Abyssal Mixing in the Laboratory
223
sink for the primary wave if they do not leave the primary beam before they can
extract substantial energy [8]. The group velocity of the primary wave is aligned
with the beam, but the group velocity of the secondary waves is definitely not, and
these secondary waves eventually leave the primary wave beam. This is a direct
consequence of the dispersion relation (1), which relates the direction of propagation
to the frequency: a different frequency, smaller for subharmonic waves, will lead to
a shallower angle. The generalization to wave beams with a finite width is presented
in detail in a recent review [9].
The Internal Wave Attractor to Enhance the Nonlinearities
In this section, we present the concept of internal wave attractor that is a key element
to provide large amplitude internal wave beams produced thanks to the focusing
properties of internal wave reflections.
Reflection of Internal Waves: A Focusing Mechanism
The dispersion relation of internal waves is very specific and leads to a very unusual
reflection on a sloping boundary that has interesting properties, central for our objective as it will be immediately clear. To be more specific, let us consider an inviscid
linearly stratified fluid of constant buoyancy frequency N and a sloping boundary,
tilted with an angle í µí»¼ with respect to the vertical, as shown in Fig. 1. Note that, if
this configuration does not seem natural for the reflection of internal waves on the
topography at the bottom of the ocean, it corresponds by symmetry to a case with
negative values of í µí»¼, but is simpler for an experimental realization.
As the pulsation of the wave is conserved during the reflection, both the incident
and reflected waves propagate with the same angle í µí¼, according to the dispersion
relation (1). It is worth to note that this is very different from the reflection in optics or
acoustics where the electromagnetic or sound waves conserve the angle with respect
to the normal to the sloping boundary, refereed usually as the classical Descartes
reflection. For internal waves, this is the angle with the gravity that is conserved.
This difference is illustrated in Fig. 1a. The reflected ray for optics or acoustics is the
dashed arrow while the one for internal waves is the solid arrow.
This non-Descartes reflection is even more intriguing, and therefore interesting,
when one considers a beam, and not only a ray. This is shown in Fig. 1b. The width
of the reflected beam is thus reduced and one gets an energy focusing for these internal waves. It is important to emphasize that this phenomenon being a direct consequence of the linear dispersion relation, one has identified here a linear transfer
toward smaller scales.
223
sink for the primary wave if they do not leave the primary beam before they can
extract substantial energy [8]. The group velocity of the primary wave is aligned
with the beam, but the group velocity of the secondary waves is definitely not, and
these secondary waves eventually leave the primary wave beam. This is a direct
consequence of the dispersion relation (1), which relates the direction of propagation
to the frequency: a different frequency, smaller for subharmonic waves, will lead to
a shallower angle. The generalization to wave beams with a finite width is presented
in detail in a recent review [9].
The Internal Wave Attractor to Enhance the Nonlinearities
In this section, we present the concept of internal wave attractor that is a key element
to provide large amplitude internal wave beams produced thanks to the focusing
properties of internal wave reflections.
Reflection of Internal Waves: A Focusing Mechanism
The dispersion relation of internal waves is very specific and leads to a very unusual
reflection on a sloping boundary that has interesting properties, central for our objective as it will be immediately clear. To be more specific, let us consider an inviscid
linearly stratified fluid of constant buoyancy frequency N and a sloping boundary,
tilted with an angle í µí»¼ with respect to the vertical, as shown in Fig. 1. Note that, if
this configuration does not seem natural for the reflection of internal waves on the
topography at the bottom of the ocean, it corresponds by symmetry to a case with
negative values of í µí»¼, but is simpler for an experimental realization.
As the pulsation of the wave is conserved during the reflection, both the incident
and reflected waves propagate with the same angle í µí¼, according to the dispersion
relation (1). It is worth to note that this is very different from the reflection in optics or
acoustics where the electromagnetic or sound waves conserve the angle with respect
to the normal to the sloping boundary, refereed usually as the classical Descartes
reflection. For internal waves, this is the angle with the gravity that is conserved.
This difference is illustrated in Fig. 1a. The reflected ray for optics or acoustics is the
dashed arrow while the one for internal waves is the solid arrow.
This non-Descartes reflection is even more intriguing, and therefore interesting,
when one considers a beam, and not only a ray. This is shown in Fig. 1b. The width
of the reflected beam is thus reduced and one gets an energy focusing for these internal waves. It is important to emphasize that this phenomenon being a direct consequence of the linear dispersion relation, one has identified here a linear transfer
toward smaller scales.