Energy transfer among internal gravity modes: Weak and strong interactions

Abstract

The general characteristics of the energy spectrum for internal gravity waves in the ocean are well known from the large body of recent experimental observations. The theoretical understanding has not developed at the same rate, perhaps due to the limitation of linear or quasi‐linear theories, which can cope only with weak interaction processes and are inadequate for representing the more violent and sporadic wave breaking processes present in nature. A detailed study of energy transfer among two‐dimensional internal gravity modes in a fully nonlinear regime was performed. Wave‐wave interactions and overturning were included in the solutions of a two‐dimensional numerical model, and the results are presented here. A background spectrum of finite amplitude, random internal gravity wave field was generated by a long time integration of a two‐dimensional model with random body forcing. Over this background field, two sets of experiments were performed: spike‐random, where energy at low, medium, and high wave numbers were introduced and integrated in time, and band‐random, where energy was introduced over a band of wave numbers instead of introducing only discrete modes. The results can be summarized as follows. Multiple triad interactions will result in a filling of the energy spectrum when energy is introduced in a particular band of wave numbers. For bands where the energy level is high enough to result in nonlinear time scales of only a few intrinsic periods, wave‐wave interactions (resonant and nonresonant) provide the mechanism for filling the spectrum. The energy transfer becomes more and more rapid with increasing energy, and no universal spectrum appears to result from these processes. As the energy input increases, energy will accumulate in high wave numbers until localized instabilities (over‐turning) occur. From that point on, these high wave numbers will remain at a saturation such that any additional energy input at the saturated band, either directly or via wave‐wave interactions, will result in localized mixing. On the other hand, additional energy input at bands other than the saturated band will result in an increase of low and medium wave band energy (via wave‐wave interactions) until an equilibrium level is achieved. The equilibrium level of any particular band will depend on the high wave number bands being saturated. For instance, any energy above the equilibrium at low wave numbers will produce localized mixing in physical space almost instantaneously. This does not mean that the low wave numbers are saturated, as their energy levels can be much lower than a saturation level. What takes place at or near an equilibrium level is that the contributions from high and low wave numbers result in localized regions in physical space where the criterion for instability is almost met. In fact, this superposition effect means that low and medium wave numbers are far from meeting any breaking criterion when taken individually, yet cannot tolerate any additional input energy when in the presence of a satura...

The spectrum of atmospheric velocity fluctuations at 8 km and 86 km

“…For example, an f-Y• slope has also been postulated to arise from interacting gravity waves on the basis of dimensional arguments (Dewan, 1979) and by a detailed study of wave-wave interactions in the ocean (McComas and Muller, 1981). In the gravity-wave case, the energy cascade would also be toward both higher and lower frequencies (Orlanski and Cerasoli, 1981).…”

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

“…One is an indirect determination based on oceanic measurements of vertical temperaturegradient variance spectra [Gregg, 1977] and vertical velocitygradient variance spectra [Gargett et al, 1981]. The other is obtained from direct numerical simulation by Orlanski and Cerasoli [1981] and Frederiksen and Bell [1983].…”

“…The subgrid diffusive-dissipative process, however, is modeled here with a biharmonic operator [Holland, 1978], rather than the usual Laplacian operator, to limit the dissipation to the smallest scales of motion possible. We also note that no special parameterizations of dissipation, such as those used in Orlanski and Cerasoli's [1981] model, are invoked within the density overturn region; in our simulation the dissipation there is determined by the equations of motion. Finally, the forcing function F is specified in the wave number space for the stream function only; this will be discussed shortly below.…”

“…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.…”

“…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.…”