Nonlinear Dispersion Relation Research Articles

Study of the parametric dependence of linear and nonlinear microtearing modes in conventional tokamak discharges

A reduced transport model for microtearing modes is developed for use in integrated predictive modeling studies, employing a unified fluid/kinetic approach to derive the nonlinear dispersion relation. This approach advances the kinetic description and allows the inclusion of nonlinear effects due to magnetic fluctuations. In this numerical study, the dependence of the microtearing mode real frequency and growth rate on plasma parameters and on DIII-D like L-mode and H-mode plasma profiles is examined. The magnetic fluctuation strength as well as electron thermal diffusivity due to microtearing modes is computed. The saturated amplitude of the magnetic fluctuations is calculated utilizing numerically determined microtearing mode eigenvalues in the nonlinear microtearing modes envelope equation. It is found that the electron temperature gradient in the presence of moderate collision frequency is required for the microtearing mode to become unstable. The effects of small and large collisionality and small and large wavenumbers on microtearing modes are found to be stabilizing, while the effects of density gradient, plasma beta, low current density, and large magnetic shear are found to be destabilizing. The microtearing mode growth rate, magnetic fluctuation strength, as well as electron thermal diffusivity is found to be larger in the H-mode plasma than in the L-mode plasma.

Viscous Overstability in Saturn’s Rings: Influence of Collective Self-gravity

Abstract We investigate the influence of collective self-gravity forces on the nonlinear, large-scale evolution of the viscous overstability in Saturn’s rings. We numerically solve the axisymmetric nonlinear hydrodynamic equations in the isothermal and non-isothermal approximation, including radial self-gravity and employing transport coefficients derived by Salo et al. We assume optical depths to model Saturn’s dense rings. Furthermore, local N-body simulations, incorporating vertical and radial collective self-gravity, are performed. Vertical self-gravity is mimicked through an increased frequency of vertical oscillations, while radial self-gravity is approximated by solving the Poisson equation for an axisymmetric thin disk with a Fourier method. Direct particle–particle forces are omitted, which prevents small-scale gravitational instabilities (self-gravity wakes) from forming, an approximation that allows us to study long radial scales and to compare directly the hydrodynamic model and the N-body simulations. Our isothermal and non-isothermal hydrodynamic model results with vanishing self-gravity compare very well with results of Latter & Ogilvie and Rein & Latter, respectively. In contrast, for rings with radial self-gravity we find that the wavelengths of saturated overstable waves settle close to the frequency minimum of the nonlinear dispersion relation, i.e., close to a state of vanishing group velocities of the waves. Good agreement is found between non-isothermal hydrodynamics and N-body simulations for moderate and strong radial self-gravity, while the largest deviations occur for weak self-gravity. The resulting saturation wavelengths of viscous overstability for moderate and strong self-gravity ( ) agree reasonably well with the length scales of axisymmetric periodic microstructure in Saturn’s inner A ring and the B ring, as found by Cassini.

The privileged spectrum of cnoidal ion holes and its extension by imperfect ion trapping

The fundamental properties of nonlinear ion hole modes propagating in current-driven collisionless plasmas are derived. Making use of Schamel's alternative method their spatial structure ϕ(x) and phase velocities u0 are analyzed and found to depend crucially on the used trapped ion distribution fit. A regular fit represents a continuous spectrum, which is called privileged or perfect since it yields a definite u0 and appears most realistic. A singular fit, on the other hand, involving jumps and moderate slope singularities at the separatrix, does reveal further classes of hole equilibria at the cost, however, of a well-defined u0. This explains why Bernstein, Greene, Kruskal (BGK)-solutions of the Vlasov–Poisson system, exhibiting a strong slope singularity of their derived trapped particle distribution, can principally not provide definite u0s. The nonlinear dispersion relation (or u0) of privileged ion holes, on the other hand, is equivalent with that of cnoidal electron holes, i.e. in addition to the ordinary ion acoustic branch there exists a correspondence to the “Langmuir” branch and to the multiple “slow electron acoustic” branches, reflecting different trapping scenarios.

Analysis of Stem Wave due to Long Breakwaters at the Entrance Channel

Recently, a new port reserves deep water depth for safe navigation and mooring, following the trend of larger ship building. Larger port facilities include long and huge breakwaters, and mainly adopt vertical type considering low construction cost. A vertical breakwater creates stem waves combining inclined incident waves and reflected waves, and this causes maneuvering difficulty to the passing vessels, and erosion of shoreline with additional damages to berthing facilities. Thus, in this study, the researchers have investigated the response of stem waves at the vertical breakwater near the entrance channel and applied numerical models, which are commonly used for the analysis of wave response at the harbor design. The basic equation composing models here adopted both the linear parabolic approximation adding the nonlinear dispersion relationship and nonlinear parabolic approximation adding a linear dispersion relationship. To analyze the applicability of both models, the research compared the numerical results with the existing hydraulic model results. The gap of serial breakwaters and aligned angles caused more complicated stem wave generation and secondary stem wave was found through the breakwater gap. Those analyzed results should be applied to ship handling simulation studies at the approaching channels, along with the mooring test.

Universal horizons and Hawking radiation in nonprojectable 2d Hořava gravity coupled to a nonrelativistic scalar field

In this paper, we study the non-projectable 2d Ho\v{r}ava gravity coupled with a non-relativistic scalar field, where the coupling is in general non-minimal and of the form $f(\phi)R$, where $f(\phi)$ is an arbitrary function of the scalar field $\phi$, and $R$ denotes the 2d Ricci scalar. In particular, we first investigate the Hamiltonian structure, and show that there are two-first and two-second class constraints, similar to the pure gravity case, but now the local degree of freedom is one, due to the presence of the scalar field. Then, we present various exact stationary solutions of this coupled system, and find that some of them represent black holes but now with universal horizons as their boundaries. At these horizons, the Hawking radiations are thermal with temperatures proportional to their surface gravities, which normally depend on the non-linear dispersion relations of the particles radiated, similar to the (3+1)-dimensional case.

Parametric instabilities in a quantum magnetoplasma with electron exchange-correlations

Relying on the density functional theory, we have examined the impact of electron exchange-correlations on the nonlinear dispersion relations and associated parametric instabilities induced by nonlinear couplings of high-frequency quantum upper-hybrid waves (QUHWs) with different low-frequency waves, like quantum lower-hybrid waves (QLHWs), quantum ion-cyclotron waves (QICWs), and quantum Alfvén waves (QAWs) in a dense quantum magnetoplasma. For theoretical description of waves, we make use of quantum hydrodynamic equations to account for the electron exchange-correlation and Bohm potentials, strongly dependent on the density fluctuations. At quantum scales, nonlinear dispersion equations are derived for QUHWs, QICWs, QLHWs, and QAWs and then Fourier transformed for obtaining the nonlinear dispersion relations and growth rates involving the three wave decay and modulational instabilities in dense quantum magnetoplasmas. The relevance to nonlinear interactions due to high- and low-frequency waves in perspective of electron exchange-correlations is highlighted.

Effect of nonthermal ions on (3+1)-dimensional envelope solitary wave in magnetized PLD dusty plasma

This work investigates the envelope solitary wave (ESW) in magnetized power law distribution dusty plasma consisting of free electrons and nonthermal ions. A (3+1)-dimensional nonlinear Schr $$\ddot{o}$$ dinger equation (NLSE) through the reductive perturbation technique is derived. The nonlinear dispersion relation $$\omega =\omega (k )$$ and coefficients of NLSE are analyzed in detail by the numerical simulation method under the effect of nonthermal ions. It is indicated that the frequency increases as the number density of nonthermal ions increase. It can be seen that the frequency with higher unperturbed ion number density is larger than that with lower case. Moreover, the effects of amplitude and width on ESW are discussed with different plasma parameters. The present results can be important in understanding the characteristics of ESW for a magnetized three component plasma system.

Topological properties of the chiral magnetic effect in multi-Weyl semimetals

We compute the chiral magnetic effect (CME) in multi-Weyl semimetals (multi-WSMs) based on the chiral kinetic theory. Multi-WSMs are WSMs with multiple monopole charges that have nonlinear and anisotropic dispersion relations near Weyl points, and we need to extend conventional computation of CME in WSMs with linear dispersion relations. Topological properties of CME in multi-WSMs are investigated in details for not only static magnetic fields but also time-dependent (dynamic) ones. We propose an experimental setup to measure the multiple monopole charge via the topological nature hidden in the dynamic CME.

Relativistic self-focusing of an intense laser pulse with hot magnetized plasma in the presence of a helical magnetostatic wiggler

In this paper, we study the nonlinear interaction of a circularly polarized laser pulse propagating through a hot magnetized plasma in the presence of a helical magnetostatic wiggler. A non-linear equation that describes the spot-size of the laser beam for both left- and right-hand polarizations has been derived. Non-linear dispersion relation describing the evolution of the laser frequency propagating through the hot magnetized plasma has been obtained. The effect of the wiggler magnetic field strength on the evolution of the laser spot-size has been discussed. The results indicate that for the right-hand polarization with increasing wiggler magnetic field strength, the laser spot-size decreases and the laser pulse becomes more focused. On the contrary, for the left-hand polarization, the self-focusing decreases with increasing wiggler magnetic field strength. Besides, it was found that in the right-hand polarization, the laser spot-size increases with the increasing plasma temperature, and the laser beam becomes more defocused. Furthermore, for the left-hand polarization, the laser self-focusing increases with the decreasing plasma temperature. Further numerical results revealed that by increasing the wiggler field strength, the normalized laser power increases significantly.

Wronskian and linear superposition solutions to generalized KP and BKP equations

We aim to find exact solutions to some generalized KP- and BKP-type equations by the Wronskian technique and the linear superposition principle. By applying Wronskian identities of the bilinear KP hierarchy, three Wronskian formulations are first furnished, with all generating functions for matrix entries satisfying a system of combined linear partial differential equations. Second, we apply the linear superposition principle to exponential traveling wave solutions of the introduced generalized KP and BKP equations. Each exponential wave in the Wronskian and N-wave solutions satisfies the corresponding nonlinear dispersion relation.

Study of plasmonic slot waveguides with a nonlinear metamaterial core: semi-analytical and numerical methods

Two distinct models are developed to investigate the transverse magnetic stationary solutions propagating in one-dimensional anisotropic nonlinear plasmonic structures made from a Kerr-type nonlinear metamaterial core embedded between two semi-infinite metal claddings. The first model is semi-analytical, in which we assume that the anisotropic nonlinearity depends only on the transverse component of the electric field and that the nonlinear refractive index modification is small compared to the linear one. This method allows us to derive analytically the field profiles and nonlinear dispersion relations in terms of the Jacobi elliptical functions. The second model is fully numerical and is based on the finite element method in which all the components of the electric field are considered in the Kerr-type nonlinearity, with no presumptions as to the nonlinear refractive index change. Our finite-element-based model is valid beyond the weak nonlinearity regime and generalizes the well-known single-component fixed power algorithm that is usually used. Examples of the main cases are investigated, including those with strong spatial nonlinear effects at low power. Loss issues are reduced through the use of a gain medium in the nonlinear metamaterial core. Using anisotropic nonlinear FDTD simulations, we provide some results for the properties of the main solution.

Terahertz pulses in a periodic plasma channel via modulated electron beam

The proposed scheme offers a promising way to study the effect of modulated relativistic electron beam in a periodic plasma channel (wiggler). The fluid equations model is used to find out the numerical behaviour of the radiation wave. In this scheme, a modulated relativistic electron beam makes possible to generate and amplify the high power terahertz (THz) radiation pulses. Using this model, we have analyzed and calculated the nonlinear dispersion relation of the radiation wave. The radiation frequency of THz wave can be tuned by varying the wiggler wavelength and beam energy. The excitation of the radiation wave is being developed with the resonance between the modulated frequency of electron beam and radiation wave in the periodic plasma channel (wiggler field) via Cerenkov interaction along the axial direction of channel, which provides the maximum enhancement in the growth rate of the radiation wave. Additionally, the growth rate of the unstable mode of the radiation wave depends on beam energy and modulation index, and becomes larger as modulation index approaches unity.

Self-focusing property of a laser beam interacting with a lattice of nanoparticles in the presence of a planar magnetostatic wiggler

In this paper, we study the nonlinear interaction of a laser beam with a periodic lattice of nanoparticles in the presence of a planar magnetostatic wiggler. The static magnetic field of the wiggler can couple with the electric field of the laser wave and change the electric field intensity of the pumped wave, leading to the formation of a nonlinear force. In consequence, the nonlinear force enhances plasmonic oscillations of the electronic cloud of each nanoparticle causing electron density modulation, which improves self-focusing property of the laser beam propagating through a periodic lattice of nanoparticles. By manipulating a classical microscopic approach into plasmonic oscillations of electronic clouds of the nanoparticles and the well–known perturbative method, a nonlinear dispersion relation describing the evolution of the laser amplitude propagating through the nanoparticle lattice has been obtained. The effect of the wiggler magnetic strength on the evolution of the laser transverse profile has been discussed. It was found that by increasing the wiggler strength, the transverse profile bandwidth shrinks and laser focusing is enhanced. In addition, further numerical results indicated that by increasing the wiggler field strength, the cut-off frequency of the body waves increases.

On the nonlinear trapping nature of undamped, coherent structures in collisionless plasmas and its impact on stability

An outstanding notion for collisionless plasmas is the essential nonlinear character of their coherent structures, which in the stationary, weak amplitude limit are described by a continuum of cnoidal electron and ion hole modes governed by a multiparametric nonlinear dispersion relation. The well-known discrete structure of undamped linear plasma modes is seamlessly embedded in this nonlinear continuum as the microscopic texture of plasma begins to reveal itself in the high temperature collisionless plasma limit. This transforms the linear-threshold-based operating mechanism of plasma turbulence into a fundamental nonlinear, multifaceted one. Based on a comprehensive three-level description of increasing profundity, a proof of this novel dictum is presented, which makes use of the joint properties of such structures, their coherency and stationarity, and uses in succession a fluid, linear Vlasov and a full Vlasov description. It unifies discrete and continuum limits by resolving the inevitable resonant region and shows that coherent electrostatic equilibria are generally controlled by kinetic particle trapping and are hence fundamentally nonlinear. By forging a link between damped and growing wave solutions, these modes render plasma stability complex and difficult to evaluate due to the entangled pattern of the stability boundary in function and parameter space, respectively. A direct consequence is the existence of negative energy modes of arbitrarily small amplitudes in the subcritical region of the two-stream instability as well as the failure of linear Landau (Vlasov, van Kampen) theory, whenever resonant particles are involved, in addressing the onset of instability in a current-carrying plasma. Responsible for this subtle phase space behavior is hence the thresholdless omnipresence of the trapping nonlinearity originating from coherency. A high resolution, exact-mass-ratio, multispecies, and collisionless plasma simulation is employed to illustrate exemplarily how tiny seed fluctuations in phase-space can act as a triggering agent for a subcritical plasma excitation verifying an access to these modes in the noisy, collisionless plasma limit.

Influence of non perfect impedance boundary on the bistable region in thermoacoustic interactions

We investigate the influence of non perfect impedance boundary on the bistable zone in thermoacoustic interactions of a horizontal Rijke tube. A wave based approach is used to obtain the nonlinear dispersion relation with frequency dependent impedance boundary condition. The location and the time delay in the response of the heater are considered as bifurcation parameters to obtain the stability boundaries. In the presence of non perfect impedance boundary condition, we find that the extent of globally unstable regime reduces and the bistable zone significantly increases. The quantitative changes in the stability boundaries and the bistable zone are investigated for different time lags. However, the nature of bifurcation remains sub critical and unaltered for the range of time delays considered in the present study.

Spirals and heteroclinic cycles in a spatially extended Rock-Paper-Scissors model of cyclic dominance

Spatially extended versions of the cyclic-dominance Rock-Paper-Scissors model have traveling wave (in one dimension) and spiral (in two dimensions) behaviour. The far field of the spirals behave like traveling waves, which themselves have profiles reminiscent of heteroclinic cycles. We compute numerically a nonlinear dispersion relation between the wavelength and wavespeed of the traveling waves, and, together with insight from heteroclinic bifurcation theory and further numerical results from 2D simulations, we are able to make predictions about the overall structure and stability of spiral waves in 2D cyclic dominance models.

Nonlinear frequency shift in Raman backscattering and its implications for plasma diagnostics

Raman backscattered radiation of intense laser pulses in plasmas is investigated for a wide range of intensities relevant to laser wakefield acceleration. The weakly nonlinear dispersion relation for Raman backscattering predicts an intensity and density dependent frequency shift that is opposite to that suggested by a simple relativistic consideration. This observation has been benchmarked against experimental results, providing a novel diagnostic for laser-plasma interactions.

Multiple propagating modes of nanowire plasmonics

We investigate electromagnetic waves propagating along a metallic wire with a closer look at multiple propagating modes. To this goal, metallic loss is examined for its influence on a hybrid wave. The number of resonance modes is found to strongly depend on the rotational azimuthal mode indices. Based on the highest quality factor, selections are made among the multiple modes. We captured both divisions and mergers of the wave-number extents of the residual functions signifying the nonlinear dispersion relation. In addition, the migrations of the groups of local minima for multiple modes are illustrated from a viewpoint of quasi-temporal evolution. Furthermore, we illustrate collective behaviors of nanowire plasmonics in terms of two constituent waves and their interactions.

Hawking radiation of charged Einstein-aether black holes at both Killing and universal horizons

We study analytically quantum tunneling of relativistic and non-relativistic particles at both Killing and universal horizons of Einstein–Maxwell-aether black holes, after high-order curvature corrections are taken into account, for which the dispersion relation of the particles becomes nonlinear. Our results at the Killing horizons confirm the previous ones, i.e., at high frequencies the corresponding radiation remains thermal and the nonlinearity of the dispersion does not alter the Hawking radiation significantly. In contrary, non-relativistic particles are created at universal horizons and are radiated out to infinity. The radiation also has a thermal spectrum, and the corresponding temperature takes the form, TUHz=2κUH(z−1)/(2πz), where z (z≥2) denotes the power of the leading term in the nonlinear dispersion relation, κUH is the surface gravity of the universal horizon, defined by peering behavior of ray trajectories at the universal horizon. We also study the Smarr formula by assuming that: (a) the entropy is proportional to the area of the universal horizon, and (b) the first law of black hole thermodynamics holds, whereby we derive the Smarr mass, which in general is different from the total mass obtained at infinity. This indicates that one or both of these assumptions must be modified.

Extended Elliptic Mild Slope Equation Incorporating the Nonlinear Shoaling Effect

Abstract The transformation during wave propagation is significantly important for the calculations of hydraulic and coastal engineering, as well as the sediment transport. The exact wave height deformation calculation on the coasts is essential to near-shore hydrodynamics research and the structure design of coastal engineering. According to the wave shoaling results gained from the elliptical cosine wave theory, the nonlinear wave dispersion relation is adopted to develop the expression of the corresponding nonlinear wave shoaling coefficient. Based on the extended elliptic mild slope equation, an efficient wave numerical model is presented in this paper for predicting wave deformation across the complex topography and the surf zone, incorporating the nonlinear wave dispersion relation, the nonlinear wave shoaling coefficient and other energy dissipation factors. Especially, the phenomenon of wave recovery and second breaking could be shown by the present model. The classical Berkhoff single elliptic topography wave tests, the sinusoidal varying topography experiment, and complex composite slopes wave flume experiments are applied to verify the accuracy of the calculation of wave heights. Compared with experimental data, good agreements are found upon single elliptical topography and one-dimensional beach profiles, including uniform slope and step-type profiles. The results indicate that the newly-developed nonlinear wave shoaling coefficient improves the calculated accuracy of wave transformation in the surf zone efficiently, and the wave breaking is the key factor affecting the wave characteristics and need to be considered in the nearshore wave simulations.