Growth Rate Of Waves Research Articles

Effect of Coriolis force on the linear stability of subaqueous dunes with erodible and non-erodible beds

Dunes, are wavy-shaped structures, that can be found on river shores and underwater. We aim to investigate the influence of Coriolis effect on the linear stability of subaqueous dunes using a simple model based on the three-dimensional shallow water equations written in rotating frame of reference. The analysis is performed for both erodible and non-erodible beds. Our analytical results in the absence of Coriolis force are verified with published results. The wave speed and growth rate of disturbances are presented for different Coriolis parameter, Froude number and erodibility. A new mode is observed, which becomes unstable beyond a threshold value of Coriolis parameter.

Wave Excitation by Power-law-Distributed Energetic Electrons with Pitch-angle Anisotropy in the Solar Corona

Abstract Radio waves from the Sun are emitted, as a rule, due to energized electrons. Observations infer that the related energized electrons follow (negative) power-law velocity distributions above a break velocity U b . They might also distribute anisotropically in the pitch-angle space. To understand radio wave generation better, we study the consequences of anisotropic power-law-distributed energetic electrons in current-free collisionless coronal plasmas utilizing 2.5-dimensional particle-in-cell simulations. We assume that the velocity distribution f u of the energized electrons follows a plateau (∂f u /∂u = 0) and a power-law distribution with spectral index α for velocities below and above U b , respectively. In the pitch-angle space, these energized electrons are spread around a center μ c = 0.5. We found that the energetic plateau-power-law electrons can more efficiently generate coherent waves if the anisotropy of their pitch-angle distribution is sufficiently strong, i.e., a small pitch-angle spread μ s . The break velocity U b affects the excitation dominance between the electrostatic and electromagnetic waves: for larger U b electrostatic waves are mainly excited, while intermediate values of U b are required for an excitation dominated by electromagnetic waves. The spectral index α controls the growth rate, efficiency, saturation, and anisotropy of the excited electromagnetic waves as well as the energy partition in different wave modes. These excited electromagnetic waves are predominantly right-handed polarized, in X- and Z-modes, as observed, e.g., in solar radio spikes. Additionally about 90% of the kinetic energy loss of the energetic electrons is dissipated, heating the ambient thermal electrons. This may contribute to the coronal heating.

STABILITY ANALYSIS OF VISCOELASTIC LIQUID DROPLETS EXCITED BY RADIAL OSCILLATIONS

When a liquid drop is periodically excited by an external radial oscillation force, the instability of standing wave mode will be formed on its surface, which is known as the spherical Faraday instability problem. The oscillation frequency of the instability surface wave will render as a harmonic or sub-harmonic mode according to the different fluid physical parameters and the forced excitation conditions. Based on the linear small perturbation theory, this paper studies the instability behavior of the viscoelastic droplet surface wave subjected to the radial oscillating force. The oscillating radial force causes the momentum equations to be Mathieu equations which included time period coefficients. Therefore, the system becomes a parametric instability problem, which can be solved by Floquet theory. In this model, the characteristics of viscoelasticity are treated as an effective viscosity which related to the rheological model of the fluid, which simplifies the solving process of the problem. Based on the analysis of the neutral stability curve and growth rate of the surface wave, the influence of viscoelastic parameters on the stability of droplets were studied. The results showed that the increase of zero-shear viscosity (μ0) as well as deformation retardation time (λ2) can inhibit the growth of droplet surface wave, therefore increased the excitation amplitude which made the droplet unstable at a harmonic mode.With the increase of oscillation amplitude, the regions of unstable growth rate decrease, and as the oscillation frequency increase, the value of droplet surface wave growth rate decrease. Through the analysis of the growth rate, it can be concluded that the increase of the stress relaxation time (λ1) increases the growth rate, thereby promoting the growth of surface wave on the droplet.

How Negative Energy and Kelvin–Helmholtz Instabilities Grow by Longitudinal Waves in Solar Atmospheric Jets

Abstract We model the propagation of slow magnetoacoustic body waves in solar jets in the course of negative energy wave excitation in the context of magnetohydrodynamic theory. Explicit approximate expressions are provided for the dispersion relation of slow body waves, providing insight into the influence of the steady flow speed, radiative cooling, and plasma-β at a glance. Analytic expressions are provided regarding critical speeds in the presence of backward waves, negative energy wave speeds, and instabilities. The buildup of the Kelvin–Helmholtz instability above the negative energy wave instability is expressed through analytic expressions that provide insight into the interplay of equilibrium conditions and dispersive effects as they affect the instability growth rate of slow body waves at various altitudes. As slow magnetoacoustic waves propagate with the same speed in the long-wavelength limit, slow body kink waves experience stronger dispersion than sausage waves. Backward waves are also probable at lower steady flow speeds for medium wavelengths when the jet hosts slow body kink waves that provide greater domains for dissipative processes. Slow body sausage waves grow faster while nearing the long-wavelength limit, while the internal plasma-β decreases the instability growth rate. The seismological aspect is that energy transfer to the external medium is observed on various timescales. The observational aspect is that slow body kink waves may exist at higher altitudes as energy has already been extracted to the external medium due to negative energy unstable slow body sausage waves in earlier stages contributing toward coronal heating.

A “Trap‐Release‐Amplify” Model of Chorus Waves

AbstractWhistler mode chorus waves are quasi‐coherent electromagnetic emissions with frequency chirping. Various models have been proposed to understand the chirping mechanism, which is a long‐standing problem in space plasmas. Based on analysis of effective wave growth rate and electron phase space dynamics in a self‐consistent particle simulation, we propose a phenomenological model called the “Trap‐Release‐Amplify” (TaRA) model for chorus. In this model, phase space structures of correlated electrons are formed by nonlinear wave particle interactions, which mainly occur in the downstream of equator. When released from the wave packet in the upstream, these electrons lead to selective amplification of new emissions which satisfy the phase‐locking condition to maximize wave power transfer, resulting in frequency chirping. The phase‐locking condition at the release point gives a chirping rate that is, fully consistent with the one by Helliwell in case of a nonuniform background magnetic field. The nonlinear wave particle interaction part of the TaRA model results in a chirping rate that is, proportional to wave amplitude, a conclusion originally reached by Vomvoridis et al. Therefore, the TaRA model unifies two different results from seemingly unrelated studies. Furthermore, the TaRA model naturally explains fine structures of chorus waves, including subpackets and bandwidth, and their evolution through dynamics of phase‐trapped electrons. Finally, we suggest that this model could be applied to explain other related phenomena, including frequency chirping of chorus in a uniform background magnetic field and of electromagnetic ion cyclotron waves in the magnetosphere.

Simulating the effects of warm O[formula omitted] ions on the growth of electromagnetic ion cyclotron (EMIC) waves

Electromagnetic ion cyclotron (EMIC) waves are believed to play a crucial role in the dynamics of the Earth’s magnetosphere. It has been widely accepted that plasma compositions can influence the growth rate of EMIC waves in the inner magnetosphere, but how warm O+ ions change the wave growth rate is not well known. In this study, we investigate the impact of ring current O+ concentration on the EMIC growth rate during a specific storm when the O+ ion flux significantly increases. We calculate the growth rate of EMIC waves by using physics-based ion distributions output from the Ring current–Atmosphere interactions Model with Self-Consistent magnetic (B) and Electric (E) fields (RAM-SCBE) model. The percentage of warm O+ is parametrically varied to examine how the maximum EMIC growth rate changes over time and how the global distribution of the maximum EMIC growth rate is affected. We found that the maximum growth rate of H-band appears in the dusk-to-midnight sector near the plasmapause, while O-band is excited at a slightly outer region. The maximum growth rate of He-band is closely related to the cold plasma density. With the increase of warm O+ density, the maximum growth rate of H-band and He-band EMIC wave is reduced, while that of O-band EMIC wave is increased, and the region with this wave excitation is widened. Such variation in the maximum EMIC growth rate implies a potential impact on the associated wave–particle interactions and change of the decay rate in the ring current.

Formation of rogue waves on the periodic background in a fifth-order nonlinear Schrödinger equation

We construct rogue wave solutions of a fifth-order nonlinear Schrödinger equation on the Jacobian elliptic function background. By combining Darboux transformation and the nonlinearization of spectral problem, we generate rogue wave solution on two different periodic wave backgrounds. We analyze the obtained solutions for different values of system parameter and point out certain novel features of our results. We also compute instability growth rate of both dn and cn periodic background waves for the considered system through spectral stability problem. We show that instability growth rate decreases (increases) for dn-(cn) periodic waves when we vary the value of the elliptic modulus parameter.

Oblique-mode breakdown in hypersonic and high-enthalpy boundary layers over a blunt cone

The nonlinear analyses of the hypersonic and high-enthalpy boundary-layer transition had received little attention compared with the widely-studied linear instabilities. In this work, the oblique-mode breakdown, as one of the most available transition mechanisms, is studied using the nonlinear parabolized stability equations (NPSE) with consideration of the thermal-chemical non-equilibrium effects. The flow over a blunt cone is computed at a free-stream Mach-number of 15. The rope-like structures and the spontaneous radiation of sound waves are observed in the schlieren-like picture. It is also illustrated that the disturbances of the species mass and vibrational temperature near the wall are mainly generated by the product term of the wall-normal velocity disturbance and the mean-flow gradient. In comparison to the CPG flow, the TCNE effects destabilize the second mode and push upstream the N factor envelope. The higher growth rate of the oblique wave leads to stronger growth of the streamwise vortices and harmonic waves.

Energy growth in subcritical viscoelastic pipe flows

This work studies the dynamics of linear non-modal waves in viscoelastic pipe flows of FENE-P fluids using transient growth and resolvent analyses. We particularly focus on the time evolution of the amplification of the disturbance energy (using the 4th-order Runge–Kutta method) and discuss the dynamical traits of the Orr and the critical-layer mechanisms in the conformation tensor field when the transient energy increases. The helical mode undergoes a larger energy growth than the axisymmetric mode. The effects of various flow parameters have been investigated on the growth rate and energy amplification of the non-modal waves and the optimal flow structures. It is found that when the elastic effect is stronger, the amplitude of the optimal conformation tensor in the pipe centre region becomes greater from a non-modal perspective. • Orr mechanism in the conformation tensor is at play when transient energy increases. • Conformation tensor in the pipe centre becomes greater when elasticity is stronger. • Helical modes, though stable, undergo larger energy growth than the axisymmetric mode.

Characteristics of Resonant Electrons Interacting With Whistler Waves in the Nearest Dipolarizing Magnetotail

AbstractThe Magnetospheric Multiscale spacecraft observations in the burst mode allow the determination of the characteristics of resonant electrons interacting with quasi‐parallel whistler waves during prolonged dipolarizations in the near Earth magnetotail at −22 RE < X ≤ −8 RE. We have detected 163 whistler bursts observed during 48 dipolarization events. The bursts were registered within ∼13 min following the dipolarization onset when the burst mode observations were available. In the majority of events, electrons with energies Wres ≥ 10 keV and pitch angles αres ∼ 100°–130° and αres ∼ 50°–80°made the maximum positive contribution to the growth rate of whistler waves propagating quasi‐parallel and antiparallel to the ambient magnetic field, respectively. Our analysis shows that electrons with Wres ∼ 10–20 keV could potentially be scattered into the loss cone by the low frequency whistler waves with fw ∼ (0.05–0.2)fce (fw is the wave frequency and fce is the electron gyrofrequency). The electrons that are scattered into the loss cone can contribute to electron precipitation within ∼13 min following the dipolarization onset. We suggest that whistler waves that are excited due to cyclotron instability driven by the temperature anisotropy of suprathermal electrons (≥2 keV) may, in turn, affect electron distribution in this energy range. Specifically, lower energy resonant electrons transfer a part of their kinetic energy to waves, while more energetic electrons absorb wave energy increasing their kinetic energy. This may lead to the transformation of higher‐energy part of electron distribution from Maxwellian to the power law shape.

A Study on Instabilities and Nonlinear Wave Energy Exchange Process in Terrestrial Ionospheric Plasma

From ground and satellite based observatories, various forms of nonlinear effects and plasma waves instabilities are observed in the Earth’s ionosphere. In this paper we present a brief review on nonlinear wave-particle interaction energy exchange mode plasma maser instability and on probable growth rate of high frequency non-resonant waves at different regions of the Earth’s ionosphere.

Trapped upper hybrid waves as eigenmodes of non-monotonic background density profiles

Non-monotonic plasma density structures such as blobs and magnetic islands give rise to trapped upper hybrid (UH) waves. Trapped UH waves which satisfy Bohr–Sommerfeld quantization can be thought of as eigenmodes of a cavity. Using fully kinetic particle-in-cell simulations, we verify the existence of these UH eigenmodes and demonstrate their significance as only eigenfrequencies become unstable to three-wave interactions. The eigenmodes can be excited through parametric decay instabilities (PDIs) of an X-mode pump wave at approximately twice the UH frequency, as could be the case for a gyrotron beam traversing a blob in a magnetically confined fusion plasma. We derive a closed expression for the wavenumber of UH waves, which is accurate both close to the UH layer and to the electron cyclotron resonance. This allows for fast analysis of eigenmodes in a non-monotonic structure. An expression for the amplification of PDI daughter waves in an inhomogeneous plasma is extended to a decay region where the first several derivatives vanish. From the amplification in a convective PDI, we estimate the growth rate of the absolute PDI involving the trapped waves. We show that the excitation of eigenmodes through PDIs in our simulations are indeed absolute rather than convective due to the trapping of the daughter waves. Additionally, we show that only eigenmodes get excited through the PDIs, and that we are able to predict the growth rates of the daughter waves and how they scale with the pump wave intensity. This is evidence supporting a fundamental assumption of analytical theory describing low threshold strong scattering observed in magnetically confined fusion experiments during second harmonic electron cyclotron resonance heating (ECRH). Such low threshold instabilities can degrade ECRH performance but also offer novel uses for ion heating or as diagnostics.

Numerical Investigation of Nonlinear Boundary-Layer Transition for Cones at Mach 6

Direct numerical simulations were carried out for a straight and a flared cone at Mach 6 and zero angle of attack to investigate the effects of geometry on the linear and nonlinear stages of the laminar–turbulent transition process. The cone geometries and the flow conditions of the simulations are chosen to closely match those of the experiments conducted at the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University. In the linear (primary) instability regime the flared cone resulted in larger integrated growth rates ( factors) and a higher dominant second mode frequency. Investigations of the secondary instability regime revealed that the flared cone also leads to a stronger fundamental resonance, characterized by larger integrated growth rates of the secondary instability waves after resonance onset. For both cases, the so-called controlled fundamental breakdown simulations were carried out, which showed the development of similar patterns of streamwise hot streaks on the surface of the cones. Streamwise streaks were also observed in the Purdue experiments (for the flared cone only) using temperature-sensitive paint. A detailed comparison of the results obtained for both geometries indicated that the streak development is caused by the same nonlinear mechanisms.

An Unexpected Whistler Wave Generation Around Dipolarization Front

AbstractDipolarization front (DF)—a sharp boundary separating hot tenuous plasmas from cold dense plasmas—is followed by a strong Bz region, which is termed flux pileup region (FPR) or dipolarizing flux bundle (DFB). Using 9‐year (2001–2009) Cluster data, we show that the FPR hosts whistler waves because of the pancake distribution of electrons, whereas the DF boundary hosts lower hybrid drift waves due to the strong density and magnetic gradients statistically. Different from statistical results, we report an unexpected case: whistler waves are generated at the DF boundary rather than inside the FPR. Using Cluster data, we observed strong whistler waves at DF boundaries but no whistlers inside FPRs, although flux tubes inside the FPRs are significantly compressed and suprathermal electrons there are perpendicularly anisotropic. We calculate wave growth rates and successfully explain the generation/damping of these whistlers. We find that 1–4 keV electrons are responsible for generation/damping of these whistlers.

The role of Cairns–Gurevich distributed electrons on obliquely propagating ion‐acoustic waves

AbstractWe used a new distribution of electrons in a two‐component magnetized plasma to study the non‐linear ion‐acoustic solitary structures. The distribution called “Cairns–Gurevich distribution” describes simultaneously the evolution of the energetic electrons and those trapped in the plasma potential well. A modified KdV equation describing the non‐linear comportment of the ion‐acoustic wave (IAW) was found by using the standard reductive perturbation technique and the appropriate independent variables. The behaviour of the soliton by changing the plasma parameters has been investigated, and we demonstrated that by decreasing the non‐thermality parameter, the soliton solution amplitude is enhanced. In addition, we have discussed the growth rate of the solitary waves by calculating the instability criterion. Through discussion, we have conferred how different plasma parameters, such as the trapping, non‐thermality, Mach number, obliqueness via the angle of propagation, and magnetic field via the ion‐cyclotron frequency, can affect the solitary wave structures. This kind of theoretical studies can be relevant to understand the non‐linear propagation of IA solitary waves plasmas of electrons and particles in laser‐plasma interaction, pulsar magnetosphere, the auroral zone, and the upper ionosphere, where plasma with trapped and energetic electrons are often present.

Van Allen Probe Observations of Disappearance, Recovery and Patchiness of Plasmaspheric Hiss Following Two Consecutive Interplanetary Shocks: First Results

AbstractWe present, for the first time, a plasmaspheric hiss event observed by the Van Allen probes in response to two successive interplanetary (IP) shocks occurring within an interval of ∼2 h on December 19, 2015. The first shock arrived at 16:16 UT and caused disappearance of hiss for ∼30 min. Combined effect of plasmapause crossing, significant Landau damping by suprathermal electrons and their gradual removal by magnetospheric compression led to the disappearance of hiss. Calculation of electron phase space density and linear wave growth rates showed that the shock did not change the growth rate of whistler waves within the core frequency range of plasmaspheric hiss (0.1–0.5 kHz) during this interval making conditions unfavorable for the generation of hiss. The recovery began at ∼16:45 UT which is attributed to an enhancement in local plasma instability initiated by the first shock‐induced substorm and additional possible contribution from chorus waves. This time, the wave growth rate peaked within the core frequency range (∼350 Hz). The second shock arrived at 18:02 UT and generated patchy hiss persisting up to ∼19:00 UT. It is shown that an enhanced growth rate and additional contribution from shock‐induced poloidal Pc5 mode (periodicity ∼240 s) ultralow frequency (ULF) waves resulted in the excitation of hiss waves during this period. The hiss wave amplitudes were found to be additionally modulated by background plasma density and fluctuating plasmapause location. The investigation highlights the important roles of IP shocks, substorms, ULF waves, and background plasma density in the variability of plasmaspheric hiss.

Wave trapping and E × B staircases

A model of E × B staircases is proposed based on a wave kinetic equation coupled to a poloidal momentum equation. A staircase pattern is idealized as a periodic radial structure of zonal shear layers that bound regions of propagating wave packets, viewed as avalanches. Wave packets are trapped in shear flow layers due to refraction. In this model, an E × B staircase motif emerges due to the interaction between propagating wave packets (avalanches) and trapped waves in the presence of an instability drive. Amplitude, shape, and spatial period of the staircase E × B flow are predicted as functions of the background fluctuation spectrum and the growth rate of drift waves. The zonal flow velocity radial profile is found to peak near its maxima and to flatten near its minima. The optimum configuration for staircase formation is a growth rate, that is, maximum at zero radial wave number. A mean shear flow is responsible for a preferential propagation speed of avalanches. It is not a mandatory condition for the existence of staircase solutions, but has an impact on their spatial period.

2D Parametric Model for Surface Wave Development Under Varying Wind Field in Space and Time

AbstractA fully consistent 2D parametric model of wave development under spatially and/or time varying winds is developed. Derived coupled equations are written in their characteristic form to provide practical means to rapidly assess how the energy, frequency and direction of dominant surface waves are developing and distributed under varying wind forcing conditions. For young waves, nonlinear interactions drive the peak frequency downshift, and the wind energy input and wave breaking dissipation are governing the wave energy evolution. With a prescribed wind wave growth rate, proportional to (u*/c) squared, wave breaking dissipation must follow a power‐function of the dominant wave slope. For uniform wind conditions, this choice for the growth rate imposes solutions to follow fetch laws, with exponents q = −1/4, p = 3/4 correspondingly. This set of exponents recovers the Toba's laws, and imposes the wave breaking exponent equal to 3. A varying wind direction can then drive spectral peak direction changes, leading to the occurrence of focusing/defocusing wave groups over localized areas where wave‐rays merge and cross. Significant (but finite) local variations of the energy are then expected under varying wind forcing. Propagating away from a stormy area, wave rays generally diverge, leading to dispersive swell systems. Examples of practical applications of this model are provided in (Kudryavtsev et al., 2021, companion paper).

Nonlocal Theory of Excitation of Electron Bernstein Waves by a Relativistic Electron Beam in Plasma with Loss-Cone Distribution of Electron

A nonlocal theory of excitation of electron Bernstein waves in a magnetized plasma slab by a relativistic electron beam with loss-cone distribution of electrons is presented. Kinetic theory is employed to obtain the susceptibility of plasma electrons whereas fluid theory is used to obtain the susceptibility of beam electrons. The mode structure and growth rate of the electron Bernstein wave are obtained. The growth rate in the case of Cerenkov and slow cyclotron interactions are proportional to $${{\omega }_{b0}}^{1/3}$$ and $${\omega }_{b0}$$ , respectively. For typical parameter, the normalized growth rate attains maximum value at $$b\approx 1.743$$ with different normalized beam velocities. These parameters control the growth rate due to nonlocality which rapidly decreases. The present theoretical scheme may be possible application in the field of heating of spherical torus plasmas.

Stability of Capillary Waves of an Arbitrary Symmetry on a Jet in a Uniform Electrostatic Field

The dispersion relation for capillary waves of an arbitrary azimuthal symmetry on an elliptic cross-section jet in a uniform electrostatic field which is perpendicular to the axis of symmetry of the jet is derived. The stability of the first three azimuthal modes is investigated. It is shown that the stability of capillary waves on the jet reduces with increase in the external electrostatic field strength and the related eccentricity of transverse jet cross-section, as well as with increase in the polarization charge on the jet surface. Comparison with a jet in a radial electrostatic field is carried out. In the case of the transverse electrostatic field the instability growth rates of the capillary waves of any symmetry on the jet surface are turned out to be significantly higher than those in the case of the radial electrostatic field.