A. R. Barakat

TRANSPORT EQUATIONS FOR MULTICOMPONENT ANISOTROPIC SPACE PLASMAS: A REVIEW

The authors attempt to present a unified approach to the study of transport phenomena in multicomponent anisotropic space plasmas. In particular, a system of generalized transport equations is presented that can be applied to widely different plasma flow conditions. The generalized transport equations can describe subsonic and supersonic flows, collision-dominated and collisionless flows, plasma flows in rapidly changing magnetic field configurations, multicomponent plasma flows with large temperature differences between the interacting species and plasma flows that contain anisotropic temperature distributions. In addition, if Maxwells equations of electricity and magnetism are added to the system of transport equations, they can be used to model electrostatic shocks, double layers, and magnetic merging processes. These transport equations also contain terms which act to regulate both the heat flow and temperature anisotropy, processes which appear to be operating in the solar wind. Also, the authors show that with the appropriate assumptions, the system of generalized transport equations reduces to each of the other major systems of transport equations for anisotropic plasmas that have been derived to date.

Monte Carlo Study of the transition region in the polar wind: An improved collision model

A Monte Carlo simulation was used to study the steady state flow of the polar wind protons through a background of O+ ions. The simulation region included a collision-dominated region (barosphere), a collisionless region (exosphere), and the transition layer embedded between these two regions. Special attention was given to using an accurate collision model, i.e., the Fokker-Planck expression was used to represent H+ - O+ collisions. The model also included the effects of gravity, the polarization electric field, and the divergence of the geomagnetic field. For each simulation, 105 particles were monitored, and the collected data were used to calculate the H+ velocity distribution function fnof;H+, density, drift velocity, parallel and perpendicular temperatures, and heat fluxes for parallel and perpendicular energies at different altitudes. The transition region plays a pivotal role in the behavior of the H+ flow. First, the shape of the distribution function is very close to a slowly drifting Maxwellian in the barosphere, while a “kidney bean” shape prevails in the exosphere. In the transition region, the shape of fnof;H+ changes in a complicated and rapid manner from Maxwellian to kidney bean. Second, the flow changes from subsonic (in the barosphere) to supersonic (in the exosphere) within the transition region. Third, the H+ parallel and perpendicular temperatures increase with altitude in the barosphere due to frictional heating, while they decrease with altitude in the exosphere due to adiabatic cooling. Both temperatures reach their maximum values in the transition region. Fourth, the heat fluxes of the parallel and perpendicular energies are positive and increase with altitude in the barosphere, and they change rapidly from their maximum (positive) values to their minimum (negative) values within the transition region. The results of this simulation were compared with those found in previous work in which a simple (Maxwell-molecule) collision model was adopted. It was found that the choice of the collision model can alter the results significantly. The effect of the body forces was also investigated. It was found that they can also alter the results significantly. Both the body forces and collision model have a large effect on the heat flux, while they have only a small quantitative effect on the lower-order moments (density, drift velocity, and temperature).

Comparison of transport equations based on Maxwellian and bi-Maxwellian distributions for anisotropic plasmas

Because of the relevance to space physics and aeronomy, the authors studied the extent to which transport equations based on both Maxwellian and bi-Maxwellian series expansions can describe plasma flows characterised by non-Maxwellian velocity distributions, with emphasis given to modelling the anisotropic character of the distribution function. For the Maxwellian expansion the authors considered Grads 5-, 13-, and 20-moment approximations, while for the bi-Maxwellian expansion they considered the 6- and 16-moment approximations. To determine the adequacy of a given series expansion, they selected a simple plasma flow problem which possessed an analytic solution so that the velocity distribution functions obtained from the different series expansions could be compared with the exact solution. The comparisons indicate that the Maxwellian-based 20-moment approximation is a reasonable approximation for temperature anisotropies up to T/sub ////Tperpendicular to approximately 2-3, while the bi-Maxwellian-based 16-moment approximation can describe temperature anisotropies as large as T/sub ////Tperpendicular to approximately 20.

Effect of centrifugal acceleration on the polar wind

The ionospheric convection electric fields that occur at high latitudes cause plasma to drift across the cusp region and the polar cap. Since the magnetic field at high latitudes is close to vertical, pointing downward (upward) in the northern (southern) hemisphere, the convecting plasma experiences a centrifugal acceleration as it crosses the polar region because of the diverging magnetic field geometry. The centrifugal force is directly proportional to the mass of the plasma particles, and it is reasonable to ask whether this force has an effect on polar plasma outflow, particularly for the more massive ion O+. To date, a number of studies have addressed this question, but the theoretical models used in these studies were either overly simplified (i.e., neglected processes known to be important in the polar ionosphere) or else did not use appropriate boundary conditions or take account of the time variability of the problem. The results of these prior investigations were often contradictory. In order to overcome the limitations of these earlier studies, we have used a macroscopic particle-in-cell (PIC) code, which is sophisticated in the sense that a broad range of physical processes are incorporated in its description, in conjunction with time-varying boundary conditions obtained from a time-dependent, three-dimensional, hydrodynamic model of the polar ionosphere. This enables us to properly account for the variation of boundary conditions along a flux tube trajectory. Initially, our macroscopic PIC model was solved for steady state conditions. This allowed us to compare results from our code with those of a prior study of centrifugal acceleration that uses a PIC formulation. Also, by obtaining steady state solutions for both low and high electron temperatures, we have been able to directly compare the effects of electron temperature and centrifugal force on the polar plasma outflow, a comparison that a time-dependent simulation might obscure. Then time-dependent PIC solutions were obtained for the plasma in a convecting flux tube, using solutions to a time-dependent, three-dimensional, hydrodynamic model to provide realistic boundary values for the electron and ion temperatures and the H+ and O+ densities and drift velocities along a flux tube trajectory. Both steady state and time-dependent solutions indicate that centrifugal acceleration does not significantly contribute to the loss of plasma from the polar ionosphere.

The effect of wave-particle interactions on the polar wind O+

The escape of the polar wind plasma is an important element in the ionosphere-magnetosphere coupling. Both theory and observations indicate that the wave-particle interactions (WPI) play a significant role in the dynamics of ion outflow along open geomagnetic field lines. A Monte Carlo simulation was developed in order to include the effect of the WPI in addition to the factors that are traditionally included in the ‘classical‘ polar wind (i.e. gravity, electrostatic field, and divergence of geomagnetic field lines). The ion distribution function (fj), as well as the profiles of its moments (density, drift velocity, temperature, etc.) were found for different levels of WPI, that is, for different values of the normalized diffusion rate in the velocity space (D∼⊥j). Although the model included O+, H+ and electrons, we presented only the results related to the O+ ion. We found that (1) both the density and drift velocity of O+ increased with the WPI strength, and consequently, the O+ escape flux was enhanced by a factor of up to 105; (2) The O+ ions could be energized up to a few electron volts; (3) for moderate and high levels of WPI (D∼⊥(O+)>∼1), the distribution function f(O+) displayed very pronounced conic features at altitudes around 3Re. Finally, the interplay between the downward body force, the upward mirror force, and the perpendicular heating resulted in the formation of the “pressure cooker” effect. This phenomena explained some interesting features of our solution, such as, the peak in the O+ temperature, and the formation of “ears” and conics for f(O+) around 2.5Re.

Momentum and energy exchange collision terms for interpenetrating bi-Maxwellian gases

For application to aeronomy and space physics problems involving strongly magnetised plasma flows, the authors derived momentum and energy exchange collision terms for interpenetrating bi-Maxwellian gases. Collision terms were derived for Coulomb, Maxwell molecule, and constant collision cross-section interaction potentials. The collision terms are valid for arbitrary flow velocity differences and temperature differences between the interacting gases as well as for arbitrary temperature anisotropies. The collision terms had to be evaluated numerically and the appropriate coefficients are presented in tables. However, the collision terms were also fitted with simplified expressions, the accuracy of which depends on both the interaction potential and the temperature anisotropy. In addition, the authors derived the closed set of transport equations that are associated with the momentum and energy collision terms.

Trapped particles in the polar wind

The flow of plasma along open field lines at high latitudes is highly variable and depends both on conditions in the underlying ionosphere and thermosphere and on the transport of particles and energy from the magnetosphere. Past attempts to model this time variability have, for the most part, examined the response of the plasma on a stationary field line to certain prespecified boundary conditions and heat sources. While such prespecified conditions may bear some resemblance to what occurs naturally, they are artificial and cannot be expected to yield a truly quantitative understanding of the various physical processes that interact to produce the dynamic polar wind. The present study is one in a series of studies that attempts to eliminate this artificiality by coupling the mathematical description of the polar wind to a three-dimensional time-dependent model of the high-latitude ionosphere. In this study, an individual flux tube of plasma is followed as it moves under the influence of combined corotation and convection electric fields. Boundary conditions at the lower end of the flux tube are obtained from the ionosphere model, which takes into account all significant particle species, chemical reactions, and heat sources that contribute to the state of the ionosphere. A multi-ion macroscopic particle-in-cell code is used to model the plasma in the flux tube. A description of the behavior of H+ and O+ for the altitude range from 2000 km to about 8 Earth radii is obtained as the flux tube moves along the trajectory, which traverses regions of the subauroral ionosphere, dayside and nightside ovals, and polar cap. The goal of the study is to determine the extent to which ion trapping can occur in the polar wind and the effects that collisions, wave-particle interactions, centrifugal acceleration, and varying ionospheric conditions have on the trapped ions. The main conclusion of the study is that O+ trapping is important and it acts to increase the O+ density at high altitudes.

Dynamic features of the polar wind in the presence of hot magnetospheric electrons

A time-dependent macroscopic particle-in-cell (mac-PIC) model was used to study the temporal evolution of the polar wind under the influence of a hot electron population. First, the steady state results of the mac-PIC model were found for a wide range of hot/cold electron temperature ratios and compared with the results of the well-established time-independent semikinetic model, and excellent agreement was found. Second, simulations were conducted to study the temporal evolution of a plasma that was originally in a steady state condition, and then a hot electron population was suddenly introduced. The profiles of the plasma moments again displayed discontinuities, which oscillated with a decreasing amplitude until they reached their steady state values. As the hot electron temperature increased, the oscillation amplitude increased, and the altitude of the discontinuity decreased, while the period of oscillation and decay rate remained essentially unchanged. Third, simulations were conducted for plasma flux tubes as they drifted across the subauroral, cusp, polar cap, and auroral regions. It was found that as soon as the plasma entered the polar cap, the signatures of the hot electrons were observed. The strength of these signatures varied with time owing to the variation in the instantaneous values of the density and temperature of the thermal electrons. After the plasma exited the polar cap the signatures of the hot electrons persisted for a while, and a density bump formed. For more energetic hot electrons the signatures of the hot electrons became more pronounced in the polar cap and persisted longer after the flux tube left the polar cap. The results of this study were shown to explain some interesting features of the polar wind that were observed by the POLAR satellite.

Comparison of Maxwellian and bi-Maxwellian expansions with Monte Carlo simulations for anisotropic plasmas

The authors studied the extent to which Maxwellian and bi-Maxwellian series expansions can describe plasma flows characterised by non-Maxwellian velocity distributions. The problem they considered was the steady-state flow of a weakly ionised plasma subjected to homogeneous electric and magnetic fields, and both polarisation and hard-sphere collision models were used. Monte Carlo simulations provided a basis for determining the adequacy of a given series expansion. They found that, in general, the bi-Maxwellian-based 16-moment expansion for the velocity distribution function is better suited to describing anisotropic plasmas than the Maxwellian-based 20-moment expansion. They also found that for large electric fields the distribution function displays toroidal and bean-shaped characteristics, depending on the collision-to-cyclotron frequency ratio.

Effects of wave–particle interactions on the dynamic behavior of the generalized polar wind

Abstract The effects of wave–particle interactions (WPI) on the plasma outflows at high latitudes was the subject of several studies. Previous attempts to address this problem, for the most part, modeled the response of the plasma on a “stationary” field line to certain pre-specified boundary conditions. However, the horizontal plasma drift across the different regions at high latitudes (i.e., the cusp, the polar cap, the nightside aurora, and the subauroral regions) results both in rapid changes in conditions and in a coupling of the different regions. Here, we used a time-dependent macroscopic particle-in-cell (mac-PIC) model to investigate the dynamic behavior of the “generalized” polar wind with horizontal plasma convection taken into account. A representative magnetic flux tube, extending from 2000 km to ∼8RE, was followed as it crossed the different high latitude regions. The lower boundary conditions were adopted from a 3-D time-dependent hydrodynamic model, while the WPI levels were adopted from observations. The behavior of the generalized polar wind both with and without the WPI, and with and without the presence of magnetospheric electrons was investigated. The comparison of the results for these four cases elucidated the effects of WPI, and their relative importance with respect to those of the magnetospheric electrons. It was found that the WPI influence is more pronounced on the ion temperature, and progressively less apparent on the vertical drift, u(H+), and the density, n(H+), respectively. Also, the effect of the WPI was strongest in the cusp region. In contrast, the presence of magnetospheric electrons was the dominant factor for the evolution of n(H+) and u(H+) in the polar cap. It was also found that the n(H+) and u(H+) rates of change during the transition from the subauroral to the cusp regions were higher than the rates of change during the transition from the cusp to the polar cap.