Wave Number Of Perturbation Research Articles

Transient response of relativistic electron bunches to wave-number selected perturbations near the micro-bunching instability threshold

Many spatio-temporal systems can undergo instabilities, leading to the spontaneous formation of spatial structures (patterns). However, a range of cases exist for which the pattern itself is not directly visible because of technical or fundamental reasons. This is the case for the spontaneous formation of millimeter-scale patterns appearing inside relativistic electron bunches of accelerators. We demonstrate in this case how the study of responses to sine external perturbations can be used as a ‘probe’ to deduce the characteristic wavenumber of the pattern formation process. Experiments are performed in the UVSOR-II electron storage ring when the electron bunch is subjected to so-called microbunching instability, and the sine perturbations are provided by an external laser. The response is constituted of pulses of coherent synchrotron radiation, whose amplitude depends on the perturbation wavenumber. Experimental results on the dynamics are compared to numerical calculations obtained using a Vlasov–Fokker–Planck model.

The radial–azimuthal instability of accretion disk with anomalous viscosity

The stability of the accretion disk is solved by numerical simulations when the radial and azimuthal perturbations are considered, where we adopt the anomalous viscosity model, which is close to real accretion disks. The results are discussed in the inner, intermediate, and outer regions of the accretion disk, respectively. With the increase of viscosity, α, the thermal mode and the viscous mode, as well as the acoustic modes, become more unstable in the disk dominated by radiation pressure (inner region). The instability is also influenced by the azimuthal perturbation wavenumber, n. With the increase of n, the thermal mode becomes more unstable, while the in-mode and out-mode become more stable no matter if the disk is dominated by radiation pressure or by gas pressure (intermediate and outer regions). There are many differences between our results and others’ results, especially in the inner region of the disk, when the anomalous viscosity is considered.

Dendrite Growth in Lithium Metal Batteries – Stability Analysis of Electrodeposition Across Tethered Anion Electrolytes

Lithium metal batteries have reliability issues due to formation of dendrites during the recharge cycle. These dendrites grow due to electrodeposition of lithium on their tips and eventually short the battery internally. An analysis by Chazalviel [1]showed that the cause lay in the migration of anions away from the cathode leading to the formation of large electric fields in its vicinity. The electric field lines converge on dendrite tips leading to preferential deposition on the tips and dendrite growth.We propose a structured electrolyte with a fraction of the anions tethered in place to a solid matrix to prevent their migration away from the cathode (Fig 1). This is expected to modify the transport within the battery and increase the stability of electrodeposition. The fraction of fixed anions is considered as a parameter in our analysis. We model the electrolyte with diffusion-migration equations for the cation and the mobile anion for varying current densities. Solving the governing equations with appropriate boundary conditions shows that two regimes of transport are present depending on the current passing through the cell. At low current densities, the transport is governed almost entirely by diffusion of the mobile ions. At higher current densities, the mobile anions pile up against the anode and are depleted at the cathode leading to the formation of a zone possessing only the fixed anions and the concentration of cations required to balance their charge. This electromigration dominated region grows with increasing current density and facilitates the passage of currents beyond the diffusion limit. It ensures that the conductivity remains finite throughout the electrolyte and reduces the electric field at the cathode (Fig 2).We further perform a stability analysis of the electrodeposition incorporating the effect of surface tension as a stabilizing agent. Surface tension alters the chemical equilibrium at the cathode to slow the reaction at the crests and accelerate it at the troughs. The stability analysis is performed in both the high and low current density regimes. For the high current density regime, we consider a two-region model wherein part of the inter-electrode spacing has diffusion-dominated transport while the remainder has migration-dominated transport. In the low current density regime, diffusion is the only mode of transport in the electrolyte. Two parameters are extracted from the analysis, viz. the critical wavenumber of perturbations at which transport is rendered stable by surface tension and the growth rate of the most unstable perturbation mode (Fig 3) as metrics of the stability of electrodeposition.The stability analysis shows that both the critical wavenumber and the growth rate of the most unstable mode can be reduced by tethering anions, thereby increasing the range of stable wavenumbers. Even a moderate amount of tethered anions as small as 10% is effective in stabilizing the electrodeposition at high currents. At low currents, however, the effect is less dramatic. For higher fractions of tethered anions, approaching the limit of single ion conductors, the findings show that the factors affecting dendrite growth are much more mitigated than conventional salt-based electrolytes. However, the proposed electrolyte does not completely stop dendrite growth.

Crossover of Varicose and Whipping Instabilities in Electrified Microjets

In electrified liquid jets, varicose instability leads to jet breakup into droplets while whipping instability is responsible for jet stretching. We show that the coupling and relative importance of these two instabilities dictates the outcome for jet breakup. The codevelopment of transverse and radial perturbations lead to remarkable breakup modes linked to initial perturbation magnitude, perturbation wave numbers, and jet charge levels.

Stability of ion acoustic solitary waves in a magnetized plasma consisting of warm adiabatic ions and non-thermal electrons having vortex-like velocity distribution

AbstractSchamel's modified Korteweg-de Vries–Zakharov–Kuznetsov (S-ZK) equation, governing the behavior of long wavelength, weak nonlinear ion acoustic waves propagating obliquely to an external uniform static magnetic field in a plasma consisting of warm adiabatic ions and non-thermal electrons (due to the presence of fast energetic electrons) having vortex-like velocity distribution function (due to the presence of trapped electrons), immersed in a uniform (space-independent) and static (time-independent) magnetic field, admits solitary wave solutions having a sech4 profile. The higher order stability of this solitary wave solution of the S-ZK equation has been analyzed with the help of multiple-scale perturbation expansion method of Allen and Rowlands (Allen, M. A. and Rowlands, G. 1993 J. Plasma Phys. 50, 413; 1995 J. Plasma Phys. 53, 63). The growth rate of instability is obtained correct to the order k2, where k is the wave number of a long wavelength plane wave perturbation. It is found that the lowest order (at the order k) instability condition is strongly sensitive to the angle of propagation (δ) of the solitary wave with the external uniform static magnetic field, whereas at the next order (at the order k2) the solitary wave solutions of the S-ZK equation are unstable irrespective of δ. It is also found that the growth rate of instability up to the order k2 for the electrons having Boltzmann distribution is higher than that of the non-thermal electrons having vortex-like distribution for any fixed δ.

Nonlinear Interaction of 3D Kinetic Alfvén Waves and Ion Acoustic Waves in Solar Wind Plasmas

We have investigated the nonlinear interaction between a 3D kinetic Alfven wave (KAW) and an ion acoustic wave (IAW) in solar wind plasmas. A set of dimensionless equations was developed that describes the pump KAW perturbed by a low-frequency ion acoustic wave. The dependence of the growth rate of the modulational instability on the perturbation wave number was studied. We simulated numerically the dynamical equation of KAW with a pseudo-spectral method, taking ponderomotive nonlinearity into account. The 3D KAW itself propagates in the form of a vortex beam in a magnetised plasma, which manifests the presence of orbital angular momentum of the wave eigenmodes. We discuss the evolution of these vortex structures. Our results reveal that the Kolmogorov scaling is followed by a steeper scaling of power spectra, which is consistent with the solar wind observations by the Cluster spacecraft. We discuss the relevance of our investigation for solar wind plasmas.

Rayleigh–Taylor instability at ionization fronts: perturbation analysis

The linear growth rate of the Rayleigh-Taylor instability (RTI) at ionization fronts is investigated via perturbation analysis in the limit of incompressible fluids. In agreement with previous numerical studies is found that absorption of ionizing radiation inside the HII region due to hydrogen recombinations suppresses the growth of instabilities. In the limit of a large density contrast at the ionization front the RTI growth rate has the simple analytical solution n=-nur+(nur^2+gk)^(1/2), where nur is the hydrogen recombination rate inside the HII region, k is the perturbation's wavenumber and g is the effective acceleration in the frame of reference of the front. Therefore, the growth of surface perturbations with wavelengths lambda >> lambda_{cr} = 2\pi g/nur^2 is suppressed by a factor (lambda_{cr}/4lambda)^(1/2) with respect to the non-radiative incompressible RTI. Implications on stellar and black hole feedback are briefly discussed.

Numerical simulations of circular vortex sheets by using two regularization models

We perform numerical simulations for circular vortex sheets by using two regularization models and present the late-time evolution of unstable interfaces. We also give a linear stability analysis for the circular vortex sheet and find that the sheet is always susceptible to the Kelvin-Helmholtz instability. The numerical results show that the motion of the circular sheet is similar to that of the periodic vortex sheet at an early time, but the secondary instability at a late time depends on the wavenumber of the initial perturbation. The interface of the initial two-modes evolves two primary and two secondary roll-ups while the interface of the initial four-modes has four symmetric roll-ups until late times and does not form a secondary instability. The non-symmetric interface of the single-mode develops to a structure more complex than that of the symmetric interfaces, having continuously evolved roll-ups and a strong interaction of spirals. We also give quantitative comparisons of the Krasny model and the Beale-Majda model. The solutions of the two models are similar on a large scale, but are different on a small scale. The solution of the Beale-Majda model exhibits somewhat irregular features for a small regularization parameter while the Krasny model gives well-regularized solutions with fine resolution.

Biaxial extensional motion of an inertially driven radially expanding liquid sheet

We consider the inertially driven, time-dependent biaxial extensional motion of inviscid and viscous thinning liquid sheets. We present an analytic solution describing the base flow and examine its linear stability to varicose (symmetric) perturbations within the framework of a long-wave model where transient growth and long-time asymptotic stability are considered. The stability of the system is characterized in terms of the perturbation wavenumber, Weber number, and Reynolds number. We find that the isotropic nature of the base flow yields stability results that are identical for axisymmetric and general two-dimensional perturbations. Transient growth of short-wave perturbations at early to moderate times can have significant and lasting influence on the long-time sheet thickness. For finite Reynolds numbers, a radially expanding sheet is weakly unstable with bounded growth of all perturbations, whereas in the inviscid and Stokes flow limits sheets are unstable to perturbations in the short-wave limit.

Linear relaxation to planar travelling waves in inertial confinement fusion

In this paper we study the linear stability of planar travelling waves for a scalar reaction–diffusion equation with nonlinear anisotropic diffusion. The mathematical model is derived from the full thermo-hydrodynamical model describing the process of inertial confinement fusion. We show that solutions of the Cauchy problem with physically relevant initial data become planar exponentially fast with the rate s(ε′, k) > 0, where ε′ = (Tmin/Tmax) ≪ 1 is a small temperature ratio and k ≫ 1 is the transversal wrinkling wavenumber of perturbations. We rigorously recover in some particular limit (ε′, k) → (0, +∞) a dispersion relation s(ε′, k) ∼ γ0kα previously computed heuristically and numerically in some physical models of inertial confinement fusion.

On the non-Gaussian correlation of the primordial curvature perturbation with vector fields

We compute the three-point cross-correlation function of the primordial curvature perturbation generated during inflation with two powers of a vector field in a model where conformal invariance is broken by a direct coupling of the vector field with the inflaton. If the vector field is identified with the electromagnetic field, this correlation would be a non-Gaussian signature of primordial magnetic fields generated during inflation. We find that the signal is maximized for the flattened configuration where the wave number of the curvature perturbation is twice that of the vector field and in this limit, the magnetic non-linear parameter becomes as large as |bNL| ∼ \U0001d4aa(103). In the squeezed limit where the wave number of the curvature perturbation vanishes, our results agree with the magnetic consistency relation derived in arXiv:1207.4187.

Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns

A dark solitary wave, in one space dimension and time, is a wave that is bi-asymptotic to a periodic state, with a phase shift, and with localized modulation in between. The most well-known case of dark solitary waves is the exact solution of the defocusing nonlinear Schrödinger equation. In this paper, our interest is in developing a mechanism for the emergence of dark solitary waves in general, and not necessarily integrable, Hamiltonian PDEs. The focus is on the periodic state at infinity as the generator. It is shown that a natural mechanism for the emergence is a transition between one periodic state that is (spatially) elliptic and another one that is (spatially) hyperbolic. It is shown that the emergence is governed by a Korteweg–de Vries (KdV) equation for the perturbation wavenumber on a periodic background. A novelty in the result is that the three coefficients in the KdV equation are determined by the Krein signature of the elliptic periodic orbit, the curvature of the wave action flux and the slope of the wave action, with the last two evaluated at the critical periodic state.

Dynamical stability of infinite homogeneous self-gravitating systems and plasmas: application of the Nyquist method

We complete classical investigations concerning the dynamical stability of an infinite homogeneous gaseous medium described by the Euler-Poisson system or an infinite homogeneous stellar system described by the Vlasov-Poisson system (Jeans problem). To determine the stability of an infinite homogeneous stellar system with respect to a perturbation of wavenumber k, we apply the Nyquist method. We first consider the case of single-humped distributions and show that, for infinite homogeneous systems, the onset of instability is the same in a stellar system and in the corresponding barotropic gas, contrary to the case of inhomogeneous systems. We show that this result is true for any symmetric single-humped velocity distribution, not only for the Maxwellian. If we specialize on isothermal and polytropic distributions, analytical expressions for the growth rate, damping rate and pulsation period of the perturbation can be given. Then, we consider the Vlasov stability of symmetric and asymmetric double-humped distributions (two-stream stellar systems) and determine the stability diagrams depending on the degree of asymmetry. We compare these results with the Euler stability of two self-gravitating gaseous streams. Finally, we determine the corresponding stability diagrams in the case of plasmas and compare the results with self-gravitating systems.

Linear analysis of the Richtmyer-Meshkov instability in shock-flame interactions

Shock-flame interactions enhance supersonic mixing and detonation formation. Therefore, their analysis is important to explosion safety, internal combustion engine performance, and supersonic combustor design. The fundamental process at the basis of the interaction is the Richtmyer-Meshkov instability supported by the density difference between burnt and fresh mixtures. In the present study we analyze the effect of reactivity on the Richtmyer-Meshkov instability with particular emphasis on combustion lengths that typify the scaling between perturbation growth and induction. The results of the present linear analysis study show that reactivity changes the perturbation growth rate by developing a pressure gradient at the flame surface. The baroclinic torque based on the density gradient across the flame acts to slow down the instability growth of high wave-number perturbations. A gasdynamic flame representation leads to the definition of a Peclet number representing the scaling between perturbation and thermal diffusion lengths within the flame. Peclet number effects on perturbation growth are observed to be marginal. The gasdynamic model also considers a finite flame Mach number that supports a separation between flame and contact discontinuity. Such a separation destabilizes the interface growth by augmenting the tangential shear.

Nonlinear interaction of kinetic Alfvén waves with slow Alfvén waves and application to solar wind

This paper presents the nonlinear coupling between kinetic Alfvén waves (KAWs) and slow Alfvén waves (SWs) in high-β plasmas (β ≫me /mi), which is applicable to solar wind plasma. The pump KAW is perturbed by a low-frequency SW. When the ponderomotive nonlinearities are incorporated into the KAW and SW dynamics, the model equations of KAW and SW turn out to be the modified Zakharov system of equations. The growth rate of the instability has been calculated and its dependence on the perturbation wave number has also been presented. The relevance of these investigations for solar wind plasma has been discussed.

Convection in stable and unstable fronts

Density gradients across a reaction front can lead to convective fluid motion. Stable fronts require a heavier fluid on top of a lighter one to generate convective fluid motion. On the other hand, unstable fronts can be stabilized with an opposing density gradient, where the lighter fluid is on top. In this case, we can have a stable flat front without convection or a steady convective front of a given wavelength near the onset of convection. The fronts are described with the Kuramoto-Sivashinsky equation coupled to hydrodynamics governed by Darcy's law. We obtain a dispersion relation between growth rates and perturbation wave numbers in the presence of a density discontinuity accross the front. We also analyze the effects of this density change in the transition to chaos.

Dark Solitons, Dispersive Shock Waves, and Transverse Instabilities

The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrödinger/Gross–Pitaevski (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev–Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields (1) the maximal growth rate and associated wavenumber of unstable perturbations and (2) the separatrix between convective and absolute instabilities. The instability properties of the dark soliton are directly related to those of oblique dispersive shock wave (DSW) solutions. Stationary and nonstationary oblique DSWs are constructed analytically and investigated numerically by direct simulations of the NLS/GP equation. It is found that stationary and nonstationary oblique DSWs have the same jump conditions in the shallow and hypersonic regimes. These results have application to controlling nonlinear waves in dispersive media.

Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability

The Benjamin-Feir instability describes the instability of a uniform oscillatory wave train in an irrotational flow subject to small perturbation of wave number, amplitude and frequency. Their instability analysis is based on the perturbation around the second order Stokes wave which satisfies the dynamic and kinematic free-surface boundary conditions up to the second order. In the same irrotational flow and perturbation framework of the Benjamin-Feir analysis, the perturbation in the present paper is around a nonlinear oscillatory wave train which solves exactly the dynamic free-surface boundary condition and satisfies the kinematic free-surface boundary condition up to the third order. It is shown that the nonlinear oscillatory wave train is stable with respect to the perturbation when the irrotational flow involves small Rayleigh energy dissipation.

Dispersion and the dissipative characteristics of surface waves in the generalized S-transform domain

Wavenumber, group velocity, phase velocity, and frequency-dependent attenuation characterize the propagation of surface waves in dispersive, attenuating media. We use a mathematical model based on the generalized [Formula: see text] transform to simultaneously estimate these characteristic parameters for later use in joint inversion for near-surface shear wave velocity. We use a scaling factor in the generalized S transform to enable the application of the method in a highly dispersive medium. We introduce a cost function in the [Formula: see text]-domain to estimate an optimum value for the scaling factor. We also use the cost function to generalize the application of the method for noisy data, especially data with a low signal-to-noise ratio at low frequencies. In that case, the estimated wavenumber is perturbed. As a solution, we estimate wavenumber perturbation by minimizing the cost function, using Simulated Annealing. We use synthetic and real data to show the efficiency of the method for the estimation of the propagation parameters of highly dispersive and noisy media.

High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

In this paper, we demonstrate the detailed numerical studies of three classical two dimensional detonation waves by solving the two dimensional reactive Euler equations with species with the fifth order WENO-Z finite difference scheme (Borges et al. in J. Comput. Phys. 227:3101–3211, 2008) with various grid resolutions. To reduce the computational cost and to avoid wave reflection from the artificial computational boundary of a truncated physical domain, we derive an efficient and easily implemented one dimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the two dimensional unsteady reactive Euler equation when one of the directions of domain is periodical and inflow/outflow in the other direction. The numerical comparison among characteristic, free stream, extrapolation and PML boundary conditions are conducted for the detonation wave simulations. The accuracy and efficiency of four mentioned boundary conditions are verified against the reference solutions which are obtained from using a large computational domain. Numerical schemes for solving the system of hyperbolic conversation laws with a single-mode sinusoidal perturbed ZND analytical solution as initial conditions are presented. Regular rectangular combustion cell, pockets of unburned gas and bubbles and spikes are generated and resolved in the simulations. It is shown that large amplitude of perturbation wave generates more fine scale structures within the detonation waves and the number of cell structures depends on the wave number of sinusoidal perturbation.