An Improved Parameterization for Energy Exchange from Wind to Stokes Waves

We investigate the effects of the background dipole magnetic field and cold electron number density on the linear and nonlinear growth of whistler-mode chorus waves for a region of relatively small anisotropy (A T ) in Saturn’s inner magnetosphere. The linear and nonlinear features of wave growth rate and associated frequency at L = 6 are presented in detail. Although a large anisotropy is generally in favor of linear and nonlinear wave growth, the nonlinear wave growth for a small anisotropy can still be generated. All cases show a small threshold for wave amplitudes , which compromises the requirement to trigger the nonlinear wave growth, but the comparisons also clearly indicate the important transition process from the linear phase to the nonlinear phase. After checking the variation of the calculation time steps depending on the chosen electron number density N c and background magnetic field B c , respectively, a large N c can promote the nonlinear wave growth, but a large B c works against it. Our results present how these parameters really affect the generations of linear and nonlinear wave growth quantitatively. This could be significant to further understand the monumental importance of whistler-mode chorus waves and the corresponding wave–particle interactions in the planetary magnetosphere.

In this review we consider the propagation of nonlinear waves in magnetically structured plasmas. We start from studying nonlinear surface waves on magnetic interfaces and give a derivation of the equation governing propagation of these waves in incompressible plasmas. We present the results of a numerical solution of this equation. Then we discuss various generalizations of this equation.We proceed to study slow nonlinear waves in magnetic slabs. We present the Benjamin-Ono equation describing nonlinear slow sausage surface waves in magnetic slabs, and discuss its solutions in the form of algebraic solitons. Then we consider nonlinear slow body waves and write down the governing equation for these waves. We discuss its numerical solution which shows that the evolution of an initial perturbation results in a gradient catastrophe.After that we consider nonlinear slow sausage surface and slow body waves in magnetic tubes, which are similar to the corresponding counterparts in magnetic slabs. We present a system of two equations describing both types of waves. We briefly discuss the derivation of the Leibovich-Roberts equation for slow sausage surface waves from this system of equations, and present the results of its numerical study. Then we give a brief description of the derivation of the governing equation for slow body waves, which is very similar to its counterpart for slow body waves in magnetic slabs and has very similar properties. We complete this part of the review with presenting the governing equation for non-axisymmetric surface waves in a thin magnetic tube in an incompressible plasma.In the last part of the paper we consider the damping of nonlinear waves in weakly dissipative magnetically structured plasmas. We briefly discuss the application of the obtained results to the problem of heating of the solar corona.KeywordsSurface WaveSolitary WaveSolar PhysMagnetic TubeKink ModeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into sine waves, Stokes waves, and phase-locked Stokes waves otherwise known as breather trains. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over cycle times of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrodinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the nonlinear spectral representation. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its nonlinear coherent structures (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of multidimensional, quasi-periodic Fourier series, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height Hs, the peak period Tp, and the length of the time series T. The time series analyzed here have 8192 points and T =1677.72 s = 27.96 min. Near the peak of the storm, we find Hs ≈ 0.55 m, Tp ≈ 2.4 s so that for the wave steepness of a near Gaussian process, \({S} = \left (\pi ^{5/2}/g\right )H_{s}/{T}_{p}^{2}\), we find S ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, \({I_{BF}} = \left (\pi ^{5/2}/g \right ) H_{s} T/{T}_{p}^{3}\), and we find IBF ≈ 119. Since the BF parameter describes the nonlinear behavior of the modulational instability, leading to the formation of breather packets in a measured wave train, we find the IBF for these storm waves to be a surprisingly high number. This is because IBF, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a nonlinear spectral point of view, the presence of large numbers of breather trains and we determine many of their properties, including the rise time for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of breather turbulence, highly nonlinear integrable turbulence theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

Abstract. A numerical model for long-term simulation of gravity surface waves is described. The model is designed as a component of a coupled Wave Boundary Layer/Sea Waves model, for investigation of small-scale dynamic and thermodynamic interactions between the ocean and atmosphere. Statistical properties of nonlinear wave fields are investigated on a basis of direct hydrodynamical modeling of 1-D potential periodic surface waves. The method is based on a nonstationary conformal surface-following coordinate transformation; this approach reduces the principal equations of potential waves to two simple evolutionary equations for the elevation and the velocity potential on the surface. The numerical scheme is based on a Fourier transform method. High accuracy was confirmed by validation of the nonstationary model against known solutions, and by comparison between the results obtained with different resolutions in the horizontal. The scheme allows reproduction of the propagation of steep Stokes waves for thousands of periods with very high accuracy. The method here developed is applied to simulation of the evolution of wave fields with large number of modes for many periods of dominant waves. The statistical characteristics of nonlinear wave fields for waves of different steepness were investigated: spectra, curtosis and skewness, dispersion relation, life time. The prime result is that wave field may be presented as a superposition of linear waves is valid only for small amplitudes. It is shown as well, that nonlinear wave fields are rather a superposition of Stokes waves not linear waves. Potential flow, free surface, conformal mapping, numerical modeling of waves, gravity waves, Stokes waves, breaking waves, freak waves, wind-wave interaction.

This paper focuses on the study of linear and non-linear surface and internal waves, in a complete setting, using a two-layer model of a stratified fluid. The respective Korteweg-de Vries evolutionary equations have been obtained, analysed, and compared with the ‘rigid lid’ model data. Boussinesq-type equations have been derived for the interacting modes pertaining to one type and to different types. It is shown that in addition to the known mechanisms of interaction between internal and surface waves, interaction between long non-linear baroclinic modes and barotropic modes, propagating in the same direction, is likely in such a system.

The nonlinear propagation of different types of DANW acoustic dust nonlinear waves has been investigated in a magnetized dusty plasma consisting of negatively charged dust particles, Maxwellian electrons, and ions. Application of the standard reductive perturbation theory is used to derive the corresponding three-dimensional nonlinear a complex Ginzburg–Landau (3D-CGLE) equation which governs the dynamics of the dust-acoustic wave packets. The stationary analytical solutions of the CGLE are numerically analysed where the effect of the physical parameters of the dusty plasma model on the wave’s propagation is taken into account. It has been found that there can be a relationship between the appearance of soliton waves and electromagnetic waves, as well as between shock-like waves and periodic travelling waves. Expression of the importance of these findings is the cornerstone of explaining the true relationship between the propagation of nonlinear waves in the physics of space, for example, the Earth’s magnetic field.

We use large-eddy simulation (LES) and theoretical analyses to study the turbulent flow over fast opposing water waves. A novel nonlinear viscous model for airflow perturbations induced by Stokes waves is developed, which can account for wave-perturbation viscous stress and the nonlinear forcing by multimode interactions of wave-correlated quantities in the wind field. Overall, the nonlinear viscous model can describe the wave-coherent airflow in the LES results for high-order Stokes waves, which demonstrates the negligible effects of wave-perturbation turbulent stress in the wind. According to the model, the dominant components of the fundamental mode of the airflow perturbation primarily result from the linear response of the wind to the wave and, thus, are not substantially affected by the nonlinear forcing. However, the weak components of the fundamental mode, which produce the form drag on the wave, are created by the combined effects of the nonlinear forcing and the wave-perturbation viscous stress. We found that the main mechanism for generating the nonlinear forcing is the interaction between the second harmonic and the fundamental mode of the wave-correlated quantities in the air. In this mechanism, wave nonlinearity exerts its effects mainly through the second harmonic of the wave surface, instead of the second harmonic of the wave kinematics. Therefore, it is further demonstrated that a second-order Stokes wave is sufficient to capture the wave nonlinearity effects on the form drag.

Excitation of nonlinear surface waves is studied at the SBN-75 crystal — air interface. The SBN-75 crystal is characterised by the diffusion-type photorefractive nonlinearity, the surface wave being excited in the surface layer and the depth of its penetration into the crystal being determined by the nonlinear crystal parameters. Because of a large refraction coefficient in SBN-75, the penetration depth of the surface wave into the air is small; therefore, the nonlinear surface wave is localised in the surface crystal layer several micrometers in thickness. The oscillating intensity distribution of the surface wave at the output end of the crystal is measured. The oscillation period is determined by the angle of incidence of the exciting beam. The nonlinear wave is excited not only at the crystal — air interface but also in the case when the active surface of the crystal is covered by an electrode, for example, an aquadag layer. This circumstance opens up a possibility of studying new properties of the surface waves when an external electric field is applied to the crystal.

On a base of developed hydrodynamic models the reflecting capacities of edges, cracks and ice field breaks are studied when the surface waves pass through them. It was investigated the ice flexural rigidity and compression effects on the spatial distribution of wave disturbance amplitudes and bend stresses of the ice cover on both sides of the examined horizontal non-uniformity. Assessment of the ice breaking possibility within the edge-side zones was made. The surface wave propagation from the basin deep-water area through the bottom's step into the area of the finite depth is investigated. The spatial and temporal features of the formed dynamic frontal zones on both sides of the step are revealed. The dependence of the basin surface displacement amplitudes and wave current velocities on the ice thickness, incident wave periods, and depth of the step occurrence, is studied. The three-dimensional bending vibrations of floating ice cover induced by a moving vortex formation are studied. Dependence of the amplitude-phase vibration characteristics on the ice rigidity in bending, angular velocity and forward speed of the vortex is investigated. Spatial distribution of the flexural-gravity disturbances both in front of the moving pressure region and in the back wake is analyzed. A mathematical model of the non-linear dynamics of surface waves of finite amplitude in the ice floes seas was developed involving the method of multiscale expansions. Using the model, it was studiedthe dependence of the amplitude-phase structure of disturbances, formed by the propagating non-linear periodic waves, on the basin depth, ice thickness and non-linearity of its vertical acceleration, frequency and steepness of initial wave harmonics. The ice Stokes drift velocity was estimated as well as the ice induced non-linear mass transport analyzed.&nbsp

Surface waves in a nonlinear magnetoelastic conductor of finite electric conductivity

Foreword. I: Theoretical problems concerning propagation of surface wave in elastic anisotropic media. On the role of anisotropy in crystalloacoustics V.I. Alshits. Surface waves of non-Rayleigh type S.V. Kuznetsov. Nonlinearity in elastic surface waves acts nonlocally D.F. Parker. Explicit secular equations for surface waves in an anisotropic elastic half space from Rayleigh to today T.C.T. Ting. II: Bending and edge waves in plates and shells. 'Nongeometrical phenomena' in propagation of elastic surface waves V.M. Babich, A.P. Kiselev. Complex rays and internal diffraction at the cusp edge M. Deschamps, O. Poncelet. Edge waves in the fluid beneath an elastic sheet with linear nonhomogeneity R.V. Goldstein, A.V. Marchenko. On continuum modelling of wave propagation in layered medium: bending waves K.B. Ustinov. Edge-localised bending waves in anisotropic media: energy and dispersion III: Experimental, numerical and semi-analytical methods for analysis of surface waves. Surface electromagnetic perturbations induced by unsteady-state subsurface flow P.M. Adler, V.M. Entov. Resonant waves in a structured elastic halfspace M.V. Ayzenberg-Stepanenko. Mumerical analysis of Rayleigh waves in anisotropic media A.V. Kaptsov. Guided waves in anisotropic media: applications R.A. Kline. A general purpose computer model for calculating elastic waveguide properties, with application to Non-Destructive Testing M.J.S. Lowe, P. Cawley, B.N. Pavlakovic. The influence of the initial stresses on the dynamic instability of an anisotropic cone Y.R. Rossikhin, M.V. Shitikova. Embedding theorem and mutual relation for the interface and shear wavespeeds I.V. Simonov. IV: Applications of surface waves to analysis of fracture and damage. The non-uniqueness of constant velocity crack propagation K.B. Broberg. Embedding formulae for planar cracks R.V. Craster, A.V. Shanin. Wave propagation and crack detection in layered structures E.V. Glushkov, N.V. Glushkova, A.V. Ekhlakov. Author Index.

The authors investigate the propagation of nonlinear high frequency TM surface waves on a thin unmagnetized plasma layer bounded by vacuum. It is found that envelope surface wave solitons from the modulation of finite amplitude electron waves by slow ion acoustic motion exist. The non-existence of solitons with purely electronic modulation is also discussed.

The propagation of monochromatic nonlinear symmetric hybrid surface waves in a cylindrical nonlinear anisotropic inhomogeneous metal-dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. The setting is reduced to a new type of nonlinear eigenvalue problem and an analytical method of its solution is developed. The propagation modes are found that have not been previously reported in the literature. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

The contact of a medium characterized by Kerr nonlinearity and a medium characterized by a stepwise nonlinearity, where the dielectric permittivity depends on the amplitude of the electric field strength, is considered. It is found that an optical domain of finite thickness is formed. Two new types of surface waves are described. The influence of the switching field on the intensity distribution across the interface and on total power flow is analyzed. The intensity at the interface in the case of contact with a nonlinear focusing medium can be maximal, in contrast to the case of contact with a linear medium where the intensity at the interface could not be maximal. The total power flow monotonically decreases with an increase in the nonlinearity parameter and it takes the minimum value at a specific value of the switching amplitude. The largest fraction of the power flow is concentrated on the right side of the optical domain for small values of the switching field starting from a certain value.

In this paper, we study the propagation and stability of nonlinear sound waves in accretion disks. Using the shearing box approximation, we derive the form of these waves using a semi-analytic approach and go on to study their stability. The results are compared to those of numerical simulations performed using finite difference approaches such as employed by ZEUS as well as Godunov methods. When the wave frequency is between Omega and two Omega (where Omega is the disk orbital angular velocity), it can couple resonantly with a pair of linear inertial waves and thus undergo a parametric instability. Neglecting the disk vertical stratification, we derive an expression for the growth rate when the amplitude of the background wave is small. Good agreement is found with the results of numerical simulations performed both with finite difference and Godunov codes. During the nonlinear phase of the instability, the flow remains well organised if the amplitude of the background wave is small. However, strongly nonlinear waves break down into turbulence. In both cases, the background wave is damped and the disk eventually returns to a stationary state. Finally, we demonstrate that the instability also develops when density stratification is taken into account and so is robust. This destabilisation of freely propagating nonlinear sound waves may be important for understanding the complicated behaviour of density waves in disks that are unstable through the effects of self-gravity or magnetic fields and is likely to affect the propagation of waves that are tidally excited by objects such as a protoplanet or companion perturbing a protoplanetary disk. The nonlinear wave solutions described here as well as their stability properties were also found to be useful for testing and comparing the performance of different numerical codes.