Growth Of Wave Amplitude Research Articles

Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction

This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.

Liquid sloshing and wave breaking in circular and square-base cylindrical containers

Near resonance sloshing in containers, filled with a liquid to a given depthh, depends on three parameters, which are the viscous damping, the frequency offset that contains the forcing amplitude and the fluid depth. Experiments have been conducted with low-viscosity liquids mainly in circular cylindrical containers of radiusRsubjected to harmonic horizontal forcing; complementary experiments on wave breaking have been performed in a square-base container. The fluid depth was kept large (h/R> 1) so that it was no longer a variable parameter. The bounds of existence of the different wave regimes, namely planar waves, swirling waves, chaotic sloshing as well as breaking waves, have been determined as a function of forcing frequencies relative to the lowest natural frequency ω1and for a wide range of forcing amplitudes. It is shown that when the forcing frequency ω is slightly larger than the lowest natural frequency ω1, planar wave motion bifurcates to a swirling wave mode at finite wave amplitude, the value of which depends on the offset parameter. The swirl wave amplitude grows exponentially and saturates at a certain value. The swirl has a hard-spring behaviour, is very robust and can generate a vortical flow of the liquid column. Chaotic sloshing and wave breaking occur quasi-periodically: growth of planar wave amplitude at a rate depending on the forcing amplitude, collapse, irregular swirl and again growth of planar wave amplitude. The details and periodicity of the chaotic behaviour and breaking depend on the frequency-offset parameter. Close to the natural frequency, chaotic wave motion is possible without breaking. Planar wave breaking is, in general, associated with spilling caused by the encounter of nearly freely falling lumps of fluid with the upward moving wave crest, in a way demonstrated previously in two-dimensional wave breaking. In three dimensions, the wave crest is destabilized in the crosswise direction so that spilling is not uniform along the wave crest and an irregular swirl is generated following breaking; free fall of fluid lumps occurs over many wave periods. The complementary experiments, performed in a square-base container of base dimensionL, show four different wave patterns of wavelengthsLandL/2 crosswise to the primary wave. This cross-wave instability is interpreted in terms of parametric instability.

Influence of thermocapillary effects on wave characteristics of the heated liquid film

Wave characteristics of the water film flow over a vertical plate with a heater were studied. Without a heat flux, data obtained perfectly coincide with results of other authors obtained for an isothermal liquid film. When heating the falling liquid, thermocapillary forces form the rivulets and thin film between them. It was found out that an increase in the heat flux causes a rise of the phase velocity and decreases the frequency of 3D waves. It is shown that in contrast to the known data for an isothermal liquid film, the average relative wave amplitude increases with a rise of dimensionless complex Reloc/Ka loc 1/11 at relatively high heat flux densities in the interrivulet. The growth of wave amplitude under the action of thermocapillary forces was revealed in the interrivulet, what corresponds to calculations.

Statistical and dynamical analyses of generation mechanisms of solitary internal waves in the northern South China Sea

This study analyzes the effects of thermocline shoaling on the ocean internal wave (IW) generation in the north South China Sea (NSCS). Seven years of satellite synthetic aperture radar (SAR) images from 1995 to 2001 are used for the statistical analysis of IW occurrence, and field measurements of sea surface wind, sea state, and vertical temperature profiles are used for analyzing IW generation and SAR imaging conditions. Latitudinal distribution of IW packets shows that 22% of IW packets distributed in the east of 118°E obviously originate from the Luzon Strait, and 78% of IW packets west of 118°E may propagate from the east or evolve into the solitons originating from the east boundary owing to the fission effect of shoaling thermocline. The yearly distribution of IW occurrence frequencies reveals an interannual variability, implying that there are long‐term and large‐scale processes modifying the SAR‐observed IW occurrence. The monthly SAR‐observed IW occurrence frequencies show that the high frequencies are distributed from April to July and reach a peak in June with a maximum frequency of 20%. The low occurrence frequencies are distributed in winter from December to February of next year with a minimum frequency of 1.5% in February. This study proposes that the IW generation needs the necessary and sufficient conditions: initial disturbance formation and wave amplitude growth. Owing to the dissipation effect on the initial disturbance, only fully grown waves have a chance to radiate out of the source region. A physical model and PKdV equation are adopted for analyzing the sufficient conditions for solitary IW amplitude growth. The results indicate that the thermocline shoaling provides the forcing to soliton amplitude growth, so that the soliton amplitude growth ratio (SAGR) serves as a decisive factor for the IW occurrence frequency. Theoretical analysis predicts a linear relation between the two. Application of theoretical models to field measurements in the Luzon Strait gives a correlation coefficient as high as 0.845 with a confidence level of 99% for months from March to November. The linear regression gives a correlation coefficient (R2) of 0.6519 and a SAGR threshold (minimum) value of 0.90 for IW occurrence. According to the theoretical solutions, the eastward propagating disturbances have no chance to grow up, so that they hardly appear on the east side of the submarine ridges in the eastern Luzon Strait.

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.

RF propagation in finite thickness unidirectional magnetic photonic crystals

This paper presents an analysis of a new class of magnetic photonic crystals (MPCs) constructed from periodic arrangements of available (possibly anisotropic) homogeneous material layers. Earlier, analytical studies of semi-infinite versions of these crystals demonstrated that they exhibit the phenomena of minimal reflection at their interface, large amplitude growth of the harmonic wave within the crystal, and concurrent group velocity slow-down. These characteristics are associated with the so called frozen mode and occur at a specific frequency associated with a stationary inflection point within the Bloch diagram. In this paper, we present a characterization of these phenomena for a practical, finite thickness crystal slab and propose a realizable combination of materials consisting of available ferrite and dielectric media. The existence of significant wave amplitude growth and slow down are verified for materials with realistic losses. In addition, we identify and characterize the bandwidth of the magnetic photonic crystals and examine its relationship to the amplitude growth.

Effect of ionization on ion acoustic solitary waves in a collisional dusty plasma

The nonlinear wave propagation characteristics of ion acoustic (IA) solitary waves in a dusty plasma, i.e. dust ion acoustic (DIA) solitary waves in the presence of ionization, collisions of ions with dust grains and with the background neutral gas, have been investigated by employing a reductive perturbation technique. It has been found that the nonlinear DIA wave is governed by Korteweg–de Vries (KdV) equation with a positive or negative linear damping term. It has also been found that the ionization instability leads to the exponential growth of the DIA solitary wave amplitude with time, whereas ion–dust and ion–neutral collisions reduce the growth rate.

Receptivity of boundary layers to distributed wall vibrations

Linear three-dimensional receptivity of boundary layers to distributed wall vibrations in the large Reynolds number limit (Re-->infinity) is studied in this paper. The fluid motion is analysed by means of the multiscale asymptotic technique combined with the method of matched asymptotic expansions. The body surface is assumed to be perturbed by small-amplitude oscillations being tuned in resonance with the neutral Tollmien-Schlichting wave at a certain point on the wall. The characteristic length of the resonance region is found to be O(Re-3/16), which follows from the condition that the boundary-layer non-parallelism and the wave amplitude growth have the same order of magnitude. The amplitude equation is derived as a solvability condition for the inhomogeneous boundary-value problem. Investigating detuning effects, we consider perturbations in the form of a wave packet with a narrow O(Re-3/16) discrete or continuous spectrum concentrated near the resonant wavenumber and frequency. The boundary-layer laminarization based on neutralizing the oncoming Tollmien-Schlichting waves (or wave packets) is also discussed.

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.

A numerical study on global propagations and amplitude growths of large‐scale gravity wave packets

By using a full‐implicit‐continuous‐Eulerian (FICE) scheme and taking a set of basic atmospheric motion equations in spherical coordinates as governing equations, a fully nonlinear numerical model for the dynamics of the middle and upper atmosphere is established to numerically study the nonlinear global propagations and amplitude growths of large‐scale gravity wave packets. The simulation results show that the newly established model can successfully exhibit the essential characteristics of the global propagations of gravity waves: wave energies propagate upward along the ray paths derived from the linear gravity wave theory, and wave disturbance amplitudes increase with the increasing heights. During the upward propagation of gravity wave packets, there occurs evident enhancement of background wind along the propagation direction. Moreover, our simulation demonstrates that Earth rotation has little influences on the wave energy propagations and amplitude growths. However, the primary nonlinear curvature terms ( tan and − tanθ) play an important role in the growths of gravity wave amplitudes, which causes a evident latitudinal dependence of wave amplitude growth. In the Northern Hemisphere, with the same spatial scales and initial zonal disturbance amplitudes, the waves propagating in lower‐latitude regions have larger disturbance amplitudes.

Normal Mode Rossby Waves as Revealed byUARS/ISAMS Observations

Global behavior of the 5- and 10-day waves, which are typical modes of normal mode Rossby waves, has been investigated for the region up to the mesopause with the aid of Upper Atmosphere Research Satellite Improved Stratosphere and Mesospheric Sounder observations. It is found that wave structures are easily influenced in the mesosphere by nonuniform and strong background zonal winds and an increasing damping effect with height. The vertical growth of wave amplitudes is prohibited in the mesosphere even without critical regions, and amplitudes become constant or slightly decay near the mesopause. Nevertheless, estimated variations of the wind field in the mesosphere related to the waves are consistent with results of radar observations, and Rossby normal modes may fairly contribute to total wave variance there.

The formation of ionosphere–magnetosphere ducts over the seismic zone

A physical mechanism of whistler duct formation due to electric field growth in the ionosphere above the seismic zone is considered. The ducts are plasma irregularities, stretched along geomagnetic field, with the transverse spatial scale of the order of 10 km and with the relative plasma density variation over the background value of the order of 10%. It is shown that such structures are formed by the field-aligned currents as a result of the interaction between the ionospheric electric field and the horizontal irregularities of the ionospheric conductivity. The conductivity irregularities are excited by the instability of acoustic-gravity waves (AGW), developed in the ionosphere when the electric field exceeds a certain critical value. Since the ionosphere conductivity disturbances depend on the AGW parameters, the wave amplitude growth results in modulating the conductivity and forming the additional currents due to the ionospheric electric field. The Joule heat released by these currents amplifies the AGW amplitude and, hence, the magnitude of the conductivity variations, which resulted in a growth of the oscillations and in the exponential growth of the disturbances, associated with the acoustic-gravity wave.

Experiments on space-charge waves in electron beams propagating through a resistive-wall channel

Localized longitudinal space-charge waves in a transport channel with a resistive wall are investigated experimentally. The space-charge waves with various pulse widths are generated in electron beams with energies of 3.0–8.5 keV and currents of 30–135 mA, and they are transported through a resistive channel with a resistance of 5–10 kΩ and a length of 1 m. The resistive channel, made of a glass tube with a resistive material coating, is surrounded by a long solenoid which provides the focusing for the beam. The localized space-charge waves are measured with current monitors and energy analyzers at the entrance and exit of the resistive channel, and the wave-amplitude growth and damping rates are determined. The measured results are compared with the theory in the long-wavelength range. For localized waves with short widths, where the long-wavelength approximation is not valid any more, dispersion and distributed capacitance effects on the growth/damping rates are discussed. In addition, preliminary results on the energy width measurements of space-charge waves are presented.

Kinetics of ordering and spinodal decomposition in the pair approximation

The kinetics of atomic ordering and phase separation in inhomogeneous systems were investigated employing microscopic master equations in the pair approximation. For a homogeneous system at completely thermodynamic equilibrium, the kinetic equations produce the same equilibrium states as the cluster variation method. We studied the kinetics of both short- and long-range order and spinodal decomposition kinetics, as well as the morphological pattern formation and evolution. It is shown that the development of long-range order and growth of concentration wave amplitudes are significantly delayed due to the decreases in driving forces as a result of short-range order relaxation in the pair approximation as compared to that obtained from the point approximation with the same initial condition. It is demonstrated that the kinetics obtained by assuming the pair distribution functions always at equilibrium is found to be a good approximation for the pair approximation. The effect of bond energies on the ordering kinetics will be discussed.

Characteristics of annulus baroclinic flow structure during amplitude vacillation

Abstract An investigation is made of the mechanics of amplitude vacillation in a numerically simulated rotating annulus flow system. Amplitude vacillation is characterized by a periodic change of vertical wave structure in concert with growth and decay of wave amplitude and phase speed. The temperature wave amplitude profile for the dominant component consists of three local maxima: (1) lower boundary layer, (2) upper half layer and (3) lower half layer. The lower layer waves lead the time-dependent structural variation during vacillation. Two types of amplitude vacillation found in the experimental measurements (Buzyna et al., 1989: J. Atmos. Sci. 46, 2716–2729) can be distinguished in the temperature wave by whether the lower layer waves split from and travel behind the upper layer waves by one wave period in each cycle of vacillation. Linear eigenvalue analyses with respect to the instantaneous axisymmetric state at various points in time are performed to elucidate the simple interaction between the dominant wave and the zonal mean state. During the vacillation cycle, the zonal mean state is modified by the wave, which causes a change in growth rate and vertical structure of the linearly most unstable eigenmode. This, in turn, forces the actual changes of the nonlinear solutions.

Excitation of nonlinear Alfvén waves by an ion beam in a plasma: 1. Right‐hand polarized waves

A self‐consistent theory of MHD wave excitation in the solar wind plasma by a cold ion beam moving along the ambient magnetic field is developed. Using analytical and numerical technics, we investigate the linear stage of the exponential growth of wave amplitudes, their saturation due to trapping or pitch angle diffusion of resonant ions and the steepening of the wave profile due to plasma nonlinearity. The derivative nonlinear Schrödinger equation was used to describe the nonlinear input of plasma particles, and a hybrid method of incomplete numerical simulation was used to investigate strongly nonlinear motion of resonant particles. We have studied the temporal behavior of wave energy for waves propagating in both positive and negative directions in relation to the magnetic field, directions and we have found that in cases under consideration the steepening profile developed only for right‐hand polarized waves. The physical mechanism for wave steepening is higher harmonic generation due to plasma nonlinearity.

Characteristics of Finite Amplitude Stationary Gravity Waves in the Atmosphere of Venus

This paper extends the study of stationary gravity waves generated near the surface of Venus reported previously by Young et al. to include finite amplitude effects associated with large amplitude waves. Waves are forced near the surface of Venus by periodic forcing. The height-dependent profiles of static stability and mean wind in the Venus atmosphere play a very important role in the evolution of the nonlinear behavior of the waves, just as they do in the linear wave solutions. Certain wave properties are qualitatively consistent with linear wave theory, such as wave trapping, resonance, and wave evanescence for short horizontal wavelengths. However, the finite amplitude solutions also exhibit many other interesting features. In particular, for forcing amplitudes representative of those that could be expected in mountainous regions such as Aphrodite Terra, waves generated near the surface can reach large amplitudes at and above cloud levels, with clear signatures in the circulation pattern. At still higher levels, the waves can reach large enough amplitude to break, unless damping rates above the clouds are sufficient to limit wave amplitude growth. Well below cloud levels the waves develop complex flow patterns as the result of finite amplitude wave–wave interactions, and waves are generated having considerably shorter horizontal wavelengths than that associated with the forcing near the surface. Nonlinear interactions can excite waves that are resonant with the background wind and static stability fields even when the primary surface forcing does not, and these waves can dominate the wave spectrum near cloud levels. A global map of Venus topographic slopes derived from Magellan altimetry data shows that slopes of magnitude comparable to or exceeding that used to force the model are ubiquitous over the surface.

Inertio-Gravity Waves in the Atmosphere of Neptune

A refined analysis of Voyager 2 radio occultation data has revealed distinct, quasi-periodic temperature variations in the troposphere and stratosphere of Neptune. Temperature oscillations with amplitudes of 0.1-1 K and vertical wavelengths of 1-10 km are present at latitudes 61°N (ingress) and 45°S (egress). We used techniques of spectral analysis to identify the predominant periodicities of the data and applied digital filters to isolate signal perturbations within selected wavelength passbands. Basic results include detailed vertical profiles of the temperature perturbations associated with these atmospheric waves. The data also provide a weak constraint on the horizontal structure—the horizontal wavelength probably exceeds the vertical wavelength by at least an order of magnitude. We confined our interpretation to two waves in the stratosphere whose vertical structure was characterized over 9 and 15 cycles of oscillation, respectively, spanning 2 to 3 pressure scale heights. In both cases, the observations are consistent with the characteristics of inertio-gravity waves, but cannot be reconciled with a Rossby-wave interpretation. We applied standard theory to model the wave behavior and to estimate the wave contributions to the eddy mixing and momentum budget of the middle atmosphere. The exponential growth of wave amplitude with height should result in overturning, or wave breaking, at pressures less than about 0.5 mbars for one wave and about 3 mbars for the other. The process of wave dissipation, though poorly understood, may be a significant source of "eddy mixing" on Neptune, with wave-generated eddy diffusivities as large as 50 m2 sec-1. The wave forcing of the mean flow that accompanies dissipation could alter the zonal winds by as much as about 9 m sec-1 per planet rotation.

Nonlinear wave-wave interactions related to gravity wave reflection in the auroral upper F-region observed with the EISCAT radar

Some observational evidences of nonlinear wave-wave interactions associated with TIDs in the quiet daytime auroral F-region are presented. The dynamic frequency spectra in the upper F-region and the one-dimensional wavenumber spectra of TIDs show energy cascading from longer periods (larger scales) to shorter ones (smaller scales). The bispectra and the rates of change of the action density indicate that a chain of nonlinear resonant triad interactions take place. The TID-inducing AGW's wavenumbers along the magnetic field k ξ = k r ξ + jk j ξ are calculated with an inversion method developed in this paper by using simultaneous measurements of N e and V iξ . The k ξ r obtained as a function of height and period show a nearly non-dispersive character at greater heights. The observed nonlinear wave-wave interactions in the higher F-region are accompanied by negative values of − k ξ 3 and wave amplitude growth ( k ξ j> 1 2 H ), which probably imply an over-reflection. This reflection is supported by sharp meridional neutral wind shear and large incident wavelength. The weakness of the dispersion together with the reflection may enable the resonant conditions to be satisfied.

Steady‐state traveling waves on the surface of a viscous liquid film falling down on vertical wires and tubes

AbstractVarious nonlinear wavy regimes of a viscous liquid film flowing down vertical wires and tubes were calculated using the integral method. The linear stability analysis of the trivial smooth solution was compared with the results published previously. In the region of linear instability, the competition among the gravity, viscous and capillary forces formed the steady‐state traveling solutions of finite amplitudes. At least two families of waves were shown to be parameterized by the wave number for given values of external parameters (Reynolds number, cylinder radius and physical characteristics of liquid). The basic waves characteristics depended on external parameters and on wave number. The intensity of wavy processes increased with decreasing cylinder radius. The calculations show the catastrophic growth of wave amplitude, when the system flows down a vertical tube of sufficiently small radius and moves into the linear, unstable region.