234
T. Dauxois et al.
or on any range of wave vectors between k min and k max . Note that the accessible range
of wave vectors [k min , k max ] is set by the experimental PIV mesh. The integration in
Eq. (9) has been done in Fig. 9 with an integration range which represents 84% of
the energy in the entire range [k min , k max ]. The linear dispersion relation is seen to
attract the maxima of the energy spectra. Above results are convincing signatures of
a discrete wave turbulence framework for internal waves in this intermediate forcing
amplitude regime.
If we repeat the same experiment with a larger amplitude, we have indications
that the system is beyond the wave turbulence-like regime and has reached a mixing
regime. Indeed, short-scale perturbations in particular clearly escape any relation to
linear wave dynamics. This is expected to be due to overturnings, natural precursors
to mixing.
Mixing Inferred from Vorticity Distribution
An important issue is whether or not sufficiently energetic internal wave motion can
produce an irreversible energy contribution to mixing. Figure 10a presents the comparison between density profiles measured before and after experiments: while no
modification of the density (within experimental error) can be observed for the intermediate amplitude forcing that leads to wave turbulence regime described in the
previous section, one gets a clear evidence of mixing in case of a larger forcing
amplitude.
Further, differences between the regimes corresponding to low and high mixing
are clearly seen in statistics of extreme events. This statistics is obtained by the calculation of probability density functions (PDF). Since we are interested in small-scale
events destabilizing the stratification, we take the horizontal y-component of vorticity í µí¼(x, z, t) = í µí¼u∕í µí¼z − í µí¼w∕í µí¼x measured in the vertical midplane of the test tank
as a relevant quantity and consider the PDF of the dimensionless quantity í µí¼∕N. In
Fig. 10b, we present the vorticity PDFs corresponding to different wave regimes in
the attractor. In a stable attractor (see dash-dotted curve), extreme events are completely absent and the wave motion is concentrated within the relatively narrow
branches of the attractor while the rest of the fluid is quiescent. Accordingly, the
PDF has a sharp peak at zero vorticity and is fully localized between well-defined
maximum and minimum values of vorticity. For larger forcing amplitudes (dashed
and solid curves), the development of TRI increases the probability of extreme events
due to summation of primary and secondary wave components.
The occurrence of local overturning events can be viewed as a competition
between stratification and vorticity. In a two-dimensional flow, a relevant stability parameter is a version of the Richardson number, which can be introduced as
Ri í µí¼ = N 2 ∕í µí¼ 2 . For a horizontal stratified shear flow this parameter reduces to the conventional gradient Richardson number Ri = N
2 ∕(du∕dz)
2
, where du∕dz is the velocity shear. Flows with large Ri are generally stable, and the turbulence is suppressed
by the stratification. The classic Miles-Howard necessary condition for instability
T. Dauxois et al.
or on any range of wave vectors between k min and k max . Note that the accessible range
of wave vectors [k min , k max ] is set by the experimental PIV mesh. The integration in
Eq. (9) has been done in Fig. 9 with an integration range which represents 84% of
the energy in the entire range [k min , k max ]. The linear dispersion relation is seen to
attract the maxima of the energy spectra. Above results are convincing signatures of
a discrete wave turbulence framework for internal waves in this intermediate forcing
amplitude regime.
If we repeat the same experiment with a larger amplitude, we have indications
that the system is beyond the wave turbulence-like regime and has reached a mixing
regime. Indeed, short-scale perturbations in particular clearly escape any relation to
linear wave dynamics. This is expected to be due to overturnings, natural precursors
to mixing.
Mixing Inferred from Vorticity Distribution
An important issue is whether or not sufficiently energetic internal wave motion can
produce an irreversible energy contribution to mixing. Figure 10a presents the comparison between density profiles measured before and after experiments: while no
modification of the density (within experimental error) can be observed for the intermediate amplitude forcing that leads to wave turbulence regime described in the
previous section, one gets a clear evidence of mixing in case of a larger forcing
amplitude.
Further, differences between the regimes corresponding to low and high mixing
are clearly seen in statistics of extreme events. This statistics is obtained by the calculation of probability density functions (PDF). Since we are interested in small-scale
events destabilizing the stratification, we take the horizontal y-component of vorticity í µí¼(x, z, t) = í µí¼u∕í µí¼z − í µí¼w∕í µí¼x measured in the vertical midplane of the test tank
as a relevant quantity and consider the PDF of the dimensionless quantity í µí¼∕N. In
Fig. 10b, we present the vorticity PDFs corresponding to different wave regimes in
the attractor. In a stable attractor (see dash-dotted curve), extreme events are completely absent and the wave motion is concentrated within the relatively narrow
branches of the attractor while the rest of the fluid is quiescent. Accordingly, the
PDF has a sharp peak at zero vorticity and is fully localized between well-defined
maximum and minimum values of vorticity. For larger forcing amplitudes (dashed
and solid curves), the development of TRI increases the probability of extreme events
due to summation of primary and secondary wave components.
The occurrence of local overturning events can be viewed as a competition
between stratification and vorticity. In a two-dimensional flow, a relevant stability parameter is a version of the Richardson number, which can be introduced as
Ri í µí¼ = N 2 ∕í µí¼ 2 . For a horizontal stratified shear flow this parameter reduces to the conventional gradient Richardson number Ri = N
2 ∕(du∕dz)
2
, where du∕dz is the velocity shear. Flows with large Ri are generally stable, and the turbulence is suppressed
by the stratification. The classic Miles-Howard necessary condition for instability