Finite-amplitude Solitary Waves Research Articles

DAMPING OF FINITE AMPLITUDE SOLITARY WAVES IN A FLUME

Solitary wave is a permanent wave when the dissipation is ignored. Many analytical solutions have been developed for finite amplitude solitary waves. In additional to the perturbation solutions, closed form solutions are also available (e.g., McCowan 1891, Clamond and Fructus 2003, being denoted as CF from hereon), which are more accuracy, especially for larger amplitude solitary waves which were discussed in Wang and Liu (2022). In Wang and Liu (2022) ’s laboratory experiments, solitary waves are slowly damped along the wave flume, which can be attributed to the energy dissipation inside the boundary layers on the bottom and sidewalls. Keulegan (1948) first derived an approximate solution to describe the wave damping in the wave flume for small amplitude solitary wave. The basic idea of calculating the damping effect is that the rate of energy dissipation inside the boundary layers must be the same as the rate of wave energy loss. In this work, the same principle of energy balance is adopted for calculating the wave damping for finite amplitude solitary waves in a wave flume. The closed form solutions provided in CF are employed here for analyses.

Laminar boundary layers and damping of finite amplitude solitary wave in a wave flume

In this paper, the laminar viscous effects on finite amplitude solitary wave propagation in a wave flume with a rectangular cross section are investigated. The laminar viscous effects are most important in the laminar boundary layers adjacent to the bottom and sidewalls of the flume. The closed-form solutions of Clamond and Fructus [“Accurate simple approximation for the solitary wave,” C. R. Mec. 331, 727 (2003)] provide the free stream velocity to derive the boundary layer flows. The rotational velocities inside the bottom and sidewall boundary layers are first obtained analytically after linearization. Numerical solutions for the nonlinear boundary layer flows are also calculated. The importance of the nonlinearity is discussed. The laminar viscous damping rate is estimated by balancing the energy dissipation inside the boundary layers and the rate of wave energy change in the entire wave. Numerical results of the damping rate show the dependence on the viscosity of the fluid, the wave amplitude, the water depth, and the width of the wave flume. New laboratory experiments on the wave damping of finite amplitude solitary waves are also performed. The theoretical results are confirmed with the laboratory data. The methodologies introduced in this paper for obtaining boundary layer solutions and laminar viscous damping rate can be applied to other transient waves with finite amplitude.

On finite amplitude solitary waves—A review and new experimental data

The existing analytical solutions for finite amplitude solitary waves, including the perturbation solutions, based on either the nonlinearity parameter, α=H/h, or the dispersion parameter, ε=k2h2, and the closed form solutions, are reviewed. The convergence characteristics of the perturbation solutions are discussed, showing that the perturbation solutions for the velocity field diverge for large wave amplitude. The relationships between three existing closed form solutions are discussed. The analytical solutions are then compared with exact numerical solutions. The agreement is generally good for the free surface profiles, but not for the velocity field. One of the closed form solutions [Clamond, D. and Fructus, D., “Accurate simple approximation for the solitary wave,” C. R. Mec. 331, 727 (2003)] is in almost perfect agreement with the exact numerical solutions for both the free surface profiles and the velocity fields. New laboratory experiments, measuring both free surface profile and velocity field over a wide range of α values (up to 0.6) are then presented. High speed particle image velocimetry is used to measure the velocity field in the entire water column. Detailed comparisons among the experimental data, analytical theories, and numerical solutions show that for relatively small amplitude solitary waves, say, α≤0.2, all theories and numerical results agree very well with the experimental data. However, when α≥0.3 only [Clamond, D. and Fructus, D., “Accurate simple approximation for the solitary wave,” C. R. Mec. 331, 727 (2003)]'s solution and the numerical agree with the experimental data.

Numerical simulation of solitary gravity waves on deep water with constant vorticity

We present a numerical study of essentially nonlinear dynamics of surface gravity waves on deep water with constant vorticity using governing equations in conformal coordinates. The dispersion relation of surface gravity waves on shear flow is known to have two branches; one of which is weakly dispersive for long waves. Weakly nonlinear evolution of the waves of this branch can be described by the Benjamin–Ono equation, which is integrable and has soliton and multi-soliton solutions. Currently, the extent to which the properties of such solitary waves obtained within the weakly nonlinear model are preserved in the exact Euler equations is unknown. We investigate the behavior of this class of solitary waves without the restrictive assumption of weak nonlinearity by using the exact Euler equations. The evolution of localized initial perturbations leading to the formation of single or multiple solitary waves is modeled, and the properties of finite-amplitude solitary waves are discussed. We show that within the framework of the exact equations, two-soliton collisions are almost elastic, but in contrast to solutions of the Benjamin–Ono equation, the waves receive a phase shift as a result of the interaction.

High-order strongly nonlinear long wave approximation and solitary wave solution

We consider high-order strongly nonlinear long wave models expanded in a single small parameter measuring the ratio of the water depth to the characteristic wavelength. By examining its dispersion relation, the high-order system for the bottom velocity is found stable to all disturbances at any order of approximation. On the other hand, systems for other velocities can be unstable and even ill-posed, as signified by the unbounded maximum growth. Under the steady assumption, a new third-order solitary wave solution of the Euler equations is obtained using the high-order strongly nonlinear system and is expanded in an amplitude parameter, which is different from that used in weakly nonlinear theory. The third-order solution is shown to well describe various physical quantities induced by a finite-amplitude solitary wave, including the wave profile, horizontal velocity profile, particle velocity at the crest and bottom pressure. For numerical computations, the first- and second-order strongly nonlinear systems for the bottom velocity are considered. It is shown that finite difference schemes are unstable due to truncation errors introduced in approximating high-order spatial derivatives and, therefore, a more accurate spatial discretization scheme is necessary. Using a pseudo-spectral method based on finite Fourier series combined with an iterative scheme for the inversion of a non-local operator, the strongly nonlinear systems are solved numerically for the propagation of a single solitary wave and the head-on collision of two counter-propagating solitary waves of finite amplitudes, and the results are compared with previous laboratory measurements.

Planar and non-planar nucleus-acoustic solitary waves in warm degenerate multi-nucleus plasmas

A warm degenerate plasma (containing ultra-relativistically or non-relativistically warm degenerate inertia-less electron species, non-relativistically warm degenerate inertial light nucleus species and stationary heavy nucleus species) is considered. The basic features of planar and non-planar solitary structures associated with the degenerate pressure-driven nucleus-acoustic waves propagating in such a warm degenerate plasma system are investigated. The reductive perturbation method, which is valid for small- but finite-amplitude solitary waves, is used. It is found that the effects of non-planar cylindrical and spherical geometries, non- and ultra-relativistically degenerate electron species and the temperature of degenerate electron species significantly modify the basic features (i.e. speed, amplitude and width) of the solitary potential structures associated with degenerate pressure-driven nucleus-acoustic waves. The warm degenerate plasma model under consideration is applicable not only to all cold white dwarfs, but also to many hot white dwarfs, such as DQ white dwarfs, white dwarf H1504+65, white dwarf PG 0948+534, etc.

Effects of Vortex-Like Ion Distribution on Dust-Acoustic Solitary Waves in a Self-Gravitating Opposite Polarity Dusty Plasmas

A self-gravitating opposite polarity dust plasma (SGOPDP) medium (containing both positively and negatively charged dust, vortex-like distributed ions and Maxwellian electrons) has been considered in order to examine the effect of vortex-like (trapped) ion distribution on dust-acoustic (DA) solitary waves (SWs) propagating in SGOPDP medium. The reductive perturbation method, which is valid for small but finite amplitude SWs, is employed to derive a modified Korteweg-de Vries equation having stronger nonlinearity. The basic features of the DA SWs in SGOPDP medium are found to be significantly modified by the combined effect of self-gravitational field and vortex-like ion distribution. The results of this paper have many implications in space and laboratory dusty plasmas.

Resonance theory of water waves in the long-wave limit

AbstractThe instability due to resonant interactions of finite-amplitude water waves is examined in the long-wave limit. In contrast to the well-known case of a small-amplitude limit in which the resonance is considered for a flat surface, here we consider a periodic approximate of the finite-amplitude solitary wave which is the long-wave limit of the periodic wave. The resonance conditions for the corresponding perturbations yield a new family of resonance curves that are totally different from those of the small-amplitude limit obtained by Phillips and Mclean. Under these resonance conditions, we conduct a systematic asymptotic analysis for small wavenumbers to obtain the growth rates of the perturbations explicitly and clarify whether each resonance curve is associated with instability. These results are verified numerically by showing that the instability bands for finite-amplitude periodic waves in shallow water are located along these unstable resonance curves.

Effect of ion temperature on ion-acoustic solitary waves in a magnetized plasma in presence of superthermal electrons

Obliquely propagating ion-acoustic soliatry waves are examined in a magnetized plasma composed of kappa distributed electrons and fluid ions with finite temperature. The Sagdeev potential approach is used to study the properties of finite amplitude solitary waves. Using a quasi-neutrality condition, it is possible to reduce the set of equations to a single equation (energy integral equation), which describes the evolution of ion-acoustic solitary waves in magnetized plasmas. The temperature of warm ions affects the speed, amplitude, width, and pulse duration of solitons. Both the critical and the upper Mach numbers are increased by an increase in the ion temperature. The ion-acoustic soliton amplitude increases with the increase in superthermality of electrons. For auroral plasma parameters, the model predicts the soliton speed, amplitude, width, and pulse duration, respectively, to be in the range of (28.7–31.8) km/s, (0.18–20.1) mV/m; (590–167) m, and (20.5–5.25) ms, which are in good agreement with Viking observations.

Weakly nonlinear kink-type solitary waves in a fully relativistic plasma

A fully and coherent relativistic fluid model derived from the covariant formulation of relativistic fluid equations is used to study small but finite amplitude solitary waves. This approach has the characteristic to be consistent with the relativistic principle and consequently leads to a more general set of equations valid for fully relativistic plasmas with arbitrary Lorentz relativistic factor. A kink-solitary wave solution is outlined. Due to electron relativistic effect, the localized structure may experience either a spreading or a compression. This latter phenomenon (compression) becomes less effective and less noticeable as the relativistic character of the ions becomes important. Our results may be relevant to cosmic relativistic double-layers and relativistic plasma structures that involve energetic plasma flows.

Finite-amplitude solitary waves in a two-layer fluid

Second-mode nonlinear internal waves at a thin interface between homogeneous layers of immiscible fluids of different densities have been studied theoretically and experimentally. A mathematical model is proposed to describe the generation, interaction, and decay of solitary internal waves which arise during intrusion of a fluid with intermediate density into the interlayer. An exact solution which specifies the shape of solitary waves symmetric about the unperturbed interface is constructed, and the limiting transition for finite-amplitude waves at the interlayer thickness vanishing is substantiated. The fine structure of the flow in the vicinity of a solitary wave and its effect on horizontal mass transfer during propagation of short intrusions have been studied experimentally. It is shown that, with friction at the interfaces taken into account, the mathematical model adequately describes the variation in the phase and amplitude characteristics of solitary waves during their propagation.

Solitary waves in a weakly stratified two-layer fluid

A problem on stationary waves on the interface between a homogeneous fluid and an exponentially stratified fluid is considered. The density difference on the interface being assumed to have the same order of smallness as the density gradient of the fluid inside the stratified layer, an equation of the second-order shallow water approximation is derived for description of propagation of finite-amplitude solitary waves.

On the solitary wave paradigm for tsunamis

Since the 1970s, solitary waves have commonly been used to model tsunamis especially in experimental and mathematical studies. Unfortunately, the link to geophysical scales is not well established, and in this work, we question the geophysical relevance of this paradigm. In part 1, we simulate the evolution of initial rectangular‐shaped humps of water propagating large distances over a constant depth. The objective is to clarify under which circumstances the front of the wave can develop into an undular bore with a leading soliton. In this connection, we discuss and test various measures for the threshold distance necessary for nonlinear and dispersive effects to manifest in a transient wave train. In part 2, we simulate the shoaling of long smooth transient and periodic waves on a mild slope and conclude that these waves are effectively non‐dispersive. In this connection, we discuss the relevance of finite amplitude solitary wave theory in laboratory studies of tsunamis. We conclude that order‐of‐magnitude errors in effective temporal and spatial duration occur when this theory is used as an approximation for long waves on a sloping bottom. In part 3, we investigate the phenomenon of disintegration of long waves into shorter waves, which has been observed, for example, in connection with the Indian Ocean tsunami in 2004. This happens if the front of the tsunami becomes sufficiently steep, and as a result, the front turns into an undular bore. We discuss the importance of these very short waves in connection with breaking and runup and conclude that they do not justify a solitary wave model for the bulk tsunami.

Ion-acoustic solitary waves and their interaction in a weakly relativistic two-dimensional thermal plasma

This paper discusses the existence of ion-acoustic solitary waves and their interaction in a weakly relativistic two-dimensional thermal plasma. Two Korteweg–de Vries equations for small but finite amplitude solitary waves in both ξ and η directions are derived. The phase shifts and trajectories of two solitary waves after the collision with an arbitrary angle α are also obtained. The effects of parameters of the normalized ion temperature σ, the ratio of heat capacity δ, the relativistic factor Fγ, and the colliding angle α on the amplitudes, the widths and the phase shifts of both the colliding solitary waves are studied. The effects of these parameters on the new nonlinear wave created by the collision between two solitary waves are examined as well. The results suggest that these parameters can significantly influence the amplitude, the width of the newly formed nonlinear wave and the colliding solitary waves. The phase shifts of the colliding solitary waves strongly depend on the colliding angle α. Moreover, there are compressive solitary waves in such a system.

Propagation and Interaction of Ion-Acoustic Solitary Waves in a Two-Dimensional Plasma Consisting of Isothermal Electrons and Hot Ions

We study the propagation and interaction of ion-acoustic solitary waves in a simple two-dimensional plasma by using the extended Poincaré–Lighthill–Kuo perturbation method. We consider the interaction between two ion-acoustic solitary waves with different propagation directions in such a system, and obtain two Korteweg-de Vries equations for small but finite amplitude solitary waves along both ξ and η trajectories. The effects of the ratio of ion temperature σ, the ratio of heat capacity γ and the colliding angle α. on the amplitude, the width of the new nonlinear wave created by the collision between two solitary waves are studied. The effects of these parameters on both the colliding solitary waves are examined as well. It is found that all the above-mentioned parameters have significant effects on the properties of these nonlinear waves.

Free surface waves in equilibrium with a vortex

Finite amplitude solitary waves of uniform depth which interact with a stationary point vortex are considered. Waves both with and without a submerged obstacle are computed. The method of solution is collocation of Bernoulli's equation at a finite number of points on the free surface coupled with equations for equilibrium of a point vortex. The stream function and vortex location are found by computing a conformal map of the flow domain to an infinite strip. For a given obstacle the solutions are parametrized with respect to Froude number and vortex circulation. When no obstacle is present there are two families of solutions, in one of which the amplitude of the wave increases by increasing the circulation, while in the other amplitude increases by decreasing the circulation. Beyond a certain critical Froude number the maximum amplitude wave has a sharp crest with an angle of 120 degrees. Similar behavior is observed for the flow past a submerged obstacle except that there is a critical Froude number below which there is no solution at all.

Dust acoustic solitary waves and double layers in a dusty plasma with two-temperature trapped ions

The combined effects of trapped ion distribution, two-ion-temperature, dust charge fluctuation, and dust fluid temperature are incorporated in the study of nonlinear dust acoustic waves in an unmagnetized dusty plasma. It is found that, owing to the departure from the Boltzmann ion distribution to the trapped ion distribution, the dynamics of small but finite amplitude dust acoustic waves is governed by a modified Korteweg–de Vries equation. The latter admits a stationary dust acoustic solitary wave solution, which has stronger nonlinearity, smaller amplitude, wider width, and higher propagation velocity than that involving adiabatic ions. The effect of two-ion-temperature is found to provide the possibility for the coexistence of rarefactive and compressive dust acoustic solitary structures and double layers. Although the dust fluid temperature increases the amplitude of the small but finite amplitude solitary waves, the dust charge fluctuation does the opposite effect. The present investigation should help us to understand the salient features of the nonlinear dust acoustic waves that have been observed in a recent numerical simulation study.

The Boussinesq–Rayleigh approximation for rotational solitary waves on shallow water with uniform vorticity

The theoretical work reported herein studies the free-surface profile, the flow structure, and the pressure distribution of a finite-amplitude solitary wave on shallow water with uniform vorticity. The kinematic problem for the stream function is formulated employing the vertical coordinate and the free surface as the independent variables of the Poisson equation with variable coefficients that are functions of the Hamiltonian of the rotational solitary wave. The exact solution of the boundary-value kinematic problem for the stream function is derived in the form of a power series complemented by a recurrence relation. The dynamic problems for the Hamiltonian and the free surface are solved globally in the Boussinesq–Rayleigh approximation. To find angles enclosed by the branches of the solution at critical points and points of bifurcation the surface streamline is also treated locally by an exact topological solution. The complete analysis of the four-dimensional Hamiltonian maps presented in §4 specifies critical values of the Froude number and the vorticity for five flow regimes: the emergence of the solitary wave, the flow separation near the bottom, the flow separation near the crest, the critical regime for an instability, and the formation of a limiting configuration. The streamlines of the recirculating flow are obtained as a single-eddy bifurcation that preserves continuity of all derivatives on the boundary streamline. The eddy separated near the crest forms the limiting configuration by blocking the upstream current. The results are compared with weakly nonlinear theory, with numerical simulations and with field observations with satisfactory agreement.

Solitary waves of permanent form in a deep fluid with weak shear

The Benjamin–Davis–Acrivos–Ono equation is generalized to account for finite, large amplitude solitary waves in a sheared deep fluid. It is shown how fine structure of stratification and weak noncritical shear in such geophysical flows do affect length (shape), wave (phase) velocity, and even stability of finite amplitude solitary waves.

Drop formation during coating of vertical fibres

When the coating film around a vertical fibre exceeds a critical thickness hc, the interfacial disturbances triggered by Rayleigh instability can undergo accelerated growth such that localized drops much larger in dimension than the film thickness appear. We associate the initial period of this strongly nonlinear drop formation phenomenon with a self-similar intermediate asymptotic blow-up solution to the long-wave evolution equation which describes how static capillary forces drain fluid into the drop. Below hc, we show that strongly nonlinear coupling between the mean flow and axial curvature produces a finite-amplitude solitary wave solution which prevents local finite-time blow up and hence disallows further growth into drops. We thus estimate hc by determining the existence of solitary wave solutions. This is accomplished by a matched asymptotic analysis which joins the capillary outer region of a large solitary wave to the thin-film inner region. Our estimate of hc = 1.68R3H–2, where R is the fibre radius and H is the capillary length H = (σ/ρg)½, is favourably compared to experimental data.