BREAKING OF STANDING INTERNAL GRAVITY-WAVES THROUGH 2-DIMENSIONAL INSTABILITIES

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.