Analytical and numerical investigation of nonlinear internal gravity waves (original) (raw)

Nonlinear dynamics of hydrostatic internal gravity waves

Theoretical and Computational Fluid Dynamics, 2008

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, there is an asymmetry between the east-and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden-Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well.

Features of fluid flows in strongly nonlinear internal solitary waves

2014

The characteristics of highly nonlinear solitary internal waves (solitons) are calculated within the fully nonlinear numerical model of the Massachusetts Institute of Technology. The verification and adaptation of the model is based on the data from laboratory experiments. The present paper also compares the results of our calculations with the calculations performed in the framework of the fully nonlinear Bergen Ocean Model. The comparison of the computed soliton parameters with the predictions of the weakly nonlinear theory based on the Gardner equation is given. The occurrence of reverse flow in the bottom layer directly behind the soliton is confirmed in the numerical simulations. The trajectories of Lagrangian particles in the internal soliton on the surface, on the pycnocline and near the bottom are computed.

On multisolitonic decay behavior of internal gravity waves

We claim that changes of scales and fine-structure could increase from multisoliton behavior of internal waves dynamics and, further, in the so-called "wave mixing". We consider initial-boundary problems for Euler equations with a stratified background state that is valid for internal water waves. The solution of the problem we search in the waveguide mode representation for a current function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm-Liouville problem for the vertical variable. The Cauchy problem is solved for initial conditions with realistic geometry and magnitude. We choose the geometry and dimensions of the McEwan experiment with the stratification of constant buoyancy frequency. The horizontal profile is defined by numerical solutions of a coupled Korteweg-de Vries system. The numerical scheme is proved to be convergent, stable and tested by means of explicit solutions for integrable case of the system. To...

Analytical solutions of long nonlinear internal waves: Part I

Natural Hazards, 2011

The Gardner equation is an extension of the Korteweg-de Vries (KdV) equation. It exhibits basically the same properties as the classical KdV, but extends its range of validity to a wider interval of the parameters of the internal wave motion for a given environment. In this paper, we derive exact solitary wave solutions for the generalized Gardner equation that includes nonlinear terms of any order. Unlike previous studies, the exact solutions are derived without assuming their mathematical form. Illustrative examples for internal solitary waves are also provided. The traveling wave solutions can be used to specify initial data for the incident waves in internal waves numerical models and for the verification and validation of the associated computed solutions.

An experimental and numerical study of nonlinear internal waves

Physics of Fluids A: Fluid Dynamics, 1993

Nonlinear internal waves were measured on the large rotating platform at the Institut de Mécanique de Grenoble (I.M.G.). The experimental data complement the results presented in Renouard et al. [J. Fluid Mech. 177, 381 (1987)] and support the assumption that the solitary Kelvin wave is accompanied by Poincaré waves. Based on the assumption of weak nonlinear, dispersive, and rotational effects, governing equations of the Boussinesq type are derived to model the evolution of an initial disturbance in a two-layer rotating fluid. The numerical study is based on these equations which are analogous to the Boussinesq equations of shallow-water theory and are not constrained to almost unidirectional propagation. Comparison of numerical solutions of the equations and experimental results are very good for moderately nonlinear conditions. These results provide supporting evidence for the resonant interaction of nonlinear Kelvin waves and linear Poincaré waves, as described by Melville et al....

2008 Nonlinear dynamics of hydrostatic internal gravity waves

2016

Analytical study of dissipative solitary waves

Physica Scripta, 2008

In this paper, the analytical solution to a new class of nonlinear solitons is presented with cubic nonlinearity, subject to a dissipation term arising as a result of a first-order derivative with respect to time, in the weakly nonlinear regime. Exact solutions are found using the combination of the perturbation and Green's function methods up to the third order. We present an example and discuss the asymptotic behavior of the Green's function. The dissipative solitary equation is also studied in the phase space in the non-dissipative and dissipative forms. Bounded and unbounded solutions of this equation are characterized, yielding an energy conversation law for non-dissipative waves. Applications of the model include weakly nonlinear solutions of terahertz Josephson plasma waves in layered superconductors and ablative Rayleigh-Taylor instability.

On four highly nonlinear phenomena in wave theory and marine hydrodynamics

Applied Ocean Research, 2002

Some recent developments in the formation of extreme waves, kinematics of steep waves, the phenomenon of ringing and currents in the ocean induced by internal waves are reviewed. Formation of extreme waves are simulated by means of a rapid fully nonlinear model. A large wave event taking place in a wave group is characterized by an elevation being significantly larger than the initial amplitude of the group. Recurrence occurs. PIV measurements of Stokes waves exhibit an exponential velocity profile all the way up to the surface elevation (wave slope up to 0.16). The computed velocity profile under crest of an extreme wave corresponds also to an exponential profile. Experimental results of the horizontal force on a vertical circular cylinder in long and steep waves exhibit a secondary cycle of high frequency in the force history. This typically occurs for waves longer than about 10 times the cylinder diameter and a Froude number vh m = ffiffiffiffi gD p larger than about 0.4, v the wave frequency, h m the maximal elevation, g the acceleration of gravity, D the cylinder diameter. Properties of internal solitons and the induced fluid velocities are described in terms of weakly and fully nonlinear models supported by PIV measurements. A rapid scheme for fully nonlinear interfacial waves in three dimensions is derived, complementing the rapid model of free surface waves. q

Kinetic equations and stationary energy spectra of weakly nonlinear internal gravity waves

Dynamics of Atmospheres and Oceans, 2000

An ensemble of random-phase internal gravity waves is considered in the dynamical framework of the Euler-Boussinesq equations. For flows with zero mean potential vorticity, a kinetic equation for the mean spectral energy density of the waves is obtained under hypothesis of Gaussian statistics with zero correlation length. Stationary scaling solutions of this equation are found for almost vertically propagating waves. The resulting spectra are anisotropic in vertical and horizontal wave numbers. For flows with small but non-zero mean potential vorticity, under the same statistical hypothesis applied to the wave part of the flow, it is shown that the vortex part and the wave part decouple. The vortex part obeys a limiting slow dynamics equation exhibiting vertical collapse and layering which may contaminate the wave-part spectra. Relation of these results to the in situ atmospheric measurements and previous work on oceanic gravity waves is discussed.

Analytical and numerical investigation of nonlinear internal gravity waves (original) (raw)

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