Rogue waves in the basin of intermediate depth and the possibility of their formation due to the modulational instability

The properties of rogue waves in the basin of intermediate depth are discussed in comparison with known properties of rogue waves in deep waters. Based on observations of rogue waves in the ocean of intermediate depth we demonstrate that the modulational instability can still play a significant role in their formation for basins of 20 m and larger depth. For basins of smaller depth, the influence of modulational instability is less probable. By using the rational solutions of the nonlinear Schrodinger equation (breathers), it is shown that the rogue wave packet becomes wider and contains more individual waves in intermediate rather than in deep waters, which is also confirmed by observations.     

Second-Order Resonant Interaction of Ring Current Protons with Whistler-Mode Waves

We present a study on the second-order resonant interaction between the ring current protons with Whistler-mode waves propagating near the quasi electrostatic limit following the previous second-order resonant theory. The diffusion coefficients are proportional to the electric field amplitude E, much greater than those for the regular first-order resonance, which are proportional to the electric field amplitudes square E^2. Numerical calculations for the pitch angle scattering are performed for typical energies of protons Ek = 50 keV and 100 keV at locations L = 2 and L = 3.5. The timescale for the loss process of protons by the Whistler waves is found to approach one hour, comparable to that by the EMIC waves, suggesting that Whistler waves may also contribute significantly to the ring current decay under appropriate conditions.     

Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method

The wave equation for linear shallow water waves propagating over a varying bottom topography has the same form as that for p-polarized electromagnetic waves in inhomogeneous dielectric media. The role played by the dielectric permittivity in the case of electromagnetic waves is played by the inverse water depth. We apply the invariant imbedding theory of wave propagation, which has been developed mainly to study the electromagnetic wave propagation, to linear shallow water waves in the special case where the depth depends on only one coordinate. By comparing the numerical result obtained using our method, when the depth profile is linear, with an exact analytical formula, we demonstrate that our method is numerically reliable. The invariant imbedding method can be used in studying the influence of complicated bottom topography on the propagation of shallow water waves, in a numerically exact manner. We illustrate this by considering the case where a periodic modulation is added to a linear depth profile. Bragg scattering due to the periodic component competes with the tsunami effect due to the linear depth variation. This competition is seen to generate interesting physical effects. We also consider a ridge-type bottom topography and examine the resonant transmission phenomenon associated with the Fabry–Perot effect.     

Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method

The wave equation for linear shallow water waves propagating over a varying bottom topography has the same form as that for p-polarized electromagnetic waves in inhomogeneous dielectric media. The role played by the dielectric permittivity in the case of electromagnetic waves is played by the inverse water depth. We apply the invariant imbedding theory of wave propagation, which has been developed mainly to study the electromagnetic wave propagation, to linear shallow water waves in the special case where the depth depends on only one coordinate. By comparing the numerical result obtained using our method, when the depth profile is linear, with an exact analytical formula, we demonstrate that our method is numerically reliable. The invariant imbedding method can be used in studying the influence of complicated bottom topography on the propagation of shallow water waves, in a numerically exact manner. We illustrate this by considering the case where a periodic modulation is added to a linear depth profile. Bragg scattering due to the periodic component competes with the tsunami effect due to the linear depth variation. This competition is seen to generate interesting physical effects. We also consider a ridge-type bottom topography and examine the resonant transmission phenomenon associated with the Fabry-Perot effect.     

Nonlinear gravitational waves on the surface of viscous fluid: Lagrangian approach

The problems of the asymptotic theory of weakly nonlinear surface waves in viscous fluid are discussed. For standing waves on deep water, the solutions obtained in the first- and second-order approximations in a small parameter—wave steepness—are analyzed. The evolution equation for the amplitude of wave packet envelope is obtained where the inverse Reynolds number is equal to the squared steepness. It is shown that this is a nonlinear Schrödinger equation with linear dissipation.     

Variational derivation of improved KP-type of equations

The Kadomtsev-Petviashvili equation describes nonlinear dispersive waves which travel mainly in one direction, generalizing the Korteweg-de Vries equation for purely uni-directional waves. In this Letter we derive an improved KP-equation that has exact dispersion in the main propagation direction and that is accurate in second order of the wave height. Moreover, different from the KP-equation, this new equation is also valid for waves on deep water. These properties are inherited from the AB-equation (E. van Groesen, Andonowati, 2007 [1]) which is the unidirectional improvement of the KdV equation. The derivation of the equation uses the variational formulation of surface water waves, and inherits the basic Hamiltonian structure.     

Conformal variables in the numerical simulations of long-crested rogue waves

The recently suggested theoretical model for highly nonlinear potential long-crested water waves is discussed, where weak three-dimensional effects are included as small corrections to exact two-dimensional equations written in terms of the conformal variables [V.P. Ruban, Phys. Rev. E 71, 055303(R) (2005)]. Some numerical results based on this theory are presented, which describe spontaneous formation of rogue waves on the deep water for different initial conditions. In particular, the given numerical examples describe: (i) nonlinear stage of the modulational instability, (ii) breathing rogue wave in a random wave field, and (iii) freak wave in a weakly crossing sea state.     

Generation and intrinsic dynamics of ring dark solitary waves

 We present experimental results on the evolution of ring dark solitary waves generated by computer-synthesized holograms. The data obtained and the detailed comparative numerical simulations show that this approach ensures reproduction of the correct intensity and phase portrait of the dark wave at the entrance of the nonlinear medium. The transverse dynamics at both even and odd initial conditions is studied and compared with the theory and simulations. Received: 24 June 1996     

Gravitational waves generated during inflation from a 5D vacuum theory of gravity in a de Sitter expansion

In this Letter we study the generation of gravitational waves during inflation from a 5D vacuum theory of gravity. Within this formalism, on an effective 4D de Sitter background, we recover the typical results obtained with 4D inflationary theory in general relativity, for the amplitude of gravitational waves generated during inflation. We also obtain a range of values for the amplitude of tensor to scalar ratio which is in agreement with COBE observations.     

Modulation instability: The beginning

We discuss the early history of an important field of “sturm and drang” in modern theory of nonlinear waves. It is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability, a phenomenon resulting in a variety of nonlinear processes such as envelope solitons, envelope shocks, freak waves, etc. Examples from water wave hydrodynamics, electrodynamics, nonlinear optics, and convection theory are given.     

Rogue wave observation in a water wave tank

The conventional definition of rogue waves in the ocean is that their heights, from crest to trough, are more than about twice the significant wave height, which is the average wave height of the largest one-third of nearby waves. When modeling deep water waves using the nonlinear Schr?dinger equation, the most likely candidate satisfying this criterion is the so-called Peregrine solution. It is localized in both space and time, thus describing a unique wave event. Until now, experiments specifically designed for observation of breather states in the evolution of deep water waves have never been made in this double limit. In the present work, we present the first experimental results with observations of the Peregrine soliton in a water wave tank.     

Experimental Study on Wave Characteristics of Stilling Basin with a Negative Step

Stilling basin with a negative step is an important structure in hydraulic systems, because it can avoid atomization and decrease scouring problems. Although stilling basins with a negative step have attracted much attention from researchers, few researchers have focused on the wave characteristics. In this research, an experimental study on the wave characteristics of stilling basins with a negative step was carried out. The wave height, average period, wave probability density and power spectrum along the flow direction of different stilling basins with a negative step were described based on the wave theory, and the results indicate discharge and step height have a significant effect on the wave characteristics. The relationships between the different characteristic wave heights, and the empirical formula for the relative characteristic wave height are obtained. Finally, the dimensionless standard deviation at the end of the stilling basin with a negative step is linearly related to the flow-energy ratio and the relative step height under B-jump.     

Estimation of critical and viscous frequencies for Biot theory in cancellous bone

The use of Biot theory for modelling ultrasonic wave propagation in porous media involves the definition of a "critical frequency" above which both fast and slow compressional waves will, in principle, propagate. Critical frequencies have been evaluated for healthy and osteoporotic cancellous bone filled with water or marrow, using data from the literature. The range of pore sizes in bone gives rise to a critical frequency band rather than a single critical frequency, the mean of which is lower for osteoporotic bone than normal bone. However, the critical frequency is a theoretical concept and previous researchers considered a more realistic "viscous frequency" above which both fast and slow waves may be experimentally observed. Viscous frequencies in bone are found to be several orders of magnitude greater than calculated critical frequencies. Whereas two waves may well be observed at all ultrasonic frequencies for water-filled cancellous bone at 20 degrees C, it is probable megahertz frequencies would be needed for observation of two waves in vivo.     

A generalized analysis of the vibration of circular rings

A theory is presented which describes the various inextensional vibrations of a circular ring. The cross-sectional shape of the ring is assumed to be constant around its circumference, but otherwise is unrestricted. In particular, it is not assumed that the principal axes of inertia of the cross-section lie in the ring plane. The theory can predict several distinct types of ring vibration, including flexural, torsional and shear waves. Excellent agreement is found with experimental results obtained from two test rings.     

Actual Tropical Waves Identified by the Wavenumber-Frequency Spectrum Analysis during Boreal Summer

Based on daily satellite-observed outgoing longwave radiation （OLR） data, the tropical intraseasonal oscillation （ISO） associated with the Asian summer monsoon is found to be dominant during the 2005 summer. Such an ISO along with other tropical waves are identified using a wavenumber-frequency spectrum analysis. Some of Significant spectral peaks coincide well with the dispersion curves of the equatorially trapped wave modes from shallow＇water theory, with various equivalent depths of 12-200 m, suggesting that actual tropical waves can be detected from observational data. Although the formation mechanism of the ISO is more complicated than those of other tropical waves with higher frequencies, significant spectral peaks that correspond to the ISO with a period of 30-60 days distinctly appear in the OLR spectra. The selective wavenumber-frequency filtering of OLR data is thus performed to isolate the ISO, Kelvin and equatoriai Rossby waves, and the properties and signatures of such tropical waves that control tropical atmospheric variability for the 2005 summer are clearly identified.     

微小孔衍射——近场光学理论

推导了圆孔衍射公式，该公式不受孔径大小和到屏距离的限制，可以作为近场光学的理论。它满足麦克斯韦方程标量形式和基尔霍夫边界条件，包括传播波和衰减波。数值计算表明，此结果优于用Ｂｅｔｈｅ模型所得到的近场理论的结果     

Essence of Inviscid Shear Instability: a Point View of Vortex Dynamics

The essence of shear instability is reviewed both mathematically and physically, which extends the instability theory of a sheet vortex from the viewpoint of vortex dynamics. For this, the Kelvin-Arnol＇d theorem is retrieved in linear context, i.e., the stable flow minimizes the kinetic energy associated with vorticity. Then the mechanism of shear instability is explored by combining the mechanisms of both Kelvin Helmholtz instability （K-H instability） and resonance of waves. The waves, which have the same phase speed with the concentrated vortex, have interactions with the vortex to trigger the instability. The physical explanation of shear instability is also sketched by extending Batchelor＇s theory. These results should lead to a more comprehensive understanding on shear instabilities.     

气相爆轰波沿胞格的动力学机理研究

 建立了一种以子单元分析为基础,研究气相爆轰波沿胞格运动时的动力学机理的新方法。根据该子单元的性质和斜冲击波关系,首先推导了对撞前后前导冲击波沿胞格对称轴的马赫数之比和入射冲击波入射角及胞格几何性质的关系,求解了胞格结构中的三波点对撞问题。然后,采用爆炸波模拟前导冲击波的自持运动过程,求解气相爆轰波沿胞格的动力学过程,理论分析表明,气相爆轰波在胞格起点首先经历一个增长过程,然后才出现衰减。理论分析结果与实验和数值计算结果的比较表明符合得较好。     

Photo-Detachment of a Noncollinear Triatomic Anion

Photo-detachment of a noncollinear triatomic anion is investigated by considering each atom of the molecular anion as a coherent source of detached-electron waves, originating in all possible directions. The waves traveling along three different trajectories result in a quantum interference that displays on a screen placed at a very large distance from the system. To explain this quantum interference, an analytical formula of detached-electron flux is derived using a collinear three-center model recently published. The detached-electron flux versus laser photon energy expression displays molecular geometry-dependent oscillatory structures on an observation plane. This oscillatory behavior in the result can be explained using the semiclassical closed-orbit theory. The outgoing electron waves produced from one coherent center are propagated in the vicinity of the sources at other two coherent centers which cause these oscillations. It is also observed that in a particular case, the noncollinear triatomic system reduces to the collinear three-center system.     

Modelling soot formation in a premixed flame using an aromatic-site soot model and an improved oxidation rate

An updated rate of O₂ oxidation of one to four ring polyaromatic hydrocarbons in premixed flames is presented based on density function theory simulations of oxygen attack at different radical sites on various PAHs. The rate is in agreement with other rates found in the literature; however, it is several orders of magnitude lower than the currently accepted oxidation rate of multi-ring aromatic species, including soot. Simulations are presented of a premixed flame using this improved rate and a new advanced soot particle model, which is developed in this paper. This model includes unprecedented detail of the particles in the ensemble, including the aromatic content, C/H composition and primary-particle aggregate structure. The O₂ oxidation rate calculated in this paper is shown to give a better prediction of particle number density and soot volume fraction for a premixed flame. The predicted particle size distributions are shown also to describe better the experimental data. Predicted C/H ratio and PAH size distributions are shown for the flame. Computed TEM-style images are compared to experimental TEM images, which show that the aggregate structure of the particles is well predicted.