Linear Dispersion Relation Research Articles

Frequency chirping in the early stage of a near-threshold bump-on-tail instability

Abstract It has been shown that the mode amplitude and frequency evolution in the early non-linear stage of a near-threshold bump-on-tail system can be reproduced by solving the linear dispersion relationship at each time step using the non-linearly modified distribution function at an earlier time. The dispersion relationship gives two solutions with departing frequencies almost immediately after the flattening of the distribution function starts to cancel out the drive. One can therefore attribute the early onset of the chirping directly to the modification of the underlying dispersion relationship. The existence of the two waves is because of the beam branch of the beam-plasma instabilities created by the perturbed distribution function. After the two chirping branches are formed, their frequencies are locked to the location of the peaks in the nonlinear distribution function, while the peaks are pushed forwards by beating itself. The transition from the beating-and-chirping scenario to chirping with hole-clump pair creation is found to be determined by the overlapping criterion of the two phase-space islands created by the two chirping branches.

Pulsational mode stability in complex EiBI-gravitating polarized astroclouds with (r,q)-distributed electrons

The pulsational mode of gravitational collapse (PMGC) originating from the combined gravito-electrostatic interaction in complex dust molecular clouds (DMCs) is a canonical mechanism leading to the onset of astronomical structure formation dynamics. A generalized semi-analytic model is formulated to explore the effects of the Eddington-inspired Born-Infeld (EiBI) gravity, non-thermal (r,q)-distributed electrons, and dust-polarization force on the PMGC stability concurrently. The thermal ions are treated thermo-statistically with the Maxwellian distribution law and the non-thermal electrons with the (r,q)-distribution law. The constitutive partially ionized dust grains are modeled in the fluid fabric. A spherical normal mode analysis yields a generalized linear PMGC dispersion relation. Its oscillatory and propagation characteristics are investigated in a judicious numerical platform. It is found that an increase in the polarization force and positive EiBI parameter significantly enhances the instability, causing the DMC collapse and vice versa. The electron non-thermality spectral parameters play as vital stabilizing factors, and so on. Its reliability and applicability are finally outlined in light of astronomical predictions previously reported in the literature.

Impurity Parallel Velocity Gradient instability

In magnetized plasmas, a radial gradient of parallel velocity, where parallel refers to the direction of magnetic field, can destabilise an electrostatic mode called Parallel Velocity Gradient (PVG). The theory of PVG has been mainly developed assuming a single species of ions. Here, the role of impurities is investigated based on a linear, local analysis, in a homogeneous, constant magnetic field. To further simplify the analysis, the plasma is assumed to contain only two ion species - main ions and one impurity species - while our methodology can be straightforwardly extended to more species. In the cold-ion limit, retaining polarization drift for both main ions and impurity ions, and assuming Boltzmann electrons, the system is described by 4 fluid equations closed by quasi-neutrality. The linearized equations can be reduced to 2 coupled equations: one for the electric potential, and one for the effective parallel velocity fluctuations, which is a linear combination of main ion and impurity parallel velocity fluctuations. This reduced system can be understood as a generalisation of the Hasegawa-Mima model. With finite radial gradient of impurity parallel flow, the linear dispersion relation then describes a new instability: the impurity-modified PVG (i-PVG). Instability condition is described in terms of either the main ion flow shear, or equivalently, an effective flow shear, which combines main ion and impurity flow shears. Impurities can have a stabilising or destabilising role, depending on the parameters, and in particular the direction of main flow shear against impurity flow shear. Assuming a reasonable value of perpendicular wavenumber, the maximum growth rate is estimated, depending on impurity mass, charge, and concentration.

A comparison of eight weakly dispersive Boussinesq-type models for non-breaking long-wave propagation in variable water depth

Weakly dispersive Boussinesq-type models are extensively used to model long-wave propagation in coastal areas and their interaction with coastal infrastructures. Many equations falling in this category have been formulated during the last decades, but few detailed comparisons between them can be found in the literature. In this work, we investigate theoretically and with computational experiments eight variants of the most popular models used by the coastal engineering community. Both weakly nonlinear and fully nonlinear models are considered, hoping to understand better when the additional complexity of the latter class of models is necessary or justified. We provide an overview and discuss the properties of these models, including the linear dispersion relation in uniform water depth, the second-order nonlinear coupling coefficient, the shoaling gradient, and the sensitivity to wave trough instabilities. The models are then numerically discretised using the same general strategy in a single numerical code, using fourth-order methods for time and space discretisation. Their capacity to simulate coastal wave propagation and their transformation when approaching the shore is assessed on three challenging one-dimensional benchmarks. It appears that fully nonlinear models are more consistent than their weakly nonlinear counterparts, which can occasionally perform better but show different behaviours depending on the case.

A novel energy-bounded Boussinesq model and a well balanced and stable numerical discretisation

Many Boussinesq models suffer from nonlinear instabilities, especially in the context of rapid variations in the bed topography. In this work, a Boussinesq system is put forward which is derived in such a way as to be both linearly and nonlinearly energy-stable.The proposed system is designed to be robust for coastal simulations with sharply varying bathymetric features while maintaining the dispersive accuracy at any constant depth. For constant bathymetries, the system has the same linear dispersion relation as Peregrine's system ([22]). Furthermore, the system transitions smoothly to the shallow-water system as the depth goes to zero.In the one-dimensional case, we design a stable finite-volume scheme and demonstrate its robustness, accuracy and stability under grid refinement in a suite of test problems including Dingemans's wave experiment.Finally, we generalise the system to the two-dimensional case.

Nonlinear Thermoplasmonics in Graphene Nanostructures.

The linear electronic dispersion relation of graphene endows the atomically thin carbon layer with a large intrinsic optical nonlinearity, with regard to both parametric and photothermal processes. While plasmons in graphene nanostructures can further enhance nonlinear optical phenomena, boosting resonances to the technologically relevant mid- and near-infrared (IR) spectral regimes necessitates patterning on ∼10 nm length scales, for which quantum finite-size effects play a crucial role. Here we show that thermoplasmons in narrow graphene nanoribbons can be activated at mid- and near-IR frequencies with moderate absorbed energy density, and furthermore can drive substantial third-harmonic generation and optical Kerr nonlinearities. Our findings suggest that photothermal excitation by ultrashort optical pulses offers a promising approach to enable nonlinear plasmonic phenomena in nanostructured graphene that avoids potentially invasive electrical gating schemes and excessive charge carrier doping levels.

Linear pressure waves in mono- and poly-disperse bubbly liquids: Attenuation and propagation speed in slow and fast and evanescent modes

Using volumetric averaged equations from a two-fluid model, this study theoretically investigates linear pressure wave propagation in a quiescent liquid with many spherical gas bubbles. The speed and attenuation of sound are evaluated using the derived linear dispersion. Mono- and poly-disperse bubbly liquids are treated. To precisely describe the attenuation effect, some forms of bubble dynamics equations and temperature gradient models are employed. Focusing on the dissipative effect, we analyze the stop band that occurs in the linear dispersion relation. In the two-fluid model, even if the dissipation effect is considered, the inconvenience that the wavenumber diverges to infinity in the resonance frequency cannot be resolved. Additionally, the validity of terminating that wavenumber value in the middle of the frequency is demonstrated. To determine a linear dispersion relation that can exactly predict thermal conduction and acoustic radiation, wave propagation velocities and attenuation coefficients are compared with some experimental data and existing models. The results show that thermal conduction and acoustic radiation should be set appropriately to accurately predict the propagation velocity and attenuation except in the high frequency range, the phase velocity in the resonance frequency range, or the attenuation in the high frequency range.

Turbulence with associated kinetic energy budget in the region of wave-blocking formed over an obstacle

Turbulence characteristics in the region of wave-blocking over a submerged forward-facing obstacle is explored, and the results are compared with those of base-flow over the obstacle. Wave-blocking over obstacle is setup, adjusting the strong incoming flow with counter-propagating waves, and confirmed by linear dispersion relation. A variation in the flow is detected in three distinct segments: flow from the upstream, wave-blocking over the obstacle and the waves propagating from the downstream. The instantaneous velocity is recorded using a three-dimensional micro-acoustic Doppler velocimeter along the flow. Turbulence properties such as the mean velocity, Reynolds stress, turbulent kinetic energy, and associated coherent structures around the wave-blocking are discussed. Moreover, power spectral density, kinetic energy budget, and turbulence length- and time-scales are examined. The stress fractions to the total shear stress contribute higher for only flow than those of wave-blocking over obstacle. The power-spectral peaks at a fixed frequency for wave-blocking along lee side are greater than those of base-flow over obstacle. Kinetic energy budget for only flow over the obstacle is much higher than that of wave-blocking, indicating loss of energy due to wave-blocking. In the near-bed region, length-scale increases indicating the higher momentum and energy transfer. Increase in anisotropy plays significant role in the production. Results are valuable to ship sailing, design of coastal structures, and sediment transport.

Linear analysis of sound velocity and attenuation in bubbly liquids

Owing to a dispersion effect induced by bubble oscillations, the change of phase velocity and attenuation of waves in bubbly liquids is important. Based on a set of volumetric averaged equations in a two-fluid model with bubble dynamics equation and temperature gradient models, we theoretically investigate linear propagation of pressure (or ultrasonic) waves in mono- and poly-disperse bubbly liquids. By incorporating the dissipation effects, the stop band (i.e., frequency range where the waves cannot propagate) appears a moderate high frequency region in a linear dispersion relation. From the comparison of sound velocity and attenuation with previous experimental data, the effect of acoustic radiation and thermal conduction is discussed. We found that, when using a two-fluid model, even if dissipation effects are considered, the inconvenience of infinite divergence of wave numbers at resonant frequency cannot be solved. However, based on the classical dispersion relation, we discuss how to solve this problem. Furthermore, thermal conduction and acoustic radiation should be appropriately set up to accurately predict the sound velocity and attenuation except in the high-frequency region, the sound velocity in the resonant frequency region, and the attenuation in the high-frequency region.

Estimating the In Situ Stratification via Remotely Sensed Internal Wave Speeds

Abstract This work tests a methodology for estimating the ocean stratification gradient using remotely sensed, high temporal and spatial resolution field measurements of internal wave propagation speeds. The internal wave (IW) speeds were calculated from IW tracks observed using a shore-based, X-band marine radar deployed at a field site on the south-central coast of California. An inverse model, based on the work of Kar and Guha, utilizes the linear internal wave dispersion relation, assuming a constant vertical density gradient is the basis for the inverse model. This allows the vertical gradient of density to be expressed as a function of the internal wave phase speed, local water depth, and a background average density. The inputs to the algorithm are the known cross-shore bathymetry, the background ocean density, and the remotely sensed cross-shore profiles of IW speed. The estimated density gradients are then compared to the synchronously measured vertical density profiles collected from an in situ instrument array. The results show a very good agreement offshore in deeper water (∼50–30 m) but more significant discrepancies in shallow water (20–10 m) closer to shore. In addition, a sensitivity analysis is conducted that relates errors in measured speeds to errors in the estimated density gradients. Significance Statement The propagation speed of ocean internal waves inherently depends on the vertical structure of the water density, which is termed stratification. In this work, we evaluate and test with real field observations a technique to infer the ocean density stratification from internal wave propagation speeds collected from remote sensing images. Such methods offer a way to monitor ocean stratification without the need for extensive in situ measurements.

Ion temperature gradient modes modulational stability with kappa-distribution

We investigated the modulational stability and instability of the ion temperature gradient (ITG) mode in electron-ion plasma. Ions are dynamic species, whereas electrons follow a Kappa distribution. We used the reduction perturbation approach to determine the linear dispersion relation for the fluid under consideration. The nonlinear Schrodinger equation describes nonlinear features such as nonlinear modulational stability or instability in the ITG mode. The product of dissipation and nonlinear coefficients, known as LM, contains both modulational stability and instability in the ion temperature gradient mode. Theoretical results are expanded numerically, showing the influence of various plasma parameters on modulational stability and instability, particularly the superthermality coefficient κ e . The current observations may be extended to space and laboratory plasma for modification.

Complete quasilinear model for the acceleration-driven lower hybrid drift instability and a computational assessment of its validity.

A complete quasilinear model is derived for the electrostatic acceleration-driven lower hybrid drift instability in a uniform two-species low-beta plasma in which current is perpendicular to the background magnetic field. The model consists of coupled nonlinear velocity space diffusion equationsfor the volume-averaged ion and electron distribution functions. Each species' diffusion coefficient depends on a time-evolving spectral density of the electric-field energy per unit volume and a time-evolving dispersion relation. The dispersion relation is expressed analytically in integral form without the use of asymptotic limits and applies to arbitrary distribution functions, so long as they can be expressed as a function of one velocity coordinate, e.g., f(v_{y}) or f(v_{⊥}). The quasilinear model conserves energy and is complete in that it fully describes the evolution of the distribution functions, including resonant and nonresonant particle-wave interactions, while accounting for distribution-function-dependent mixed-complex frequencies. The quasilinear diffusion model is solved numerically and self-consistently using a Crank-Nicolson temporal discretization and a second-order finite-volume velocity-space discretization. Numerical solutions are compared to nonlinear fourth-order accurate continuum kinetic Vlasov-Poisson simulations. Evolution of electric-field energy, growth rates, distribution functions, and diffusion coefficients are shown to be in agreement with Vlasov simulations. The quasilinear model is shown to predict anomalous transport terms, like resistivity and heating, to within a factor of order unity. Discrepancies between the quasilinear model and Vlasov simulations are assessed and attributed primarily to lack of damping in the quasilinear description and to the use of unperturbed-orbit susceptibilities in the linear theory dispersion relation. The results illuminate the predictive accuracy of the quasilinear model, place approximate bounds on its validity, and provide much needed vetting of quasilinear theory's ability to predict the nonlinear state of a microturbulent plasma.

Magnetosonic shock waves in degenerate electron–positron–ion plasma with separated spin densities

This study investigates magnetosonic shock waves in a spin-polarized three-component quantum plasma using the quantum magnetic hydrodynamic model. We explore the influence of spin effects, specifically spin magnetization current and spin pressure, on shock wave behavior. Numerical analysis of the linear dispersion relation under varying parameters such as positron imbalance, spin polarization ratio, plasma beta, quantum diffraction, and magnetic diffusivity reveals differential impacts, with diffusion exerting significant influence on the plasma frequency. Our findings highlight the sensitivity discrepancy between the real and imaginary parts of the dispersion relation. Furthermore, nonlinear behavior of magnetosonic shock waves is examined via the Korteweg–de Vries–Burgers equation, showcasing transitions between oscillatory and monotonic wave patterns based on changes in dimensionless parameters. Notably, we observe the combined effects of spin-up and spin-down positrons with spin-up and spin-down electrons on shock wave dynamics, contributing to a deeper understanding of spin-plasma interactions with implications across various fields.

Verification and validation of electrostatic turbulence simulation for numerical Lie transform code

Abstract This paper reports the verification and validation of a new gyrokinetic (GK) code, numerical Lie transform (NLT), for electrostatic turbulence simulation in tokamak plasmas. Based on the standard cyclone base case (CBC), NLT has been verified against two typical GK codes, GK toroidal code (GTC) and GK numerical experiment of tokamak, by simulating the ion temperature gradient mode and trapped electron mode. The linear dispersion relation and mode structure agree well among all these codes. In nonlinear simulation, the time evolution of ion heat diffusivity obtained from NLT is consistent with the GTC results. Furthermore, utilizing experimental equilibria with obvious turbulence features from HL-2A shot #22388, the validation of NLT for real tokamak simulation has been carried out. The profiles of ion turbulent thermal diffusivity calculated by NLT match well with experimental results.

Modulation instability of magnetosonic waves in semiconducting quantum plasma considering Fermi degenerate pressure, exchange correlation potential, and Bohm potential

This study investigates the modulation instability of magnetosonic waves in a semiconducting quantum plasma system. Utilizing the quantum hydrodynamic model, we derive the nonlinear Schrödinger equation and its solution through the reductive perturbation technique. The growth rate of modulation instability for magnetosonic waves has been derived. This study incorporates various quantum corrections, including Fermi degenerate pressure, exchange-correlation potential, and Bohm potential. We study the bright soliton profiles of magnetosonic waves, and additionally, we conduct graphical analyses of the linear dispersion relation and the product of the dispersive coefficient (P) and nonlinear coefficient (Q) of the nonlinear Schrödinger equation. It has been found that magnetosonic waves exhibit modulation instability within specific parameter ranges of the semiconductor plasma and at certain wavelength regimes. Further, we present a comparative study between GaSb and InSb semiconducting plasma systems. The exchange-correlation potential and Fermi degenerate pressure have significantly impacted the modulation instability growth rate, whereas the effect of the Bohm potential is much lower.

Adiabatic approach to the trans-Planckian problem in loop quantum cosmology

We study the scalar modes that, being observable today, were trans-Planckian before inflation, within the context of hybrid loop quantum cosmology (LQC). We analyze the dynamics of these highly ultraviolet modes by introducing modified dispersion relations to their equations of motion and discuss the impact that these relations would introduce in the power spectrum by computing the adiabaticity coefficient. More precisely, we consider two different models compatible with observations for the standard linear dispersion relation which are based on different initial conditions for the perturbations and background. One of these models avoids the issue altogether by generating less e-folds of inflation, so that the observable modes are never trans-Planckian, whereas the other suffers (arguably softly) from the trans-Planckian problem. This shows that the existence of the trans-Planckian problem in LQC is model dependent. Published by the American Physical Society 2024

High-level Green–Naghdi model for large-amplitude internal waves in deep water

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

Ion temperature gradient mode modulational stability analysis with cairn’s distribution

In this manuscript, we have studied electron-ion plasma with inhomogeneity in equilibrium number density and temperature. Ions are the dynamic species, and lighter particles in plasma obey the cairn’s distribution. We introduce Brajinskii’s equation for dynamic species and get the linear dispersion relation and the nonlinear Schrodinger equation by the reduction perturbation method. From the linear dispersion relation, we found the mode frequency and phase velocity, while from the nonlinear Schrodinger equation, we obtained the stability and instability of the ion temperature gradient mode modulation. Findings show that phase velocity is dependent on the superthermality coefficient and other plasma parameters like ion temperature, ion density, and mode parameter η i . Further, the modulational stability and instability of the mode vary with the superthermality coefficient and other plasma parameters, especially the η i . We can apply these observations equally to the laboratory as well as to the space plasma.

Fluid simulations of resistive drift-wave turbulence with diamagnetic flow in ZPED experiments

We derive a diamagnetic resistive fluid model (DRF) and develop an associated two-dimensional fluid simulation code (DRF-2D) to explore the dynamics of resistive drift modes within the plasmas of the Zheda Plasma Experiment Device (ZPED). The validation of the linear dispersion relation for the DRF-2D code revealed a harmonious agreement between analytical theory and linear numerical simulations. Leveraging plasma parameters obtained from the ZPED experiments, we conducted a comprehensive series of nonlinear simulations using the DRF-2D code. Our simulations successfully replicate the nonlinear trends in turbulent fluctuations and transport observed in the ZPED experiments, particularly demonstrating a remarkably accurate alignment of the turning point in the magnetic field. Notably, the DRF model sheds light on the observed frequency sign reversal from the electron diamagnetic direction to the ion diamagnetic direction in the ZPED experiments. This is demonstrated through well-matched turning points in the confining magnetic field between the nonlinear simulations and ZPED experiments. The fidelity of our model in capturing these phenomena underscores its efficacy in providing valuable insights and predictive capabilities for the intricate dynamics observed in the ZPED plasmas.

Boussinesq-type equations of hydroelastic waves in shallow water

Accurate computation of hydroelastic waves in shallow water is critical because many hydroelastic wave applications are nearshores, such as sea-ice and floating infrastructures. In this paper, Boussinesq assumptions for shallow water are employed to derive nonlinear Boussinesq-type equations of hydroelastic waves, in which non-uniform distribution of structural stiffness and varying water depth are considered rigorously. Application of Boussinesq assumptions enables complicated three-dimensional problems to be reduced and formulated on the two-dimensional horizontal plane, therefore the proposed Boussinesq-type models are straightforward and versatile for a wide range of hydroelastic wave applications. Two configurations, a floating plate and a submerged plate, are studied. The first-order linear governing equations are solved analytically with periodic conditions assuming constant depth and uniform stiffness, and the linear dispersion relations are subsequently derived for both configurations. For flexural-gravity waves of a floating plate, unique behaviours of flexural-gravity waves different from shallow-water waves are discussed, and a generalized solitary wave solution is investigated. A nonlinear numerical solver is developed, and nonlinear flexural-gravity waves are found to have smaller wavelength and celerity than their linear counterparts. For hydroelastic waves of a submerged plate, dual-mode analytical solutions are discovered for the first time. Numerical computation has demonstrated that a plate with decreasing submerged depth is able to transfer wave energy from the deeper water to the surface layer.