Ring-beam Distribution Research Articles

The Transport of Charged Particles in a Flowing Medium

The propagating source method for solving the time-dependent Boltzmann equation describing particle propagation in a magnetically turbulent medium is extended to a more realistic case that includes focusing and adiabatic deceleration. The solutions correspond to beam propagation in the solar wind. Pitch-angle scattering away from 90° is described by standard quasi-linear theory (QLT), while scattering through 90° is approximated by a BGK operator representing a slow mirroring process. The detailed numerical technique for solving the Fokker-Planck equation for two particular spectra is presented. Comparisons are made between our modified QLT (MQLT) model and a BGK model, between highly anisotropic scattering and moderately anisotropic scattering, and between fast particles and slow particles. It is shown that: (1) for moderately anisotropic pitch-angle scattering, the initial ring-beam distribution finally evolves into a broad Gaussian distribution and the QLT isotropic and MQLT anisotropic models could be rather well approximated by the simple relaxation time operator. (2) For highly anisotropic pitch-angle scattering, a moving pulse with a spatially extended flat tail is formed, and there exist some differences between the MQLT and BGK models. Specifically, at a particular pitch angle, the spatial distribution from MQLT model occupies a much wider region than that in the BGK model. (3) In the highly anisotropic scattering medium, more particles are cooled by adiabatic deceleration, some particles move a little faster, and the spatial distribution at a specific pitch angle is much more dispersed than that in the case of moderately anisotropic scattering. (4) Compared with the BGK model, the anisotropy persists for a little longer and some particles move a little slower; consequently, intensity profiles have a greater amplitude at later times in the MQLT model. (5) Finally, fast and slow particles have similar distribution characteristics, except that convection is much more important for slow particles.

2D hybrid simulations of the solar wind interaction with a small scale comet in high Mach number flows

We investigate the structure and dynamics of the interaction between high Mach number solar wind flows and a cometary source that is “small scale” with respect to the cometary ion gyroscales, examples being Grigg‐Skjellerup or P/Wirtanen. A series of 2D hybrid simulations are used and we focus on IMF orientations parallel and perpendicular to the simulation plane, these permit or prohibit propagation of modes with a component of k parallel to B0. The perpendicular cases show the onset of magnetosonic turbulence as MA is increased; introducing a parallel B0 field component significantly changes this turbulence and the apparent shock structure seen on a ‘fly‐by'. This implies that the Alfvén‐ion cyclotron mode plays a critical role in the observed structure, and we suggest the relevant mechanism must be resonant growth of such waves in the presence of magnetosonic turbulence from either the unstable ring beam distribution of the pickup ions, or shock reflected solar wind protons.

Lower Hybrid Drive in Solar Magnetic Reconnection Regions: Implications for Electron Acceleration and Solar Heating

AbstractLower hybrid (LH) drive involves the resonant acceleration of electrons parallel to the magnetic field by lower hybrid waves, often driven by ions with ring or ring-beam distributions. Charge-exchange between hydrogen atoms and protons with relative motions perpendicular to the magnetic field leads to ring distributions of pickup ions, with concomitant perpedicular ion ‘heating’. This paper considers the combination of LH drive and charge-exchange in the outflow regions of magnetic reconnection sites in the solar chromosphere and lower corona, showing that the combined mechanism naturally predicts major perpendicular ion heating and parallel electron acceleration, and exploring the mechanism’s relevance to specific solar reconnection phenomena, heating of the solar atmosphere, and production of energetic electrons that generate solar radio emission. Although primarily qualitative, analysis shows that the mechanism has numerous attractive aspects, including perpendicular ion heating that increases linearly with ion mass, parallel electron acceleration, predicted ion and electron temperatures that span those of the chromosphere and lower corona, and parallel electron speeds spanning those for type III bursts. Applications to chromospheric explosive events and low-lying active regions, and to heating the chromosphere, appear particularly suitable. Sweeping of plasma frozen-in to chromospheric and coronal magnetic field lines across the neutral atmosphere due to motions of sub-photospheric fields represents an obvious and important generalisation of the mechanism away from reconnection sites. The requirements that the neutrals not be strongly collisionally coupled to the plasma and that sufficient neutrals are available for charge-exchange restricts the LH drive mechanism to above the photosphere but below where the corona is essentially fully ionised. LH drive may thus be important in heating the chromosphere and low corona while other heating mechanisms dominate at higher altitudes. Although attractive thus far, quantitative analyses of LH drive in these contexts are necessary before definitive conclusions are reached.

Galileo observations of ion cyclotron waves in the Io torus

Ion cyclotron waves generated near Io have been observed on four Galileo passes of the plasma torus, in December 1995, October 1999, November 1999, and February 2000. The waves have frequencies near the gyrofrequencies of SO + 2 and SO + ions, and are propagating at angles up to 40° to the ambient magnetic field. These waves are generated by ring-beam distributions of SO + 2 and SO + pickup ions. In some regions there are also waves with frequencies near the gyrofrequency of S + ions. This suggests that ring-beam sulfur ions can also make the plasma unstable. We perform a kinetic dispersion analysis for a plasma with ring-beam SO + 2, SO +, and S + pickup ions. We find that simultaneous growth of parallel propagating SO + 2, SO +, and S + cyclotron waves is possible and that the dominant mode depends on the density of the ring-beams.

Transport of energetic charged particles in a radial magnetic field. Part 1. Large-angle scattering

A new approach, the propagating-source method, is introduced to solve the time-dependent Boltzmann equation. The method relies on the decomposition of the particle distribution function into scattered and unscattered particles. It is assumed in this paper that the particles are transported in a constant-velocity spherically expanding supersonic flow (such as the solar wind) in the presence of a radial magnetic field. Attention too has been restricted to very fast particles. The present paper addresses only large-angle scattering, which is modelled as a BGK relaxation time operator. A subsequent paper (Part 2) will apply the propagating-source method to a small-angle quasilinear scattering operator. Initially, we consider the simplest form of the BGK Boltzmann equation, which omits both adiabatic deceleration and focusing, to re-derive the well-known telegrapher equation for particle transport. However, the derivation based on the propagating-source method yields an inhomogeneous form of the telegrapher equation; a form for which the well-known problem of coherent pulse solutions is absent. Furthermore, the inhomogeneous telegrapher equation is valid for times t much smaller than the ‘scattering time’ τ, i.e. for times t [Lt ] τ, as well as for t > τ. More complicated forms of the BGK Boltzmann equation that now include focusing and adiabatic deceleration are solved. The basic results to emerge from this new approach to solving the BGK Boltzmann equation are the following. (i) Low-order polynomial expansions can be used to investigate particle propagation and transport at arbitrarily small times in a scattering medium. (ii) The theory of characteristics for linear hyperbolic equations illuminates the role of causality in the expanded integro-differential Fokker–Planck equation. (iii) The propagating-source approach is not restricted to isotropic initial data, but instead arbitrarily anisotropic initial data can be investigated. Examples using different ring-beam distributions are presented. (iv) Finally, the numerical scheme can include both small-angle and large-angle particle scattering operators (Part 2). A detailed discussion of the results for the various Boltzmann-equation models is given. In general, it is found that particle beams that experience scattering by, for example, interplanetary fluctuations are likely to remain highly anisotropic for many scattering times. This makes the use of the diffusion approximation for charged-particle transport particularly dangerous under many reasonable solar-wind conditions, especially in the inner heliosphere.

A theory for AKR fine frequency structure

An alternative explanation of the fine frequency structure (≤1 kHz) associated with auroral kilometric radiation (AKR) is presented. It is based on a realistic model of the electron distribution function inside the AKR source region. The resulting cyclotron maser instability growth rate is shown to possess an extremely narrow bandwidth, of the order Δω/ω ∼ 10−3, which may naturally account for the AKR fine frequency structure. The results differ from many previous calculations and simulations which relied on simpler electron distribution function models such as a loss‐cone, ring‐beam or DGH distribution.

The structure of mass‐loading shocks: 1. Comets

A new multifluid model to describe the solar wind‐cometary ion plasma in the outer cometary coma is derived. This model is distinguished from previous multifluid models in that wave‐particle effects are included explicitly. By considering hydrodynamic timescales, self‐consistent dissipative terms (which correspond to the spatial diffusion of cometary ions) are derived, thereby allowing the structure of a cometary bow shock to be resolved. In the hypersonic limit, our diffusive multifluid model reveals an attractive connection to the one‐fluid model formulated by Biermann et al. (1967), a connection which serves as an important guide when investigating our more complicated nonhypersonic diffusive model. The detailed model consists of cometary ions diffusing in a warm solar wind which is heated by the dissipation of Alfvénic turbulence (in situ and generated by the initial ring beam distribution). The quasi‐parallel cometary shock is found to be at least an order of magnitude thicker than a quasi‐perpendicular cometary shock. It is furthermore found that for undisturbed solar wind (sonic) Mach numbers in excess of ∼6, the cometary bow shock (for both perpendicular and parallel cases) is smoothed completely by the cometary ions and no “proton” or thermal subshock is necessary. Finally, a detailed comparison of the model with the plasma observations made at Halley by Giotto is undertaken. It is found that the observed shock structure, thickness, location, and plasma parameters compare well with the predictions of the theoretical model, particularly in the case of the quasi‐parallel shock.

Transition from reactive to kinetic electromagnetic instabilities generated by ring-beam ions

The dispersion relation for the low-frequency electromagnetic waves propagating parallel to the ambient magnetic field is studied for a plasma comprised of energetic ring-beam distribution of ions and a background population. Detailed properties of the dispersion relation are studied for various physical parameters with particular emphasis placed on the transition from the cold case to the thermal regime. In addition, the stability characteristics are investigated for different values of α, where tan α is the ratio between the ring and the beam velocity, ranging from α=0 (pure beam) to α=90° (pure ring). The unstable modes are studied in further detail, and the dependence of the growth rate on several parameters, such as the density of the ring-beam ions and thermal speeds associated with the background and the ring-beam ions, is discussed.

Quasilinear diffusion rates of cometary ions

The freshly created ions in the upstream region of the cometary bow shock usually form a ring-beam velocity distribution in the solar wind frame of reference. These newborn ions subsequently excite a variety of instabilities which in turn cause the ions to diffuse in pitch angle space to form a quasispherical shell velocity distribution. Such a process can be described by quasilinear kinetic theory. In the present paper, the detailed properties of the initial diffusion rates associated with the cometary newborn ions are studied for various physical parameters including the so-called injection angle α (that is, the average initial pitch angle associated with the ions) ranging from 0° (pure beam distribution) to 90° (pure ring distribution). It is shown that initial diffusion rates are strongly dependent upon the angle α, such that for 0°≤α≤60° (quasiparallel regime) the pitch angle diffusion rate remains relatively small, while for 60°≤α≤90° (quasiperpendicular regime) the diffusion rate increases substantially.

Combined energy and pitch angle diffusion of pickup ions at comet Halley

It is well known that cometary pickup ions (e.g., H2O+, OH+, O+, CO+, H+) initially form a ring‐beam distribution in the solar wind reference frame, which is highly unstable to the growth of MHD waves (such as ion‐cyclotron waves). The low‐frequency magnetic fluctuations (or waves), which were observed upstream of comet Halley, cannot only pitch angle scatter the pickup ions so that the distribution becomes at least partially isotropized, but also stochastically accelerate the ions, resulting in the energetic ion populations observed in the vicinity of comet Halley. We have used numerical solutions of the quasi‐linear diffusion equation to investigate the cometary ion pickup process at comet Halley. Both pitch angle and energy diffusion are taken into account. Many quasi‐linear models of cometary pickup ions exist which involve one type of diffusion or the other but not both types at once. We find that the pitch angle scattering occurs faster than the energy diffusion, as expected. Moreover, our results demonstrate that the distribution of accelerated energetic ions is more isotropic than that of ions which have just been picked up. In fact, the ion distribution function on the initial pickup shell is quite anisotropic, even close to the comet Halley bow shock.

Nonlinear wave interactions and evolution of a ring-beam distribution of energetic electrons

A ring-beam distribution function of moderately relativistic electrons is unstable to electromagnetic and electrostatic waves. The results obtained in numerical simulations show that electromagnetic radiation corresponding to the normal modes of the background plasma is observed to grow even in the presence of a strong electrostatic instability and becomes very strong when the growth of the electrostatic Langmuir waves is minimized, and that the instability process is best described as a beam cyclotron resonance. Another strong radiation generated by the ring beam is the Z mode which is coupled to the electrostatic Langmuir wave. Under certain circumstances, these mechanisms may be significant in astrophysical situations.

Analytical solutions for oblique wave growth from a ring-beam distribution

Analytical solutions are presented for the linear growth rate of oblique plasma waves in a magnetized plasma due to resonant interactions with a model ringbeam distribution. Explicit closed-form solutions for the angular dependence are obtained in terms of modified Bessel functions of the first kind. In the limits of either quasi-longitudinal or quasi-transverse propagation the analytical solutions take the form of simple algebraic expansions, which allow an immediate comparison of the relative contributions from different harmonic resonances, and which also determine the conditions for marginal stability for any specific resonance. The results can be applied, for instance, to the growth of waves following ionization of neutrals originating from cometary, planetary, or interstellar material in the solar wind. In a weakly unstable plasma the analytical results also provide an important check on the complex numerical codes that hitherto constituted the only method available for evaluating the growth of oblique plasma waves.

Excitations of low‐frequency hydromagnetic waves by freshly created ions in the solar wind

Low‐frequency hydromagnetic waves excited by newborn ions in the solar wind plasma are studied. The freshly created ions appear in the solar wind frame with a ring beam distribution. Both Alfvén and fast magnetosonic waves are made unstable by the presence of the newborn ions. The dependence of the growth rate of both waves on the newborn ion density, the angle between the interplanetary magnetic field (IMF) and solar wind flow, and the angle of wave propagation relative to the IMF is investigated. Analytic approximations for the growth rates are presented, and numerical solutions of the dispersion equation are shown. The approximations are quite close to the numerically determined growth rates. We find that the waves grow preferentially in the direction parallel to the IMF and that the growth rates increase with both newborn ion density and the angle between the IMF and the solar wind flow.

Resonant interactions between cometary ions and low frequency electromagnetic waves

We explore the conditions for resonance between cometary pick-up ions and parallel propagating electromagnetic waves. A model ring—beam distribution for the pick-up H 2O + ions is adopted which allows a direct comparison of the source of free energy for growth from either the beam or the gyrating ring in the limit near marginal stability. Under average solar wind conditions in the inner solar system, the gyrating ring provides the dominant contribution to wave growth. The presence of a field-aligned beam is only important to allow resonance with R-mode waves which occur in two distinct frequency bands either well above or below the pick-up ion gyrofrequency. The most unstable mode is the low frequency R-mode or fast MHD wave, though higher frequency whistlers or low frequency L-mode waves may also be excited by the same source of free energy. The nature of the unstable waves is strongly influenced by the inclination α of the interplanetary field. For α ≲ 3° the rate of the low frequency R-mode growth is dramatically reduced and resonant L-mode waves should experience net ion beam damping. Conversely for α ≳ 75°, the ion beam velocity will be insufficient to allow resonant R-mode instability; L-mode waves should therefore predominate. The low frequency fast MHD mode should experience the most rapid amplification for intermediate inclination; 30° ≲ α ≲ 75°. In the frame of the solar wind such waves must propagate along the field in the direction upstream towards the Sun with a phase speed lower than the beaming velocity of the pick-up ions. The waves are consequently blown back away from the Sun and would thus be detected with a left-hand polarization by an observer in the cometary frame. We consider this the most likely mechanism to account for the interior MHD waves observed by satellites over an extended spatial region surrounding comets Giacobini-Zinner and Halley.

Amplification of electromagnetic waves by a ring–beam distribution of moderately relativistic electrons

In many cases of astrophysical interest, important radiation processes have been observed in regions where the plasma frequency is much larger than the cyclotron frequency, ω2e/Ω2e ≫1. It has been found that the induced emission of the eigenmode of the cold background plasma, i.e., ω=(ω2e +k2c2)1/2, by the presence of a ring–beam distribution of energetic electrons, becomes progressively insignificant as ωe/Ωe increases. A new and important unstable beam-cyclotron mode that is excited by the energetic electrons has been found. This instability has relatively large growth rates for oblique propagation. The group velocity of the unstable mode along the ambient magnetic field is approximately equal to the parallel beam velocity. Numerical solutions of the dispersion equation are presented and discussed.