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field with a rich multi-peak discrete frequency spectrum embedded into a continuous spectrum of weaker magnitude. Convincing evidences of a wave turbulence
framework [4] for internal waves are also provided. Spontaneous summation of the
wave-field components produces moreover a statistically significant amount of extreme
overturning events which eventually lead to a well-measurable mixing. This suggests
that such a set-up is appropriate to study abyssal mixing in the laboratory.
Let us consider a stratified fluid with an initially constant buoyancy frequency
N = [(−g∕ ̄
𝜌)(d𝜌∕dz)]
1∕2 , where 𝜌(z) is the density distribution over the vertical coordinate z, ̄
𝜌, and g the gravity acceleration. The dispersion relation for linear internal
gravity waves is given by
𝜃 = ± arcsin 𝛺,
(1)
where 𝜃 is the slope of the wave beam to the horizontal, and 𝛺 (resp. 𝜔 = 𝛺N) the
non-dimensional (resp. dimensional) frequency of oscillations. This anisotropic dispersion relation requires preservation of the slope of the internal wave beam upon
reflection at a rigid boundary. As we will recall below, in the case of a sloping boundary, this property gives a purely geometric reason for a strong variation of the width
of internal wave beams (focusing or defocusing) upon reflection. Internal wave focusing provides a necessary condition for large shear and overturning, as well as shear
and bottom layer instabilities at slopes.
In a confined fluid domain, focusing usually prevails, leading to a concentration of wave energy on a closed loop, the internal wave attractor [5]. At the level
of linear mechanisms, the width of the attractor branches is set by the competition
between geometric focusing and viscous broadening. (see [6] for a precise and interesting of this question). High concentration of energy at attractors make them prone
to triadic resonance instability which sets in as the energy injected into the system
increases [7]. Note that the particular case for which both secondary waves have a
frequency equal to half of the forcing frequency is of special interest in the oceanographic context where viscosity is negligible. In that case, the appropriate name is
parametric subharmonic instability and abbreviated as PSI. By abuse of language,
some authors have sometimes extended the use of the name PSI to cases for which
secondary waves are not corresponding to half of the forcing frequency. For the sake
of terminological consistency, we propose to abbreviate triadic resonance instability
using the acronym TRI.
The onset of instability is similar to the classic concept of triadic resonance, which
is best studied for the idealized case, with monochromatic in time and space carrier
wave as a basic state which feeds two secondary waves via nonlinear resonant interactions. The resonance occurs when temporal 𝛺 1 + 𝛺 2 = 𝛺 0 and spatial 𝐤 𝟏 + 𝐤 𝟐 = 𝐤 𝟎
conditions are satisfied (k is the wave vector while subscripts 0, 1 and 2 refer to the
primary, and two secondary waves, respectively). In a wave attractor, the wave beams
serve as a primary wave, and the resonance conditions are satisfied with a good accuracy [7], providing a consistent physical framework for the short-term behavior of
the instability.
The usual theory for the TRI does not take into account the finite width of the
experimental beam. Qualitatively, the subharmonic waves can serve as an energy