Unstable Waves Research Articles

Convolution Representation of Traveling Pulses in Reaction-Diffusion Systems

Convolution representation manifests itself as an important tool in the reduction of partial differential equations. In this study, we consider the convolution representation of traveling pulses in reaction-diffusion systems. Under the adiabatic approximation of inhibitor, a two-component reaction-diffusion system is reduced to a one-component reaction-diffusion equation with a convolution term. To find the traveling speed in a reaction-diffusion system with a global coupling term, the stability of the standing pulse and the relation between traveling speed and bifurcation parameter are examined. Additionally, we consider the traveling pulses in the kernel-based Turing model. The stability of the spatially homogeneous state and most unstable wave number are examined. The practical utilities of the convolution representation of reaction-diffusion systems are discussed.

Shear Flow Instability over a Finite Time Interval

Within the framework of a discrete quasi-geostrophic model with two vertical levels, the problem of linear stability of the flow of a stratified rotating fluid with constant vertical and horizontal velocity shifts is solved. It is shown that taking into account the horizontal shear leads to a qualitative change in the dynamics of unstable wave disturbances. The main feature is related to the effect of temporary exponential growth of unstable perturbations, i.e. growth over a finite time period. This effect manifests itself in the alternation of stages of smooth oscillating behavior (in time) with stages of exponential (explosive) growth of finite duration. A kinematic interpretation of the effect of temporal exponential growth is given, which is associated with the passage of a time-dependent perturbation wave vector through the region of exponential instability that exists in the absence of a horizontal shear. It is shown that mathematically this effect is described by solutions of a second-order differential equation containing turning points. Asymptotic solutions of the equation are given for weak horizontal shifts.

Chiral Light Amplifier with Pumped Weyl Semimetals.

Parallel electric and magnetic fields applied to Weyl semimetals pump axial charge via the axial anomaly until balanced by intervalley relaxation. The resulting nonequilibrium steady state exhibits the chiral magnetic effect as well as the anomalous Hall effect, which coupled with Maxwell's equations leads to unstable electromagnetic waves at low frequency and long wavelength. Here, we show that such chiral magnetic instability manifests itself as anomalous reflectance of the surface of pumped Weyl semimetal. Depending on electric, chiral magnetic, and anomalous Hall conductivities, the reflectance is found to exceed unity in a finite range of frequency for a circularly polarized light incident along the direction of Weyl node separation.

An experimental study on the stability of hypersonic plate boundary layer regulated by a plasma actuation array

Laminar–turbulent transition control is one of the key problems in the design of hypersonic vehicles. In particular, for air-breathing hypersonic vehicles, early onset turbulence in the boundary layer of a scramjet inlet precursor is desirable. Plasma actuation has become a popular flow control method in recent years and has made progress in regulating the stability of supersonic boundary layers. However, there have been few studies on the regulation of the stability of hypersonic boundary layers. In this paper, an experimental study on the stability of the hypersonic plate boundary layer is carried out using a spanwise plasma actuation array. The characteristics and evolution of different kinds of unstable waves in the hypersonic plate boundary layer (especially the dominant first mode wave) are analyzed based on the results of linear stability theory and high-frequency pulsation sensors. The typical morphological characteristics of the boundary layer and the macro-control effect of the plasma actuation array are explored through a high-speed schlieren method. Finally, based on grayscale mode extraction and proper orthogonal decomposition, the influences of three different actuation frequencies on the instability waves and characteristic structure of the boundary layer are studied, including the dominant frequency of the first mode wave, the half-frequency of the dominant frequency of the first mode wave, and the dominant frequency of the second mode wave, the change of characteristic structures under the regulation of plasma actuation is further discussed. The corresponding regulation rules and mechanisms are summarized. The results show that the plasma actuation array can advance the starting position of laminar discontinuities and that the induced coherent structure can excite instabilities at an earlier flow-direction position. The actuation can be used to control the stability of the boundary layer by acting on the first mode wave to break the original unstable wave spectrum characteristics. This verifies the ability of extensional array plasma actuation to regulate the stability of the hypersonic plate boundary layer and suggests it has great potential in the promotion of hypersonic boundary layer transition.

Overstable Convective Modes in a Polytropic Stellar Atmosphere

Within the convection zone of a rotating star, the presence of the Coriolis force stabilizes long-wavelength convective modes. These modes, which would have been unstable if the star lacked rotation, are called overstable convective modes or thermal Rossby waves. We demonstrate that the Sun’s rotation rate is sufficiently rapid that the lower half of its convection zone could possess overstable modes. Further, we present an analytic solution for atmospheric waves that reside within a polytropic stratification. We explore in detail the properties of the overstable and unstable wave modes that exist when the polytrope is weakly unstable to convective overturning. Finally, we discuss how the thermal Rossby waves that reside within the convection zone of a star might couple with the prograde branch of the g modes that are trapped within the star’s radiative zone. We suggest that such coupling might enhance the photospheric visibility of a subset of the Sun’s g modes.

Marginal stability constraint on runaway electron distribution

High-frequency kinetic instabilities of the strongly anisotropic runaway electrons (RE) can enhance the pitch-angle scattering of the runaways significantly. This wave-induced scattering can easily prevail over runaway scattering on high-Z impurities. In a steady state, collisional damping balances the kinetic drive of the unstable waves, keeping the RE distribution function at marginal stability. The marginal stability constraint limits the achievable RE densities and the shape of the RE distribution function. In this study, we consider whistler and compressional Alfvén waves as the primary source of enhanced elastic scattering of the runaways. By balancing the anomalous Doppler resonance drive with the collisional wave damping, we find the RE distribution function in the ultra-relativistic range of the phase space. We also derive an expression for the spectral energy density of the waves. We show that the power needed to compensate for the wave dissipation is negligible compared to the work of the electric field. The latter is in balance with the synchrotron losses of the runaway electrons.

Bump Attractors and Waves in Networks of Leaky Integrate-and-Fire Neurons

Bump attractors are wandering localised patterns observed in in vivo experiments of spatially-extended neurobiological networks. They are important for the brain's navigational system and specific memory tasks. A bump attractor is characterised by a core in which neurons fire frequently, while those away from the core do not fire. We uncover a relationship between bump attractors and travelling waves in a classical network of excitable, leaky integrate-and-fire neurons. This relationship bears strong similarities to the one between complex spatiotemporal patterns and waves at the onset of pipe turbulence. Waves in the spiking network are determined by a firing set, that is, the collection of times at which neurons reach a threshold and fire as the wave propagates. We define and study analytical properties of the voltage mapping, an operator transforming a solution's firing set into its spatiotemporal profile. This operator allows us to construct localised travelling waves with an arbitrary number of spikes at the core, and to study their linear stability. A homogeneous "laminar" state exists in the network, and it is linearly stable for all values of the principal control parameter. Sufficiently wide disturbances to the homogeneous state elicit the bump attractor. We show that one can construct waves with a seemingly arbitrary number of spikes at the core; the higher the number of spikes, the slower the wave, and the more its profile resembles a stationary bump. As in the fluid-dynamical analogy, such waves coexist with the homogeneous state, are unstable, and the solution branches to which they belong are disconnected from the laminar state; we provide evidence that the dynamics of the bump attractor displays echoes of the unstable waves, which form its building blocks.

Investigation of hypersonic cone boundary layer stability regulation with plasma actuation

Boundary layer transition has always been a frontier in the field of fluid mechanics, and hypersonic boundary layer stability experiments can help reveal the physical mechanisms behind such transitions. In particular, the regulation of unstable waves in boundary layers is critical for transition prediction and control. Plasma actuation is a popular flow control method that has made progress in moderating the stability of supersonic boundary layer. However, there have been few studies on regulating the stability of hypersonic boundary layers with plasma actuation. In this paper, wind tunnel experiments are carried out under Mach 6 flow to study the stability regulation of a hypersonic sharp cone boundary layer with nanosecond pulsed plasma actuation. First, the typical characteristic structure of the rope-like structure is captured by the high-speed schlieren method. Then, combining the sensor results and theoretical analysis, the rope-like structure and the dominant instability wave resolved by the schlieren power spectrum density method are determined to correspond to the second-mode wave. The characteristic unstable quasi-ordered structure of the boundary layer under actuation is then extracted, and the impact effect and modulation effect of this structure on the second-mode wave are analyzed. Finally, the mechanism by which actuation influences the boundary layer instability is studied using proper orthogonal decomposition. The results show that actuation can enhance boundary layer pulsation, and the coupling effect between the actuation and boundary layer can produce regular unstable quasi-ordered structures. The intrinsic mechanism works by impinging on and modulating the second-mode waves, and there are characteristic modes of the rope-like structure and the unstable structures distributed over the whole flow-direction range. This verifies the ability of plasma actuation to stimulate the instability of hypersonic cone boundary layers and provides technical support for the further development of transition control methods.

Robust poleward jet shifts in idealised baroclinic-wave life-cycle experiments with noisy initial conditions

Abstract. Idealised baroclinic-wave life-cycle experiments are a widely used tool to study fundamental characteristics of mid-latitude baroclinic instability. A typical life cycle evolves from an initialised baroclinically unstable jet through an exponential growth phase of a particular unstable wave mode, followed by wave breaking during the mature phase and wave–mean flow interaction driving a jet shift during the decay phase. Many authors distinguish between life cycles with predominantly anticyclonic (LC1) and cyclonic (LC2) wave breaking, and the transition between the two flavours is typically controlled via the strength of cyclonic meridional wind shear in the initial conditions. While baroclinic wave growth has traditionally been triggered via a specified initial perturbation with fixed zonal wave number, this study extends the concept of baroclinic-wave life cycles by analysing the influence of random initial perturbations without any preferred zonal dependency on the life-cycle evolution. We find that the growth phase shows a robust LC1–LC2 distinction as a function of initialised meridional shear, while a preference for LC1-like characteristics is observed during the decay phase for all life cycles with non-monochromatic initial perturbations. In particular, the persistent cut-off cyclones that typically form for LC2 initialisations are found to eventually become unstable, with the onset of instability coming sooner for larger noise perturbations. All non-monochromatic life cycles result in a poleward jet shift in their final state, regardless of the strength of the initial shear. Consistently, anticyclonic wave breaking tends to be predominant during the mature and decay phases, even for LC2 initialisations. Equatorward jet shifts associated with cyclonic wave breaking still exist, although purely as a transient interim state. We show that wave–wave interactions resulting from the initialised random wave spectrum play an important role during all phases of the life cycle.

Effect of the Quadric Shear Zonal Flows and Beta on the Downstream Development of Certain Unstable Baroclinic Waves

Many factors are affecting the downstream development of baroclinic waves, among which zonal shear flow is one of the factors that need to be considered. In this paper, the influence of zonal shear flow and <em>β</em> on the downstream development of unstable chaotic baroclinic waves is studied from the two-layer model in a wide channel controlled by quasi-geostrophic potential vorticity equation. Through the obtained Lorentz equation, We concentrated on the influence of zonal shear flow (the second derivative of baseline zonal flow is not zero) on the downstream development of baroclinic waves. In the absence of zonal shear flow, chaotic behavior along feature points would occur, and the amplitude would change rapidly from one feature to another, that is, it would change very quickly in space. When zonal shear flow is introduced, the influence of zonal shear flow on the downstream development of unstable baroclinic waves is examined categorically. And from Lorentz’s final equation, we’re investigating a change in his solution. It is found that the zonal shear flow smoothes the solution of the equation and reduces the instability, and with the increase of zonal shear flow, the stability in space will increase gradually. The second derivative of the zonal shear flow (the quadrical shear flow) therefore has a major influence on the stability of space.

The frequency of current fluctuations in two-valley semiconductors in an external electric and strong magnetic (

Boltzmann's kinetic equations have not been used to date to study nonequilibrium phenomena in semiconductors, and therefore, to obtain analytical expressions for the oscillation frequency inside the semiconductor and the critical external electric field, it is of theoretical interest. In this theoretical work, the frequency of oscillations occurring inside a two-valley semiconductor of the GaAs type in an external constant electric field and in an external strong magnetic field (

Theoretical Investigations of Gyrating Ion Beam-Driven Ion-Acoustic Wave Instability in Plasma Using First Order Perturbation Theory

A Gyrating ion beam traveling through a plasma cylinder comprising of electrons, positive ions K + , and heavy C 7 F 14 − negative ions, through Cerenkov interaction, drive IAWs (ion-acoustic waves) to unstable mode. In this process, two ion-acoustic (IA) unstable wave modes: a fast K + mode and a slow C 7 F 14 − mode in the presence of positive and heavy negative ions are observed in the plasma. Numerical calculations of the sound wave phase velocity, unstable mode frequency, and the growth rates have been carried out. The calculations show that the growth rate and unstable wave frequencies for both the IA wave modes increase with the increase in the relative density of used heavy negative ions. Moreover, the growth rate is measured as 1/3rd of ion beam density power and the phase velocity rises with the concentration of heavy negative ions. There is a considerable effect of an increase in the magnetic field on the growth rate. The growth rate of the fast (K+) mode increases with the magnetic field more rapidly as compared to the slow ion mode C 7 F 14 − .

Propagation characteristics of unstable detonation waves in a round tube with annular perturbation

The detonation propagation characteristics of the mixtures, 2H2 + O2+3Ar and CH4 + 2O2, were investigated. Accordingly, the mixtures were tested in round tube with inner diameter of D = 80 mm and annular tubes with widths of w = 25 mm, 15 mm, and 5 mm. The two mixtures represent stable mixture with regular cell pattern and unstable mixture with irregular cell pattern, respectively. Smoked foils were utilized to record cellular structure under various initial conditions. Subsequently, the length scale Ldsc was measured, which represents the length from the start of the test section to the position where the cellular structure changes drastically (the cell size obviously increases or the structure disappears). The results reveal that both mixtures can successfully propagate in round tube and annular tubes of 25 mm and 15 mm, but fail in 5 mm annular tube. The Ldsc value of 2H2 + O2 + 3Ar is higher than that of CH4 + 2O2 in 80 mm and 15 mm tubes, but it is opposite in 25 mm tube. Moreover, the relationship between Ldsc and hydraulic diameter DH was analysed. For a given tube, the values of Ldsc and Ldsc/DH increased when the initial pressure increased. And the variation trend of Ldsc and Ldsc/DH of CH4 + 2O2 is steeper. Furthermore, the mixtures 2H2 + O2 + 3Ar and CH4 + 2O2 resulted in over-driven detonation in 15 mm and 25 mm annular tubes, respectively. The ratio between the total reaction length (sum of the induction length and exothermic length) and the hydraulic diameter (DH/(Δi + Δe)) correspond to critical values of 18 for hydrogen-oxygen-argon and 6 for methane-oxygen, below which the detonation will fail.

Spectral transition of multiscale turbulence in the tokamak pedestal

The transition in the turbulence spectrum from ion-scale dominated regimes to multiscale transport regimes that couple ion and electron scales is studied with gyrokinetic simulations of turbulent transport. The simulations are based on DIII-D high-confinement mode (H-mode) plasma parameters in the tokamak pedestal. The transition is initiated by varying the ion temperature gradient. To our knowledge, no full multiscale simulations of pedestal-like transport have been done previously. The experimental parameters lie in a bifurcation region between the two regimes. At long wavelengths, a complex, ion-direction hybrid mode is the dominant linearly unstable drift wave, while an electron temperature gradient-driven mode is unstable at short wavelengths. In the transition from the multiscale branch to the ion-scale branch, the magnitude of the ion-scale poloidal wavenumber spectrum of the nonlinear turbulent energy flux increases and the magnitude of the high-wavenumber spectrum decreases. The decrease in the electron-scale transport is due to nonlinear mixing with ion-scale fluctuations and the ion-scale-driven zonal flows. A shift in the total energy associated with the fluctuating electrostatic potential intensity from dominantly drift kinetic energy in the multiscale regime to dominantly potential intensity in the ion-scale regime is well-correlated with the trend in the total energy flux.

Observations of Rapidly Growing Whistler Waves in Front of Space Plasma Shock due to Resonance Interaction between Fluctuating Electron Velocity Distributions and Electromagnetic Fields

The whistler-mode wave extending from the fast-magnetosonic wave branch is a fundamental perturbation of electromagnetic fields and plasmas in various environments including planetary space, laboratory, and astrophysics. The origin and evolution of the waves is a long-standing question due to the limited instrumental capability in resolving highly variable plasma and electromagnetic fields. Here, we analyze observational data with a high time resolution from the Magnetospheric Multiscale spacecraft in front of the terrestrial bow shock (e.g., foreshock). We develop a novel approach to extract the three-dimensional fluctuating electron velocity distributions (δf e( V )) from their background (f e0( V )), and have successfully captured the coherent resonance between fluctuating electrons (δf e( V )) and wavelike electromagnetic fields (δ B , δ E ) at an unprecedentedly high frequency (>1 Hz) for investigating wave–particle interactions. We provide that the unstable whistler wave grows rapidly over a timescale that is much shorter than the proton gyro-period. Regarding the energy origin for the waves, we find the ion distributions consisting of the solar wind ion flows and the ion beams reflected from the shock play crucial roles in providing the free energy and determining the eigenmode disturbances of fields and electrons. The quantification of wave growth rate and the characterization of wave–particle interactions for the instability driver can significantly advance the understandings of wave evolution and energy conversion between multisource multispecies particles and wave electromagnetic fields.

Creation of solitons and density cavities by lower hybrid waves

Formation of Korteweg-de Vries (KdV) solitons by nonlinear lower hybrid waves (LHWs) is not possible in usual electron ion plasma because a dispersive term does not have a suitable form. It is pointed out that the dispersion characteristics of electrostatic LHWs are modified in the presence of field-aligned shear flow to produce KdV solitons with negative electrostatic potential as it has been observed by satellites in the upper ionosphere. Plasma density decreases within the solitary structures. The parallel electron velocity shear ve0=ve0(x)ẑ also gives rise to unstable waves in the intermediate frequency range in linear limit whereas the ambient magnetic field B0=B0ẑ is assumed to be constant. The instability and structure size depend upon the electron parallel velocity shear parameter Se=1Ωedve0(x)dx (where Ωe is the electron gyrofrequency) and the propagation direction with respect to the ambient magnetic field B0. The theoretical model is applied to the upper ionosphere, and the estimated width of the structures turns out to be of the order of 70 m, which is closer to the observations.

Spiral wave chimeras in populations of oscillators coupled to a slowly varying diffusive environment

Chimera states are firstly discovered in nonlocally coupled oscillator systems. Such a nonlocal coupling arises typically as oscillators are coupled via an external environment whose characteristic time scale τ is so small (i.e., τ → o) that it could be eliminated adiabatically. Nevertheless, whether the chimera states still exist in the opposite situation (i.e., τ ≫ 1) is unknown. Here, by coupling large populations of Stuart—Landau oscillators to a diffusive environment, we demonstrate that spiral wave chimeras do exist in this oscillator-environment coupling system even when τ is very large. Various transitions such as from spiral wave chimeras to spiral waves or unstable spiral wave chimeras as functions of the system parameters are explored. A physical picture for explaining the formation of spiral wave chimeras is also provided. The existence of spiral wave chimeras is further confirmed in ensembles of FitzHugh—Nagumo oscillators with the similar oscillator-environment coupling mechanism. Our results provide an affirmative answer to the observation of spiral wave chimeras in populations of oscillators mediated via a slowly changing environment and give important hints to generate chimera patterns in both laboratory and realistic chemical or biological systems.

Pattern formation and front stability for a moving-boundary model of biological invasion and recession

We investigate pattern formation in a two-dimensional (2D) Fisher–Stefan model, which involves solving the Fisher–KPP equation on a compactly-supported region with a moving boundary. The boundary evolves analogously to the classical one-phase Stefan problem, such that boundary speed is proportional to the local population density gradient. By combining the Fisher–KPP and classical Stefan theory, the Fisher–Stefan model alleviates two limitations of the Fisher–KPP equation for biological populations. Unlike the Fisher–KPP equation, solutions to the Fisher–Stefan model have compact support, explicitly defining the region occupied by the population. Furthermore, the Fisher–Stefan model admits travelling wave solutions with non-negative density for all wave speeds c∈(−∞,∞), and can thus model both population invasion and population recession. In this work, we investigate whether the 2D Fisher–Stefan model predicts pattern formation, by analysing the linear stability of planar travelling wave solutions to sinusoidal transverse perturbations. Planar fronts of the Fisher–KPP equation are linearly stable. Similarly, we demonstrate that invading planar fronts (c>0) of the Fisher–Stefan model are linearly stable to perturbations of all wave numbers. However, our analysis demonstrates that receding planar fronts (c<0) of the Fisher–Stefan model are linearly unstable for all wave numbers. This is analogous to unstable solutions for planar solidification in the classical Stefan problem. Introducing a surface tension regularisation stabilises receding fronts for short-wavelength perturbations, giving rise to a range of unstable modes and a most unstable wave number. We supplement linear stability analysis with level-set numerical solutions that corroborate theoretical results. Overall, front instability in the Fisher–Stefan model suggests a new mechanism for pattern formation in receding biological populations.

Oblique Electromagnetic Instabilities Driven by Pickup Ion Ring-beam Distributions in the Outer Heliosheath. I. Linear Instability Analysis

The energetic neutral atom ribbon observed by the Interstellar Boundary Explorer spacecraft is believed to originate from the pickup ions in the outer heliosheath. The outer heliosheath pickup ions generally have a ring-beam velocity distribution at a certain pickup angle, α, the angle at which these ions are picked up by the interstellar magnetic field. The pickup ion ring-beam distributions can drive unstable waves of different propagation angles with respect to the background interstellar magnetic field, θ. Previous studies of the outer heliosheath pickup ion dynamics were mainly focused on ring-like pickup ion distributions with α ≈ 90° and/or the parallel- and antiparallel-propagating unstable waves (θ = 0° and 180°). The present study carries out linear kinetic instability analysis to investigate both the parallel and oblique unstable modes (0° ≤ θ ≤ 180°) driven by ring-beam pickup ion distributions of different pickup angles between 0° and 90°. Our linear instability analysis reveals that ring-beam pickup ion distributions can excite oblique mirror waves as well as parallel/quasi-parallel and oblique right- and left-helicity waves. The maximum growth rate among all the instabilities belongs to the parallel-propagating left-helicity waves at most pickup angles. Furthermore, the evolution of the unstable mirror waves by varying pickup angle indicates that as the pickup angle increases, the maximum growth rate of the mirror modes increases, while its propagation angle decreases.

Traveling waves for a three-species competition system with two weak aboriginal competitors

In this paper, we study the traveling wave solutions for a three species competition system with two weak aboriginal competitors and one strong alien competitor. We are concerned with the existence of traveling waves such that these two co-existence aboriginal competitors are wiped out by the invading alien strong competitor. First, we derive the existence of wave profiles based on an application of Schauder's fixed point theorem with the help of constructing suitable generalized upper-lower solutions to capture the unstable wave tail limit. Then a new method for deriving the stable wave tail limit is introduced. Finally, the minimal invading speed is characterized.