A nonlinear, collisional, two-fluid model of uniform plasma convection across a field-aligned current (FAC) sheet, describing the stationary Alfvén (StA) wave, is presented. In a previous work, Knudsen showed that, for cold, collisionless plasma [D. J. Knudsen, J. Geophys. Res. 101, 10761 (1996)], the stationary inertial Alfvén (StIA) wave can accelerate electrons parallel to a background magnetic field and cause large, time-independent plasma-density variations having spatial periodicity in the direction of the convective flow over a broad range of spatial scales and energies. Knudsen suggested that these fundamental properties of the StIA wave may play a role in the formation of discrete auroral arcs. Here, Knudsen’s model has been generalized for warm, collisional plasma. From this generalization, it is shown that nonzero ion-neutral and electron-ion collisional resistivity significantly alters the perpendicular ac and dc structure of magnetic-field-aligned electron drift, and can either dissipate or enhance the field-aligned electron energy depending on the initial value of field-aligned electron drift velocity. It is also shown that nonzero values of plasma pressure increase the dominant Fourier component of perpendicular wavenumber.
I. INTRODUCTION
The shear-mode Alfvén wave balances ion polarization current in the direction perpendicular to the background magnetic field with magnetic-field-aligned (i.e., parallel) current carried by electrons. For large wavelengths perpendicular to , the parallel electric field magnitude is insignificant, and the parallel phase speed equals the Alfvén speed (, where is the ion mass and is the plasma density). When becomes small, comparable to the electron inertial length for the case in which ( pressure normalized to magnetic field pressure) or comparable to the ion acoustic gyroradius in higher- plasmas, the parallel electric field magnitude (balanced by electron inertia for and by parallel electron pressure for ) becomes significant and the wave becomes dispersive.1,2 Waves in the low- and high- limits are termed “inertial” and “kinetic,” respectively, and together they are known as a dispersive Alfvén wave. The dispersive Alfvén wave has been studied extensively because of its possible role in accelerating electrons in the Earth’s magnetosphere, particularly the auroral ionosphere, where the inertial wave can generate transient bursts of electron energy at energies up to .3–7 The dispersive Alfvén wave also leads to electron acceleration as a result of interhemispheric field-line resonances having periods of .8,9 See the comprehensive review by Stasiewicz et al.10 The linear dispersion relations for the dispersive Alfvén wave in both the inertial and kinetic limits have been verified in laboratory experiments.11–17
The dispersive Alfvén wave can be described in the context of two-fluid theory. Linear properties are derived by neglecting the nonlinear term of the complete time-derivative operator (). Seyler et al.18 predicted that inclusion of this nonlinear term leads to Alfvén wave steepening, wave breaking, and enhanced wave dissipation. Those authors also found a solution predicting a purely spatial perturbation in plasma density in the zero-frequency limit.
In this paper, we are concerned with the limit . Knudsen19 predicted that in the presence of background, perpendicular, plasma convection through the wave fields, the term alone generates Alfvén-wave-like behavior, with electromagnetic structures evolving in space as a result of the apparent time variation seen by plasma particles convecting across static (electric and magnetic) field structures. Within otherwise large-scale sheets of homogeneous parallel current, the stationary Alfvén (StA) wave imposes structure both on plasma density and on parallel electron energy.
The term “stationary Alfvén wave” was introduced by Maltsev et al.20 to describe a stationary electromagnetic structure resulting from plasma convection past a conducting strip. The application of the term was broadened by Mallinckrodt and Carlson21 to include structuring from a moving field source in a magnetized plasma. The term “Alfvén wing” is used to describe a stationary wave resulting from moving conductors, for example in the case of Io orbiting within Jupiter’s magnetosphere22 or as a result of a conducting tether orbiting with a spacecraft.23 In these cases, the structure of the stationary wave is imposed by the source, although Chust et al.22 argue that additional filamentation can result. In contrast, the nonlinear two-fluid model described here and by Knudsen19 predicts a self-consistent StA wave whose structure arises from intrinsic properties of the current sheet in which the wave forms. In other words, the wave represents the spatial restructuring of a large-scale homogeneous current sheet into a new electromagnetic equilibrium, and does not require a spatially structured source. As with a dispersive Alfvén wave, according to two-fluid predictions, parallel electron thermal pressure and nonzero electron mass lead to a significant parallel electric field associated with the StA wave.
The purpose of the present paper is to extend the fluid theory predictions of a stationary inertial Alfvén wave19 to the finite- case, i.e., to the realm of the stationary kinetic Alfvén wave, and to study its properties in the presence of collisions.
II. STATIONARY ALFVÉN WAVE EQUATIONS
In this study, we present a two-dimensional ( Cartesian plane) wave model describing a sheetlike , stationary , electromagnetic structure embedded within a warm, collisional plasma, oriented obliquely at a small angle , to a uniform background magnetic field . This stationary Alfvén wave pattern is supported by uniform plasma flow , due to a convection electric field , across a magnetic-field-aligned current sheet. Wave equations describing StA waves are derived from the following set of isotropic, time-independent, two-fluid equations:
and
Here, denotes particle species, is the particle charge, is the fluid velocity, is the particle density, is the isotropic thermal pressure, is the particle temperature in energy units , and is the collisional resistivity operator. The set of equations (1)–(4) are closed through the quasineutrality condition . The electron and ion collisional resistivity operators are
where is the ion-neutral collision frequency, is the electron-neutral collision frequency, and is the electron-ion Coulomb collision frequency. The coefficient is a constant with , in accordance with Braginskii.24 The fluid velocity of the neutrals is assumed to be negligible with respect to the fluid velocities of the charged species.
Perpendicular ion drift is represented by
where is the initial ion drift at and represents the sum of the drift and steady-state analog of the polarization drift . The nonlinear variation in ion velocity is not necessarily a small perturbation and is, in general, comparable in magnitude to . Similarly, the perpendicular and field-aligned electron drifts are
and
Here, is the initial field-aligned electron drift at []. The perpendicular electron drift is assumed to result from the drift alone, since the electron polarization drift is smaller than that of the ions by a factor of and may be neglected.
A graphical representation of the assumed Cartesian geometry, including vector directions of background magnetic field and initial particle drifts, for our two-dimensional StA wave model is shown in Fig. 1. StA wave patterns are assumed to be nearly field-aligned, specifically, oriented at a small angle with respect to the background magnetic field such that
Conceptually, is the effective, but nonexistent, parallel wave phase speed. The subscript indicates the magnetic-field-aligned (i.e., parallel) direction with unit vector
The total magnetic field is composed of an assumed uniform background magnetic field in the direction , the magnetic field structure associated with the background current channel , and the wave field , so that . Only terms to first order in , the component of the total field, are retained. Terms of order can be shown to be of order and are consequently neglected. An important correction to the field-aligned component of electric field, proportional to [see Eq. (16)], is retained by including terms to first order in . Using Eq. (9), the differential operators and are transformed into the field-aligned direction,
and
An expression that relates the density of species to its fluid velocity is obtained from the continuity equation, Eq. (2). Equations (10a) and (10b) are used to reduce both the electron and ion continuity equations to differential equations in ,
and
Integrating over the interval yields expressions for the total particle density in terms of both the initial particle density and the components of the particle fluid velocities,
and
Normalized functions of the electron and ion fluid velocities have been defined as and , respectively. Also, is the initial condition at for the function of electron velocity . In this paper, we investigate situations in which the initial ion drift in is dominated by motion, and . Under these assumptions, the term can be neglected with respect to in Eq. (12a). Particle densities are necessarily positive for physically meaningful solutions, thus only solutions for which both and are considered.
The stationary analog of the ion polarization drift is calculated from the ion momentum equation (1). Retaining terms to first order in , the ion momentum equation can be written as
where is the ion gyrofrequency. The definitions of the functions of electron and ion fluid velocity, along with Eq. (12a) and (12b) and the quasineutrality condition, are used to solve Eq. (13) for the stationary analog of the polarization drift,
Equation (14) is written in terms of the dimensionless variables , , , , and . The factor is the initial polarization drift variable and is defined as
In this paper, we limit our study to cases in which the initial ion drift is assumed to be dominated by motion, and so we only consider cases for which .
The parallel electron motion is a response to the parallel component of the electric field,
which is calculated from the electron momentum equation (1),
Solving Eq. (17) for the parallel component of electric field, and reducing it to a one-dimensional differential equation in using Eqs. (8) and (12b) yields
Here, is the electron thermal speed. We see above that the parallel electric field, balanced by electron inertia (first term on the right-hand side), has the opposite sign of the parallel electron pressure term (last term on the right-hand side). This difference was first identified by Goertz and Boswell.1 Also, we see that the finite-electron conductivity term, expressed as in Eq. (18) (middle term on the right-hand side), is out of phase with both electron inertia and electron pressure terms. This phase difference plays a fundamental role in the spatial decay and growth of StA waves. Equation (18) is normalized, and rewritten as a function of ,
Equation (19) is expressed in terms of the normalized parameters and . As a note, representing the electron collision frequency in this manner allows for the role of parallel electron resistivity to be generalized. If we were to consider an effective electron collision frequency as the result of all resistive electron interactions, not just particle collisions, then we could equivalently use
without altering the resulting StA wave equation.
The field-aligned current density is assumed to be carried by the electrons and is given by
Using Eqs. (8) and (12b), the current density is rewritten in terms of the function ,
Here, the current density has been normalized to the current density at .
Differentiating Eq. (18) with respect to and combining Eqs. (3), (4), (10a), (10b), (12b), and (21) and the quasineutrality condition yield a second-order, nonlinear, differential equation for the function ,
Substitution of Eq. (14) into Eq. (23) yields the wave equation for a resistive stationary Alfvén wave,
Equation (24) is coupled to Eq. (14) through the polarization field and, together, represent the closed set of equations describing StA wave modification of field-aligned electron drift. It is important to note that Eqs. (24) and (14) contain the full ion and electron convective nonlinearities and are valid to first order in the magnetic perturbation . We see in Eqs. (24) and (14) that the characteristic scale length for these waves is the electron inertial length, . The characteristic time scale is identified through the dimensionality of the collision frequency and is the time required for a particle to drift the distance of one electron inertial length, .
For simplicity and without loss of generality, the numerical results presented in this paper have been obtained with the assumption that the initial perpendicular ion drift is bulk-plasma drift and thus the ion polarization parameter, Eq. (15), is for all solutions. Furthermore, we only consider solutions for strongly magnetized plasma and ignore ion gyrofrequency effects by letting . With these two assumptions, the coupled set of differential equations describing the stationary Alfvén wave, Eqs. (24) and (14), respectively, reduce to
and
III. NUMERICAL SOLUTIONS
A. General StIA wave characteristics
The general properties of the collisionless StIA wave were first presented in detail by Knudsen19 and so only a brief discussion is given here. Knudsen found that uniform plasma convection across field-aligned current sheets supports the StIA wave, which can accelerate electrons parallel to a background magnetic field and cause large, time-independent, plasma-density variations having spatial periodicity in the direction of the convective flow over a broad range of spatial scales. Figure 2 shows numerical solutions of Eq. (30) for the perpendicular wavelength associated with the dominant spectral wave number , the maximum deviation from the initial value in electron velocity , and total density as a function of the initial field-aligned electron drift speed for three different values of effective parallel wave phase speed . The top panel in Fig. 2 shows that, for a fixed effective parallel wave phase speed, the perpendicular wavelength of StIA waves spans several orders of magnitude, depending on the value of the initial parallel electron drift. It is clear in the top panel of Fig. 2 that a wave “resonance” occurs at . The middle panel shows that the parallel electric StIA wave field either accelerates or decelerates electrons in the parallel direction, depending on the magnitude of the initial parallel electron drift. For initial values of parallel electron drift smaller than the effective wave phase speed , the StIA wave accelerates electrons in the direction of their initial drift, while for , electrons are decelerated. Note that no enhancement in electron energy occurs in the absence of the field-aligned current and when . The maximum density perturbation, seen in the bottom panel of Fig. 2, shows that the StIA wave depletes the particle density for antiparallel electron drift (APED), and enhances the particle density for parallel electron drift (PED). Reference 19 characterizes the general properties of the StIA wave in more detail.
B. Collisional StIA waves
To investigate the effects of electron collisional resistivity on the StIA wave, Eqs. (25) and (26) are solved numerically for , with and . In Fig. 3, the normalized parallel electric field , parallel electron velocity , and particle density are plotted as a function of for initial electron drift and . A family of curves is shown representing solutions for (solid line), (dashed line), (dashed-dotted line), and (dotted line). We see in Fig. 3 that sufficiently frequent electron collisions damp the spatial pattern of fluid variables to dc values equal to the mean of the undamped patterns, i.e., waveforms. Likewise, spatial wave damping increases with increasing electron collisionality until the wave is critically damped spatially. For , the phase of the parallel component of electric field balanced by finite-electron conductivity leads the phase of the electron inertia contribution at , as seen in Eq. (18), causing damping of the spatial pattern and the spatial mean of the field-aligned electric field to reach more-negative values.
In contrast, for , the phase of the resistive force on the electrons lags behind that of the inertial response at , causing the spatial pattern to grow in the perpendicular direction, as seen in Fig. 4. In Fig. 4, a family of curves is shown that represent solutions for (solid line), (dashed line), (dashed-dotted line), and (dotted line). The nonlinear evolution of the phase relationship between the inertial and resistive contributions to the parallel electric field causes the total parallel electric field spatial pattern to steepen. The amplitude of the pattern increases for increasing values of until the wave crests and subsequently collapses. Figure 5 shows the spatial growth rate of the StIA pattern for several values of electron collision frequency, plotted as a function of for a fixed value of . We see, in Fig. 5, that the spatial growth rate becomes discontinuous at , corresponding to the asymptotic approach to zero of the dominant perpendicular wavelength as , as seen in Fig. 2. For antiparallel electron drift, electron collisions only damp the amplitude of the modulated fluid variables, with the growth rate exhibiting a local minimum in its dependence.
By neglecting electron collisions and retaining only the ion collisional resistivity, Eqs. (25) and (26) are solved numerically for with , , , and . Normalized , , and are plotted in Fig. 6 for (solid line), (dashed line), (dashed-dotted line), and (dotted line). We see, in Fig. 6, that ion-neutral collisions, for , damps the wave amplitude while introducing an increasing dc offset in the fluid variables and . This dc offset results from ion collisions continuously reducing the perpendicular wavenumber, broadening the -space wave spectrum. This spreading in space is most clearly visible in the top panel of Fig. 6. Unlike the case of electron collisional damping, the field-aligned component of electric field is damped to a singular negative dc value, irrespective of the ion-collision frequency. Ion-neutral collisional damping of the StIA-wave amplitude is observed for all values of and . Figure 7 is a plot of the spatial growth rate as a function of for a fixed and several values of . The growth rate increases in the negative direction for increasing ion-neutral collision frequency and exhibits a local minimum in its dependence for antiparallel electron drift.
C. Finite plasma- effects on StIA waves
The inclusion of nonzero plasma- gives important warm-plasma corrections to the behavior of the StIA wave. In this paper, we have restricted our study to low-beta plasma in the range . Numerical solutions to the StIA wave equations are obtained for and . In Fig. 8, the dominant perpendicular spectral wavelength is plotted as a function of for several values of and . We see in Fig. 8 that, for antiparallel electron drifts, increasing finite plasma- significantly reduces the dominant wavelength at small values of , whereas remains relatively unaffected at large . For increasing parallel electron drift, the dominant perpendicular wavelength also reduces, with the most significant reduction occurring as approaches the value of . The inclusion of finite plasma- was found to produce an increase in the amplitude of the nonlinear modulation of the fluid variables by less than a few percent over several orders of magnitude change in . The role of the parameter on the spatial growth and decay rates of StIA waves is beyond the scope of this paper and will be the topic of future work.
D. Kinetic StA waves
The stationary kinetic Alfvén (StKA) wave is a solution to the StA wave equations, for which (i.e., the sound gyroradius is larger than the electron inertial length ) and . In this section, the general properties of the StKA wave are discussed and several key differences between the StKA wave and the StIA wave are highlighted. Similar to the StIA wave, the StKA wave carries a nonzero parallel component of electric field. This parallel electric field is balanced by the parallel electron pressure gradient [see the last term in Eq. (18)], and can energize particles along magnetic field lines. Figure 9 shows solutions to the collisionless StA wave equations for normalized quantities , , and as a function of with . In the top panel of Fig. 9, we see that the dominant perpendicular wavelength increases with increasing and decreases with increasing . The functional dependence of on and is the opposite of what was found for the case of the StIA wave, cf. Fig. 2. The middle panel of Fig. 9 shows that the StKA wave can accelerate electrons to parallel speeds much larger than the local Alfvén speed. Unlike the StIA wave, however, the StKA wave modulates the direction of the electron flow. Thus, the StKA wave creates adjacent, alternating polarity, filamentary structures of field-aligned current. The StKA wave also produces large density depletions or enhancements, like the StIA wave, which are dependent on the direction of the initial parallel electron drift, cf. the bottom panel in Fig. 9. It is important to note that studying kinetic effects using a two-fluid model is limited in its range of applicability. For the two-fluid model to be valid for our study of the StKA wave, we require that , which is equivalent to the condition that . For all cases of the StKA wave presented here, we let . Thus, our study is restricted in parameter space to StKA wave solutions for which .
E. Collisional StKA waves
For nonzero electron collisions, numerical solutions to the StA wave equations with for , , and are plotted in Fig. 10. Solutions for (solid line), (dashed line), (dashed-dotted line), and (dotted line) are shown. It is clear from Fig. 10 that parallel electron conductivity causes the amplitude of the StKA wave pattern to grow spatially. This wave growth is being generated by the initial phase lag between the parallel electric field, balanced by finite electron conductivity, and the parallel thermal pressure. This mechanism is similar to the case for the StIA wave. The growth rate clearly increases with increasing collision frequency. Similar to a spatially growing StIA wave, StKA wave growth continues until the parallel electric field steepens and crests, at which point the wave collapses (not shown). Figure 11 shows the spatial growth rate of the StKA wave as a function of for both parallel and antiparallel electron drift. In both cases, we see that the growth rate increases with increasing while exhibiting only a weak dependence on . No wave damping is observed for nonzero values of electron collisionality.
Ion-neutral collisions can also produce growth in StKA wave amplitude, as seen in Fig. 12. In Fig. 12, electron collisions are neglected, and is set to unity. Numerical solutions to the StA wave equations for , , and are shown for (solid line), (dashed line), (dashed-dotted line), and (dotted line). As in the case of electron collisions, the StKA wave amplitude grows until the parallel component of the electric field crests, collapsing the wave. Ion-neutral collisional enhancements to the polarization current, for large in the kinetic Alfvén wave regime, can be sufficient to cause dissipation of the StKA pattern, as shown in Fig. 13. Like the case of ion-neutral collisional damping of the StIA wave, the value of the dominant perpendicular wavelength in Fig. 13 varies continuously in space, resulting in a continuously changing -space wave spectrum. For the StKA wave, however, this effect is much weaker. In Fig. 14, the growth rate of the StKA wave is plotted as a function of the normalized effective parallel wave phase speed . For antiparallel electron drift, we find that the growth rate is always positive and decreasing with increasing . The growth rate in the case of parallel electron drift, however, becomes negative at large values of . The ratio of ion to electron temperature does not change the general characteristics of the StKA wave. The effects of on the collisional StKA wave is beyond the scope of this paper, and will be the topic of future work.
IV. DISCUSSION
A. Source of free energy and wave growth
The source of free energy which is responsible for the stationary Alfvén wave is the background magnetic-field-aligned current . As seen in Figs. 2 and 9, in the absence of the field-aligned current the wave is stable and no wave perturbation forms. To show that these waves are stable in the absence of the background field-aligned current, we write Eq. (25) in a generalized form assuming that ,
where the characteristic pattern is
and the driving term on the right-hand side of Eq. (27) is
Here, we see that the driver is spatially and temporally constant and depends on the sources of free energy, i.e., the initial parallel current in the form of and the perpendicular plasma drift . Also, the characteristic pattern is influenced by the free energy source in terms of . When the initial parallel current is turned off, i.e., and hence , the driving term is zero and without an initial perturbation, the StA wave pattern will not form. Thus, in the absence of the background-parallel-current free-energy source, the StA wave is stable. This is not strictly true, however. For , an initial ion polarization current also provides a source of free energy capable of supporting the StA wave.
The damping term shows explicitly that the spatial wave-form is damped when and grows when . In order to understand how parallel electron conductivity leads to wave growth, it is useful to use the parallel electric field, Eq. (18),
For inertial waves, i.e., the case , the electric field, balanced by electron inertia, leads the resistive term when . The nonlinear relative-phase difference causes the electrons traveling along the magnetic field to spend more time in the accelerating field than in the decelerating field, leading to a spatially growing field-aligned current and a spatially growing StIA wave. Conversely, when , the electrons spend more time in the decelerating field than in the accelerating field, leading to a spatially damped parallel current. In the kinetic regime, the parallel electric field balanced by the parallel electron pressure dominates over, and is 180° out of phase with, the electric field balanced by the electron inertia. Again this phase difference was first identified by Goertz and Boswell.1 So, when , the pressure force leads the resistive force and the nonlinear relative phase difference causes the electrons to spend more time in the accelerating field, leading to a field-aligned current that grows spatially in amplitude and a spatially growing StKA wave. The source of free energy available for the spatial growth of the StKA wave amplitude is the bulk plasma drift , generated by the convection electric field .
Spatial damping and growth of the StA wave pattern by ion-neutral collisions occurs through damping/growth of the ion polarization current. The steady-state analog of the ion polarization drift is given by Eq. (14). For simplicity, we assume that , and ignore the ion thermal pressure reducing Eq. (14) to
Substitution into Eq. (23) for the case and neglecting electron collisions yields
Here, is the ion acoustic gyroradius. In the inertial regime , the ion polarization drift is in the opposite direction to the bulk plasma drift , and the ion-neutral collisional drag force damps the polarization current. This spatial damping of the ion polarization current damps the StIA wave. Hence ion-neutral collisions are purely dissipative in the inertial regime. In the kinetic regime , the ion polarization drift is in the same direction as . During the first half-period of the wave, ion-neutral collisional drag acts to damp the spatial pattern (see Fig. 12). As position increases in the direction, ion-neutral collisions inevitably reduce the net value of ion flow below changing the direction of the cross-field current, and in doing so, the direction of the field-aligned component of electric field. This results in growth during the second half-wavelength of the wave, and causes net spatial growth of the amplitude of the StKA wave-form. The free-energy source for this wave growth is the plasma drift .
B. Linear limit of StA wave equation
To illustrate the connection between our model and previous work, we compare some limiting cases of our model with related models of the dispersive Alfvén wave. First, in the cold-plasma collisionless limit, the wave equation for the stationary inertial Alfvén wave, first derived by Knudsen,19 is recovered,
In the linear, or small-amplitude , limit, retaining electron collisions and neglecting initial field-aligned electron drift , Eq. (25) reduces to
Here, is the acoustic gyroradius, with being the sound speed. By Fourier transforming Eq. (31) and substituting the relation from the Fourier transform of Eqs. (10a) and (10b), Eq. (31) can be written as
For the inertial Alfvén wave in collisionless plasma, Eq. (32) reduces to the zero-frequency limit of Eq. (5) in Drozdenko and Morales,25 describing the linear effects of cross-field plasma flow on a field-aligned current channel,
Recognizing that is the wave frequency, as observed by the convecting plasma, allows Eq. (32) to reduce to Eq. (18) of Morales and Maggs,26
If collisions are neglected and electron inertia is included, Eq. (32), with , will represent the dispersion relation for the low-frequency dispersive Alfvén wave including finite Larmor radius effects,10
Setting and , we see that the parallel wave phase speed is simply the Alfvén speed,27 i.e., .
C. Applications to auroral ionosphere
Before the StA wave model can be applied to predicting details of auroral arc formation, several important effects not represented in the present form of the StA wave model must be evaluated and possibly incorporated into the model.
One potentially important effect in both the auroral ionosphere and a laboratory experiment that remains unaddressed by the present model is wave reflection at a conducting boundary. StA wave solutions are nonlinear and, thus, wave reflection might not involve simple superposition. There is no obvious reason to expect that the interference produced by a reflected StA wave would inhibit the electron acceleration and plasma density variation induced by the wave, particularly since, in the linear limit, the parallel electric field associated with the inertial Alfvén wave may constructively interfere upon reflection from a conducting boundary such as the ionosphere.1
In addition to wave reflections, the present StA wave model does not include background parallel and perpendicular inhomogeneities. The strong inhomogeneities present in the auroral ionosphere may make the establishment of a StA wave propagating at a fixed angle to the background magnetic field nontrivial. Of particular importance is the functional dependence of the electron and ion collision frequencies with altitude. We have shown that both Coulomb and ion-neutral collisions can significantly alter the StA wave-form and so inhomogeneities in the collision frequencies would have to be addressed before making any direct applications of the collisional StA wave to the auroral zone. Predicting the details of interference and reflection as well as parallel and perpendicular inhomogeneities may be accomplished more conveniently via computer simulation.
Although an auroral arc often appears as a periodic structure, i.e., multiple parallel discrete arcs, it can also appear as a solitary structure. The solutions to the StA wave equations, presented previously, pertain to a plasma that is unbounded and predict the periodic nature of the StA wave. Solitary solutions to the StA wave equations have not been found. One explanation for the absence of a solitary wave solution is the possible restriction in the range of wave-number values imposed by the finite size of a naturally occurring current sheet. Another explanation is model incompleteness, which motivates our future efforts on the StA wave.
V. SUMMARY
In this paper, we predict how the inclusion of ion-neutral collisional resistivity, parallel electron resistivity, and nonzero plasma affects the wave properties of the stationary Alfvén wave. The parallel component of electric field associated with the stationary Alfvén wave accelerates electrons parallel to the magnetic field and creates either density enhancements or depletions depending on the direction of the initial parallel electron drift. Ion-neutral and electron-ion collisional resistivity are predicted to both significantly alter the perpendicular ac and dc structure of magnetic-field-aligned electron drift and either dissipate or enhance the field-aligned electron energy depending on the initial value of parallel electron drift velocity. Nonzero plasma is predicted both to reduce the dominant perpendicular wavelength of the StIA wave structure and to have minimal effect on the wave amplitude. With the inclusion of resistive effects and nonzero plasma , the conclusions of Knudsen’s cold, collisionless case are unchanged. However, unlike the StIA wave, the StKA wave, occurring at higher beta, modulates the direction of parallel electron flow, creating adjacent, alternating polarity, current filaments. The electric field produced by finite parallel electron conductivity supports the parallel StKA wave electric field, and enhances the parallel electron energy. Ion-neutral collisional resistivity is predicted to either enhance or dissipate the parallel electron energy, depending on the magnitude of the initial parallel electron drift.
ACKNOWLEDGMENTS
Useful discussions with S. Vincena, W. Gekelman, J. Maggs, G. Morales, and M. Goldman are gratefully acknowledged. This work is part of a dissertation to be submitted by S. M. Finnegan to the Eberly College of Arts and Sciences, West Virginia University, Morgantown, WV, in partial fulfillment of the requirements for the Ph.D. degree in physics.
This work is supported by a U.S. NSF/DOE grant at WVU (Grant No. PHYS-06). D. J. Knudsen is supported by the National Research Council of Canada. This work is part of the Space-Plasma Campaign being carried out by M. E. Koepke, S. M. Finnegan, D. J. Knudsen, R. Rankin, R. Marchand, C. Chaston, and S. Vincena at the Basic Plasma Science Facility, University of California, Los Angeles and thus receives part of the facility support from the NSF and the DOE.