Dynamics of the magnetospheric cyclotron ELF/VLF maser in the backward-wave-oscillator regime. II. The influence of the magnetic-field inhomogeneity

We study the influence of the magnetic-field inhomogeneity on the nonlinear dynamics of the absolute instability of whistler-mode waves in the Earth’s magnetosphere in the presence of a step-like deformation in the distribution function of energetic electrons. Development of this instability, implying the transition of the magnetospheric cyclotron maser to the regime of a backward-wave oscillator (BWO), was proposed earlier as a generation mechanism of magnetospheric chorus emissions. We analyze the results of numerical simulations of the simplified nonlinear equations describing the magnetospheric-BWO dynamics in the case of low efficiency of wave-particle interactions. We found that the case of an inhomogeneous magnetic field where the system length is much greater than the length characterizing the linear stage of the BWO regime has important specific features compared with the case of a homogeneous medium. The main feature of the nonlinear wave dynamics in the magnetospheric BWO in an inhomogeneous magnetic field consists in the fact that for a sufficiently large excess over the generation threshold, a sequence of separate wave packets, i.e., discrete elements, is formed. The frequency within each packet varies in time, and these discrete elements are close in their properties to the chorus elements observed in the magnetosphere. The results of calculations confirm the quantitative estimates of parameters of chorus emissions, which were performed earlier on the basis of the BWO model. ^†Deceased Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 11, pp. 977–987, November 2008.     

Generation of VLF emissions with the increasing and decreasing frequency in the magnetosperic cyclotron maser in the backward wave oscillator regime

We study the mechanisms of the formation of falling tones in the dynamic spectrum of whistler-mode waves generated by energetic electrons in the Earth’s magnetosphere when the backward-wave oscillator (BWO) regime is realized in the magnetospheric cyclotron maser. As was shown earlier, this regime allows one to explain many features of ELF/VLF chorus emissions in the magnetosphere, in particular, the generation of elements with discrete frequency spectrum, characterized by a large growth rate and a fast frequency drift. On the basis of numerical simulations of a simplified system of nonlinear equations describing the magnetospheric BWO dynamics under the assumption of small efficiency of wave-particle interactions we show that the falling tones are generated in the case where the generation region is shifted from the equatorial plane (geomagnetic-field minimum) upstream with respect to the motion of energetic electrons. In this case, the resonant electrons move towards the decreasing magnetic field in the process of generation; hence, their longitudinal velocity increases, which corresponds to a decrease in the cyclotron-resonance frequency. Two mechanisms of the shift of the generation region from the equator are considered, i.e., (i) an increase in the linear instability growth rate (e. g., due to an increase in the energetic-electron density), and (ii) persistence of the phase bunching of the particles coming back to the generation region due to the bounce oscillations. We show that both of these mechanisms can result in the formation of falling tones, but the properties of the generated emissions such as the frequency drift rate and characteristic time interval between the elements are different. The conditions of preserving the phase bunching due to the bounce oscillations are discussed. Probably, this mechanism can operate in the case where the length of the generation region along the magnetic field is close to the characteristic bounce-oscillation length of energetic electrons which is realized for a sufficiently high cold-plasma density in the generation region.     

Self-modulation oscillatory modes in a relativistic backward-wave oscillator

We present the results of numerical simulation of nonlinear dynamics in a relativistic backward-wave oscillator (BWO) over a broad range of parameters. A comprehensive study of the onset of self-modulation is carried out. It is found that the self-modulation boundary in the parameter plane has a complex shape. This is due to the competition between two different dynamical modes leading to the instability of single-frequency stationary oscillations. It is shown that the transition to chaos through period doubling bifurcations characteristic of a nonrelativistic BWO exists only for small values of the relativistic factor γ₀, while the transition through intermittency dominates for large γ₀. Higher College for Applied Sciences, State University of Saratov, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 6, pp.566–572, June 1999.     

First and second order non-equilibrium phase transition and evidence for non-extensive Tsallis statistics in Earth’s magnetosphere

In this paper strong evidence is provided for significant far from equilibrium phase transition processes in the Earth’s magnetosphere as revealed by the nonlinear analysis of in situ observations. These results constitute the solid base for the solution of the durable controversy about the chaotic or non-chaotic character of the magnetospheric dynamics. During the last two decades the concept of low dimensional chaos was supported by theoretical and experimental methods by our group in Thrace and others scientists, as an explicative paradigm of the magnetospheric dynamics including substorm processes. In parallel, the concept of self-organized criticality (SOC) and space-time intermittency was introduced as new and opposing to low dimensional chaos concepts for modeling the magnetospheric dynamics. Novel results concerning the nonlinear analysis of in situ space plasma data (magnetic-electric field, energetic particles and bulk plasma flow time series) obtained by the Geotail spacecraft presented in this paper for the first time reveal the following: (a) Coexistence of SOC and chaos states in the magnetospheric system and global phase transition from one state to the other during substorms. (b) Strong intermittent turbulent character of the magnetospheric system at the SOC or the low dimensional chaos states. (c) Clear indications for non-extensivity and q-Gaussian statistics during periods of low dimensional and chaotic dynamics of the magnetosphere. (d) Low dimensional and nonlinear space plasma dynamics in the day side magnetopause and bow shock dynamics. The dual character of the magnetospheric dynamics including low dimensional chaotic (coherent) and high dimensional turbulent states, as supported in this paper, is in agreement and verifies previous theoretical and experimental studies.     

Cyclotron Wave-Particle Interactions in the Whistler-Mode Waveguide

We study the cyclotron interaction of energetic electrons and whistler waves in plasma waveguides formed by inhomogeneous distribution of cold plasma. Such waveguides can be formed in the Earth's magnetosphere, e.g., by the plasmapause or by ducts with enhanced background-plasma density. In this paper, we consider a cylindrically symmetric model of a magnetospheric duct with enhanced cold-plasma density in a homogeneous magnetic field. The spatial structure of the eigenmodes of such a waveguide is found. We obtain a set of self-consistent quasilinear equations for cyclotron instability with eigenmode structure taken into account, thus generalizing the quasilinear theory of a magnetospheric cyclotron maser.     

Modulational instability and exact soliton solutions for a twist-opening model of DNA dynamics

We study the nonlinear dynamics of DNA which takes into account the twist-opening interactions due to the helicoidal molecular geometry. The small amplitude dynamics of the model is shown to be governed by a solution of a set of coupled nonlinear Schrödinger equations. We analyze the modulational instability and solitary wave solution in the case. On the basis of this system, we present the condition for modulation instability occurrence and attention is paid to the impact of the backbone elastic constant K. It is shown that high values of K extend the instability region. Through the Jacobian elliptic function method, we derive a set of exact solutions of the twist-opening model of DNA. These solutions include, Jacobian periodic solution as well as kink and kink-bubble solitons.     

Modulational instability in the cubic-quintic nonlinear Schrödinger equation through the variational approach

The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrödinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic-quintic nonlinear Schrödinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE’s, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.     

Coupled Klein–Gordon equations and energy exchange in two-component systems

A system of coupled Klein–Gordon equations is proposed as a model for one-dimensional nonlinear wave processes in two-component media (e.g., long longitudinal waves in elastic bi-layers, where nonlinearity comes only from the bonding material). We discuss general properties of the model (Lie group classification, conservation laws, invariant solutions) and special solutions exhibiting an energy exchange between the two physical components of the system. To study the latter, we consider the dynamics of weakly nonlinear multi-phase wavetrains within the framework of two pairs of counter-propagating waves in a system of two coupled Sine–Gordon equations, and obtain a hierarchy of asymptotically exact coupled evolution equations describing the amplitudes of the waves. We then discuss modulational instability of these weakly nonlinear solutions and its effect on the energy exchange.     

Modeling polymerization of microtubules: A semi-classical nonlinear field theory approach

In this paper, for the first time, a three-dimensional treatment of microtubules’ polymerization is presented. Starting from fundamental biochemical reactions during microtubule’s assembly and disassembly processes, we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in three dimensions. We found that the dynamics of a microtubule is mathematically expressed via a cubic-quintic nonlinear Schrödinger (NLS) equation. We show that in 3D a vortex filament, a generic solution of the NLS equation, exhibits linear growth/shrinkage in time as well as temporal fluctuations about some mean value which is qualitatively similar to the dynamic instability of microtubules. By solving equations numerically, we have found spatio-temporal patterns consistent with experimental observations.     

Dynamics of the plane and solitary waves in a Noguchi network:Effects of the nonlinear quadratic dispersion

We consider a modified Noguchi network and study the impact of the nonlinear quadratic dispersion on the dynamics of modulated waves. In the semi-discrete limit, we show that the dynamics of these waves are governed by a nonlinear cubic Schrodinger equation. From the graphical analysis of the coefficients of this equation, it appears that the nonlinear quadratic dispersion counterbalances the effects of the linear dispersion in the frequency domain. Moreover, we establish that this nonlinear quadratic dispersion provokes the disappearance of some regions of modulational instability in the dispersion curve compared to the results earlier obtained by Pelap et al.(Phys. Rev. E 91 022925(2015)). We also find that the nonlinear quadratic dispersion limit considerably affects the nature, stability, and characteristics of the waves which propagate through the system. Furthermore, the results of the numerical simulations performed on the exact equations describing the network are found to be in good agreement with the analytical predictions.     

Daytime hiss-triggered chorus emissions observed at low latitude ground station Srinagar, India

The first observation of hiss-triggered discrete chorus riser emissions recorded during daytime at our newly setup low latitude ground station Srinagar (geomag. lat., 24° 10′ N; L = 1.28), India on March 30, 2009 are reported. From the spectral analysis of these emissions, it is found that the chorus is hiss-triggered and each chorus element has the tendency to originate from the hiss band. A possible generation mechanism of these hiss-triggered chorus riser emissions is proposed.     

Instability and dynamics of thin viscoelastic liquid films

The instability, rupture, and subsequent growth of holes in a thin Jeffreys-type viscoelastic film under the influence of long-range van der Waals force are investigated using both linear stability analysis and nonlinear numerical solutions. The linear stability analysis of full governing equations valid for arbitrary wave numbers shows that although fluid rheology does not influence the dominant length scale of the instability, it significantly affects the growth rate. It is shown that neglect of inertia and solvent dynamics results in a nonphysical singularity in the growth rate beyond a critical value of relaxation time. We further carry out numerical simulations of a set of long-wave, nonlinear differential equations (also derived in Rauscher et al., Eur. Phys. J. E 17, 373 (2005)) governing the evolution of the free surface. The nonlinear simulations, in their domain of validity, confirm the results of the linear analysis. Interestingly, results from nonlinear simulations further show that both for Newtonian and viscoelastic liquids, the shape and the dewetting dynamics of a hole are identical when examined in terms of a rescaled time which depends on rheological parameters. Thus, viscoelasticity of Jeffreys type merely accelerates the growth rate, without however affecting the important morphological characteristics.     

Electrothermal Oscillatory Instability in Spherical Ferroelectric Resonators. Three-Mode Case

We consider the equations of interaction between electromagnetic oscillations and the temperature in a nonlinear dielectric resonator and study the dynamics of the oscillatory instability in the system. The threshold conditions (power and self-modulation frequency) of electrothermal excitation are calculated for microwave potassium-tantalate resonators for the case of three-mode interaction. The conditions for observing electrothermal excitation in the three-mode case are found to be quite favorable. In this case, the threshold power of excitation of temperature oscillations is smaller than that in the two-mode case and can amount to a few microwatts.     

Characteristics of nighttime VLF/ELF emissions with frequency drift and estimation of the large-scale electric field from the frequency drift observed at a low latitude Indian ground station, Jammu

Characteristics of nighttime VLF/ELF emissions are examined on the basis of the data obtained at our low latitude ground station Jammu (geomagnetic latitude, 22°26′ N; L = 1.17), India during our VLF/ELF campaign. From the detailed analysis of a huge amount of acquired VLF/ELF data at Jammu we have found three remarkable events which clearly exhibit a rise in their frequency in pre and past midnight sectors during magnetically quiet and substorm periods. Our analysis shows that the frequency drift in VLF/ELF emissions seems to be a rare phenomenon at low latitudes during magnetically quiet and substorm periods in pre and post midnight sectors. This property of temporal frequency drift (regular frequency increases with time) in VLF/ELF emissions observed at our station Jammu are interpreted in terms of a quasi-linear electron cyclotron instability model for wave excitation. The initial frequency increase is believed to be due to a combind effect of L-shell drift of energetic electrons. Further, the frequency drifts in VLF/ELF emissions observed at Jammu have been used to estimate the large scale electric field during quiet and substorm periods in pre and post midnight sectors. This investigation would be most useful for the study of the wave-particle interaction processes, magnetospheric plasma structure and particle dynamics, especially during quiet periods in premidnight sector at low latitudes.     

Explosive instability of quasi-phonon triads in antiferromagnet under frequency modulated electromagnetic field

Compensation of nonlinear frequency shift of hybrid magnetoelastic waves (quasi-phonons) in the process of parametric three-boson coupling is studied theoretically and experimentally. The singular frequency modulation of electromagnetic pumping is proposed for the observation of explosive instability of quasi-phonon triads. The explosive supercritical dynamics is simulated theoretically on the basis of strong nonlinear equations of magnetoelastic dynamics and observed in α-Fe₂O₃ magnetoacoustic resonator.     

Nonlinear dynamics of high-density electron-beam resonant instabilities. Analytical solution

We obtain an analytical solution of the nonlinear dynamics of a resonant instability in a dense electron beam. It is shown that the instability of a dense beam saturates because of a nonlinear frequency shift and a deviation from the resonance condition.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 31–35, January, 1984.The authors acknowledge useful discussions with A. A. Rukhadze.     

Interacting signal packets in a lossless nonlinear transmission network with linear dispersion

In the small amplitude limit, we use the reductive perturbation method and the continuum limit approximation to derive a coupled nonlinear Schrö dinger (CNLS) equation describing the dynamics of two interacting signal packets in a discrete nonlinear electrical transmission line (NLTL) with linear dispersion. With the help of the derived CNLS equations, we present and analyze explicit expressions for the instability growth rate of a purely growing modulational instability (MI). We establish that the phenomenon of the MI can be observed only for “small” nonzero modulation wavenumbers. Also, we point out the effects of the linear dispersive element, as well as of the frequencies of the signal packets, on the instability growth rate. It is shown that the linear dispersion and the frequencies of signal packets can be well used to control the instability domain. Through the CNLS equations, we analytically investigate the propagation of solitary waves in the network. Our analytical studies show four types of interaction of signal packets propagating in the network: bright–bright, dark–dark, bright–dark and dark–bright soliton interactions.     

Competing mechanisms of plasma transport in inhomogeneous configurations with velocity shear: the solar-wind interaction with earth's magnetosphere

Two-dimensional simulations of the Kelvin-Helmholtz instability in an inhomogeneous compressible plasma with a density gradient show that, in a transverse magnetic field configuration, the vortex pairing process and the Rayleigh-Taylor secondary instability compete during the nonlinear evolution of the vortices. Two different regimes exist depending on the value of the density jump across the velocity shear layer. These regimes have different physical signatures that can be crucial for the interpretation of satellite data of the interaction of the solar wind with the magnetospheric plasma.     

Rossby waves in the magnetic fluid dynamics of a rotating plasma in the shallow-water approximation

We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.