A numerical study of the frequency and the energetics of nonlinear internal gravity waves

Abstract

Internal gravity wave motions in a two-dimensional vertical plane of fluid having constant mean Brunt-Vaisala frequency are numerically simulated. The waves are integrated to statistical equilibrium under forcing and dissipation with forcing applied to low wave number modes. The model is used to study the variation of the internal wave energy dissipation rate with the Richardson number Ri, the effect of nonlinear wave interaction on the wave frequency, the wave number spectrum of the buoyancy flux, and the ,balance of the spectral energy flux. The energy dissipation rate is shown by our simulations to vary strongly near Ri • 1 and to increase proportional to Ri-•. The rms frequency fluctuation is found to increase approximately linearly with Ri-•/2 and with the wave number magnitude. The simulations show that the buoyancy flux varies nonuniformly with wave numbers, kinetic to potential energy conversion occurs mostly at low wave numbers, and an opposite conversion occurs in the rest of the wave number space. The mean spectral energy balance in the region where negative conversion occurs is marked by positive (gain) potential energy transfer and negative (loss) kinetic energy transfer in the intermediate wave number range and positive transfer for both kinetic and potential energy in the high wave number range; the latter is balanced by dissipation. The' spectra of kinetic and potential energy are presented. The similarity in the slope of the kinetic energy spectra between the stratified turbulence from our simulation and the two-dimensional unstratified, homogeneous turbulence is noted. small-scale waves. This in turn leads to a drastic increase of ß energy dissipation rate as well as buoyancy flux generation (kinetic to potential energy conversion) in the internal wave field. In this paper, these different effects of small-scale nonlinear wave interaction on the internal wave field are the subject of our investigation.Our first objective will be to investigate the dependence of energy dissipation rate on the strength of wave interaction.The increase of internal wave energy dissipation resulting from the strong wave interaction is believed to affect significantly the internal wave mean energy level and the evolution of internal wave energy spectrum [Munk, 1981' Orlanski andCerasoli, 1981;McComas and Muller, 1981]. The degree of influence depends on how rapidly the dissipation rate varies with the interaction intensity. It has been hypothesized that the dissipation rate increases exponentially with increasing interaction [Munk, 1981]. We shall seek to verify this possible relationship. Our second objective will be to investigate the effect of interaction on the internal wave frequency. The internal wave frequency has a tendency to fluctuate as the interac-• Now at Ocean Dynamics Branch, Paper number 5C0760. 0148-0227/86/005C-0760505.00 tion becomes strong. The fluctuation causes the frequency to depart from the dispersion surface, and this has been shown to modify the well-known resonant wave interaction mechanism...

Nonlinear interactions among internal gravity waves

“…A mechanism for such phase-dependent, preferential energy transfer is not apparent although FB remark that similar oscillatory energy growth at half the standing wave period is also characteristic of the generalized Mathieu equation model for internal wave instability [Mied, 1976;Drazin, 1977;Klostermeyer, 1982]. We now turn to Shen and Holloway [1986]…”

Instability mechanisms of a two-dimensional progressive internal gravity wave

“…Laboratory experiments usually involve standing waves (McEwan 1971;McEwan, Mander & Smith 1972;McEwan & Robinson 1975;Benielli & Sommeria 1998) or partially standing waves (Martin et al 1972;Teoh, Ivey & Imberyer 1997), which are easier to generate in laboratory installations than progressive waves. In most numerical works reported in the literature, the statistical properties of a deterministic or of a random internal wave field are investigated in a vertical plane (for instance Orlanski & Cerasoli 1981;Frederiksen & Bell 1983, 1984Chen & Holloway 1986). In Carnevale & Martin (1982) and Holloway (1979), these properties are compared to a statistical model (a review was made by Muller et al 1986).…”