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To detect the frequency triplets, we use the bispectrum analysis which measures the extent of statistical dependence among three spectral components (𝛺 k ,
𝛺 i , 𝛺 j ) satisfying the relationship 𝛺 k = 𝛺 i + 𝛺 j , with the quantity M(𝛺 i , 𝛺 j ) =
F(𝛺 i )F(𝛺 j )F ∗ (𝛺 i + 𝛺 j ), where F is the Fourier transform and ∗ denotes the complex conjugate. In practice, the bispectrum is usually normalized and considered in
form of bicoherence which is 0 for triplets with random phases and 1 for triplets
with perfect phase coupling. The bicoherence is shown in Fig. 8b. In addition to
the strong peak (0.61, 0.61) corresponding to the forcing frequency (therefore to
self-correlation), the possible triplets satisfying the definition of triadic resonance
at 𝛺 k = 𝛺 0 can be found on the line with slope −1 connecting the points (0, 0.61)
and (0.61, 0). This emphasizes that the mechanism at play is triadic. Other peaks
are also visible corresponding to other choices of 𝛺 k revealing that the instability
mechanism is repeated and leads to a cascade.
Thanks to this beautiful representation, it can therefore be attested that the energy
transfer from global to small scales in attractors operates via a hierarchy of triadic
interactions producing a complex internal wave field with a rich multi-peak discrete
frequency spectrum embedded in a continuous spectrum of weaker magnitude.
The bicoherence demonstration of the cascade is closely related to the bispectrum
shown in Fig. 8b. Both axis represent frequencies, and the color on the bispectrum
diagram is proportional to the product of the amplitudes. From this picture it can be
also be easily seen that the amplitude of the daughter waves tend to lie on the antidiagonals. On very long time intervals the picture shown in Fig. 8b can be changed
significantly due to slow evolution of the nonlinear interactions to a completely new
regime.
A Route Towards Wave Turbulence
It is important to emphasize that the final stage is non-trivial since these phenomena
are beyond the domain of pure wave-wave interactions: it corresponds to a regime
usually called wave turbulence [4]. A similar situation takes place for surface waves,
where the flourishing literature gives a fully consistent description of energy cascades between components of wave spectra, only in the case of weakly nonlinear
processes, while experimental reality deals with cascades significantly “contaminated” by effects of a finite size fluid domain, wave breaking, wave cusps, nonlinear
dispersion, viscous damping of wave-field components, etc. The very specific dispersion relation for internal waves introduces additional complications. For instance,
in rotating fluids, which have a dispersion relation analogous to stratified fluids, the
usefulness of the formalism of wave turbulence as a basis for the studies in rotating
turbulence has been reported for experiments only recently [21]. The three dimensional structure of wave attractor and transition to wave turbulence in a rotating annular frustum was recently described in [22]. For internal waves, the question is still
fully open, from both experimental and numerical points of view.