Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

Bhat p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p85 Breaking Waves in Deep and Intermediate Waters Marc Perlin, Wooyoung Choi, and Zhigang Tian p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 115 Balance and Spontaneous Wave Generation in Geophysical Flows J.

An experimental study of energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model to simulate the dissipation due to wave breaking are reported in this paper. Measured wave surface elevations are used to examine the characteristic time and length scales associated with wave groups and local breaking waves, and to estimate and parameterize the energy dissipation and dissipation rate due to wave breaking. Numerical tests using the eddy viscosity model are performed and we find that the numerical results well capture the measured energy loss. In our experiments, three sets of characteristic time and length scales are defined and obtained: global scales associated with the wave groups, local scales immediately prior to breaking onset and post-breaking scales. Correlations among these time and length scales are demonstrated. In addition, for our wave groups, wave breaking onset predictions using the global and local wave steepnesses are found based on experimental results. Breaking time and breaking horizontal length scales are determined with high-speed imaging, and are found to depend approximately linearly on the local wave steepness. The two scales are then used to determine the energy dissipation rate, which is the ratio of the energy loss to the breaking time scale. Our experimental results show that the local wave steepness is highly correlated with the measured dissipation rate, indicating that the local wave steepness may serve as a good wave-breaking-strength indicator. To simulate the energy dissipation due to wave breaking, a simple eddy viscosity model is proposed and validated with our experimental measurements. Under the small viscosity assumption, the leading-order viscous effect is incorporated into the free-surface boundary conditions. Then, the kinematic viscosity is replaced with an eddy viscosity to account for energy loss. The breaking time and length scales, which depend weakly on wave breaking strength, are applied to evaluate the magnitude of the eddy viscosity using dimensional analysis. The estimated eddy viscosity is of the order of 10 −3 m 2 s −1 and demonstrates a strong dependence on wave breaking strength. Numerical simulations with the eddy viscosity estimation are performed to compare to the experimental results. Good agreement as regards energy dissipation due to wave breaking and surface profiles after wave breaking is achieved, which illustrates that the simple eddy viscosity model functions effectively. † Email address for correspondence: perlin@umich.edu 218 Z. Tian, M. Perlin and W. Choi

We derive general evolution equations for two-dimensional weakly nonlinear waves at the free surface in a system of two fluids of different densities. The thickness of the upper fluid layer is assumed to be small compared with the characteristic wavelength, but no restrictions are imposed on the thickness of the lower layer. We consider the case of a free upper boundary for its relevance in applications to ocean dynamics problems and the case of a non-uniform rigid upper boundary for applications to atmospheric problems. For the special case of shallow water, the new set of equations reduces to the Boussinesq equations for two-dimensional internal waves, whilst, for great and infinite lower-layer depth, we can recover the well-known Intermediate Long Wave and Benjamin-Ono models, respectively, for one-dimensional uni-directional wave propagation. Some numerical solutions of the model for one-dimensional waves in deep water are presented and compared with the known solutions of the uni-directional model. Finally, the effects of finite-amplitude slowly varying bottom topography are included in a model appropriate to the situation when the dependence on one of the horizontal coordinates is weak. u/: Choi and R. Camassa fact that each of these models is valid only for a certain depth. Let hlo and hzo be the undisturbed depth of the upper and lower layers, respectively, and let L be a characteristic wavelength. The KdV and the Boussinesq models are valid for hlo/L+l and h20/h10 = 0(1) while the ILW equation is for hl0/L<1 and h20/h10+1. Therefore, there is no theory in between the KdV equation and the ILW equation which can cover the whole range of the ratio hz0/hI0. Moreover, since all previous models, with the exception of the Boussinesq equations, are for uni-directional waves (or weakly two-dimensional waves with a preferred direction of propagation), general wave propagation cannot be properly described by these models. This happens, for instance, whenever reflected waves need to be taken into account, like in the case of internal waves propagating over a non-uniform sea bed in the ocean or over a hill in the atmosphere. Also, the uni-directional equations model the propagation of internal wave modes only, and nonlinear interaction of waves from different modes such as the interaction between surface and internal waves is neglected. It is therefore desirable to have a general model valid for two-dimensional waves in a fluid of arbitrary and non-uniform depth for real applications. This model should still afford the remarkable simplification over the original Euler equations achieved by the previously known models, yet it should be able to handle the more realistic situations mentioned above.Recently, for the case of a homogeneous fluid layer, much progress has been made in this direction. Evolution equations for surface waves correct up to the third-order non-linearity in wave slope for a fluid of finite depth have been derived by Matsuno (1992) for one-dimensional waves and Choi (1995) for two-dimensional waves....