Evolution Of Nonlinear Waves Research Articles

Identifying ship-wakes in a shallow estuary using machine learning

Ship wakes generated in relatively shallow estuaries are subjected to complex dynamic processes such as nonlinear wave evolution and wave–wave interaction. Identifying ship-wakes in coastal waters is important for maritime and coastal management, including waterway planning, shoreline protection, and navigation-related problems in busy waters. In this study, a machine learning framework was developed to identify ship-wakes using supervised training algorithms. Wave data were measured at two stations in the Pearl River Estuary of South China. The two stations are 1,500 m apart, one is at near-field of the navigation channel, and the other is at far-field. Two machine learning techniques, namely, the multilayer perceptron (MLP) model and support vector machine (SVM) model, are employed. We used two data formats in the training and testing processes; one is the digital data of wave time series, the other is the spectrogram derived from the time series. Tests suggested that spectrogram is a more appropriate format for both models versus the time-series format. The SVM model has higher cross-validation scores and higher computational costs compared to the MLP model. The trained models using the near-field data can be used for predictions at the far-field location with high accuracy. Sensitivity tests revealed that long primary waves and primary wake chirps are the most critical components in the spectrogram among overall ship-wake characteristics for ship-wake recognition. The time and frequency resolutions in the spectrogram only have minor effects on model performance.

IMPACT OF COLLISIONS ON THE DYNAMICS OF WAVES OF FINITE AMPLITUDE IN A PLASMA

The problem of determining wave dynamics, considering the evolution of nonlinear waves of finite amplitude in a weakly collisional, Maxwellian plasma, is the focus of this article. Considering this medium with a certain interaction potential associated with a certain limited core of pairwise interaction, the dynamics of the distribution function was studied using the Vlasov equation. Collisions of electrons with neutral particles are described using the collision integral in the Bhatnagar - Gross- Krook form. Having constructed its particular solution and the equilibrium distribution function in the form of a series, an equation was obtained that makes it possible to determine the potential function. Considering the case of a Maxwellian plasma, an integral potential equation was obtained. Based on it, an equation was constructed that determines the perturbation of the potential relative to the spatially homogeneous one. At the same time, this perturbation appears due to the existence of some spatial-temporal stable structure. Based on this, a dispersion relation was obtained, which makes it possible to estimate the spatial scales of the coherent structure.

Estimating ocean wave directional spreading using wave following buoys: a comparison of experimental buoy and gauge data

Directional spreading of ocean waves plays an important role in various aspects of ocean engineering, such as wave-induced loads, nonlinear wave evolution, and wave breaking. Wave following buoys, which are widely deployed across the oceans, offer the potential to measure directional wave properties. To assess the accuracy of directional spreading estimates made using buoy measurements, we compared estimates based on experimentally obtained buoy and wave gauge measurements, first presented in McAllister and van den Bremer (J Phys Oceanogr 50:399–414, 2020) Buoy and gauge measurements were recorded at the same locations and in identical sea states, allowing for a like-for-like comparison. We examine experiments with both following (unimodal in direction) and crossing (bimodal in direction) sea states. In addition to this, we use synthetic wave data to investigate the effects of wave generation and nonlinearity on spreading estimates. Our results show that while directional estimates produced using buoy measurements are reasonably accurate in following sea states, they struggle to identify distinct directional peaks in crossing sea states. We find that spreading estimates made using buoy measurements tend to underestimate the degree of directional spreading by approximately 9-14% in following sea states, which is most apparent in narrowly spread conditions.

Reduced order model for nonlinear multi-directional ocean wave propagation

In this paper, we introduce a reduced order model (ROM) for the propagation of nonlinear multi-directional ocean wave-fields. The ROM relies on Galerkin projection of Zakharov equations embedded in the high-order spectral (HOS) method, which describes the evolution of nonlinear waves. The dominant flow features of wave evolution are computed from proper orthogonal decomposition (POD) and these modes are used for the projection. The HOS scheme to compute the vertical velocity is treated in a novel way for an efficient implementation of POD-based ROM. We refer to this alternative formalism of HOS as HOS-simple. The final reduced order model (ROM) is derived from the Galerkin projection of HOS-simple. For the case of irregular waves, where the number of modes required are in the range of 200, the ROM has no significant advantage since both HOS and HOS-simple are much faster than real-time. The real advantage is demonstrated in multi-directional (or short-crested) irregular waves, where the ROM is the only model capable of achieving real-time computation, a major improvement to the standard HOS method. The potential use of the ROM in propagating short-crested waves from far-field to near-field for real-world applications involving wave probes in a wave tank/controlled environment as well as X-band radar in open ocean is also demonstrated.

Multipoint Analysis of Source Regions of EMIC Waves and Rapid Growth of Subpackets

AbstractElectromagnetic Ion Cyclotron (EMIC) rising tone emissions are important in understanding the nonlinear wave evolution and interaction with the energetic particles. We present observations of rising tone emissions of EMIC waves by THEMIS A, D, and E spacecraft in the outer magnetosphere. These emissions with subpacket structures in the proton band were observed during the time interval 14:20 UT to 14:30 UT on September 9, 2010. The observed waves are left‐handed polarized with wave normal angles less than . THEMIS A was closest to the equator and was in a higher L‐shell than THEMIS E and D. The smallest radial separation is 2,000 km between THEMIS E and D spacecraft. This configuration of THEMIS allows us to investigate the subpackets of rising tone EMIC waves observed simultaneously at 14:23 UT by three spacecraft. Hilbert Huang Transformation (HHT) is applied to show the variations of the instantaneous frequency and the observed wave amplitude. The direction of energy flow is determined from the analysis of the Poynting flux. There is a rapid nonlinear growth of the EMIC subpackets within one wavelength. Subpackets are dynamic in nature as their structure changes within one wave period, which is further supported by the nonlinear wave growth theory. Optimum and threshold amplitudes for the EMIC wave growth are calculated beside the nonlinear transition time (). Observed ion energies and pitch angle spectra of the ion fluxes are consistent with the energy associated with the Landau and cyclotron resonance conditions.

Foreshock cavitons and spontaneous hot flow anomalies: a statistical study with a global hybrid-Vlasov simulation

Abstract. The foreshock located upstream of Earth's bow shock hosts a wide variety of phenomena related to the reflection of solar wind particles from the bow shock and the subsequent formation of ultra-low-frequency (ULF) waves. In this work, we investigate foreshock cavitons, which are transient structures resulting from the non-linear evolution of ULF waves, and spontaneous hot flow anomalies (SHFAs), which are thought to evolve from cavitons as they accumulate suprathermal ions while being carried to the bow shock by the solar wind. Using the global hybrid-Vlasov simulation model Vlasiator, we have conducted a statistical study in which we track the motion of individual cavitons and SHFAs in order to examine their properties and evolution. In our simulation run where the interplanetary magnetic field (IMF) is directed at a sunward–southward angle of 45∘, continuous formation of cavitons is found up to ∼11 Earth radii (RE) from the bow shock (along the IMF direction), and caviton-to-SHFA evolution takes place within ∼2 RE from the shock. A third of the cavitons in our run evolve into SHFAs, and we find a comparable amount of SHFAs forming independently near the bow shock. We compare the properties of cavitons and SHFAs to prior spacecraft observations and simulations, finding good agreement. We also investigate the variation of the properties as a function of position in the foreshock, showing that transients close to the bow shock are associated with larger depletions in the plasma density and magnetic field magnitude, along with larger increases in the plasma temperature and the level of bulk flow deflection. Our measurements of the propagation velocities of cavitons and SHFAs agree with earlier studies, showing that the transients propagate sunward in the solar wind rest frame. We show that SHFAs have a greater solar wind rest frame propagation speed than cavitons, which is related to an increase in the magnetosonic speed near the bow shock.

Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

Trapped particle instability in : I homogeneous Vlasov plasmas

Kinetic instability due to trapped particles in non-linear evolution of large amplitude electrostatic waves i.e BGK (Bernstein-Greene-Kruskal) mode is investigated in a one dimensional, collisionless, homogeneous Vlasov-Poisson plasma with immobile ions and kinetic electrons. Considering a cosinusoidal density perturbation with non-linear perturbation amplitude α = 0.05 as starting point, we have demonstrated that long time BGK solutions with scale (where corresponds to the longest length scale in the system) are not stable due to Landau growth in the secondary sideband modes. This growth in the sideband modes is primarily related to trapped particles in the potential well of the perturbation which leads to trapped particle instability when the amplitude of the sidebands become comparable to that of the primary nonlinear mode. In the past, to explain trapped particle instability, two physical mechanisms were held responsible, namely, wave particle interaction phenomenon at resonance location v ϕ = ω/k and mode coupling interaction phenomenon between primary and secondary sideband modes. We have demonstrated and emphasized both the phenomena in our study. In addition, effect of primary mode perturbation amplitude on trapped particle instability is investigated and growth rates corresponding to secondary sideband modes are reported alongwith a new scaling law for detrapping time as a function of perturbation amplitude. Also, scaling of growth rates corresponding to lower and upper sideband modes with respect to perturbation amplitude is obtained which demonstrates the existence of a transition point at α = 0.075 between small and large perturbation amplitudes and growth rate scales as γ ∼ α 2 for α < 0.075 whereas γ ∼ α 1/2 for α > 0.075.

Planet-driven density waves in protoplanetary discs: Numerical verification of non-linear evolution theory

ABSTRACT Gravitational coupling between protoplanetary discs and planets embedded in them leads to the emergence of spiral density waves, which evolve into shocks as they propagate through the disc. We explore the performance of a semi-analytical framework for describing the non-linear evolution of the global planet-driven density waves, focusing on the low planet mass regime (below the so-called thermal mass). We show that this framework accurately captures the (quasi-)self-similar evolution of the wave properties expressed in terms of properly rescaled variables, provided that certain theoretical inputs are calibrated using numerical simulations (an approximate, first principles calculation of the wave evolution based on the inviscid Burgers equation is in qualitative agreement with simulations but overpredicts wave damping at the quantitative level). We provide fitting formulae for such inputs, in particular, the strength and global shape of the planet-driven shock accounting for non-linear effects. We use this non-linear framework to theoretically compute vortensity production in the disc by the global spiral shock and numerically verify the accuracy of this calculation. Our results can be used for interpreting observations of spiral features in discs, kinematic signatures of embedded planets in CO line emission (‘kinks’), and for understanding the emergence of planet-driven vortices in protoplanetary discs.

A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

Numerical Study on Weakly Nonlinear Evolution of Pressure Waves in Water Flows Containing Many Translational Bubbles Acting a Drag Force

Weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flow uniformly containing many spherical gas bubbles is numerically investigated with a special attention to a drag force acting bubbles and translation of bubbles. The gas and liquid phases are flowing with initially independent velocities. Drag force and virtual mass force are introduced as interfacial momentum transports. Translation and spherically symmetric oscillations are considered as bubble dynamics. In this paper, under these assumptions, we numerically solve the KdVB (Korteweg-de Vries-Burgers) equation previously derived by ourselves (Yatabe et al., Phys. Fluids, 33 (2021), 033315) from basic equations based on a two-fluid model. The main results are summarized as follows: (i) The drag force acting on bubbles increases a dissipation effect of waves and drastically changes the phase and amplitude of waves. (ii) Although the translation of bubbles increases the nonlinear effect of waves, its contribution to waveform is quantitatively small. (iii) The effect of the drag force decreases with decreasing the initial void fraction and with increasing the initial bubble radius. That of the translation decreases with decreasing the initial void fraction, and is almost independent of the initial bubble radius. (iv) The spatiotemporal evolution of two type of dissipation effects (i.e., dissipation terms) due to the acoustic radiation and to the drag force is different tendency.

Generation of Shoreward Nonlinear Internal Waves South of the Hainan Island: Synthetic Aperture Radar Observations and Numerical Simulations

AbstractThe generation of shoreward nonlinear internal waves (NLIWs) on the continental shelf south of the Hainan Island (SHI) is investigated based on spaceborne synthetic aperture radar (SAR) observations and numerical simulations. Two types of shoreward NLIWs are identified from SAR images according to their distinct geographic distribution. One type of NLIWs, named Type‐N NLIWs, is distributed in the northern part of SHI, and the other one is named Type‐S NLIWs, distributed in the southern part of SHI. The SAR‐observed wave occurrence frequency during the spring and neap tides, combined with the calculated body force, suggests that the Type‐N NLIWs originate from the Xisha Islands, whereas the Type‐S NLIWs originate from both the Xisha Islands and the continental shelf break, and the shelf break has a larger contribution. The synergistic analyses of the internal tidal raypath, gamma parameter and earliest SAR‐observed NLIWs suggest that the Type‐N NLIWs are caused by the impingement of the diurnal internal tidal beams emanating from the Xisha Islands on the near‐surface pycnocline close to the continental shelf. Based on the realistic shelf‐slope topography and tidal forcing, the two‐dimensional numerical simulations using the MITgcm suggest that the Type‐S NLIWs result from the nonlinear disintegration of a mode‐1 diurnal internal tide, which develops from a depression formed at the continental shelf break. Furthermore, numerical experiments show that the background current can greatly affect the nonlinear evolution of the internal waves generated at the shelf break.

Nonlinear Wave Evolution in Interaction with Currents and Viscoleastic Muds

A numerical model is extended to investigate the nonlinear dynamics of surface wave propagation over mud in the presence of currents. A phase-resolving frequency-domain model for wave-current interaction is improved to account for wave modulations due to viscoelastic mud of arbitrary thickness. The model compares well with published laboratory data and performs slightly better than the model with viscous mud-induced wave damping mechanism. Monochromatic and random wave simulations are conducted to examine the combined effect of currents, mud-induced wave dissipation and modulation, and nonlinear wave-wave interactions on surface wave spectra. Results indicate that current effects on wave damping over viscoelastic mud is not as straightforward as that over viscous mud. For example, while opposing currents consistently increase damping of random waves over viscous mud, they can decrease damping over viscoelastic mud due to high variations in frequency-dependent damping stemming from mud’s elasticity. It is shown that a model that assumes the mud layer to be thin for simplification can overestimate wave damping over thick mud layers.

Freak wave in high-order weakly nonlinear wave evolution with bottom topography change

In deep-water, quasi-resonant four-wave interaction gives rise to the occurrence probability of extreme wave height, which leads to freak wave as a natural disaster. As water wave enters shallow water, four-wave interaction becomes weak and wave shoaling makes this effect become complicated. This study conducts a Monte Carlo simulation in modified Nonlinear Schrödinger equation considering bottom topography change, and gives spatial evolution of high-order nonlinearity in wave train from statistical moment. The result indicates, slope angle plays an important role in a shallow water depth, and steep slope brings about a significant enhancement of extreme event in distribution of maximum wave height and surface elevation.

Weakly nonlinear evolution of pressure waves in a polydisperse bubbly liquid

Weakly nonlinear propagation of plane pressure waves in initially quiescent liquids containing spherical bubbles with a small initial polydispersity of the bubble radius and bubble number density is theoretically studied to clarify the effect of the polydispersity on wave propagation. The wavelength is comparable with the bubble radius and the incident frequency of waves is with an eigenfrequency of the bubble. We consider two different cases of the size of the polydispersity (i.e., small and large polydispersities) and assume that the polydispersity appears at a field far from a sound source. The polydispersity is formulated into a set of perturbation expansions. Bubble oscillations are spherically symmetric, and bubbles do not coalesce, break up, appear, and disappear. The basic set is composed of the conservation laws of mass and momentum for gas and liquid phases based on a two-fluid model, equation of motion for bubble wall, and so on. From the perturbation analysis up to the third order of approximation, two cases of nonlinear wave equation for a long-range propagation of waves were derived. We concluded that the polydispersity affects the advection of waves and is expressed as a variable coefficient.

Linear and Nonlinear Plasma Processes in Ionospheric HF Heating

Featured observations of high frequency (HF) heating experiments are first introduced; the uniqueness of each observation is presented; the likely cause and physical process of each observed phenomenon instigated by the HF heating are discussed. A special point in the observations, revealed through the ionograms, is the competition between the Langmuir parametric instability and upper hybrid parametric instability excited in the heating experiments and the impact of the natural cusp at foE (the peak plasma frequency of the ionospheric E region) on the competition. The ionograms also infer the generation of Langmuir and upper hybrid cavitons. Ray tracing theory is formulated. With and without the appearance of large-scale field-aligned density irregularities in the background ionosphere, ray trajectories of the ordinary mode (O-mode) and extraordinary mode (X-mode) sounding pulses are calculated numerically. The results explain the artificial Spread-F recorded by the digisondes in the heating experiments. Parametric instabilities, which are the directly relevant processes to achieve effective heating of the ionospheric F region, are formulated and analyzed. The threshold fields and growth rates of Langmuir and upper hybrid parametric instabilities are derived as the theoretical basis of many radar observations and electron-plasma wave interactions. Harmonic cyclotron resonance interaction processes between electrons and upper hybrid waves are introduced. Formulation and analysis are presented. The numerical results show that ultra-energetic electrons are generated. These electrons enhance airglow at 777.4 nm as well as cause ionization. Physical processes leading to the generation of artificial ionization layers are discussed. The nonlinear Schrodinger equation governing the nonlinear evolution of Langmuir waves and upper hybrid waves are derived and solved. The nonlinear periodic and solitary solutions of the equations are obtained. The localized Langmuir and upper hybrid waves generated by the HF heater form cavitons near the HF reflection layer and near the upper hybrid resonance layer, which induce bumps in the virtual height spread of the ionogram trace similar to that induced by the density cusp at E-F1 transition layer; the down-going Langmuir waves and upper hybrid waves evolve into nonlinear periodic waves propagating along the magnetic field, which backscatter incoherently the sounding pulses to cause downward virtual height spread.

Преобразование Дарбу для т-симметричной нелокальной комплексной модифицированной системы уравнений Кортевега-де Фриза

onlinear partial differential equations are widely used as models to describe physical phenomena in various fields of sciences such as fluid mechanics, solid-state physics, plasma physics, chemical physics, condensed matter physics, optical fibers, biology, and geochemistry. One of the nonlinear partial differential equations is the complex modified Korteweg-de Vries equation. This equation has been proposed as a model for the nonlinear evolution of plasma waves and is the physical model that incorporates the propagation of transverse waves in a molecular chain model, and in a generalized elastic solid. In this paper, we study the T-symmetry nonlocal complex modified Korteweg-de Vries system of equations. This nonlocal system is obtained by Ablowitz-Musslimani type of reduction and is respectively T-symmetric nonlocal cmKdV system of equations. The Ablowitz - Musslimani type of reductions arises from remarkably simple symmetry reductions of general AKNS scattering problems. The method of the Darboux transformation is applied to obtain exact solutions.

Theoretical analysis of equatorial near-inertial solitary waves under complete Coriolis parameters

An investigation of equatorial near-inertial wave dynamics under complete Coriolis parameters is performed in this paper. Starting from the basic model equations of oceanic motions, a Korteweg de Vries equation is derived to simulate the evolution of equatorial nonlinear near-inertial waves by using methods of scaling analysis and perturbation expansions under the equatorial beta plane approximation. Theoretical dynamic analysis is finished based on the obtained Korteweg de Vries equation, and the results show that the horizontal component of Coriolis parameters is of great importance to the propagation of equatorial nonlinear near-inertial solitary waves by modifying its dispersion relation and by interacting with the basic background flow.

ТОЧНЫЕ РЕШЕНИЯ УРАВНЕНИЯ ХИРОТА С ПОМОЩЬЮ МЕТОДА СИНУС–КОСИНУС

Nonlinear partial differential equations of mathematical physics are considered to be major subjects in physics. The study of exact solutions for nonlinear partial differential equations plays an important role in many phenomena in physics. Many effective and viable methods for finding accurate solutions have been established. In this work, the Hirota equation is examined. This equation is a nonlinear partial differential equation and is a combination of the nonlinear Schrödinger equation and the complex modified Korteweg–de Vries equation. The nonlinear Schrödinger equation is the physical model and occurs in various areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. The complex modified Korteweg–de Vries equation has been applied as a model for the nonlinear evolution of plasma waves and represents the physical model that incorporates the propagation of transverse waves in a molecular chain model and in a generalized elastic solid. To find exact solutions of the Hirota equation, the sine-cosine method is applied. This method is an effective tool for searching exact solutions of nonlinear partial differential equations in mathematical physics. The obtained solutions can be applied when explaining some of the practical problems of physics.

Variety interaction solutions comprising lump solitons for the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation

<abstract> This paper deals with localized waves in the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation in the incompressible fluid. Based on Hirota's bilinear method, N-soliton solutions related to CDGKS equation are constructed. Taking the special reduction, the exact expression of multiple localized wave solutions comprising lump soliton(s) are obtained by using the long wave limit method. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear localized waves. </abstract>