Wave Vector Of Perturbation Research Articles

Explosive Growth of Asymmetric Disturbances in a Flow with Vertical Shear

The classical problem of geophysical hydrodynamics is the problem of the instability of a zonal geostrophic flow with a vertical velocity shear. At present, the instability with respect to symmetric perturbations that do not depend on the coordinate along the flow has been most thoroughly studied. For a symmetric instability to arise, the two-dimensional perturbation wave vector must lie inside a certain sector in the vertical plane of the wave numbers. In this paper, we study the instability with respect to asymmetric perturbations oriented at an angle to the flow. Fundamentally new features of the temporal dynamics of the amplitudes of such perturbations are found. The main feature is associated with the existence of a stage of exponential explosive growth of finite duration. A kinematic interpretation of this stage is given, which is related to the passage of the vertical projection of the perturbation wave vector through the sector of symmetric instability.

Numerical study of the effect of a magnetic field on Rayleigh-Taylor instability with different density disturbances

Rayleigh-Taylor instability (RTI) is a fundamental physical process in fluids and plasmas. RTI is ubiquitous and must be considered in the field of high-energy-density physics, such as in space physics, astrophysics, and inertial confinement fusion. In this work, two-dimensional numerical simulations of laser-driven RTI with different density perturbations are performed using a radiation magnetohydrodynamic simulation program (FLASH). The effect of the applied magnetic field on the evolution of RTI at different Atwood numbers is systematically discussed. The results show that RTI evolves freely without an external magnetic field, and it is accompanied by the generation of secondary Kelvin–Helmholtz instability. Reducing the Atwood number weakens the mixing of fluids and has a strong stabilizing effect on the RTI. Introducing an external magnetic field parallel to the perturbation wave vector further inhibits the development of RTI and Kelvin–Helmholtz instability, with magnetic pressure playing a dominant role. The study results are important to gaining an in-depth understanding of the mixing of magnetic fluids and the magnetic field evolution at the instability interface and provide a reference for subsequent experimental studies on the related magnetization RTI.

Hydrodynamic instability of premixed tubular flame stabilized in stretched rotating flow

The paper analyzes the hydrodynamic instability of a cylindrical flame stabilized in a converging rotating gas flow simulating the flow in a cyclone combustion chamber. The analysis was carried out within the framework of a two-dimensional model describing the gas flow with radial and tangential supply of combustible gas through the walls of a cylindrical chamber. In the stationary case, the gas flow in combustion products and the combustible mixture is described by the solution for a rotating tornado-shaped vortex. The rotation of the gas creates a centrifugal force, which has a stabilizing effect on the flame perturbations. The dependences of the perturbations growth rate on the wave vector of perturbations, the radial and tangential inlet gas velocities, and other parameters of the problem are obtained. The analytical criterion of neutral stability separating regions of stable flame with respect of any perturbations and flame instability in the plane of radial and tangential velocities is obtained. In the absence of rotation and at the large flame radii, the model converts into the Landau- Darrieus model of flame hydrodynamic instability.

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.

Electric and chiral response to a pseudoelectric field in Weyl materials

One of the unique features of Weyl and Dirac materials is the possibility to realize effective pseudo-electromagnetic (axial) fields by perturbing the band structure. These fields couple with a different sign to quasiparticles of different chirality. Here, the authors show that a dynamical pseudoelectric field allows for a Hall-like electric current response resembling the celebrated anomalous Hall effect. The conductivity for this nondissipative response is determined by the wave vector of perturbation, for example, sound.

The competition between the hydrodynamic instability from noise and magnetorotational instability in the Keplerian disks

We venture for the comparison between growth rates for magnetorotational instability (MRI) and hydrodynamics instability in the presence of an extra force in the local Keplerian accretion flow. The underlying model is described by the Orr–Sommerfeld and Squire equations in the presence of rotation, magnetic field, and an extra force, plausibly noise with a nonzero mean. We obtain MRI using the Wentzel–Kramers–Brillouin approximation without extra force for a purely vertical magnetic field and vertical wavevector of the perturbations. Expectedly, MRI is active within a range of magnetic field, which changes depending on the perturbation wavevector magnitude. Next, to check the effect of noise on the growth rates, a quartic dispersion relation has been obtained. Among those four solutions for the growth rate, the one that reduces to the MRI growth rate at the limit of vanishing mean of noise in the MRI active region of the magnetic field is mostly dominated by MRI. However, in the MRI inactive region, in the presence of noise, the solution turns out to be unstable, which is almost independent of the magnetic field. Another growth rate, which is almost complementary to the previous one, leads to stability at the limit of vanishing noise. The remaining two growth rates, which correspond to the hydrodynamical growth rates at the limit of the vanishing magnetic field, are completely different from the MRI growth rate. More interestingly, the latter growth rates are larger than that of the MRI. If we consider viscosity, the growth rates decrease depending on the Reynolds number.

Modulational instability of the coupled waves between magnetosonic wave and ion acoustic wave

Present paper studies the interaction between the high frequency magnetosonic wave and the ion acoustic wave which is excited by the high frequency magnetosonic wave due to the ponderomotive force. The dispersion relation and the growth rate of the perturbation wave are obtained. The instability region is given in the present paper. It is found that the growth rate depends on the background magnetic field, the frequency of the magnetosonic wave as well as the perturbation wave vector. The results show that the long wavelength perturbation wave which propagates perpendicular to the magnetic field is stable; the larger the plasma thermal pressure, the smaller the growth rate. Our results may be quite important in understanding magnetic turbulence in the laser plasma and astrophysical phenomenon.

Modulational Instability of the Coupled Waves Between High-Frequency Magnetosonic Wave and Low-Frequency Magnetosonic Wave

It is well known that the ponderomotive force of a relatively high-frequency magnetosonic wave (HMSW) can excite the low-frequency magnetosonic wave (LMSW). By using the coupled system of dynamical equations of LMSW in the presence of a ponderomotive force of HMSW (pump wave), we have studied the modulational instability of the coupled waves between the HMSW and the LMSW. It is found that there are instabilities for the coupled waves between the HMSW and the LMSW. It is also found that the growth rate depends on several system parameters, such as the frequency of HMSW, the perturbation wave vector, and the background magnetic field.

Theoretical and numerical analysis of density perturbation development induced by high velocity impact

The problem of high velocity impact between two solid plates where one of them has a non-uniformly disturbed density field is studied. The nature of an initial perturbation here differs from one considered in the classical Richtmyer–Meshkov instability (RMI). We consider the instability that develops from the initial perturbations of the density field with a flat interface between plates, while RMI is triggered by a shock passing through the corrugated interface. The structure of perturbation fields generated in the plates due to impact and the interface evolution are studied via the analytic linear and nonlinear models for normal modes using the Euler equations for compressible fluids and appropriate boundary conditions. Such analysis reveals three different regimes in which the generated disturbances can develop depending on the direction of the perturbation wave vector. The obtained theoretical findings are in good quantitative agreement with our detailed numerical simulations.

On MHD Oscillations of a Plasma Column of Finite Conductivity with a Jump of Parameters in the Near-Wall Region

In a straight-cylinder model in a strong longitudinal magnetic field, the MHD oscillations of a plasma of a finite conductivity are considered in a situation when there is no resonant surface in the plasma on which the component of the perturbation wave vector vanishes along the field. At high conductivity, there are large-scale oscillations with a radial wavelength on the order of the column radius, which in the entire volume differ little from those described by an ideal MHD. It is shown that with a jump in the density and electron temperature on a certain magnetic surface near the wall, another MHD mode may exist along with such oscillations due to the finite conductivity. In this mode, the perturbations are large-scale and almost ideal in most of the plasma volume, but small-scale in the radial coordinate and not ideal in the wall region.

The effect of magnetic field on perturbation evolution in homogeneously sheared flows

The goal of this paper is to identify and characterize the fundamental mechanisms that contribute toward flow stabilization in sheared plasma flows. Toward that end, we investigate the evolution of velocity and magnetic field perturbations in homogeneously sheared magnetohydrodynamic (MHD) flows subjected to an imposed streamwise magnetic field. The influences of magnetic field strength ($B_{0}$) and perturbation wavevector orientation ($\unicode[STIX]{x1D6FD}$) are characterized using linear analysis and direct numerical simulations. The linear analysis of ideal MHD indicates that the perturbation evolution is governed by four processes: pressure redistribution, kinetic energy production, kinetic–magnetic energy exchange and magnetic energy production due to magnetic stretching. The interplay between these processes can be characterized by the ratio of shear-to-magnetic timescales ($R_{A}\equiv V_{A}\unicode[STIX]{x1D705}/S$) and $\unicode[STIX]{x1D6FD}$, where $V_{A}$, $\unicode[STIX]{x1D705}$ and $S$ are Alfvén wave speed, initial wavenumber and mean flow shear, respectively. For cases with low values of $\boldsymbol{R}_{\boldsymbol{A}}~(\ll \mathbf{1})$, a three-stage perturbation evolution is seen. At the first stage, pressure redistribution and production dominate leading to hydrodynamic-type behaviour. In the second stage, the onset of magnetic stretching process leads to an increase in magnetic energy. At late stages, production subsides and the dynamics is dominated by harmonic exchange between velocity and magnetic fields. For cases of $\boldsymbol{R}_{\boldsymbol{A}}\approx \mathbf{1}$, the magnetic field reacts rapidly enough that hydrodynamic and magnetic production stages occur simultaneously followed by harmonic exchange. In the case of $\boldsymbol{R}_{\boldsymbol{A}}\gg 1$, all three stages occur simultaneously leading to harmonic exchange between kinetic and magnetic energies without any perturbation growth. For all cases considered, the late stage harmonic exchange results in equipartition between perturbation magnetic and kinetic energies. For a given $R_{A}$, the effect of increasing $\unicode[STIX]{x1D6FD}$ is to reduce the intensity of coupling and progressively slow down the three stages of evolution. For spanwise wavevector perturbations, the velocity–magnetic field interaction mechanism vanishes and there is no effect of pressure or magnetic field on individual velocity components.

Rayleigh–Taylor instability of a magnetic tangential discontinuity in the presence of oscillating gravitational acceleration

We study the magnetic Rayleigh–Taylor (MRT) instability of a magnetohydrodynamic interface in an infinitely conducting incompressible plasma in the presence of oscillating gravity acceleration. We show that the evolution of the interface shape is described by the Mathieu equation. Written in the dimensionless form this equation contains two parameters, a and q. The parameter q can be considered as the dimensionless wavenumber. The two parameters are related by a = Kq2, where K, in turn, depends on the ratio of densities at the two sides of the interface, ζ, the parameter s determining the relative magnitude of the gravity acceleration, the magnetic shear angle α, and the angle ϕ determining the direction of the perturbation wave vector. We calculate the dependence of the instability increment on q at fixed K, and the dependence on K of the maximum value of the increment with respect to q. We apply the theoretical results to the stability of a part of the heliopause near its apex point. Using the typical values of plasma and magnetic field parameters near the heliopause we obtain that the instability growth time is comparable with the solar cycle period.

Interface Instability during Ion Intercalation Process

Misfit stress generated by lattice mismatch between the lithiated and delithiated phases upon ion (de)intercalation plays a key role in cycling-induced mechanical degradation and capacity fading of battery electrodes. As a general phenomenon, misfit stress also leads to interface instability. A well-known example is the Stranski-Krastanov mode or Asaro-Tiller-Grinfeld instability [1] of epitaxial thin film growth (Figure 1(a)), where the misfit strain between thin film and substrate promotes the undulation of thin film and the transition to the island growth behavior. In this work, we provide the first theoretical analysis to predict that an analogous phenomenon can also occur during the ion (de)intercalation process, in which the misfit stress triggers the morphological instability of the moving phase boundary and results in non-uniform (de)intercalation, Figure 1(a). Such a phenomenon is generally applicable to battery compounds that undergo phase transformations with volume change upon ion insertion/extraction. It could have important implications for battery performance and degradation, such as increasing the kinetic resistance to (de)interclation and enhancing stress concentration that damages electrodes and accelerates capacity fading. Our analysis offers useful insights on potential practical strategy to suppress the onset of stress-induced interface instability during electrode charge/discharge. In the theoretical treatment, we consider a general intercalation system in two dimensions in which ion intercalation proceeds through a first-order phase transformation with isotropic misfit strain and elasticity. The phase boundary between the Li-rich and Li-poor phases is assumed to maintain (partial) coherency. We consider interface-limited phase boundary migration, i.e. the interface velocity is proportional to the Li chemical potential difference between the Li-rich and Li-poor phases at the interface, which includes contribution from coherency stress. Linear stability analysis is carried out to determine the stability of a small periodic perturbation applied to the originally flat interface, Figure 1(b). The analysis results in the following main findings: 1) The phase boundary is unstable against perturbations with wave vector below a critical value kc, or equivalently, when perturbation wavelength is larger than the critical wave length λc = 2*π/kc. As shown in Figure 1(c), there exists a fastest growing wave vector km, at which the perturbation amplitude has the maximum growth rate. Both kc and km are proportional to A/(E*e0 2), where A is interfacial energy, E is the Young’s modulus and e0 is the misfit strain. With typical values of these properties for intercalation compounds (A ~ 0.1 J/m^2, E ~ 100 GPa, e0~ 0.01), the critical and fastest growing wave lengths of interface perturbation are in the range of 10 – 100 nm. Interface stability is hence expected in nano- and micro-sized particles. 2) Interface is most susceptible to instability development at the beginning stage of the (de)intercalation process. As shown in Figure 1(d), the fastest growing wave vector of interface perturbation depends on the depth of the interface from the particle surface, z0. While km and kcare the largest when interface is near the electrode surface, they decay with interface moving into the interior of the particle. 3) Interface instability is suppressed in the interior of intercalation compounds below a critical particle size. Figure 1(d) illustrates that the fastest growing wave vector drops to 0, i.e. no perturbation with finite wave length can grow, when the system dimension in the main intercalation direction is smaller than a critical thickness L*, which is on the order of 100 nm. Therefore, nanoscaling electrodes not only benefits intercalation kinetics but also improves interface stability. Complementing the linear stability analysis, we also performed phase-field simulation of the phase boundary evolution during delithiation of LiFePO4 cathode. Consistent with the theoretical prediction, interface instability is observed in the simulation (Figure 1(e)), which reveals the details of the interface morphological evolution at later stage. Notably, similar interface morphology is observed in a recent operando hard x-ray microscopy experiment [2]. Based on the findings from this study, we identify two useful approaches to stabilize phase boundary during ion (de)intercalation: i) Reduce one dimension of the electrode particles to the sub-micron regime, i.e. nanoplate morphology is desirable. ii) Apply large current pulses at the beginning of discharge/charge to have interface rapidly traverse the near-surface region where instability is easiest to develop.

THE CONVERGENCE OF PARTICLE-IN-CELL SCHEMES FOR COSMOLOGICAL DARK MATTER SIMULATIONS

ABSTRACT Particle methods are a ubiquitous tool for solving the Vlasov–Poisson equation in comoving coordinates, which is used to model the gravitational evolution of dark matter (DM) in an expanding universe. However, these methods are known to produce poor results on idealized test problems, particularly at late times, after the particle trajectories have crossed. To investigate this, we have performed a series of one- and two-dimensional “Zel’dovich pancake” calculations using the popular particle-in-cell (PIC) method. We find that PIC can indeed converge on these problems provided that the following modifications are made. The first modification is to regularize the singular initial distribution function by introducing a small but finite artificial velocity dispersion. This process is analogous to artificial viscosity in compressible gas dynamics, and, as with artificial viscosity, the amount of regularization can be tailored so that its effect outside of a well-defined region—in this case, the high-density caustics—is small. The second modification is the introduction of a particle remapping procedure that periodically reexpresses the DM distribution function using a new set of particles. We describe a remapping algorithm that is third-order accurate and adaptive in phase space. This procedure prevents the accumulation of numerical errors in integrating the particle trajectories from growing large enough to significantly degrade the solution. Once both of these changes are made, PIC converges at second order on the Zel’dovich pancake problem, even at late times, after many caustics have formed. Furthermore, the resulting scheme does not suffer from the unphysical, small-scale “clumping” phenomenon known to occur on the pancake problem when the perturbation wavevector is not aligned with one of the Cartesian coordinate axes.

Stabilizing action of pressure in homogeneous compressible shear flows: effect of Mach number and perturbation obliqueness

AbstractCompressibility exerts a stabilizing influence on a variety of high-speed shear flows such as turbulent mixing layers, transitioning boundary layers and homogeneously sheared turbulence. An important stabilizing feature that is common amongst all shear flows is the velocity–pressure interaction dynamics. In this study, velocity–pressure interactions of individual perturbation or fluctuation modes are investigated using direct numerical simulations and linear analysis in high-Mach-number homogeneous shear flow. For a given perturbation wave mode, the action of pressure is shown to depend on two important factors: the orientation of the perturbation wavevector with respect to the shear plane and the Mach number. It is shown that the streamwise perturbation wave mode rapidly develops a high level of kinetic energy but is self-limiting owing to the action of pressure. On the other hand, the energy of spanwise perturbation wave modes grows unaffected by pressure or Mach number. Oblique modes combine spanwise and streamwise characteristics and are shown to be chiefly responsible for stabilizing effects seen in shear flows. Three regimes of obliqueness of different linear stability characteristics are identified. The critical role of perturbation obliqueness on stabilization is established.

Cross modulation instability in normal-dispersion fibre lasers and amplifiers

This paper examines instability resulting from cross phase modulation interaction between signal and pump waves in normal-dispersion fibre lasers and amplifiers. From analysis of the dispersion relation for the wave vector of small harmonic perturbations, we derive dynamic spectral characteristics of the modulation gain coefficient. Its dependence on the group velocity difference between signal and pump waves is investigated. Analytical results are compared to numerical simulation data. We discuss how the general relationships obtained in this study can be used in practical applications.

Quantum instability of two non-parallel flows: Parallel wave propagation

Abstract The streaming instability for two non-paralleled plasma shells propagating through a background quantum plasma has been studied. The dispersion relation is calculated, assumed that the perturbation wave vector is in the direction of one of the plasma shell, by linearizing and solving fluid-Maxwell equations. The resulting dispersion equation is analyzed numerically in a wide range of system parameters. Investigation of cut-off wave number and growth rate are shown that, the instability for a wide range of wave number when the velocity of shells reaches to a specific value, β = β c .

Magnetic-diffusion-driven shear instability of solar flux tubes

Macroscopic gas motions are widespread throughout the solar atmosphere and shearing motions couple to the non--ideal effects, destabilising low frequency fluctuations in the medium. The origin of this non-ideal magnetohydrodynamic instability lies in the collisional coupling of the neutral particles to the magnetized plasma in the presence of a sheared background flow. Unsurprisingly, the maximum growth rate and most unstable wavenumber depend on the flow gradient and ambient diffusivities. The orientation of the magnetic field, velocity shears and perturbation wave vector play a crucial role in assisting the instability. When the magnetic field and wave vector are both vertical, ambipolar and Ohm diffusion can be combined as Pedersen diffusion and cause only damping; in this case only Hall drift in tandem with shear flow drives the instability. However, for non-vertical fields and oblique wave vectors, both ambipolar diffusion and Hall drift are destabilizing. We investigate the stability of magnetic elements in the network and internetwork regions. The shear scale is not yet observationally determined, but assuming a typical shear flow gradient $\sim 0.1 \,\mbox{s}^{-1}$ we show that the magnetic diffusion shear instability grows on a time scale of one minute. Thus, it is plausible that network--internetwork magnetic elements are subject to this fast growing, diffusive shear instability, which could play an important role in driving low frequency turbulence in the plasma in the solar photosphere and chromosphere.

Alfvén waves in coronal holes: formation of discontinuities in inhomogeneous magnetic fields

Context. Solar wind fluctuations are characterized by discontinuities. The nature and properties of these structures have been largely studied in the literature, and different mechanisms have been proposed to explain their formation.Aims. We investigate the evolution of Alfvenic perturbations propagating in the inhomogeneous magnetic field of a coronal open-field region, in order to study both the way that small-scale structures are generated and the possible formation of discontinuities.Methods. We constructed a model for the equilibrium magnetic field in a coronal hole. The model represents a potential field with a complex structure: regions of opposite polarity or of only the dominant polarity are present at low or high altitudes, respectively. The evolution of small-amplitude Alfven waves in the inhomogeneous structure is studied by employing a WKB approach that describes how the perturbation wavevector and the wave phase vary along magnetic lines.Results. We find that small-scale structures form in the perturbation at relatively low altitudes (~3 × 104 km) above the coronal base. An initially monochromatic perturbation develops a steep power-law spectrum with slope α ≃ 2.3, which is strongly anisotropic with a predominance of quasi-perpendicular wavevectors. Small-scale structures are localized around separatrices of the magnetic structures. In many cases they contain quasi-perpendicular rotational discontinuities that can propagate to the upper corona, eventually reaching the solar wind. Conclusions. The considered mechanism could be responsible for forming a fraction of the population of discontinuities detected in the solar wind.

Modulational instability for an induced field in the far-wake region of a space vehicle

The behavior of the induced field and the generation of density cavitons in the far-wake region () of a space vehicle can be described by a set of nonlinear coupling equations. Modulational instability of the induced field is investigated on the basis of the nonlinear equations. The results show that the induced field is modulationally unstable and will collapse into spatial localized structures; meanwhile, density cavitons will be generated. The characteristic scale and the maximum growth rate of the induced field depend not only on the angle between the amplitude of pump waves E0 and the perturbation wave vector k, but also on the energy density of pump waves |E0|2.