Resonat and Non-Resonant Wave-Wave Interactions for Internal Gravity Waves

Abstract

A detailed study of energy transfer among two-dimensional internal gravity modes in a fully non-linear regime was performed. A number of techniques were used: They were (i) solutions of the gyroscopic equations with three and four waves, (ii) integration of a finite difference numerical model, and (iii) laboratory experiments. The solution of the four wave gyroscopic equations differed dramatically from the three wave case, and the four wave solutions were aperiodic. In both cases, the non-linear interaction time scale, T μ , was found to be inversely proportional to the square root of the total wave energy (T μ ∼ E-l/2), even when the weak interaction assumption was violated. Integration of a finite difference numerical model showed that triad evolution was greatly affected when many waves other than the primary triad components could be excited. Initial condition experiments for triad evolution were performed with either a quiescent background state or a random field of waves, and the final states were similar, although the time to reach steady state was short when a background field was present. The numerical model was used to simulate surface forced, resonant modes and results were compared to laboratory experiments. Good agreement was found, not only in the initial wave evolution but also in energy level of the final states. An equilibrium state was achieved in both types of experiments, and wave-wave interactions and wave breaking were important in the energy distribution. The numerical model was used to create a random, finite amplitude internal wave field, and a set of experiments whereby this basic state initially perturbed was performed. In these experiments energy was introduced over bands of low, medium and high wavenumbers. The results show that when the basic state energy is low and non-linear time scales are much greater than intrinsic wave periods, multiple triad interactions account for the distribution of any input energy. As the energy level increases, the high wavenumbers become saturated and localized overturning provides the dissipation mechanism. Additional energy input to low and medium wavenumbers will eventually result in an equilibrium state, whereby any extra energy input will result in very rapid, localized overturning. This equilibrium level depends on the presence of saturated high wave-numbers and once achieved, the system is very inefficient at transferring energy via wave-wave interactions while very efficient at dissipating energy via localized overturning.