Nonlinear Growth Rate Research Articles

Effect of Beam Deceleration in the Two-Stream Instability

Two-stream instability induced by the interaction between a proton beam and a plasma is investigated analytically in consideration of the beam deceleration by the nonlinear effect. Expression of beam deceleration is derived from a modification of the orbit of the beam particle due to the wave potential. Nonlinear dispersion relation is obtained from a set of equations which includes the term of the beam deceleration in addition to the perturbation of the beam velocity. Saturation level of the wave potential is determined from the nonlinear growth rate. Result indicates that the beam deceleration due to the nonlinear interaction provides the saturation level which is almost equal to one obtained by the analyses based on the trapping of beam particles.

Nonlinear growth rate of wind water waves and their excitation near the stability threshold

The amplitude dependences of the growth rate of water waves generated by turbulent wind are theoretically investigated within the framework of a quasilinear approximation.

Parametric excitation of a lower-hybrid drift wave in the ionosphere

Parametric excitation of a lower-hybrid drift (LHD) wave has been studied in the ionosphere due to the propagation of a high-frequency electromagnetic (EM) pump wave (with wave number k0≊0) in the presence of the electron collisions. It is considered to be a four wave interaction process of LHD, electromagnetic pump wave, and two electron plasma waves as sidebands. Nonlinear growth rate of a LHD wave is found to be higher for 1<krLe<1.5 as compared to the case krLe<1, where k and rLe, respectively, are the wave number of the LHD wave and electron Larmor radius. Wavelengths of the excited lower-hybrid drift waves lie in the range of the various irregularities observed by Jicamarca [Farley et al., J. Geophys. Res. 75, 7199 (1970)], and Altair and Tradex radars [R. T. Tsunoda, Geophys. Res. Lett. 7, 848 (1980)]. It is seen that the nonlinear growth rate drops significantly in the presence of the electron collisions.

A model of physiologically structured population dynamics with a nonlinear individual growth rate

In this article we consider a size structured population model with a nonlinear growth rate depending on the individual's size and on the total population. Our purpose is to take into account the competition for a resource (as it can be light or nutrients in a forest) in the growth of the individuals and study the influence of this nonlinear growth in the population dynamics. We study the existence and uniqueness of solutions for the model equations, and also prove the existence of a (compact) global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, we obtain sufficient conditions for the convergence to a stationary size distribution when the total population tends to a constant value, and consider some simple examples that allow us to know something about their global dynamics.

Nonlinear growth rate of Langmuir waves in the case of a wide spectrum

A nonlinear equation for the evolution of a Langmuir harmonic in a wide spectrum is derived in a consistent manner. This equation substitutes the usual quasilinear growth rate of the weak plasma turbulence theory. The derivation is based on the determination of the resonant particle distribution through the solution of the exact nonlinear equations of motion in a given field, composed of all effectively acting harmonics. The method requires a slow variation of the harmonic amplitudes over the nonlinear characteristic time of resonant interaction, and small enough amplitudes for the averaging method to be applicable. Two nonlinear terms appear in this equation: one is proportional to the particular harmonic considered, the other is a contribution from the whole effective spectrum around this harmonic. From this last term, it follows that a wave with negligible initial amplitude can grow spontaneously through the nonlinear wave–particle interaction.

Jeffreys' Drag Instability Applied to Waves in the Lower Atmosphere: Linear and Nonlinear Growth Rates

In a geostrophically balanced region of the atmosphere the horizontal pressure gradients do no work since they are aligned at right angles to the winds, but near the earth's surface where friction destroys geostrophic balance these large-scale pressure gradients can feed energy into smaller-scale flows. One mechanism through which this occurs is Jeffreys' drag instability, a mechanism normally associated with the formation of waves on a sloping sheet of water but equally capable of generating a variety of waves in the lower atmosphere. This study examines Jeffeys' drag instability, as a source of large amplitude boundary layer waves that fall off exponentially above a shallow surface layer, and large amplitude gravity wave modes ducted in the overlying troposphere. jeffreys' formulation is modified to the atmospheric context and modeled to compute conditions that support the linear and nonlinear growth of various waves. Several regimes of instability are identified, but the greatest growth rates are associated with waves propagating upstream in boundary layer jets, a wind form that is characteristic of the Ekman layer. Analysis of some observed waves of remarkable amplitude indicates that they occurred under conditions that strongly favored Jeffreys' drag mechanism. It is also suggested that the drag mechanism allows waves in the boundary layer to generate turbulence and drive mixing at the expense of the available potential energy in the larger scales.

A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage

Finite-difference calculations with random and single-mode perturbations are used to study the three-dimensional instability of vortex rings. The basis of current understanding of the subject consists of a heuristic inviscid model (Widnall, Bliss & Tsai 1974) and a rigorous theory which predicts growth rates for thin-core uniform vorticity rings (Widnall & Tsai 1977). At sufficiently high Reynolds numbers the results correspond qualitatively to those predicted by the heuristic model: multiple bands of wavenumbers are amplified, each band having a distinct radial structure. However, a viscous correction factor to the peak inviscid growth rate is found. It is well described by the first term, 1 – α1(β)/Res, for a large range of Res. Here Res is the Reynolds number defined by Saffman (1978), which involves the curvature-induced strain rate. It is found to be the appropriate choice since then α1(β) varies weakly with core thickness β. The three most nonlinearly amplified modes are a mean azimuthal velocity in the form of opposing streams, an n = 1 mode (n is the azimuthal wavenumber) which arises from the interaction of two second-mode bending waves and the harmonic of the primary second mode. When a single wave is excited, higher harmonics begin to grow successively later with nonlinear growth rates proportional to n. The modified mean flow has a doubly peaked azimuthal vorticity. Since the curvature-induced strain is not exactly stagnation-point flow there is a preference for elongation towards the rear of the ring: the outer structure of the instability wave forms a long wake consisting of n hairpin vortices whose waviness is phase shifted π/n relative to the waviness in the core. Whereas the most amplified linear mode has three radial layers of structure, higher radial modes having more layers of radial structure (hairpins piled upon hairpins) are excited when the initial perturbation is large, reminiscent of visualization experiments on the formation of a turbulent ring at the generator.

Nonlinear interface stability analysis of alloy solidification including effects of surface energy

The morphological stability of the solid-liquid interface in the solidification of a dilute binary alloy is analyzed using linear and nonlinear perturbation analyses. The energy-balance condition at the interface is modified to incorporate the effects of surface excess energy. The Landau coefficient, which represents the nonlinear growth rate, is calculated using the direct method of undetermined coefficients. The results are applied to practical metallic and nonmetallic systems. The proposed surface energy correction is found to be significant at large growth velocities and low frequencies of interface perturbation: Linear stability is obtained over a larger range of frequencies and velocities; while the correction restricts the frequency range of nonlinear stability, the Landau coefficients obtained are larger and indicate greater stability over this range.

Parametric decay instability of an upper-hybrid wave in a two-temperature plasma

Parametric decay of an upper-hybrid pump into another upper-hybrid wave and a low-frequency lower-hybrid mode is considered in a two-electron temperature plasma. Expressions for the nonlinear dispersion relation and growth rate are obtained. It is found that the growth rate is quite sensitive to the hot-electron temperature and the density ratio of the hot and the cold components only when the side-band frequency is close to the second or third harmonic of the cyclotron frequency. The relevance of our investigation to Q machines and the ELMO bumpy torus is pointed out.

High‐resolution radar observations of daytime kilometer‐scale wave structure in the equatorial electrojet

Using a new antenna configuration and new data processing hardware, we probed the electrojet with unprecedented spatial resolution in January 1991 at Jicamarca. The altitude resolution was 125 m, and the east‐west antenna beamwidth was about 0.6°, or about 1 km at electrojet altitudes. The high spatial resolution allowed us to observe, at certain altitudes and times, pronounced wavelike structures with vertical wavelengths of the order of 1 km and periods of tens of seconds. These observations closely resemble the predictions of recent nonlocal, nonlinear numerical simulations of large‐scale turbulence in the equatorial electrojet (Ronchi et al., 1991). The comparison demonstrated the validity of the theory and confirmed the idea that the Doppler power spectra of type 2 echoes at 50 MHz (due to 3‐m plasma waves generated by the gradient‐drift instability and a nonlinear cascade of energy from long‐wavelength modes to short) are controlled primarily by the large‐scale dynamics and not by the nonlinear local growth and decay rates of the 3‐m waves themselves.

A self‐consistent fully turbulent theory of auroral E region irregularities

We present a nonlinear theory of high‐latitude E region irregularities that is based solely on the effects of mode coupling on the turbulent state of the plasma. We have been using a nonlinear equation based on a two‐fluid model governing the evolution of ion acoustic turbulence in the absence of k∥ = k · B0/B0 effects. We have first shown that the linear eigenmodes cannot be used as a zeroth‐order approximation to describe the often‐seen strongly turbulent high‐latitude E region plasma waves. However, with the assumption of weakly growing or decaying nonlinear modes with broad frequency spectra (quasi‐steady state turbulence), we have been able to obtain a set of constraints on the mean frequency and spectral width of both linearly stable and linearly unstable modes. With the use of an “eddy damped quasi‐normal Markovian” closure scheme for the spectral density and a self‐consistent expression for the frequency broadening and the nonlinear growth rate, we have been able, as a result, to relate the Doppler width of radar spectra to the broadband density fluctuation level. Our estimates have led us to believe that strongly turbulent conditions are associated with broadband fluctuation levels of the order of 5 to 10%, no more, in the situations of interest. Our closure scheme also produces a specific set of nonlinear coupled equations that could be solved numerically in order to derive spectral laws pertinent to high‐latitude E region turbulence.

Plasma-maser effect and evolution of resonant waves in turbulent plasmas

The evolution of resonant waves due to the nonlinear interaction associated with the plasma-maser effect is considered, and the nonlinear growth rate obtained. It is shown that the latter contains a "polarization" (only for the resonant waves) as well as a "direct third-order" contribution. In particular, the effect of the plasma-maser mechanism on the beam instability caused by Landau growth is investigated. It is shown that this effect can significantly modify the behavior of the instability. The conditions for the dominance of the plasma-maser effect nonlinearity over that of the usual quasilinear mechanism are given.

Growth rate of second-harmonic generation in glass

We study the growth rate of the light-induced second-order nonlinearity in germanium-doped optical fiber preforms. We seed the glass with both infrared light and green light (Nd:YAG and doubled Nd:YAG) to create the second-order nonlinearity and measure its formation rate while varying the intensity of either the fundamental or the second-harmonic seeding beams. We find that the formation rate varies as a power law of the intensities, but with an exponent larger than predicted by recent models.

Weakly Nonlinear Instability of a Liquid Jet in a Viscous Gas

The weakly nonlinear instability of a viscous liquid jet emanated into a viscous gas contained in a coaxial vertical circular pipe is investigated as an initial-value problem. The linear stability theory predicted that the jet may become unstable either due to capillary pinching or due to interfacial stress fluctuation. The results of nonlinear stability analysis shows no tendency of supercritical stability for both of the linearly unstable modes. In fact, the nonlinear growth rate of the disturbance is faster than the exponential growth rate of the linear normal mode disturbance for the same flow parameters. Moreover, the most amplified linear normal mode disturbance evolves nonlinearly into a nonsinusoidal wave of shorter wavelength. No nonlinear instability is found for the linearly stable disturbances. Thus, while the linear theory is adequate for the prediction of the onset of jet breakup, nonlinear theory is required to describe the outcome of the jet breakup.

Parametric Excitation of Hybrid Modes in Multispecies Plasmas

AbstractNonlinear decay of an electromagnetic wave into lower‐hybrid and upper‐hybrid wave in a plasma containing two types of ions and two temperature electrons has been analytically investigated. Hydrodynamical model of the plasma is used. Nonlinear dispersion relation and growth rates are calculated for parametric decay, modulational and filamentation instabilities. As an application of the investigation growth rates are calculated for typical parameters of both laboratory and space plasmas. Effect of addition of second species of electrons and ions is discussed.

Parametric decay of an electron-Bernstein wave excited by the optical mixing of two microwave beams

Nonlinear theory of an electron-Bernstein wave at lower-hybrid frequency is developed. Fluid theory is applied for the excitation of a lower-hybrid wave whereas a kinetic model is used for the electron-Bernstein mode as fluid theory breaks down for short-wavelength perturbations, and the propagation of these waves depends upon the cyclotron motion of the electrons about the field lines. The nonlinearity arises because of electrons motion, introduced through ponderomotive force. The electron-Bernstein wave is assumed to be excited by optical mixing of two microwave beams. The nonlinear dispersion relation and growth rates are calculated for three-wave and four-wave parametric interaction processes. Since the electron-Bernstein wave is nonlinearly excited by two intense beams, there is no power threshold and the excited power increases proportionally to the product of the incident powers. The investigation appears to be of great potential because of recent significant developments in gyrotron technology.

Island Growth during the Crystallization of Amorphous Ge48Te52 Films

Microstructural TEM studies have been combined with temperature-dependent measurements of laser reflection and transmission to study crystallization in amorphous films of near-stoichiometric germanium telluride. The transformation proceeds by rapid growth of crystalline islands through the film thickness followed by two-dimensional growth in the film plane. Isothermal studies find an Avrami exponent of 4.5 suggesting a non-linear crystal growth rate.75nm of Ge48Te52 was deposited by co-evaporation from separate Knudsen cells onto Si, carbon-coated mica, and glass substrates. Compositions were determined by x-ray fluorescence from films deposited on silicon.

Modulational and filamentation instabilities in ELMO bumpy torus

An analytical investigation of modulational and filamentation instabilities of an upper-hybrid wave due to electron-acoustic density perturbations in a two-electron temperature plasma has been made. A hydrodynamical model of the plasma is used. Nonlinear dispersion relation and growth rates are calculated. As an application of the investigation, the variation of the growth rate with the concentration ratio of hot to cold electrons, temperature ratio (Tc/Th) and magnetic field for typical parameters of laser-plasma interaction in ELMO bumpy torus is studied and is shown graphically.

Nonlinear evolution of magnetic island in a sheared magnetic field with uniform plasma background

The paper is devoted to the nonlinear evolution of the single tearing type perturbation in a plasma configuration with magnetic shear. The primary effect in the study of the nonlinear dynamics of the magnetic perturbation is the modification of the orbits of resonant particles caused by the chain of magnetic islands, moving along the magnetic surface due to drift effects. The structure of a parallel electric field and its influence on ion dynamics, including the so-called 'integral' effects dealing with the finiteness of the Larmor radius, are incorporated in the nonlinear analysis. Nonlinear growth rates and saturation widths of the magnetic islands are evaluated. The drift tearing mode is found to have metastable properties: the perturbations of finite amplitude which are damped in the linear approximation appear to be nonlinearly unstable. The authors discuss the energetics of such a hard regime of instability onset and the nonlinear stabilization mechanisms. The influence of a uniform stationary plasma background (BG) is considered. The presence of the cold dense BG slows down the growth of aperiodical instability and decreases the threshold of its nonlinear saturation. The field-aligned oscillation of BG electrons has a strong stabilizing effect on the drift tearing mode, even if their density is relatively small. When the density of the BG plasma exceeds the density of the current-carrying plasma, drift tearing modes cannot exist. The presence of the BG plasma strengthens the metastable properties of configurations with magnetic shear. The nonlinear growth rates and thresholds of linear and nonlinear stabilization are evaluated for a wide range of parameters (densities, temperatures) of current-carrying and BG plasmas.

Transport due to ηi modes in the presence of radiofrequency fields

The influence of a radiofrequency (RF) wave in the ion cyclotron range of frequencies (ICRF) on the toroidal ion temperature gradient driven drift mode (ηi mode) is presented. The four-wave parametric process involving a fast magnetosonic source wave and ion cyclotron sidebands is considered. Explicit expressions for the non-linear growth rate and the associated ion thermal conductivity are derived. The stability and transport properties are found to be sensitive to the sign of, where and are the pump and the ηi mode wave vectors, respectively.