Magnetized Cold Plasma Research Articles

Reflection of nonlinear solitary waves (mKdV solitons) at critical density of negative ions in a magnetized cold plasma

The reflection of ion acoustic solitons in a magnetized inhomogeneous plasma having positive ions, negative ions and electrons is studied theoretically at the critical density of negative ions, where both compressive and rarefactive solitons are found to exist. Relevant modified Korteweg–deVries (mKdV) equations corresponding to two oppositely moving solitons are coupled to obtain a variable coefficient coupled mKdV equation, which is further solved to obtain the expression of amplitude and width of the reflected solitons. The effect of magnetic field B0, its obliqueness θ (angle between the directions of magnetic field and wave propagation) and negative to positive ion density ratio d0 on the amplitude and width of the reflected solitons together with the reflection coefficient are investigated. A comparison is made between the incident and reflected solitons and it is found that the compressive (rarefactive) solitons are downshifted (upshifted) after the reflection.

Modified Korteweg-deVries soliton evolution at critical density of negative ions in an inhomogeneous magnetized cold plasma

Soliton propagation at critical density of negative ions is studied for weakly inhomogeneous magnetized cold plasma having positive ions, negative ions, and electrons. A general phase velocity relation is obtained and possible modes are studied for different cases involving different constituents of the plasma. Two types of modes (fast and slow) are found to propagate for the equal mass of the positive and negative ions. However, a limit on the obliqueness of magnetic field is obtained for the propagation of slow mode. For both types of modes, a variable coefficient modified Korteweg-deVries equation with an additional term arisen due to the density gradient is realized, which admits solutions for compressive solitons and rarefactive solitons of the same amplitudes at critical negative ion density. The propagation characteristics of these solitons are studied under the effect of densities of ions, magnetic field, and its obliqueness. The amplitudes of fast and slow wave solitons show their opposite behavior with the negative ion concentration, which is consistent with the variation of phase velocities with the negative ion density.

The three-wave coupling coefficients for a cold magnetized plasma

The resonant interaction between three waves in a uniform magnetized plasma is considered. Using the somewhat inaccessible result of the general theory, we deduce the explicit expressions for the coupling coefficients of a cold magnetized two-component plasma.

Adaptive finite elements method for the solution of the Maxwell equations in inhomogeneous magnetized plasma

The wave propagation in magnetized cold plasma inhomogeneous in one direction (the slab plasma model) involves solution of the set of the second order ODE’s. Several problems have to be treated in the numerical solution. An initial value approach cannot be used for the boundary problem where exponentially growing solution exists. The equations are stiff (the O and X modes have different wavelengths) and there is also, in the cold plasma model, a singularity of the solution in the upper hybrid resonance. For the boundary problem of this type, the finite elements method fits well. To remove the singularity at UHR, we introduce small ad hoc collisions. In warm plasma, the O and X waves are converted in the UHR region to the electron Bernstein waves. It is well known that the power of the O and X waves absorbed in UHR region in a weakly collisional cold plasma represents the power converted to EBW. To minimize the computational requirements of the algorithm, an adaptive mesh, based on the local error estimation, is constructed. The code is used to analyze electron cyclotron emission (ECE) from the spherical tokamak MAST.

Enhancement of Low-Energy Electron Ion Recombination in a Magnetic Field

Electron-ion recombination in cold magnetized plasmas shows a dramatic enhancement of the radiative recombination rate for bare highly charged ions relative to what standard radiative recombination rates predict. To understand the mechanism of this enhancement we investigate the classical chaotic dynamics of an electron in the combined Coulomb field of the ion and the magnetic field in the electron cooler. An increased flux of electrons visiting the vicinity of the target ion leads to an enhancement of the recombination process.

Three‐dimensional electromagnetic wave polarizations in a plasma

It is known that electromagnetic waves propagating in a magnetized cold plasma are not necessarily purely transverse waves. The existence of a longitudinal component implies that the normal to the polarization plane is not, in general, parallel to the propagation vector. The offset angle between these two directions is computed for each characteristic wave. Our investigation shows that the offset angle can be large under certain conditions. For a wave field that is a superposition of two characteristic waves, its polarization plane wobbles in space as the field propagates while its normal precesses. Several examples are considered, computed, and given to illustrate the 3D nature of wave polarizations.

Mode conversion in a weakly magnetized plasma with a longitudinal density profile

Motivated by conditions in the terrestrial foreshock, the mode conversion process in the inner region for a weakly magnetized plasma with a longitudinal linear density profile is examined. Coupled differential equations are employed to describe the wave energy coupling and the mode structure of electrostatic Langmuir and upper-hybrid branches, and electromagnetic O, X, and Z modes. For an incident electrostatic wave (the “inverse” problem), the coupled equations are solved using Green’s function method, and the mode conversion coefficient is determined from the asymptotic solutions. In the limit of a vanishing magnetic field, solutions converge to previous unmagnetized solutions. It is found that the magnetic field narrows the window of conversion and reduces the angles of propagation of electromagnetic waves with respect to the magnetic field, in agreement with results from previous solutions (of the “direct” problem) for a magnetized cold plasma. The narrowing of the window occurs at angles of propagation larger than the angle corresponding to the maximum conversion.

Nonlinear self-interaction of plasma electromagnetic pulses propagating obliquely to a very strong ambient magnetic field

We study nonlinear self-interactions including modulational instability in the case of a plane electromagnetic pulse propagating through a magnetized cold plasma at an arbitrary oblique angle to the external magnetic field. For intended applications to pulsar magnetospheres, the magnetic field is so large that both the electron- and ion-cyclotron frequencies are enormous compared with the plasma frequency or the frequency ω of the wave itself. The plasma is assumed to contain two singly charged species, either electrons and positrons or electrons and ions. (No approximation is made with respect to the mass ratio.) We restrict ourselves to the case eE0/mω ≪ 1 (i.e. the wave amplitude E0 excites the electrons to weakly, but not fully, relativistic velocities). We consider a pulse whose linear polarization is in the plane of the wave vector and the magnetic field. (The orthogonal polarization is purely electromagnetic, and induces no motion along magnetic field lines.) The pulse is assumed to be modulated along the direction of the group velocity vector. We show, using a self-consistent multiple-scales solution, that the envelope obeys the nonlinear Schrödinger equation, and from the coefficients of this equation we derive the conditions for modulational instability. Computation of the nonlinear coefficients requires detailed consideration of ponderomotive, relativistic and harmonic effects, all of which, in the ‘weakly relativistic’ case considered here, enter at the same order in the approximation scheme. Unlike the case of propagation parallel to a strong magnetic field, in oblique propagation we find a wide parameter range for modulational instability and soliton formation on time scales appropriate for pulsar micropulses.

Intense propagation in a magnetized cold plasma: modulational instabilities of fully relativistic electromagnetic waves

This paper studies modulational instabilities of intensely propagating, circularly polarized, plane electromagnetic plasma waves in the presence of an external magnetic field pointing exactly in the direction of propagation. By ‘intense propagation’, we mean that eEo/mw (where Eo is the amplitude of the wave's transverse electric field) is comparable to unity, so that the problem is fully nonlinear and cannot be solved by regular perturbation methods. The positive component may consists of either positrons or singly charged ions. The plasma is assumed to be fully ionized and cold. Modulated intensely propagating electromagnetic waves couple (in general) to longitudinal motions via the ponderomotive force. For given wavenumber, the frequency of the wave depends on the amplitude not only via the obvious mechanism of relativistic mass increase but also via the density inhomogeneities induced by the ponderomotive force. This coupling effectively involves derivatives of the envelope, and the effect of longitudinal motions is comparable to that of relativistic nonlinearities. For this reason, a proper expansion procedure requires verification that one has obtained a self-consistent solution of all the field equations up to the appropriate order, including the longitudinal equations. This is best accomplished using the well-known two-timing approach. Modulational instability arises in the case of an ion-electron plasma with relativistic electrons but non-relativistic ions. However, for parameter values appropriate to pulsar magnetospheres, where the electron-cyclotron frequency is much larger than the frequency of the wave, there is no instability on the time scale of a micropulse.

Transient analysis of a magnetized plasma in three-dimensional space

The time-domain formation, in terms of unified nodal equations, of a magnetized cold plasma in three-dimensional space using the Bergeron method is described. The validity of this treatment is shown by simulations of both transverse propagation and longitudinal propagation. In the formulation of the characteristic equation in the time domain, the authors introduce variables corresponding to the first and second derivates of the electric polarization. Thus the iterative computation can be performed by using only the values obtained at the previous time step. This procedure very closely matches the architecture for high-speed computation in the vector processor of a supercomputer. This property can render practical the simulation of three-dimensional fields involving gyroelectric anisotropic media and expand the generality of the numerical vector analysis method in the time domain.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Calculation of coupling to slow and fast waves in the LHRF from phased waveguide arrays

A previously reported algorithm for solving the problem of coupling electromagnetic energy in the LHRF from a phased array of identical rectangular waveguides to a plane-stratified, magnetized cold plasma is numerically implemented. The resulting computer codes are sufficiently general to allow for an arbitrary number of waveguides with finite dimensions in both poloidal and toroidal directions and are thus capable of computing coupling to both slow and fast waves in the plasma. Some of the details of the implementation and the extension of the algorithm to allow study of the Fourier spectrum of slow and fast waves launched by the array are discussed. Good agreement is found with previously reported, less general work for the slow wave launching case. The effect of phasing multirow arrays in the poloidal direction is studied, and an asymmetric between phasing ‘up’ and ‘down’ is found that persists in the case where the plasma adjacent to the array is uniform. A 4 × 3 array designed to launch fast waves of high phase velocity is studied. By using the optimal poloidal phasing, low reflection coefficients (|R|2 ⪅ 20%) are found under some not unrealistic edge plasma conditions, but most of the input power is trapped in the outermost layer of the plasma. Implications of our results for fast wave current drive experiments are discussed.

Electromagnetic wave propagation in relativistic magnetized plasmas

An improved mathematical technique and a new code for deriving the conductivity tensor for collisionless plasmas have been developed. The method is applicable to a very general case, including both hot (relativistic) and cold magnetized plasmas, with only isotropic equilibrium distributions being considered here. The usual derivation starts from the relativistic Vlasov equation and leads to an integration over an infinite sum of Bessel functions which has to be done numerically. In the new solution the integration is carried out over a product of two Bessel functions only. This reduces the computing time very significantly. An added advantage over existing codes is our capability to perform the computations for waves propagating obliquely to the magnetic field. Both improvements greatly facilitate investigations of properties of the plasma under conditions hitherto unexplored.

Properties of electromagnetic waves in ferrites

In this paper we present a derivation of the general wave equation governing the propagation of electromagnetic waves in ferrites. This is shown to be equivalent to the usual Helmholtz form obtained by introducing a permeability tensor. Its real underlying structure is somewhat obscured by the Helmholtz form because in this form the time dependence is submerged in the frequency-dependent elements of the permeability tensor. However, the main purpose of the paper is to provide a systematic account of the properties of electromagnetic waves in ferrites and this is done by making use of the close analogy that exists between wave propagation in ferrites and that in cold magnetized plasmas. This facilitates the construction of a so-called ‘C. M. A. diagram’ for a ferrite, which classifies the waves according to the various topologies of the refractive index and ray surfaces (along with the associated wave polarizations) in a two-dimensional parameter space that relates the wave frequency to the precessional frequencies intrinsic to the magnetization of the ferrite. The insight into the behaviour of waves in ferrites that this approach provides is illustrated in connection with the phenomena of cut-offs, resonances and wave-focusing, which give rise to caustics.

A broadband VLF burst associated with ring current electrons

During the periods after the maximum phase of geomagnetic storms on 2 April 1972 a broadband VLF emission burst has been observed by the equatorially orbiting satellite S³‐A (Explorer 45) along its inbound plasmapause crossings. The frequency band broadening takes place just outside of the nighttime plasmasphere, where the density of cold plasma has been known to be very low during the later phase of a geomagnetic storm. Instead of the gradual broadening of several hours duration, a burst type broadening of VLF emission lasting less than ten minutes was observed in the similar location. The magnetic field component of this emission is very weak and the frequency spreads below the local half electron cyclotron frequency. Corresponding enhancement of the anisotropic ring‐current electrons is also very sudden and limited below the order of 10 keV without significant velocity dispersion, in contrast to the gradual broadening events. The cause of this type of emission band spreading can be attributed to the generation of the quasi‐electrostatic whistler mode emissions of short wavelength by hot bi‐maxwellian electrons surging into the domain of relatively low density magnetized cold plasma. The lack of energy dispersion in the enhanced electrons indicates that the inner edge of the plasma sheet, the source of these hot electrons, is not far from the location of this event.

Wave Propagation in a Moving Cold Magnetized Plasma

Abstract Polarization relations and dispersion equations are derived for media electrically anisotropic in the comoving frame. The obtained results are discussed mainly for cold magnetized plasmas, briefly for uniaxial dielectric crystals. Special directions of wave propagation are considered.

Remarks concerning the dispersion equation of electromagnetic waves in a magnetized cold plasma

The dispersion formula due to Lassen (1927) and Försterling is discussed and compared with other dispersion formulas. If the corresponding expressions for the conductivity eigenvalues are inserted into this formula one obtains as special cases Appleton's (1928) as well as Sen and Wyller's (1960) dispersion formulas for an electron plasma with weak as well as strong absorption, respectively. Also S eeger's (1961) dispersion formula for the inclusion of the ion motion appears as a particular case. A simple reasoning is presented for the omission of the Lorentz polarization term in the case of radio wave propagation, which is erroneously included in the dispersion formulas of Hartree (1929, 1931) and Appleton(1932).

Dispersion and stability of waves in plasmas in the presence of a coriolis force

The dispersion law for the propagation of waves in cold magnetized plasmas is derived for arbitrary directions of the rotation axis with respect to the static magnetic field. The waves are shown to be stable, not only in the case of a cold plasma, but in any plasma case which yields hermitian mobility tensors. An interesting special case is when the rotation and magnetization axes are parallel, because then for suitable values of rotation and external magnetic field the two effects can cancel each other, though only for one plasma species at the time. The rotation thus decisively affects and shifts the number and width of the existing pass- and stop-bands in a magnetized plasma. The inclusion of thermal effects through a scalar barotropic pressure is not nearly as significant.

Three-wave interaction in cold magnetized plasmas

The coupled equations for the resonant nonlinear interaction between three waves in cold magnetized plasmas are derived and written in a comparatively simple and symmetric form. The deficiencies of previous papers are pointed out and the possibility of explosive interactions in the solar corona is suggested.

On the asymptotic decay of coupling for infinite phased arrays

The asymptotic decay of coupling with distance from a single excited element it an infinite linear array has been analyzed for several special cases and is shown to behave according to exp (-jkr)/r3/2where r is the distance from the excited element. This behavior is also characteristic of propagation over a lossy surface. By geometric extrapolation one would expect an asymptotic decay of coupling for planar arrays to behave according to exp (-jkr)/r2. This extension is carried out for an arbitrary infinite periodic planar current sheet with the expected result. In addition, this behavior is shown to be valid for an infinite parallel plate array immersed in a magnetized cold plasma. In all cases, the asymptotic behavior is shown to depend on the form of the singularity of the derivative of the reflection coefficient with scan angle. This singularity occurs at grazing incidence of the radiated beams.

Instabilities in a Plasma-Beam System Immersed in a Magnetic Field

The interaction of an electron or an ion beam moving with velocity $\ensuremath{\beta}c$ through a stationary plasma in the presence of a static magnetic field is investigated theoretically under the assumption that the plasma is cold, the intensity of the beam is very small, and that the beam, as well as the waves resulting from the interaction, are aligned along the direction of the magnetic field. It is well known that an electron beam moving with velocity exceeding the phase velocity of electromagnetic waves in a stationary plasma is capable of exciting such waves and these waves (circularly polarized) rotate in the same direction and at the same angular frequency as the gyroelectrons of the beam. It is shown in this investigation that there is an apparent reversal in the direction of rotation of these gyroelectrons as seen by the stationary observer. Consequently, the excited wave has a circular motion in the same direction as perturbed stationary ions. The frequency $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}$ of the excited wave satisfies the inequality $|\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}|l{\ensuremath{\Omega}}_{i}$, where ${\ensuremath{\Omega}}_{i}$ is the gyrofrequency of perturbed stationary ions. Similarly, a wave excited by an ion beam has a circular motion in the same direction as perturbed stationary electrons. The frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}^{i}$ of the latter wave satisfies the inequality $|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}^{i}|l{\ensuremath{\Omega}}_{e}$, where ${\ensuremath{\Omega}}_{e}$ is the gyrofrequency of perturbed stationary electrons. It is shown that an electron (ion) beam moving in the direction of the magnetic field can excite only a wave having negative (positive) helicity. The reverse situation occurs for a beam moving against the magnetic field. General relationships are formulated and illustrated graphically for determining the frequency and growth rate of waves which can be excited by an electron or an ion beam in any magnetized cold plasma. A particular case is illustrated in which an incident ion beam interacts with a relative dense plasma such as occurring in thermonuclear discharges, ionosphere, and interstellar space. It is shown that in such cases there is an excitation of hydromagnetic waves for a wide range of velocities of the ion beam.