Three-dimensional Gravity Waves Research Articles

Hamiltonian Dysthe Equation for Three-Dimensional Deep-Water Gravity Waves

Hamiltonian Dysthe Equation for Three-Dimensional Deep-Water Gravity Waves

A three-dimensional model of wave interactions with permeable structures using the lattice Boltzmann method

To simulate the three-dimensional interaction of waves and porous structures using the lattice Boltzmann method, a novel multi-relaxation-time lattice Boltzmann scheme corresponding to the volume-averaged Navier-Stokes equations incorporating porous flow is developed. The porosity is introduced into the equilibrium distribution function, and the frictional forces produced by the porous media are added by the discrete force model. Through the Maxwell iteration, the relation between the mesoscopic lattice Boltzmann scheme and the macroscopic governing equations is established. Large eddy simulation and the single-phase volume-of-fluid techniques are modified to take porous media into account. The friction parameters in this model are calibrated using the experimental data of two-dimensional dam-break waves interacting with porous media composed of crushed rocks. Validations are carried out by comparing the simulation result of the proposed model with the laboratory data of three-dimensional dam-break waves impacting a prism and three-dimensional surface gravity waves (solitary waves and cnoidal waves) interacting with a vertical permeable breakwater. Furthermore, the interaction of regular waves and a rubble mound breakwater is used to test the capability of the model to handle spatially varying porous media. The different simulation results strongly agree with the experimental data, which prove that the lattice Boltzmann model has the ability to simulate complex wave motions near porous structures.

Three-dimensional surface gravity waves of a broad bandwidth on deep water

A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

On the breaking inception of unsteady water wave packets evolving in the presence of constant vorticity

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Rapid spectral evolution of steep surface wave groups with directional spreading

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Crest speeds of unsteady surface water waves

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Three-Dimensional Internal Gravity Waves in a Stratifield Free-Surface Flow

Three-Dimensional Internal Gravity Waves in a Stratifield Free-Surface Flow

Process of the Formation of Internal Waves Initiated by the Start of Motion of a Body in a Stratified Viscous Fluid

The mechanism of formation of three-dimensional internal gravity waves initiated by the start of motion of a disk of given diameter and finite thickness in the horizontal direction along the disk symmetry axis from right to left at a given uniform velocity in an incompressible viscous linearly density-stratified fluid is first considered in detail. The consideration is carried out on the basis of numerical solution of the system of Navier—Stokes equations in the Boussinesq approximation and visualization of the three-dimensional vortex structure of the flow calculated. The obtained fields of the velocity vectors and pressure perturbations possess horizontal and vertical symmetry planes passing through the disk symmetry axis. The process of formation of flow in the upper half-space caused by the shear and gravitational instabilities is described. In this process, two horizontal vortex filaments are initially formed between the back disk face and the place of pulsed start and then transformed into legs of the hairpin vortex loop whose head is located to right of the start point. Thereafter, vortex rings are periodically formed above the start point during half the buoyancy period of fluid. The left-hand halves of the rings are transformed into half-waves occupying space between the disk and the start point.

Dynamics of Orographic Gravity Waves Observed in the Mesosphere over the Auckland Islands during the Deep Propagating Gravity Wave Experiment (DEEPWAVE)

Abstract On 14 July 2014 during the Deep Propagating Gravity Wave Experiment (DEEPWAVE), aircraft remote sensing instruments detected large-amplitude gravity wave oscillations within mesospheric airglow and sodium layers at altitudes z ~ 78–83 km downstream of the Auckland Islands, located ~1000 km south of Christchurch, New Zealand. A high-altitude reanalysis and a three-dimensional Fourier gravity wave model are used to investigate the dynamics of this event. At 0700 UTC when the first observations were made, surface flow across the islands’ terrain generated linear three-dimensional wave fields that propagated rapidly to z ~ 78 km, where intense breaking occurred in a narrow layer beneath a zero-wind region at z ~ 83 km. In the following hours, the altitude of weak winds descended under the influence of a large-amplitude migrating semidiurnal tide, leading to intense breaking of these wave fields in subsequent observations starting at 1000 UTC. The linear Fourier model constrained by upstream reanalysis reproduces the salient aspects of observed wave fields, including horizontal wavelengths, phase orientations, temperature and vertical displacement amplitudes, heights and locations of incipient wave breaking, and momentum fluxes. Wave breaking has huge effects on local circulations, with inferred layer-averaged westward flow accelerations of ~350 m s−1 h−1 and dynamical heating rates of ~8 K h−1, supporting recent speculation of important impacts of orographic gravity waves from subantarctic islands on the mean circulation and climate of the middle atmosphere during austral winter.

On three-dimensional internal gravity wave beams and induced large-scale mean flows

The three-dimensional propagation of internal gravity wave beams in a uniformly stratified Boussinesq fluid is discussed, assuming that variations in the along-beam and transverse directions are of long length scale relative to the beam width. This situation applies, for instance, to the far-field behaviour of a wave beam generated by a horizontal line source with weak transverse dependence. In contrast to the two-dimensional case of purely along-beam variations, where nonlinear effects are minor even for beams of finite amplitude, three-dimensional nonlinear interactions trigger the transfer of energy to a circulating horizontal time-mean flow. This resonant beam–mean-flow coupling is analysed, and a system of two evolution equations is derived for the propagation of a small-amplitude beam along with the induced mean flow. This model explains the salient features of the experimental observations of Bordes et al. (Phys. Fluids, vol. 24, 2012, 086602).

Stability analysis from fourth order evolution equation for counter-propagating gravity wave packets in the presence of wind flowing over water

Fourth order nonlinear evolution equations are derived for two counter-propagating surface gravity wave packets in deep water in the presence of wind flowing over water. The resulting equations are asymptotically exact and nonlocal. Stability analysis is made for a uniform standing surface gravity wave train for longitudinal perturbation on the basis of these equations. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness for some different values of dimensionless wind velocity. Significant deviations are noticed between the results obtained from third order and fourth order nonlinear evolution equations. This paper has an application in rough waves. References T. Brooke Benjamin and J. E. Feir. The disintegration of wave trains on deep water part 1. theory. Journal of Fluid Mechanics , 27:417–430, 1967. doi:10.1017/S002211206700045X Sudebi Bhattacharyya and K. P. Das. Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline. The ANZIAM Journal , 39:214–229, 1997. doi:10.1017/S033427000000881X K. P. Das. On evolution equations for a three-dimensional surface gravity wave packet in a two-layer fluid. Wave Motion , 8(2):191–204, 1986. doi:10.1016/0165-2125(86)90024-7 A. Davey and K. Stewartson. On three-dimensional packets of surface waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 338(1613):101–110, 1974. doi:10.1098/rspa.1974.0076 Suma Debsarma and K. P. Das. Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water. The ANZIAM Journal , 43:513–524, 2002. doi:10.1017/S1446181100012116 A. K. Dhar and K. P. Das. A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water. Physics of Fluids A: Fluid Dynamics , 2(5), 1990. A. K. Dhar and K. P. Das. Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water. Physics of Fluids A: Fluid Dynamics , 3(12), 1991. A. K. Dhar and K. P. Das. Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear. The ANZIAM Journal , 35:348–365, 1994. doi:10.1017/S0334270000009346 V. D. Djordjevic and L. G. Redekopp. On two-dimensional packets of capillary-gravity waves. Journal of Fluid Mechanics , 79:703–714, 1977. doi:10.1017/S0022112077000408 J. C. Dungey and W. H. Hui. Nonlinear energy transfer in a narrow gravity-wave spectrum. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 368(1733):239–265, 1979. doi:10.1098/rspa.1979.0126 K. B. Dysthe. Note on a modification to the nonlinear schrodinger equation for application to deep water waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 369(1736):105–114, 1979. doi:10.1098/rspa.1979.0154 Tetsu Hara and Chiang C. Mei. Frequency downshift in narrow banded surface waves under the influence of wind. Journal of Fluid Mechanics , 230:429–477, 1991. doi:10.1017/S002211209100085X Tetsu Hara and Chiang C. Mei. Wind effects on the nonlinear evolution of slowly varying gravity-capillary waves. Journal of Fluid Mechanics , 267:221–250, 1994. doi:10.1017/S0022112094001163 S. J. Hogan. The fourth-order evolution equation for deep-water gravity-capillary waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 402(1823):359–372, 1985. doi:10.1098/rspa.1985.0122 Peter A. E. M. Janssen. On a fourth-order envelope equation for deep-water waves. Journal of Fluid Mechanics , 126:1–11, 1983. doi:10.1017/S0022112083000014 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water. I. superharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 360(1703):471–488, 1978. doi:10.1098/rspa.1978.0080 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water II. subharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 360(1703):489–505, 1978. doi:10.1098/rspa.1978.0081 R. D. Pierce and E. Knobloch. On the modulational stability of traveling and standing water waves. Physics of Fluids , 6(3), 1994. Michael Stiassnie. Note on the modified nonlinear Schrodinger equation for deep water waves. Wave Motion , 6(4):431–433, 1984. doi:10.1016/0165-2125(84)90043-X M. A. Weissman. Nonlinear wave packets in the Kelvin–Helmholtz instability. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 290(1377):639–681, 1979. doi:10.1098/rsta.1979.0019 V. E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics , 9(2):190–194, 1968. doi:10.1007/BF00913182

A nonoverlapping heterogeneous domain decomposition method for three-dimensional gravity wave impact problems

A 3D heterogeneous flow model which combines incompressible viscous flow and potential flow both with free-surface boundaries is developed and solved numerically, using finite element method and boundary element method, respectively. The coupled model is solved based on a nonoverlapping domain decomposition method, which reduces the problem to the coupling interface. At the interface, the problem is equivalent to Bernoulli’s equation consisting of mappings from potential/velocity to velocity/pressure. These mappings enable us to design the scheme following the Dirichlet–Neumann method for nonoverlapping domain decomposition, and lead to a staggered scheme. We also discuss data transfer as well as free surface reconstruction at the interface. Errors introduced by the surface reconstruction technique are examined and discussed in detail. The errors from the first-order staggered scheme is also studied. A comparison of numerical prediction with experimental data that the proposed method performs efficiently and accurately for wave impact problems.

Generation mechanisms of convectively induced internal gravity waves in a three-dimensional framework

The generation mechanisms of convective gravity waves in the stratosphere are investigated in a three-dimensional framework by conducting numerical simulations of four ideal storms under different environmental conditions: one un-sheared and three constant low-level sheared basic-state winds with the depth of the shear layer of 6 km and the surface wind speeds (Us) of 8, 18, and 28 m s−1, using the Advanced Regional Prediction System (ARPS) model. The storms simulated under the un-sheared (Us = 0 m s−1), weakly sheared (Us = 8 and 18ms−1), and strongly sheared (Us = 28ms−1) basicstate winds are classified into single-cell, multicell, and supercell storms, respectively. For each storm, the wave perturbations in a control simulation, including nonlinearity and microphysical processes, are compared with those in quasi-linear dry simulations forced by diabatic forcing and nonlinear forcing that are obtained from the control simulation. The gravity waves generated by the two forcing terms in the quasi-linear dry simulations are out of phase with each other for all of the storms. The gravity waves in the control simulation are represented by a linear sum of the wave perturbations generated by the nonlinear forcing and diabatic forcing. This result is consistent with the results of previous studies in a two-dimensional framework. This implies that both forcing mechanisms are important for generating the convective gravity waves in the three-dimensional framework as well. The characteristics of the three-dimensional gravity waves in the stratosphere were determined by the spectral combination of the forcing terms and the wave-filtering and resonance factor that is determined from the basic-state wind and stability as well as the vertical structure of the forcing.

Retrieval of the three-dimensional wave structure of gravity waves from multi-position airglow measurements

The aim of this work is to develop mathematical foundation of a method which can be used to infer the three-dimensional gravity wave characteristics from multi position airglow observations from space. This work derives a one-dimensional Fredholm integral equation of the first kind, which describes the relations between the gravity wave spectrum and spatial structure of wave perturbations registered by a space-based airglow imager. It is shown that the solution of this equation belongs to the central slice through a three-dimensional gravity wave spectrum, whose plane is perpendicular to the optical axis of the airglow imager. Thus, in order to retrieve the three-dimensional gravity wave characteristics from the airglow observations performed from space, it is needed to obtain the set of images of a local emission layer area from different imager positions. Then this data must be processed using the developed mathematical techniques to obtain a set of the central slices of the three-dimensional gravity wave spectrum. Applying the technique, for a superposition of three individual waves, amplitude and wave vector can be determined.

Acoustic gravity tornadoes in the atmosphere

It is shown that three-dimensional (3D) acoustic gravity waves (AGWs) in the atmosphere can appear in the form of acoustic gravity tornadoes (AGTs) characterized by twisted density structures or density ropes carrying orbital angular momentum. For our purposes, we use a previously obtained 3D wave equation for AGWs, and show that this equation in the paraxial approximation admits solutions in the form of Laguerre–Gauss acoustic gravity vortex beams or AGTs/AG whirls with twisted density structures supporting the dynamics of the AGTs.

Three-dimensional periodic interfacial gravity waves: Analytical and numerical results

The properties of fully three-dimensional gravity waves propagating on the interface between two finite layers of different densities are investigated. Solutions are calculated via a computer-generated perturbation expansion in wave steepness. Series solutions are analytically computed to third-order for general wave parameters, and numerically to 27th-order for five specific values of μ = 0 , 0.001 , 0.1 , 0.5 and 0.99, where μ is the ratio of the density of the upper fluid to that of the lower fluid. For near limiting waves, the series of frequency, kinetic energy and potential energy are summed using Padé approximants. For both two and three-dimensional cases, the present theory is found to coincide with previous theories such as two-dimensional interfacial standing waves, two-dimensional interfacial progressive waves and three-dimensional surface gravity waves respectively, showing the validity and general applicability of the solutions. The numerical results demonstrate the influence of the ratio density and thicknesses of the two fluids on the wave profile and wave frequency bifurcation. Particular attention is paid to the harmonic resonances where multiple solutions are possible.

Asymptotic Shallow Water Models for Internal Waves in a Two-Fluid System with a Free Surface

In this paper, we derive asymptotic models for the propagation of two and three-dimensional gravity waves at the free surface and the interface between two layers of immiscible fluids of different densities, over an uneven bottom. We assume the thickness of the upper and lower fluids to be of comparable size, and small compared to the characteristic wavelength of the system (shallow water regimes). Following a method introduced by Bona, Lannes and Saut based on the expansion of the involved Dirichlet-to-Neumann operators, we are able to give a rigorous justification of classical models for weakly and strongly nonlinear waves, as well as interesting new ones. In particular, we derive linearly well-posed systems in the so called Boussinesq/Boussinesq regime. Furthermore, we establish the consistency of the full Euler system with these models, and deduce the convergence of the solutions.

Existence of a directional Stokes drift in asymmetrical three-dimensional travelling gravity waves

We consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid. The pattern is non-symmetric with respect to the propagation direction of the waves and we consider a general non-resonant situation. Defining a couple of amplitudes ε 1 , ε 2 along the basis of wave vectors which satisfy the dispersion relation, following Iooss and Plotnikov (2009), travelling waves exist with an asymptotic expansion in powers of ε 1 , ε 2 , for nearly all pair of angles made by the basic wave vectors with the critical propagation direction, and for values of the couple ( ε 1 2 , ε 2 2 ) in a subset of the plane, with asymptotic full measure at the origin. We prove the remarkable property that on the free surface, observed in the moving frame, the propagation direction of the waves differs from the asymptotic direction taken by fluid particles, by a small angle which is computed. To cite this article: G. Iooss, P. Plotnikov, C. R. Mecanique 337 (2009).

Paralinearization of the Dirichlet to Neumann Operator, and Regularity of Three-Dimensional Water Waves

This paper is concerned with a priori C ∞ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a Diophantine condition are automatically C ∞. In particular, we prove that the solutions defined by Iooss and Plotnikov are C ∞. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.

Statistics and Dynamics of Aircraft Encounters of Turbulence over Greenland

Abstract Historical records of aviation turbulence encounters above Greenland are examined for the period from 2000 to 2006. These data identify an important flow regime that contributes to the occurrence of aircraft turbulence encounters, associated with the passage of surface cyclones that direct easterly or southeasterly flow over Greenland’s imposing terrain. The result of this incident flow is the generation of mountain waves that may become unstable through interactions with the background directional wind shear. It is shown that this regime accounted for approximately 40% of the significant turbulent events identified in the 7-yr database. In addition, two specific cases from the database are examined in more detail using a high-resolution mesoscale model. The model simulations highlight the important role of three-dimensional gravity wave–critical level interactions and demonstrate the utility of high-resolution forecasts in the prediction of such events.