Kinetic modeling of the polar wind

ARTICLE IN PRESS

Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) 1984–2027 www.elsevier.com/locate/jastp

Kinetic modeling of the polar wind S.W.Y. Tama,b,, T. Changa, V. Pierrardc a

Massachusetts Institute of Technology, Kavli Institute for Astrophysics and Space Research, Cambridge, MA 02139, USA Plasma and Space Science Center, College of Sciences, National Cheng Kung University, 1 Ta-Hsueh Road, Tainan 70101, Taiwan c Belgian Institute for Space Aeronomy, 3 av. Circulaire, B-1180 Brussels, Belgium

b

Accepted 31 July 2007 Available online 19 August 2007

Abstract The polar wind, a plasma outflow along open geomagnetic field lines from the Earth’s ionosphere, has been the subject of many theoretical studies. Since the proposal of its existence more than 30 years ago, the polar wind has been described by various theoretical modeling approaches. In this paper, we first describe the two traditional modeling approaches for the classical polar wind: moment-based models and collisionless kinetic calculations; and discuss how other models of the outflow were formed by combining these two traditional approaches. We then discuss how kinetic calculations have been improved to take into account the effects of Coulomb collisions, which may contribute to the formation of double-peaked H+ velocity distributions in the polar wind. As observations of escaping O+ fluxes have guided theoreticians to go beyond the classical polar wind description, we discuss how models were modified or constructed in order to take into account the various non-classical effects, including those due to photoelectrons, which may also provide an explanation for the satellite observations of day–night asymmetries in addition to the acceleration of the O+ ions. r 2007 Elsevier Ltd. All rights reserved. Keywords: Kinetic models; Fluid models; Hybrid models; Coulomb collisions; Ion acceleration; Photoelectrons

1. Introduction Plasma flows along open magnetic field lines are a very common occurrence in space. General examples of these include galactic jets, stellar winds (the solar wind in particular) and magnetospheric outflows originating from the polar regions of planetary ionospheres. For the Earth, the existence of such a plasma outflow from the polar ionosphere Corresponding author. Plasma and Space Science Center,

College of Sciences, National Cheng Kung University, 1 Ta-Hsueh Road, Tainan 70101, Taiwan. Tel.: +886 6 2383399x512. E-mail addresses: [email protected], [email protected]. edu.tw (S.W.Y. Tam). 1364-6826/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jastp.2007.08.006

was first proposed by Axford (1968) and Banks and Holzer (1968). This outflow was termed the ‘‘polar wind’’ in analogy with the solar wind, which had just been theorized and observed (Parker, 1958; Bonetti et al., 1962, 1963; Neugebauer and Snyder, 1962; Snyder et al., 1963). The purpose of this paper is to review some of the theoretical models in polar wind research. Overall, there have been numerous theoretical studies on the topic, many of which motivated by observations in the polar cap region. Observations of the polar wind are reviewed by Yau et al. (2007) and Lemaire et al. (2007) in this issue. Here, we highlight some of the observations that revealed the characteristics of the polar wind in order for the reader to understand

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the motivations behind some of the models to be discussed. The polar wind was postulated to be a supersonic outflow of light ions. However, the supersonic nature of the phenomenon could not be confirmed until the measurements of H+ ions by the Dynamics Explorer (DE) 1 satellite (Nagai et al., 1984). The role of the O+ in the polar wind was unclear for a long time. Traditionally, the heavy ions were believed to be gravitationally bound. But suprathermal O+ ions with supersonic speed were observed in the polar cap magnetosphere by the DE-1 satellite (Gurgiolo and Burch, 1982; Waite et al., 1985). However, the observations could not account for the source of those O+ ions; they could be from the polar ionosphere, or from elsewhere. It was not until the observations by the Akebono satellite that the role of the heavy ions in the polar wind was resolved. Measurements by the satellite revealed that the polar wind O+ ions attain supersonic speed, and dominated over the H+ as the major ion species even at altitudes as high as 11,000 km (Abe et al., 1993). The satellite also observed asymmetric outflow velocities for both the H+ and O+ in regard to the dayside and nightside sectors, where the speeds are significantly higher in the sunlit region. As for electron observations, measurements by the DE-1 and -2 satellites indicated that photoelectrons have a significant impact on the electric potential along polar wind magnetic field lines (Winningham and Gurgiolo, 1982). Because the electric potential also governs the outflow of the ions, the observations implied that the existence of photoelectrons may strongly affect the dynamics of the polar wind. More recently electron measurements by the Akebono satellite also provided evidence that was consistent with the idea of photoelectrons playing an important role in the polar wind. Such observations included a net upward electron heat flux in the dayside polar wind (Yau et al., 1995), and a day–night asymmetry—in addition to that of the ion outflow velocities mentioned above—of the existence of an electron anisotropy: On the dayside, the upward-moving electron population was observed to have a higher temperature in the field-aligned direction compared with its downward-moving counterpart (Yau et al., 1995); such an up–down anisotropy, however, was seemingly absent on the nightside (Abe et al., 1996). Early theoretical studies of the polar wind by Banks and Holzer (1968, 1969b) examined the outflow under ‘‘classical’’ conditions, where the

1985

cold plasma (2000 K) in the ionosphere provided the only particle source for the steady-state, quasineutral, current-free outflow along open divergent magnetic field lines. Based on the results generated from a system of hydrodynamic equations, coupled by a field-aligned electric field and frictional forces among the species, Banks and Holzer (1968) showed that the H+ and He+ ions could reach supersonic speeds and escape the polar ionosphere. They attributed the ion acceleration to hydrodynamic frictions. However, they also pointed out that in the presence of heavier ions such as the O+, the electric field would increase due to the larger charge separation between the ions and the electrons. (More accurately speaking, the electric field was present to counteract the effect of the gravitational force, keeping the ions and the electrons from separating.) The enhanced electric field would also contribute to the acceleration of the lighter ions. The O+, due to their heavier mass, however, would not gain enough energy to overcome the gravitational potential barrier according to their calculations; its density was assumed to follow a profile in hydrostatic equilibrium. Banks and Holzer’s hydrodynamic model and their interpretation of the ion acceleration were soon challenged. Dessler and Cloutier (1968), believing that their evaporative theory describing the outflow of neutral hydrogen atoms would also apply for the H+, argued against the validity of the hydrodynamic model above the exobase, the altitude at which the Coulomb collision mean free path of H+ equals the density scale height of the O+. They also disagreed with Banks and Holzer by suggesting that the escape of the H+ and He+ ions out of the polar ionosphere was due to the field-aligned electric field, rather than the frictional forces. In a letter of criticism of the hydrodynamic formulation of the polar wind, Dessler and Cloutier (1969) presented an alternative model for the outflow. Their model, considered the first polar wind model based on a kinetic approach, assumed that O+ was the dominant ion species in the ionosphere. A polarization electric potential was calculated based on the charge separation between the electrons and the O+ ions, under the assumption that the two species had the same temperature. When calculating the velocities of the light ions (H+ and He+), the individual velocities of the particles were ignored; one bulk velocity was used to approximate each of the entire light ion species— an assumption strongly objected by Banks and

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Holzer (1969a) in their reply to Dessler and Cloutier’s letter. In the low-altitude collision-dominated regime, the upward drift velocities of the light ions due to the net action of the electric field and gravity were estimated based on the 901 deflection time for their Coulomb collisions with the O+. Above the exobase, the ions would be collisionless and their bulk velocities were affected only by the polarization electric field and gravity. (For a more quantitative review of this model, we refer the reader to the article by Lemaire et al. (2007) in this issue.) The results of Dessler and Cloutier (1969), like those by Banks and Holzer (1968), showed the transition of H+ from subsonic to supersonic speeds. Dessler and Cloutier claimed that in both models the transition and most of the H+ acceleration occurred above the exobase, where Coulomb collisions were not important; thus the frictional forces could not be the major mechanism for the acceleration. Based on that argument, they further claimed that their approach was more correct and more straightforward. Banks and Holzer (1969a), however, refuted their argument by showing that the majority of the H+ acceleration in the hydrodynamic calculations in fact took place below the exobase, reasserting the idea that frictions were the main acceleration mechanism of the polar wind. The debate between Banks & Holzer and Dessler & Cloutier signified the division of traditional polar wind modeling into two schools of thought: the moment approach exemplified by the hydrodynamic model, and the collisionless kinetic approach demonstrated by the treatment of the H+ ions above the exobase in Dessler and Cloutier’s calculations. In Section 2, we shall discuss these two fundamental approaches, and review the formulations of some classical polar wind models that utilized one or both of these approaches. Coulomb collisions are one of the important mechanisms that affect the transport and dynamics of the polar wind. However, their effects are obviously missing in the collisionless kinetic description. In Section 3, we shall discuss how some models have managed to incorporate the effects of Coulomb interactions into a kinetic description. As mentioned earlier, measurements by the DE satellites (Winningham and Gurgiolo, 1982) have shown that the existence of photoelectrons has significant effect on the electric potential, which may in turn affect the transport and dynamics of the polar wind. In fact, Axford (1968) had suggested that the presence of photoelectrons might lead to

the increase of ion flux, including that of the O+. However, except for the work by Lemaire (1972a) and Lemaire and Scherer (1972b), this proposed acceleration mechanism had received little attention from polar wind theoreticians before the DE observations regarding photoelectrons. In this paper, besides the discussion of the development of classical polar wind models, we shall also review some of the modeling efforts for non-classical polar wind processes such as photoelectron effects. We would like to note that the results of some of the theoretical studies in our discussion have been reviewed in articles by Schunk (1986) and Ganguli (1996). However, unlike the previous polar wind review articles, which focused on the results of those theoretical studies, our emphasis will also be on the models; we will discuss the modeling approach, assumptions and methodology. In particular, regarding the methodology of the models, we will focus on those which were fully or partially based on a kinetic approach.

2. Theoretical modeling of the classical polar wind Most of the early polar wind theories were on the classical polar wind, in which the cold plasma from the ionosphere provides the only particle source for the steady-state outflow along the open divergent geomagnetic field lines. The plasma is quasi-neutral and the flow without a field-aligned electrical current. It consists mainly of O+, H+ and electrons. At altitudes above 500 km, the densities of the neutral particles are low enough so that it is justifiable to neglect ‘‘chemical’’ reactions such as photoionization, recombination and charge exchange. Although there were polar wind models that considered the outflow from below 500 km and thus included those ‘‘chemical’’ reactions, we shall limit our discussion here, unless otherwise noted, to altitudes where those reactions are negligible, rendering Coulomb interactions as the dominant type of collisions. The magnitude of the geomagnetic field for the polar wind is sufficiently strong that the gyration periods and Larmor radii of all the particle species are much smaller than the relevant transition time scales and scale lengths of the plasma. Therefore, the guiding center approximation can be applied when modeling the polar wind. In addition, the polar magnetic field lines are often assumed to be radial, with the profile of the field following that of a dipole configuration, i.e., Bs3, where B is the

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magnetic field and s is the geocentric distance along the field line. It is assumed that only the transport and dynamics along the geomagnetic field lines is important in the classical polar wind; the perpendicular drift is not considered. Because the polar cap, in general, is a relatively quiescent region, wave– particle interactions (WPI), whose effects are assumed negligible, are considered non-classical mechanisms in polar wind studies. The acceleration of the ions is directly related to the electric field component EJ, which is parallel to the geomagnetic field. For the polar wind, the physical effects that influence EJ can be categorized into two different kinds. First, the particle distributions in the polar wind, in general, are not isotropic. With anisotropic particle species streaming in an inhomogeneous magnetic field, a parallel electric field is required to maintain quasi-neutrality throughout the flow. As will be shown in Section 5, this ‘‘ambipolar’’ effect that arises due to the anisotropy of the particle distributions can be a major contributor to EJ when photoelectrons are present in the polar wind. In the case where the particle distributions are isotropic, a parallel electric field still exists because of the wellknown ‘‘polarization’’ effect, which occurs when there is a force that would otherwise produce a separation between the ions and the electrons. In the polar wind, the large disparity of the gravitational force experienced by the ions and the electrons contributes to this polarization effect. Because of the heavier mass of the O+ ions, when their concentration is higher, this effect is expected to result in a larger upward electric field. As both of the effects discussed above can contribute to the polar wind electric field, we generalize the term ‘‘ambipolar electric field’’ to describe EJ that is due to a combination of both the ambipolar and polarization effects; that is, when particle anisotropy exists. We reserve the term ‘‘polarization electric field’’ specifically for models without particle anisotropy. Based on the assumptions and properties of the classical polar wind discussed above, the spatial dimensions in the formulation of the outflow can be reduced to include only the direction along the geomagnetic field line. Moreover, under the gyrotropic approximation, the velocity space transverse to the magnetic field is characterized only by v?, the magnitude of the perpendicular velocity. With these approximations, the phase-space variables reduce to (s, vJ, v?), where s is the distance along the field line

1987

and vJ is the velocity parallel to the magnetic field. The time-dependent distribution function f j ðt; s; vk ; v? Þ for a given particle species j in the outflow is governed by the following collisional kinetic equation:    df j qj q q q þ vk  g  E k  qt qs qvk dt mj  0  df j B q vk q , ð1Þ v2?  fj ¼ 2B qvk v? qv? dt where qj and mj are the electric charge and mass of the species, respectively, g is the gravitational acceleration, B0  dB=ds is the gradient of the magnetic field and df j =dt represents the rate of change of the distribution function due to collisions. Eq. (1) is a Boltzmann equation whose terms take into account the various major forces or accelerations a particle experiences as it travels along the geomagnetic field line. The terms with g and EJ describe, respectively, the accelerations due to the gravitational and field-aligned electric fields. The two terms that follow take into account the effect due to the mirror force, which causes a particle with velocity (vJ, v?) to experience changes in   those quantities according to dvk =dt ¼  v2? =2 B0 =B and dv? =dt ¼ ðvk v? =2ÞðB0 =BÞ. The term on the right-hand side of Eq. (1) is associated with the effects due to collisions. Such effects on the particle distribution may also be expressed in term of a collisional operator C on fj, i.e.: df j df j ¼ ¼ Cf j . (2) dt dt If we consider the polar wind only at altitudes above 500 km, then the ‘‘chemical’’ reactions among the particles would be negligible, and C would reduce to a Coulomb collision operator. To obtain a complete solution of the polar wind requires solving the following set of equations: one for each particle species based on Eqs. (1) or (2), coupled by the equations for the constraints of quasi-neutrality and current-free flow, i.e.: X qj nj ¼ 0 (3) j

and X

qj nj uj ¼ 0,

(4)

j

where nj and uj are the density and average flow velocity of the species j. Eqs. (1) or (2) for the different species are further coupled by EJ and the

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fj’s in the collision terms characterizing the interrelationship among the particle species. Obviously, such a solution, even with a simplified steady-state version of the equations, is difficult to obtain. Further approximations thus seem necessary. The two fundamental approaches of modeling the polar wind—moment equations (e.g. Banks and Holzer, 1968; Schunk, 1977) and collisionless kinetic calculations (e.g. Dessler and Cloutier, 1969; Lemaire and Scherer, 1970)—represent different types of approximations to the kinetic collisional approach embodied in Eq. (1). 2.1. Fundamental approaches 2.1.1. Moment approach Moment-based models describe the particle transport using a few quantities of the species, such as the density, velocity, temperature and heat flux. The equations that describe these quantities are obtained by taking velocity moments of Eq. (1). Due to the convective term vk q=qs in Eq. (1), the left-hand side of each moment equation contains a velocity moment term of the next higher order. An approximation needs to be made in order to close the system of equations. Depending on how the closure assumption is made, moment-based models can be sub-divided into two different classes: hydrodynamic and generalized transport models. In a hydrodynamic model, the system, whose variables are described by the ‘‘standard’’ transport equations such as the continuity, momentum and energy equations, is closed by imposition of a relationship between the variables. For instance, the early applications of the hydrodynamic approach to the polar wind by Banks and Holzer (1968, 1969b) and Marubashi (1970) included the steady-state continuity and momentum equations (q=qt  0) while closing the system with an assumption of isothermal profiles for the particle species. The steady-state hydrodynamic model by Holzer et al. (1971) consisted of additional transport equations for the parallel and perpendicular temperatures of the ion species (H+ and O+); the system was closed with imposed expressions of the heat fluxes in terms of lower-order moments, based on estimations of energy transfer under the consideration of the collision mean free path. Gombosi and his collaborators (Gombosi et al., 1985, 1992; Gombosi and Killeen, 1987; Gombosi and Nagy, 1988, 1989; Gombosi and Schunk, 1988) closed the system of continuity, momentum and energy equations for the

isotropic H+, O+ and electrons with the assumption that the heat flow was due to thermal conduction, i.e., opposite and proportional to the temperature gradient. The system of equations was mainly intended to investigate a non-classical timedependent polar wind, but also had application in the classical polar wind (Gombosi et al., 1992). The models mentioned above, with the exception of that by Holzer et al. (1971), which was intended to investigate the collisionless regime of the polar wind, had lower boundaries at altitudes where ion production and loss processes are important. Those studies, therefore, considered such ‘‘chemical’’ processes in addition to the effects of Coulomb collisions. The reaction rates of some of the most typical ion production and loss processes in the ionosphere are summarized in Gombosi (1998). The Coulomb collisional effects on the moment quantities were expressed in terms of friction between the particle species, characterized by a number of collision frequencies (Holt and Haskell, 1965; Burgers, 1969). A simplified collisionless hydrodynamic model, consisting of the continuity and momentum equations and closed by an equation of state relating the temperature and density, has also been applied to study the temporal behavior under a density perturbation in the polar wind (Singh and Schunk, 1985; Gombosi and Schunk, 1988). The generalized transport approach, unlike the hydrodynamic models, is based on the formulation proposed by Grad (1949). The formulation utilizes the idea that at zeroth-order approximation the distribution function is a drifting Maxwellian distribution whose lowest-order moments corresponded to the actual density, n(r, t), average velocity, u(r, t) and temperature, T(r, t), of the particles; furthermore, it is postulated in Grad’s as well as Chapman–Enskog expansion that the higher-order expansion of the velocity distributions do not affect the values of these lower-order moments n, u and T. Higher-order terms are added to the zeroth-order approximation as modifications of the distribution function. These higher-order modifications are small correction terms based on the infinite series of the Hermite polynomials, which is assumed to span a complete set in functional space. These modification terms do not change the values of n, u and T. Closure of a generalized transport system relies on truncation of the series. By matching with the ‘‘standard’’ transport equations, coefficients for the terms in the truncated series are determined,

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giving rise to velocity distribution functions that depend on a number of parameters representing the various moments of the particle species. Depending on where the series is truncated, these velocity distributions consist of a different number of moments. By taking the moments of Eq. (1) based on these prescribed velocity distribution functions, one obtains a closed set of transport equations for the parameters, where the collision terms are characterized by velocity-averaged collision frequencies (Burgers, 1969) that depend on those parameters themselves. The most basic generalized transport model consists of five moments (density, three components of the drift velocity and the isotropic temperature), approximating the velocity distribution function in the form of a Maxwellian. More sophisticated generalized transport models based on a slight deviation from a Maxwellian distribution were discussed in detail by Schunk (1977). These models, depending on the approximation, may consist of 8, 10, 13 or 20 moments although it was pointed out that the system for the 20-moment approximation might be too large for practical use. However, with the gyrotropic property of the polar wind, the number of variables in these models can be reduced dramatically. For instance, the 13-moment approximation (Schunk and Watkins, 1981, 1982; Mitchell and Palmadesso, 1983), in practice, consists of only five independent, non-vanishing moments for each species: density, flow speed, parallel and perpendicular temperatures, and heat flow. A detailed discussion of the 13-moment transport equations can be found in Schunk and Nagy (2000). The 13-moment approximation was first applied to the polar wind by Schunk and Watkins (1981) in order to describe the anisotropy of the electrons. Because the model was based on a zeroth-order isotropic drifting Maxwellian distribution, pressure or temperature anisotropy could only be introduced through the higher-order modification terms to the velocity distribution function. Thus, it was natural to develop a generalized transport model based on a drifting bi-Maxwellian velocity distribution, as anisotropy is already built in at the zeroth-order approximation. That led to the applications of the bi-Maxwellian-based 16-moment approximation to the polar wind (Ganguli et al., 1987; Demars and Schunk, 1989, 1994). In this generalized transport system, the velocity distribution function for a particle species is assumed to be of the following form (Demars and Schunk, 1979; Barakat and

1989

Schunk, 1982): f ¼ f B ½1 þ C,

(5)

where the subscript j denoting the particle species has been dropped to simplify the notation henceforth, !

"

2

mðv  uÞ2? mðv  uÞk exp  fB ¼ n  p ffiffiffiffiffiffi 2T ? 2T k ð2pÞ3=2 T ? T k m3=2

#

(6) is the zeroth-order bi-Maxwellian distribution function, n, u, TJ and T? are parameters interpreted as the density, drift velocity, parallel and perpendicular temperatures of the species respectively, and the term C, whose expression is given in Appendix A, contains parameters that are related to the heat fluxes in addition to those above. Because all the velocity moments higher than the heat flux are absent in Eqs. (5) (6) and (A.10), closure of the system is achieved when no new variable is introduced in the heat flux equations. Notice that because of the gyrotropic nature of polar wind transport, the 16-moment system reduces to only six ^ parameters: n, u, TJ, T?, qJ and q?, where u ¼ u  b, ^ ^ ^ qk ¼ qk  b and q? ¼ q?  b, with b defined as the unit vector in the direction of the magnetic field and qk;? defined in Eqs. (A.4) and (A.5). We note that u is physically interpreted as the field-aligned component of the drift velocity, and qk;? as the parallel and perpendicular heat flows, or the parallel and perpendicular heat flux per unit density. The physical quantities that correspond to the interpretations of the six parameters have the following definitions: Z n  dvf , (7) u  hvk i,

(8)



T k  mðvk  uÞ2 ,

(9)

,

(10)



qk  ðvk  uÞmðvk  uÞ2 ,

(11)



q?  ðvk  uÞ12mv2? ,

(12)

T? 

1

2 2mv?

where / S denotes the average of the quantity inside the brackets according to Eq. (A.6). If one uses the velocity distribution function f based on Eqs. (5), (6) and (A.10), and uses the averaging scheme (A.6) to find the right-hand side of

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Eqs. (7)–(12), the results would be exactly equal to the parameters that represent the physical quantities on the left-hand side of the corresponding equations. Therefore, the physical interpretation of the parameters in the assumed velocity distribution function f is entirely consistent with their definitions. Equations that describe the transport of these physical quantities are obtained by substituting Eqs. (5), (6) and (A.10) into (1) and taking velocity moments. The resulting transport equations for those quantities defined in Eqs. (7)–(12) can be found in Ganguli et al. (1987). Here, we express those transport equations for the 16-moment polar wind model in an equivalent form, based on the notations defined earlier. The zeroth-order moment would give the continuity equation, which describes the transport of density, n: qn q nu þB ¼ 0. (13) qt qs B Higher orders in velocity moment result in: the momentum equation (first order)   qu qu 1 qðnT k Þ GM qE k B0 T k  T ? du  ¼ , þu þ þ 2  m B m qt qs nm qs s dt

(14) the parallel and perpendicular energy equations (second order) qT k qT k 2 qðnqk Þ 2B0 quk dT k þu þ ðq  q? Þ þ 2T k ¼ ,  n qs qt qs B k qs dt

(15)

species. The collisional effects on one species due to another are characterized by a velocity-averaged collisional frequency, which depends on the density, velocity and temperature moments of both species. Other generalized transport models are based on a similar idea: the closure assumption approximates the distribution function by adding sophisticated modification terms to the zeroth-order drifting Maxwellian or bi-Maxwellian distribution. Consequently, like a hydrodynamic model, these models consist of systems of moment equations for all the species. When applied to the classical polar wind, the steady-state moment equations for all the species are coupled by Eqs. (3) and (4), the quasineutrality and current-free conditions. We note that some generalized transport models do not include anisotropy (e.g. the 5- and 8-moment models), or do not distinguish between the parallel and perpendicular heat flows (e.g. the 8- and 13-moment models). (A detailed discussion of those models can be found in Schunk (1977).) In order to compare generalized transport models with such differences (e.g. Blelly and Schunk, 1993), the parallel and perpendicular temperatures and heat flows are combined into single quantities: the average temperature, T, and the total heat flow, qtotal, where T ¼ 13T k þ 23T ?

(19)

and qtotal ¼ 12qk þ q? .

(20)

qT ? qT ? 1 qðnq? Þ B0 dT ?  ð2q? þ uT ? Þ ¼ þu þ n qs qt qs B dt (16)

A number of generalized transport models have been applied to the classical polar wind and solved with different methods. The systems of 13-moment equations by Schunk and Watkins (1981, 1982) were and the heat flow equations for parallel and solved by first removing all the time-derivative perpendicular energies (third order) terms, thus reducing the systems to a steady-state qqk qqk qu 3T k qT k dqk description. Ganguli et al. (1987) and Demars and þu þ 3qk þ ¼ , (17) qs 2m qs qt qs dt Schunk (1989, 1994) solved the 16-moment system    qq? qq qu T k qT ? B0 T?  dq þ u ? þ q? þ  uq? þ Tk  T? (18) ¼ ?, qs m qs qt qs B m dt where G is the gravitational constant, M is the mass of the Earth and terms with d/dt, whose approximate (quasi-linear) expansion can be found in Appendix A, represent changes of the moments due to collisional effects (Burgers, 1969). In general, the collision terms consist of separate components, each arising from collisions with a different particle

in a similar fashion. These calculations all had a relatively high boundary of 1500 km at the bottom of the flux tube. Their solutions for the ions were either entirely subsonic or entirely supersonic throughout the altitude ranges of the models. Mitchell and Palmadesso (1983) calculated the evolution of the time-dependent 13-moment system,

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1991

Fig. 1. Steady-state H+ velocity (left panel) and temperature (right panel) profiles generated by time-dependent moment-based models. The solid line is for the standard 5-moment equation set, the dashed line for the 8-moment, the dotted line for the 13-moment and the dash-dotted line for the 16-moment (Blelly and Schunk, 1993). All these solutions feature a transition from subsonic to supersonic flow as the altitude increases. This can be easily inferred from the plots based on the fact that the sonic speed for H+ temperature of 10,000 K is 12.8 km/s, and that at high altitudes all four solutions had lower temperatures and higher speeds compared with the corresponding reference values.

obtaining a classical polar wind solution when the system evolved into a steady state. Blelly and Schunk (1993) used a similar time-dependent method to find solutions of the classical polar wind in their comparative studies of the 5-, 8-, 13- and 16-moment transport formulation. In contrast to the steady-state systems, these polar wind solutions generated by the time-dependent method generally featured a transition from subsonic to supersonic H+ outflow (see Fig. 1). The formulations of moment-based polar wind models generally take into account the transfers of moments between particle species due to Coulomb collisions. Such moment transfers are characterized by collision frequencies, which are velocity-averaged quantities based on drifting Maxwellian or biMaxwellian particle velocity distributions and are expressed in terms of a finite number of moments of the species. The moment approach, therefore, greatly simplifies the description of Coulomb collisional effects. But as we shall discuss below, one should be cautious about such a collisional description for suprathermal particles. Nevertheless, the moment approach is particularly useful when the particle species are close to thermodynamic equilibrium such that their velocity distributions are well approximated by a drifting Maxwellian, as it is the case in collision-dominated plasmas. However, the moment approximation may break down when particle velocity distributions exhibit non-thermal features. As discussed, moment-based

models use a finite number of transport quantities to characterize the behavior of velocity distributions, which in principle, may consist of the contributions from an infinite number of moments. Non-thermal particle features cause the velocity distributions to deviate significantly from a drifting Maxwell distribution. It is then questionable whether the distributions can be sufficiently well characterized by only a finite number of lower-order moments. Moreover, the velocity-averaged approach by moment-based models in describing the Coulomb collisional effects is inappropriate for suprathermal particles. The Coulomb collision mean free path, l, for a charged particle is strongly dependent on V, its velocity relative to the background (Ichimaru, 1973). The dependence is approximately lV4. With long mean free paths, suprathermal particles (such as photoelectrons) thus may not become thermalized even in the collision-dominated regime. For this reason, the velocity-averaged Coulomb collisional effects based on thermal particle velocity distributions are not good approximations for these particles. A system of moment equations, due to its complexity and nonlinear nature, is very difficult to be solved analytically. Therefore, moment-based models are generally solved with numerical means. Because of the intrinsically stiff nature of moment equation systems, due essentially to the high ion-toelectron mass ratios, even numerical solutions are not straightforward to obtain. These systems

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generally exhibit a number of singularities that arise from the truncation or the assumption used to close the system (Yasseen and Retterer, 1991). In the polar wind application, the steady-state 16-moment system, for instance, exhibits singularities near the ion sonic point, i.e., the altitude at which the ion thermal speed equals its flow velocity, and cannot provide transonic solutions easily. (Due to the current-free condition, the flow velocity of the electrons is of the same order as that of the ions, which is much lower than the electron thermal speed.) However, such transonic solutions are required for the appropriate description of the polar wind because observations indicate that both H+ and O+ are supersonic at high altitudes (Abe et al., 1993). Time-dependent moment systems, however, do not have this difficulty concerning the ion sonic point. Transonic classical polar wind solutions can be obtained by iterations of the time-dependent system until a steady state is reached (e.g. Gombosi et al., 1985; Blelly and Schunk, 1993). 2.1.2. Collisionless kinetic approach A different type of approximation to Eq. (1) is to neglect the collision term on the right-hand side. In steady state (q/qt0), the Liouville theorem can be applied to such a collisionless formulation to describe the evolution of the particle distributions along a geomagnetic field line. The solution would be a function of the constants of motion, which are determined by the boundary conditions. For example, let v and v0 denote the velocity of a particle along the same trajectory at geocentric distances s and s0, respectively. The solution is then given by f ðs; vÞ ¼ f ðs0 ; v0 Þ,

(21)

where the time dependence of the distribution has been dropped due to the steady-state consideration. The velocities are related through the constants of motion: the total energy E ¼ 12mv2 þ qFE ðsÞ þ mFG ðsÞ ¼ 12mv20 þ qFE ðs0 Þ þ mFG ðs0 Þ

(22) and the magnetic moment m ¼ 12mv2? =B ¼ 12mv2?0 =B0 ,

(23)

where B and B0 are the magnetic field at s and s0, respectively, FE is the electric potential and related to the field-aligned electric field by E k ¼ dFE =ds

(24)

and FG ðsÞ ¼ GM=s is the gravitational potential. By imposing a distribution function f ðs0 ; vÞ for each particle species at the reference geocentric distance s0, one uses Eqs. (21)–(23) to obtain f(s, v), which is a function of FE ðsÞ. The density n(s) and velocity u(s), obtained from Eqs. (7) and (8), depend on FE ðsÞ. To determine the electric potential, the local quasi-neutrality and current-free flow conditions, Eqs. (3) and (4), shall be used. Note that the escape fluxes n(s)u(s) for individual particle species are proportional to the local magnetic field (c.f. Eq. (13) with q/qt0). The zero-current condition needs to be imposed at one altitude only, such as s0, and it will be maintained due to the property of the escape fluxes. Once the electric potential profile is determined, the various quantities of the particle species can be evaluated by taking moments of the distributions. Obviously, such a modeling approach is valid only when collisions are not important. This validity issue is closely related to the concepts of exobase and baropause. The exobase is the altitude above which Coulomb collisions are negligible. The region above the exobase is known as the exosphere. It is where the collisionless kinetic approach is applicable. Thus, models based on the collisionless kinetic approach have also been referred to as exospheric models in the literature of the polar wind (e.g. Lemaire, 1972a). A reasonable lower boundary for such models is the exobase. The baropause is the altitude at which the Knudsen number (Kn), defined as the ratio of the Coulomb collision mean free path and the electron density scale height, equals 1; it is by definition the upper boundary of the barosphere. In the ionosphere, Kn increases rapidly with altitude from values much smaller than one to values larger than one (Lemaire, 1972b), primarily due to the rapid increase of the collision mean free path associated with the downward density gradient. Because of the rapid increase of Kn, the barosphere and the exosphere are presumably separated by only a narrow altitudinal range. Collisionless kinetic models usually approximate the transition between the two regions as a surface. The exobase is assumed to be just above the surface and the baropause just below, with the two essentially sharing the same altitude. The boundary distributions at the exobase are usually provided by the plasma conditions at the baropause. The model by Dessler and Cloutier (1969), which was discussed in Section 1, was a simplified version of this approach, in that the electric field and thus

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the electric potential was assumed. The study calculated the acceleration of the H+ and He+ using the Pannekoek–Rosseland formula (Pannekoek, 1922; Rosseland, 1924) for the polarization electric field: Ek ¼

mo g , 2e

(25)

where mo is the mass of the O+, e is the ionic charge and g is magnitude of the gravitational acceleration. Such an electric field, together with the gravitation field, would result in approximately equal forces for the individual electrons and O+ ions. The Pannekoek–Rosseland electric field, Eq. (25), was later shown by Lemaire and Scherer (1970) to be consistent with an exosphere composed of only the O+ and electrons, under the assumption that both species were distributed in a Maxwellian with the same temperature and no field-aligned bulk velocity at the reference altitude. Thus, the H+ and He+ were essentially test particles in the primitive model by Dessler and Cloutier (1969). Moreover, the imposed distribution functions for these ions were Dirac d-functions with zero dispersion around a single velocity value. For more complicated distribution functions, however, Lemaire and Scherer (1970, 1971, 1972a, 1974) have demonstrated that one must recognize the nature of the particles’ trajectories before being able to obtain analytical forms for the density, velocity and other moments of the species. By assuming a monotonic potential energy profile for each particle species, they derived formulae to separate the entire phase space for the exospheric particles into four regimes according to the categories of the particle trajectories (Lemaire and Scherer, 1971): (1) the ballistic particles, which emerge from the barosphere and are reflected in the exosphere; (2) the escaping particles, which leave the barosphere and have sufficient kinetic energy and proper pitch angle to be lost into the magnetosphere or the interplanetary medium; (3) the trapped particles, which bounce between two mirror points in the exosphere; and (4) the incoming particles, which originate from the magnetosphere, and enter the barosphere as well as those which may be reflected in the exosphere. For species such as the O+ and the electrons, which have a positive total potential energy (electric+gravitational) above the baropause, all four classes of trajectories may exist. However, there are no ballistic or trapped H+ ions

1993

due to their monotonically decreasing exospheric potential energy with altitude. Lemaire and Scherer’s (1970, 1971, 1974) selfconsistent exospheric models were based on the approach outlined by Eqs. (21)–(23), (7) and (8), with Eqs. (3) and (4) coupling the formulation of all the species. The reference level s0 was set at the baropause or exobase, assumed to be a surface at 2000 km altitude. In particular, Lemaire and Scherer (1970) examined a few cases where the models differed by the ion composition and by the classes of particle trajectories included. Among the cases studied, the most realistic and comprehensive model consisted of the species of O+, H+ and electron, and particles with escaping, ballistic and trapped trajectories. The field line of interest in that calculation, instead of open and radial, was at 801 latitude at the baropause under the assumption of a dipole geomagnetic field configuration. However, Lemaire and Scherer (1970) noted that the results did not significantly differ among highlatitude field lines. The boundary velocity distribution functions at the exobase were truncated Maxwellian based on the baropause density and temperature, n0 and T0 respectively, i.e.: f ðs0 ; vÞ ¼ n0 ðm=2pT 0 Þ3=2 exp½ðm=2T 0 Þv2 ,

(26)

but with the velocity space that corresponded to incoming particles truncated. Such a regime for the incoming particles at the boundary generally had to be self-consistently determined along with the electric potential profile. Lemaire and Scherer’s assumption of monotonic total (electric+gravitational) potential energy profiles somewhat simplified the criteria for incoming particles. The truncated regime in the velocity space at the exobase boundary was simply vk o0 for species whose total potential energy monotonically decreased with altitude. For species with monotonically increasing total potential energy, however, the corresponding regime of truncation would depend on the electric potential (see Lemaire and Scherer, 1971), which means that the remaining portion of the species’ boundary velocity distribution (26) was part of the outputs of the self-consistent model. The particle velocity distributions in the exosphere were related to those at the exobase by Eq. (21). The velocity space in the exosphere, (vJ, v?), was mapped to that at the exobase, (vJ0, v?0), based on Eqs. (22) and (23), the equations characterizing the trajectories of the particles in the phase space. For incoming particles

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1994

that were able to reach the exobase, such a mapping would render a zero value for their velocity distribution in the exosphere. Some of the incoming particles, however, could not reach the exobase; the mapping did not apply for them. But they were assumed to be absent in the exosphere and have zero contribution to the velocity distributions as well. For escaping and ballistic particles, one can combine Eqs. (21)–(23) and (26) to show that they were distributed according to f ðs; vÞ ¼ n0 ðm=2pT 0 Þ3=2 exp½ðm=2T 0 Þðv2 þ RÞ;

s4s0 ,

(27) where RðsÞ ¼

2q ½FE ðsÞ  FE ðs0 Þ þ 2½FG ðsÞ  FG ðs0 Þ. m (28)

As the trapped particles do not reach the exobase, Eq. (26) does not apply to them. However, in the model, they were assumed to be distributed in the exosphere according to Eq. (27). We note that the quantity mR(s)/2 is the total potential energy of a particle in the exosphere relative to the exobase, which we shall refer to as the potential energy difference. Lemaire and Scherer (1970, 1971, 1974) performed a change of variable for the electric potential in the formulation: s0 FE ðsÞ ¼ j0 ðsÞ . (29) s With Eq. (29), the zero level of the electric potential was assigned to be at infinity, i.e., FE ! 0 as s ! 1. In addition, the electric potential difference between the exobase, s0, and infinity would simply be j0(s0). Eq. (28) could be written as RðsÞ ¼ 2FG0 ½1 þ a  ð1 þ bÞy,

(30)

where FG0  FG ðs0 Þ ¼ GM=s0 , a¼

q j ðs0 Þ, mFG0 0

(31)

b¼

q j ðsÞ, mFG0 0

(32)

and y ¼ s0 =s. The numerical results based on the new variable j0 were discussed in detail by Lemaire and Scherer (1971). Fig. 2 shows the potential energy difference

Fig. 2. Profiles of the normalized potential energy difference relative to the exobase as found in a collisionless kinetic calculation along a magnetic field line. The exobase was at 2000 km altitude. The baropause parameter T0 was assumed to be 3000 K for each of the particle species. The potential energy for each particle species was a monotonic function. The dashed lines are the asymptotes for the normalized potential energies of the species: top dashed line for the O+; middle for the electrons; bottom for the H+. (Adapted from Lemaire and Scherer (1971) with permission. Copyright 1971 American Institute of Physics.)

for all the particle species in their model, normalized by a constant value associated with the baropause temperature. It is clear from the figure that the potential energies of all three species were monotonic functions of the altitude, justifying their derived formulae for the classification of the particles. As expected, the potential energy difference for each species approached an asymptotic value at infinity. Lemaire and Scherer (1971) noted that the asymptote for the electrons, in particular, was given by the result ð1 þ ae ÞFG0 me ¼ 1:94 eV. Because the gravitational potential energy for the electrons was negligible compared to such an asymptotic value, one can deduce that j0(s0) was essentially 1.94 eV at the exobase altitude of 2000 km as indicated in the bottom panel of Fig. 3. The field-aligned polarization electric field profile, obtained by using Eqs. (24) and (29), is shown in the top panel of Fig. 3. Based on the signs and the monotonic relationship of the potential energy difference (see Fig. 2), the magnitude of the electric force was larger than the gravitational force for the H+ at all altitudes, but the opposite was true for the O+. As a result, the average velocity of the H+ ions rapidly increased above the exobase and

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Fig. 3. Profiles of the electric field (top panel) and the electric potential j0 (see Eq. (29) for definition) associated with the normalized potential energies in Fig. 2. (Reprinted from Lemaire and Scherer (1971) with permission. Copyright 1971 American Institute of Physics.)

their field-aligned flow became supersonic, in qualitative agreement with the results of the hydrodynamic calculations by Banks and Holzer (1968). The average O+ flow speed, despite the net downward combined force of the electric and gravitational fields, increased slightly with altitude. This kind of counterintuitive increase in the outflow velocity was later explained by Tam et al. (1998b) to be due to the kinetic effect of particle reflection. The increase in the O+ velocity, however, was very small, only up to a few cm/s in this early kinetic polar wind model by Lemaire and Scherer (1971). Because of the reflection of the O+ ions by the net combined force, the major ion constituent changed from the O+ to the H+ above 5500 km. Lemaire and Scherer (1972a) performed similar numerical calculations with the H+ distribution at the exobase replaced by truncated drifting Maxwellian, i.e.: f ðs0 ; vÞ ¼ n0 ðm=2pT 0 Þ3=2 exp½ðm=2T 0 Þðv  u0 Þ2 , (33) ^ ^ a where u0 ¼ u0 b is a velocity vector parallel to b, unit vector in the direction of the magnetic field. As in their earlier studies, the truncation of the distribution in Eq. (33) was due to the absence of

1995

incoming particles in the model. The regime vJo0 was truncated in the H+ velocity distribution at the exobase based on the assumption of a monotonic total potential energy profile for the species. It was found that the choice of u0 did not affect the overall results qualitatively. But with a larger u0, the change of the major ion species from O+ to H+ occurred at a lower altitude. In these models, because of the absence of incoming particles, the resulting distributions at the exobase were not a complete Maxwellian or drifting Maxwellian. Thus, the resulting densities, velocities and temperatures of the species at the boundary were not the same as n0, u0 and T0. In particular, the temperature obtained just above the exobase boundary would be lower than T0 due to the truncation of the distribution. However, unlike the moment approach, the collisionless kinetic approach has the capability of describing non-thermal particle features. For instance, Pierrard and Lemaire (1996) introduced an exospheric model based on Lorentzian velocity distribution functions. Such a model can take into account the effect due to the presence of suprathermal tails in the distributions at altitudes where Coulomb collisions are not important. Effects of suprathermal tails on the H+ polar wind were also studied with Monte Carlo simulations (Barghouthi et al., 2001). 2.2. Combinations of fundamental approaches 2.2.1. Simplified collisionless kinetic approach/ hybrid approach A modeling approach that simplifies the formulation discussed in Section 2.1.2 provides a different treatment between the electrons and the ions: it retains the kinetic formulation for the ions while treating the electrons as a fluid. Such a modeling approach is very common in plasma physics. It also has applications in space physics for phenomena besides the polar wind. The approach is conventionally known as ‘‘hybrid’’ in the literature of plasma physics and space physics, signifying the entirely different treatments of the ions and the electrons (e.g. Hamasaki et al., 1977; Hewett and Seyler, 1981; Papadopoulos et al., 1988; Akimoto and Winske, 1990; Porteous and Graves, 1991; Savoini et al., 1994; Swift, 1995; Boeuf and Garrigues, 1998; Pritchett, 2000; Belova et al., 2004). More recently, the usage of this terminology has been broadened to include situations where parts of the plasma are treated kinetically and the

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remainder as fluids (or by moment equations). (See, for example, Wang et al. (1994); Fu and Park (1995); Lin and Chen (2001); and the discussion by Winske and Yin (2001).) However, most of the publications on the polar wind do not follow such a convention of nomenclature. Holzer et al. (1971) were the first to apply such a ‘‘hybrid’’ treatment to the polar wind particles; they referred to their model as ‘‘kinetic’’. Although such a label de-emphasized the partially kinetic nature of their approach, they did describe their study as a ‘‘simple kinetic theory’’. Barakat and Schunk (1983) were the first authors in the polar wind literature to label such an approach with a term that acknowledges its partially kinetic nature; they called it ‘‘semi-kinetic’’. This term was later adopted by several other authors on the polar wind in describing this modeling approach (e.g. Li et al., 1988; Demars and Schunk, 1991, 1992; Wilson, 1992, 1994; Ho et al., 1992, 1993; Horwitz et al., 1994; Su et al., 1998). In the review article on the history of the polar wind in this issue, Lemaire et al. (2007) also refer to this modeling approach as ‘‘semi-kinetic’’. In this article, our discussion of the polar wind studies focuses on the modeling approaches and techniques. Because some of the modeling approaches for the polar wind are also applicable to researches in plasma physics and other areas of space physics, we shall conform to the nomenclature that is more widely accepted. We shall hereafter use the term ‘‘hybrid’’ to refer to the approach with kinetic ions and fluid electrons, or its generalization as discussed above.1 In most hybrid models of the polar wind, such as that by Holzer et al. (1971), the electrons are treated as a massless quasi-neutralizing fluid. Such a treatment neglects the contribution of the bulk velocity moment of the electrons, extremely simplifying the description of their transport. Later (in Section 3.1.2 for instance) we shall discuss hybrid models with more rigorous treatments for the electrons. In order to distinguish such a difference, we shall further classify the hybrid models by specifically referring to those which neglect the 1 However, we should note that the term ‘‘semi-kinetic’’ is used in this article to refer to a different modeling approach, which will be discussed in Section 2.2.2. That approach applies the fully kinetic (where the Knudsen number Kn is larger than unity) and moment formulations (where Kn is less than one) each at a different altitude range. In each regime, the ions and electrons are treated similarly.

contribution of the electron bulk velocity as ‘‘simplified kinetic’’. Such a term not only emphasizes that the dynamics and transport properties of the plasma are primarily determined in the kinetic part of those models, it is also consistent with the usage of the term ‘‘simple kinetic theory’’ by Holzer et al. (1971). In simplified collisionless kinetic models, the treatment of the ions is generally the same as in the full kinetic version; the ions are described by Eqs. (21)–(23). However, for the electrons, their density is usually assumed to follow a specific relationship that depends on the electric potential. A very common choice for this kind of assumption is the Boltzmann distribution:

 (34) ne ¼ ne0 exp eðFE  FE0 Þ=kT e , where ne is the electron density, ne0 is the electron density at the reference altitude at which the electric potential is FE0 , e is the magnitude of the electronic charge, k is the Boltzmann constant and Te is the electron temperature, whose profile is assumed. Eqs. (21)–(23) and (7) for the ions, Eq. (34), and the quasi-neutrality condition (Eq. (3)) together form a self-consistent system that includes the electric potential. We should note that under the assumption of a constant Te, the Boltzmann electron distribution corresponds to the steady-state solution of the electron momentum equation used by Banks and Holzer (1968, 1969b): 1 qðPe Þ eE k þ ¼ 0, re qs me

(35)

where Pe ¼ ne T e is the electron pressure and re ¼ ne me . Eq. (35) can be considered as an approximation of Eq. (14) when the latter is applied to the electrons under steady-state consideration. Due to the light mass of the electrons, the gravitational force term in their momentum Eq. (14) is negligible. In addition, the inertia term is generally small compared with the pressure gradient term in the equation, particularly under the condition of a subsonic electron flow. If one makes the assumptions that the temperature of the electrons is isotropic and that the Coulomb collisional effects on their flow velocity (or momentum) are negligible, then the steady-state electron momentum equation would be dominated by the pressure gradient term and the electrostatic force term, reducing the equation to the approximation (35). Because Eq. (34) is a solution of Eq. (35) for constant Te, it is conceivable that under the

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additional assumptions discussed above, hybrid models with relatively rigorous fluid treatment— such as Eq. (14)—for the electrons can be transformed into simplified kinetic models that are based on the Boltzmann electron distribution (34). Holzer et al. (1971) were the first to apply the simplified collisionless kinetic approach to the classical polar wind. The exobase in their calculations was chosen to be at 4500 km altitude, where the truncated H+ and O+ distributions were in the form of Eq. (33) but composed only of the escaping and ballistic populations (no trapped ions). The electron temperature was assumed to be isothermal at 3000 K. The model generated a solution for the polar wind by iterating the numerical results of ne, FE and the densities of the H+ and O+ ions until those results converged. By comparing the ion transport quantities of this exospheric model with those of their hydrodynamic calculations, these authors showed that the two approaches gave good agreement for the profiles of densities and bulk velocities (Fig. 4). The ion temperatures from both calculations were somewhat different, but their profiles agreed in terms of qualitative behavior. Other authors have also compared the results of simplified collisionless kinetic calculations for the classical polar wind with those from moment models (Demars and Schunk, 1991, 1992; Ho et al., 1993). Generally, the two approaches were found to be in agreement when describing the collisionless regime of the polar wind. Simplified collisionless kinetic models have also been used to examine non-classical effects due to elevated ion or electron temperature in the ionosphere, and/or the existence of magnetospheric electrons (Barakat and Schunk, 1983; Li et al., 1988; Wilson et al., 1990). These studies will be discussed in a later section. However, we should note here that Wilson et al. (1990) introduced a different technique for the kinetic ion calculations in their approach. Instead of identifying the different populations of the ions based on their trajectories, they followed the temporal evolution of the ions along the flux tube with particle-in-cell (PIC) simulations. Ions constituting the upward portion of the boundary distributions were injected into the simulations at the lower boundary. Steady-state results for the ions were reached when their transport quantities became approximately constant in time. Because of the simulation technique, it was not necessary to categorize the particle phase space by trajectories. The precision of the PIC simulation technique

1997

Fig. 4. Comparison of the particle densities, ion velocities and ion temperatures generated by the collisionless kinetic calculations and the hydrodynamic equations. (Adapted from Holzer et al. (1971) with permission. Copyright 1971 American Geophysical Union.)

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regarding the sizes of the time steps, grid cells, etc. was thoroughly discussed by Barakat et al. (1998a). 2.2.2. Semi-kinetic approach As discussed in Section 2.1, both the moment and collisionless kinetic approaches have their own advantages and shortcomings. However, as pointed out by Lemaire and Scherer (1973), these two approaches are complementary for the polar wind. The classical hydrodynamic descriptions are expected to be appropriate only in the low-altitude collision-dominated region, whereas the exospheric kinetic calculations are expected to be zeroth-order approximations in the high-altitude collisionless domain only. There have been modeling efforts to combine these two approaches or approximations, each describing a different range of altitudes (Lemaire, 1972b; Lemaire and Pierrard, 2001). Strictly speaking, there is a finite altitude region where the transition from collision-dominated to collisionless regimes takes place. But for modeling purpose, the transition region separating the baropause and the exobase was reduced to a surface in these semikinetic studies (see the footnote in Section 2.2.1 for the meaning of the term ‘‘semi-kinetic’’ in this article). The baropause/exobase surface then became an obvious choice for the boundary between the domains of moment and collisionless kinetic calculations. In order for the density and bulk velocity profiles of the particle species to be continuous throughout the entire altitude range, Lemaire (1972b) devised a scheme to match those quantities obtained separately from the exospheric kinetic calculations (Lemaire and Scherer, 1970, 1971) and the hydrodynamic equations (Banks and Holzer, 1968, 1969b) at the exobase or baropause. In his exospheric model, the velocity distributions for the particle species at the exobase, f(s0, v), were in the form of Eq. (26), except for the truncation of the incoming particle regime vJo0 and v4vescape , where vescape was the escape speed of the particle species at s0. T0 was assumed to be equal to the isothermal species temperature in the hydrodynamic calculation, which described the plasma from the lower boundary up to the baropause. The density and velocity for the species at the exobase could be obtained by Eqs. (7) and (8). These values were then matched with the corresponding results from the hydrodynamic calculation at the baropause. For species like the O+ ions and electrons, with a total potential energy increasing monotonically with

Fig. 5. Matching the boundary H+ velocity between the hydrodynamic and the exospheric calculations at the baropause in a semi-kinetic model. The solid lines on the left-hand side correspond to a family of hydrodynamic solutions for five different values of the upward H+ bulk velocity at the reference level of 950 km. The ion and electron temperatures were equal at 2500 K. The baropause or exobase altitudes are marked by the intersections of the vertical lines with the curves. Only one of these solutions fits the kinetic solution shown by the solid line on the right-hand side. The dotted line corresponds to the solution when the ion and electron temperatures were at 3000 K. For comparison the dashed line corresponds to the result of a collisionless kinetic model whose baropause was at 950 km and ion and electron temperature at 3000 K. (Reprinted from Lemaire (1972b) with permission. Copyright 1972 Elsevier.)

altitude, the exobase density and field-aligned flow speed depend on vescape . For the H+, whose total potential decreases with altitude, vescape ¼ 0, rendering the exobasepdensity nh ðs0 Þ ¼ nh0 =2, and outflow ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi speed uh ðs0 Þ ¼ 2T h0 =pmh . Because of the specific matching conditions, not all hydrodynamic solutions could fit with a kinetic solution. Fig. 5 demonstrates such a restriction on the admissibility of the lower boundary conditions for the hydrodynamic calculations in this modeling approach. 3. Collisional kinetic calculations For collisionless kinetic calculations, discontinuity across the baropause/exobase surface is intrinsic. Even with the matching scheme for the density and

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velocity moments in the semi-kinetic model (Lemaire, 1972b), the temperatures of the particle species as well as the intensity of the electric field in the calculation results were discontinuous. Such a discontinuity arises mainly because Maxwellian or bi-Maxwellian velocity distributions, despite being assumed at the exobase boundary in all exospheric calculations, cannot be obtained in the solution due to a lack of downward particles. Without a magnetospheric particle source, the only downward particles in collisionless kinetic models are those reflected by the potential energy barrier. However, the H+ ions, for instance, have a potential energy that decreases with altitude, and cannot be reflected. Even if one imposes a magnetospheric source for the H+, the incoming ions will all have speeds that exceed the local escape velocity, and cannot fill the low-energy downward portion of the velocity distribution. Thus, their velocity distribution at the exobase is far different from the assumed Maxwellian distribution. Khoyloo et al. (1991) discussed the issue of discontinuity by considering the total potential energy for the H+ ions. Based on their simplified collisionless kinetic model of an H+-electron classical polar wind, they showed that a boundary velocity distribution significantly different from a Maxwellian is necessary for the potential energy profile to be continuous. However, because the exobase velocity distribution and the potential profile are self-consistently related, it is not a simple task to predict precisely what boundary distribution has to be imposed to ensure continuity of the solution. In reality, the separation between the collisiondominated and collisionless regimes is not a baropause/exobase surface, but a transition region of finite extent in altitude. Thus, the change in the particle velocity distributions with altitude should be gradual, and not as dramatic as that described in typical exospheric calculations where the transition is discontinuous across the model boundary. In order to reduce the gap of discontinuity in the kinetic solution, one may lower the altitude of the boundary of the transition region and place it deeper into the collision-dominated regime, where Maxwellian distributions are more reasonable approximations for the particle velocity distributions. However, the kinetic calculations should then include the effects of Coulomb collisions. Some kinetic models of the polar wind mimicked the effects of the Coulomb collisions by using simplified operators to represent the collision term on the right-hand side of Eq. (1). For example,

1999

Barakat et al. (1991) adopted the Maxwell molecule collision model (Davison, 1957; Barakat and Lemaire, 1990), which is known for simulating non-resonant ion-neutral interactions, to study the collisions between the H+ and a static background of O+ ions in the polar wind despite the fact that such a model does not really provide an accurate description for Coulomb interactions. Their Monte Carlo simulation of the H+ ions was based on a test-particle assumption: the assumed profiles of the electric potential and the density of O+ background ions were unaffected by the H+ ions. Wilson (1992) modified a simplified collisionless PIC simulation (Wilson et al., 1990) by incorporating self-collisions between the H+ ions, as well as their interaction with a static O+ ion background, which was assumed to be unaffected by these H+-O+ collisions. The model randomly paired up particles and performed binary collisional scattering in random directions. This randomized binary collisional scheme was later applied to model the self-collisions of the O+ ions in other studies (Wilson, 1994; Ho et al., 1997). 3.1. Fokker– Planck equation The effect of collisions in the polar wind can be more rigorously incorporated into Eq. (1) with a Coulomb collision operator representing the term df j =dt (Landau, 1936; Rosenbluth et al., 1957; Balescu, 1960; Lenard, 1960). One way is to express the equation in the Fokker–Planck form:   df j df j q 1 q dðDj f j Þ , ¼ ¼  d aj f j  (36) qv 2 qv dt dt where aj is the dynamic friction vector:  X q2j q2b ln L  mj 1þ aj ðt; s; vÞ ¼  4p mb m2j b Z 0 vv  dv0 f b ðt; s; v0 Þ ; jv  v0 j3

ð37Þ

Dj is the velocity diffusion tensor: Dj ðt; s; vÞ ¼ 4p

X q2j q2b ln L b

Z 

m2j dv0 f b ðt; s; v0 Þ



 I ðv  v0 Þðv  v0 Þ  ; jv  v0 j jv  v0 j3

ð38Þ I is the unit dyadic and df j =dt is defined in Eq. (1). In Eqs. (37) and (38), ln L is the Coulomb

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logarithm, a parameter containing the screening of the Coulomb field. Each term in the summation represents the effects of the collisions of the particle species j with a background species b. The summation also includes the species j itself, representing the effects of self-collisions. Eq. (36) is nonlinear in fj due to these self-collisions. It is difficult to solve the system of equations which consists of Eqs. (36)–(38) for each particle species. The Coulomb collision operator for each particle species depends nonlinearly on its velocity distribution. Moreover, these collision terms depend on the velocity distributions of all the other species. Besides such an inter-relationship among all the particles, the species are also coupled by an ambipolar electric field EJ, as well as the quasineutrality and current-free conditions (Eqs. (3) and (4)), making a complete, self-consistent kinetic solution of the system even more difficult to achieve. 3.1.1. Test-particle calculations For a minority species, however, it may be justified to neglect the self-collisions due to its relatively low density. Such an approximation allows one to treat the minority species as test particles for the purpose of collisions, in which case the terms with b ¼ j are dropped from the summations in Eqs. (37) and (38), removing the nonlinearity in fj. A number of authors have studied the Coulomb collisional effects on steady-state velocity distributions of minority species in the polar wind using this test-particle approximation (Yasseen et al., 1989; Barghouthi et al., 1993; Barakat et al., 1995; Lie-Svendsen and Rees, 1996; Leer et al., 1996; Pierrard and Lemaire, 1998). In all those testparticle studies, the electric field was prescribed and the velocity distributions of the major species with which the test particles collide were assumed to be Maxwellian, reducing the description of the test species to a linearized Fokker–Planck equation. Despite these common set of assumptions, the methodology for solving the Fokker–Planck equation differed in all of those studies. By assuming the background velocity distributions to be drifting Maxwellian, one may simplify the expressions for aj and Dj (see Appendix B). The collisional friction and diffusion characterized by these two parameters were incorporated into the Monte Carlo simulations by Yasseen et al. (1989). Their description of the global kinetic evolution of photoelectrons included their Coulomb interaction with the thermal electrons. Monte Carlo simulations

based on the same principle were carried out by Barghouthi et al. (1993) and Barakat et al. (1995), except that the minor species was the H+ and the major species that provided the collisional background was the O+. Such a test-particle model involving the polar wind H+ and its Coulomb collisions with the O+ has also been solved numerically with a finite difference technique by Lie-Svendsen and Rees (1996), who performed a change of variable for the steady-state linearized Fokker–Planck equation. Their collisional kinetic calculations were actually part of a semikinetic model composed of three different approaches, and were applied only to the transition region. In the collision-dominated regime at lower altitudes, the model was described by moment equations. Above the transition region, the collisionless kinetic approach was applied. The test-particle semi-kinetic model has also been used to describe the He+ ions, taking into consideration their Coulomb interaction with the O+ (Leer et al., 1996). An alternative technique for obtaining a solution for the steady-state linearized Fokker–Planck equation was demonstrated by Pierrard and Lemaire (1998). The technique was based on a specialized spectral method. The velocity distribution of the test species, H+, was expanded in terms of three sets of orthogonal functions: Legendre polynomials for vJ/v, i.e., the cosine of the pitch angle; speed polynomials (Shizgal, 1979) for v, and Legendre polynomials for s. The method of the polynomial expansion used to find the solution of the Fokker–Planck equation is briefly explained in Appendix C. The coefficients of these terms consistent with the steady-state linearized Fokker–Planck equation that included an O+ background were determined numerically. Test-particle studies of the H+ based on the Coulomb collisional kinetic approach, though carried out with different calculation techniques, have rendered similar qualitative results (Barakat et al., 1995; Lie-Svendsen and Rees, 1996; Pierrard and Lemaire, 1998): the H+ velocity distribution evolves from a thermal, Maxwellian-like distribution to one with double humps as the altitude increases. The reason for the double-humped distribution is due to the ‘‘velocity filtration effect’’ (Scudder and Olbert, 1979), which arises from the velocity dependence of the Coulomb collision mean free path and the non-local nature of the outflow. Particles with low velocities relative to the background have short mean free paths; they tend to thermalize and relax towards a nearly isotropic

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Maxwellian. In contrast, suprathermal particles, with their long mean free paths, are able to stream through the background with little influence of the Coulomb interactions. In addition to the thermal core, this ‘‘velocity filtration effect’’ leads to the formation of a non-thermal tail in the H+ distribution, which resembles a mushroom cap in the velocity space, as the altitude increases. Figs. 6 and 7 show the evolution of the H+ distribution at various altitudes of the polar wind based on the results of test-particle calculations (Barakat et al., 1995; Pierrard and Lemaire, 1998), which indicated that the tail population develop a secondary peak at about 1000–2000 km altitude. According to those studies, while the H+ ions in the non-thermal tail continue to be accelerated upward by the electrostatic potential, those in the core portion are dragged by the background O+ ions under the velocity-dependence effect of Coulomb collisions.

Fig. 6. Contour plots of the H+ velocity distributions at six different altitudes in the collisional kinetic test-particle calculations by Barakat et al. (1995). The velocities, V~ k and V~ ? , are normalized by the O+ thermal speed. The dotted line shows the H+drift velocity. (Reprinted from Barakat et al. (1995) with permission. Copyright 1995 American Geophysical Union.)

Fig. 7. Contour plots of the dimensionless H+ velocity distributions at three different altitudes in the collisional kinetic test-particle calculations by Pierrard and Lemaire (1998). The vector quantity y is the H+ velocity normalized by the O+ thermal speed. (Reprinted from Pierrard and Lemaire (1998) with permission. Copyright 1998 American Geophysical Union.)

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This creates a secondary peak in the velocity distribution function in the transition region. As the altitude increases, the density of the O+ background decreases and its influence on the H+ ions via collisions diminishes. Fewer H+ ions are dragged by the O+ background. At further higher altitudes, the low-speed H+ population appears to vanish as Figs. 6f and 7a suggest.

3.1.2. Nonlinear collisional calculations: self-consistent hybrid model The previously described test-particle calculations for the Fokker–Planck Eq. (31) of the H+ ions neglected the self-collisions among themselves, linearizing the Coulomb collision operator in terms of their velocity distribution. With an assumed velocity distribution for the major particle species, the nonlinearity intrinsic to Coulomb collisions was completely removed from these test-particle models. As a matter of fact, a fully kinetic model with a rigorous, nonlinear description of the Coulomb collisions has yet to exist. However, such a nonlinear calculation of the Coulomb collisions has been carried out for the polar wind with a self-consistent hybrid model by Tam et al. (1995b, 1998a, b). The model aimed to describe the global kinetic collisional effects of photoelectrons in the polar wind, a topic to be discussed in Section 5, but it also took into account the nonlinearity of the Coulomb interactions. The model was hybrid in that the H+ and O+ ions as well as the photoelectrons were described by Monte Carlo simulations based on a collisional kinetic approach (Eq. (36)), while the thermal electron transport properties were determined with a fluid calculation. We note that unlike those hybrid/simplified kinetic models discussed in Section 2.2.1, which treated the electrons as a massless neutralizing fluid, Tam et al. (1995b, 1998a, b) considered the momentum and energy transport of the thermal electrons, including the transfer of those quantities due to Coulomb collisions. The fluid part of their model consisted of equations for the entire electron population, where the thermal electrons were assumed isotropic: B

  q ne T e þ ne me u2e þ ns T sk þ ns me u2s qs B   qFG qFE B0 þ ðne þ ns Þ me e þ ðne T e þ ns T s? Þ qs qs B dM e dM s ¼ þ , ð39Þ dt dt

   q ne ue 5 me u2e Q Te þ þ ws B qs B 2 2 B i dE n u ns us dEs e e e þ þ , þ ðme FG  eFE Þ ¼ B B dt dt ð40Þ where the subscripts e and s stand for the thermal and photoelectrons respectively, Z Qw ¼ dvðmv2 =2Þvk f (41) is the energy flux, and dM e;s =dt and dEe;s =dt represent respectively the momentum and energy transfers to the thermal and suprathermal electrons due to Coulomb collisions. As will be explained later, the collisional transfer values are supplied by the kinetic component of the model. The kinetic part of the model describes the evolution of the particle velocity distributions along the magnetic field line based on Eq. (36), with df j =dt as defined in Eq. (1). The calculations for each particle species were individually performed, similar to the global kinetic collisional test-particle simulation by Yasseen et al. (1989) where the electric field and the collisional background was pre-determined. The background species for Coulomb collisions included the thermal electrons, O+ and H+, whose velocity distributions were assumed to be drifting Maxwellian. Photoelectrons, due to their relatively low density, were treated as test particles for the consideration of Coulomb collisions and not included in the background. The model took into account the self-consistent nonlinear Coulomb interactions, including selfcollisions, with an iterative scheme between the fluid and kinetic calculations. In each iteration, the Monte Carlo calculations provided updated velocity distributions for the ions and photoelectrons, from which density, velocity and temperature profiles were extracted. The quasi-neutrality and currentfree conditions (Eqs. (3) and (4)) then provided the density and velocity profiles of the thermal electrons. The momentum and energy transfer values from each background species to the ions and photoelectrons due to the Coulomb interactions were also found from the Monte Carlo simulation. Because momentum and energy are conserved for Coulomb collisions, dM e;s =dt and dEe;s =dt could be determined from the results of the kinetic calculations. With all this information, profiles of the thermal electron temperature and the ambipolar electric field were determined by Eqs. (39) and (40).

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The updated electric field and thermal electron results, as well as the ion results from the kinetic calculations then provided the background for the Monte Carlo simulations in the next iteration. When the results of the fluid and kinetic calculations converge, the iteration approach gives a selfconsistent polar wind solution that takes into account the nonlinear dependence of the Coulomb collisional operators in Eq. (36) for all the ions.

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As shown in Fig. 8, the velocity distributions of the H+ ions in the polar wind, as determined by the self-consistent hybrid model, also featured double peaks, in agreement with the collisional test-particle calculations discussed earlier (Barakat et al., 1995; Lie-Svendsen and Rees, 1996; Pierrard and Lemaire, 1998). However, because the selfcollisions among the H+ ions, which tend to relax their velocity distribution to drifting Maxwellian,

Fig. 8. Contour plots of H+(left panels) and O+(right panels) velocity distributions at altitudes of 500 km (bottom panels), 5000 km (middle panels) and 10,000 km (top panels) of the dayside polar wind based on Tam et al. (1998b). Note the formation of double peaks of the H+contours at 10,000 km. At 5000 km, the double peaks of the H+contours are barely visible.

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were included in the self-consistent hybrid model as well as the H+–O+ collisions, it takes a longer distance for the distribution to evolve into a double hump. Note that unlike in Figs. 6 and 7, the velocity distribution of thermal H+ ions has a peak at a non-vanishing upward directed velocity. This peak value is close to the peak of the O+ velocity distribution, which was implicitly postulated as a drifting Maxwellian in all fluid models of the polar wind. Using simulation as the methodology for solving the kinetic equations, the self-consistent hybrid model circumvented a major difficulty that would be present if the equations were to be solved analytically or numerically. As discussed in Section 2.1.2, it is necessary to identify the various regimes of the particle trajectories in the phase space in order to solve the kinetic equations numerically (or analytically). In Section 5.3.3, we shall see that this is not a trivial task even for collisionless calculations. For collisional calculations, one expects such a task to be a lot more difficult. Simulations, on the other hand, follow the evolution of the particles along the flux tube, making it unnecessary to distinguish the types of trajectories in the phase space. The iteration approach of the self-consistent hybrid model, in principle, can be applied to fully kinetic models, in which the thermal electrons are also included in the simulations, to generate collisional solutions of the polar wind. However, such collisional solutions are, in practice, almost impossible to achieve. The reason is due to the restriction on the simulation time step. The time step must be smaller than the shortest time scale among all the particle species. In the polar wind and most other non-turbulent applications, that would correspond to the collision time scales for the thermal electrons due to their relatively high density and low mass. Because of the large mass difference between the ions and electrons, their time scales are very different. The effort of simulations to trace the ions with time steps comparable to the thermal electron time scales therefore would be exhaustingly inefficient. For applications that involve a macroscopic or mesoscopic scale, such as the polar wind, whose spatial scale is thousands of kilometers, the simulation process, being slowed down by the thermal electron time scales, is exceedingly timeconsuming and highly impractical even with the advancement in computational resources over the recent years. This was the reason why Tam et al.

(1995b, 1998b) considered the thermal electrons as a fluid whose density, bulk velocity and temperature profiles satisfy Eqs. (39) and (40). 4. Polar wind O+ acceleration Theoretical studies of the classical polar wind, though varying in modeling approaches, have always featured a significant upward acceleration of the H+ ions. For such models that describe the ion species with kinetic calculations, or those which are moment-based but rely on a time-dependent technique to obtain steady-state results, the H+ outflow is usually found to undergo a transition from subsonic at low altitudes to supersonic at high altitudes. The results of those polar wind models, therefore, are consistent with the observations of supersonic H+ in the polar cap by the DE-1 (Nagai et al., 1984) and Akebono satellites (Abe et al., 1993). The O+ ions, in contrast, play an inactive role in the classical polar wind theory. The transport and dynamics of the O+ have been neglected in most early calculations; it was usually assumed that the ions were unable to overcome their large gravitational potential barrier due to their heavier mass and relatively small thermal speed. As a consequence, the O+ density profile was assumed to be almost in hydrostatic equilibrium. There have been attempts that included the field-aligned flux of O+ ions in self-consistent description of the classical polar wind (e.g. Lemaire and Scherer, 1970, 1971, 1972a; Holzer et al., 1971; Schunk and Watkins, 1981, 1982; Li et al., 1988). However, the results of all those studies concluded that the escape flux of the O+ should be negligibly small and that the dominant ion species changed from O+ to H+ beyond an altitude of several thousand kilometers. These theoretical conjectures were not supported, however, by the observations reported in the next subsection. 4.1. Observations of escaping O+ ions There is experimental evidence of escaping O+ ions in the polar cap. Based on the data from the DE-1 High Altitude Plasma Instrument (HAPI) at altitudes around 20,000 km, Gurgiolo and Burch (1982) reported that the observed polar wind ions— later confirmed to be O+ by Gurgiolo and Burch (1985)—consisted of an unheated and a heated component. They interpreted the unheated component

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as a constituent of the classical polar wind, and attributed the heated component, which consisted of ions with energy as high as 8 eV, to ion perpendicular heating in regions at lower altitudes of the polar wind. However, Green and Waite (1985) argued that those ions probably originated from the cusp, where they were energized by a mechanism beyond the description of the classical polar wind, and entered the polar cap by convection. Waite et al. (1985) reported a number of events of outflowing O+ ions observed by the Retarding Ion Mass Spectrometer (RIMS) aboard DE-1. Those ions had energies up to 10 eV, and were observed at magnetospheric altitudes throughout the polar cap. But those authors suggested that the outflowing O+ ions might have been energized in the topside ionosphere near the dayside polar cap boundary, and spreading throughout the polar cap region by convection. Thus, despite the observations of the suprathermal O+ ions in the polar cap, experimentalists could not draw a conclusion based

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on the DE-1 data with regard to the flow of those O+ ions that originated from the polar ionosphere. Nevertheless, as we shall discuss later, investigators of the polar wind were motivated by these observations to explore possible ion acceleration mechanisms beyond the description of the classical theory. The role of the O+ ions in the polar outflow was much more clearly revealed by the observations of the Suprathermal Mass Spectrometer (SMS) aboard the Akebono satellite (Abe et al., 1993). The instrument measured a significant mean outflow velocity for the O+ at altitudes as low as 5000 km (see Fig. 9). Similar to the H+, the observed O+ outflow velocity monotonically increased with altitude, attaining a supersonic flow at the apogee of the satellite (10,000 km). Moreover, contrary to the theory of the classical polar wind, the O+ was found to be the dominant ion species even at such a high altitude. Thus, there was a significant amount of escape flux for the O+. At altitudes below 5000 km, although the O+ outflow velocity

Fig. 9. Akebono observations of the parallel ion velocities of the polar wind as a function of altitude, averaged over all magnetic local times, all Kp levels, and all invariant latitudes above 801. Left: average ion velocities; right: standard velocity deviation. The dotted curve in the right panel indicates the number of samples in each 100-km altitude bin. (Reprinted from Abe et al. (1993) with permission. Copyright 1993 American Geophysical Union.)

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averaged over a number of data samples was low, its standard deviation was about 2 km/s (see Fig. 9), suggesting that there were a considerable number of events in which the O+ ions attained significant upward velocities in the polar ionosphere. 4.2. Modeling of non-classical polar wind effects The DE-1 and Akebono observations of escaping O+ ions in the polar cap indicated that the classical polar wind theory may not be an adequate description for the outflow. Theoretical studies have examined a number of ‘‘non-classical’’ effects that, via various mechanism(s), may lead to the acceleration of O+ ions in the polar wind. One of the acceleration mechanisms considered was ion heating. Gombosi et al. (1985) included external heating terms in the energy equations of their timedependent hydrodynamic model, and showed that the O+ outflow could become supersonic with a significant amount of density flux. In a subsequent study (Gombosi and Killeen, 1987), the ion heating terms were modeled after the horizontal frictional heating due to the ion-neutral collisions. It was shown that a short duration of the heating could generate an O+ flux bulge that propagated upward along the flux tube, consisting of oxygen ions with a subsonic average velocity of about 2 km/s. A different type of ion heating, provided by WPI, has also been suggested as a possible explanation for the escape of polar wind O+ ions. Barakat and Barghouthi (1994) incorporated the effect of WPI (the type of WPI was unspecified) on the polar wind ions into their steady-state collisionless kinetic calculations. They employed an iterative approach in order to reach a self-consistent solution that accounted for both the WPI and the ions’ kinetic behavior. Similar to an earlier study of the resonant interaction between electromagnetic ion cyclotron waves and the O+ in the auroral zone (Retterer et al., 1987), the effect of the WPI in the polar wind study was expressed in the form of an operator for perpendicular diffusion, and was taken into account by Monte Carlo simulations. It had been well understood that transverse heating would lead to parallel acceleration as the ions travel upward, because of the mirror force transferring their perpendicular energy to the field-aligned direction (Retterer et al., 1987). The results of the polar wind study indicated that both the density and outflow velocity of the O+ ions are strongly related to the level of the wave energy; the escape flux of these

heavy ions could be enhanced by a factor of 105 with strong WPI. Pierrard and Barghouthi (2006) have also studied the effects of the WPI on the double-hump H+ ion velocity distribution function in the polar wind. WPI have also been suggested to lead to crossfield transport of the polar wind plasma. Ganguli and Gavrishchaka (2001) showed that transverse inhomogeneities in the field-aligned ion outflow velocity could excite the D’Angelo instability (Kelvin–Helmholtz instability under the special conditions considered by D’Angelo, 1965), causing diffusion of the plasma across the magnetic field lines. The polar wind study was mainly based on a self-consistent three-dimensional fluid model, but also included a kinetic test-particle simulation to confirm the results of the fluid calculation, which described the transport of the H+ ions and the electrons in a uniform magnetic field. Recently, Gavrishchaka et al. (2007) have updated the model with a dipole magnetic field and extended its application to include also the transport of the O+ ions. Their new study also showed a significant cross-field diffusion induced by the transverse inhomogeneities of the field-aligned ion flow. Cross-field transport has been suggested as a possible mechanism for accelerating high-latitude O+ ions; such an acceleration mechanism involves convection and the geometry of the geomagnetic field. Test-particle calculations by Cladis (1986) showed that the E  B convection could produce a parallel acceleration of polar ions originating near the cusp, allowing them to escape to the magnetosphere. Such an effect can also be viewed as a centrifugal acceleration in the convecting plasma frame of reference. This parallel acceleration effect on the polar wind ions was examined by several authors based on PIC simulations that treated electrons as a massless neutralizing fluid (Horwitz et al., 1994; Demars et al., 1996a; Ho et al., 1997). In those studies, the parallel ion acceleration associated with the convection was taken into account in a similar fashion. The models were a one-dimensional flux tube. The simulations characterized the field-aligned component of the acceleration due to the E  B convection with a parallel centrifugal force. The ions were followed as they were influenced by the combined action of this force and the standard forces (electric, gravitational and mirror) in the polar wind. Horwitz et al. (1994) further simplified the description of the centrifugal acceleration in their collisionless calculations by

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neglecting the latitude dependence, approximating the parallel centrifugal force with the corresponding value at 901 magnetic latitude. Their model was later extended by Ho et al. (1997), who incorporated the collisional modeling technique by Wilson (1994) into the PIC simulations (see also Section 3) and obtained similar results. Horwitz et al. (1994) mainly focused on the steady-state results for constant convection electric fields under cool and warm plasma temperatures. They emphasized the strong association between the O+ velocities and the magnitude of the convection electric field, which resulted in an increase in the O+ escape flux by two orders of magnitude when the electric field changed from 0 to 100 mV/m under cool plasma conditions. However, Demars et al. (1996a) argued that the boundary conditions used by Horwitz et al. (1994) were not representative of the ionosphere under most conditions, and investigated the collisionless steady-state problem themselves with different boundary conditions. The two steady-state studies agreed qualitatively in terms of the association between the convection electric field and the density and velocity of the O+ ions. However, Demars et al. (1996a) interpreted the results with a different conclusion, suggesting that electron temperature rather than centrifugal acceleration was a more critical factor for the O+ outflow flux. They further supported that conclusion based on the results of their simulations with time-dependent boundary conditions, which was intended to mimic the changing plasma conditions encountered by the convecting flux tube. We note that other than the associated effect of centrifugal acceleration, convection may provide an alternative explanation for the existence of escaping O+ ions in the polar wind. Schunk and his collaborators have considered convection as a means of transporting O+ ions that are energized elsewhere to the polar wind. Beginning with the work by Schunk and Sojka (1989), they have performed a series of investigations with threedimensional time-dependent models of the ‘‘generalized’’ polar wind, which includes the polar wind as well as other high-latitude regions such as the auroral oval and the cusp, taking into account the relevant physical mechanisms in each region (e.g. Schunk and Sojka, 1997; Barakat and Schunk, 2001; Demars and Schunk, 2002; Schunk, 2007; and references contained therein). These models consisted of a large number of convecting flux tubes covering the entire high-latitude region. Each

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individual flux tube was essentially a one-dimensional fluid system whose boundary conditions were time-dependent due to the convection changing the plasma conditions it encountered. There have also been simplified kinetic studies of the generalized polar wind by means of PIC simulations (Demars et al., 1996b, 1998; Barakat and Schunk, 2001; Barakat et al., 2003). But those studies considered only a single convecting flux tube. Because the O+ energization in generalized polar wind models relied on physical mechanisms in other high-latitude regions, the details of those studies are beyond the scope of this review. They are discussed by Schunk (2007) in this issue. We note that recently Banerjee and Gavrishchaka (2007) have also performed a study based on the original idea by Schunk and Sojka (1997). As mentioned earlier, Demars et al. (1996a) argued that the electron temperature is a significant factor governing the outflow of the polar wind O+ ions. Such an idea had been suggested by Barakat and Schunk (1983) based on their parametric study of the electron temperature with a simplified collisionless kinetic model. The study compared the ion profiles of the steady-state polar wind solutions under isothermal electron temperatures ranging from 1000 to 10,000 K. The O+ outflow velocities at the high altitudes of several RE were found to increase from less than 1 to a few km/s as the electron temperature parameter increased. A similar variation trend was also found for the H+ velocity. In fact, the relations between the ion velocities and the electron temperature as demonstrated by Barakat and Schunk (1983) can be understood qualitatively as follows. The model assumed the electric field to be Ek ¼ 

kT e dne , ne ds

(42)

which, based on the definition in Eq. (24), is equivalent to Eq. (34). As the electrons were a massless neutralizing fluid, the factor ð1=ne Þdne =ds; where ne was the sum of the O+ and H+ densities, generally varied relatively weakly compared with the change in the parameter Te. The electric field in Eq. (42) was thus expected to follow the variation trend of Te, leading to stronger ion acceleration for a higher electron temperature. Using a model very similar to that by Barakat and Schunk (1983), Li et al. (1988) compared the relative significance of the electron temperature and ion heating as an O+ driving and acceleration

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mechanism. These authors mimicked the effect of ion heating by imposing high temperatures for the H+ and O+ distributions at the exobase of their steady-state simplified collisionless kinetic calculations, and showed that with either the perpendicular or the parallel temperature elevated from 3000 to 10,000 K, both the O+ density and outflow velocity would dramatically increase, resulting in an O+ escape flux four or five orders of magnitude larger. By comparing their results for different values of the assumed electron temperature, they agreed with Barakat and Schunk (1983) that for low exobase ion temperature, an increase in the electron temperature from 3000 to 10,000 K could dramatically increase the O+ escape flux. They also pointed out that, however, if the exobase ion thermal energies exceeded about 3 eV, such an increase in the electron temperature would produce a negligible change in the O+ flux. The models by Barakat and Schunk (1983) and Li et al. (1988) treated the entire electron population as a massless neutralizing fluid, described only by a single parameter for its isothermal temperature. There have also been studies based on such a simplified collisionless kinetic approach but with modified assumption for the electron temperature in order to mimic the coexistence of cold ionospheric and hot magnetospheric electron populations in the steadystate polar outflow (Barakat and Schunk, 1984; Ho et al., 1992). Barakat and Schunk (1984) assumed that each of the two electron populations was itself a fluid with its own isotropic temperature and obeyed its own Boltzmann relation (c.f. Eq. (34)), rendering a total electron density, nt, of the form: nt ¼ nc0 expðeFE =kT c Þ þ nh0 expðeFE =kT h Þ,

(43)

where nco (nh0) is the cold (hot) electron density at the exobase, and Tc (Th) is the cold (hot) electron temperature, which was assumed to be constant. Using Eq. (43) to relate the electric potential and the electron density (which was the sum of the O+ and H+ densities), Barakat and Schunk (1984) studied the effect of the hot magnetospheric electrons on the polar wind by examining their results based on the variation of two parameters: the hot/cold electron temperature ratio, Z ¼ T h =T c , and the hot/total electron density at the exobase, m0 ¼ nh0 =ðnh0 þ nc0 Þ. They found that when both parameters were relatively small (for example, Z ¼ 10 and m0 ¼ 0.001), the polar wind solutions were qualitatively similar to those obtained by their previous model (Barakat and Schunk, 1983).

But for higher values of Z or m0, the solution would feature a discontinuity, which was interpreted as a contact surface between the hot and cold electrons. The discontinuity of the electric potential would represent a double-layer that reflected the cold ionospheric electrons and prevented their escape. As for the effect on the O+ ions, the hot magnetospheric electrons reduced the potential barrier for their escape, thereby enhancing the outflow flux and velocity for these heavy ions (see Fig. 10). Ho et al. (1992) focused on the effect of the temperature gradient associated with the presence of hot magnetospheric electrons. In their collisionless PIC simulations, the massless neutralizing electron fluid was assumed to be isotropic everywhere along a polar wind flux tube. A family of parameterized hyperbolic tangent functions was used to model the temperature profile of the electron fluid, which included the hot magnetospheric and the cold ionospheric electron populations. Those functions shared common values at the boundaries of the model, and differed only by a parameter governing the gradient at the mid-point of the simulation range (see Fig. 11). The electric field in the calculations followed that derived by Hultqvist (1971) in connection with the precipitation of plasma sheet hot ions into the ionosphere under the steady-state condition: k a dT e ðsÞ kT e ðsÞ dne Ek ¼  1 þ  , (44) e 2 ds ne e ds where a, the thermal diffusion coefficient was taken to be 1.4. The electric field in Eq. (44), in addition to the usual term derived from the Boltzmann relation, features a contribution from the electron temperature gradient. With the electron temperature profile assumed by Ho et al. (1992) (see Fig. 11), the temperature gradient introduced a downward electric field contribution, which was opposite to the contribution from the last term of Eq. (44) or the usual direction of the polar wind electric field. The steady-state results by Ho et al. (1992) indicated that the sharp electron temperature gradients could locally lead to a positive gradient in the H+ potential energy, in contrast to the monotonically decreasing profile commonly found in classical polar wind solutions. The electron temperature gradients were also shown to introduce some nontrivial variations in the overall O+ potential as shown in Fig. 12a. In general, the variation of the O+ potential could be summarized as follows: (1) the sharper the electron temperature gradient,

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Fig. 10. The effect of the hot/cold electron temperature ratio Z on the electric potential and the O+ velocity, as found in the calculations by Barakat and Schunk (1984). The dotted curves represent the case for Z ¼ 1. The O+ potentials on the left panels are the sum of the electric and gravitational potentials normalized by kT0(O+), where k is the Boltzmann constant and T0(O+) is the O+ temperature at the baropause, assumed to be 3000 K. Note that the left panels are plotted with different scales. The solid and dashed curves in the bottom right panel represent the solutions for Z ¼ 1000 when the discontinuity occurs at the highest and lowest possible altitudes, respectively. (Reprinted from Barakat and Schunk (1984) with permission. Copyright 1984 American Geophysical Union.)

the higher a potential barrier it would introduce, and due to the O+ ions being reflected by the potential barrier, the lower the O+ density above the temperature gradient (Fig. 12b); (2) the O+ potential energy could significantly decrease with altitude under high electron temperatures and small

electron temperature gradients. Ho et al. (1992) found that for the most extreme case considered in their study, where the electron temperature profile featured a very large gradient (profile 5 in Fig. 11), despite the large potential barrier introduced, the overall O+ velocity could still reach 6 km/s in the

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simulation due to the large altitude range of the high electron temperature, as opposed to a 1 km/s velocity if the electrons were cold and isothermal. Besides the hot magnetospheric electrons, there is another electron population, namely the photoelec-

Fig. 11. The assumed electron temperature profiles in the calculations by Ho et al. (1992). Profiles 1–5 correspond to maximum electron temperature gradients at 4.8RE of 8000, 11,000, 15,000, 30,000 and 130,000 K/RE. Profile 0 corresponds to the isothermal case with electron temperature of 4400 K. (Reprinted from Ho et al. (1992) with permission. Copyright 1992 American Geophysical Union.)

trons, whose existence has been associated with the acceleration of the polar wind ions. As a matter of fact, in his historical paper where the term ‘‘polar wind’’ was invented, Axford (1968) postulated that photoelectrons would enhance the electric field and even lead to the escape of heavy ions such as the O+. He did not consider that the thermal (cold) ionospheric electrons play a similar role especially in the nightside part of the polar cap where photoelectrons are absent. In the next section, we shall discuss the physical mechanism that associates photoelectrons with ion acceleration, the experimental motivation and the methodology of various modeling efforts that focused on the effects of photoelectrons in the polar wind. Most of the acceleration mechanisms discussed in this section could operate in the steady-state polar outflow. We conclude this section, however, by noting that there have been studies on the response of the polar wind ions to transient effects. For example, Gombosi and Nagy (1988, 1989) investigated return current generated polar wind transients with their time-dependent hydrodynamic model. They found that return currents might generate significant downward heavy ion flow in the topside ionosphere. But when the return current ceased, the polar ionosphere would rapidly return to its current-free state, resulting in an upward propagating heavy ion transient characterized by a short O+ upwelling event. Such a transient effect may also provide an explanation for O+ ions with considerable upward speed. 5. Effects of photoelectrons in the polar wind The effects of photoelectron flux in the polar wind were first modeled by Lemaire (1972a) and Lemaire and Scherer (1972b) with a collisionless kinetic

Fig. 12. Bulk parameters of O+ corresponding to the assumed electron temperature profiles of Fig. 11. (Reprinted from Ho et al. (1992) with permission. Copyright 1992 American Geophysical Union.)

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approach. The exospheric model consisted of the H+, O+, thermal electrons and photoelectrons. The study compared the steady-state solutions of the model under different amounts of photoelectron flux at the exobase, while the boundary conditions for the other particle species were unchanged. It was found that with a larger photoelectron flux, the ambipolar electric field would increase, accelerating the ions to higher velocities and significantly increasing the O+ escape flux. The flux, the bulk velocity and the parallel and perpendicular temperatures of the thermal electrons were also strongly dependent on the photoelectron flux; an increase in the photoelectron flux would reduce the flux and velocity of the thermal electrons, but enhance their temperatures. The effect of the photoelectron flux on the densities of the other particle species, however, was found to be insignificant, as the photoelectron density remained negligible compared with that of the thermal electrons even though the suprathermal population carried most of the electron flux. Similar conclusions regarding the effects of the photoelectron flux were reached by Lemaire (1972a) based on an exospheric model (see Section 2.2.2) in which the photoelectrons were incorporated into the collisionless kinetic calculations above the exobase boundary. Tam et al. (1995b) explained the association between the ambipolar electric field and the photoelectrons by considering an energy flux mechanism. They pointed out that in a steady-state collisionless plasma outflow along divergent magnetic field lines, every particle species is governed by the following mass and energy conservation equations: q nu ¼ 0, (45) qs B   q 1 ½Q þ nuðmFG þ qFE Þ ¼ 0. (46) qs B w Eqs. (45) and (46) can be derived by taking the zeroth- and second-order (energy) moments of Eq. (1) and neglecting the collisional and timedependent terms. In particular, Eq. (46) expresses a direct relation between the particle energy flux Qw and the electric potential. Due to the two classical polar wind conditions, quasi-neutrality and currentfree flow, Eq. (46) for different species are coupled, such that a larger upward electron energy flux would be consistent with a larger electric potential drop. Because photoelectrons carry a large amount of upward energy flux, their presence is expected to

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enhance the ambipolar electric field, and, therefore, to increase the outflow velocities of the ions. 5.1. Photoelectron signatures in the polar wind Experimentally, there has been increasingly convincing evidence that photoelectrons play a significant role in the dynamics of the polar wind. Early polar cap measurements obtained by the ISIS1 satellite showed evidence of ‘‘anomalous’’ fieldaligned photoelectron fluxes in both upward and downward directions, where the downgoing (return) fluxes were considerably smaller than the outgoing fluxes above a certain energy (Winningham and Heikkila, 1974). Such non-thermal features were confirmed by the DE-1 and -2 satellites (Winningham and Gurgiolo, 1982); outgoing field-aligned electron fluxes in the photoelectron energy range were observed by the HAPI on DE-1 and the Low Altitude Plasma Instrument (LAPI) on DE-2, while evidence of return fluxes was found in the lowaltitude electron distributions measured by the LAPI. These electron fluxes were considered anomalous because their existence could not be related to the idea of thermal conductivity and temperature gradient in classical fluid theories. Similar to the ISIS-1 measurements, the return fluxes observed by DE-2 were comparable to the outgoing fluxes below some truncation energy, but considerably smaller above that. Winningham and Gurgiolo (1982) suggested that the existence of the return fluxes may be due to reflection of electrons by the ambipolar electric field along the geomagnetic field line above the satellite. The truncation energy, obtained by comparing the outgoing and the return electron fluxes, would thus provide an estimate for the potential drop due to the electric field. They found that the truncation energy ranged from 5 to 60 eV, and thus deduced the electric potential drop above the altitude of the DE-2 satellite to be within a similar range, which generally exceeded the predictions by classical polar wind theories. Winningham and Gurgiolo (1982) also associated the variation of the truncation energy with changes in the solar zenith angle at the production layer below the satellite. The solar zenith angle is related to the photoionization rate, which itself is related to the local ionospheric photoelectron density (Jasperse, 1981). These DE satellite observations therefore implied a relationship between the ionospheric photoelectron density below the satellite and the electric potential along the field line above it, and

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were consistent with the idea that the photoelectrons may significantly affect the ambipolar electric field. 5.2. Photoelectron impact on polar wind characteristics While the observations by the DE satellites suggested that photoelectrons may contribute to the dynamics of the polar wind, more recent experimental results from the Akebono satellite have revealed some characteristics of the polar wind, which could be associated with the effects of photoelectrons. In Section 4.1, we have already discussed some observations by the Akebono satellite regarding the unexpectedly significant role of the O+ ions in the polar wind. But perhaps equally unexpected was the day–night asymmetries in the ion and electron features revealed by the insitu measurements of the satellite. Based on the satellite data between 5000 and 9000 km altitudes, Abe et al. (1993) reported remarkably higher outflow velocities for the H+ and O+ ions in the sunlit region than on the nightside. For example, the velocity for the H+ was found to be about 12 km/s on the dayside, but only about 5 km/s on the nightside. Similarly, the O+ velocity in the sunlit region (7 km/s) was about twice of that in the midnight sector (3 km/s). The Akebono satellite also observed a day–night asymmetry in the electron behavior. Yau et al. (1995) separated the electrons observed in the dayside polar wind into two distinct populations according to their traveling direction along the geomagnetic field lines. They found that the temperature of the upstreaming population was greater than that of downstreaming population, i.e., T e;up 4T e;down , indicative of an upwardly directed heat flux. Abe et al. (1996) performed a similar analysis for the nightside electrons, but found that in contrast to the dayside, such up-down anisotropy did not exist. Because photoelectrons exist primarily in the sunlit ionosphere, they are the natural candidate to account for the marked day–night asymmetries observed in the polar wind. The exospheric calculations by Lemaire and Scherer (1972b) and Lemaire (1972a) showed that escaping photoelectron flux could lead to an electric field enhancement and an increase in ion outflow velocities. These results, along with the relationship between the electric potential drop and the solar zenith angle as

observed by Winningham and Gurgiolo (1982) based on DE satellite data, and the day–night asymmetries in ion outflow velocities (Abe et al., 1993) were all consistent with the idea of photoelectrons providing an important acceleration mechanism for the polar wind. 5.3. Modeling of the photoelectron effects Because the ambipolar electric field may depend strongly on the photoelectron flux, it is important to include the mechanisms that influence the dynamics of these suprathermal electrons when modeling their effects on the polar wind. Coulomb collisions are one of such mechanisms operating in the polar wind, especially at low and mid-altitudes. The collisions may transfer the energy of the photoelectrons to other particle components, thus reducing the escaping photoelectron flux. When taking into account the effects of Coulomb collisions, however, one must realize the velocity dependence of the collision frequencies and mean free path, which has been discussed in Section 2.1.1. Because the energies of the individual photoelectrons cover a very wide range, typically from a few eV to tens of eV (Lee et al., 1980), the collision frequencies for these suprathermal electrons are vastly different. Thus, the velocity-averaged approach of moment-based models in describing the Coulomb collisional effects seems inadequate for the photoelectrons. A kinetic approach is necessary for the collisional description of photoelectrons. In addition, the photoelectron distributions in the polar wind are subject to the ‘‘velocity filtration effect’’, which, as discussed in Section 3.1.1, is associated with the velocity dependence of the Coulomb collision frequencies and the non-local nature of the outflow. Scudder and Olbert (1979) first discussed the velocity filtration effect in their steady-state study of the halo electrons in the solar wind, a space plasma setting similar to the polar outflow. Using a simplified collisional operator, their calculations demonstrated that a Maxwellian electron distribution could transform into a distribution with non-thermal tails as the heliocentric distance increased. The non-thermal tails, which resembled the halo electron population, formed because electrons with lower velocities were preferentially thermalized by the collisions; the effect of the preferential thermalization accumulated along the solar wind outflow, leading to the change in the velocity distribution. The photoelectron distributions

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in the polar ionosphere are far from Maxwellian, but their evolution along the polar wind is influenced by the velocity filtration effect as the photoelectrons of lower energies are more subject to thermalization compared with those with higher energies. In order to take into account the velocity filtration effect, an accurate description of the polar wind photoelectrons must also consider the nonlocal or global nature of the outflow. 5.3.1. Global kinetic collisional test-particle calculations Yasseen et al. (1989) provided a global kinetic collisional description of the photoelectrons in the polar wind. Their model, adapting a Monte Carlo simulation technique developed by Retterer et al. (1987) for WPI, treated the photoelectrons as test particles. The study used a boundary photoelectron distribution that was consistent with the observed polar wind data at low altitudes, a background electron density profile based on a fluid simulation and an assumed electric field consistent with DE satellite measurements (Winningham and Gurgiolo, 1982). The effect of the Coulomb collisions between the photoelectrons and the background electrons was taken into account as described in Section 3.1.1. The results of their steady-state calculations indicated that photoelectrons not only could provide an explanation for the observed anomalous fieldaligned energy fluxes at the high-altitude polar wind, but also could be responsible for the downward electron distribution observed at low altitudes (Winningham and Gurgiolo, 1982). Tam et al. (1995a) examined the significance of the photoelectron heat flux in the polar wind with global kinetic collisional test-particle calculations similar to those by Yasseen et al. (1989). However, in Tam et al.’s study, the assumed electric field and the density, velocity and temperature profiles of the background thermal species were obtained by regeneration of a polar wind solution with the 16-moment model by Ganguli et al. (1987). The background species included the thermal electrons, the H+ and an O+ population that was in hydrostatic equilibrium, assumed bound by the terrestrial gravitational field. The electric field in the background only amounted to a potential drop of less than 1 V, which was much smaller than those observed by the DE satellites (Winningham and Gurgiolo, 1982). By comparing the heat flux of photoelectrons in their test-particle calculations with that of the thermal electrons in the back-

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ground, Tam et al. (1995a) concluded that the photoelectrons dominated their thermal counterpart in terms of the heat flux, and that consequently, the direction of the overall electron heat flux was upward, as dictated by the photoelectron contribution. They also estimated that due to the presence of the photoelectrons, a significantly larger electric field would be required to maintain the quasi-neutrality and current-free conditions of the polar wind. Although the conclusion by Tam et al. (1995a) regarding the effect of photoelectrons on the electric field agreed with that drawn from collisionless studies (Lemaire and Scherer, 1972b; Lemaire, 1972a), the collisional model did not represent a complete polar wind description. Aside from the lack of self-consistency due to the test-particle approach, the assumption of a static O+ population may no longer be valid under a significant increase in the electric field. After all, as discussed in Section 4.1, data from the Akebono satellite (Abe et al., 1993) had indicated that the polar wind O+ ions have higher outflow velocities on the dayside, the sector where photoelectrons are present, as compared with the nightside. Nevertheless, the theoretical study provided a motivation as well as a stepping stone for the development of a selfconsistent polar wind theory that included the global kinetic collisional physics of the photoelectrons. 5.3.2. Global kinetic collisional calculations: self-consistent hybrid model The aforementioned issues of self-consistency and non-static O+ ions were later resolved in a steadystate collisional model developed by Tam et al. (1995b). The model, known as the self-consistent hybrid model, has been discussed in Section 3.1.2, where we describe its kinetic and fluid components and its iterative scheme to obtain self-consistent collisional solutions. Here, we emphasize the treatment of photoelectrons in the model. As mentioned before, the model described the dynamics of the photoelectrons with global kinetic collisional calculations based on a Monte Carlo simulation technique. Because the photoelectrons in the polar wind have a relatively low density compared with the thermal electrons, they were treated as test particles in the model, but only for the purpose of Coulomb collisions. Their density and velocity were taken into account in the quasi-neutrality and current-free polar wind conditions (Eqs. (3) and (4)). Moreover, their various transport properties, including their

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2

(10-4 J m-2 s-1)

Electron Heat Flux

1.5 1 Qe,suprathermal 0.5 0 Qe,thermal

Qe,thermal

⏐

-0.5 50

Qe,suprathermal

change in momentum and energy due to collisions with other particle species, were factored into the determination of the thermal electron temperature and the self-consistent ambipolar electric field in the fluid calculations (see Eqs. (39) and (40)). The self-consistent hybrid model signified two breakthroughs in polar wind modeling. First, it was the first model to successfully incorporate the global kinetic collisional photoelectron effects into a selfconsistent polar wind description. Second, due to its collisional kinetic treatment of the ions, it was the first steady-state model to generate a global selfconsistent polar wind solution that spanned continuously from a collisional subsonic regime at low altitudes to a collisionless supersonic regime at high altitudes. The self-consistent hybrid model has been used to examine the role of photoelectrons in the driving and acceleration of the polar wind. The polar wind model was first applied to the sunlit region where photoelectrons primarily exist (Tam et al., 1995b). It was later extended to simulate the nighttime conditions in order to provide a comparison between the dayside and nightside polar wind (Tam et al., 1998a, b). For the dayside polar wind, Tam et al. (1995b, 1998a, b) found that the total electron heat flux was upward, in agreement with observations by the Akebono satellite (Yau et al., 1995). As shown in Fig. 13, the photoelectrons were responsible for such a result as their contribution to the heat flux dominated that from the thermal electrons in terms of magnitude. By comparing the results based on dayside and nightside polar wind conditions, Tam et al. (1998a, b) showed a considerable difference in the self-consistent ambipolar electric potential drop across the altitude range of their simulations: the electric potential drop from 500 km to 2RE altitudes was only about 2 V on the nightside where there were no photoelectrons, but was found to vary from 5 to 12 V depending on the photoelectron concentration (density ratio between the photoelectrons and the thermal electrons) in the dayside ionosphere. Their results showing the increase of the self-consistent electric potential drop with the photoelectron concentration were apparently consistent with the DE observations by Winningham and Gurgiolo (1982), who pointed out the association between the potential drop and the solar zenith angle. The increase in the electric potential drop, or equivalently, the ambipolar electric field due to higher relative photoelectron densities led to an

⏐

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40 30 20 10 0

2

4

6

8

10

12

Altitude (103 km) Fig. 13. Comparison of the electron heat fluxes by Tam et al. (1998b). Top panel: heat fluxes carried by the thermal electrons ðQe;thermal Þ and photoelectrons ðQe;suprathermal Þ. The heat flux contribution by the thermal electrons was in the downward direction (negative sign) while that by the photoelectrons was upwardly directed (positive sign). Bottom panel: the ratio of the magnitudes of the two heat fluxes.

increase in the ion outflow velocities in the selfconsistent calculations by Tam et al. (1998a, b). The remarkable difference in the O+ outflow velocity due to different ionospheric photoelectron concentrations can be seen in Fig. 14, which includes the results for the nightside polar wind solutions as well as three dayside solutions with different boundary relative photoelectron densities. These results not only addressed the Akebono observation of a day–night asymmetry in ion outflow velocities (Abe et al., 1993), but also supported the idea of a photoelectron-driven polar wind. The photoelectron-driven polar wind theory also addressed the Akebono observation of the day– night asymmetry regarding the electron anisotropy (Yau et al., 1995; Abe et al., 1996; see also Section 5.2). Tam et al. (1998a, b) found that the presence of photoelectrons in the sunlit polar wind, due to their upward heat flux contribution, led to a

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Fig. 14. Profiles of the O+ outflow velocity as found by Tam et al. (1998b). The four cases had identical boundary conditions, except for the density ratio of photoelectrons to thermal electrons (ns/ne) at the lower end of the calculations, which have the following values: (a) 0; (b) 5  104; (c) 103; (d) 1.5  103.

Fig. 15. Parallel temperatures for upward and downwardmoving electrons in the self-consistent hybrid calculations by Tam et al. (1998a). The dayside temperatures are represented by the solid and dot–dashed lines; the nightside temperatures, which were virtually equal for the two electron populations, are represented by the dashed line. (Reprinted from Tam et al. (1998a) with permission. Copyright 1998 American Geophysical Union.)

larger parallel temperature for the upward electron population compared with its downward counterpart. On the nightside, where photoelectrons are absent, no such anisotropy was found in their selfconsistent studies (see Fig. 15). 5.3.3. Self-consistent collisionless hybrid calculations Khazanov et al. (1997) investigated the upper limit of the potential drop that should be expected in the polar wind due to the presence of photoelectrons. The study was based on a steady-state selfconsistent collisionless hybrid model. These authors pointed out that the exclusion of collisional

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processes would lead to easier escape of the photoelectrons and therefore to the formation of the largest possible field-aligned electrostatic potential drop over the polar cap in the scenario of a photoelectron-driven polar outflow. The model was similar to that by Tam et al. (1995b, 1998a, b) as it also relied on an iterative approach between kinetic and fluid calculations to generate self-consistent solutions. The kinetic approach was used for the O+, H+ and photoelectron distributions, and the fluid approach was to describe the thermal electrons and the electric field. The ions and the electrons were coupled based on the quasi-neutrality and current-less conditions through the self-consistent electric field. Unlike Tam et al.’s model, whose collisional kinetic approach was based on particle simulations, the kinetic calculations by Khazanov et al. (1997) relied on identifying the regimes for the various categories of particle trajectories in the phase space, a solution method feasible for the collisionless approximation (see Section 2.1.2). By assuming a monotonic total potential energy for each particle species and the absence of incoming particles from the outermost regions of the model, the collisionless kinetic calculations took into account the particles that had enough energy to escape the Earth’s gravitational field, as well as ballistic particles that could not escape and would be reflected to the ionospheric boundary. Because the plasma was considered collisionless, a particle’s magnetic moment and total energy is conserved under such an assumption; the calculations of the particle trajectories are then described by Eqs. (21)–(23). For a given electric potential profile, the model then determines, based only on the total energy of a particle, whether the particle would eventually escape or be reflected, and in the case for a ballistic particle, where the reflection occurs. That allows the determination of the evolution of the particle distribution functions from the lower, ionospheric boundary to any arbitrary position along a magnetic field line. The densities and particle fluxes can then be found by solving the integrals that represent the corresponding moments of the distribution functions. The fluid approach of the self-consistent collisionless calculations (Khazanov et al., 1997) was very similar to that by Tam et al. (1995b, 1998a, b), which are based on Eqs. (3), (4), (39) and (40). The idea of the fluid approach in both models was to determine an updated electric field or electric

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potential profile that could be used by the kinetic calculations in the iterative procedure. Both models determined the thermal electron density and particle flux (or equivalently, density and velocity) by using the results of the kinetic calculations for the ions and photoelectrons, and the quasi-neutrality and current-less conditions, Eqs. (3) and (4). The thermal electron density and particle flux were then utilized in the coupled momentum and energy equations to find a new electric potential profile as well as the thermal electron temperature, which was assumed isotropic. Unlike Tam et al. (1995b, 1998a, b), whose momentum and energy equations (Eqs. (39) and (40) respectively) included the contributions from the entire electron population, Khazanov et al. (1997) removed the photoelectron component from the fluid equations by using the kinetic equation for those suprathermal electrons. Neglecting the effects of Coulomb collisions and the gravitational force for the electrons, their fluid equations were qðne T e Þ 1 qðAme ne u2e Þ   ene E k ¼ 0, qs A qs   q Qwe B  ene E k ue ¼ 0, qs B



(47)

(48)

where AB1 was the cross-sectional area of the flux tube, and Qwe, the thermal electron energy flux (c.f. Eq. (41)), was separated into the convective particle motion and the heat flux Qe:   (49) Qwe ¼ ne ue 52T e þ 12me u2e þ Qe . In particular, the formal definition of Qe, the thermal electron heat flux, is Z Qe  dv 12mðv  ue Þ2 ðvk  ue Þf e , (50) where fe is the distribution of the thermal electrons (the entire electron population with the exclusion of the photoelectrons), and ue is their mean velocity. By comparing Eqs. (47)–(49) with Eqs. (39) and (40), we note that besides the consideration of the Coulomb collisional effects, the only other difference between these two systems of fluid equations is the closure assumption involving Qe, the thermal electron heat flux. Tam et al. (1995b, 1998a, b) recognized that the photoelectron population, whose distribution had already been determined in the kinetic calculations, dominated the electron heat flux. They assumed that the thermal electron heat flux was negligible in comparison with its suprather-

mal counterpart; or in other words, the skewness of the thermal electron distribution was insignificantly small when compared with that for the entire electron population. With such an assumption, they approximated the thermal electron distribution with a drifting Maxwellian, rendering a zero value for Qe, a term that would otherwise exist in Eq. (40). Khazanov et al. (1997) interpreted Qe in a different way, treating the heat flux as a thermal conductivity flux with the following expression (Banks and Kockarts, 1973): Qe ¼ w0 T 5=2 e

qT e , qs

(51)

where w0 was a constant associated with the thermal conductivity. Khazanov et al. (1997) applied the self-consistent collisionless hybrid calculations to study a number of cases with the photoelectron concentrations ranging from 0% to 1% at 500 km altitude. The electric potential drop along the polar wind was the main emphasis of their study. Because of the exclusion of collisional processes, the values determined in their study were considered the upper limits for the potential drop. As in the study by Tam et al. (1998b), Khazanov et al. (1997) showed that the electric potential drop generally increased with the photoelectron concentration. Although the collisionless study considered an altitude range from 500 km to 5 RE, it was found that most of the potential drop occurred within the initial 2–3RE. Therefore, it is feasible to compare the calculation results for the electric potential with those by Tam et al. (1995b, 1998b), who examined the polar wind from 500 km to 2RE. Tam et al. (1995b, 1998b) examined different polar wind cases, some with low boundary ion temperatures around 3000 K, and some near 10,000 K. Boundary ion temperatures of 2000 K were used in the study by Khazanov et al. (1997) who pointed out that doubling these boundary values decreased the electric potential drop below 2RE by 20%. Although the boundary ion temperatures, particularly that of the O+ ions, have a nonnegligible effect on the electric potential, such an effect is not as significant as that due to the boundary photoelectron concentration (Tam, 1996). We shall neglect the difference due to the boundary ion temperatures in the discussion below, which considers the cases of low ion temperatures. For these cases, both groups of authors have

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published their results of the electric potential for boundary photoelectron concentrations of 0.05% and 0.1%. Khazanov et al. (1997) showed that the electric potential drops were close to 6 V in the former case, and 6.47 V for the latter in their selfconsistent collisionless study. Tam et al. (1998b) found an electric potential drop of about 5 V with a 0.05% photoelectron concentration at the boundary. Considering that the value provided by the collisionless study was an upper limit, one may argue that the results from the two models in this case seemed to be consistent. However, with a boundary photoelectron concentration of 0.1%, Tam et al. (1995b) showed that the electric potential drop was about 12 V, considerably larger than the upper limit found by the collisionless calculations. Khazanov et al. (1997) also compared some other results between the two models for the case of 0.1% photoelectron concentration, as shown in Fig. 16. Because of the relatively large electric potential drop obtained by Tam et al. (1995b), their results featured significantly higher ion outflow velocities. In addition, their thermal electron temperature was

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also much higher than that obtained from the collisionless calculations. Coulomb collisions may provide part of the explanation for the discrepancy in the thermal electron temperature between the two models. Thermal electrons are expected to gain energy through their Coulomb interaction with the photoelectrons. In fact, besides such direct interaction, there is another way in which Coulomb collisions may contribute to the heating of the thermal electrons when photoelectrons are present. As discussed earlier, photoelectrons enhance the ambipolar electric field, and in turn, lead to higher ion outflow velocities. When the ions are being accelerated by the photoelectron-enhanced ambipolar electric field, they would become more energetic when they collide with the thermal electrons. As a result, the amount of energy transferred from the ions to the thermal electrons due to their Coulomb interactions tends to increase. The Coulomb collisional heating for the thermal electrons, whether through their interactions with the photoelectrons or with the ions, is stronger when the photoelectron

Fig. 16. Comparison of the results of Khazanov et al.’s (1997) collisionless calculations and with those of Tam et al.’s (1995b) collisional calculations: (a) O+ and H+ densities; (b) ion velocities; (c) electric potential; (d) thermal electron temperature Te. The percentages in the labels indicate the photoelectron concentration in the collisionless calculations. The case with 0.1% photoelectron concentration is plotted with solid and dotted lines; that with 0% plotted with dashed and dot-dashed lines. T95 (solid and dotted lines in bold) represents the case considered in Tam et al. (1995b), in which the photoelectron concentration was 0.1%. The expression ‘‘a0ne’’ in one of the labels in (d) represents the case where a source term of such an expression was added as an energy input for the thermal electrons to mimic the effect of their collisions with the photoelectrons. The curve for such a case with a0 ¼ 5  104 eV s1 essentially overlaps with the solid line (.1%) in (d). (Reprinted from Khazanov et al. (1997) with permission. Copyright 1997 American Geophysical Union.)

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concentration is larger. The thermal electron temperature was sensitive to the photoelectron concentration according to the calculation results by Tam et al. (1998b). The discrepancy between the results of Tam et al. (1995b, 1998a, b) and Khazanov et al. (1997), particularly in the case of high photoelectron concentration, perhaps can ultimately be explained by the work of Khazanov et al. (1998). In that study, the authors discussed the constraints on the field-aligned potential energy structure in the traditional collisionless kinetic calculation approach. A common trait of models with such an approach is that the accessibility of a particle to an arbitrary position in space, by assumption, depends only on its total energy. The assumption introduces constraints on the functional dependence of the potential energy with respect to the magnetic field, B. These constraints were first recognized and discussed by Chiu and Schulz (1978), who found two constraints in their consideration of a potential energy that involved only the field-aligned electric field. Khazanov et al. (1998) generalized the potential energy (P) to include the gravitational field in the two constraints: (1) dP/dB must be monotonic and (2) d2P/dB240 if dP/dBo0. These constraints must be satisfied for every population in the collisionless kinetic calculations throughout the entire spatial domain. The reason for these constraints is that the total energy alone does not determine the actual accessibility of a particle along magnetic field lines. One must also consider its magnetic moment. For example, there are situations where the total energy of a particle is high enough to overcome the escape potential, but because of conservation of its magnetic moment, its parallel velocity may drop to zero somewhere along the magnetic field line, and the particle will actually be reflected because vJ vanishes there. Khazanov et al. (1998) discussed some of the factors that could cause the violation of the two constraints for the O+ ions in the polar wind. In particular, the increase in the electric potential difference along the field line due to the increase in photoelectron concentration could be a factor. The study concluded that even a photoelectron concentration as high as 0.03% would cause the O+ potential energy profile to violate the constraints of the collisionless kinetic calculations by Khazanov et al. (1997). Because the kinetic calculations by Tam et al. (1995b, 1998a, b) were based on simulations, which did not require identifying the different trajectory regimes in the

phase space, their studies were not subjected to such an issue. 5.3.4. Discontinuous solutions Wilson et al. (1997) considered how a zero electric current could be achieved in the steady-state polar wind when there were large upward photoelectron fluxes in the ionosphere, accompanied by the presence of downward streaming magnetospheric particles. They examined the possibility of a large localized electric potential barrier reflecting a majority of the photoelectrons, thus reducing their escape flux. Their study was based on a simplified collisionless kinetic approach in which the thermal electrons were assumed to be a stationary massless neutralizing fluid. The model consisted of two particle sources, one at each end of a flux tube that covered the altitude range of 500 km—9RE. The ionospheric boundary was a source for the H+, O+ and photoelectrons, while the magnetospheric particle source provided hot, downward streaming electrons and H+ ions. In order to simulate a localized potential barrier for the upward streaming electrons, these authors looked for solutions that featured a discontinuity in the electric potential. Thermal electrons were assumed to follow the Boltzmann relation at altitudes below the discontinuity, thus relating the electric potential with their charge neutralizing density. Above the discontinuous boundary, thermal electrons were assumed to be absent as all of them were supposedly reflected by the localized electric field. Self-consistent solutions of the model were found by iterations on the electric potential until charge neutrality was achieved. The current-free condition of the polar wind was not applied to the iterative calculations. However, for a given set of boundary conditions, multiple solutions were possible; only solutions that turned out to have a negligible electric current were feasible for the study. The simplified collisionless kinetic model by Wilson et al. (1997) was later modified by Su et al. (1998) to combine steady-state calculations of three different approaches in order to describe the polar wind at different altitudes. Solutions of the model were obtained by iterations among the results in the three altitude ranges. In the altitude range of 2–9RE, the calculations by Su et al. (1998) were based on a simplified collisionless kinetic approach, same as that for the model by Wilson et al. (1997). Between the altitudes of 800 km and 2RE, the model used simplified kinetic PIC simulations to describe the

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ions and photoelectrons while the thermal electrons remained a massless neutralizing fluid. The PIC simulations included the H+ and O+ self-collisions, and the influence by the O+ on the H+ due to their collisions (but not vice versa) with a randomized collisional scheme (Ho et al., 1997; see also Section 3), while the photoelectrons were collisionless. Some ion heating terms were also incorporated into the calculations in this altitude range to mimic the effects of the collisions between the ions and the thermal electrons. From 120–800 km altitudes, the calculations were based on a modified and truncated version of the field line interhemispheric plasma (FLIP) model, a fluid model that included also the interactions involving the neutral particles and was originally applied to closed magnetic field lines (Richards and Torr, 1985). Both Wilson et al. (1997) and Su et al. (1998) considered photoelectron fluxes that represented the solar minimum and solar maximum cases, and obtained discontinuous solutions, which featured a large electric potential drop of the order of 40–60 V. As Su et al. (1998) pointed out, the discontinuities were at high altitudes; always above 3RE. Su et al. (1998) also compared their

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photoelectron-driven polar wind results with the case equivalent to the classical polar wind, in which both the photoelectrons and the magnetospheric particle populations were neglected. They found that the solutions for the two cases were similar at altitudes below the discontinuity; but at higher altitudes, the results differed in that the solution was continuous when the photoelectrons and the magnetospheric particles were absent (see Fig. 17). The study seemed to raise the interesting idea that in terms of their influence on the electric field and the ions, the effects of the photoelectrons and the magnetospheric particles would cancel out each other at low altitudes; when these populations were all present, the ambipolar electric field, except at the altitude where the discontinuity occurred, was not significantly different from that of the classical polar wind anywhere else. The theory therefore suggested that the acceleration of the O+ ions in the presence of photoelectrons and magnetospheric particles was primarily due to the discontinuity in the electric potential, which corresponded to an electrostatic double layer. We should note that discontinuous solutions of the polar wind had been predicted by Barakat and

Fig. 17. Comparison of the electric potential (left) and the ion bulk velocities (right) by Su et al. (1998) for the cases where photoelectrons and hot magnetospheric electrons were both present (solid and dot–dashed lines) and both absent (long-dashed and short-dashed lines). The horizontal dotted lines indicate the interface between the collisional and collisionless regimes. (Reprinted from Su et al. (1998) with permission. Copyright 1998 American Geophysical Union.)

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Schunk (1984), whose results were briefly discussed in Section 4.2 and Fig. 10. However, the discontinuity in their solutions was shown to occur within a given altitude range rather than a specific altitude, as the electric potential profile featured double values over a finite range of altitudes (see the middle and bottom panels of Fig. 10). Barakat et al. (1998b) performed similar simplified kinetic calculations with a PIC technique that was first applied to the polar wind by Wilson et al. (1990) (see Section 2.2.1), and were able to find steady-state solutions with discontinuity at a distinct altitude. Unlike the studies by Wilson et al. (1997) and Su et al. (1998), the models by Barakat and Schunk (1984) and Barakat et al. (1998b) did not include photoelectrons and magnetospheric ions. The electrons common in those models were a hot population originating from the magnetosphere and a cold population from the ionosphere (see Eq. (43)). Despite the different sets of particle populations considered, those studies shared similar results in terms of the magnitude of the double-layer electric potential (tens of eV), the direction of the doublelayer electric field (upward), and the altitudes at which the discontinuity occurred (a few RE). In consideration of these similarities, it is reasonable to conclude that the existence of photoelectrons is irrelevant to the formation of discontinuous electrostatic potential and double-layer electric field in those theoretical results, although such features have yet to be actually observed in the polar wind. If such features do exist in the polar wind, the mixing of hot magnetospheric electrons with their cold ionospheric counterparts appears to be a possible mechanism to account for them. 6. Summary Since Axford (1968) and Banks and Holzer (1968) proposed the existence of the polar wind, there have been many different models of such an outflow along open, divergent field lines. Early polar wind theories were mostly on the classical polar wind, in which cold plasma from the ionosphere provides the only particle source for the steady-state quasineutral current-free outflow. However, even for the description of the classical polar wind, there were two traditional schools of thought: one favoring the moment approach, and the other favoring the collisionless kinetic approach, which has also been known as the exospheric approach in the literature. As we have discussed, each of the two

traditional approaches has its own advantages and drawbacks. Thus, models that combined these two approaches began to appear. Such models could generally be categorized into two new classes: the simplified collisionless kinetic or hybrid approach in which the ions were described by kinetic calculations and electrons were assumed to follow the Boltzmann distribution; and the semi-kinetic approach where moment equations described the outflow at low altitudes while kinetic calculations were applied at high altitudes. We have discussed all these modeling approaches for the classical polar wind. In the literature, there have been studies comparing simplified collisionless kinetic calculations with moment-based models. It was generally found that results based on the two different approaches would agree in the collisionless regime of the classical polar wind. An apparent drawback of the collisionless kinetic approach is the lack of collisions in its description. Later generations of kinetic calculations have attempted to remedy such a drawback. Over the years, the formulations of collisional effects have become more and more sophisticated. Early efforts approximated the Coulomb interactions with simplified collision operators in the kinetic equations. In later studies, linearized versions of Coulomb collision operators were used. Such an approximation resulted in a test-particle approach for examining the effects of Coulomb collisions. Recently, a self-consistent hybrid approach based on an iterative scheme was able to take into account the nonlinear effects of the Coulomb collision operators. There have been a number of non-classical polar wind theories focusing on the acceleration of the O+ ions evidenced by satellite observations of escaping O+ fluxes. Non-classical effects such as those due to WPI, centrifugal acceleration, hot magnetospheric electrons and photoelectrons have been suggested as acceleration mechanisms for the O+ ions in studies of the steady-state polar wind outflow. In addition to those steady-state studies, we have discussed the modeling efforts for nonclassical effects, such as flux tube convection and electric current, in a time-dependent polar wind. These time-dependent effects have also been shown to affect the dynamics of the O+ ions. Among the various non-classical effects, we have emphasized our discussion on those due to photoelectrons, as the existence of these suprathermal electrons primarily in the sunlit region provides a natural explanation for the day–night asymmetric

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features of the polar wind ions and electrons observed by the Akebono satellite. The photoelectron concentration in the ionosphere is believed to be a significant factor that may affect the ion outflow velocity. A higher density ratio of photoelectrons to thermal electrons can enhance the ambipolar electric field, which in turn accelerates the ions to larger outflow velocities. Acknowledgments The authors are indebted to Joseph Lemaire for useful discussions and valuable suggestions for improving the quality of this article. We also thank George Khazanov for discussions. V. Pierrard thanks the Belgian Science Policy SPP for the grant Action 1 MO/35/010. The portion of this work performed at MIT was partially supported by NSF, NASA and AFOSR. Appendix A. The 16-moment equations A.1. The 16 moments and the approximation of the distribution functions The ‘‘generalized transport’’ approach describes a given particle species by a number of its macroscopic (velocity-averaged) physical quantities. Each of these physical quantities is related to a velocity moment. In the 16-moment model, these physical quantities are: density (zeroth-order moment, 1 scalar value) Z n ¼ dvf , (A.1) average drift velocity (first-order moment, 3 scalar values) u ¼ hvi,

(A.2)

pressure tensor (second-order moment, 6 distinct scalar values) P ¼ nmhcci,

(A.3)

and heat flux vectors for parallel and perpendicular energies per unit density (third-order moment, 3 scalar values each) D E qk ¼ m c2k c , (A.4) 1

q? ¼ m c2? c , (A.5) 2 where m is the mass of the species, f is its distribution function, c  v  u and / S denotes the average of

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the quantity inside the brackets, i.e., for an arbitrary quantity (scalar, vector or tensor) A Z 1 dvf A: (A.6) hAi ¼ n Each of these physical quantities is treated as a parameter in the model. Note that other physical quantities up to the third order of the velocity moments can be expressed in terms of combinations of these 16 parameters. For example, the parallel and perpendicular temperatures, and the stress tensor are, respectively, related to the pressure tensor by 1 T k ¼ P : e3 e3 , n

(A.7)

1 P : ðe1 e1 þ e2 e2 Þ, 2n

(A.8)

s ¼ P  nT ? I  nðT k  T ? Þe3 e3 ,

(A.9)

T? ¼

where I is the unit dyadic, and (e1, e2, e3) is a set of orthogonal unit vectors with e3 in the direction parallel to the magnetic field. The system of 16-moment equations uses Eqs. (5) and (6) as the assumption for the distribution function, where the expression for C consists of the independent variable v, as well as parameters that represent the various physical quantities. Specifically, C¼

b? ½b ðc2  c22 Þðs : e1 e1 Þ þ 2b? ðs : e1 e2 Þc1 c2 þ 2bk ðs : c? ck Þ 2nm ? 1     b? bk b2 b c2 b c2 1  ? ? q?  ck  ? 1  ? ? q?  c?  m 4 m 2 ! b2k bk c2k b? bk 1 ð1  bk c2k Þqk  c? ,  qk  ck  2m 3 2m

ðA:10Þ where c1;2 ¼ c  e1;2 ; bk ¼ m=T k and b? ¼ m=T ? (Barakat and Schunk, 1982). Note that the parametric dependence of the distribution function defined in Eqs. (5), (6) and (A.10) is consistent with the definitions of the physical quantities in Eqs. (A.1)–(A.5) and (A.7)–(A.9). In the polar wind application, due to the gyrotropic nature of the transport, the set of 16-moment equations reduces to six different parameters, whose transport properties are described by Eqs. (13)–(18). It is interesting to point out that the special mathematical function adopted for C or for highorder approximation of f, is such that the zerothorder moment (density), the first-order moments (u) and the parallel and perpendicular temperatures (TJ, T?) calculated with f B ð1 þ C þ   Þ are the

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same as those calculated with fB, the zeroth-order approximation of the velocity distribution function. In other words, these six lower-order moments are already fully determined by the zeroth-order approximation fB. In fact, the same is true for the lower-order moments in other generalized transport models based on Grad’s (1949) expansion of the velocity distribution; these lower-order moments, as in the case of the Chapman–Enskog expansion, are fully determined by the zeroth-order approximation. It is important to note that solution of the Fokker–Planck equation described in Appendix C are expanded in orthogonal polynomials, but that the values of the lower-order moments depend on the degree of the polynomial expansion, i.e., on the number of terms used to approximate the velocity distribution function.

Moment-based expressions for collisional terms were derived by Burgers (1969) for various microscopic interactions under the approximation of small drift velocity differences between the interacting species. The Coulomb collisional terms in Eqs. (14)–(18) (used by Ganguli et al., 1987) are as follows, where the subscript a labels the species under consideration and the running subscript b labels the background particle species: (A.11)

  X ma nab 6 dT ak 4mb 4 T bk  2 þ T ak þ T b? ¼ 5 5 dt ðm þ m Þ 5m a b a b      4mb 6mb T a gab , þ T a? þ 2T b þ 4 þ 5ma ma

ðA:12Þ X ma nab dT a? ¼ dt ðma þ mb Þ b

 1 dT ak ,  3ðT b  T a Þ þ mb ðub  ua Þ2 ð1 þ gab Þ  2 dt

ðA:13Þ X dqak ¼ nab qak , dt b X dqa? ¼ nab qa? , dt b

vab ¼

nb ð32pÞ1=2 e2a e2b ðma þ mb Þðln LÞ expðw2ab Þ 3m2a mb a3ab (A.16)

is the Coulomb collision frequency, ln L is the Coulomb logarithm, ea and eb are the algebraic electric charge for species a and b respectively, and T j ¼ 13T jk þ 23T j? ; j ¼ a; b,

(A.17)

a2ab ¼

2T a 2T b þ , ma mb

(A.18)

w2ab ¼

ðub  ua Þ2 , a2ab

(A.19)

4 4 8 6 wab þ 315 wab . gab ¼ 25w2ab þ 35

A.2. Collisional terms

dua X ¼ nab ðub  ua Þð1 þ gab Þ, dt b

where

(A.14)

(A.20)

Appendix B. Test-particle formulations of the Fokker–Planck equation The effects of Coulomb collisions on a species j can be expressed in the Fokker–Planck form, Eqs. (36)–(38). In this appendix, we consider a special case where test-particle approximation applies to the species j, and the background species for the collisions are distributed as drifting Maxwellian. The test-particle approximation allows the terms with b ¼ j to be dropped from the summations in Eqs. (37) and (38). The distributions of the background species, by assumption, are nb f b ðvÞ ¼ 3=2 3 exp½ðv  ub Þ2 =v2b , (B.1) p vb where ub is the bulk velocity of the background species b. The dependence of fb on t and s has been dropped for simplicity, but the formulae in this discussion also apply when there is temporal and spatial dependence provided the density, bulk velocity and temperature of the background species do not vary in time and space faster than the time scale of the collision frequency and more steeply than the collisional mean free path. With (B.1), Eqs. (37) and (38) become: X aj ðvÞ ¼  njb (B.2) s cb , b

(A.15)

Dj ðvÞ ¼

X b

cb cb Djb k c2b

þ

Djb ?

  cb cb I 2 , cb

(B.3)

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where I is the unit dyadic, cb  v  ub ,   8pnb q2j q2b ln L mj jb ns ðvÞ ¼ 1þ Gðcb =vb Þ, mb m2j cb v2b Djb k ðvÞ ¼

Djb ? ðvÞ ¼

8pnb q2j q2b ln L m2j cb 4pnb q2j q2b ln L m2j cb

Gðcb =vb Þ,

(B.4)

(B.5)

½Fðcb =vb Þ  Gðcb =vb Þ, (B.6)

FðxÞ is the error function Z x 2 2 FðxÞ  pffiffiffi dyey , p 0

(B.7)

FðxÞ  xF0 ðxÞ , GðxÞ ¼ 2x2

(B.8)

with F0 ðxÞ  dF=dx. Eq. (B.2) indicates that the Coulomb collisional friction is in the direction opposite to cb, the velocity of the particle relative to the drifting background. Eqs. (B.3)–(B.5) describe the diffusion due to the Coulomb collisions, with Djb k denoting the diffusion coefficient in the direction parallel to cb, and Djb ? in directions normal to that. We note that for a non-drifting Maxwellian background (i.e., ub ¼ 0 and cb ¼ v), the expressions for aj and Dj reduce to those given in Hinton (1983), for example.

The spectral method used in Pierrard and Lemaire (1998) to solve the Fokker–Planck equation is based on the expansion of the solution in polynomials: f ðs; v=vth ; mÞ ¼ expðv

where y ¼ v=vth . For numerical integration of moments of typical Maxwellian velocity distribution functions over the half-infinite range, the method of solution is very efficient with the natural weight function y2 expðy2 Þ. The recursion relation of these speed polynomials can be found in Shizgal (1979). The spectral method based on the speed polynomials permits the expressions of the moments of the velocity distribution function as a simple linear combination of the expansion coefficients. For instance, the number density:

n1 N1 X X

(C.3)

It is important to point out that this particular type of polynomial expansions of the velocity distribution function as in Eq. (C.1) is quite different from that used in the Chapman–Enskog or Grad’s (1949) or Schunk’s (1977) expansions. There are no a priori closure relations imposed to the coefficients alr ðsÞ, and the density, flux, average velocity and temperatures of the particle is not only determined by fB, the choice of the zeroth-order approximation of the velocity distribution function. In other words, all the moments of the velocity distribution function, even the lower-order ones (e.g. density, average velocity and temperatures), are affected when the number of terms of the polynomial expansion (C1) is increased. References

Appendix C. Polynomial expansion to solve the Fokker–Planck equation

=v2th Þ

(C.2)

0

nðsÞ ¼ 2pp1=4 a00 ðsÞ.

and

2

of orthonormal functions: Z 1 y2 expðy2 ÞS s ðyÞSr ðyÞdy ¼ dsr ,

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! alr ðsÞPl ðmÞSr ðv=vth Þ ,

l¼0 r¼0

(C.1) where alr ðsÞ are the coefficients that depend on altitude, Pl ðmÞ are Legendre polynomials of the variable m, the cosine of the pitch angle and S r ðv=vth Þ are speed polynomials, with vth being the thermal speed. These speed polynomials are eigenfunctions for a Fokker–Planck operator (Shizgal and Blackmore, 1984). They form a complete basis

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