被均匀流缓慢调制的有限水深毛细重力波

非线性的存在会产生高次谐波,这些谐波又反作用于原来的低次谐波,使波幅发生缓慢变化,从而产生缓慢调制现象．这里从考虑均匀流作用下的毛细重力水波基本方程出发,在不可压缩、无旋、无黏条件假设下,使用多重尺度分析方法推导出了在均匀流影响下有限深水毛细重力波振幅所满足的非线性Schr?dinger方程(NLSE)．分析了NLSE解的调制不稳定性．给出了毛细重力波调制不稳定的条件和钟型孤立波产生的条件．分析了无量纲最大不稳定增长率随无量纲水深和表面张力的变化趋势．同时给出了无量纲不稳定增长率随无量纲微扰动波数变化的曲线,呈现出了先增大后减小的趋势．最后指出均匀顺流减小了无量纲不稳定增长率及最大增长率,逆流增大了它们．由表面张力作用产生的毛细波及重力与表面张力共同作用产生的毛细重力波,与流的相互作用可以改变海表粗糙度和海洋上层流场结构,进而影响海气界面动量、热量及水汽的交换．了解海表这些短波动力机制,对卫星遥感的精确测量、海气相互作用的研究及海气耦合模式的改进等有重要意义．      

Two-dimensional modulation and instabilities of flexural waves of a thin plate on nonlinear elastic foundation

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. It is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear Schrödinger equation. The latter equation allows us to study the modulational instability conditions leading to different zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.     

Discrete breathers and modulational instability in a discrete $$\varvec{\phi ^{4}}$$ nonlinear lattice with next-nearest-neighbor couplings

The properties of discrete breathers and modulational instability in a discrete \(\phi ^{4}\) nonlinear lattice which includes the next-nearest-neighbor coupling interaction are investigated analytically. By using the method of multiple scales combined with a quasi-discreteness approximation, we get a dark-type and a bright-type discrete breather solutions and analyze the existence conditions for such discrete breathers. It is found that the introduction of the next-nearest-neighbor coupling interactions will influence the existence condition for the bright discrete breather. Considering that the existence of bright discrete breather solutions is intimately linked to the modulational instability of plane waves, we will analytically study the regions of discrete modulational instability of plane carrier waves. It is shown that the shape of the region of modulational instability changes significantly when the strength of the next-nearest-neighbor coupling is sufficiently large. In addition, we calculate the instability growth rates of the \(q=\pi \) plane wave for different values of the strength of the next-nearest-neighbor coupling in order to better understand the appearance of the bright discrete breather.     

Ginzburg-Landau system of complex modulation equations for distributed nonlinear dissipative transmission lines

The envelope modulation of a monoinductance transmission line is reduced to generalized coupled Ginzburg-Landau equations, from which a single cubic-quintic Ginzburg-Landau equation containing derivatives with respect to the space variable in the cubic terms is deduced. We investigate the modulational instability of the space wave solutions of both the system and the single equation. For the generalized coupled Ginzburg-Landau system, we consider only the zero wave numbers of the perturbations whose modulational instability conditions depend only on the coefficients of the system and the wave numbers of the carriers. In this case, a modulational instability criterion is established, which depends on both the perturbation wave numbers and the carrier. We also study the coherent structures of the generalized coupled Ginzburg-Landau system and present some numerical results. Published in Neliniini Kolyvannya, Vol. 9, No. 4, pp. 451–489, October–December, 2006.     

Maximum amplitude of modulated wavetrain

For water wave predictions, it is of practical importance to know the size of the biggest wave that emerges from a given uniform wavetrain when it experiences Benjamin-Feir's modulational instability. As far as we are aware, the series of experiments performed by Su and Green [5] are the only work that has pursued this problem so far. In this work their experimental result is compared with theoretical predictions given by the nonlinear Schrödinger equation as well as with the numerical simulation of the fully nonlinear water wave equation. The result of the fully nonlinear simulation shows that the height of the biggest wave reaches more than 3.5 times the initial amplitude of the wavetrain at the maximum modulation when the initial steepness a₀k₀ of the wavetrain is 0.125.     

Stability of bichromatic gravity waves on deep water

The stability of bichromatic gravity waves with small but finite amplitudes propagating in two directions on deep water is considered. Starting from the Zakharov equation, elementary quartet interactions are isolated and stability criteria are formulated. Results are illustrated for various combinations of bichromatic wave trains, from long-crested to standing waves. Two generic mechanisms operate: the first one is a modulational instability of one of the two components of the bichromatic wave train; the second mechanism is a modulation which couples both components of the wave train. However a third mechanism eventually comes into play: the resonant interaction of Phillips and Longuet-Higgins which leads initially to the linear growth of a third wave. When this latter is active, in particular for wave trains with wave vectors close together, it is shown by numerical integration that the long-time recurrence is destroyed.     

Dynamics of Rossby solitary waves with time-dependent mean flow via Euler eigenvalue model

The investigation on the fluctuations of nonlinear Rossby waves is of great importance for the understanding of atmospheric or oceanic motions. The present paper mainly deals with the well-known atmospheric blocking phenomena through the nonlinear Rossby wave theories and the corresponding methods. Based on the equivalent barotropic potential vorticity model in the β-plane approximation underlying a weak time-dependent mean flow, the multiscale technique and perturbation approximated methods are adopted to derive a new forced Korteweg-de Vries model equation with varied coefficients (vfKdV) for the Rossby wave amplitude. For a further analytical treatment of the obtained model problem, a special kind of basic flow is adopted. The evolution processes of atmospheric blocking are well discussed according to the given parameters according to the dipole blocking theory. The effects of some physical factors, especially the mean flow, on the propagation of atmospheric blocking are analyzed.

     

Irregularity and singular vector growth of the differentially heated rotating annulus flow

The problem of weather prediction is to a large part determined by the large-scale atmospheric flow. This flow is irregular but not fully turbulent. The irregularity can be related to instability, nonlinear wave–wave interactions, or randomly excited singular vectors. In the present study, the latter mechanism is investigated numerically and experimentally. For this purpose, a baroclinic quasigeostrophic low-order model is adjusted to a differentially heated rotating annulus experiment, an established laboratory analog to the atmospheric circulation. We linearize the low-order model about a time-mean state to study the growth of singular vectors and compare those with singular vectors that have been derived empirically from the experimental data. Qualitative agreement of the numerically and experimentally derived singular vectors form the basis for a deeper analysis of the low-order model. In particular, for a broad range of annulus rotation frequencies and radial temperature differences, we compute the maximal growth rates of the low-order model. These growth rates are displayed as a function of the Taylor and thermal Rossby number, a frame widely used in studies of annulus regime transitions. We can show that most regime transitions defined by E.N. Lorenz in the sixties have their counterpart in the Taylor-Rossby singular value diagram. Most striking is the fact that for the irregular regime by far, the largest growth rates can be found. We suppose that irregularity in the transition region to geostrophic turbulence might for some part result from randomly excited singular vectors with unusual large growth rates. In concert with nonlinear wave–wave coupling, this process might explain the gradual broadening of the wave spectrum that has been found for the route to geostrophic turbulence in annulus flows.     

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

The influence of topography on the nonlinear interaction of Rossby waves in the barotropic atmosphere

ＴＨＥＩＮＦＬＵＥＮＣＥＯＦＴＯＰＯＧＲＡＰＨＹＯＮＴＨＥＮＯＮＬＩＮＥＡＲＩＮＴＥＲＡＣＴＩＯＮＯＦＲＯＳＳＢＹＷＡＶＥＳｉＮＴＨＥＢＡＲＯＴＲＯＰＩＣＡＴＭＯＳＰＨＥＲＥＸｉｏｎｇＪｉａｎ－ｇａｎｇ（熊建刚）ＹｉＦａｎ（易帆）ＬｉＪｕｎ（李钧）（...     

Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation

The time evolution of a uniform wave train with a small modulation which grows is computed with a fully nonlinear irrotational flow solver. Many numerical runs have been performed varying the initial steepness of the wave train and the number of waves in the imposed modulation. It is observed that the energy becomes focussed into a short group of steep waves which either contains a wave which becomes too steep and therefore breaks or otherwise having reached a maximum modulation then recedes until an almost regular wave train is recovered. This latter case typically occurs over a few hundred time periods. We have also carried out some much longer computations, over several thousands of time periods in which several steep wave events occur. Several features of these modulations are consistent with analytic solutions for modulations using weakly nonlinear theory, which leads to the nonlinear Schrödinger equation. The steeper events are shorter in both space and time than the lower events. Solutions of the nonlinear Schrödinger equation can be transformed from one steepness to another by suitable scaling of the length and time variables. We use this scaling on the modulations and find excellent agreement particularly for waves that do not grow too steep. Hence the number of waves in the initial modulation becomes an almost redundant parameter and allows wider use of each computation. A potentially useful property of the nonlinear Schrödinger equation is that there are explicit solutions which correspond to the growth and decay of an isolated steep wave event. We have also investigated how changing the phase of the initial modulation effects the first steep wave event that occurs.     

Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

Characterizing JONSWAP rogue waves and their statistics via inverse spectral data

Rogue waves in random sea states modeled by the JONSWAP power spectrum are high amplitude waves arising over non-uniform backgrounds that cannot be viewed as small amplitude modulations of Stokes waves. In the context of Nonlinear Schrödinger (NLS) models for waves in deep water, this poses the challenge of identifying appropriate analytical solutions for JONSWAP rogue waves, investigating possible mechanisms for their formation, and examining the validity of the NLS models in these more realistic settings. In this work we investigate JONSWAP rogue waves using the inverse spectral theory of the periodic NLS equation for moderate values of the period. For typical JONSWAP initial data, numerical experiments show that the developing sea state is well approximated by the first few dominant modes of the nonlinear spectrum and can be described in terms of a 2- or 3-phase periodic NLS solution. As for the case of uniform backgrounds, proximity to instabilities of the underlying 2-phase solution appears to be the main predictor of rogue wave occurrence, suggesting that the modulational instability of 2-phase solutions of the NLS is a main mechanism for rogue wave formation and that heteroclinic orbits of unstable 2-phase solutions are plausible models of JONSWAP rogue waves. To support this claim, we correlate the maximum wave strength as well as the higher statistical moments with elements of the nonlinear spectrum. The result is a diagnostic tool widely applicable to both model or field data for predicting the likelihood of rogue waves. Finally, we examine the validity of NLS models for JONSWAP data, and show that NLS solutions with JONSWAP initial data are described by non-Gaussian statistics, in agreement with the TOPEX field studies of sea surface height variability.     

Bifurcation mechanism of interfacial electrohydrodynamic gravity-capillary waves near the minimum phase speed under a horizontal electric field

We investigate the evolution of interfacial gravity-capillary waves propagating along the interface between two dielectric fluids under the action of a horizontal electric field. There is a uniform background flow in each layer, and the relative motion tends to induce Kelvin–Helmholtz(KH) instability. The combined effects of gravity, surface tension and electrically induced forces are all taken into account. Under the short-wave assumption, the expansion and truncation method of Dirichlet-Neumann(DN) operators is applied to derive a reduced dynamical model. When KH instability is suppressed linearly by a considerably large electric field, our numerical results reveal that in certain regions of parameter space, nonlinear symmetric traveling wave solutions can be found near the minimum phase speed. Additionally, the detailed bifurcation structures are presented together with typical wave profiles.     

Forced dissipative Boussinesq equation for solitary waves excited by unstable topography

In the paper, the effects of topographic forcing and dissipation on solitary Rossby waves are studied. Special attention is given to solitary Rossby waves excited by unstable topography. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a forced dissipative Boussinesq equation. By using the modified Jacobi elliptic function expansion method and the pseudo-spectral method, the solutions of homogeneous and inhomogeneous dissipative Boussinesq equation are obtained, respectively. With the help of these solutions, the evolutional character of Rossby waves under the influence of dissipation and unstable topography is discussed.     

The nonlinear dynamics of elastic tubes conveying a fluid

The Kirchhoff equations for elastic tubes are modified to include the effect of fluid flow. Using the techniques of linear and nonlinear analysis specially developed for the Kirchhoff equations, the effect of the fluid flow on the basic twist-to-writhe instability is investigated. The results suggest an intriguing modification of the bifurcation threshold due to the flow. Beyond threshold the buckled tube acquires a slight curvature which modifies the flow rate and results in a correction to nonlinearity of the amplitude equation governing the deformation dynamics.     

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

Some exact expressions for the temporal evolution of long Rossby waves on a beta-plane

Some exact expressions are derived to describe the temporal evolution of forced Rossby waves in a two-dimensional beta-plane configuration where the background flow has constant zonal-mean velocity. The meridional length scale of the problem is assumed to be small relative to the zonal length scale and so the long-wave limit of zero aspect ratio is taken. In the case where the background flow velocity is zero, an exact solution is obtained in terms of generalized hypergeometric functions. A late-time asymptotic approximation is obtained and it shows that the solution oscillates with time and its amplitude goes to zero in the limit of infinite time. In the case of a non-zero background flow velocity, the solution is evaluated using two different procedures which give two equivalent expressions in terms of different generalized hypergeometric functions. The late-time asymptotic behaviour is investigated and it is found that the solution approaches a steady state in the limit of infinite time.We also derive a solution in the form of an asymptotic series expansion for the more general situation where a Rossby wave packet is generated by a zonally-localized boundary condition comprising a continuous spectrum of wavenumbers or Fourier modes. The exact solutions found here can be used as leading-order solutions in weakly-nonlinear analyses and other studies involving more realistic configurations for time-dependent Rossby waves or wave packets.     

An example of nonlinear toxic mass spreading

A one-dimensional, nonlinear problem of reproductive toxic mass spreading is studied in this paper. The nonlinearity is due to the difference of the reproduction rates in the toxic region and the nontoxic region. Multiple steady state solutions are found and their stability and instability are proved. Due to the instability, there may exist turning points (also called saddle-node bifurcation points), at which an infinitesimal perturbation of some parameters may cause a catastrophic change in the location of the steady state toxic front (the interface of the toxic region and the nontoxic region). For the time dependent case, the propagation of the toxic front is considered. An integral equation is derived to determine the propagation of the toxic front. Some numerical results are found for a specific example.     

A general nonlocal nonlinear Schrödinger equation with shifted parity,charge-conjugate and delayed time reversal

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a \(\beta \)-plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.