P. Bouruetaubertot et al., BREAKING OF STANDING INTERNAL GRAVITY-WAVES THROUGH 2-DIMENSIONAL INSTABILITIES, Journal of Fluid Mechanics, 285, 1995, pp. 265-301
The evolution of an internal gravity wave is investigated by direct nu
merical computations. We consider the case of a standing wave confined
in a bounded (square) domain, a case which can be directly compared w
ith laboratory experiments. A pseudo-spectral method with symmetries i
s used. We are interested in the inertial dynamics occurring in the li
mit of large Reynolds numbers, so a fairly high spatial resolution is
used (129(2) or 257(2)), but the computations are limited to a two-dim
ensional vertical plane. We observe that breaking eventually occurs, w
hatever the wave amplitude: the energy begins to decrease after a give
n time because of irreversible transfers of energy towards the dissipa
tive scales. The life time of the coherent wave, before energy dissipa
tion, is found to be proportional to the inverse of the amplitude squa
red, and we explain this law by a simple theoretical model. The wave b
reaking itself is preceded by a slow transfer of energy to secondary w
aves by a mechanism of resonant interactions, and we compare the resul
ts with the classical theory of this phenomenon: good agreement is obt
ained for moderate amplitudes. The nature of the events leading to wav
e breaking depends on the wave frequency (i.e. on the direction of the
wave vector); most of the analysis is restricted to the case of fairl
y high frequencies. The maximum growth rate of the inviscid wave insta
bility occurs in the limit of high wavenumbers. We observe that a well
-organized secondary plane wave packet is excited. Its frequency is ha
lf the frequency of the primary wave, corresponding to an excitation b
y a parametric instability. The mechanism of selection of this remarka
ble structure, in the limit of small viscosities, is discussed. Once t
his secondary wave packet has reached a high amplitude, density overtu
rning occurs, as well as unstable shear layers, leading to a rapid tra
nsfer of energy towards dissipative scales. Therefore the condition of
strong wave steepness leading to wave breaking is locally attained by
the development of a single small-scale parametric instability, rathe
r than a cascade of wave interactions. This fact may be important for
modelling the dynamics of an internal wave field.