Coupling of surface and internal gravity waves: a mode coupling model (original) (raw)
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INTERNAL GRAVITY WAVES: From Instabilities to Turbulence
Annual Review of Fluid Mechanics, 2002
▪ We review the mechanisms of steepening and breaking for internal gravity waves in a continuous density stratification. After discussing the instability of a plane wave of arbitrary amplitude in an infinite medium at rest, we consider the steepening effects of wave reflection on a sloping boundary and propagation in a shear flow. The final process of breaking into small-scale turbulence is then presented. The influence of those processes upon the fluid medium by mean flow changes is discussed. The specific properties of wave turbulence, induced by wave-wave interactions and breaking, are illustrated by comparative studies of oceanic and atmospheric observations, as well as laboratory and numerical experiments. We then review the different attempts at a statistical description of internal gravity wave fields, whether weakly or strongly interacting.
Nonlinear interaction of internal and surface gravity waves in a two-layer fluid with free surface
Journal of Mathematical Sciences, 2010
UDC 532.59 A new nonlinear model of the propagation of wave packets in the system "liquid layer with solid bottom-liquid layer with free surface" is considered. With the use of the method of multiple-scale expansions, the first three linear approximations of the nonlinear problem are obtained. Solutions of problem of the first approximation are constructed and analyzed in detail. It is shown that there exist internal and surface components of the wave field, and their interaction is analyzed.
A Note on internal gravity wave spectra
Journal of Geophysical Research, 1972
Several measurements of the slope of internal wave spectra have given wave number k dependencies of the form k-5/8 or k-•. The first of these is sometimes discarded because it is thought that Kolmogoroff's ideas of small-scale isotropic turbulence should not apply to internal gravity waves. However, it is suggested in this note that an empirically determined k-5/8 dependence is quite divorced from Kolmogoroff physics. Simple dimensional arguments show that k-• spectra are possible for long internal waves, but that k-a spectra might be more appropriate for short waves even though microstructure effects may obscure the results.
On a model system for the oblique interaction of internal gravity waves
ESAIM: Mathematical Modelling and Numerical Analysis, 2000
We give local and global well-posedness results for a system of two Kadomtsev-Petviashvili (KP) equations derived by R. Grimshaw and Y. Zhu to model the oblique interaction of weakly nonlinear, two dimensional, long internal waves in shallow fluids. We also prove a smoothing effect for the amplitudes of the interacting waves. We use the Fourier transform restriction norms introduced by J. Bourgain and the Strichartz estimates for the linear KP group. Finally we extend the result of [3] for lower order perturbation of the system in the absence of transverse effects.
An analytical and numerical study of internal gravity waves forced by isolated topography
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2008 Nonlinear dynamics of hydrostatic internal gravity waves
2016
Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, ther...
Ocean Engineering, 2008
When the wave spectrum is sufficiently narrow-banded and the wave steepness is sufficiently high, the modulational instability can take place and waves can be higher than expected from second-order wave theory. In order to investigate these effects on the statistical distribution of long-crested, deep water waves, direct numerical simulations of the Euler equations have been performed. Results show that, for a typical design spectral shape, both the upper and lower tails of the probability density function for the surface elevation significantly deviate from the commonly used second-order wave theory. In this respect, the crest elevation is observed to increase up to 18% at low probability levels. It would furthermore be expected that wave troughs become shallower due to nonlinear effects. Nonetheless, the numerical simulations show that the trough depressions tend to be deeper than in second-order theory.