencies and wavelengths in wide intervals, not necessarily in resonance with an external source. It can be observed in the experiments with a high amplitude of shaking that initially the waves appear which are in resonance, then 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 an inverse cascade of 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 inertail interval is important.
The subject of DWT, first introduced in , 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 corresponding NR-diagram (nonlinear resonance diagram) first introduced in.
In some wave turbulent systems both discrete and statistical layers of turbulence are observed simultaneously, this wave turbulent regime have been first discovered in and 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. Their interrelation is described in.
Notes
References
Category:Non-linear systems
Category:Water waves
Category:Oceanography
