We study the dynamics of internal gravity waves excited by parametric
instability in a
stably stratified medium, either at the interface between a water and a
kerosene layer,
or in brine with a uniform gradient of salinity. The tank has a rectangular
section, and
is narrow to favour standing waves with motion in the vertical plane. The
fluid
container undergoes vertical oscillations, and the resulting modulation
of the apparent
gravity excites the internal waves by parametric instability.Each internal wave mode is amplified for an excitation frequency close
to twice its
natural frequency, when the excitation amplitude is sufficient to overcome
viscous
damping (these conditions define an ‘instability tongue’ in
the parameter space
frequency-amplitude). In the interfacial case, each mode is well separated
from the
others in frequency, and behaves like a simple pendulum. The case of a
continuous
stratification is more complex as different modes have overlapping instability
tongues.
In both cases, the growth rates and saturation amplitudes behave as predicted
by the
theory of parametric instability for an oscillator. However, complex friction
effects are
observed, probably owing to the development of boundary-layer instabilities.In the uniformly stratified case, the excited standing wave is unstable
via a secondary
parametric instability: a wave packet with small wavelength and half the
primary wave
frequency develops in the vertical plane. This energy transfer toward a
smaller scale
increases the maximum slope of the iso-density surfaces, leading to local
turning and
rapid growth of three-dimensional instabilities and wave breaking. These
results
illustrate earlier stability analyses and numerical studies. The combined
effect of the
primary excitation mechanism and wave breaking leads to a remarkable intermittent
behaviour, with successive phases of growth and decay for the primary wave
over long
timescales.
Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows
