Nonlinear Gravity-capillary Waves Research Articles

Parasitic waves and micro-breaking on highly nonlinear gravity–capillary waves in a convergent channel

Gravity–capillary waves are waves influenced by both the effects of surface tension and gravity; these waves are at small scales with wavelength range from approximately 10 cm to less than 1 cm. Gravity–capillary waves play a significant role in air–sea interactions, and they exhibit much different features compared with gravity waves. They can be observed widely on the sea surface. Parasitic waves (capillary waves generated by and that ride on gravity and/or gravity–capillary waves) and micro-breaking can be observed on the water surface with winds; however, the presence of wind makes it difficult to analyse the mechanisms of the wave itself. In this paper, parasitic waves and micro-breaking on gravity–capillary waves are examined experimentally, both in the absence of wind. Parasitic waves and axisymmetric micro-breaking waves are generated mechanically in a convergent channel, where energy density increases due to spatial convergence. Three experimental techniques are used to measure wave properties: planar laser-induced fluorescence, particle image velocimetry and shadowgraphs. The wave profile evolution and vortices beneath the parasitic waves are studied. The micro-breaking of gravity–capillary waves is observed on a surface with added surfactant. The surfactant increases the Bond number, and makes breaking possible in these small-scale waves. Energy dissipation of parasitic waves and micro-breaking is quantified, and the enhanced dissipation caused by parasitic waves is identified through the experiments. In this study, mechanically generated breaking waves with wavelengths less than 10 cm are studied for the first time, without the effect of wind. The results yield insight into wave characteristics and energy dissipation on the air–sea interface at small scales.

Ondas rotacionales de gravedad-capilaridad generadas por el paso de una perturbación

Nonlinear gravity-capillary waves generated by the passage of a pressure distribution over a sheared channel with constant vorticity are investigated. The problem is modeled using the full Euler equations. The harmonic part of the velocity field is formulated in a canonical domain through the use of the conformal mapping, which flattens the fluid domain onto a strip. The Froude number is considered to be nearly-critical and the Bond number critical. The shear effect changes drastically the pattern of the generated waves for large times. Moreover, depending on the intensity of the vorticity, the wave solutions can become smoother with small amplitudes.

Dynamics of interfacial gravity–capillary waves in three-dimensional fluids of great depth

This paper proposes a new isotropic bidirectional model for weakly nonlinear gravity–capillary waves propagating between two incompressible, inviscid, and immiscible fluids of different densities. The newly developed equation is a generalization of the celebrated two-dimensional Benjamin equation. It is derived based on the nonlocal formulation of water wave (i.e., the Ablowitz–Fokas–Musslimani formulation) and computed with the modified exponential time-differencing method. It is found that horizontally two-dimensional, fully localized traveling waves (known as lumps) exist in the model equation, and plane solitary waves are unstable subject to long transverse perturbations, which evolve into groups of lumps in the long-term dynamics. When considering topographical disturbances on the rigid upper boundary, the nonlinear effect becomes important when a uniform flow passes beneath a locally confined topography with a near-critical speed, and the phenomenon of time-periodic shedding of lumps occurs. Unlike the near-critical situation, the subcritical flows usually generate steady elevation waves, while the supercritical ones produce a V-shaped pattern of wake lines. Furthermore, it is shown that a uniformly accelerated motion can also generate lumps provided that the flow stays in the transcritical regime for a considerably long time.

Discontinuous shock solutions of the Whitham modulation equations as zero dispersion limits of traveling waves

Whitham modulation theory describes the zero dispersion limit of nonlinear disperesive partial differential equations (PDEs) by a system of conservation laws for the parameters of modulated periodic traveling waves (TWs). In this work, admissible, discontinuous, weak solutions of the Whitham modulation equations—termed Whitham shocks—are identified with zero dispersion limits of TW solutions to higher order dispersive PDEs. The far-field behavior of the TW solutions satisfies the Rankine–Hugoniot jump conditions for the Whitham modulation equations. Generally, the numerically computed traveling waves represent heteroclinic connections between two periodic orbits of an ordinary differential equation. The focus here is on the fifth order Korteweg–de Vries equation where three admissible one-parameter families of Whitham shocks are identified as solution components to the generalized Riemann problem for the Whitham modulation equations. Admissible KdV5–Whitham shocks are generally undercompressive, i.e., all characteristic families pass through the shock. The heteroclinic TWs that limit to admissible Whitham shocks are found to be ubiquitous in numerical simulations of smoothed step initial conditions for other higher order dispersive equations including the Kawahara equation (with third and fifth order dispersion) and a nonlocal model of weakly nonlinear gravity-capillary waves with full dispersion. Whitham shocks are linked to recent studies of nonlinear higher order dispersive waves in optics and ultracold atomic gases. The approach presented here provides a novel method for constructing new TW solutions to dispersive nonlinear wave equations and a framework to identify physically relevant, admissible shock solutions of the Whitham modulation equations.

Gravity-capillary waves on the free surface of a liquid dielectric in a tangential electric field

Processes of propagation and interaction of nonlinear gravity-capillary waves on the free surface of a deep non-conducting liquid with high dielectric constant under the action of a tangential electric field are numerically simulated. The computational method is based on the time-dependent conformal transformation of the region occupied by the fluid into a half-plane. In the limit of a strong electric field, when the gravitational and capillary forces are negligibly small, there exists an exact analytical solution of the electro-hydrodynamic equations describing propagation without distortions of nonlinear surface waves along (or against) the electric field direction. In the situation where gravity and capillarity are taken into account, the results of numerical simulations indeed show that, for large external field, the waves traveling in a given direction tend to preserve their shape. In the limit of a strong electric field, the interaction of counter-propagating waves leads to the formation of regions, where the electrostatic and dynamic pressures undergo a discontinuity, and the curvature of the surface increases infinitely. The Fourier spectrum of the surface perturbations tends to the power-law distribution (k^-2). In the case of a finite electric field, the wave interaction results in a radiation of massive cascade of small-scale capillary waves that causes the chaotic behavior of the system. The investigated mechanism of interaction between oppositely-traveling waves can enhance development of the capillary turbulence of the fluid surface.

FIFTH-ORDER EVOLUTION EQUATION OF GRAVITY–CAPILLARY WAVES

We extend the evolution equation for weak nonlinear gravity–capillary waves by including fifth-order nonlinear terms. Stability properties of a uniform Stokes gravity–capillary wave train is studied using the evolution equation obtained here. The region of stability in the perturbed wave-number plane determined by the fifth-order evolution equation is compared with that determined by third- and fourth-order evolution equations. We find that if the wave number of longitudinal perturbations exceeds a certain critical value, a uniform gravity–capillary wave train becomes unstable. This critical value increases as the wave steepness increases.

Energy spectrum of the ensemble of weakly nonlinear gravity-capillary waves on a fluid surface

We consider nonlinear gravity-capillary waves with the nonlinearity parameter ɛ ∼ 0.1–0.25. For this nonlinearity, time scale separation does not occur and the kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in [15]. We for the first time compute an analytic expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function of the ratio of surface tension to the gravity acceleration. We show that its two limits—pure capillary and pure gravity waves on a fluid surface—coincide with the previously obtained results. We also discuss relations of the D-cascade model with a few known models used in the theory of nonlinear waves such as Zakharov’s equation, resonance of modes with nonlinear Stokes-corrected frequencies, and the Benjamin-Feir index. These connections are crucial in understanding and forecasting specifics of the energy transport in a variety of multicomponent wave dynamics, from oceanography to optics, from plasma physics to acoustics.

ENERGY SPECTRUM OF THE ENSEMBLE OF WEAKLY NONLINEAR GRAVITY--CAPILLARY WAVES ON A FLUID SURFACE

In this Letter we regard nonlinear gravity-capillary waves with parameter of nonlinearity being $\varepsilon \sim 0.1 ÷0.25$. For this nonlinearity time scale separation does not occur and kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in \emph{Kartashova, \emph{EPL} \textbf{97} (2012), 30004.} We compute for the first time an analytical expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function depending on the ratio of surface tension to the gravity acceleration. It is shown that its two limits - pure capillary and pure gravity waves on a fluid surface - coincide with the previously obtained results. We also discuss relations of the model of D-cascade with a few known models used in the theory of nonlinear waves such as Zakharov's equation, resonance of the modes with nonlinear Stokes corrected frequencies and Benjamin-Feir index. These connections are crucial in the understanding and forecasting specifics of the energy transport in a variety of multi-component wave dynamics, from oceanography to optics, from plasma physics to acoustics.

Tank study of radar backscattering from strongly nonlinear water waves

Wave tank experiments were performed to study features of microwave radar backscattering by strongly nonlinear gravity-capillary waves (GCWs). Backscattering by centimeter and partly decimeter GCW was found to be related to the microbreaking of GCWs and the generation of parasitic capillary ripples. The scattering of meter-scale GCWs was attributed to the effects of strong breaking, i.e., to the overturning of wave crests and the excitation of quasilinear (free) capillary ripples, while the scatterers for intermediate decimeter GCWs were both free and parasitic ripples.

On the stability of lumps and wave collapse in water waves

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg-de Vries solitary waves, the present study is concerned with a distinct class of gravity-capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney-Roskes-Davey-Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev-Petviashvili equation, a model for weakly nonlinear gravity-capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.

Numerical and analytical studies of non-linear gravity capillary waves in fluid layers under normal electric fields

Non-linear gravity-capillary waves travelling at constant speed are considered in the presence of a normal electric field. The fluid, which is assumed to be inviscid, irrotational and a perfect dielectric, is bounded below by a solid plate electrode held at constant voltage, and the region above the free surface is a hydrodynamically passive perfect dielectric, e.g. air. A second parallel flat plate electrode is placed laterally far away and drives a uniform normal electric field there. Electrohydrodynamic coupling occurs at the free surface through the Maxwell stresses which act to modify the normal stress balance and consequently the Bernoulli equation boundary condition there. Three harmonic problems in deforming domains need to be solved, one for the hydrodynamics and one each for the electrostatics above and below the free surface, respectively. We derive and implement an accurate boundary integral method to compute travelling waves of arbitrary wavelength and amplitude. In addition, we consider a long-wave non-linear model of the full problem and compare solutions with the direct simulations. In both cases, we establish the existence of multiple families of solutions extending the classical theory of gravity-capillary waves. An asymptotic theory is also developed to construct periodic waves with ripples. These are used in comparisons with the numerical calculations and the agreement is very good.

Waves induced by instability in falling films of finite thickness

Nonlinear gravity-capillary waves in falling viscous films of finite thickness are here investigated. Mathematical modelling of this problem is based on an evolution system with a single similarity parameter δ , which depends on the mean film thickness, gravity and liquid properties. To investigate spatially periodic structures, we apply the Galerkin procedure to obtain a dissipative dynamical system which contains disturbance wavenumber in addition to the parameter δ . Two-parametric manifold of its limit cycles is numerically constructed. Attracting invariant solutions, limit cycles and invariant torii, developing from different initial data are revealed. Parallel with the analysis of solutions to the dynamical system, the corresponding periodic waves in physical space are analysed and their mechanical meaning is discussed. The dominating waves that correspond to the attracting limit cycles have successfully been compared with experimental data.

Experiments on nonlinear gravity–capillary waves

Benjamin–Feir instability of nonlinear gravity–capillary waves is studied experimentally. The experimental results are compared with computations performed for values of wavelength and steepness identical to those employed in the experiments. The theoretical approach is based on the Zakharov nonlinear equation which is modified here to incorporate weak viscous dissipation. Experiments are performed in a wave ume which has an accurately controlled wavemaker for generation of the carrier wave, as well as an additional independent conical wavemaker for generation of controlled three-dimensional disturbances. The approach adopted in the present experimental investigation allows therefore the determination of the actual boundaries of the instability domain, and not just the most unstable disturbances. Instantaneous surface elevation measurements are performed with capacitance-type wave gauges. Multipoint measurements make it possible to determine the angular dependence of the amplitude of the forced and unforced disturbances, as well as their variation along the tank. The limits of the instability domains obtained experimentally for each set of carrier wave parameters agree favourably with those computed numerically using the model equation. The numerical study shows that application of the Zakharov equation, which is free of the narrow-band approximation adopted in the derivation of the nonlinear Schrödinger (NLS) equation, may lead to qualitatively different results regarding the stability of nonlinear gravity–capillary waves. The present experiments support the results of the numerical investigation.

Nonlinear gravity–capillary waves with forcing and dissipation

We present a study of nonlinear gravity–capillary waves with surface forcing and viscous dissipation. Based on a viscous boundary layer approximation near the water surface, the theory permits the efficient calculation of steady gravity–capillary waves with parasitic capillary ripples. To balance the viscous dissipation and thus achieve steady solutions, wind forcing is applied by adding a surface pressure distribution. For a given wavelength the properties of the solutions depend upon two independent parameters: the amplitude of the dominant wave and the amplitude of the pressure forcing. We find two main classes of waves for relatively weak forcing: Class 1 and Class 2. (A third class of solution requires strong forcing and is qualitatively different.) For Class 1 waves the maximum surface pressure occurs near the wave trough, while for Class 2 it is near the crest. The Class 1 waves are associated with Miles' (1957, 1959) mechanism of wind-wave generation, while the Class 2 waves may be related to instabilities of the subsurface shear current. For both classes of waves, steady solutions are possible only for forcing amplitudes greater than a certain threshold. We demonstrate how parasitic capillary ripples affect the dissipative and dispersive properties of the solutions. In particular, there may be a significant deviation from the linear phase speed for gravity–capillary waves. Also, wave damping is strongly enhanced by the parasitic capillaries (by as much as two orders of magnitude when compared to the case with no capillary waves). Preliminary experimental results from a wind-wave channel give good agreement with the theory. We find a sharp cut-off in the wavenumber spectra of the solutions which is similar to that observed in laboratory measurements of short gravity–capillary waves. Finally, for large wave amplitudes we find a sharp corner in the wave profile which may separate an overhanging wave crest from a train of parasitic capillaries.

Nonlinear gravity-capillary waves instability in superposed magnetic fluids influenced by a vertical magnetic field and time-dependent acceleration

This paper deals with the effect of a periodic forcing on nonlinear modulation of interfacial gravity-capillary waves propagating between two magnetic fluids of infinite depth under the influence of a constant vertical magnetic field. Based on the method of multiple scales expansion for a small amplitude of periodic force, two parametric nonlinear Schrödinger equations with explicit expressions of coefficients are derived in the resonance case. A classical nonlinear Schrödinger equation is derived in the non-resonance case. The stability of the uniform time-dependent solution is analyzed. Theoretical analysis and numerical calculations show that the resonance point is affected by the magnetic field and the applied frequency. The linear stability shows that the periodic force has a destabilizing influence in the stability criterion. It is observed that the vertical field plays the same role, and that the acceleration frequency plays a dual role in the nonlinear stability criterion. Instability was revealed in the system for large values of the applied magnetic field, but the small values of the field redistribute the stable areas.

Three-dimensional instabilities of nonlinear gravity-capillary waves

Linear three-dimensional instabilities of nonlinear two-dimensional uniform gravitycapillary waves are studied using numerical methods. The eigenvalue system for the stability problem is generated using a Galerkin method and differs in detail from techniques used to study the stability of pure gravity waves (McLean 1982) and pure capillary waves (Chen & Saffman 1985). It is found that instabilities develop in the neighbourhood of the linear (triad, quartet and quintet) resonance curves. Further, both sum and difference triad ressonances are unstable for sufficiently steep waves in consequence of which Hasselmann's (1967) theorem is restricted to weakly nonlinear waves. The appearance of a superharmonic two-dimensional instability and bifurcation to three-dimensional waves are noted.

On the Accelerations in the Free Surface of Nonlinear Gravity-Capillary Waves

Abstract The real and apparent accelerations in the free surface of nonlinear gravity-capillary waves are calculated over a wide range of wavelengths. It is shown that surface tension has hardly any effect on the results previously obtained for pure gravity waves, for wavelengths down to 20 cm in length, over the range of steepness considered. At the other extreme, we present for pure capillary waves, exact analytic results valid for all steepnesses up to and including the highest wave. This yields some intriguing conclusions including the fact that the horizontal component of the real acceleration has the opposite sign over most of the surface of the highest wave to that which it does over the surface of linear waves. We also consider the effect of gravity on this solution as well as the case when both restoring forces are of equal importance. For short waves considered here, the behavior of all the accelerations in the nonlinear wave is nonsinusoidal with maximum values considerably in excess of those p...

The fourth-order evolution equation for deep-water gravity-capillary waves

The stability of a train of nonlinear gravity-capillary waves on the surface of an ideal fluid of infinite depth is considered. An evolution equation is derived for the wave envelope, which is correct to fourth order in the wave steepness. The derivation is made from the Zakharov equation under the assumption of a narrow band of waves, and including the full form of the interaction coefficient for gravity-capillary waves. It is assumed that conditions are away from subharmonic resonant wavelengths. Just as was found by K. B. Dysthe ( Proc. R. Soc. Lond . A 369 (1979)) for pure gravity waves, the main difference from the third-order evolution equation is, as far as stability is concerned, the introduction of a mean flow response. There is a band of waves that remains stable to fourth order. In general the mean flow effects for pure capillary waves are of opposite sign to those of pure gravity waves. The second-order corrections to first-order stability properties are shown to depend on the interaction between the mean flow and the envelope frequency-dispersion term in the governing equation. The results are shown to be in agreement with some recent computations of the full problem for sufficiently small values of the wave steepness.

Particle trajectories in nonlinear gravity–capillary waves

The trajectories of surface and subsurface particles of nonlinear gravity–capillary waves are calculated. Surface tension is found to have a small effect on the trajectories and drift velocity of pure gravity waves (down to 20 cm in length). On the other hand, pure capillary wave trajectories can be considerably altered even when the influence of gravity is small (waves of up to 2 cm in length). When the restoring forces are of comparable magnitude, some remarkable trajectories are possible, containing one or more subloops. Overall, the influence of increased surface tension is to increase the relative horizontal distance travelled by a particle, as well as the magnitude of the time-averaged drift velocity ratio at the surface and, as far as short waves are concerned, its penetration depth.These results can have implications for steep waves where parasitic capillaries are generated and for observations of the wind-drift current.